symmetry and phaselocking in chains of weakly coupled oscillators weakly coupled chains of oscillators with nearest neighbor interactions are analyzed for phaselocked solutions it is shown that the symmetry properties of the coupling affect the qualitative form of the phaselocked solutions and the scaling behavior of the system as the number of oscillators grows without bound it is also shown that qualitative behavior of these solutions depends on whether the coupling is diffusive or i synaptic terms defined in the paper the methods include the demonstration that the equations for phaselocked solutions can be approximated by a singularly perturbed two point continuum boundary value problem that is easier to analyze the issue of convergence of the phaselocked solutions to solutions of the continuum equation is closely related to questions involving numerical entropy in computation schemes for a conservation law an application to the neurophysiology of motor behavior is discussed briefly

competition for synchronization in a phase oscillator system abstract a system of phase oscillators with a central oscillator co and a set of n peripheral oscillators pos is considered feed forward and feedback connections between the co and pos are determined by two interaction functions which are assumed to be smooth odd and periodic to describe the competition of pos for synchronization with the co we study the asymptotic stability of fixed points corresponding to in phase synchronization of a group of k pos while other pos are in anti phase with the co it is shown that stability conditions can be formulated in terms of four parameters that describe the slopes of the interaction functions at zero and half period points analytical description of stability in terms of the regions in 4 dimensional parameter space is given combining stability analysis with the detailed study of geometry of invariant manifolds the bifurcations of fixed points are investigated we show that various dynamical regimes such as multistability heteroclinic orbits and chaos are possible analytical stability conditions for global synchronization of pos with the co are formulated for the systems with local connections between pos it is shown that synchronization in a large system with local connections becomes unstable even under weak desynchronizing influence from the co the application of the results to modeling in neuroscience and in particular for modeling visual attention is discussed

emergent behaviors of a holonomic particle system on a sphere we study sufficient conditions for the asymptotic emergence of synchronous behaviors in a holonomic particle system on a sphere which was recently introduced by lohe non abelian kuramoto model and synchronization j phys a math theor 42 395101 395126 2009 these conditions depend only on the coupling strength and initial position diameter for identical particles we show that the position diameter approaches zero asymptotically under these sufficient conditions i e all particles approach to the same position for non identical particles the particle positions do not shrink to one point but can be squeezed into some small region whose diameter is inversely proportional to the coupling strength when the coupling strength is large we also provide several numerical results to confirm our analytical findings

order in two dimensional oscillator lattices we study phase and frequency ordering in a square lattice of limit cycle oscillators with randomly distributed natural frequencies we measure the cluster distribution in lattices of increasing size and we find that macroscopic frequency ordering is possible in two dimensions in contrast with previous results

convergence analysis of the two preconditioned iterative methods for m matrix linear systems in this paper we present the two preconditioners i s and i s r for solving m matrix linear systems and discuss the convergence of the two preconditioned iterative methods meanwhile we obtain comparison theorems between the two preconditioned iterative methods and consider the solution of m matrix linear systems by preconditioned krylov subspace methods numerical experiments are given to validate the performance of the preconditioners

block gauss elimination followed by a classical iterative method for the solution of linear systems in the last two decades many papers have appeared in which the application of an iterative method for the solution of a linear system is preceded by a step of the gauss elimination process in the hope that this will increase the rates of convergence of the iterative method this combination of methods has been proven successful especially when the matrix a of the system is an m matrix the purpose of this paper is to extend the idea of one to more gauss elimination steps consider other classes of matrices a e g p cyclic consistently ordered and generalize and improve the asymptotic convergence rates of some of the methods known so far

collective synchronization in spatially extended systems of coupled oscillators with random frequencies we study collective behavior of locally coupled limit cycle oscillators with random intrinsic frequencies spatially extended over d dimensional hypercubic lattices phase synchronization as well as frequency entrainment are explored analytically in the linear strong coupling regime and numerically in the nonlinear weak coupling regime our analysis shows that the oscillator phases are always desynchronized up to d 4 which implies the lower critical dimension d l p 4 for phase synchronization on the other hand the oscillators behave collectively in frequency phase velocity even in three dimensions d 3 indicating that the lower critical dimension for frequency entrainment is d l f 2 nonlinear effects due to the periodic nature of limit cycle oscillators are found to become significant in the weak coupling regime so called runaway oscillators destroy the synchronized ordered phase and there emerges a fully random disordered phase critical behavior near the synchronization transition into the fully random phase is unveiled via numerical investigation collective behavior of globally coupled oscillators is also examined and compared with that of locally coupled oscillators

improving jacobi and gauss seidel iterations abstract when convergent jacobi or gauss seidel iterations can be applied to solve systems of linear equations a natural question is how convergence rates are affected if the original system is modified by performing some gaussian elimination we prove that if the initial iteration matrix is nonnegative then such elimination improves convergence our results extend those contained in 4

improving jacobi methods for nonnegative h matrices linear systems abstract in this paper we consider a preconditioned iterative method for solving linear system ax b which is a generalization of method proposed by zhang et al y zhang t z huang x p liu modified iterative methods for nonnegative matrices and m matrices linear systems comput math appl 50 2005 1587 1602 and prove its convergence for the case when a is a nonnegative h matrix numerical examples are also given which show the effectiveness of our method

collective synchronization in the presence of reactive coupling and shear diversity we analyze the synchronization dynamics of a model obtained from the phase reduction of the mean field complex ginzburg landau equation with heterogeneity we present exact results that uncover the role of dissipative and reactive couplings on the synchronization transition when shears and natural frequencies are independently distributed as it occurs in the purely dissipative case an excess of shear diversity prevents the onset of synchronization but this does not hold true if coupling is purely reactive in this case the synchronization threshold turns out to depend on the mean of the shear distribution but not on all the other distribution s moments

quantum integrability in systems with finite number of levels we consider the problem of defining quantum integrability in systems with finite number of energy levels starting from commuting matrices and construct new general classes of such matrix models with a given number of commuting partners we argue that if the matrices depend on a real parameter one can define quantum integrability from this feature alone leading to specific results such as exact solvability poissonian energy level statistics and to level crossings

convergence of parallel multisplitting iterative methods for m matrices abstract despite the major recent activity in parallel processing few effective new algorithms designed exclusively for multiprocessors have been put forth one such is the multisplitting iterative algorithm suggested by o leary and white although they have given some sufficient conditions for convergence a general convergence theory has not been developed even for the classical situation where the coefficient matrix a is an m matrix or is symmetric positive definite in this paper we study the m matrix case in detail the multisplitting process for a rn n is recast as an ordinary iterative process for a certain block matrix a r kn kn where k is the number of processors and standard convergence results are used to develop a convergence theory for multisplitting iterative methods where a is an m matrix comparison results between multisplitting methods are established in terms of monotonic norms and for the case where a is irreducible in terms of the asymptotic convergence rate a key observation here is that in certain cases the rate of global convergence of these parallel iterative methods is inherent in the splitting of a and is independent of the manner in which the work is distributed among the processors thus in general one can distribute the work for load balancing purposes without affecting the convergence rate

more on modifications and improvements of classical iterative schemes for m matrices in the last four decades many articles have been devoted to the modifications and improvements of classes of preconditioners for linear systems whose matrix coefficient is an m matrix in order to improve on the convergence rates of the classical iterative schemes jacobi gauss seidel etc the present work is a contribution towards the generalization of the most common preconditioners used so far

exact results for the kuramoto model with a bimodal frequency distribution we analyze a large system of globally coupled phase oscillators whose natural frequencies are bimodally distributed the dynamics of this system has been the subject of long standing interest in 1984 kuramoto proposed several conjectures about its behavior ten years later crawford obtained the first analytical results by means of a local center manifold calculation nevertheless many questions have remained open especially about the possibility of global bifurcations here we derive the system s stability diagram for the special case where the bimodal distribution consists of two equally weighted lorentzians using an ansatz recently discovered by ott and antonsen we show that in this case the infinite dimensional problem reduces exactly to a flow in four dimensions depending on the parameters and initial conditions the long term dynamics evolves to one of three states incoherence where all the oscillators are desynchronized partial synchrony where a macroscopic group of phase locked oscillators coexists with a sea of desynchronized ones and a standing wave state where two counter rotating groups of phase locked oscillators emerge analytical results are presented for the bifurcation boundaries between these states similar results are also obtained for the case in which the bimodal distribution is given by the sum of two gaussians

comparison theorems of preconditioned gauss seidel methods for m matrices in this paper a new preconditioner for the gauss seidel method is proposed for solving linear systems whose coefficient matrix is an m matrix several comparison theorems are shown for the proposed method with several preconditioners it follows from the comparison results that our preconditioner is one of the best preconditioners in the sense of convergence rate finally numerical examples are given to illustrate our theoretical results two conjectures are proposed as well based on our numerical tests

letter to the editor comment on a comparison theorem of the sor iterative method in this work we point out that there are incorrect assertions in the article by li ying sun li ying sun a comparison theorem for the sor iterative method j comput appl math 181 2005 336 341 moreover we answer an open problem

positive cones and convergence conditions for iterative methods based on splittings the study of convergence conditions to solve large and sparse linear systems ax b by iterative methods has been discussed by many authors in this paper by using the partial order induced by positivity cone of matrices and conditions on the matrices and splittings we obtain the convergence of the iterative method the usual partial orders of nonnegative matrices and nonnegative definite matrices satisfy the conditions presented for this new order and hence we unify the theory moreover we also show that the new conditions apply to other partial orders as well thus extending the usual convergence theory to a new situation

onset of synchronization in complex networks of noisy oscillators we study networks of noisy phase oscillators whose nodes are characterized by a random degree counting the number of its connections both these degrees and the natural frequencies of the oscillators are distributed according to a given probability density replacing the randomly connected network by an all to all coupled network with weighted edges allows us to formulate the dynamics of a single oscillator coupled to the mean field and to derive the corresponding fokker planck equation from the latter we calculate the critical coupling strength for the onset of synchronization as a function of the noise intensity the frequency distribution and the first two moments of the degree distribution our approach is applied to a dense small world network model for which we calculate the degree distribution numerical simulations prove the validity of the made replacement we also test the applicability to more sparsely connected networks and formulate homogeneity and absence of correlations in the degree distribution as limiting factors of our approach

integrable time dependent quantum hamiltonians we formulate a set of conditions under which the nonstationary schr odinger equation with a time dependent hamiltonian is exactly solvable analytically the main requirement is the existence of a non abelian gauge field with zero curvature in the space of system parameters known solvable multistate landau zener models satisfy these conditions our method provides a strategy to incorporate time dependence into various quantum integrable models while maintaining their integrability we also validate some prior conjectures including the solution of the driven generalized tavis cummings model

on the rate of convergence of the preconditioned conjugate gradient method we derive new estimates for the rate of convergence of the conjugate gradient method by utilizing isolated eigenvalues of parts of the spectrum we present a new generalized version of an incomplete factorization method and compare the derived estimates of the number of iterations with the number actually found for some elliptic difference equations and for a similar problem with a model empirical distribution function

erratum to a note on the preconditioned gauss seidel method for m matrices j comput appl math 219 1 2008 59 71 in this note some errors in a recent article by niki et al h niki t kohno m morimoto the preconditioned gauss seidel method faster than the sor method j comput appl math 219 2008 59 71 are pointed out and a new proof for the corresponding result is presented

scheduled relaxation jacobi method elliptic partial differential equations epdes appear in a wide variety of areas of mathematics physics and engineering typically epdes must be solved numerically which sets an ever growing demand for efficient and highly parallel algorithms to tackle their computational solution the scheduled relaxation jacobi srj is a promising class of methods atypical for combining simplicity and efficiency that has been recently introduced for solving linear poisson like epdes the srj methodology relies on computing the appropriate parameters of a multilevel approach with the goal of minimizing the number of iterations needed to cut down the residuals below specified tolerances the efficiency in the reduction of the residual increases with the number of levels employed in the algorithm applying the original methodology to compute the algorithm parameters with more than 5 levels notably hinders obtaining optimal srj schemes as the mixed non linear algebraic differential system of equations from which they result becomes notably stiff here we present a new methodology for obtaining the parameters of srj schemes that overcomes the limitations of the original algorithm and provide parameters for srj schemes with up to 15 levels and resolutions of up to 215 points per dimension allowing for acceleration factors larger than several hundreds with respect to the jacobi method for typical resolutions and in some high resolution cases close to 1000 most of the success in finding srj optimal schemes with more than 10 levels is based on an analytic reduction of the complexity of the previously mentioned system of equations furthermore we extend the original algorithm to apply it to certain systems of non linear epdes we compute new optimal parameters of the scheduled relaxation jacobi method the new parameters are calculated for srj schemes with p 6 to p 15 levels we reduce the stiffness in the computation of optimal srj parameters analytically we provide a grid of optimal parameters for different p and numerical resolutions we benchmark srj methods against other algorithms to solve linear systems

comparison results on preconditioned sor type iterative method for z matrices linear systems in this paper we present some comparison theorems on preconditioned iterative method for solving z matrices linear systems comparison results show that the rate of convergence of the gauss seidel type method is faster than the rate of convergence of the sor type iterative method

lower critical dimension for populations of oscillators with randomly distributed frequencies a renormalization group analysis it is argued by way of a renormalization group analysis that the lower critical dimension of macroscopic mutual entrainment in a class of populations of oscillators satisfies a certain inequality which is sensitive to the tail of the distribution of native frequencies this result is supported in part by numerical simulations as well as a proof of the absence of long range order in one dimension

phase transition to frequency entrainment in a long chain of pulse coupled oscillators a long chain of pulse coupled oscillators was studied the oscillators interacted via a phase response curve similar to those obtained from pacemaker cells in the heart the natural frequencies were random numbers from a distribution with finite bandwidth stable frequency entrained states were shown always to exist above a critical coupling strength below the critical coupling the probability to have such states was shown to be zero if the number of oscillators is infinite this discontinuity establishes the existence of a phase transition in the thermodynamic limit for weak coupling clusters of frequency entrained oscillators emerged the cluster sizes were exponentially distributed even when the critical coupling was approached at this coupling the mean cluster size diverged to infinity according to a power law the standard deviation of the distribution of mean frequencies in the chain converged to zero also according to a power law

the dynamics of chimera states in heterogeneous kuramoto networks abstract we study a variety of mixed synchronous incoherent chimera states in several heterogeneous networks of coupled phase oscillators for each network the recently discovered ott antonsen ansatz is used to reduce the number of variables in the partial differential equation pde governing the evolution of the probability density function by one resulting in a time evolution pde for a variable with as many spatial dimensions as the network bifurcation analysis is performed on the steady states of these pdes the results emphasise the commonality of the dynamics of the different networks and provide stability information that was previously inferred

phase locking and critical phenomena in lattices of coupled nonlinear oscillators with random intrinsic frequencies abstract we study phase locking in a network of coupled nonlinear oscillators with local interactions and random intrinsic frequencies the oscillators are located at the vertices of a graph and interact along the edges they are coupled by sinusoidal functions of the phase differences across the edges and their intrinsic frequencies are independent and identically distributed with finite mean and variance we derive an exact expression for the probability of phase locking in a linear chain of such oscillators and prove that this probability tends to zero as the number of oscillators grows without bound however if the coupling strength increases as the square root of the number of oscillators the probability of phase locking tends to a limiting distribution the kolmogorov smirnov distribution this latter result is obtained by showing that the phase locking problem is equivalent to a discretization of pinned brownian motion the results on chains of oscillators are extended to more general graphs in particular for a hypercubic lattice of any dimension the probability of phase locking tends to zero exponentially fast as the number of oscillators grows without bound we also consider a less stringent type of synchronization characterized by large clusters of oscillators mutually entrained at the same average frequency it is shown that if such clusters exist they necessarily have a sponge like geometry

synchronization of coupled rotators josephson junction ladders and the locally coupled kuramoto model we show that the resistively shunted junction rsj equations describing a ladder array of overdamped critical current disordered josephson junctions that are current biased along the rungs of the ladder can be mapped onto a kuramoto model with nearest neighbor sinusoidal couplings this result is obtained by an averaging method in which the fast dynamics of the rsj equations are integrated out leaving the dynamics which describe the time scale over which neighboring junctions along the rungs of the ladder phase and frequency synchronize we quantify the degree of frequency synchronization of the rung junctions by calculating the standard deviation of their time averaged voltages ensuremath sigma ensuremath omega and the phase synchronization is quantified by calculating the time average of the modulus of the kuramoto order parameter r we test the results of our averaging process by comparing the values of ensuremath sigma ensuremath omega and r for the original rsj equations and our averaged equations we find excellent agreement for dc bias currents of i b i c ensuremath gtrsim 3 where i c is the average critical current of the rung junctions and critical current disorders of up to 10 we also study the effects of thermal noise on the synchronization properties of the overdamped ladder finally we find that including the effects of junction capacitance can lead to a discontinuous synchronization transition as the strength of the coupling between neighboring junctions is smoothly varied

on modified jacobi linear operators abstract by means of successive partial substitutions new fixed point linear equations can be obtained from old ones the jacobi method applied to a system in the sequence thus obtained constitutes a partial gauss seidel method applied to the original one and we analyze the behavior of the sequence of spectral radii of the successive iteration matrices the modified jacobi operators we do this under the assumption that the starting operator is nonnegative with respect to a proper cone and has spectral radius less or greater than 1 our main result is that if the jacobi operator obtained after k substitutions is irreducible then the following one either is the same or has strictly smaller or greater spectral radius this result implies that the whole sequence of spectral radii is monotone

letter to the editor a note on the preconditioned gauss seidel gs method for linear systems wen li j comput appl math 182 2005 81 90 asserted that there are some errors in article by hiroshi niki kyouji harada munenori morimoto and michio sakakihara j comput appl math 164 165 2004 587 600 and li presented a new proof for the corresponding results in h niki et al in this paper we point out some errors in li s assertion moreover we show that a new proof presented by li is imperfect

collective synchronisation in lattices of nonlinear oscillators with randomness the authors study mutual synchronisation in a model of interacting limit cycle oscillators with random intrinsic frequencies it is shown rigorously that the model exhibits no long range order in one dimension and that in higher dimensional lattices large clusters of synchronised oscillators necessarily have a sponge like structure surprisingly the phase locking behaviour of the mean field model is completely different from that of any finite dimensional lattice indicating that d infinity is the upper critical dimension for phase locking

two new modified gauss seidel methods for linear system with m matrices in 2002 h kotakemori et al proposed the modified gauss seidel mgs method for solving the linear system with the preconditioner p i s m a x h kotakemori k harada m morimoto h niki a comparison theorem for the iterative method with the preconditioner i s m a x j comput appl math 145 2002 373 378 since this preconditioner is constructed by only the largest element on each row of the upper triangular part of the coefficient matrix the preconditioning effect is not observed on the nth row in the present paper to deal with this drawback we propose two new preconditioners the convergence and comparison theorems of the modified gauss seidel methods with these two preconditioners for solving the linear system are established the convergence rates of the new proposed preconditioned methods are compared in addition numerical experiments are used to show the effectiveness of the new mgs methods

exact solution of generalized tavis cummings models in quantum optics quantum inverse methods are developed for the exact solution of models which describe n two level atoms interacting with one mode of the quantized electromagnetic field containing an arbitrary number of excitations m either a kerr type nonlinearity or a stark shift term can be included in the model and it is shown that these two cases can be mapped from one to the other the method of solution provides a general framework within which many related problems can similarly be solved explicit formulae are given for the rabi splitting of the models for some n and m on and off resonance it is also shown that the solution of the pure tavis cummings model can be reduced to solving a homogeneous ordinary differential equation of second order generalization of the method to the case of several cavity modes is indicated

a note on the preconditioner pm i sm kotakemori et al h kotakemori k harada m morimoto h niki a comparison theorem for the iterative method with the preconditioner i s m a x journal of computational and applied mathematics 145 2002 373 378 have reported that the convergence rate of the iterative method with a preconditioner p m i s m was superior to one of the modified gauss seidel method under the condition these authors derived a theorem comparing the gauss seidel method with the proposed method however through application of a counter example wen li wen li a note on the preconditioned gaussseidel gs method for linear systems journal of computational and applied mathematics 182 2005 81 91 pointed out that there exists a special matrix that does not satisfy this comparison theorem in this note we analyze the reason why such a to counter example may be produced and propose a preconditioner to overcome this problem

comparison results for solving preconditioned linear systems in this paper we present some comparison theorems between two different modified gauss seidel mgs methods the second preconditioning based on the first preconditioning is also discussed in this paper

an extended gs method for dense linear systems davey and rosindale k davey i rosindale an iterative solution scheme for systems of boundary element equations internat j numer methods engrg 37 1994 1399 1411 derived the gsor method which uses an upper triangular matrix w in order to solve dense linear systems by applying functional analysis the authors presented an expression for the optimum w moreover davey and bounds k davey s bounds a generalized sor method for dense linear systems of boundary element equations siam j comput 19 1998 953 967 also introduced further interesting results in this note we employ a matrix analysis approach to investigate these schemes and derive theorems that compare these schemes with existing preconditioners for dense linear systems we show that the convergence rate of the gauss seidel method with preconditioner p g is superior to that of the gsor method moreover we define some splittings associated with the iterative schemes some numerical examples are reported to confirm the theoretical analysis we show that the egs method with preconditioner p g c o p t produces an extremely small spectral radius in comparison with the other schemes considered

a class of parameter dependent commuting matrices we present a novel class of real symmetric matrices in arbitrary dimension d linearly dependent on a parameter x the matrix elements satisfy a set of nontrivial constraints that arise from asking for commutation of pairs of such matrices for all x and an intuitive sufficiency condition for the solvability of certain linear equations that arise therefrom this class of matrices generically violates the wigner von neumann non crossing rule and is argued to be intimately connected with finite dimensional hamiltonians of quantum integrable systems

a comparison theorem for the sor iterative method in 1997 kohno et al have reported numerically that the improving modified gauss seidel method which was referred to as the imgs method is superior to the sor iterative method in this paper we prove that the spectral radius of the imgs method is smaller than that of the sor method and gauss seidel method if the relaxation parameter w 0 1 as a result we prove theoretically that this method is succeeded in improving the convergence of some classical iterative methods some recent results are improved

structure and stability of the equilibrium set in potential driven flow networks abstract in this paper we address local bifurcation properties of a family of networked dynamical systems specifically those defined by a potential driven flow on a directed graph these network flows include linear consensus dynamics or kuramoto models of coupled nonlinear oscillators as particular cases as it is well known for consensus systems these networks exhibit a somehow unconventional dynamical feature namely the existence of a line of equilibria following from a well known property of the graph laplacian matrix in connected networks with positive weights negative weights which arise in different contexts e g in consensus models in signed graphs or in kuramoto models with antagonistic actors may on the one hand lead to higher dimensional manifolds of equilibria and on the other be responsible for bifurcation phenomena in this direction we prove a saddle node bifurcation theorem for a broad family of potential driven flows in networks with one or more negative weights the goal is to state the conditions in structural terms that is in terms of the expressions defining the flowrates and the graph theoretic properties of the network not only the eigenvalue requirements but also the nonlinear transversality assumptions supporting the bifurcation motivate an analysis of independent interest concerning the rank degeneracies of nodal matrices arising in the linearized dynamics this analysis is performed in terms of the contraction deletion structure of spanning trees and uses several results from matrix analysis different examples illustrate the results some linear problems including signed graphs are aimed at illustrating the analysis of nodal matrices whereas in a nonlinear framework we apply the characterization of saddle node bifurcations to networks with a sinusoidal kuramoto like flow

the noisy voter model under the influence of contrarians abstract the influence of contrarians on the noisy voter model is studied at the mean field level the noisy voter model is a variant of the voter model where agents can adopt two opinions optimistic or pessimistic and can change them by means of an imitation herding and an intrinsic noise mechanisms an ensemble of noisy voters undergoes a finite size phase transition upon increasing the relative importance of the noise to the herding form a bimodal phase where most of the agents share the same opinion to a unimodal phase where almost the same fraction of agent are in opposite states by the inclusion of contrarians we allow for some voters to adopt the opposite opinion of other agents anti herding we first consider the case of only contrarians and show that the only possible steady state is the unimodal one more generally when voters and contrarians are present we show that the bimodal unimodal transition of the noisy voter model prevails only if the number of contrarians in the system is smaller than four and their characteristic rates are small enough for the number of contrarians bigger or equal to four the voters and the contrarians can be seen only in the unimodal phase moreover if the number of voters and contrarians as well as the noise and herding rates are of the same order then the probability functions of the steady state are very well approximated by the gaussian distribution

the link between integrability level crossings and exact solution in quantum models we investigate the connection between energy level crossings in integrable systems and their integrability i e the existence of a set of non trivial integrals of motion in particular we consider a general quantum hamiltonian linear in the coupling u h u t uv and require that it has the maximum possible number of nontrivial commuting partners also linear in u we demonstrate how this commutation requirement alone leads to 1 an exact solution for the energy spectrum and 2 level crossings which are always present in these hamiltonians in violation of the wigner von neumann non crossing rule moreover we construct these hamiltonians explicitly by resolving the above commutation requirement and show their equivalence to a sector of gaudin magnets central spin hamiltonians in contrast fewer than the maximum number of conservation laws does not guarantee level crossings

conformists and contrarians in a kuramoto model with identical natural frequencies we consider a variant of the kuramoto model in which all the oscillators are now assumed to have the same natural frequency but some of them are negatively coupled to the mean field these contrarian oscillators tend to align in antiphase with the mean field whereas the positively coupled conformist oscillators favor an in phase relationship the interplay between these effects can lead to rich dynamics in addition to a splitting of the population into two diametrically opposed factions the system can also display traveling waves complete incoherence and a blurred version of the two faction state exact solutions for these states and their bifurcations are obtained by means of the watanabe strogatz transformation and the ott antonsen ansatz curiously this system of oscillators with identical frequencies turns out to exhibit more complicated dynamics than its counterpart with heterogeneous natural frequencies

study on the preconditioners i sm kotakemori et al 2002 2 have reported that the convergence rate of the iterative method with a preconditioner p m i s m a x was superior to one of the modified gauss seidel methods under a special condition the authors derived a theorem comparing the gauss seidel method to remove the requirement for this condition morimoto et al 2004 4 have proposed the preconditioner p s m i s s m however it is pointed out that there exists a special matrix that does not satisfy this comparison theorem to overcome this problem kohno et al 2009 3 have proposed some preconditioners in this note we present a new preconditioner and from numerical results we show that the convergence rate of the proposed method is better than that of the gauss seidel method with other preconditioners in addition we presented the comparison theorem for the proposed preconditioner we succeeded to overcome two drawbacks mentioned above

dynamics of a semiconductor laser array with delayed global coupling we study the dynamics of an array of single mode semiconductor lasers globally but weakly coupled by a common external feedback mirror and by nearest neighbor interactions we seek to determine the conditions under which all lasers of the array are in phase whether in a steady periodic quasiperiodic or chaotic regime in order to maximize the output far field intensity we show that the delay may be a useful control parameter to achieve in phase synchronization for the in phase steady state there is a competition between a delay induced hopf bifurcation leading to an in phase periodic regime and a delay independent hopf bifurcation leading to an antiphased periodic regime both regimes are described analytically and secondary hopf bifurcations to quasiperiodic solutions are found close to the stable steady state the array is described by a set of kuramoto equations for the phases of the fields above the first hopf bifurcation these equations are generalized by the addition of second and third order time derivatives of the phases

integrability and level crossing manifolds in a quantum hamiltonian system we consider a two spin model represented classically by a nonlinear autonomous hamiltonian system with two degrees of freedom and a nontrivial integrability condition and quantum mechanically by a real symmetric hamiltonian matrix with invariant blocks of dimensionalities k frac 1 2 l l 1 l 1 2 dots in the six dimensional parameter space of this model classical integrability is satisfied on a five dimensional hypersurface and level crossings occur on four dimensional manifolds that are completely embedded in the integrability hypersurface except for some lower dimensional submanifolds under mild assumptions the classical integrability condition can be reconstructed from a purely quantum mechanical study of level degeneracies in finite dimensional invariant blocks of the hamiltonian matrix our conclusions are based on rigorous results for k 3 and on numerical results for k 6 10

off shell bethe ansatz equations and n point correlators in the su 2 wznw theory we prove that the wavevectors of the off shell bethe ansatz equation for the inhomogeneous su 2 lattice vertex model render in the quasiclassical limit the solutions of the knizhnik zamolodchikov equation

lu decomposition of m matrices by elimination without pivoting abstract it is shown that if a or a is a singular m matrix satisfying the generalized diagonal dominance condition y t a 0 for some vector y 0 then a can be factored into a lu by a certain elimination algorithm where l is a lower triangular m matrix with unit diagonal and u is an upper triangular m matrix the existence of lu decomposition of symmetric permutations of a and for irreducible m matrices and symmetric m matrices follow as colollaries this work is motivated by applications to the solution of homogeneous systems of linear equations ax 0 where a or a is an m matrix these applications arise e g in the analysis of markov chains input output economic models and compartmental systems a converse of the theorem metioned above can be established by considering the reduced normal form of a

a comparison theorem for the iterative method with the preconditioner i s max in 1991 gunawardena et al linear algebra appl 154 156 1991 123 have reported the modified gauss seidel method with a preconditioner i s in this article we propose to use a preconditioner i smax instead of i s here smax is constructed by only the largest element at each row of the upper triangular part of a by using the lemma established neumann and plemmons linear algebra appl 88 89 1987 559 we get the comparison theorem for the proposed method simple numerical examples are also given

renormalization of oscillator lattices with disorder a real space renormalization transformation is constructed for lattices of nonidentical oscillators with dynamics of the general form dvarphi k dt omega k g summation operator l f lk varphi l varphi k the transformation acts on ensembles of such lattices critical properties corresponding to a second order phase transition toward macroscopic synchronization are deduced the analysis is potentially exact but relies in part on unproven assumptions numerically second order phase transitions with the predicted properties are observed as g increases in two structurally different two dimensional oscillator models one model has smooth coupling f lk varphi l varphi k phi varphi l varphi k where phi x is nonodd the other model is pulse coupled with f lk varphi l varphi k delta varphi l phi varphi k lower bounds for the critical dimensions for different types of coupling are obtained for nonodd coupling macroscopic synchronization cannot be ruled out for any dimension d 1 whereas in the case of odd coupling the well known result that it can be ruled out for d 3 is regained less

improving the modified gauss seidel method for z matrices abstract in 1991 a d gunawardena et al reported that the convergence rate of the gauss seidel method with a preconditioning matrix i s is superior to that of the basic iterative method in this paper we use the preconditioning matrix i s if a coefficient matrix a is an irreducibly diagonally dominant z matrix then i s a is also a strictly diagonally dominant z matrix it is shown that the proposed method is also superior to other iterative methods

s matrix for generalized landau zener problem the schrodinger equation i psi a bt psi with constant hermitian matrices a and b is studied in the form where b is diagonalized this equation is a generalization of the landau zener problem to an arbitrary number of crossing energy levels an approach the independent crossing approximation leading to a partial understanding of the general case in terms of the two level problem is introduced it is found that certain s matrix elements and thus the corresponding transition probabilities for the general problem are exactly given by formulae of unexpected simplicity suggesting that some kind of general analytic solution might exist the full s matrix is calculated for a solvable special case of the problem the solution of another special case previously discovered for systems with three states is generalized to any number of states the asymptotic behaviour of the system is discussed in general and given explicitly to lowest order in 1 t

modified iterative methods for consistent linear systems abstract in order to solve a linear system ax b certain elementary row operations are performed on a before applying the gauss seidel or jacobi iterative methods it is shown that when a is a nonsingular m matrix or a singular tridiagonal m matrix the modified method yields considerable improvement in the rate of convergence for the iterative method it is also shown that in some cases this method is superior to certain other modified iterative methods the performance of this modified method on some matrices other than m matrices is also investigated

the preconditioned gauss seidel method faster than the sor method in recent years a number of preconditioners have been applied to linear systems a d gunawardena s k jain l snyder modified iterative methods for consistent linear systems linear algebra appl 154 156 1991 123 143 t kohno h kotakemori h niki m usui improving modified gauss seidel method for z matrices linear algebra appl 267 1997 113 123 h kotakemori k harada m morimoto h niki a comparison theorem for the iterative method with the preconditioner i s m a x j comput appl math 145 2002 373 378 h kotakemori h niki n okamoto accelerated iteration method for z matrices j comput appl math 75 1996 87 97 m usui h niki t kohno adaptive gauss seidel method for linear systems internat j comput math 51 1994 119 125 10 since these preconditioners are constructed from the elements of the upper triangular part of the coefficient matrix the preconditioning effect is not observed on the nth row of matrix a in the present paper in order to deal with this drawback we propose a new preconditioner in addition the convergence and comparison theorems of the proposed method are established simple numerical examples are also given and we show that the convergence rate of the proposed method is better than that of the optimum sor

comparisons of weak regular splittings and multisplitting methods comparison results for weak regular splittings of monotone matrices are derived as an application we get upper and lower bounds for the convergence rate of iterative procedures based on multisplittings this yields a very simple proof of results of neumann plemmons on upper bounds and establishes lower bounds which has in special cases been conjectured by these authors

algebraic relaxation of an order parameter in randomly coupled limit cycle oscillators in their recent paper phys rev e 58 1789 1998 stiller and radons sr study following our earlier work phys rev lett 68 1073 1992 the behavior of globally and randomly coupled phase oscillators with distributed intrinsic frequencies they claim that their simulation results do not confirm the power law behavior of an order parameter found numerically by the author attributing its cause to the poor precision of the author s integration scheme here demonstrated is that the power law survives even for a scheme better than sr s provided that finite size effects are properly taken into account as was done in our previous work

exact analytical solution of the n level landau zener type bow tie model the non stationary schrodinger equation in a finite basis of states is considered for the hamiltonian matrix linearly depending on time exact analytical solutions of asymptotic transition probabilities are obtained for a bow tie model in which an arbitrary number of linear time dependent diabatic potential curves cross at one point and only a particular horizontal curve has interactions with the others based on the contour integral method used some mathematical aspects such as a possible generalization of the whittaker functions are also briefly discussed

dynamics of majority rule in two state interacting spin systems we introduce a two state opinion dynamics model where agents evolve by majority rule in each update a group of agents is specified whose members then all adopt the local majority state in the mean field limit where a group consists of randomly selected agents consensus is reached in a time that scales ln n where n is the number of agents on finite dimensional lattices where a group is a contiguous cluster the consensus time fluctuates strongly between realizations and grows as a dimension dependent power of n the upper critical dimension appears to be larger than 4 the final opinion always equals that of the initial majority except in one dimension

on optimal improvements of classical iterative schemes for z matrices many researchers have considered preconditioners applied to linear systems whose matrix coefficient is a z or an m matrix that make the associated jacobi and gauss seidel methods converge asymptotically faster than the unpreconditioned ones such preconditioners are chosen so that they eliminate the off diagonal elements of the same column or the elements of the first upper diagonal milaszewicz laa 93 1987 161 170 gunawardena et al laa 154 156 1991 123 143 in this work we generalize the previous preconditioners to obtain optimal methods good jacobi and gauss seidel algorithms are given and preconditioners that eliminate more than one entry per row are also proposed and analyzed moreover the behavior of the above preconditioners to the krylov subspace methods is studied

glassy synchronization in a population of coupled oscillators a phase model for a population of oscillators with random excitatory and inhibitory mean field coupling and subject to external white noise random forces is proposed and studied in the thermodynamic limit different stable phases for the oscillator population may be found i an incoherent state where all possible values of an oscillator phase are equally probable ii a synchronized state where the population has a nonzero collective phase iii a glassy phase where the global synchronization is zero but the oscillators are in phase with the random disorder and iv a mixed phase where the oscillators are partially synchronized and partially in phase with the disorder these predictions are based upon bifurcation analysis of the reduced equation valid at the thermodynamic limit and confirmed by brownian simulation

comparison theorems for weak splittings of bounded operators comparison theorems for weak splittings of bounded operators are presented these theorems extend the classical comparison theorem for regular splittings of matrices by varga the less known result by wo nicki and the recent results for regular and weak regular splittings of matrices by neumann and plemmons elsner and lanzkron rose and szyld the hypotheses of the theorems presented here are weaker and the theorems hold for general banach spaces and rather general cones hypotheses are given which provide strict inequalities for the comparisons it is also shown that the comparison theorem by alefeld and volkmann applies exclusively to monotone sequences of iterates and is not equivalent to the comparison of the spectral radius of the iteration operators

the quest for solvable multistate landau zener models recently integrability conditions ics in mutistate landau zener mlz theory were proposed 1 they describe common properties of all known solved systems with linearly time dependent hamiltonians here we show that ics enable efficient computer assisted search for new solvable mlz models that span complexity range from several interacting states to mesoscopic systems with many body dynamics and combinatorially large phase space this diversity suggests that nontrivial solvable mlz models are numerous in addition we refine the formulation of ics and extend the class of solvable systems to models with points of multiple diabatic level crossing

convergence analysis of the preconditioned gauss seidel method for h matrices in 1997 kohno et al toshiyuki kohno hisashi kotakemori hiroshi niki improving the modified gauss seidel method for z matrices linear algebra appl 267 1997 113 123 proved that the convergence rate of the preconditioned gauss seidel method for irreducibly diagonally dominant z matrices with a preconditioner i s a is superior to that of the basic iterative method in this paper we present a new preconditioner i k b which is different from the preconditioner given by kohno et al toshiyuki kohno hisashi kotakemori hiroshi niki improving the modified gauss seidel method for z matrices linear algebra appl 267 1997 113 123 and prove the convergence theory about two preconditioned iterative methods when the coefficient matrix is an h matrix meanwhile two novel sufficient conditions for guaranteeing the convergence of the preconditioned iterative methods are given

averaging of globally coupled oscillators abstract we study a specific system of symmetrically coupled oscillators using the method of averaging the equations describe a series array of josephson junctions we concentrate on the dynamics near the splay phase state also known as the antiphase state ponies on a merry go round or rotating wave we calculate the floquet exponents of the splay phase periodic orbit in the weak coupling limit and find that all of the floquet exponents are purely imaginary in fact all the floquet exponents are zero except for a single complex conjugate pair thus nested two tori of doubly periodic solutions surround the splay phase state in the linearized averaged equations we numerically integrate the original system and find startling agreement with the averaging results on two counts the observed ratio of frequencies is very close to the prediction and the solutions of the full equations appear to be either periodic or doubly periodic as they are in the averaged equations such behavior is quite surprising from the point of view of generic dynamical systems theory one expects higher dimensional tori and chaotic solutions we show that the functional form of the equations and not just their symmetry is responsible for this nongeneric behavior

a note on the preconditioned gauss seidal gs method for linear systems in this note recent comparison results for preconditioned gauss seidel gs methods are discussed a new strict comparison result between two different preconditioned gs methods is given some errors in a recent article by niki et al j comput appl math 164 165 2004 587 are pointed out and a new proof for the corresponding results in niki et al is presented

modified gauss seidel type methods and jacobi type methods for z matrices abstract in this paper we present the convergence analysis for some modified gauss seidel and jacobi type iterative methods and provide a comparison of spectral radius among the gauss seidel iterative method and these modified methods some recent results are improved

classification of parameter dependent quantum integrable models their parameterization exact solution and other properties we study general quantum integrable hamiltonians linear in a coupling constant and represented by finite n n real symmetric matrices the restriction on the coupling dependence leads to a natural notion of nontrivial integrals of motion and classification of integrable families into types according to the number of such integrals a type m family in our definition is formed by n m nontrivial mutually commuting operators linear in the coupling working from this definition alone we parameterize type m operators i e resolve the commutation relations and obtain an exact solution for their eigenvalues and eigenvectors we show that our parameterization covers all type 1 2 and 3 integrable models and discuss the extent to which it is complete for other types we also present robust numerical observation on the number of energy level crossings in type m integrable systems and analyze the taxonomy of types in the 1d hubbard model

the survey of preconditioners used for accelerating the rate of convergence in the gauss seidel method several preconditioned iterative methods reported in the literature have been used for improving the convergence rate of the gauss seidel method in this article on the basis of nonnegative matrix comparisons between some splittings for such preconditioned matrices are derived simple numerical examples are also given

constants of motion for superconducting josephson arrays abstract we show that series arrays of n identical overdamped josephson junctions have extremely degenerate dynamics in particular we prove that such arrays have n 3 constants of motion for all n 3 the analysis is based on a coordinate transformation that reduces the governing equations to an n 3 parameter family of low dimensional systems in the weak coupling limit the reduced equations can be analyzed completely either all solutions approach the synchronous state or they converge to a continuous family of incoherent oscillations depending on a certain parameter value at the transitional value of this parameter the system becomes completely integrable then the phase space is foliated by invariant two dimensional tori for any n 3 the infinite n limit of the system is an integro partial differential equation with rigorously low dimensional dynamics it supports solitons in the integrable case and chaotic waves in the non integrable case

quantum integrability in the multistate landau zener problem we analyze hamiltonians linear in the time variable for which the multistate landau zener lz problem is known to have an exact solution we show that they either belong to families of mutually commuting hamiltonians polynomial in time or reduce to the lz problem which is considered trivially integrable the former category includes the equal slope bow tie and generalized bow tie models for each of these models we explicitly construct the corresponding families of commuting matrices the equal slope model is a member of an integrable family that consists of the maximum possible number for a given matrix size of commuting matrices linear in time the bow tie model belongs to a previously unknown similarly maximal family of quadratic commuting matrices we thus conjecture that quantum integrability understood as the existence of nontrivial parameter dependent commuting partners is a necessary condition for the lz solvability descendants of the lz hamiltonian are e g general su 2 and hamiltonians time dependent linear chain linear nonlinear and double oscillators we explicitly obtain solutions to all these lz problems from the case

perturbation analysis of the kuramoto phase diffusion equation subject to quenched frequency disorder the kuramoto phase diffusion equation is a nonlinear partial differential equation which describes the spatiotemporal evolution of a phase variable in an oscillatory reaction diffusion system synchronization manifests itself in a stationary phase gradient where all phases throughout a system evolve with the same velocity the synchronization frequency the formation of concentric waves can be explained by local impurities of higher frequency which can entrain their surroundings concentric waves in synchronization also occur in heterogeneous systems where the local frequencies are distributed randomly we present a perturbation analysis of the synchronization frequency where the perturbation is given by the heterogeneity of natural frequencies in the system the nonlinearity in the form of dispersion leads to an overall acceleration of the oscillation for which the expected value can be calculated from the second order perturbation terms we apply the theory to simple topologies like a line or sphere and deduce the dependence of the synchronization frequency on the size and the dimension of the oscillatory medium we show that our theory can be extended to include rotating waves in a medium with periodic boundary conditions by changing a system parameter the synchronized state may become quasidegenerate we demonstrate how perturbation theory fails at such a critical point