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

generalised meshes for quantum mechanical problems a new method to discretise schrodinger equations on a mesh is described this method is based on an accurate approximation of a variational calculation the regularly spaced mesh and meshes based on the zeros of orthogonal polynomials are studied in detail it is shown that with each type of mesh is associated a particular kinetic energy operator and an optimal formula for its discretized form the applications to some simple potential problems show that the method is very accurate as well as very simple applications to many body problems indicate that the accuracy of the results is improved by an order of magnitude with respect to conventional mesh calculations

emergent dynamics of a generalized lohe model on some class of lie groups we introduce a lohe group which is a new class of matrix lie groups and present a continuous dynamical system for the synchronization of group elements in a lohe group the lohe group includes classical lie groups such as the orthogonal unitary and symplectic groups and since lohe groups need not be compact global existence of odes may fail the proposed dynamical system generalizes the lohe model lohe in j phys a 43 465301 2010 lohe in j phys a 42 395101 395126 2009 itself a nonabelian generalization of the kuramoto model and alongside we also generalize the analytical framework ha and ryoo in j stat phys 163 411 439 2016 of emergent and unique phase locked states for the construction of the phase locked states we introduce lyapunov functions measuring the ensemble diameter and the dissimilarity between two lohe flows and derive gronwall type differential inequalities for them the global existence of solutions then become a consequence of the boundedness of these lyapunov functions our sufficient framework for the emergent dynamics is formulated in terms of coupling strength and initial states and it leads to the global existence of solutions and the formation and uniqueness of a phase locked asymptotic state as a concrete example we demonstrate how our theory can show emergent phenomenon on the heisenberg group where all initial configurations tend to a unique phase locked state exponentially fast

cucker smale flocking under hierarchical leadership a mathematical theory on flocking serves the foundation for several ubiquitous multiagent phenomena in biology ecology sensor networks and economics as well as social behavior like language emergence and evolution directly inspired by the recent fundamental works of cucker and smale on the construction and analysis of a generic flocking model we study the emergent behavior of cucker smale flocking under hierarchical leadership the rates of convergence towards asymptotically coherent group patterns in different scenarios are established the consistent convergence towards coherent patterns may well reveal the advantages and necessities of having leaders and leadership in a complex biological technological economic or social system with sufficient intelligence

generalising the kuramoto model for the study of neuronal synchronisation in the brain abstract in this article we have generalised the kuramoto model to allow one to model neuronal synchronisation more appropriately the generalised version allows for different connective arrangements time varying natural frequencies and time varying coupling strengths to be realised within the framework of the original kuramoto model by incorporating the above mentioned features into the original kuramoto model one can allow for the adaptive nature of neurons in the brain to be accommodated extensive tests using the generalised kuramoto model were performed on a n 4 coupled oscillator network examination of how different connective arrangements time varying natural frequencies and time varying coupling strengths affected synchronisation separately and in combination are reported the effects on synchronisation for large n are also reported

galois groups of trinomials abstract galois groups of irreducible trinomials xn axs b z x are investigated assuming the classification of finite simple groups we show that under some simple yet general hypotheses bearing on the integers n s a and b only very specific groups can occur for instance if the two integers nb and as n s are coprime and if s is a prime number then already the galois group of f x is either the alternating group an or the symmetric group sn this significantly extends work of osada

self organization in systems of self propelled particles we investigate a discrete model consisting of self propelled particles that obey simple interaction rules we show that this model can self organize and exhibit coherent localized solutions in one and in two dimensions in one dimension the self organized solution is a localized flock of finite extent in which the density abruptly drops to zero at the edges in two dimensions we focus on the vortex solution in which the particles rotate around a common center and show that this solution can be obtained from random initial conditions even in the absence of a confining boundary furthermore we develop a continuum version of our discrete model and demonstrate that the agreement between the discrete and the continuum model is excellent

on the emergence and orbital stability of phase locked states for the lohe model we study the emergence and orbital stability of phase locked states of the lohe model which was proposed as a non abelian generalization of the kuramoto phase model for synchronization lohe introduced a first order system of matrix valued ordinary differential equations for quantum synchronization and numerically observed the asymptotic formation and orbital stability of phase locked states of the lohe model in this paper we provide an analytical framework to confirm lohe s observations of emergent phase locked states this extends earlier special results on lower dimensions to any finite dimension for the construction and orbital stability of phase locked states we introduce lyapunov functions to measure the ensemble diameter and dissimilarity between two lohe flows and using the time evolution estimates of these lyapunov functions we present an admissible set of initial states and show that an admissible initial state leads to a unique phase locked asymptotic state

emergent dynamics of cucker smale flocking particles in a random environment abstract we present a new kinetic cucker smale fokker planck cs fp type equation with a degenerate diffusion which describes the dynamics for an ensemble of infinitely many cucker smale particles in a random environment the asymptotic dynamics of the cs fp equation exhibits a threshold like phenomenon depending on the relative strength between the coupling strength and the noise strength in the small coupling regime the noise effect becomes dominant which induces the velocity variance to increase to infinity exponentially fast in contrast the velocity alignment effect is strong in the large coupling regime and the velocity variance tends to zero exponentially fast we present the global existence of classical solutions to the cs fp equation for a sufficiently smooth initial datum without smallness in its size for the kinetic cs fp equation with a metric dependent communication weight we provide a uniform in time mean field limit from the stochastic cs model to the kinetic cs fp equation without convergence rate

the unexplained accuracy of the lagrange mesh method the lagrange mesh method is an approximate variational calculation which resembles a mesh calculation because of the use of a gauss quadrature in order to analyze its accuracy four different lagrange mesh calculations based on the zeros of laguerre polynomials are compared with exact variational calculations based on the corresponding laguerre basis the comparison is performed for three solvable radial potentials the morse harmonic oscillator and coulomb potentials the results show that the accuracies of the energies obtained for different partial waves with the different mesh approximations are very close to the variational accuracy even in the presence of the centrifugal singularity the same property holds for the approximate wave functions this striking accuracy remains unexplained

noise suppresses or expresses exponential growth in this paper we will show that noise can make a given system whose solutions grow exponentially become a new system whose solutions will grow at most polynomially on the other hand we will also show that noise can make a given system whose solutions are bounded become a new system whose solutions will grow exponentially in other words we reveal that the noise can suppress or expresses exponential growth

time delay effect on the flocking in an ensemble of thermomechanical cucker smale particles abstract we study the communication time delay effect on the emergent dynamics of thermodynamic cucker smale tcs ensemble with temperature as an internal state in the presence of time delay conservation laws for momentum and energy do not hold so that the analytic tool based on the energy method cannot be applied for the tcs model as it is we show that temperature stays positive in the presence of small time delay as long as temperatures stay positive the tcs model is well posed by direct application of cauchy lipschitz theory when the size of time delay and initial data are sufficiently small we can derive differential inequalities for velocity and temperature diameters using these differential inequalities we show that temperature and velocity diameters tend to zero exponentially fast whereas the position diameter is uniformly bounded we also present several numerical simulations and compare them with analytic results

from discrete cucker smale model to continuous cucker smale model in a temperature field we study emergent dynamics of the discrete and continuous cucker smale c s models in a self consistent temperature field and provide refined monocluster flocking estimates more precisely we present a sufficient framework leading to the monocluster flocking toward the constant common temperature and velocity exponentially fast this improves the authors earlier work where existence of common temperature and velocity are not identified our proposed sufficient framework is formulated in terms of the initial data and system parameters using the refined flocking we also provide a uniform in time convergence from the discrete model to the continuous model under some suitable compatibility condition on the asymptotic velocities for the discrete and continuous model as the time step tends to zero we also provide several numerical examples and compare them with analytical results we study emergent dynamics of the discrete and continuous cucker smale c s models in a self consistent temperature field and provide refined monocluster flocking estimates more precisely we present a sufficient framework leading to the monocluster flocking toward the constant common temperature and velocity exponentially fast this improves the authors earlier work where existence of common temperature and velocity are not identified our proposed sufficient framework is formulated in terms of the initial data and system parameters using the refined flocking we also provide a uniform in time convergence from the discrete model to the continuous model under some suitable compatibility condition on the asymptotic velocities for the discrete and continuous model as the time step tends to zero we also provide several numerical examples and compare them with analytical results

on the relaxation dynamics of lohe oscillators on some riemannian manifolds we study the collective relaxation dynamics appearing in weakly coupled lohe oscillators in a large coupling regime the lohe models on the unit sphere and unitary group were proposed as a nonabelian generalization of the kuramoto model on the unit circle and their emergent dynamics has been extensively studied in previous literature for some restricted class of initial data based on the lyapunov functional approach and order parameter approach in this paper we extend the previous partial results to cover a generic initial configuration via the detailed analysis on the order parameter measuring the modulus of the centroid in particular we present a detailed relaxation dynamics and structure of the resulting asymptotic states for the lohe sphere model we also present new gradient flow formulations for the lohe matrix models with the same one body hamiltonians on some group manifolds as a direct application of this new formulation we show that every bounded lohe flow which originated from any initial configuration converges asymptotically

resolution of the gross pitaevskii equation with the imaginary time method on a lagrange mesh the lagrange mesh method is an approximate variational calculation which has the simplicity of a mesh calculation combined with the imaginary time method it is applied to the iterative resolution of the gross pitaevskii equation two variants of a fourth order factorization of the exponential of the hamiltonian and two types of mesh lagrange hermite and lagrange sinc are employed and compared the accuracy is checked with the help of these comparisons and of the virial theorem the lagrange hermite mesh provides very accurate results with short computing times for values of the dimensionless parameter of the nonlinear term up to 104 for higher values up to 107 the lagrange sinc mesh is more efficient examples are given for anisotropic and nonseparable trapping potentials 2010 the american physical society

integrals of lagrange functions and sum rules exact values are derived for some matrix elements of lagrange functions i e orthonormal cardinal functions constructed from orthogonal polynomials they are obtained with exact gauss quadratures supplemented by corrections in the particular case of lagrange laguerre and shifted lagrange jacobi functions sum rules provide exact values for matrix elements of 1 x and 1 x2 as well as for the kinetic energy from these expressions new sum rules involving laguerre and shifted jacobi zeros and weights are derived

non abelian kuramoto models and synchronization we describe non abelian generalizations of the kuramoto model for any classical compact lie group and identify their main properties these models may be defined on any complex network where the variable at each node is an element of the unitary group u n or a subgroup of u n the nonlinear evolution equations maintain the unitarity of all variables which therefore evolve on the compact manifold of u n synchronization of trajectories with phase locking occurs as for the kuramoto model for values of the coupling constant larger than a critical value and may be measured by various order and disorder parameters limit cycles are characterized by a frequency matrix which is independent of the node and is determined by minimizing a function which is quadratic in the variables we perform numerical computations for n 2 for which the su 2 group manifold is s3 for a range of natural frequencies and all to all coupling in order to confirm synchronization properties we also describe a second generalization of the kuramoto model which is formulated in terms of real m vectors confined to the m 1 sphere for any positive integer m and investigate trajectories numerically for the s2 model this model displays a variety of synchronization phenomena in which trajectories generally synchronize spatially but are not necessarily phase locked even for large values of the coupling constant

flocks herds and schools a quantitative theory of flocking we present a quantitative continuum theory of flocking the collective coherent motion of large numbers of self propelled organisms in agreement with everyday experience our model predicts the existence of an ordered phase of flocks in which all members of even an arbitrarily large flock move together with the same mean velocity stackrel ensuremath rightarrow v ensuremath ne 0 this coherent motion of the flock is an example of spontaneously broken symmetry no preferred direction for the motion is picked out a priori in the model rather each flock is allowed to and does spontaneously pick out some completely arbitrary direction to move in by analyzing our model we can make detailed quantitative predictions for the long distance long time behavior of this broken symmetry state the goldstone modes associated with this spontaneously broken rotational symmetry are fluctuations in the direction of motion of a large part of the flock away from the mean direction of motion of the flock as a whole these goldstone modes mix with modes associated with conservation of bird number to produce propagating sound modes these sound modes lead to enormous fluctuations of the density of the flock far larger at long wavelengths than those in e g an equilibrium gas our model is similar in many ways to the navier stokes equations for a simple compressible fluid in other ways it resembles a relaxational time dependent ginsburg landau theory for an n d component isotropic ferromagnet in spatial dimensions dg4 the long distance behavior is correctly described by a linearized theory and is equivalent to that of an unusual but nonetheless equilibrium model for spin systems for dl4 nonlinear fluctuation effects radically alter the long distance behavior making it different from that of any known equilibrium model in particular we find that in d 2 where we can calculate the scaling exponents exactly flocks exhibit a true long range ordered spontaneously broken symmetry state in contrast to equilibrium systems which cannot spontaneously break a continuous symmetry in d 2 the mermin wagner theorem we make detailed predictions for various correlation functions that could be measured either in simulations or by quantitative imaging of real flocks we also consider an anisotropic model in which the birds move preferentially in an easy e g horizontal plane and make analogous but quantitatively different predictions for that model as well for this anisotropic model we obtain exact scaling exponents for all spatial dimensions including the physically relevant case d 3

stochastic lohe matrix model on the lie group and mean field limit we propose a lohe matrix model in a random environment where each oscillator can be regarded as an element of a general matrix lie group g in order to make the stochastic system stays on g for all time we introduce suitable noise terms so that the underlying manifold g is positively invariant under the stochastic system then we formally derive the fokker planck type equation defined on g times mathfrak g in which mathfrak g denotes the lie algebra corresponding to g after identifying the target fokker planck equation we especially consider the unitary group g mathbf u d and show that the equation on mathbf u d admits a global unique solution and that it can be rigorously derived using a stochastic mean field limit procedure with a convergence rate of order o 1 sqrt n finally we restrict our concern to g mathbf su 2 to provide explicit calculation and present the nonlinear stability of an incoherent state for the fokker planck equation depending on the relation between parameters

stochastic flocking dynamics of the cucker smale model with multiplicative white noises we present a strong asymptotic stochastic flocking estimate for the stochastically perturbed cucker smale model we characterize a form of multiplicative white noises and present sufficient conditions on the control parameters to guarantee the almost sure exponential convergence toward constant equilibrium states when the strength of multiplicative noises is sufficiently large we show that the strong stochastic flocking occurs even for negative communication weights

emergent behavior of a second order lohe matrix model on the unitary group we study a second order extension to the first order lohe matrix model on the unitary group which can be reduced to the second order kuramoto model with inertia as a special case for the proposed second order model we present several sufficient frameworks leading to the emergence of the complete and practical synchronizations in terms of the initial data and the system parameters for the identical hamiltonians we show that the complete synchronization emerges asymptotically in contrast for the non identical hamiltonians the practical synchronization occurs for some class of initial data when the product of the coupling strength and inertia is sufficiently small

emergent behavior in flocks we provide a model for both continuous and discrete time describing the evolution of a flock our model is parameterized by a constant beta capturing the rate of decay which in our model is polynomial of the influence between birds in the flock as they separate in space our main result shows that when beta 1 2 convergence of the flock to a common velocity is guaranteed while for betages1 2 convergence is guaranteed under some condition on the initial positions and velocities of the birds only

swarming patterns in a two dimensional kinematic model for biological groups we construct a continuum model for the motion of biological organisms experiencing social interactions and study its pattern forming behavior the model takes the form of a conservation law in two spatial dimensions the social interactions are modeled in the velocity term which is nonlocal in the population density and includes a parameter that controls the interaction length scale the dynamics of the resulting partial integrodifferential equation may be uniquely decomposed into incompressible motion and potential motion for the purely incompressible case the model resembles one for fluid dynamical vortex patches there exist solutions which have constant population density and compact support for all time numerical simulations produce rotating structures which have circular cores and spiral arms and are reminiscent of naturally observed phenomena such as ant mills the sign of the social interaction term determines the direction of the rotation and the interaction length scale affects the degree o