emergent dynamics of winfree oscillators on locally coupled networks abstract the winfree model is the first mathematical model for synchronization of weakly coupled oscillators compared to the well known kuramoto model the winfree model does not conserve the total phase this leads to rich dynamic features compared to those produced by other phase models in this paper we study the emergent dynamics of the winfree model on a locally coupled static network a randomly chosen phase configuration undergoes several dynamic phase transitions such as incoherence partial locking complete locking partial oscillator death and complete oscillator death as the coupling strength increases we provide several rigorous analytical results on the emergence of these dynamic features we also provide several numerical simulations and compare their results to the analytical results

synthesis of phase locking patterns in networks with cyclic group symmetries a theoretical model for the circular arrays of neurons that are found in cortical barrel structures is presented if the connection stencil is permutation invariant then the network has cyclic group symmetries the properties of cyclic groups are used to formulate an algorithm that synthesizes coupling stencils that are shown using the rotation vector method to guarantee specified patterns of phase locking in both symmetric and asymmetric arrays such patterns are related to stable combinations of firing frequencies and to memory storage in neural networks

lyapunov function for the kuramoto model of nonlinearly coupled oscillators a lyapunov function for the phase locked state of the kuramoto model of non linearly coupled oscillators is presented it is also valid for finite range interactions and allows the introduction of thermodynamic formalism such as ground states and universality classes for the kuramoto model a minimum of the lyapunov function corresponds to a ground state of a system with frustration the interaction between the oscillators xy spins is ferromagnetic whereas the random frequencies induce random fields which try to break the ferromagnetic order i e global phase locking the ensuing arguments imply asymptotic stability of the phase locked state up to degeneracy and hold for any probability distribution of the frequencies special attention is given to discrete distribution functions we argue that in this case a perfect locking on each of the sublattices which correspond to the frequencies results but that a partial locking of some but not all sublattices is not to be expected the order parameter of the phase locked state is shown to have a strictly positive lower bound r 1 2 so that a continuous transition to a nonlocked state with vanishing order parameter is to be excluded

asymptotic formation and orbital stability of phase locked states for the kuramoto model abstract we discuss the asymptotic formation and nonlinear orbital stability of phase locked states arising from the ensemble of non identical kuramoto oscillators we provide an explicit lower bound for a coupling strength on the formation of phase locked states which only depends on the diameters of natural frequencies and initial phase configurations we show that when the phases of non identical oscillators are distributed over the half circle and the coupling strength is sufficiently large the dynamics of kuramoto oscillators exhibits two stages transition and relaxation stages in a transition stage initial configurations shrink to configurations whose diameters are strictly less than 2 in a finite time and then the configurations tend to phase locked states asymptotically this improves previous results on the formation of phase locked states by chopra spong 2009 26 and ha ha kim 2010 27 where their attention were focused only on the latter relaxation stage we also show that the kuramoto model is l 1 contractive in the sense that the l 1 distance along two smooth kuramoto flows is less than or equal to that of initial configurations in particular when two initial configurations have the same averaged phases the l 1 distance between them decays to zero exponentially fast for the configurations with different phase averages we use the method of average adjustment and translation invariant of the kuramoto model to show that one solution converges to the translation of the other solution exponentially fast this establishes the orbital stability of the phase locked states our stability analysis does not employ any standard linearization technique around the given phase locked states but instead we use a robust l 1 metric functional as a lyapunov functional in the formation process of phase locked states we estimate the number of collisions between oscillators and lower upper bounds of the transversal phase differences

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

a decision theoretic evaluation of early stopping rules we consider the decision theoretic evaluation of early stopping rules for clinical trials we describe a hypothetical phase iii two arm trial with interim analysis modelling it as a decision problem we present methods for constructing a group sequential test that is optimal for a given utility and prior distribution and for determining a utility and prior under which a given test is optimal using our methods we construct optimal tests for utilities and priors representing a variety of perspectives a view that seeks to maximize the long run response rate a view that seeks to maximize the chance of a correct decision and a personal view elicited from a medical oncologist characteristics of the optimal rules are sensitive to the input utilities and priors and their critical values and sampling properties can differ markedly from those of group sequential tests we also compute utility functions and priors under which some standard group sequential stopping rules are optimal our results suggest that the group sequential tests considered as decision procedures imply a symmetry in the decision problem that is inconsistent with the response rate perspective we conclude that decision theory is a useful device for illuminating the sometimes conflicting goals of clinical research and that the use of decision theory in designing clinical trials could lead to marked changes in statistical criteria for early termination

synchronization of globally coupled phase oscillators singularities and scaling for general couplings abstract the onset of collective behavior in a population of globally coupled oscillators with randomly distributed frequencies is studied for phase dynamical models with arbitrary coupling the effect of a stochastic temporal variation in the frequencies is also included the fokker planck equation for the coupled langevin system is reduced to a kinetic equation for the oscillator distribution function instabilities of the phase incoherent state are studied by center manifold reduction to the amplitude dynamics of the unstable modes depending on the coupling the coefficients in the normal form can be singular in the limit of weak instability when the diffusive effect of the noise is neglected a detailed analysis of these singularities to all orders in the normal form expansion is presented physically the singularities are interpreted as predicting an altered scaling of the entrained component near the onset of synchronization these predictions are verified by numerically solving the kinetic equation for various couplings and frequency distributions

statistical inference for self designing clinical trials with a one sided hypothesis in the process of monitoring clinical trials it seems appealing to use the interim findings to determine whether the sample size originally planned will provide adequate power when the alternative hypothesis is true and to adjust the sample size if necessary in the present paper we propose a flexible sequential monitoring method following the work of fisher 1998 in which the maximum sample size does not have to be specified in advance the final test statistic is constructed based on a weighted average of the sequentially collected data where the weight function at each stage is determined by the observed data prior to that stage such a weight function is used to maintain the integrity of the variance of the final test statistic so that the overall type i error rate is preserved moreover the weight function plays an implicit role in termination of a trial when a treatment difference exists finally the design allows the trial to be stopped early when the efficacy result is sufficiently negative simulation studies confirm the performance of the method

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

complete aggregation of the lohe tensor model with the same free flow the lohe tensor model is a first order tensor valued continuous time dynamical system for the aggregation of tensors with the same rank and size it reduces to the well known aggregation models such as the kuramoto model the lohe sphere model and the lohe matrix model as special cases for low rank tensors we present conditions for the solution splitting property asymptotic convergence toward a set of equilibria and analyze two possible asymptotic states completely aggregate state and bi polar state which can emerge from a set of initial data moreover we provide a sufficient framework leading to the aforementioned two asymptotic states in terms of initial data and system parameters

on the inefficiency of the adaptive design for monitoring clinical trials summary adaptive designs which allow the sample size to be modified based on sequentially computed observed treatment differences have been advocated recently for monitoring clinical trials although such methods have a great deal of appeal on the surface we show that such methods are inefficient and that one can improve uniformly on such adaptive designs using standard group sequential tests based on the sequentially computed likelihood ratio test statistic

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

group sequential clinical trials a classical evaluation of bayesian decision theoretic designs abstract bayesian decision theoretic designs for a clinical trial comparing two treatments for a disease with binary outcomes are developed and evaluated the probability of successful outcome with treatment i is denoted by pi i 1 2 and prior knowledge regarding each pi is assumed to follow a beta distribution the pi are assumed to be independent to facilitate comparison with frequentist clinical trial designs we take a hypothesis testing approach the null hypothesis is 0 where 0 is the minimum treatment effect sought by the trial and p 2 p 1 is the true treatment difference we use a simple terminal loss function reflecting the hypothesis testing goal of the trial and the total cost of the trial is the final sample size plus the terminal loss function the stopping and decision rules that minimize the expectation of the total cost are determined by backward induction monte carlo simulation is used to compare bayesian and frequentist erro

emergent behaviors of lohe tensor flocks we present a new aggregation model on the space of rank m tensors with the same size and study emergent dynamics of the proposed model our proposed new aggregation model is general enough to encompass lohe type models such as the kuramoto model the lohe sphere model and the lohe matrix models for the ensemble of real rank 0 rank 1 and rank 2 tensors respectively in this regard we call our proposed model as the lohe tensor model for rank m tensors with the same size for emergent dynamics of the proposed model we employ the ensemble diameter as a lyapunov functional to derive a riccati type differential inequality as a direct application of these differential inequalities we provide two sufficient frameworks leading to the emergent dynamics for homogeneous and heterogeneous ensembles in terms of system parameters and initial data

on sample size and inference for two stage adaptive designs proschan and hunsberger 1995 biometrics 51 1315 1324 proposed a two stage adaptive design that maintains the type i error rate for practical applications a two stage adaptive design is also required to achieve a desired statistical power while limiting the maximum overall sample size in our proposal a two stage adaptive design is comprised of a main stage and an extension stage where the main stage has sufficient power to reject the null under the anticipated effect size and the extension stage allows increasing the sample size in case the true effect size is smaller than anticipated for statistical inference methods for obtaining the overall adjusted p value point estimate and confidence intervals are developed an exact two stage test procedure is also outlined for robust inference

asymmetric group sequential boundaries for monitoring clinical trials a group sequential method is proposed for the periodic analysis of accumulating data in clinical trials which are essentially one sided as in treatment placebo studies the proposed boundaries retain the early stopping properties of the procedure described previously by demets ware 1980 when treatment appears to be ineffective or harmful but incorporate the more stringent requirements of the model proposed by o brien fleming 1979 for early rejection of the null hypothesis in favour of treatment

nonlinear stability of incoherence and collective synchronization in a population of coupled oscillators a mean field model of nonlinearly coupled oscillators with randomly distributed frequencies and subject to independent external white noises is analyzed in the thermodynamic limit when the frequency distribution isbimodal new results include subcritical spontaneous stationary synchronization of the oscillators supercritical time periodic synchronization bistability and hysteretic phenomena bifurcating synchronized states are asymptotically constructed near bifurcation values of the coupling strength and theirnonlinear stability properties ascertained

optimal asymmetric one sided group sequential tests summary we extend the optimal symmetric group sequential tests of eales jennison 1992 to the broader class of asymmetric designs two forms of asymmetry are considered involving unequal type i and type ii error rates and different emphases on expected sample sizes at the null and alternative hypotheses we discuss the properties of our optimal designs and use them to assess the efficiency of the family of tests proposed by pampallona tsiatis 1994 and two families of one sided tests defined through error spending functions we show that the error spending designs are highly efficient while the easily implemented tests of pampallona tsiatis are a little less efficient but still not far from optimal our results demonstrate that asymmetric designs can decrease the expected sample size under one hypothesis but only at the expense of a significantly larger expected sample size under the other hypothesis

complete synchronization of kuramoto oscillators with finite inertia abstract we present an approach based on gronwall s inequalities for the asymptotic complete phase frequency synchronization of kuramoto oscillators with finite inertia for given finite inertia and coupling strength we present admissible classes of initial configurations and natural frequency distributions which lead to the complete phase frequency synchronization asymptotically for this we explicitly identify invariant regions for the kuramoto flow and derive second order gronwall s inequalities for the evolution of phase and frequency diameters our detailed time decay estimates for phase and frequency diameters are independent of the number of oscillators we also compare our analytical results with numerical simulations

higher dimensional generalizations of the watanabe strogatz transform for vector models of synchronization we generalize the watanabe strogatz ws transform which acts on the kuramoto model in d 2 dimensions to a higher dimensional vector transform which operates on vector oscillator models of synchronization in any dimension for the case of identical frequency matrices these models have conserved quantities constructed from the cross ratios of inner products of the vector variables which are invariant under the vector transform and have trajectories which lie on the unit sphere s d 1 application of the vector transform leads to a partial integration of the equations of motion leaving independent equations to be solved for any number of nodes n we discuss properties of complete synchronization and use the reduced equations to derive a stability condition for completely synchronized trajectories on s d 1 we further generalize the vector transform to a mapping which acts in and in particular preserves the unit ball and leaves invariant the cross ratios constructed from inner products of vectors in this mapping can be used to partially integrate a system of vector oscillators with trajectories in and for d 2 leads to an extension of the kuramoto system to a system of oscillators with time dependent amplitudes and trajectories in the unit disk we find an inequivalent generalization of the mobius map which also preserves but leaves invariant a different set of cross ratios this time constructed from the vector norms this leads to a different extension of the kuramoto model with trajectories in the complex plane that can be partially integrated by means of fractional linear transformations

robustness and asymptotic stability for the winfree model on a general network under the effect of time delay we study the robustness and asymptotic stability of the unique equilibrium for the winfree model on a general network under the effect of small time delay time delay is a ubiquitous mechanism in classical and quantum many body systems and it is due to the finite speed of communication in this paper we further investigate the robustness of the equilibrium under the effects of small time delay and general network topology for this we present several sufficient frameworks formulated in terms of the network topology coupling strength time delay and initial configuration in the proposed framework we present a positively invariant arc which contains the unique equilibrium independent of strength of time delay by using energy estimates we show that the unique equilibrium in the invariant arc is asymptotically stable we also provide several numerical simulations and compare them with analytical results we study the robustness and asymptotic stability of the unique equilibrium for the winfree model on a general network under the effect of small time delay time delay is a ubiquitous mechanism in classical and quantum many body systems and it is due to the finite speed of communication in this paper we further investigate the robustness of the equilibrium under the effects of small time delay and general network topology for this we present several sufficient frameworks formulated in terms of the network topology coupling strength time delay and initial configuration in the proposed framework we present a positively invariant arc which contains the unique equilibrium independent of strength of time delay by using energy estimates we show that the unique equilibrium in the invariant arc is asymptotically stable we also provide several numerical simulations and compare them with analytical results

the asymptotic distribution of sums of extreme values from a regularly varying distribution on donne une description complete de la distribution asymptotique de la somme des k n valeurs extremes superieures x n 1 kd n n x n 2 kd n n x n n

the mathematics of asymptotic stability in the kuramoto model now a standard in nonlinear sciences the kuramoto model is the perfect example of the transition to synchrony in heterogeneous systems of coupled oscillators while its basic phenomenology has been sketched in early works the corresponding rigorous validation has long remained problematic and was achieved only recently this paper reviews the mathematical results on asymptotic stability of stationary solutions in the continuum limit of the kuramoto model and provides insights into the principal arguments of proofs this review is complemented with additional original results various examples and possible extensions to some variations of the model in the literature

on the global well posedness of bv weak solutions to the kuramoto sakaguchi equation abstract the kuramoto model is a prototype phase model describing the synchronous behavior of weakly coupled limit cycle oscillators when the number of oscillators is sufficiently large the dynamics of kuramoto ensemble can be effectively approximated by the corresponding mean field equation namely the kuramoto sakaguchi ks equation this ks equation is a kind of scalar conservation law with a nonlocal flux function due to the mean field interactions among oscillators in this paper we provide a unique global solvability of bounded variation bv weak solutions to the kinetic ks equation for identical oscillators using the method of front tracking in hyperbolic conservation laws moreover we also show that our bv weak solutions satisfy local in time l 1 stability with respect to bv initial data for the ensemble of identical kuramoto oscillators we explicitly construct an exponentially growing bv weak solution generated from bv perturbation of incoherent state for any positive coupling strength this implies the nonlinear instability of incoherent state in a positive coupling strength regime we provide several numerical examples and compare them with our analytical results

on the schrodinger lohe hierarchy for aggregation and its emergent dynamics the lohe hierarchy is a hierarchy of finite dimensional aggregation models consisting of the kuramoto model the complex lohe sphere model the lohe matrix model and the lohe tensor model in contrast the schrodinger lohe model is the only known infinite dimensional lohe aggregation model in literature in this paper we provide an explicit connection between the schrodinger lohe model and the complex lohe sphere model and then by exploiting this explicit relation we construct infinite dimensional liftings of the lohe matrix and the lohe tensor models in this way we establish the schrodinger lohe hierarchy which corresponds to the infinite dimensional extensions of the lohe hierarchy for the proposed hierarchy we provide sufficient frameworks leading to the complete aggregation in terms of coupling strengths and initial configurations

on the finiteness of collisions and phase locked states for the kuramoto model synchronization phenomenon is ubiquitous in our complex systems and many phenomenological models have been proposed and studied analytically and numerically among them the kuramoto model serves as a prototype model for the phase synchronization of weakly coupled oscillators in this paper we study the finiteness of collisions crossings among kuramoto oscillators in the relaxation process toward the phase locked states and the total number of phase locked states with positive kuramoto order parameters for identical oscillators it is well known that collisions between distinct oscillators cannot occur in finite time and we show that there are only a finite number of phase locked states with positive order parameters however for non identical oscillators oscillators with different natural frequencies can cross each other in their relaxation process and estimating the total number of phase locked states is a nontrivial matter we show that for the non identical case asymptotic phase locking is equivalent to the finiteness of collisions and the total number of phase locked states with positive order parameters is bounded above by 2 n where n is the number of oscillators

stability of an n 2 dimensional invariant torus in the kuramoto model at small coupling abstract when the natural frequencies are allocated symmetrically in the kuramoto model there exists an invariant torus of dimension n 2 1 n is the population size a global phase shift invariance allows us to reduce the model to n 1 dimensions using the phase differences and doing so the invariant torus becomes n 2 dimensional by means of perturbative calculations based on the renormalization group technique we show that this torus is asymptotically stable at small coupling if n is odd if n is even the torus can be stable or unstable depending on the natural frequencies and both possibilities persist in the small coupling limit

on the synchronization theory of kuramoto oscillators under the effect of inertia abstract we investigate the synchronized collective behavior of the kuramoto oscillators with inertia effect both the frequency synchronization for nonidentical case and the phase synchronization for identical case are in view as an application of our general theory we show the global frequency synchronization for the three nonidentical oscillator case

stability of incoherence in a population of coupled oscillators we analyze a mean field model of coupled oscillators with randomly distributed frequencies this system is known to exhibit a transition to collective oscillations for small coupling the system is incoherent with all the oscillators running at their natural frequencies but when the coupling exceeds a certain threshold the system spontaneously synchronizes we obtain the first rigorous stability results for this model by linearizing the fokker planck equation about the incoherent state an unexpected result is that the system has pathological stability properties the incoherent state is unstable above threshold butneutrally stable below threshold we also show that the system is singular in the sense that its stability properties are radically altered by infinitesimal noise

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

adaptive group sequential designs for clinical trials combining the advantages of adaptive and of classical group sequential approaches a general method is presented integrating the concept of adaptive interim analyses into classical group sequential testing this allows the researcher to represent every group sequential plan as an adaptive trial design and to make design changes during the course of the trial after every interim analysis in the same way as with adaptive designs the concept of adaptive trial designing is thereby generalized to a large variety of possible sequential plans

relative stability of trimmed sums this paper gives extensions of mori s strong law for r s n s n x n 1 x n r where s n x1 x2 x n x i are iidrv s and x i n is x i arranged in decreasing order of absolute magnitude the methods differ from mori s continuity of the distribution of the x i is assumed throughout necessary and sufficient conditions for relative stability r s n b n 1 a s for some b n including a generalised condition of spitzer s and a dominated ergodic theorem are proved a one sided version of the relative stability results is also given a theorem of kesten s is generalised to show that if r s n a n b n is bounded almost surely for constants a n b n then r s n alpha n b n xrightarrow p 0 for some n a corollary to this is that if r s n b n is bounded away from 0 and a s then r s n is relatively stable this generalises a result of chow and robbins apart from the continuity assumption

complete entrainment of kuramoto oscillators with inertia on networks via gradient like flow abstract we study the asymptotic complete entrainment of kuramoto oscillators with inertia on symmetric and connected network we provide several sufficient conditions for the asymptotic complete entrainment in terms of initial phase frequency configurations strengths of inertia and coupling and natural frequency distributions for this purpose we reinterpret the kuramoto oscillators with inertia as a second order gradient like flow and adopt analytical methods based on several lyapunov functions to apply the convergence estimate studied by haraux and jendoubi 21 our approach does not require any spectral information of the graph associated with the given network structure

bayesian adaptive designs for clinical trials summary a bayesian adaptive design is proposed for a comparative two armed clinical trial using decision theoretic approaches a loss function is specified based on the cost for each patient and the costs of making incorrect decisions at the end of a trial at each interim analysis the decision to terminate or to continue the trial is based on the expected loss function while concurrently incorporating efficacy futility and cost the maximum number of interim analyses is determined adaptively by the observed data we derive explicit connections between the loss function and the frequentist error rates so that the desired frequentist properties can be maintained for regulatory settings the operating character istics of the design can be evaluated on frequentist grounds extensive simulations are carried out to compare the proposed design with existing ones the design is general enough to accommodate both continuous and discrete types of data we illustrate the methods with an animal study evaluating a medical treatment for cardiac arrest

a shocking display of synchrony abstract this article explores the kuramoto model describing the synchronization of a population of coupled oscillators two versions of this model are considered a discrete version suitable for a population with a finite number of oscillators and a continuum model found in the limit of an infinite population when the strength of the coupling between the oscillators exceeds a threshold the oscillators partially synchronize we explore the transition in the continuum model which takes the form of a bifurcation of a discrete mode from a continuous spectrum we use numerical methods and perturbation theory to study the patterns of synchronization that form beyond transition and compare with the synchronization predicted by the discrete model there are similarities with instabilities in ideal plasmas and inviscid fluids but these are superficial

systems of matrix riccati equations linear fractional transformations partial integrability and synchronization we partially integrate a system of rectangular matrix riccati equations which describe the synchronization behavior of a nonlinear complex system of n globally connected oscillators the equations take a restricted form in which the time dependent matrix coefficients are independent of the node we use linear fractional transformations to perform the partial integration resulting in a system of reduced size which is independent of n generalizing the well known watanabe strogatz reduction for the kuramoto model for square matrices the resulting constants of motion are related to the eigenvalues of matrix cross ratios which we show satisfy various properties such as symmetry relations for square matrices the variables can be regarded as elements of a classical lie group not necessarily compact satisfying the matrix riccati equations trajectories lie either within or on the boundary of a classical domain and we show by numerical example that complete synchronization can occur even for the mixed case provided that certain unitarity conditions are satisfied we extend the definition of cross ratios to rectangular matrix systems and show that again the eigenvalues are conserved special cases are models with real vector unknowns for which trajectories lie on the unit sphere in higher dimensions with well known synchronization behavior and models with complex vector wavefunctions that describe synchronization in quantum systems possibly infinite dimensional we partially integrate a system of rectangular matrix riccati equations which describe the synchronization behavior of a nonlinear complex system of n globally connected oscillators the equations take a restricted form in which the time dependent matrix coefficients are independent of the node we use linear fractional transformations to perform the partial integration resulting in a system of reduced size which is independent of n generalizing the well known watanabe strogatz reduction for the kuramoto model for square matrices the resulting constants of motion are related to the eigenvalues of matrix cross ratios which we show satisfy various properties such as symmetry relations for square matrices the variables can be regarded as elements of a classical lie group not necessarily compact satisfying the matrix riccati equations trajectories lie either within or on the boundary of a classical domain and we show by numerical example that complete synchronization can occur even for the mixed ca

sequential monitoring of response adaptive randomized clinical trials clinical trials are complex and usually involve multiple objectives such as controlling type i error rate increasing power to detect treatment difference assigning more patients to better treatment and more in literature both response adaptive randomization rar procedures by changing randomization procedure sequentially and sequential monitoring by changing analysis procedure sequentially have been proposed to achieve these objectives to some degree in this paper we propose to sequentially monitor response adaptive randomized clinical trial and study it s properties we prove that the sequential test statistics of the new procedure converge to a brownian motion in distribution further we show that the sequential test statistics asymptotically satisfy the canonical joint distribution defined in jennison and turnbull 2000 therefore type i error and other objectives can be achieved theoretically by selecting appropriate boundaries these results open a door to sequentially monitor response adaptive randomized clinical trials in practice we can also observe from the simulation studies that the proposed procedure brings together the advantages of both techniques in dealing with power total sample size and total failure numbers while keeps the type i error in addition we illustrate the characteristics of the proposed procedure by redesigning a well known clinical trial of maternal infant hiv transmission

on the relaxation dynamics of the kuramoto oscillators with small inertia for the kuramoto oscillators with small inertia we present several quantitative estimates on the relaxation dynamics and formational structure of a phase locked state pls for some classes of initial configurations in a super critical regime where the coupling strength is strictly larger than the diameter of natural frequencies we present quantitative relaxation dynamics on the collision numbers and the structure of pls in a critical coupling regime where the coupling strength is exactly the diameter of natural frequencies we provide a sufficient condition for an asymptotically pls solution in particular we show the existence of slow relaxation to a pls when there are exactly two natural frequencies this generalizes the earlier results of choi et al asymptotic formation and orbital stability of phase locked states for the kuramoto model physica d 241 735 754 2012 10 1016 j physd 2011 11 011 choi et al complete synchronization of kuramoto oscillators with finite inertia physica d 240

contractivity of transport distances for the kinetic kuramoto equation we present synchronization and contractivity estimates for the kinetic kuramoto model obtained from the kuramoto phase model in the mean field limit for identical kuramoto oscillators we present an admissible class of initial data leading to time asymptotic complete synchronization that is all measure valued solutions converge to the traveling dirac measure concentrated on the initial averaged phase in the case of non identical oscillators we show that the velocity field converges to the average natural frequency proving that the oscillators move asymptotically with the same frequency under suitable assumptions on the initial configuration if two initial radon measures have the same natural frequency density function and strength of coupling we show that the wasserstein p distance between corresponding measure valued solutions is exponentially decreasing in time this contraction principle is more general than previous l 1 contraction properties of the kuramoto phase model

an improved method for deriving optimal one sided group sequential tests summary we derive one sided group sequential tests for normal responses which minimize expected sample size minimization is at a single value of the normal mean or integrated with respect to a normal density the methods employed are much faster and also numerically more stable than those of jennison 1987 they provide solutions for cases with as many as 200 groups and for small numbers of groups facilitate optimization over the choice of group sizes we present a new method for constructing near optimal tests when group sizes are unpredictable based on interpolating the error spending function of an optimal test with a large number of groups

emergent behaviors of the schrodinger lohe model on cooperative competitive networks abstract we present several sufficient frameworks leading to the emergent behaviors of the coupled schrodinger lohe s l model under the same one body external potential on cooperative competitive networks the s l model was first introduced as a possible phenomenological model exhibiting quantum synchronization and its emergent dynamics on all to all cooperative networks has been treated via two distinct approaches lyapunov functional approach and the finite dimensional reduction based on pairwise correlations in this paper we further generalize the finite dimensional dynamical systems approach for pairwise correlation functions on cooperative competitive networks and provide several sufficient frameworks leading to the collective exponential synchronization for small systems consisting of three and four quantum subsystem we also show that the system for pairwise correlations can be reduced to the lotka volterra model with cooperative and competitive interactions in which lots of interesting dynamical patterns appear e g existence of closed orbits and limit cycles

formation of phase locked states in a population of locally interacting kuramoto oscillators abstract in this paper we present an asymptotic formation of phase locked states from the ensemble of kuramoto oscillators with a symmetric and connected interaction topology for a limited interaction topology that does not have an all to all interaction lyapunov type approaches based on phase and frequency diameters do not work due to the lack of completeness thus we employ an energy method together with the connectedness of underlying interaction topologies to determine the complete synchronization estimates our synchronization estimation method consists of two parts first we establish that the uniform boundedness of fluctuations yields the asymptotic formation of phase locked states using ojasiewicz gradient inequality second we show that for the initial configurations lying in the half circle the uniform boundedness of fluctuations can be derived by a comparison with solutions to the linear gronwall s differential inequality for the total phase variance

strong limit theorems for weighted quantile processes on donne une description complete du comportement presque sur des versions ponderees du processus quantile uniforme

on the relationship between the almost sure stability of weighted empirical distributions and sums of order statistics we shall disclose a relationship between the almost sure stability of weighted empirical distribution functions and sums of order statistics first we obtain an extension of a theorem due to csaki on the almost sure stability of the standardized uniform empirical distribution function this result is then shown to be an essential tool to derive a characterization of the almost sure stability of the sum of k nupper order statistics from a sample of n independent observations from a distribution with positive support in the domain of attraction of a non normal stable law where 1 k n n and k n as n

the spectrum of the locked state for the kuramoto model of coupled oscillators we analyze the linear stability of the phase locked state in the kuramoto model of coupled oscillators the main result is the first rigorous characterization of the spectrum and its associated eigenvectors for any finite number of oscillators all but two of the eigenvalues are negative and merge into a continuous spectrum as the number of oscillators tends to infinity one eigenvalue is always zero by rotational invariance the final eigenvalue corresponding to a collective mode determines the stability of the locked state

adaptive sample size calculations in group sequential trials a method for group sequential trials that is based on the inverse normal method for combining the results of the separate stages is proposed without exaggerating the type i error rate this method enables data driven sample size reassessments during the course of the study it uses the stopping boundaries of the classical group sequential tests furthermore exact test procedures may be derived for a wide range of applications the procedure is compared with the classical designs in terms of power and expected sample size

sample size reestimation for clinical trials with censored survival data a flexible design with updating of sample size in clinical trials is proposed for censored survival data patients enter trials serially and are subject to random loss to follow up the statistical inference is based on a weighted average of the linear rank statistics where the weight function at each look depends on prior observed data a stopping rule is devised to allow early termination and acceptance of the null hypothesis when the experimental treatment offers no advantage or is inferior to the control the null hypothesis that two survival distributions are equal may be rejected only at the last step when the weight function is used up the overall type i error rate is preserved independent increments for the sequential linear rank statistic is key to deriving asymptotic properties of the test statistic some extensive simulations have been carried out to compare the operating characteristics of the method under different scenarios and for a comparison with the usual log rank test with fixed sam

practical quantum synchronization for the schrodinger lohe system we present a practical synchronization method for the schrodinger lohe s l system distinct potentials the s l model describes the spatial temporal evolution of the wave functions of quantum lohe oscillators on a network with lohe couplings when the potential effects are ignored complete wave function synchronization cwfs can emerge in the sense that the l 2 distance between wave functions exponentially approaches zero for a class of initial wave functions in contrast when the lohe oscillators are under the effect of potential forces cwfs cannot occur in this study we employ a weaker concept of quantum synchronization for discussing the asymptotic collective behavior of the s l model this concept leads to practical synchronization of the s l model in practical synchronization the l 2 distance between wave functions can be upper bounded by the inverse power of the square root of the coupling strength as the coupling strength increases thus the l 2 discrepancy between wave functions arbitrarily decreases as the coupling strength increases we present a sufficient analytical framework for this practical synchronization which is a generalization of the earlier result in choi s h and ha s y 2014 j phys a math theor 47 355104

on the complete synchronization of the kuramoto phase model abstract we discuss the asymptotic complete phase frequency synchronization for the kuramoto phase model with a finite size n we present sufficient conditions for initial configurations leading to the exponential decay toward the completely synchronized states our new sufficient conditions and decay rate depend only on the coupling strength and the diameter of initial phase and natural frequency configurations but they are independent of the system size n hence they can be used for the mean field limit for the complete synchronization estimates we estimate the time evolution of the phase and frequency diameters for configurations the initial phase configurations for identical oscillators located on the half circle will converge to the complete synchronized states exponentially fast in contrast for the non identical oscillators the complete frequency synchronization will occur exponentially fast for some restricted class of initial phase configurations our estimates are based on the monotonicity arguments of extremal phase and frequencies which do not employ any linearization procedure of nonlinear coupling terms and detailed information on the eigenvalue of the linearized system around the complete synchronized states we compare our analytical results with numerical simulations

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

partial entrainment in the finite kuramoto sakaguchi model abstract although modifications of the kuramoto model have been the subject of extensive research the model itself is not yet fully understood we offer several results and observations some analytic others through simulations we derive a sufficient condition for the existence of a solution exhibiting partial entrainment with respect to a given subset of oscillators the result also implies persistence of the entrainment behavior under perturbations the critical values of the coupling strength defining the transitions between different forms of partial entrainment are predicted by an analytical approximation based on the fact that oscillators with large differences in their natural frequencies have little influence on each other s entrainment behavior the predictions agree with the actual values obtained by simulations we indicate by simulations that entrainment can disappear with increasing coupling strength and that in arrays of josephson junctions a similar phenomenon can be observed where it is also possible that a junction leaving one entrained subset joins another entrained subset

on exponential synchronization of kuramoto oscillators in this technical note we study the problem of exponential synchronization for one of the most popular models of coupled phase oscillators the kuramoto model we consider the special case of finite oscillators with distinct bounded natural frequencies our first result derives a lower bound on the coupling gain which is necessary for the onset of synchronization this bound improves the one derived by jadbabaie we then calculate a lower bound on the coupling gain that is sufficient to guarantee oscillator synchronization and derive further sufficient conditions to ensure exponential synchronization of the angular frequencies of all oscillators to the mean natural frequency of the group we also characterize the coupling gain that is sufficient for the oscillator phase differences to approach any desired compact set in finite time

designed extension of studies based on conditional power we propose a flexible method of extending a study based on conditional power the possibility for extension when the p value at the planned end is small but not statistically significant is built in to the design of the study the significance of the treatment difference at the planned end is used to determine the number of additional observations needed and the critical value necessary for use after accruing those additional observations it may therefore be thought of as a two stage procedure even though the observed treatment difference at stage 1 is used to make decisions the type i error rate is protected

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

on the łojasiewicz exponent of kuramoto model the ojasiewicz inequality and ojasiewicz exponent reveal a fundamental relation between a potential function and its gradient in this paper we explore the ojasiewicz exponent of kuramoto model and prove that the exponent is exactly 12 for equilibria located inside a quarter of circle this implies that the convergence towards such a phase locked state must be exponentially fast in contrast we give an example to see the exponent can be less than 12 for other equilibriums more precisely we prove that the exponent for the bi cluster equilibrium which is located on the boundary of a quarter of circle is 13 this gives an insight for the occurrence of exponential and algebraic convergence of kuramoto model we also present a general theorem for exponential convergence of second order gradient like system by which a criterion for the kuramoto model with inertia is established