stability of runge kutta methods for linear impulsive delay differential equations with piecewise constant arguments abstract this paper is concerned with a class of linear impulsive delay differential equations with piecewise continuous argument the conditions for stability of the exact solution are obtained under these conditions it is proven that the method preserves stability of the equations furthermore stability of runge kutta methods for this kind of equations is studied some numerical examples are given to confirm the theoretical results

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

stability of runge kutta methods in the numerical solution of equation u t au t a 0 u t this paper is concerned with the stability analysis of the runge kutta methods for the equation u t au t a0u t the stability regions for the runge kutta methods are determined the conditions that the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given

stability of solutions of nonlinear neutral differential equations with piecewise constant delay and their discretizations this paper is devoted to stability analysis of solutions of nonlinear neutral differential equations with piecewise constant delay these results form the basis for obtaining insight into the analogous properties of numerical solutions generated by bn f stable runge kutta methods it is showed theoretically and numerically that two classes of runge kutta methods considered in literature are contractive and asymptotically stable numerical experiments also demonstrate that a class of runge kutta methods can preserve their original convergence order for odes but not the other class of runge kutta methods

the quantum group structure of 2d gravity and minimal models i on the unit circle an infinite family of chiral operators is constructed whose exchange algebra is given by the universalr matrix of the quantum groupsl 2 q this establishes the precise connection between the chiral algebra of two dimensional gravity or minimal models and this quantum group the method is to relate the monodromy properties of the operator differential equations satisfied by the generalized vertex operators with the exchange algebra ofsl 2 q the formulae so derived which generalize an earlier particular case worked out by babelon are remarkably compact and may be entirely written in terms of q deformed factorials and binomial coefficients

stability of runge kutta methods in the numerical solution of linear impulsive differential equations this paper deals with the stability analysis of the analytic and numerical solutions of linear impulsive differential equations the numerical method with variable stepsize is defined the conditions that the numerical solutions preserve the stability property of the analytic ones are obtained and some numerical experiments are given

runge kutta methods for first order periodic boundary value differential equations with piecewise constant arguments in this paper we deal with the numerical solutions of runge kutta methods for first order periodic boundary value differential equations with piecewise constant arguments the numerical solution is given by the numerical green s function it is shown that runge kutta methods preserve their original order for first order periodic boundary value differential equations with piecewise constant arguments we give the conditions under which the numerical solutions preserve some properties of the analytic solutions e g uniqueness and comparison theorems finally some experiments are given to illustrate our results

convergence and asymptotic stability of galerkin methods for a partial differential equation with piecewise constant argument the paper deals with the convergence and asymptotic stability of galerkin methods for a partial differential equation with piecewise constant argument the optimal convergence orders are obtained for the semidiscrete and full discrete backward euler methods respectively both the discrete solutions are proved to be asymptotically stable under the condition that the analytical solution is asymptotically stable

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

stability analysis of runge kutta methods for unbounded retarded differential equations with piecewise continuous arguments abstract this paper deals with the asymptotical stability of the analytic solutions for the unbounded retarded differential equations with piecewise continuous arguments epca u t f t u t u t n t 0 u 0 u 0 where n z a sufficient condition that the differential equations are asymptotically stable is derived this paper is also concerned with the stability analysis of the runge kutta methods for equation u t au t bu t n t 0 u 0 u 0 where n z the conditions that the numerical solutions preserve the stability of the analytic solutions are obtained and some numerical experiments are given

asymptotic stability of runge kutta methods for nonlinear differential equations with piecewise continuous arguments this paper deals with the asymptotic stability of numerical solutions for differential equations with piecewise continuous arguments epcas the necessary and sufficient condition is given for non confluent runge kutta methods to preserve the stability of nonlinear scalar epcas as for systems we prove that some algebraically stable methods can preserve the asymptotic stability

localization transition on complex networks via spectral statistics the spectral statistics of complex networks are numerically studied the features of the anderson metal insulator transition are found to be similar for a wide range of different networks a metal insulator transition as a function of the disorder can be observed for different classes of complex networks for which the average connectivity is small the critical index of the transition corresponds to the mean field expectation when the connectivity is higher the amount of disorder needed to reach a certain degree of localization is proportional to the average connectivity though a precise transition cannot be identified the absence of a clear transition at high connectivity is probably due to the very compact structure of the highly connected networks resulting in a small diameter even for a large number of sites

uncovering disassortativity in large scale free networks mixing patterns in large self organizing networks such as the internet the world wide web and social and biological networks are often characterized by degree degree dependencies between neighboring nodes in this paper we propose a new way of measuring degree degree dependencies one of the problems with the commonly used assortativity coefficient is that in disassortative networks its magnitude decreases with the network size we mathematically explain this phenomenon and validate the results on synthetic graphs and real world network data as an alternative we suggest to use rank correlation measures such as spearman s rho our experiments convincingly show that spearman s rho produces consistent values in graphs of different sizes but similar structure and it is able to reveal strong positive or negative dependencies in large graphs in particular we discover much stronger negative degree degree dependencies in web graphs than was previously thought rank correlations allow us to compare the assortativity of networks of different sizes which is impossible with the assortativity coefficient due to its genuine dependence on the network size we conclude that rank correlations provide a suitable and informative method for uncovering network mixing patterns

stability of θ methods for advanced differential equations with piecewise continuous arguments abstract this paper deals with the stability analysis of numerical methods for the solution of advanced differential equations with piecewise continuous arguments we focus on the behaviour of the one leg method and the linear method in the solution of the equation x t ax t a 0 x t a 1 x t 1 with real a a 0 a 1 and designates the greatest integer function the stability regions of two methods are determined the conditions under which the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given

stability of runge kutta methods in the numerical solution of equation u t au t a0u t a1u t 1 this paper is concerned with the stability analysis of the runge kutta methods for the equation u t au t a 0u t a 1u t 1 the stability regions for the runge kutta methods are determined the conditions that the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given

q analogues of clifford and weyl algebras spinor and oscillator representations of quantum enveloping algebras we introduceq analogues of clifford and weyl algebras using these we construct spinor and oscillator representations of quantum enveloping algebras of typean 1 bn cn dn andan 1 1 also we discuss the irreducibility and the unitarity of these representations

complex networks renormalization flows and fixed points recently it has been claimed that some complex networks are self similar under a convenient renormalization procedure we present a general method to study renormalization flows in graphs we find that the behavior of some variables under renormalization such as the maximum number of connections of a node obeys simple scaling laws characterized by critical exponents this is true for any class of graphs from random to scale free networks from lattices to hierarchical graphs therefore renormalization flows for graphs are similar as in the renormalization of spin systems an analysis of classic renormalization for percolation and the ising model on the lattice confirms this analogy critical exponents and scaling functions can be used to classify graphs in universality classes and to uncover similarities between graphs that are inaccessible to a standard analysis

preservation of oscillations of the runge kutta method for equation x t ax t a1x t 1 0 the paper deals with the preservation of oscillations of the runge kutta method for equation x t ax t a 1x t 1 0 it is proved that oscillations of the analytic solution are preserved by the runge kutta method special interpolation functions of the numerical solutions are given it turns out that zeros of the interpolation function of the numerical solution converge to ones of the analytic solution with the same order of accuracy as that of the corresponding runge kutta method some numerical experiments are presented

power law distribution of degree degree distance a better representation of the scale free property of complex networks whether real world complex networks are scale free or not has long been controversial recently in broido and clauset a d broido a clauset nat commun 10 1017 2019 it was claimed that the degree distributions of real world networks are rarely power law under statistical tests here we attempt to address this issue by defining a fundamental property possessed by each link the degree degree distance the distribution of which also shows signs of being power law by our empirical study surprisingly although full range statistical tests show that degree distributions are not often power law in real world networks we find that in more than half of the cases the degree degree distance distributions can still be described by power laws to explain these findings we introduce a bidirectional preferential selection model where the link configuration is a randomly weighted two way selection process the model does not always produce solid power law distributions but predicts that the degree degree distance distribution exhibits stronger power law behavior than the degree distribution of a finite size network especially when the network is dense we test the strength of our model and its predictive power by examining how real world networks evolve into an overly dense stage and how the corresponding distributions change we propose that being scale free is a property of a complex network that should be determined by its underlying mechanism e g preferential attachment rather than by apparent distribution statistics of finite size we thus conclude that the degree degree distance distribution better represents the scale free property of a complex network

stability analysis of runge kutta methods for systems u t lu t mu t this paper deals with the stability of runge kutta methods applied to the complex linear system u t lu t mu t the condition under which the numerical solution is asymptotically stable is presented which is stronger than a stability and weaker than a f stability furthermore in the case of 2 norm and l being a real symmetric matrix by using email protected approximation and order star theory it is proved that for a stable runge kutta methods suppose whose stability function is given by the r s email protected approximation to e x the numerical solution is asymptotically stable if and only if r is even

tensor operators for quantum groups and applications tensor operators are defined for hopf algebras if the hopf algebra is quasitriangular the properly defined tensor product of tensor operators is a tensor operator in one application the problem of defining invariant correlation functions for the uq sl 2 symmetric spin 1 2 heisenberg chain is considered in another application it is shown that suitably modified bosonic and fermionic q creation and annihilation operators are tensor operators of uq sl n