construction of high order runge kutta methods which preserve delay dependent stability of ddes this paper is concerned with the construction of some high order runge kutta methods which preserve delay dependent stability of delay differential equations the methods of the first kind are developed by extending the ideas of brugnano et al while the methods of the second kind are developed according to the structure of the stability matrix we show that the derived methods are 0 stable for delay differential equations meanwhile the runge kutta methods can own the same order of the accuracy as the radau methods or gauss methods if the parameters are adequately defined these results not only improve the order of accuracy of the methods investigated by huang but also open an interesting route of finding new 0 stable runge kutta methods finally numerical experiments are proposed to illustrate the theoretical results

exponential stability of impulsive systems with application to uncertain sampled data systems we establish exponential stability of nonlinear time varying impulsive systems by employing lyapunov functions with discontinuity at the impulse times our stability conditions have the property that when specialized to linear impulsive systems the stability tests can be formulated as linear matrix inequalities lmis then we consider lti uncertain sampled data systems in which there are two sources of uncertainty the values of the process parameters can be unknown while satisfying a polytopic condition and the sampling intervals can be uncertain and variable we model such systems as linear impulsive systems and we apply our theorem to the analysis and state feedback stabilization we find a positive constant which determines an upper bound on the sampling intervals for which the stability of the closed loop is guaranteed the control design lmis also provide controller gains that can be used to stabilize the process we also consider sampled data systems with constant sampling intervals and provide results that are less conservative than the ones obtained for variable sampling intervals

on the asymptotic stability ofθ methods for delay differential equations stability regions of methods for the linear delay differential test equations begin gathered y t py t qy t tau t 0 hfill y t varphi t t in tau 0 hfill end gathered where is a positive constant are presented in the case thatp andq are real constant coefficients necessary and sufficient conditions on the stepsize for the stability of a method are obtained furthermore whenp andq are complex coefficients sufficient conditions for the stability of the methods are also given

convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled data systems under dwell time constraints stability analysis and control of linear impulsive systems is addressed in a hybrid framework through the use of continuous time time varying discontinuous lyapunov functions necessary and sufficient conditions for stability of impulsive systems with periodic impulses are first provided in order to set up the main ideas extensions to the stability of aperiodic systems under minimum maximum and ranged dwell times are then derived by exploiting further the particular structure of the stability conditions the results are non conservatively extended to quadratic stability analysis of linear uncertain impulsive systems these stability criteria are in turn losslessly extended to stabilization using a particular yet broad enough class of state feedback controllers providing then a convex solution to the open problem of robust dwell time stabilization of impulsive systems using hybrid stability criteria relying finally on the representability of sampled data systems as impulsive systems the problems of robust stability analysis and robust stabilization of periodic and aperiodic uncertain sampled data systems are straightforwardly solved using the same ideas several examples are discussed in order to show the effectiveness and reduced complexity of the proposed approach

analytical and numerical stability of neutral delay integro differential equations and neutral delay partial differential equations this paper is concerned with the analytical and numerical stability of neutral delay integro differential equations ndides and neutral delay partial differential equations ndpdes we study the delay dependent stability of the real coefficient linear test equations for ndides furthermore we prove that the trapezium rule can preserve the delay dependent stability of the test equations considered we also discuss the delay dependent stability of the continuous problems the semi discrete problems and the fully discrete problems of linear ndpdes some numerical experiments are given to confirm the theoretical results

on the asymptotic stability properties of runge kutta methods for delay differential equations in this paper asymptotic stability properties of runge kutta r k methods for delay differential equations ddes are considered with respect to the following test equation begin eqnarray left begin array ccl y t a y t b y t tau 0 3cm y t g t end array right hskip 0 3cm begin array r t 0 0 3cm tau leq t leq 0 end array end eqnarray where a b in bbb r tau 0 and g t is a continuous real valued function in the last few years stability properties of r k methods applied to ddes have been studied by numerous authors who have considered regions of asymptotic stability for any positive delay and thus independent of the specific value of tau

technical communique stability and stabilization of nonuniform sampling systems this paper is concerned with nonuniform sampling systems where the sampling interval is time varying within a certain known bound the system is transformed into a time varying discrete time system where time varying parts due to the sampling interval variation are treated as norm bounded uncertainties using robust control techniques to reduce conservatism arising from modeling time varying parts as a single uncertainty the time varying parts are modeled as n uncertainties with larger n a less conservative stability condition is derived at sacrifice of more computation it is shown through a numerical example that the proposed stability condition is better than existing stability conditions

corrigendum to polytopes of high rank for the symmetric groups adv math 228 2011 3207 3222 abstract in lemma 13 of fernandes and leemans 2011 1 it is shown that if is a rank n 2 string c group isomorphic to s n then each of the maximal parabolic subgroups i is not a transitive imprimitive group however there is a flaw in the arguments showing that no maximal parabolic subgroup can be embedded in s 2 s n 2 this lemma is essential to the proof of theorem 2 of 1 which states that for n 7 there exists up to isomorphism and duality a unique regular n 2 polytope having a group s n as its automorphism group this paper serves to correct the previous arguments and thus complete the proof of one of the main theorems about polytopes of high rank for the symmetric groups

unconditionally stable difference methods for delay partial differential equations this paper is concerned with the numerical solution of parabolic partial differential equations with time delay we focus in particular on the delay dependent stability analysis of difference methods that use a non constrained mesh i e the time step size is not required to be a submultiple of the delay we prove that the fully discrete system unconditionally preserves the delay dependent asymptotic stability of the linear test problem under consideration when the following discretization is used a variant of the classical second order central differences to approximate the diffusion operator a linear interpolation to approximate the delay argument and finally the trapezoidal rule or the second order backward differentiation formula to discretize the time derivative we end the paper with some numerical experiments that confirm the theoretical results

numerical solution of constant coefficient linear delay differential equations as abstract cauchy problems summary in this paper we present an approach for the numerical solution of delay differential equations begin equation left begin array l y prime left t right ly left t right my left t tau right t geq 0 y left t right varphi left t right tau leq t leq 0 end array right end equation where tau 0 l m in mathbb c m times m and varphi in c left left tau 0 right mathbb c m right different from the classical step by step method we restate 1 as an abstract cauchy problem and then we discretize it in a system of ordinary differential equations the scheme of discretization is proved to be convergent moreover the asymptotic stability is investigated for two significant classes of asymptotically stable problems 1

an improved stabilization method for sampled data control systems with control packet loss this technical note presents a new method for stability analysis and stabilization of sampled data systems with control packet loss it is assumed that if the control packet from the controller to the actuator is lost then the actuator input to the plant is set to zero the new method is based on a novel construction of piecewise differentiable lyapunov functionals by using an impulsive system representation of sampled data systems a significant feature of the new lyapunov functionals is that they are continuous at impulse times but not necessarily positive definite inside the impulse intervals applying the new lyapunov functionals to sampled data systems with control packet loss improved criteria for stability and stabilization are derived the new criteria are proved theoretically to be less conservative than the existing results illustrative examples are given which substantiate the usefulness of the proposed method

sampled data h8 state feedback control of systems with state delays in the present paper we adopt the second approach to the nite horizon sampled data state delay h 1 control problem note that in the case of state delays the lifting technique leads to complicated discrete system with in nite dimensional state variable we obtain the piecewise constant state feedback controllers by solving coupled rpdes between sampling and updating the terminal conditions at the sampling instants we derive an asymptotic approximation to the solution of these rpdes by expanding it in the powers of the largest delay the resulting approximation is obtained by solving uncoupled low order partial dia erential equations the performance of the system with the controller that has been obtained using the zero approximation the one that corresponds to zero delay is analysed when the open loop system possesses a non zero delay bounded real lemmas for systems with state delays and jumps are obtained 2 problem formulation

stability of networked control systems with asynchronous renewal links an impulsive systems approach we consider networked control systems in which sensors actuators and controller transmit through asynchronous communication links each introducing independent and identically distributed intervals between transmissions we model these scenarios through impulsive systems with several reset maps triggered by independent renewal processes i e the intervals between jumps associated with a given reset map are identically distributed and independent of the other jump intervals for linear dynamic and reset maps we establish that mean exponential stability is equivalent to the spectral radius of an integral operator being less than one we also prove that the origin of a non linear impulsive system is locally stable with probability one if its local linearization about the zero equilibrium is mean exponentially stable which justifies the importance of studying the linear case the applicability of the results is illustrated through an example using a linearized model of a batch reactor

order stars and stability for delay differential equations we consider runge kutta methods applied to delay differential equations y t ay t by t 1 with real a and b if the numerical solution tends to zero whenever the exact solution does the method is called tau 0 stable using the theory of order stars we characterize high order symmetric methods with this property in particular we prove that all gauss methods are tau 0 stable furthermore we present sufficient conditions and we give evidence that also the radau methods are tau 0 stable we conclude this article with some comments on the case where a andb are complex numbers

a generalization of the burnside basis theorem abstract a b group is a group such that all its minimal generating sets with respect to inclusion have the same size we prove that the class of finite b groups is closed under taking quotients and that every finite b group is solvable via a complete classification of frattini free finite b groups we obtain a general structure theorem for finite b groups applications include new proofs for the characterization of finite matroid groups

brief paper input to state stability for networked control systems via an improved impulsive system approach this paper presents a novel impulsive system approach to input to state stability iss analysis of networked control systems ncss with time varying sampling intervals and delays this approach is based upon the new idea that an ncs can be viewed as an interconnected hybrid system composed of an impulsive subsystem and an input delay subsystem a new type of time varying discontinuous lyapunov krasovskii functional which makes full use of the information on the piecewise constant input and the bounds of the network delays is introduced to analyze the iss property of ncss linear matrix inequality based sufficient conditions are derived for iss of ncss with respect to external disturbances when applied to the approximate tracking problem for ncss the derived iss result provides bounds on the steady state tracking error numerical examples are provided to show the efficiency of the proposed approach

natural continuous extensions of runge kutta methods the present paper develops a theory of natural continuous extensions nces for the discrete approximate solution of an ode given by a runge kutta process these nces are defined in such a way that the continuous solutions furnished by the one step collocation methods are included

delay dependent stability and hybrid l2 l2 gain analysis of linear impulsive time delay systems a continuous timer dependent lyapunov like functional approach abstract this paper presents a new approach for analyzing delay dependent stability and hybrid l 2 l 2 gain performance of linear impulsive delay systems the new approach is inspired by the delay partitioning method the timer dependent lyapunov functional method and the looped functional method in the delay partitioning framework a new type of timer dependent lyapunov functional is constructed which depends on the partition on impulse intervals and also on impulse dynamics different from the previous discontinuous lyapunov functionals the introduced lyapunov functional is continuous along the trajectories of the considered impulsive delay system consequently two different problems of exponential stability and hybrid l 2 l 2 gain performance are tackled by using the same class of lyapunov functionals it is shown that the positive definiteness of this lyapunov functional inside impulse intervals is not necessary for proving exponential stability by use of new integral inequalities based techniques delay dependent criteria for exponential stability and finite hybrid l 2 l 2 gain are established in terms of linear matrix inequalities numerical examples are provided to illustrate the efficiency of the new approach

stability analysis of linear impulsive delay dynamical systems via looped functionals this article investigates the asymptotic stability of impulsive delay dynamical systems idds by using the lyapunovkrasovskii method and looped functionals the proposed conditions reduce the conservatism of the results found in the literature by allowing the functionals to grow during both the continuous dynamics and the discrete dynamics sufficient conditions for asymptotic stability in the form of linear matrix inequalities lmi are provided for the case of impulsive delay dynamical systems with linear and time invariant lti base systems non impulsive actions several numerical examples illustrate the effectiveness of the method

c groups of high rank for the symmetric groups abstract we give presentations for the c groups of rank n 1 of the symmetric group s n we also classify c groups of rank n 2 for s n we show that all these c groups correspond to regular hypertopes that is thin residually connected flag transitive geometries therefore we generalise some similar results obtained in the framework of string c groups that are in one to one correspondence with abstract regular polytopes

wirtinger based integral inequality application to time delay systems in the last decade the jensen inequality has been intensively used in the context of time delay or sampled data systems since it is an appropriate tool to derive tractable stability conditions expressed in terms of linear matrix inequalities lmis however it is also well known that this inequality introduces an undesirable conservatism in the stability conditions and looking at the literature reducing this gap is a relevant issue and always an open problem in this paper we propose an alternative inequality based on the fourier theory more precisely on the wirtinger inequalities it is shown that this resulting inequality encompasses the jensen one and also leads to tractable lmi conditions in order to illustrate the potential gain of employing this new inequality with respect to the jensen one two applications on time delay and sampled data stability analysis are provided

stability analysis of two step runge kutta methods for delay differential equations abstract we investigate stability properties of two step runge kutta methods with respect to the linear test equation y t ay t by t t 0 y t g t t 0 where a and b are complex parameters it is known that the solution y t to this equation tends to zero as t if b a we will show that under some conditions this property is inherited by any a stable two step runge kutta method applied on a constrained mesh to delay differential equations i e that the corresponding method is p stable

numerical treatment of delay differential equations by hermite interpolation a class of numerical methods for the treatment of delay differential equations is developed these methods are based on the wellknown runge kutta fehlberg methods the retarded argument is approximated by an appropriate multipoint hermite interpolation the inherent jump discontinuities in the various derivatives of the solution are considered automatically problems with piecewise continuous right hand side and initial function are treated too real life problems are used for the numerical test and a comparison with other methods published in literature

stability properties of numerical methods for solving delay differential equations abstract stability properties of numerical methods for delay differential equations are considered some suitable definitions for the stability of the numerical methods are included and runge kutta type methods satisfying these properties are tested on a numerical example

instability of runge kutta methods when applied to linear systems of delay differential equations this paper investigates the stability of runge kutta methods when they are applied to the complex linear system of delay differential equations y prime left t right ly left t right my left t 1 right where l m in mathbb c d times d we prove that no runge kutta method preserves asymptotic stability

highest rank of a polytope for an this research was supported by a marsden grant uoa1218 of the royal society of new zealand and by the portuguese foundation for science and technology fct fundacao para a ciencia e a tecnologia through cidma center for research and development in mathematics and applications within project uid mat 04106 2013

stabilization of sampled data control systems with control inputs missing this note is concerned with the stabilization problem for sampled data control systems with control inputs missing a sufficient condition is derived for the existence of exponentially stabilizing state feedback controllers via the switched system approach the obtained condition establishes several quantitative relations among some system parameters such as the sampling period and the exponential decay rate the actual data missing rate and the admissible data missing rate bound two illustrative examples are given to show the effectiveness of the proposed results

some remarks on the use of time varying delay to model sample and hold circuits this note revises the so called input delay approach to the control of sampled data systems with nonuniform sampling in which the sample and hold circuit is embedded into an analog system with a time varying input delay it is shown that the conservatism in calculating the maximal admissible sampling period is reduced by about 57 if the time varying delay embedding step is omitted

stability and l2 gain analysis for impulsive delay systems an impulse time dependent discretized lyapunov functional method abstract the problems of exponential stability and l 2 gain for a class of time delay systems with impulsive effects are studied the main tool used is the construction of an impulse time dependent complete lyapunov functional by dividing the impulse interval and delay interval into several segments the matrix functions of this functional are chosen to be continuous piecewise linear moreover an impulse time dependent weighting factor is introduced to coordinate the dynamical behavior of the nondelayed and integral terms of this functional along the trajectories of the system by applying this functional delay dependent sufficient conditions for exponential stability and l 2 gain are derived in terms of linear matrix inequalities as by products new delay independent sufficient conditions for the same problems are also derived the efficiency of the proposed results is illustrated by numerical examples

stabilizing transmission intervals for nonlinear delayed networked control systems in this note we consider a nonlinear process with delayed dynamics to be controlled over a communication network in the presence of disturbances and study robustness of the resulting closed loop system with respect to network induced phenomena such as sampled distorted delayed and lossy data as well as scheduling protocols for given plant controller dynamics and communication network properties e g propagation delays and scheduling protocols we quantify the control performance level in terms of mathcal l p gains as the transmission interval varies maximally allowable transfer interval mati labels the greatest transmission interval for which a prespecified mathcal l p gain is attained the proposed methodology combines impulsive delayed system modeling with lyapunov razumikhin techniques to allow for matis that are smaller than the communication delays other salient features of our methodology are the consideration of variable delays corrupted data and employment of model based estimators to prolong matis the present stability results are provided for the class of uniformly globally exponentially stable uges scheduling protocols the well known round robin rr and try once discard tod protocols are examples of uges protocols finally a nonlinear example is provided to demonstrate the benefits of the proposed approach

delay dependent stability of high order runge kutta methods this paper is concerned with the study of the delay dependent stability of runge kutta methods for delay differential equations first a new sufficient and necessary condition is given for the asymptotic stability of analytical solution then based on this condition we establish a relationship between 0 stability and the boundary locus of the stability region of numerical methods for ordinary differential equations consequently a class of high order runge kutta methods are proved to be 0 stable in particular the 0 stability of the radau iia methods is proved

asymptotic stability in the numerical solution of linear pure delay differential equations as abstract cauchy problems abstract in bellen and maset preprint an approach is presented for solving the delay differential equation dde y t ly t my t t 0 y t t t 0 where 0 l m c m m and c 0 c m passing through a reformulation of as an abstract cauchy problem and its discretization in a system of ordinary differential equations odes in this paper we investigate the asymptotic stability of this approach on the class of ddes with l 0

volterra integral approach to impulsive renewal systems application to networked control we analyze impulsive systems with independent and identically distributed intervals between transitions our approach involves the derivation of novel results for volterra integral equations with positive kernel we highlight several applications of these results and show that when applied to the analysis of impulsive systems they allow us to i provide necessary and sufficient conditions for mean square stability stochastic stability and mean exponential stability which can be equivalently tested in terms of a matrix eigenvalue computation an lmi feasibility problem and a nyquist criterion condition ii assess performance of the impulsive system by computing a second moment lyapunov exponent the applicability of our results is illustrated in a benchmark problem considering networked control systems with stochastically spaced transmissions for which we can guarantee stability for inter sampling times roughly twice as large as in previous papers

on some sets of generators of finite groups abstract in several papers finite groups with fixed cardinalities of sets of generators are investigated and sometimes are named b groups groups with all subgroups being b groups are called groups with the basis property in the first part of this paper we correct and generalize some characterizations of groups with the basis property in the second part we consider groups satisfying analogous conditions but only for sets of generators of prime power orders the class of groups introduced here is much larger then the class of groups with the basis property for example it contains all nilpotent groups

brief paper stability and stabilization of aperiodic sampled data control systems using robust linear matrix inequalities stability analysis of an aperiodic sampled data control system is considered for application to networked and embedded control the stability condition is described in a linear matrix inequality to be satisfied for all possible sampling intervals although this condition is numerically intractable a tractable sufficient condition can be constructed with the mean value theorem special attention is paid to tightness of the sufficient condition for less conservative stability analysis a region dividing technique for the reduction of conservatism and generalization to stabilization are also discussed an example demonstrates the efficacy of the approach

estimation of sampling period for stochastic nonlinear sampled data systems with emulated controllers this note focuses on determining the value of the maximum allowable sampling period masp that guarantees the stability of stochastic nonlinear sampled data systems with emulated controllers two methods are proposed to compute masp one based on sufficient lyapunov like conditions on stabilization of stochastic nonlinear systems in mean square and the other from applying barbalat s lemma comparison studies are made and illustrated for the two methods by a numerical example

walks on generating sets of groups we study a markov chain on generating n tuples of a fixed group which arises in algorithms for manipulating finite groups the main tools are comparison of two markov chains on different but related state spaces and combinatorics of random paths the results involve group theoretical parameters such as the size of minimal generating sets the number of distinct generating k tuples for different k s and the maximal diameter of the group

stability analysis of sampled data systems via free matrix based time dependent discontinuous lyapunov approach in this paper a new time dependent discontinuous lyapunov functional namely free matrix based time dependent discontinuous fmbtdd lyapunov functional is introduced for stability analysis of sampled data systems first a modified free matrix based integral inequality mfmbii is derived based on the existing free matrix based integral inequality 1 and it is applied to develop a stability criterion for sampled data systems and then inspired by mfmbii fmbtdd term is established that leads to efficient stability conditions four numerical examples are given to demonstrate the effectiveness of the proposed methods

exact and nonstandard numerical schemes for linear delay differential models abstract delay differential models present characteristic dynamical properties that should ideally be preserved when computing numerical approximate solutions in this work exact numerical schemes for a general linear delay differential model as well as for the special case of a pure delay model are obtained based on these exact schemes a family of nonstandard methods of increasing order of accuracy and simple computational properties is proposed dynamic consistency of the new nonstandard methods are proved and illustrated with numerical examples for asymptotic stability positive preserving properties and oscillation behaviour

independent generating sets and geometries for symmetric groups abstract julius whiston showed that the size of an independent generating set in the symmetric group s n is at most n 1 we determine all sets meeting this bound we also give some general remarks on the maximum size of an independent generating set of a group and its relationship to coset geometries for the group in particular we determine all coset geometries of maximum rank for the symmetric group s n for n 6

observer synthesis under time varying sampling for lipschitz nonlinear systems in this work the problem of observation of continuous time nonlinear lipschitz systems under time varying discrete measurements is considered this class of systems naturally occurs when continuous processes are observed through digital sensors and information is sent via a network to a computer for state estimation since the network introduces variations in the sampling time the observer must be designed so that it takes them into account here impulsive observers which make instantaneous correction when information is received are investigated moreover we consider time varying observer gains adapting to the varying sampling interval in order to deal with both continuous time and discrete time dynamics a new hybrid model is used to state the problem and establish the convergence of the proposed observer first generic conditions are provided using a hybrid lyapunov function then a restriction of the generic lyapunov function is used to establish tractable conditions that allows the analysis and synthesis of an impulsive gain

string c groups as transitive subgroups of sn abstract if is a string c group which is isomorphic to a transitive subgroup of the symmetric group s n other than s n and the alternating group a n then the rank of is at most n 2 1 with finitely many exceptions which are classified it is conjectured that only the symmetric group has to be excluded

stability in linear multistep methods for pure delay equations abstract the stability regions of linear multistep methods for pure delay equations are compared with the stability region of the delay equation itself a criterion is derived stating when the numerical stability region contains the analytical stability region this criterion yields an upper bound for the integration step conditional q stability these bounds are computed for the adams bashforth adams moulton and backward differentiation methods of orders 8 furthermore symmetric adams methods are considered which are shown to be unconditionally q stable finally the extended backward differentiation methods of cash are analysed

p stability properties of runge kutta methods for delay differential equations recently the author and others have studied a class of runge kutta methods for ddes these methods make use of certain constrained meshes which allow to get optimal order results in this paper we study their asymptotic stability properties using the test equation begin gathered y t ay t by t tau t 0 hfill y t g t for tau leqq t leqq 0 hfill end gathered wherea b 0 andg t is continuous and complex valued it is known thaty t 0 ast if b re a in particular we show that any astable one step collocation method for odes inherits the same property when it is applied to ddes with a constrained mesh i e it is p stable

method of lines approximations of delay differential equations we consider a special type of numerical methods for delay differential equations ddes by introducing a new independent variable an initial value problem for ddes is converted into an initial boundary value problem for the convection equation thus it is also possible to get an approximate solution to the dde problem by solving the initial boundary value problem with a suitable numerical method instead of solving the original problem in this paper we study a family of method of lines mol approximations to the problem which is obtained by applying runge kutta rk methods for space discretization and prove their convergence under the assumption that the rk methods satisfy a condition known as an algebraic characterization of a stability the result is also confirmed by numerical experiments moreover we show that the condition derives several stability properties of the mol approximations

asymptotic stability barriers for natural runge kutta processes for delay equations this paper investigates the asymptotic stability properties of a class of numerical methods for delay differential equations ddes the so called natural runge kutta methods we first examine the behavior of these methods when applied to the neutral model equation y t a y t b y t 1 c y t 1 with a b c in mathbb r we also consider the case when a b c in mathbb c and provide a suitable geometric characterization of their asymptotic stability regions then by means of the obtained results in conjunction with those given in n guglielmi and e hairer ima j numer anal 21 2001 pp 439 450 we are able to give a final answer concerning the possible preservation of asymptotic stability of the considered class of methods when applied to systems of linear ddes of the form y t l y t m y t 1 with l m in mathbb r d times d d 1 we are interested here in methods that produce stable numerical solutions for all values of the parameters a b and c in the first equation and l and m in the second one for which the exact solution tends to zero to this aim we direct our attention to a novel stability setting recently introduced and investigated for the scalar nonneutral case see n guglielmi numer math 77 1997 pp 467 485 n guglielmi ima j numer anal 18 1998 pp 399 418 n guglielmi and e hairer numer math 83 1999 pp 371 383 n guglielmi and e hairer ima j numer anal 21 2001 pp 439 450 s maset numer math 87 2000 pp 355 371

stability of runge kutta methods for linear delay differential equations this paper investigates the stability of runge kutta methods when they are applied to the complex linear scalar delay differential equation y prime left t right ay left t right by left t 1 right this kind of stability is called tau stability we give a characterization of tau stable runge kutta methods and then we prove that implicit euler method is tau stable

delay dependent stability of symmetric schemes in boundary value methods for ddes we consider the symmetric schemes in boundary value methods bvms applied to delay differential equations y t ay t by t t with real coefficients a and b if the numerical solution tends to zero whenever the exact solution does the symmetric scheme with k 1 m k 2 boundary conditions is called t k 1 k 2 0 stable three families of symmetric schemes namely the extended trapezoidal rules of first etrs and second etr 2s kind and the top order methods toms are considered in this paper by using the boundary locus technology the delay dependent stability region of the symmetric schemes are analyzed and their boundaries are found then by using a necessary and sufficient condition the considered symmetric schemes are proved to be t n n 1 0 stable

stability properties of interpolants for runge kutta methods in solving initial value problems for ordinary differential equations by means of a discrete method we often need to know the solution on a set of points that differs from the grid systems with driving equation dense output discontinuity problems delay equations etc the nonnodal approximations are generally obtained by local interpolation and different features can be requested from the interpolants according to the problem treated in this paper we introduce the concept of stable interpolant for runge kutta methods roughly speaking a stable interpolant maintains the stability properties of the runge kutta method itself the lack of such stability makes the continuous extension not reliable especially in solving stiff equations

a recognition algorithm for special linear groups contain th speciase l linea sl rfr grou q pamplifying the problem he said he would be happy to have a program that couldbe used fo r genera l purpose practical computatio in th range e rf n 30 charle ssims suggested tha onte should be more ambitiou ass to degree sa y d 70 bu trestrict onesel tfo the case q 2 in this pape wr e shall propos ae method whichwe believ teo be practica fol r value osf d up t somewhero e aroun 60d or 70 andwhich is no particularlt y sensitiv to thee size of q w muse t emphasize howeverthat our ideas hav noe t yet at th time e of writing been implemente andd tested until adequate evidenc ha accumulates e d such claim nost mus b takee t n veryseriously ou algorithr ims partly probabilistic it whes respons n ies positiv iettells us that the group generate bdy x certainly does contai sl rfn q but if theoutput is negative that assertion will usually b onle mady e with probabilit 1y e where e is a pre determined small positive number the analogous proble fomr permutatio n group is tso decide fo r a given se t xof permutation osf a set o sizf e n whethe thr e subgroup generate bdy x containsthe alternating group alt n an elegan t probabilistic algorith fo recognizinr m galternating and symmetric group hass been in use fo manr y years le t u cals l anelement of th symmetrie c group sym n if on primae of cycle its has prims elength p non oef the othe r cycles have length divisibl by p ane d ps n 3 themethod exploit thse following facts i if g i alt zs or sym n the an good proportio onf th elemente s o f g areprima ii if g does no contait n alt n and if g i primitivs a e permutation groupthen g contains no such elements the forme irs no hart d to prove see fo exampler 20 lectur e 11 th lattee rassertion comes from a well known old theorem of jordan see fo exampler 30 theorem 13 9 accordingl the ide ya i s to tes it f g i primitives whic cahn bedone pretty quickl byy fo exampler a method of atkinson i f no thet n certainlyg is neither alt n no sym n r i f g i primitives th e crucial ste isp to make asuitable numbe orf independent random selection of elements osf g no mant yare needed see fo r exampl 20 e som if e prima permutation show up thens gis identified correctl as alt ny o r sym n i f not the n with high probabilit g ydoes not contain alt n thus ther is a smale l probability tha the algoritht m willfail to identify alt n or sym n this procedur wa firsts e programme ford

polytopes of high rank for the symmetric groups abstract in the atlas of abstract regular polytopes for small almost simple groups by leemans and vauthier the polytopes whose automorphism group is a symmetric group s n of degree 5 n 9 are available two observations arise when we look at the results 1 for n 5 the n 1 simplex is up to isomorphism the unique regular n 1 polytope having s n as automorphism group and 2 for n 7 there exists up to isomorphism and duality a unique regular n 2 polytope whose automorphism group is s n we prove that 1 is true for n 4 and 2 is true for n 7 finally we also prove that s n acts regularly on at least one abstract polytope of rank r for every 3 r n 1

an analysis of delay dependent stability for ordinary and partial differential equations with fixed and distributed delays this paper is concerned with the study of the stability of ordinary and partial differential equations with both fixed and distributed delays and with the study of the stability of discretizations of such differential equations we start with a delay dependent asymptotic stability analysis of scalar ordinary differential equations with real coefficients we study the exact stability region of the continuous problem as a function of the parameters of the model next it is proved that a time discretization based on the trapezium rule can preserve the asymptotic stability for the considered set of test problems in the second part of the paper we study delay partial differential equations the stability region of the fully continuous problem is analyzed first then a semidiscretization in space is applied it is shown that the spatial discretization leads to a reduction of the stability region when the standard second order central difference operator is employed to approximate the diffusion operator finally we consider the delay dependent stability of the fully discrete problem where the partial differential equation is discretized both in space and in time some numerical examples and further discussions are given

retarded differential equations retarded differential equations rdes are differential equations having retarded arguments they arise in many realistic models of problems in science engineering and medicine where there is a time lag or after effect numerical techniques for such problems may be regarded as extensions of dense output methods for ordinary differential equations odes but scalar rdes are inherently infinite dimensional with a richer structure than their ode counterparts we give background material develop a theoretical foundation for the basic numerics and give some results not previously published