stability properties of implicit runge kutta methods new stability concepts bs stability and bsi stability internal bs stability are introduced bs stability is a modification and extension of b stability and enables the derivation of order results for runge kutta methods applied to general nonlinear stiff initial value problems these order results b consistency b convergence are not affected by stiffness several classes of implicit runge kutta methods are shown to be bs stable gauss radau ia and radau iia schemes

symmetric green s functions on complete manifolds on demontre d une facon constructive que l operateur de laplace admet une fonction de green symetrique g x y sur une variete de riemann complete

on the algebraic equations in implicit runge kutta methods this paper is concerned with the system of nonlinear algebraic equations which arise in the application of implicit runge kutta methods to stir initial value problems without making the classical assumption that the stepsize h 0 is small we derive transparent conditions on the method that guarantee existence and uniqueness of solutions to the equations besides we discuss the sensitivity of the runge kutta procedure with respect to perturbations in the algebraic equations

linear growth harmonic functions on a complete manifold soit m une variete de riemann a n dimensions a courbure de ricci non negative on suppose que le volume des boules geodesiques de rayon r centrees en un point x m satisfait v b x r o r h alors la dimension de l espace des fonctions harmoniques a croissance lineaire sur m doit etre k

polynomial growth harmonic functions on groups of polynomial volume growth we consider harmonic functions of polynomial growth of some order d on cayley graphs of discrete groups of polynomial volume growth of order d w r t the word metric and prove the optimal estimate for the dimension of the space of such harmonic functions more precisely the dimension of this space of harmonic functions is at most of order d d 1 as in the already known riemannian case this estimate is polynomial in the growth degree more generally our techniques also apply to graphs roughly isometric to cayley graphs of groups of polynomial volume growth

order results for implicit runge kutta methods applied to stiff systems an extensive order analysis is given for the local and global errors of certain classes of runge kutta methods applied to general nonlinear stiff initial value problems it turns out that there are three order levels for linear problems y ay the classical orders apply for general nonlinear problems however two types of order reductions occur there a distinction has to be made between stiff problems where only f y is large and stiff problems where also some of the other derivatives f t f tt f yy cdots are large for these two classes of problems order results can be guaranteed which differ only by one power of the stepsize parameter h the order levels for both classes of problems are considerably lower than the respective conventional orders these order results are optimal in the sense that higher orders would contradict numerical observations made with particular initial value problems

extending convergence of bdf methods for a class of nonlinear strongly stiff problems abstract in 1995 a new approach was presented to overcome the drawbacks of existing convergence theory for stiff initial value problems by auzinger et al bit 36 4 1996 635 652 and illustrated by means of a convergence argument for the implicit euler scheme in the present paper our intention is to obtain extending quantitive convergence results of bdf methods for a class of nonlinear strongly stiff problems and to illustrate further this new approach some numerical examples confirm our results

stability and b convergence properties of multistep runge kutta methods this paper continues earlier work by the same author concerning the stability and b convergence properties of multistep runge kutta methods for the numerical solution of nonlinear stiff initial value problems in a hilbert space a series of sufficient conditions and necessary conditions for a multistep runge kutta method to be algebraically stable diagonally stable b or optimally b convergent are established by means of which six classes of high order algebraically stable and b convergent multistep runge kutta methods are constructed in a unified pattern these methods include the class constructed by burrage in 1987 as special case and most of them can be regarded as extension of the gauss radauia radauiia and lobattoiiic runge kutta methods we find that the classes of multistep runge kutta methods constructed in the present paper are superior in many respects to the corresponding existing one step runge kutta schemes

b convergence properties of multistep runge kutta methods by using the theory of b convergence for general linear methods to the special case of multistep runge kutta methods a series of b convergence results for multistep runge kutta methods is obtained and it is proved that the family of algebraically stable r step s stage multistep runge kutta methods with parameters al a2 ar presented by burrage in 1987 is optimally bconvergent of order at least s and b convergent of order s 1 provided that r s and aj o j 1 2 r furthermore this family of methods is optimally b convergent of order s 1 if some other additional conditions are satisfied

polynomial growth harmonic functions on connected sums of complete riemannian manifolds let m be a connected sum of complete riemannian manifolds satisfying the volume doubling condition and the poincare inequality we prove that the space of polynomial growth harmonic functions on m is finite dimensional whenever m has finitely many ends and satisfies the finite covering condition on each end this result directly generalizes that of tam and it also partially generalizes that of colding and minicozzi ii

on the metric dimension of corona product graphs given a set of vertices s v 1 v 2 v k of a connected graph g the metric representation of a vertex v of g with respect to s is the vector r v s d v v 1 d v v 2 d v v k where d v v i i 1 k denotes the distance between v and v i s is a resolving set for g if for every pair of distinct vertices u v of g r u s r v s the metric dimension of g dim g is the minimum cardinality of any resolving set for g let g and h be two graphs of order n 1 and n 2 respectively the corona product g h is defined as the graph obtained from g and h by taking one copy of g and n 1 copies of h and joining by an edge each vertex from the ith copy of h with the ith vertex of g for any integer k 2 we define the graph g kh recursively from g h as g kh g k 1h h we give several results on the metric dimension of g kh for instance we show that given two connected graphs g and h of order n 1 2 and n 2 2 respectively if the diameter of h is at most two then dim g kh n 1 n 2 1 k 1dim h moreover if n 2 7 and the diameter of h is greater than five or h is a cycle graph then dim g kh n 1 n 2 1 k 1dim k 1 h

resolvability and fault tolerant resolvability structures of convex polytopes abstract in this paper we study resolvability and fault tolerant resolvability of convex polytopes and related geometric graphs imran et al 2010 18 raised an open problem asserting that whether or not every family of convex polytope has a constant metric dimension raza et al 2018 28 studied the fault tolerant metric dimension of certain families of convex polytopes they also concluded their study with an open problem asking whether or not every family of convex polytope has a constant fault tolerant metric dimension in this paper we provide negative answers to both of the aforementioned open problems we answer the first question by constructing a family of convex polytopes with an unbounded metric dimension by proving a result between resolvability and fault tolerant resolvability structures of a graph we show that the family of convex polytope with an unbounded metric dimension also possesses an unbounded fault tolerant resolvability structure moreover we construct three more infinite families of graphs which are closely related to convex polytopes having an unbounded metric dimension

high order algebraically stable multistep runge kutta methods in this paper we examine some of the stability properties of a special class of multivalue methods called in k burrage order properties of multivalue methods ima j numer anal submitted multistep runge kutta methods and construct families of algebraically stable methods of arbitrarily high order

d convergence of general linear methods for stiff delay differential equations abstract this paper is concerned with the numerical solution of delay differential equations ddes we focus on the error behaviour of general linear methods for stiff ddes a restricted type of interpolation procedure is considered for general linear methods d convergence properties of general linear methods with this interpolation procedure are investigated some extant results are unified

b convergence properties of defect correction methods i b convergence properties of defect correction methods based on the implicit euler and midpoint schemes are discussed the property ofb convergence means that there exist global error bounds for nonlinear stiff problems independent of their stiffness it turns out that the orders ofb convergence of these methods coincide with the conventional orders of small wherel is a lipschitz constant of the right hand side in part i these assertions are reduced to the validity of the so called hypothesisa which is discussed in greater detail in part ii numerical experiments confirming the theoretical analysis are also given in part ii

on the fault tolerant metric dimension of convex polytopes abstract a convex polytopes is a polytope that is also a convex set of points in the n dimensional euclidean space r n by preserving the same adjacency relation between vertices of a convex polytope its graph is constructed the metric dimension problem has been extensively studied for convex polytopes and other families of graphs in this paper we study the fault tolerant metric dimension problem for convex polytopes by using a relation between resolving sets and fault tolerant resolving sets of graphs we prove that certain infinite families of convex polytopes are the families of graphs with constant fault tolerant metric dimension we conclude the paper with some open problems

general linear methods for volterra integro differential equations with memory a new class of numerical methods for volterra integro differential equations with memory is developed the methods are based on the combination of general linear methods with compound quadrature rules sufficient conditions that guarantee global and asymptotic stability of the solution of the differential equation and its numerical approximation are established numerical examples illustrate the convergence and effectiveness of the numerical methods

stability properties for a general class of methods for ordinary differential equations the concepts of g stability for linear multistep methods and b stability for runge kutta methods are combined in a unified approach to nonlinear stability for a class of methods general enough to include these as special cases recent work on these developments by kevin burrage and the author is reviewed and to exemplify the new ideas and techniques an algebraic stability analysis is presented for the backward differentiation methods of orders 1 2 and 3

equilibrium attractive properties of a class of multistep runge kutta methods the main purpose of this paper is to discuss the equilibrium attractive properties of a class of multistep runge kutta methods for initial value problems of ordinary differential equations some algebraic conditions insuring the equilibrium attractivity are given and some methods satisfying these algebraic conditions are constructed some numerical examples confirm our results

computing the metric dimension of wheel related graphs abstract an ordered set w w 1 w k v g of vertices of g is called a resolving set or locating set for g if every vertex is uniquely determined by its vector of distances to the vertices in w a resolving set of minimum cardinality is called a basis for g and this cardinality is the metric dimension or location number of g denoted by g in this paper we study the metric dimension of certain wheel related graphs namely m level wheels an infinite class of convex polytopes and antiweb gear graphs denoted by w n m q n and awj 2 n respectively we prove that these infinite classes of wheel related graphs have unbounded metric dimension the study of an infinite class of convex polytopes generated by wheel denoted by q n also gives a negative answer to an open problem proposed by imran et al 2012 in 8 open problem is it the case that the graph of every convex polytope has constant metric dimension it is natural to ask for characterization of graphs with unbounded metric dimension

stability and b convergence of general linear methods abstract this paper continues earlier work by the same author concerning the stability and b convergence of general linear methods for the numerical solution of nonlinear stiff initial value problems in a hilbert space in a previous paper we have proved that bh consistency resp bh consistency together with bh stability implies optimal b convergence resp b convergence in this paper by means of bs and bsi stability properties the sufficient conditions for a method to be b bh bh or bh consistent bh or weakly bh stable b or optimally b convergent respectively are further established

on families of convex polytopes with constant metric dimension let g be a connected graph and d x y be the distance between the vertices x and y a subset of vertices w w 1 w 2 w k is called a resolving set for g if for every two distinct vertices x y v g there is a vertex w i w such that d x w i d y w i a resolving set containing a minimum number of vertices is called a metric basis for g and the number of vertices in a metric basis is its metric dimension dim g a family g of connected graphs is a family with constant metric dimension if dim g is finite and does not depend upon the choice of g in g in this paper we study the metric dimension of some classes of convex polytopes which are obtained by the combinations of two different graph of convex polytopes it is shown that these classes of convex polytopes have the constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of these classes of convex polytopes

minimal doubly resolving sets and the strong metric dimension of some convex polytopes abstract in this paper we consider two similar optimization problems on graphs the strong metric dimension problem and the problem of determining minimal doubly resolving sets we prove some properties of strong resolving sets and give an integer linear programming formulation of the strong metric dimension problem these results are used to derive explicit expressions in terms of the dimension n for the strong metric dimension of two classes of convex polytopes d n and t n on the other hand we prove that minimal doubly resolving sets of d n and t n have constant cardinality for n 7

convergence of runge kutta methods for neutral delay integro differential equations the present paper is concerned with the convergence of numerical methods to solve initial value problems of neutral delay integro differential equations ndides the error estimation of runge kutta methods for a class of nonlinear systems of ndides are derived some numerical experiments are given which confirm the theoretical results

the concept of b convergence a useful concept of convergence for nonlinear stiff ode s will be developed which permits the derivation of uniform global error bounds independent of the stiffness of the considered problem this concept will be discussed for three simple methods for stiff systems the implicit euler scheme the implicit midpoint rule and the implicit trapezoidal rule