superconvergent explicit two step peer methods we consider explicit two step peer methods for the solution of nonstiff differential systems by an additional condition a subclass of optimally zero stable methods is identified that is superconvergent of order p s 1 where s is the number of stages the new condition allows us to reduce the number of coefficients in a numerical search for good methods we present methods with 4 7 stages which are tested in fortran90 and compared with dopri5 and dop853 the results confirm the high potential of the new class of methods

representations of the symmetric group are reducible over simply transitive subgroups if the characteristic of f is zero this problem has been solved by saxl 16 an important feature of saxl s result is that a groupg as in the problem above is either 2 transitive or fixes a point i e is contained in some n 1 the caseg n 1 can then be settled using the branching rule and induction on the other hand an explicit list of k transitive groups for k 2 is available which can be used to complete the proof in characteristic zero see 16 for more details from now on we assume that p 0 in this case the problem is important for determining maximal subgroups of finite classical groups 1 13 however the situation is nowmore complicated for example to determine the pairs g d as above withg an one needs the mullineux conjecture 7 2 ifg is intransitive then up to a conjugation it is contained in a standard young subgroup of the form n k k the irreducible restrictions from n to n 1 have been described in 14 see also 11 6 in 11 it is also

on the durfee size of kronecker products of characters of the symmetric group and its double covers abstract we investigate kronecker products of characters for s n and spin products for the double covers s n we give bounds on the durfee size of these products and provide information on the existence of special constituents also all products of durfee size at most 2 are explicitly described

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

strong stability preserving explicit peer methods in this paper we study explicit peer methods up to order p 13 which have the strong stability preserving ssp property this class of general linear methods has the favourable property of a high stage order the effective ssp coefficient is maximized by solving a nonlinear constraint optimization problem numerically to high precision the coefficient matrices of the optimized methods are sparse in a very structured way linear multistep methods are obtained as a special case of only one stage

highly efficient strong stability preserving runge kutta methods with low storage implementations strong stability preserving ssp runge kutta methods were developed for time integration of semidiscretizations of partial differential equations ssp methods preserve stability properties satisfied by forward euler time integration under a modified time step restriction we consider the problem of finding explicit runge kutta methods with optimal ssp time step restrictions first for the case of linear autonomous ordinary differential equations and then for nonlinear or nonautonomous equations by using alternate formulations of the associated optimization problems and introducing a new more general class of low storage implementations of runge kutta methods new optimal low storage methods and new low storage implementations of known optimal methods are found the results include families of low storage second and third order methods that achieve the maximum theoretically achievable effective ssp coefficient independent of stage number as well as low storage fourth order methods that are more efficient than current full storage methods the theoretical properties of these methods are confirmed by numerical experiment

the control of parasitism in g symplectic methods g symplectic general linear methods are designed to approximately preserve symplectic invariants for hamiltonian systems in this paper the properties of g symplectic methods are explored computationally and theoretically good preservation properties are observed over long times for many parameter ranges but for other parameter values the parasitic behavior to which multivalue methods are prone corrupts the numerical solution by the growth of small perturbations two approaches for alleviating this effect are considered first compositions of methods with growth parameters of opposite signs can be used to cancel the long term effect of parasitism second methods can be constructed for which the growth parameters are zero by design each of these remedies is found to be successful in eliminating parasitic behavior in long term simulations using a variety of test problems

construction of high order quadratically stable second derivative general linear methods for the numerical integration of stiff odes theory of general linear methods glms for the numerical solution of autonomous system of ordinary differential equations of the form y f y is extended to include the second derivative y g y f y f y this extension of glms is called second derivative general linear methods sglms in this paper we will construct two stage a and l stable sglms of order p and stage order q p up to six with low error constants we will show the efficiency of the proposed methods by implementing on some well known stiff problems

on the hamiltonian structure of normal forms at elliptic equilibria of reversible vector fields in r4 abstract this paper addresses the question whether normal forms of smooth reversible vector fields in r 4 at an elliptic equilibrium possess a formal hamiltonian structure in the non resonant case we establish a formal conjugacy between reversible and hamiltonian normal forms in the case of non semi simple 1 1 resonance and p q resonance with p q 2 we establish a weaker form of equivalence namely that of a formal orbital equivalence to a hamiltonian normal form that involves an additional time reparametrization of orbits moreover in case p q 3 we show that no formal conjugacy to a hamiltonian normal form exists

on the zero stability of variable stepsize multistep methods the spectral radius approach in this paper we illustrate a novel approach for studying the asymptotic behaviour of the solutions of linear difference equations with variable coefficients in particular we deal with the zero stability of the 3 step bdf method on grids with variable stepsize for the numerical solution of ivps for odes our approach is based on the theory of the spectral radius of a family of matrices and yields almost optimal results which give a slight improvement to the best results already known from the literature the success got on the chosen example suggests that our approach has a good potential for more general and harder stability analyses of numerical methods

construction of highly stable two step w methods for ordinary differential equations we describe the construction of linearly implicit two step w methods of high stage order and desirable stability properties for the numerical solution of stiff differential systems the presented methods are demonstrated to be a and l stable when the stepsize is held constant they also preserve good stability properties in a variable stepsize environment under quite demanding conditions imposed on the stepsize pattern

irreducible restrictions of representations of symmetric and alternating groups in small characteristics abstract building on reduction theorems and dimension bounds for symmetric groups obtained in our earlier work we classify the irreducible restrictions of representations of the symmetric and alternating groups to proper subgroups such a classification is known when the characteristic of the ground field is greater than 3 but the small characteristics cases require a substantially more delicate analysis and new ideas our results fit into the aschbacher scott program on maximal subgroups of finite classical groups

on the asymptotic properties of a family of matrices in this paper we consider bounded families f of complex n n matrices after introducing the concept of asymptotic order we investigate how the norm of products of matrices behaves as the number of factors goes to infinity in the case of defective families f u sing the asymptotic order allows us to get a deeper knowledge of the asymptotic behaviour than just considering the so called generalized spectral radius with reference to the well known finiteness conjecture for finite families we also introduce the concepts of spectrum maximizing product and limit spectrum maximizing product showing that for finite families of 2 2matrices defectivity is equivalent to the existence of defective such limit products 2001 elsevier science inc all rights reserved

stepsize restrictions for the total variation diminishing property in general runge kutta methods much attention has been paid in the literature to total variation diminishing tvd numerical processes in the solution of nonlinear hyperbolic differential equations for special runge kutta methods conditions on the stepsize were derived that are sufficient for the tvd property see e g shu and osher j comput phys 77 1988 pp 439 471 and gottlieb and shu math comp 67 1998 pp 73 85 various basic questions are still open regarding the following issues 1 the extension of the above conditions to more general runge kutta methods 2 simple restrictions on the stepsize which are not only sufficient but at the same time necessary for the tvd property and 3 the determination of optimal runge kutta methods with the tvd property in this paper we propose a theory by means of which we are able to clarify the above questions moreover by applying our theory we settle analogous questions regarding the related strong stability preserving ssp property see e g gottlieb shu and

numerical treatment of o d es the theory ofa methods almost all commonly used methods for o d es and their most miscellaneous compositions area methods i e they can be reduced toz o z j az j 1 h x j 1 z j 1 z j h z j s a s s j 1 1 m this paper presents a general theory fora methods and discusses its practical consequences an analysis of local discretization error l d e accumulation results in a general order criterium and reveals which part of the l d e effectively influences the global error this facilitates the comparison of methods and generalizes considerably the concept of error constants it is shown as a consequence that the global error cannot be safely controlled by the size of the l d e and that the conventional error control may fail in important cases furthermore butcher s effective order methods the concept of nordsieck forms and gear s interpretation of lineark step schemes as relaxation methods are generalized the stability of step changing is shortly discussed

implementation of two step runge kutta methods for ordinary differential equations we investigate the potential for efficient implementation of two step runge kutta methods tsrk a new class of methods introduced recently by jackiewicz and tracogna for numerical integration of ordinary differential equations the implementation issues addressed are the local error estimation changing stepsize using nordsieck technique and construction of interpolants the numerical experiments indicate that the constructed error estimates are very reliable in a fixed and variable stepsize environment

construction of algebraically stable dimsims the class of general linear methods for ordinary differential equations combines the advantages of linear multistep methods high efficiency and runge kutta methods good stability properties such as a l or algebraic stability while at the same time avoiding the disadvantages of these methods poor stability of linear multistep methods high cost for runge kutta methods in this paper we describe the construction of algebraically stable general linear methods based on the criteria proposed recently by hewitt and hill we also introduce the new concept of e algebraic stability and investigate its consequences examples of e algebraically stable methods are given up to order p 4

stepsize conditions for general monotonicity in numerical initial value problems for runge kutta methods and linear multistep methods much attention has been paid in the literature to special nonlinear stability properties indicated by the terms total variation diminishing tvd strong stability preserving ssp and monotonicity stepsize conditions guaranteeing these properties were studied e g by shu and osher j comput phys 77 1988 pp 439 471 gottlieb shu and tadmor siam rev 43 2001 pp 89 112 hundsdorfer and ruuth monotonicity for time discretizations dundee conference report na 217 2003 university of dundee dundee uk 2003 pp 85 94 higueras j sci comput 21 2004 pp 193 223 and siam j numer anal 43 2005 pp 924 948 spiteri and ruuth siam j numer anal 40 2002 pp 469 491 gottlieb j sci comput 25 2005 pp 105 128 and ferracina and spijker siam j numer anal 42 2004 pp 1073 1093 and math comp 74 2005 pp 201 219 in the present paper we obtain a special stepsize condition guaranteeing the above properties for a generic numerical process this condition is best possible in a well defined and natural sense it is applicable to the important class of general linear methods and it can also be used to answer some open questions for methods of which the above stability properties were studied earlier

two step runge kutta methods implicit two step runge kutta methods are studied it will be shown that these methods require fewer stages to achieve the same order as one step runge kutta methods which means the two step methods are potentially more efficient than one step methods order conditions are derived and examples of two step one stage methods of order 2 and two step two stage methods of order 4 are presented stability properties of these methods with respect to y ay are studied and a stable two step methods of order 2 are characterized two step two stage methods of order 4 which are a stable are found by an extensive computer search semi implicit two stage methods of order 4 were also constructed this is in contrast to the situation encountered in the runge kutta theory where the unique two stage method of order 4 is not semi implicit

an extension and analysis of the shu osher representation of runge kutta methods in the context of solving nonlinear partial differential equations shu and osher introduced representations of explicit runge kutta methods which lead to stepsize conditions under which the numerical process is total variation diminishing tvd much attention has been paid to these representations in the literature in general a shu osher representation of a given runge kutta method is not unique therefore of special importance are representations of a given method which are best possible with regard to the stepsize condition that can be derived from them several basic questions are still open notably regarding the following issues 1 the formulation of a simple and general strategy for finding a best possible shu osher representation for any given runge kutta method 2 the question of whether the tvd property of a given runge kutta method can still be guaranteed when the stepsize condition corresponding to a best possible shu osher representation of the method is violated 3 the generalization of the shu osher approach to general possibly implicit runge kutta methods in this paper we give an extension and analysis of the original shu osher representation by means of which the above questions can be settled moreover we clarify analogous questions regarding properties which are referred to in the literature by the terms monotonicity and strong stability preserving ssp

efficient a stable peer two step methods abstract a stability is a desirable property for stiff integrators however it may be difficult to prove or implement in an automated search for multi stage multi step methods requiring eigenvalue computations on complex sets of parameters in this paper we apply a purely algebraic criterion for a stability of peer two step methods requiring only the solution of algebraic equations and combine it with a new criterion for zero stability of peer methods on general time grids both criteria lead to formulations solvable by standard numerical methods and are easily verifiable by anyone if sufficient data are provided several 4 stage methods of order 3 and 4 are constructed and compared with existing methods on a few standard test problems

a new theoretical approach to runge kutta methods runge kutta rk methods are treated here as linear multistage methods within the author s concept of a methods as a consequence the traditional difference in treatment of rk methods and linear multistep methods vanishes in particular it is not necessary to consider elementary differentials it is shown how the order of a rk method depends on the error constants of its stages and the conditions nonlinear equations for order p are generated from a very simple recursion

representations of runge kutta methods and strong stability preserving methods over the last few years a great effort has been made to develop monotone high order explicit runge kutta methods by means of their shu osher representations in this context the stepsize restriction to obtain numerical monotonicity is normally computed using the optimal representation in this paper we extend the shu osher representations for any runge kutta method giving sufficient conditions for monotonicity we show how optimal shu osher representations can be constructed from the butcher tableau of a runge kutta method the optimum stepsize restriction for monotonicity is given by the radius of absolute monotonicity of the runge kutta method l ferracina and m n spijker siam j numer anal 42 2004 pp 1073 1093 and hence if this radius is zero the method is not monotone in the shu osher representation methods with zero radius require negative coefficients and to deal with them an extra associate problem is considered in this paper we interpret these schemes as representations of perturbed runge kutta methods we extend the concept of radius of absolute monotonicity and give sufficient conditions for monotonicity optimal representations can be constructed from the butcher tableau of a perturbed runge kutta method

exponential integrators for large systems of differential equations we study the numerical integration of large stiff systems of differential equations by methods that use matrix vector products with the exponential or a related function of the jacobian for large problems these can be approximated by krylov subspace methods which typically converge faster than those for the solution of the linear systems arising in standard stiff integrators the exponential methods also offer favorable properties in the integration of differential equations whose jacobian has large imaginary eigenvalues we derive methods up to order 4 which are exact for linear constant coefficient equations the implementation of the methods is discussed numerical experiments with reaction diffusion problems and a time dependent schrodinger equation are included

the runge kutta theory in a nutshell in an earlier paper albrecht siam j numer anal 24 1987 pp 391 406 runge kutta methods were treated as composite linear methods the resulting linear theory is elementary yielding the order conditions as orthogonal relations from a recursion this paper further extends this approach and discusses its implications it is extended to fehlberg forms and the global as well as the stage errors are given in great detail including a new presentation of the principal error function that simplifies error minimization the orthogonal structure of the order conditions has many advantages it provides in particular a powerful strategy for existence proofs and facilitates the calculation of rk methods the recursion which originally was designed to generate the order conditions is considerably generalized by mappings and becomes a major tool also for the classical theory it can be used for a recursive generation of rooted trees also butcher s elementary differentials and weights as well as his order conditions can be obtained recursively to this purpose the elementary weights are reformulated and the rooted trees are complemented by blossoms the whole approach extends to rosenbrock methods

high order linear multistep methods with general monotonicity and boundedness properties we consider linear multistep methods that possess general monotonicity and boundedness properties strict monotonicity in terms of arbitrary starting values for the multistep schemes is only valid for a small class of methods under very stringent step size restrictions this makes them uncompetitive with the strong stability preserving ssp runge kutta methods by relaxing these strict monotonicity requirements a larger class of methods can be considered including many methods of practical interest in this paper we construct linear multistep methods of high order up to six that possess relaxed monotonicity or boundedness properties with optimal step size conditions numerical experiments show that the new schemes perform much better than the classical monotonicity preserving multistep schemes moreover there is a substantial gain in efficiency compared to recently constructed ssp runge kutta ssprk methods

representations of the symmetric group which are irreducible over subgroups let f be an algebraically closed field of characteristic p and n be the symmetric group on n letters in this paper we classify all pairs g d where d is an irreducible f n module of dimension greater than 1 and g is a proper subgroup of n such that the restriction d g is irreducible provided p 3

irreducible restrictions of representations of symmetric groups in small characteristics reduction theorems we study irreducible restrictions of modules over symmetric groups to subgroups we get reduction results which substantially restrict the classes of subgroups and modules for which this is possible such results are known when the characteristic of the ground field is greater than 3 but the small characteristics cases require a substantially more delicate analysis and new ideas this work fits into the aschbacher scott program on maximal subgroups of finite classical groups

numerical search for algebraically stable two step almost collocation methods we investigate algebraic stability of the new class of two step almost collocation methods for ordinary differential equations these continuous methods are obtained by relaxing some of the interpolation and collocation conditions to achieve strong stability properties together with uniform order of convergence on the whole interval of integration we describe the search for algebraically stable methods using the criterion based on the nyquist stability function proposed recently by hill this criterion leads to a minimization problem in one variable which is solved using the subroutine fminsearch from matlab examples of algebraically stable methods in this class are also presented

determinacy of sufficiently differentiable maps variants of the algebraic conditions of mather are shown to be sufficient for the k determinacy of cu maps with respect to j flat contact or right cr equivalence relations where u k k the algebraic conditions are seen to be necessary

strong stability preserving transformed dimsims abstract in this paper we investigate the strong stability preserving ssp property of transformed diagonally implicit multistage integration methods dimsims within this class examples of ssp methods of order p 1 2 3 and 4 and stage order q p have been determined and also suitable starting and finishing procedure have been constructed the numerical experiments performed on a set of test problems have shown that ssp transformed dimsims can be more accurate and competitive with ssp runge kutta methods of the same order

kronecker products characters partitions and the tensor square conjectures abstract we study the remarkable saxl conjecture which states that tensor squares of certain irreducible representations of the symmetric groups s n contain all irreducibles as their constituents our main result is that for sufficiently large n they contain representations corresponding to young diagrams of hooks two row and diagrams obtained from hooks and two rows by adding a finite number of squares for that we develop a new sufficient condition for the positivity of kronecker coefficients in terms of characters and apply this tool using combinatorics of rim hook tableaux in combination with known results on unimodality of certain partition functions we also present connections and speculations on random characters of s n

a new approach to error estimation for general linear methods we present a simple accurate and reliable approach to the estimation of the local discretization error for general linear methods for ordinary differential equations in this approach the input vector for the next step from x n to x n 1 x n h is rescaled and modified accordingly to compensate for the change of stepsize from h to h r h

on finite determinacy of formal vector fields it is widely understood that finite determinacy plays an important role for studies of various singularities for instance deformation theory of singularities of map germs depends deeply on the algebraic characterization of finitely determined map germs in 1 we defined finite determinacy for singularities of formal vector fields in the same way as for map germs under some conditions we gave characterizations of finitely determined formal vector fields first we state briefly the results of 1 let k be the field of real numbers r or complex numbers c a formal vector field x is a derivation of o k x 1 x n i e x is a k linear mapping of into itself which satisfies x f g x f g f x g where f ge we say that two formal vector fields x and y are equivalent if there is a k algebra automorphism q of z such that q y 0 1 yq x a formal vector field x is called k determined if any formal vector field y with the same k jet as x is equivalent to x a formal vector field x is called finitely determined if there is a positive integer k such that x is k determined a k jet z is called wild if for any formal vector field x such that the k jet of x equals z x is not finitely determined now we state several conditions on the eigenvalues for a 1 jet x

convergence and order reduction of runge kutta schemes applied to evolutionary problems in partial differential equations we address the question of convergence of fully discrete runge kutta approximations we prove that under certain conditions the order in time of the fully discrete scheme equals the conventional order of the runge kutta formula being used however these conditions which are necessary for the result to hold are not natural as a result in many problems the order in time will be strictly smaller than the conventional one a phenomenon called order reduction this phenomenon is extensively discussed both analytically and numerically as distinct from earlier contributions we here treat explicit runge kutta schemes although our results are valid for both parabolic and hyperbolic problems the examples we present are therefore taken from the hyperbolic field as it is in this area that explicit discretizations are most appealing

optimal explicit strong stability preserving general linear methods this paper constructs strong stability preserving general linear time stepping methods that are well suited for hyperbolic pdes discretized by the method of lines these methods generalize both runge kutta rk and linear multistep schemes they have high stage orders and hence are less susceptible than rk methods to order reduction from source terms or nonhomogeneous boundary conditions a global optimization strategy is used to find the most efficient schemes that have low storage requirements numerical results illustrate the theoretical findings

exploiting structure in the construction of dimsims we describe the structure of the nonlinear system for the coefficients of diagonally implicit multistage integration methods for ordinary differential equations this structure is then utilized in the search for methods of high order and stage order with a given stability function new methods were obtained by solving the nonlinear systems by state of the art software based on least squares minimization

partially implicit peer methods for the compressible euler equations when cut cells are used for the representation of orography in numerical weather prediction models this leads to very small cells on one hand this results in very harsh time step restrictions for explicit methods due to the cfl criterion on the other hand cut cells only appear in a small region of the domain therefore we consider a partially implicit method in cut cells the jacobian incorporates advection diffusion and acoustics while in the full cells of the free atmosphere only the acoustic part is used i e the method is linearly implicit in the cut cell regions and semi explicit in the free regions and computes with time step sizes restricted only by the cfl condition in the free atmosphere furthermore we use a simplified jacobian in the cut cell regions in order to save storage and gain computational efficiency while the method retains the order independently of the jacobian we present a linear stability theory which takes the effects of the simplifications of the jacobian on stability into account the presented method is as stable and accurate as the underlying split explicit method but furthermore it can compute with cut cells with nearly no additional effort

implementation of diagonally implicit multistage integration methods for ordinary differential equations we investigate the implementation of diagonally implicit multistage integration methods dimsims the implementation issues addressed are the local error estimation changing stepsize using the nordsieck technique and the construction of continuous interpolants numerical experiments with a method of order three indicate that the error estimates that have been constructed are very reliable in both a fixed and a variable stepsize environment

representations of the general linear groups which are irreducible over subgroups we classify all triples g v h such that sl n q leq g leq gl n q v is a representation of g of dimension greater than one over an algebraically closed field bbb f of characteristic prime to q and h is a proper subgroup of g such that the restriction v downarrow h is irreducible this problem is a natural part of the aschbacher scott program on maximal subgroups in finite classical groups

on quasi consistent integration by nordsieck methods in this paper we study advantages of numerical integration by quasi consistent nordsieck formulas all quasi consistent numerical methods possess at least one important property for practical use which has not attracted attention yet i e the global error of a quasi consistent method has the same order as its local error this means that the usual local error control will produce a numerical solution for the prescribed accuracy requirement if the principal term of the local error dominates strongly over remaining terms in other words the global error control can be as cheap as the local error control in the methods under discussion here we apply the above mentioned idea to nordsieck adams moulton methods which are known to be quasi consistent moreover some nordsieck adams moulton methods are even super quasi consistent the latter property means that their propagation matrices annihilate two leading terms in the defect expansion of such methods in turn this can impose a strong relation between the local and global errors of the numerical solution and allow the global error to be controlled effectively by a local error control we also introduce implicitly extended nordsieck methods such that in some sense they form pairs of embedded formulas with their source nordsieck methods this facilitates the local error control in quasi consistent nordsieck schemes numerical examples presented in this paper confirm clearly the power of quasi consistent integration in practice

construction of two step runge kutta methods with large regions of absolute stability we describe the construction of explicit two step runge kutta methods of order p and stage order q p 1 or q p with large regions of absolute stability this process is illustrated for the methods of order p 2 and 3 and leads to new methods which are competitive with explicit runge kutta methods of the same order

regularity properties of multistage integration methods abstract the numerical method for ordinary differential equations is regular if it has the same set of finite asymptotic values as the underlying differential system this paper examines the regularity and strong regularity properties of diagonally implicit multistage integration methods dimsims introduced recently by j c butcher a sufficient condition for regularity and strong regularity of such methods of any order is given and it is proved that this condition is also necessary for two step two stage dimsims of order greater than or equal to two it is also demonstrated that there exist regular schemes in the class of explicit dimsims this is in contrast to explicit runge kutta methods with more than one stage which are always irregular

nordsieck methods with computationally verified algebraic stability abstract we describe the search for algebraically stable nordsieck methods of order p s and stage order q p where s is the number of stages this search is based on the theoretical criteria for algebraic stability proposed recently by hill and hewitt and hill for general linear methods for ordinary differential equations these criteria which are expressed in terms of the non negativity of the eigenvalues of a hermitian matrix on the unit circle are then verified computationally for the derived nordsieck methods of order p 2

explicit two step peer methods we present a new class of explicit two step peer methods for the solution of nonstiff differential systems a construction principle for methods of order p s s the number of stages with optimal zero stability is given two methods of order p 6 found by numerical search are tested in matlab on several representative nonstiff problems the comparison with ode45 confirms the high potential of the new class of methods

elements of a general theory of composite integration methods this contribution is a survey of results on general linear methods after an introduction into the concept of a methods which is a slight generalization of general linear methods and its main results rk methods are presented in the a method notation which offers the following advantages a the order conditions are obtained from a simple recursion without relying on rooted trees or elementary differentials b they appear in a transparent form as orthogonality relations in dependence of the error constants of the stages of the rk method c their orthogonal structure permits elegant existence proofs and yields new auxiliary conditions d the error accumulation within the stages of the method is completely understood the proof of the main result on rk methods theorem 5 is simplified

order conditions for general linear methods we describe the derivation of order conditions without restrictions on stage order for general linear methods for ordinary differential equations this derivation is based on the extension of the albrecht approach proposed in the context of runge kutta and composite and linear cyclic methods this approach was generalized by jackiewicz and tracogna to two step runge kutta methods by jackiewicz and vermiglio to general linear methods with external stages of different orders and by garrappa to some classes of runge kutta methods for volterra integral equations with weakly singular kernels this leads to general order conditions for many special cases of general linear methods such as diagonally implicit multistage integration methods nordsieck methods and general linear methods with inherent runge kutta stability exact coefficients for several low order methods with some desirable stability properties are presented for illustration

on restrictions of modular spin representations of symmetric and alternating groups let f be an algebraically closed field of characteristic p and h be an almost simple group or a central extension of an almost simple group an important problem in representation theory is to classify the subgroups g of h and fh modules v such that the restriction v g is irreducible for example this problem is a natural part of the program of describing maximal subgroups in finite classical groups in this paper we investigate the case of the problem where h is the schur s double cover n or s n

on the inversion of the vandermonde matrix abstract the inversion of the vandermonde matrix has received much attention for its role in the solution of some problems of numerical analysis and control theory this work deals with the problem of getting an explicit formula for the generic element of the inverse we derive two algorithms in o n2 and o n3 and compare them with the parker traub and the bjorck pereyra algorithms

representations of the alternating group which are irreducible over subgroups ii nsf dms 1161094 dms 0901241 dms 1201374 humboldt foundation simons foundation 204181 simons foundation fellowship 305247

ring theoretic aspects of quandles abstract we associate to every quandle x and an associative ring with unity k a nonassociative ring k x following 3 the basic properties of such rings are investigated in particular under the assumption that the inner automorphism group i n n x acts orbit 2 transitively on x a complete description of right or left ideals is provided the complete description of right ideals for the dihedral quandles r n is given it is also shown that if for two quandles x and y the inner automorphism groups act 2 transitively and k x is isomorphic to k y then the quandles are of the same partition type however we provide examples when the quandle rings k x and k y are isomorphic but the quandles x and y are not isomorphic these examples answer some open problems in 3

a general class of two step runge kutta methods for ordinary differential equations a general class of two step runge kutta methods that depend on stage values at two consecutive steps is studied these methods are special cases of general linear methods introduced by butcher and are quite efficient with respect to the number of function evaluations required for a given order general order conditions are derived using the approach proposed recently by albrecht and examples of methods are given up to the order 5 these methods can be divided into four classes that are appropriate for the numerical solution of nonstiff or stiff differential equations in sequential or parallel computing environments

computation of optimal monotonicity preserving general linear methods monotonicity preserving numerical methods for ordinary differential equations prevent the growth of propagated errors and preserve convex boundedness properties of the solution we formulate the problem of finding optimal monotonicity preserving general linear methods for linear autonomous equations and propose an efficient algorithm for its solution this algorithm reliably finds optimal methods even among classes involving very high order accuracy and that use many steps and or stages the optimality of some recently proposed methods is verified and many more efficient methods are found we use similar algorithms to find optimal strong stability preserving linear multistep methods of both explicit and implicit type including methods for hyperbolic pdes that use downwind biased operators

multiplicity free kronecker products of characters of the symmetric groups we provide a classification of multiplicity free inner tensor products of irreducible characters of symmetric groups thus confirming a conjecture of bessenrodt concurrently we classify all multiplicity free inner tensor products of skew characters of the symmetric groups we also provide formulae for calculating the decomposition of these tensor products

p stable general nystrom methods for y f y t we focus our attention on the family of general linear methods glms for the numerical solution of second order ordinary differential equations odes these are multivalue methods introduced in d ambrosio et al 2012 3 with the aim to provide an unifying approach for the analysis of the properties of accuracy of numerical methods for second order odes our investigation is addressed to providing the building blocks useful to analyze the linear stability properties of glms for second order odes thus we present the extension of the classical notions of stability matrix stability polynomial stability and periodicity interval a stability and p stability to the family of glms special attention will be given to the derivation of highly stable glms whose stability properties depend on the stability polynomial of indirect runge kutta nystrom methods based on gauss legendre collocation points which are known to be p stable in this way we are able to provide p stable glms whose order of convergence is greater than that of the corresponding rkn method without heightening the computational cost we finally provide and discuss examples of p stable irreducible glms satisfying the mentioned features

on monotonicity and boundedness properties of linear multistep methods in this paper an analysis is provided of nonlinear monotonicity and boundedness properties for linear multistep methods instead of strict monotonicity for arbitrary starting values we shall focus on generalized monotonicity or boundedness with runge kutta starting procedures this allows many multistep methods of practical interest to be included in the theory in a related manner we also consider contractivity and stability in arbitrary norms

search for efficient general linear methods for ordinary differential equations we analyze implicit general linear methods with s internal stages and r s 1 external stages of order p s 1 and stage order q s or q s 1 these methods might eventually lead to more efficient formulas than the class of dimsims and the class of general linear methods with inherent runge kutta stability we analyze also error propagation and estimation of local discretization errors examples of such methods which are a and l stable are derived up to the stage order q 3 or q 4 and order p 4

a nonlinear optimization approach to the construction of general linear methods of high order we describe the construction of diagonally implicit multistage integration methods of order and stage order p q 7 and p q 8 for ordinary differential equations these methods were obtained using state of the art optimization methods particularly variable model trust region least squares algorithms

monotonicity preserving linear multistep methods in this paper we provide an analysis of monotonicity properties for linear multistep methods these monotonicity properties include positivity and the diminishing of total variation we also pay particular attention to related boundedness properties such as the total variation bounded tvb property in the analysis the multistep methods are considered in combination with suitable starting procedures this allows for monotonicity statements for classes of methods which are important and often used in practice but which were thus far not covered by theoretical results

on almost extreme components in kronecker products of characters of the symmetric groups abstract using a recursion formula due to dvir we obtain information on maximal and almost maximal components in kronecker products of characters of the symmetric groups this is applied to confirm a conjecture made by bessenrodt and kleshchev in 1999 which classifies all such kronecker products with only three or four components

strong stability preserving explicit runge kutta methods of maximal effective order we apply the concept of effective order to strong stability preserving ssp explicit runge kutta methods relative to classical runge kutta methods methods with an effective order of accuracy are designed to satisfy a relaxed set of order conditions but yield higher order accuracy when composed with special starting and stopping methods we show that this allows the construc tion of four stage ssp methods with effective order four such methods cannot have classical order four however we also prove that effective order five methods like classical order five methods require the use of nonpositive weights and so cannot be ssp by numerical optimization we construct explicit ssp runge kutta methods up to effective order four and establish the optimality of many of them numerical experiments demonstrate the validity of these methods in practice