two step hybrid methods for periodic initial value problems we consider a family of two step methods for the numerical integration of periodic initial value problems where the required solution has an unknown period we derive efficient p stable formulas of orders 4 and 6 which are particularly appropriate for nonlinear systems as their newton iteration matrices can be expressed as a perfect square and a perfect cube respectively

a conditionally p stable fourth order exponential fitting method for y f f y abstract we search for the modification of the stability properties of a p stable multistep algorithm with an order greater than two when this is reformulated on the basis of the exponential fitting we effectively construct the exponential fitting version of the fourth order two step algorithm of chawla bit 21 1981 190 193 in order to characterize the stability properties of the new version we first introduce the concept of the conditional p stability of a family of exponential fitting methods and the parameter max to be associated to a conditionally p stable family we then show that this version is conditionally p stable indeed with max 3 4 this means that the method can be used for stiff problems provided some typically nonsevere restrictions are imposed on the stepsize h

free probability for pairs of faces i we consider a notion of bi freeness for systems of non commutative random variables with two faces one of left variables and another of right variables this includes bi free convolution operations bi free cumulants the bi free central limit and bi freeness with amalgamation over an algebra b

high orderp stable formulae for the numerical integration of periodic initial value problems recently there has been considerable interest in the approximate numerical integration of the special initial value problemy f x y for cases where it is known in advance that the required solution is periodic the well known class of stormer cowell methods with stepnumber greater than 2 exhibit orbital instability and so are often unsuitable for the integration of such problems an appropriate stability requirement for the numerical integration of periodic problems is that ofp stability however lambert and watson have shown that ap stable linear multistep method cannot have an order of accuracy greater than 2 in the present paper a class of 2 step methods of runge kutta type is discussed for the numerical solution of periodic initial value problems p stable formulae with orders up to 6 are derived and these are shown to compare favourably with existing methods

distributionally robust optimization under moment uncertainty with application to data driven problems stochastic programming can effectively describe many decision making problems in uncertain environments unfortunately such programs are often computationally demanding to solve in addition their solution can be misleading when there is ambiguity in the choice of a distribution for the random parameters in this paper we propose a model that describes uncertainty in both the distribution form discrete gaussian exponential etc and moments mean and covariance matrix we demonstrate that for a wide range of cost functions the associated distributionally robust or min max stochastic program can be solved efficiently furthermore by deriving a new confidence region for the mean and the covariance matrix of a random vector we provide probabilistic arguments for using our model in problems that rely heavily on historical data these arguments are confirmed in a practical example of portfolio selection where our framework leads to better performing policies on the true distribution underlying the daily returns of financial assets

a class of rosenbrock type schemes for second order nonlinear systems of ordinary differential equations abstract we develop a class of generalized rosenbrock type schemes for second order nonlinear systems of ordinary differential equations we convert the second order systems to equivalent first order form and then employ the square of the jacobian these methods when applied to a linear time invariant system u u au 0 reproduce a class of schemes given by baker and bramble that are derived from a particular class of rational approximations to the exponential with denominators of the form 1 2 z 2 s for an s stage method for our problem then an s stage scheme requires the solution of 2 s linear algebraic systems at each time step with the same real matrix we employ the theory of butcher 1 4 series to develop order conditions and then present specific examples of fourth order methods which are unconditionally stable by appropriate choice of parameter 2 numerical results confirming the rate of convergence are presented

analysis of a class of multi stage multistep runge kutta methods abstract in this paper a class of methods that numerically solve initial value problems for second order ordinary differential equations of the form y f x y x is investigated methods in this class are two step implicit runge kutta methods with s internal stages that do not require an update of y there are many examples in the literature of methods which conform to our format using a type of nystrom tree and a corresponding special type of nystrom series the order conditions for this method are developed with this technique of putting order conditions in terms of trees we obtain a set of simplifying conditions that serve as a framework for generating and analyzing higher order methods our analysis affords the development of a two parameter family of eight order methods the issue of maximum obtainable order for unconditionally stable s stage methods is investigated for s 1 2 when implemented these methods in general require at each step the solution of an algebraic equation of the form y m im f y y r n where m is an s 1 s 1 matrix to facilitate solving this equation we develop a method where m is lower triangular

mixed collocation methods for y f x y inline equatio abstract the collocation methods introduced here are based on linear combinations of trigonometric functions and powers the motivation is to provide better approximations for oscillatory solutions of initial value problems for differential equations of the special form y f x y the resulting methods for two or more collocation points are implicit runge kutta nystrom methods with coefficients which depend on both the fitted angular frequency and the steplength algebraic and trigonometric order conditions are considered and the stability properties of some methods are examined particular mixed collocation methods and other methods for the same class of problems are compared by applying them to a variety of test problems

a quartic c 3 spline collocation method for solving second order initial value problems abstract this paper introduces and analyzes a new formula for integrating special second order periodic and nonperiodic initial value problems in ordinary differential equations the presented formula is based on quartic c3 splines as an approximation to the exact solution of the initial value problem y x f x y y 0 y0 y 0 y 0 it turns out that the proposed method is a continuous extension of the well known fourth order numerov s method and hence possesses nonvanishing intervals of periodicity and absolute stability

stage reduction on p stable numerov type methods of eighth order we present an implicit hybrid two step method for the solution of second order initial value problem it costs only six function evaluations per step and attains eighth algebraic order the method satisfy the p stability property requiring one stage less we conclude dealing with implementation issues for the methods of this type and give some first pleasant results from numerical tests

unconditionally stable methods for second order fredholm integro differential equations in this article second order fredholm integro differential equations with initial conditions are written as a system of first order equations and solved numerically by finite difference methods conditions for convergence are presented and a fourth order method is applied to a test equation in connection with examining the asymptotic stability of such problems

pgscm a family of p stable boundary value methods for second order initial value problems in this paper we introduce a family of linear multistep methods used as boundary value methods for the numerical solution of initial value problems for second order ordinary differential equations of special type we rigorously prove that these schemes are p stable in a generalized sense of arbitrarily high order this overcomes the barrier that lambert and watson established in lambert and watson 1976 1 on linear multistep methods used in the classic way that is as initial value methods we call the new methods pgscms an acronym for p n stable generalized stormer cowell methods numerical illustrations which confirm the theoretical results of the paper are finally given

modified explicit runge kutta methods for the numerical solution of the schrodinger equation a new procedure for constructing runge kutta methods for an efficient integration of the radial schrodinger equation is developed in this paper the new modified runge kutta methods are of algebraically order four the asymptotic expressions of the local errors for large energies are discussed numerical results obtained for the widely used woods saxon potential show the efficiency of the new methods compared with other special optimized fourth order runge kutta methods the error analysis will be clearly confirmed by the resonance problem

an alternating moment condition for bi freeness abstract in this note we demonstrate an equivalent condition for bi freeness inspired by the well known vanishing of alternating centred moments condition from free probability we show that all products satisfying a centred condition on maximal monochromatic intervals have vanishing moments if and only if the family of pairs of faces they come from is bi free and show that similar characterisations hold for the amalgamated and conditional settings in addition we construct a bi free unitary brownian motion and show that conjugation by this process asymptotically creates bi freeness these considerations lead to another characterisation of bi free independence

a class of two step p stable methods for the accurate integration of second order periodic initial value problems abstract in this paper we consider a two parameter family of two step methods for the accurate numerical integration of second order periodic initial value problems by applying the methods to the test equation y 2y 0 we determine the parameters so that the phase lag frequency distortion of the method is minimal the resulting method is a p stable method with a minimal phase lag 6h6 42000 the superiority of the method over the other p stable methods is illustrated by a comparative study of the phase lag errors and by illustrating with a numerical example

families of methods for ordinary differential equations based on trigonometric polynomials abstract we consider the construction of methods based on trigonometric polynomials for the initial value problems whose solutions are known to be periodic it is assumed that the frequency w can be estimated in advance the resulting methods depend on a parameter hw where h is the step size and reduce to classical multistep methods if 0 gautschi 4 developed adams and stormer type methods in our paper we construct nystrom s and milne simpson s type methods numerical experiments show that these methods are not sensitive to changes in w but require the jacobian matrix to have purely imaginary eigenvalues

a p stable complete in phase obrechkoff trigonometric fitted method for periodic initial value problems in this paper we derive a p stable trigonometric fitted obrechkoff method with phase lag frequency distortion infinity it is easy to see from numerical results presented that the new method is much more accurate than previous methods

two step fourth order p stable methods with phase lag of order six for y t y abstract we present a new family of two step fourth order methods which when applied to the test equation y 2y 0 are at once p stable and have a phase lag of order h6 h h h is the step size

distributionally robust joint chance constraints with second order moment information we develop tractable semidefinite programming based approximations for distributionally robust individual and joint chance constraints assuming that only the first and second order moments as well as the support of the uncertain parameters are given it is known that robust chance constraints can be conservatively approximated by worst case conditional value at risk cvar constraints we first prove that this approximation is exact for robust individual chance constraints with concave or not necessarily concave quadratic constraint functions and we demonstrate that the worst case cvar can be computed efficiently for these classes of constraint functions next we study the worst case cvar approximation for joint chance constraints this approximation affords intuitive dual interpretations and is provably tighter than two popular benchmark approximations the tightness depends on a set of scaling parameters which can be tuned via a sequential convex optimization algorithm we show that the approximation becomes essentially exact when the scaling parameters are chosen optimally and that the worst case cvar can be evaluated efficiently if the scaling parameters are kept constant we evaluate our joint chance constraint approximation in the context of a dynamic water reservoir control problem and numerically demonstrate its superiority over the two benchmark approximations

characterisation of a class of p stable methods for differential equations of second order abstract some theorems in a recent paper by chawla and neta give sufficient but not necessary conditions for p stability within particular classes of numerical methods all p stable methods in the relevant families are identified here

analytic aspects of the bi free partial r transform abstract it is shown that the bi freely infinitely divisible laws and only these laws can be used to approximate the distributions of sums of identically distributed bi free pairs of commuting faces furthermore the necessary and sufficient conditions for this approximation are found bi free convolution semigroups of measures and their levy khintchine representations are also studied here from an infinitesimal point of view the proofs depend on the harmonic analysis machinery we developed for integral transforms of two variables without reference to the combinatorics of moments and bi free cumulants

harmonic analysis for the bi free partial s transform abstract we develop an analytic machinery to study voiculescu s bi free partial s transform and then use the results to characterize the multiplicative bi free infinite divisibility it is shown that the class of infinitely divisible distributions coincides with the class of limit distributions for products of bi free pairs of left and right infinitesimal unitaries where the pairs are not required to be identically distributed but all left variables are assumed to commute with all right variables furthermore necessary and sufficient conditions for convergence to a given infinitely divisible distribution are found

addition of certain non commuting random variables soit g le groupe libre non commutatif sur deux generateurs et soit u j j 1 2 les unites dans l 2 g correspondant a la translation a gauche par les generateurs et on considere l 2 g la fonction g g e soient x j des operateurs de la forme x j j u j les operateurs x j peuvent etre consideres comme des variables aleatoires de moments ou de facon equivalente avec des distributions donnees par les fonctionnelles analytiques j ou j f avec ces conventions la distribution de x 1 x 2 depend seulement des distributions de x 1 et x 2 on explique cette relation

an improved trigonometrically fitted p stable obrechkoff method for periodic initial value problems in this paper we present an improved p stable trigonometrically fitted obrechkoff method with phase lag frequency distortion infinity compared with the previous p stable trigonometrically fitted obrechkoff method developed by simos our new method is simpler in structure and more stable in computation we have also improved the accuracy of the first order derivative formula from the numerical illustration presented we can show that the new method is much more accurate than the previous methods

families of two step fourth order p stable methods for second oder differential equations this paper deals with a class of symmetric hybrid two step fourth order p stable methods for the numerical solution of special second order initial value problems such methods were proposed independently by cash 1 and chawla 3 and normally require three function evaluations per step the purpose of this paper is to point out that there are some values of the free parameters available in the methods proposed which can reduce this work we study two classes of such methods the first is the class of economical methods see definition 3 1 which reduce this work to two function evaluations per step and the second is the class of efficient methods see definition 3 2 which reduce this work with respect to implementation for nonlinear problems we report numerical experiments to illustrate the order acuracy and implementational aspects of these two classes of methods

on operator valued bi free distributions abstract in this paper operator valued bi free distributions are investigated given a subalgebra d of a unital algebra b it is established that a two faced family z is bi free from b b op over d if and only if certain conditions relating the b valued and d valued bi free cumulants of z are satisfied using this we verify that a two faced family of matrices is r cyclic if and only if they are bi free from the scalar matrices over the scalar diagonal matrices furthermore the operator valued bi free partial r s and t transforms are constructed new proofs of results from free probability are developed in order to facilitate many of these bi free results

free probability for pairs of faces iii 2 variables bi free partial s and t transforms abstract we introduce two 2 variables transforms the partial bi free s transform and the partial bi free t transform these transforms are the analogues for the bi multiplicative and respectively for the additive multiplicative bi free convolution of the 2 variables partial bi free r transform in our previous paper in this series