differentiable interpolants for high order runge kutta methods for a particular family of pairs of explicit runge kutta methods of orders p 1 and p sets of efficient continuously differentiable interpolants of several orders up to p are characters zed algorithmically in terms of several arbitrary parameters the approach can be applied in an obvious way to yield interpolants for other types of explicit runge kutta methods as well as families of other types of explicit and implicit methods derivative evaluations required for each pair of methods are reused and additional derivative evaluations are selected in an attempt to minimize the total number of stages required the analysis provides a lower bound on the number of stages required and indicates for example why twelve stages are required to provide interpolants for eight stage pairs of orders 5 and 6 in contrast to the eighteen stages used to obtain a known interpolant of order 7 for a pair of methods of orders 6 and 7 only sixteen stages are required by the proposed derivation details for interpola

approximating the exponential from a lie algebra to a lie group consider a differential equation y a t y y y 0 y0 with y 0 g and a r g g where g is a lie algebra of the matricial lie group g every b g can be mapped to g by the matrix exponential map exp tb with t r most numerical methods for solving ordinary differential equations odes on lie groups are based on the idea of representing the approximation y n of the exact solution y t n t n r by means of exact exponentials of suitable elements of the lie algebra applied to the initial value y 0 this ensures that y n g when the exponential is difficult to compute exactly as is the case when the dimension is large an approximation of exp tb plays an important role in the numerical solution of odes on lie groups in some cases rational or polynomial approximants are unsuitable and we consider alternative techniques whereby exp tb is approximated by a product of simpler exponentials in this paper we present some ideas based on the use of the strang splitting for the approximation of matrix exponentials several cases of g and g are considered in tandem with general theory order conditions are discussed and a number of numerical experiments conclude the paper

perturbed collocation and runge kutta methods it is well known thatsome implicit runge kutta methods are equivalent to collocation methods this fact permits very short and natural proofs of order anda b an bn stability properties for this subclass of methods see 9 and 10 the present paper answers the natural question ifall rk methods can be considered as a somewhat perturbed collocation after having introduced this notion we give a proof on the order of the method and discuss their stability properties much of known theory becomes simple and beautiful

estimating species richness by a poisson compound gamma model we propose a poisson compound gamma approach for species richness estimation based on the denseness and nesting properties of the gamma mixture we fix the shape parameter of each gamma component at a unified value and estimate the mixture using nonparametric maximum likelihood a least squares crossvalidation procedure is proposed for the choice of the common shape parameter the performance of the resulting estimator of n is assessed using numerical studies and genomic data

galerkin lie group variational integrators based on unit quaternion interpolation abstract lie group variational integrators of arbitrary order are developed using the galerkin method based on unit quaternion interpolation to our knowledge quaternions have not been used before for this purpose though they allow a very simple and efficient way to perform the interpolation the resulting integrators are symplectic and structure preserving in the sense that certain symmetries in the lagrangian of the mechanical system are carried over to the discrete setting which leads to the preservation of the corresponding momentum maps the integrators furthermore exhibit a very good long time energy behavior i e energy is neither dissipated nor gained artificially at the same time the lie group structure is preserved by carefully defining the variations the interpolation method and by solving the non linear system of equations directly on the manifold rather than constraining it in a surrounding space using lagrange multipliers as a consequence we are able to show that lie group variational integrators based on the special orthogonal group are equivalent to the variational integrators for constrained systems using the discrete null space method employed e g in dmocc discrete mechanics and optimal control of constrained systems we show new numerical results on the convergence rates which are substantially higher than the known theoretical bounds and on the relation between accuracy and computational cost

interpolating high order runge kutta formulas abstract a runge kutta formula becomes inefficient when the step size must be reduced often to produce answers at specified points recently a lot of effort has been devoted to providing runge kutta formulas with an interpolation capability so that answers can be produced inexpensively throughout the span of a step though quite successful at low and moderate orders interpolation of high order formulas is still unsatisfactory a new approach to the task is presented in the context of interpolating an important pair of orders 7 and 8 due to prince and dormand j comput appl math 7 67 75 1981

a fifth order interpolant for the dornand and prince runge kutta method abstract a family of fifth order interpolants for the fifth order solution provided by the dormand and prince runge kutta pair rk5 4 7m which requires two additional function evaluations per step is presented an optimal interpolant in this family has been determined by choosing the parameters to minimize the leading coefficients of the local truncation error of the continuous solution some numerical experiments with the nonstiff detest problems show that the proposed optimal method has a good interpolatory behavior

estimating the population size for capture recapture data with unequal catchability a point estimator and its associated confidence interval for the size of a closed population are proposed under models that incorporate heterogeneity of capture probability real data sets provided in edwards and eberhardt 1967 journal of wildlife management 31 87 96 and carothers 1973 journal of animal ecology 42 125 146 are used to illustrate this method and to compare it with other estimates the performance of the proposed procedure is also investigated by means of monte carlo experiments the method is especially useful when most of the captured individuals are caught once or twice in the sample for which case the jackknife estimator usually does not work well numerical results also show that the proposed confidence interval performs satisfactorily in maintaining the nominal levels

interpolation in lie groups we consider interpolation in lie groups based on points on the manifold together with tangent vectors at some of these points we construct hermite interpolation polynomials if the points and tangent vectors are produced in the process of integrating an ordinary differential equation in terms of lie algebra actions we use the truncated inverse of the differential of the exponential mapping and the truncated baker campbell hausdorff formula to relatively cheaply construct an interpolation polynomial much effort has lately been put into research on geometric integration i e the process of integrating differential equations in such a way that the configuration space of the true solution is respected by the numerical solution some of these methods may be viewed as generalizations of classical methods and we investigate the construction of intrinsic dense output devices as generalizations of the continuous runge kutta methods

morse theory for lagrangian intersections soit p une variete symplectique compacte et soit l p une sous variete lagrangienne avec 2 p l 0 pour un diffeomorphisme exact de p avec la propriete que l coupe l transversalement on demontre une inegalite de morse reliant l ensemble l l a la cohomologie de l

an algorithm for starting multistep methods the present paper deals with the problem of starting multistep methods we take into consideration an adams bashforth moulton pece pair with the predictor of order q and the corrector of order q 1 to start this method q 1 starting values are necessary in addition to y0 a well known result from theory says that the order of convergence of the whole integration is q 1 if all those starting values are accurate of that same order present production codes start with a predictor of order 1 and a corrector of order 2 at the first step and then proceed step by step each time raising the order by 1 until all the necessary starting values have been obtained but in this manner all the starting errors keep of order 3 and so the whole integration converges no faster than that order this drawback is normally compensated for by taking very small step sizes in the starting phase the general algorithm we propose furnishes at a reasonably low cost the necessary number q 1 of starting values each of the appropriate order q 1 whatever q might be it is independent of the particular multistep formula considered and is mainly designed to be used for high values of q q 10 where the alternative strategies are too expensive or do not exist at all the numerical results reported show the validity of our approach 2003 elsevier science ltd all rights reserved

order barriers for continuous explicit runge kutta methods in this paper we deal with continuous numerical methods for solving initial value problems for ordinary differential equations the need for which occurs frequently in applications whereas most of the commonly used multistep methods provide continuous extensions by means of an interpolant which is available without making extra function evaluations this is not always the case for one step methods we consider the class of explicit runge kutta methods and provide theorems used to obtain lower bounds for the number of stages required to construct methods of a given unifonn order p these bounds are similar to the butcher barriers known for the discrete case and are derived up to order p 5 as far as we know the examples we present of 8 stage continuous runge kutta methods of uniform order 5 are the first of their kind

scattered manifold valued data approximation we consider the problem of approximating a function f from an euclidean domain to a manifold m by scattered samples f xi i i in mathcal i f i i i where the data sites xi i i in mathcal i i i i are assumed to be locally close but can otherwise be far apart points scattered throughout the domain we introduce a natural approximant based on combining the moving least square method and the karcher mean we prove that the proposed approximant inherits the accuracy order and the smoothness from its linear counterpart the analysis also tells us that the use of karcher s mean dependent on a riemannian metric and the associated exponential map is inessential and one can replace it by a more general notion of center of mass based on a general retraction on the manifold consequently we can substitute the karcher mean by a more computationally efficient mean we illustrate our work with numerical results which confirm our theoretical findings

nonparametric lower bounds for species richness and shared species richness under sampling without replacement summary a number of species richness estimators have been developed under the model that individuals or sampling units are sampled with replacement however if sampling is done without replacement so that no sampled unit can be repeatedly observed then the traditional estimators for sampling with replacement tend to overestimate richness for relatively high sampling fractions ratio of sample size to the total number of sampling units and do not converge to the true species richness when the sampling fraction approaches one based on abundance data or replicated incidence data we propose a nonparametric lower bound for species richness in a single community and also a lower bound for the number of species shared by multiple communities our proposed lower bounds are derived under very general sampling models they are universally valid for all types of species abundance distributions and species detection probabilities for abundance data individuals detectabilities are allowed to be heterogeneous among species for replicated incidence data the selected sampling units e g quadrats need not be fully censused and species can be spatially aggregated all bounds converge correctly to the true parameters when the sampling fraction approaches one real data sets are used for illustration we also test the proposed bounds by using subsamples generated from large real surveys or censuses and their performance is compared with that of some previous estimators

exact steady state distributions of multispecies birth death immigration processes effects of mutations and carrying capacity on diversity stochastic models that incorporate birth death and immigration also called birth death and innovation models are ubiquitous and applicable to many problems such as quantifying species sizes in ecological populations describing gene family sizes modeling lymphocyte evolution in the body many of these applications involve the immigration of new species into the system we consider the full high dimensional stochastic process associated with multispecies birth death immigration and present a number of exact and asymptotic results at steady state we further include random mutations or interactions through a carrying capacity and find the statistics of the total number of individuals the total number of species the species size distribution and various diversity indices our results include a rigorous analysis of the behavior of these systems in the fast immigration limit which shows that of the different diversity indices the species richness is best able to distinguish different types of birth death immigration models we also find that detailed balance is preserved in the simple noninteracting birth death immigration model and the birth death immigration model with carrying capacity implemented through death surprisingly when carrying capacity is implemented through the birth rate detailed balance is violated

smoothness equivalence properties of manifold valued data subdivision schemes based on the projection approach interpolation of manifold valued data is a fundamental problem which has applications in many fields the linear subdivision method is an efficient and well studied method for interpolating or approximating real valued data in a multiresolution fashion a natural way to apply a linear subdivision scheme overline s to interpolate manifold valued data is to first embed the manifold at hand into an euclidean space and construct a projection operator p that maps points from the ambient space to a closest point on the embedded surface and then consider the nonlinear subdivision operator s p circ overline s when applied to symmetric spaces such as s n 1 so n sl n se n g n k the projection method can also be carried out in such a way that the resulting schemes enjoy natural coordinate invariance properties and robust numerical implementations despite such nice features the mathematical analysis of such nonlinear subdivision schemes is in its infancy in this article we attack the so called smoothness equivalence conjecture which asserts that the smoothness property of s is exactly the same as that of overline s we show that in the cases of s n 1 so n and related manifolds we have a proximity condition of the form s overline s y infty lesssim sum i 1 p 1 delta i y infty delta p i y infty where p is the accuracy order of overline s armed with this proximity condition and other known approximation theoretic results we can establish the result that the ho lder smoothness exponent of s is always as high as that of overline s no matter how high the latter is

estimating the number of classes via sample coverage abstract assume that a random sample is drawn from a population with unknown number of classes and possibly unequal class probabilities a nonparametric estimation technique is proposed to estimate the number of classes using the idea of sample coverage which is defined as the sum of the cell probabilities of the observed classes since expected sample coverage can be well estimated we were motivated to find its role in the estimation of the number of classes this work generalizes the result of esty to a nonparametric approach and extends darroch and ratcliff to incorporate the heterogeneity of the class probabilities the coefficient of variation of the class sizes is shown to play an important role in the recommended estimation procedures the performance of the proposed estimators is investigated by means of monte carlo simulations

geodesic finite elements for cosserat rods we introduce geodesic finite elements as a new way to discretize the non linear configuration space of a geometrically exact cosserat rod these geodesic finite elements naturally generalize standard one dimensional finite elements to spaces of functions with values in a riemannian manifold for the special orthogonal group our approach reproduces the interpolation formulas of crisfield and jelenic geodesic finite elements are conforming and lead to objective and path independent problem formulations we introduce geodesic finite elements for general riemannian manifolds discuss the relationship between geodesic finite elements and coefficient vectors and estimate the interpolation error then we use them to find static equilibria of hyperelastic cosserat rods using the riemannian trust region algorithm of absil et al we show numerically that the discretization error depends optimally on the mesh size copyright 2009 john wiley sons ltd

collocation and relaxed collocation for the fer and the magnus expansions we consider the fer and the magnus expansions for the numerical solution of the nonlinear matrix lie group ode y gamma t y y y 0 y 0 where y evolves in a matrix lie group g and gamma t y is in the lie algebra hbox fr g ghe departing from a geometrical approach that distinguishes between those operations performed in the group and those performed in the tangent space we construct lie group invariant methods based on collocation we prove that as long as the two expansions are correctly truncated the collocation nodes c 1 c 2 ldots c nu yield numerical methods whose order is the same as in the classical setting we also relax the collocation conditions thereby devising explicit methods of order 3 to conclude we discuss the proposed methods in a numerical experiment that arises in hamiltonian mechanics comparing the results with projection techniques

nearby lagrangians with vanishing maslov class are homotopy equivalent we prove that the inclusion of every closed exact lagrangian with vanishing maslov class in a cotangent bundle is a homotopy equivalence we start by adapting an idea of fukaya seidel smith to prove that such a lagrangian is equivalent to the zero section in the fukaya category with integral coefficients we then study an extension of the fukaya category in which lagrangians equipped with local systems of arbitrary dimension are admitted as objects and prove that this extension is generated in the appropriate sense by local systems over a cotangent fibre whenever the cotangent bundle is simply connected this generation statement is used to prove that every closed exact lagrangian of vanishing maslov index is simply connected finally we borrow ideas from coarse geometry to develop a fukaya category associated to the universal cover allowing us to prove the result in the general case

a duality exact sequence for legendrian contact homology we establish a long exact sequence for legendrian submanifolds l p r where p is an exact symplectic manifold which admit a hamiltonian isotopy that displaces the projection of l to p off of itself in this sequence the singular homology h maps to linearized contact cohomology ch which maps to linearized contact homology ch which maps to singular homology in particular the sequence implies a duality between ker ch h and ch im h furthermore this duality is compatible with poincare duality in l in the following sense the poincare dual of a singular class which is the image of a ch maps to a class ch such that a 1 the exact sequence generalizes the duality for legendrian knots in r3 see 26 and leads to a refinement of the arnold conjecture for double points of an exact lagrangian admitting a legendrian lift with linearizable contact homology first proved in 7

legendrian contact homology in p times mathbb r a rigorous foundation for the contact homology of legendrian submanifolds in a contact manifold of the form p x r where p is an exact symplectic manifold is established the class of such contact manifolds includes 1 jet spaces of smooth manifolds as an application contact homology is used to provide smooth isotopy invariants of submanifolds ofr and more generally invariants of self transverse immersions into r n up to restricted regular homotopies when n 3 this application is the first step in extending and providing a contact geometric underpinning for the new knot invariants of ng

exact lagrangian immersions with one double point revisited we study exact lagrangian immersions with one double point of a closed orientable manifold k into mathbb c n we prove that if the maslov grading of the double point does not equal 1 then k is homotopy equivalent to the sphere and if in addition the lagrangian gauss map of the immersion is stably homotopic to that of the whitney immersion then k bounds a parallelizable n 1 manifold the hypothesis on the gauss map always holds when n 2k or when n 8k 1 the argument studies a filling of k obtained from solutions to perturbed cauchy riemann equations with boundary on the image f k of the immersion this leads to a new and simplified proof of some of the main results of ekholm and smith exact lagrangian immersions with a single double point 2011 which treated lagrangian immersions in the case n 2k by applying similar techniques to a lagrange surgery of the immersion as well as to an extension of these results to the odd dimensional case

unitary integrators and applications to continuous orthonormalization techniques in this paper the issue of integrating matrix differential systems whose solutions are unitary matrices is addressed such systems have skew hermitian coefficient matrices in the linear case and a

log linear models for capture recapture summary log linear models are developed for capture recapture experiments and their advantages and disadvantages discussed ways in which they can be extended sometimes with only partial success to open populations subpopulations trap dependence and long chains of recapture periods are presented the use of residual patterns and analysis of subsets of data to identify behavioural patterns and acceptable models is emphasised and illustrated with two examples 1 capture recapture the biological problem with which we are concerned is that of estimating the parameters of a free living population of unknown size from a sequence of s samples in each of which any individual either is or is not observed the models in this paper require that each animal be marked naturally or artificially in such a way that we can observe the numbers of animals which have any particular pattern of being or not being observed through the sequence of samples there are t 2s 1 different observable patterns which we shall refer to as capture histories although there is no requirement for physical capture and denote each pattern by a sequence of l s and 2 s representing respectively seen or not seen in a particular sample a general pattern of capture history will be denoted a with ra the number of individuals observed to have this pattern thus for example with s 3 samples and a 121 r 2 is the number of animals seen in the first and third samples but not in the second the pattern with all 2 s is by definition unobservable in general the population will be subject to demographic change by birth or immigration and by death or emigration the simplest assumption for the response of an animal to the observer one which underlies the classical petersen schnabel or jolly seber estimators is that every animal alive in the population at a sampling time has the same probability of being observed this probability may vary from one sample to another or be constant over all or a number of samples if this assumption fails it may do so for either or both of two reasons the first is that of trap dependence the probability of an individual being observed in a sample depending on its past history of capture the second is that of heterogeneity between individuals who because of differences in their behaviour or in where they live inherently have different probabilities of being observed in any sample the basic idea of heterogeneity refers to properties of individuals which remain constant over time but individual animals may also have time varying behaviour the aspect most commonly considered being temporary departure of an individual from the population various excellent texts describe theoretical and practical aspects of capture recapture notably the comprehensive one by seber 1982 the introduction by begon 1979 the

high order embedded runge kutta formulae abstract the criteria to be satisfied by embedded runge kutta pairs of formulae are reviewed two new formulae of orders 6 and 8 are presented together with tests on their efficiency relative to other high order formulae in current use

global error estimation with runge kutta triples abstract the applications of runge kutta rk interpolation to global error estimation using the zadunaisky and related techniques are considered it is shown that the pseudo problem can be based on dense output values within any one step and reliable global error estimates can be obtained at the integration mesh points by using special rk formulae some special formulae of orders 2 6 are presented together with numerical results

a multi step transversal linearization mtl method in non linear structural dynamics an implicit family of multi step transversal linearization mtl methods is proposed for efficient and numerically stable integration of nonlinear oscillators of interest in structural dynamics the presently developed method is a multi step extension and further generalization of the locally transversal linearization ltl method proposed earlier by roy proceedings of the academy of the royal society london 457 2001 539 566 roy and ramachandra journal of sound and vibration 41 2001a 653 679 international journal for numerical methods in engineering 51 2001b 203 224 and roy international journal of numerical methods in engineering 61 2004 764 the mtl based linearization is achieved through a non unique replacement of the nonlinear part of the vector field by a conditionally linear interpolating expansion of known accuracy whose coefficients contain the discretized state variables defined at a set of grid points in the process the nonlinear part of the vector field becomes a conditionally determinable equivalent forcing function the mtl based linearized differential equations thus become explicitly integrable based on the linearized solution a set of algebraic constraint equations are so formed that transversal intersections of the linearized and nonlinearized solution manifolds occur at the multiple grid points the discretized state vectors are thus found as the zeros of the constraint equations simple error estimates for the displacement and velocity vectors are provided and in particular it is shown that the formal accuracy of the mtl methods as a function of the time step size depends only on the error of replacement of the nonlinear part of the vector field presently only two different polynomial based interpolation schemes are employed for transversal linearization viz the taylor like interpolation and the lagrangian interpolation while the taylor like interpolation leads to numerical ill conditioning as the order of interpolation increases the lagrangian interpolation is shown to overcome this numerical problem finally the family of mtl methods is illustrated through limited numerical results for a couple of harmonically driven workhorse oscillators viz the hardening duffing and the duffing holmes oscillators comparisons with results obtained via a sixth order runge kutta method with adaptive time step sizes are also provided

estimating the number of species a review how many kinds are there suppose that a population is partitioned into c classes in many situations interest focuses not on estimation of the relative sizes of the classes but on estimation of c itself for example biologists and ecologists may be interested in estimating the number of species in a population of plants or animals numismatists may be concemed with estimating the number of dies used to produce an ancient coin issue and linguists may be interested in estimating the size of an author s vocabulary in this article we review the problem of statistical estimation of c many approaches have been proposed some purely data analytic and others based in sampling theory in the latter case numerous variations have been considered the population may be finite or infinite if finite samples may be taken with replacement multinomial sampling or without replacement hypergeometric sampling or by bernoulli sampling if infinite sampling may be multinomial or bernoulli or the sample may be th

superconvergent interpolants for the collocation solution of boundary value ordinary differential equations a long standing open question associated with the use of collocation methods for boundary value ordinary differential equations is concerned with the development of a high order continuous solution approximation to augment the high order discrete solution approximation obtained at the mesh points which subdivide the problem interval it is well known that the use of collocation at gauss points leads to solution approximations at the mesh points for which the global error is o h2k where k is the number of collocation points used per subinterval and h is the subinterval size this discrete solution is said to be superconvergent the collocation solution also yields a c0 continuous solution approximation that has a global error of o hk 1 in this paper we show how to efficiently augment the superconvergent discrete collocation solution to obtain c1 continuous superconvergent interpolants whose global errors are o h2k the key ideas are to use the theoretical framework of continuous runge kutta schemes and to augment the collocation solution with inexpensive monoimplicit runge kutta stages specific schemes are derived for k 1 2 3 and 4 numerical results are provided to support the theoretical analysis

forward and adjoint sensitivity analysis with continuous explicit runge kutta schemes abstract we study the numerical solution of tangent linear first and second order adjoint models with high order explicit continuous runge kutta pairs the approaches currently implemented in popular packages such as sundials or daspkadjoint are based on linear multistep methods for adaptive time integration of nonlinear models interpolation of the forward model solution is required during the adjoint model simulation we propose to use the dense output mechanism built in the continuous runge kutta schemes as a highly accurate and cost efficient interpolation method in the inverse problem run we implement our approach in a fortran library called denserks which is found to compare well to other similar software on a number of test problems

an improved nonparametric lower bound of species richness via a modified good turing frequency formula summary it is difficult to accurately estimate species richness if there are many almost undetectable species in a hyperdiverse community practically an accurate lower bound for species richness is preferable to an inaccurate point estimator the traditional nonparametric lower bound developed by chao 1984 scandinavian journal of statistics 11 265 270 for individual based abundance data uses only the information on the rarest species the numbers of singletons and doubletons to estimate the number of undetected species in samples applying a modified good turing frequency formula we derive an approximate formula for the first order bias of this traditional lower bound the approximate bias is estimated by using additional information namely the numbers of tripletons and quadrupletons this approximate bias can be corrected and an improved lower bound is thus obtained the proposed lower bound is nonparametric in the sense that it is universally valid for any species abundance distribution a similar type of improved lower bound can be derived for incidence data we test our proposed lower bounds on simulated data sets generated from various species abundance models simulation results show that the proposed lower bounds always reduce bias over the traditional lower bounds and improve accuracy as measured by mean squared error when the heterogeneity of species abundances is relatively high we also apply the proposed new lower bounds to real data for illustration and for comparisons with previously developed estimators