robustness of choice of number of doses for maximum likelihood estimation of the ed50 in bioassay a number of studies have looked at the choice of number of doses for estimating the median effective dose ed50 when the form of a dose response curve is correctly assumed see for example muller and schmitt 1990 here we address the question of the robustness of number of doses chosen using muller and schmitt s method when a logistic dose response curve is incorrectly assumed the underlying true dose response curves considered include the probit cubic logistic and aranda ordaz asymmetric models the simulation results show that for the above true dose response curves and the uniform design density with doses spaced symmetrically or slightly asymmetrically around the assumed ed50 choosing as many doses as possible either minimizes the mean squared error mse or produces a mse chosen to the minimum copyright 2002 john wiley sons ltd

mutations of puzzles and equivariant cohomology of two step flag varieties we introduce a mutation algorithm for puzzles that is a three direction analogue of the classical jeu de taquin algorithm for semistandard tableaux we apply this algorithm to prove our conjectured puzzle formula for the equivariant schubert structure constants of two step flag varieties this formula gives an expression for the structure constants that is positive in the sense of graham thanks to the equivariant version of the quantum equals classical result our formula specializes to a littlewood richardson rule for the equivariant quantum cohomology of grassmannians

structure constants for chern classes of schubert cells a formula for the structure constants of the multiplication of schubert classes is obtained in rebecca and allen arxiv preprint arxiv 1909 05283 2019 in this note we prove analogous formulae for the chern schwartz macpherson csm classes and segre schwartz macpherson ssm classes of schubert cells in the flag variety by the equivalence between the csm classes and the stable basis elements for the cotangent bundle of the flag variety a formula for the structure constants for the latter is also deduced

designs for generalized linear models with several variables and model uncertainty standard factorial designs sometimes may be inadequate for experiments that aim to estimate a generalized linear model for example for describing a binary response in terms of several variables a method is proposed for finding exact designs for such experiments that uses a criterion allowing for uncertainty in the link function the linear predictor or the model parameters together with a design search designs are assessed and compared by simulation of the distribution of efficiencies relative to locally optimal designs over a space of possible models exact designs are investigated for two applications and their advantages over factorial and central composite designs are demonstrated

equivariant quantum schubert polynomials abstract we establish an equivariant quantum giambelli formula for partial flag varieties the answer is given in terms of a specialization of universal double schubert polynomials along the way we give new proofs of the presentation of the equivariant quantum cohomology ring as well as graham positivity of the structure constants in equivariant quantum schubert calculus

schubert polynomials for the affine grassmannian of the symplectic group we study the schubert calculus of the affine grassmannian gr of the symplectic group the integral homology and cohomology rings of gr are identified with dual hopf algebras of symmetric functions defined in terms of schur s p and q functions an explicit combinatorial description is obtained for the schubert basis of the cohomology of gr and this is extended to a definition of the affine type c stanley symmetric functions a homology pieri rule is also given for the product of a special schubert class with an arbitrary one

optimal designs for binary data abstract this article is concerned with the problem of selecting the values of the explanatory variable in logistic regression to obtain likelihood based confidence regions of minimum area one way of finding analytic solutions is by replacing the log likelihood by a quadratic surface around the maximum approximating in this way the likelihood regions by ellipses the optimal allocation for this local approximation is derived without unnecessary restrictions since the implementation of the optimal allocation requires accurate initial estimates a two stage procedure that uses the second stage to complement the first is recommended an alternative approach is to deal directly with the likelihood regions in this case the identification of the globally optimal allocation calls for numerical integration and optimization the results presented here attempt to strengthen those introduced by abdelbasit and plackett 1983 who derived optimal allocation for the local criterion under the restriction of symmet

schubert polynomials and classes of hessenberg varieties abstract regular semisimple hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory numerical analysis representation theory algebraic geometry and combinatorics we give a giambelli formula expressing the classes of regular semisimple hessenberg varieties in terms of chern classes in fact we show that the cohomology class of each regular semisimple hessenberg variety is the specialization of a certain double schubert polynomial giving a natural geometric interpretation to such specializations we also decompose such classes in terms of the schubert basis for the cohomology ring of the flag variety the coefficients obtained are nonnegative and we give closed combinatorial formulas for the coefficients in many cases we introduce a closely related family of schemes called regular nilpotent hessenberg schemes and use our results to determine when such schemes are reduced

integer valued minimax robust designs for estimation and extrapolation in heteroscedastic approximately linear models abstract we present our findings on a new approach to robust regression design this approach differs from previous investigations into this area in three respects the use of a finite design space the use of simulated annealing to carry out the numerical minimization problems and in our search for integer valued rather than continuous designs we present designs for the situation in which the response is thought to be approximately polynomial we also discuss the cases of approximate first and second order multiple regression in each case we allow for possible heteroscedasticity and also obtain minimax regression weights the results are extended to cover extrapolation of the regression response to regions outside of the design space a case study involving dose response experimentation is undertaken the optimal robust designs which protect against bias as well as variance can be roughly described as being obtained from the classical variance minimizing designs by replacing replicates with clust

the application of the annealing algorithm to the construction of exact optimal designs for linear regression models this article reports the application of the annealing algorithm to the construction of exact d i and g optimal designs for polynomial regression of degree 5 on the interval 1 l and for the second order model in two factors on the design space 1 l 1 1 details of the perturbation scheme and the annealing schedules used are given and the method of implementation is illustrated by means of a simple example the algorithm is assessed by comparing its performance in terms of computer time and effkiency with the modified fedorov procedure and it is shown to be particularly effective in finding g optimal designs the salient features of the exact designs constructed in this study are also summarized

equivariant pieri rules for isotropic grassmannians we give a pieri rule for the torus equivariant cohomology of submaximal grassmannians of lie types b c and d to the authors best knowledge our rule is the first manifestly positive formula beyond the equivariant chevalley formula we also give a simple proof of the equivariant pieri rule for the ordinary type a grassmannian

kernel and probit estimates in quantal bioassay abstract a kernel method for the estimation of quantal dose response curves is considered in contrast to parametric modeling this local smoothing method does not require any assumptions beyond smoothness of the dose response curve and in this sense is nonparametric in finite sample situations the kernel estimate of the dose response curve is not necessarily monotone and therefore if the additional assumption of monotonicity of the dose response curve is made a monotonized version is discussed bias variance asymptotic normality and uniform consistency including rates of convergence of kernel estimates are derived and applied to establish consistency and limiting distribution of kernel estimates of the ed here ed is the effective dose at level that is the dose where 100 of the subjects show a response the properties of the kernel estimated ed are compared with the corresponding properties of the maximum likelihood estimator assuming the probit model practical application of the k

equivariant littlewood richardson skew tableaux we give a positive equivariant littlewood richardson rule also discovered independently by molev our proof generalizes a proof by stembridge of the classical littlewood richardson rule we describe a weight preserving bijection between our indexing tableaux and trapezoid puzzles which restricts to a bijection between positive indexing tableaux and knutson tao puzzles

pieri rule for the affine flag variety abstract we prove the affine pieri rule for the cohomology of the affine flag variety conjectured by lam lapointe morse and shimozono we study the cap operator on the affine nilhecke ring that is motivated by kostant and kumar s work on the equivariant cohomology of the affine flag variety we show that the cap operators for pieri elements are the same as pieri operators defined by berg saliola and serrano this establishes the affine pieri rule we also discuss properties of cap operators which are not necessarily affine a type

robust discrimination designs we study the construction of experimental designs the purpose of which is to aid in the discrimination between two regression models each of which might be only approximately specified a rough description of our approach is that we impose neighbourhood structures on each regression response and determine the members of these neighbourhoods which are least favourable in the sense of minimizing the kullback leibler divergence designs are obtained which maximize this minimum divergence both static and sequential approaches are studied we then consider sequential designs whose purpose is initially to discriminate but which move their emphasis towards efficient estimation or prediction as one model becomes favoured over the other

oscillation analysis of advertising capital model analytical and numerical studies abstract this paper mainly deals with the oscillation of nonlinear delay differential equation which is used to describe advertising capital model analytically and numerically firstly the condition of oscillation of the analytic solution is presented by the technique of the theory of characteristic secondly the asymptotic behavior of non oscillatory analytic solution is verified thirdly the methods are applied to the mentioned equation some conditions under which the numerical solution oscillates are obtained moreover it is proved that every non oscillatory numerical solution tends to the steady state of the model finally some numerical simulations for verifying the theoretical findings are provided

littlewood richardson polynomials abstract we introduce a family of rings of symmetric functions depending on an infinite sequence of parameters a distinguished basis of such a ring is comprised by analogues of the schur functions the corresponding structure coefficients are polynomials in the parameters which we call the littlewood richardson polynomials we give a combinatorial rule for their calculation by modifying an earlier result of b sagan and the author the new rule provides a formula for these polynomials which is positive in the sense of w graham we apply this formula for the calculation of the product of equivariant schubert classes on grassmannians which implies a stability property of the structure coefficients the first manifestly positive formula for such an expansion was given by a knutson and t tao by using combinatorics of puzzles while the stability property was not apparent from that formula we also use the littlewood richardson polynomials to describe the multiplication rule in the algebra of the casimir elements for the general linear lie algebra in the basis of the quantum immanants constructed by a okounkov and g olshanski

robustness of design in dose response studies summary we construct experimental designs for dose response studies the designs are robust against possibly misspecified link functions for this they minimize the maximum meansquared error of the estimated dose required to attain a response in 100p of the target population here p might be one particular value p d 0 5 corresponds to ed50 estimation or it might range over an interval of values of interest the maximum of the mean squared error is evaluated over a kolmogorov neighbourhood of the fitted link both the maximum and the minimum must be evaluated numerically the former is carried out by quadratic programming and the latter by simulated annealing

on equivariant quantum schubert calculus for g p we show a z2 filtered algebraic structure and a quantum to classical principle on the torus equivariant quantum cohomology of a complete flag variety of general lie type generalizing earlier works of leung and the second author we also provide various applications on equivariant quantum schubert calculus including an equivariant quantum pieri rule for partial flag variety fln1 nk n 1 of lie type a

gromov witten invariants for and pontryagin product for ω we give an explicit formula for t equivariant 3 pointed genus zero gromov witten invariants for g b we derive it by finding an explicit formula for the pontryagin product on the equivariant homology of the based loop group k

schubert polynomials for the affine grassmannian let g be a complex simply connected simple group and k a maximal compact subgroup let f c t denote the field of formal laurent series and o c t the ring of formal power series the quotient qr g f g 0 is known as the affine grassmannian of g it is known that qr is homotopy equivalent to the space of based loops k into the compact group k see 9 30 the multiplication on qk endows the homology h qr and cohomology h qr with structures of dual hopf algebras these hopf algebras were first identified by bott 4 another descrip tion is given by ginzburg 10 which we shall not use the bruhat decomposition of g f induces a stratification of qr by schubert cells ilw indexed by grassman nian elements w of the affine weyl group waffwe denote by aw g h qr and aw g h qr the schubert classes in homology and cohomology when g sl n c bott identifies h qr and h qr with a subring an and a quotient an of the ring of symmetric functions our main theorem theorem 7 1 identifies the schubert classes aw g qr and aw g h qr as explicit symmet ric functions under this isomorphism these symmetric functions are combinatori ally defined in homology the aw g h qr are represented by lascoux lapointe morse s fc schur functions sw x g an and in cohomology the aw g h qr are represented by the dual fc schur functions or affine schur functions fw x g an see 21 24 17 our theorem was originally conjectured by mark shimozono the conjecture was made explicit in the cohomology case by jennifer morse kostant and kumar 14 studied the topology of homogeneous spaces for arbi trary kac moody groups and in particular calculated the structure constants of h qr in the schubert basis using the algebraic construction of the nilhecke ring our connection between the topology and the combinatorics proceeds via the study of three subalgebras of the nilhecke ring the first subalgebra is the nilcoxeter al gebra ao which is the algebra generated by the divided difference operators the second algebra is a certain centralizer algebra aaff s which we call the peterson subalgebra peterson 28 constructs an isomorphism j from the t equivariant ho mology hr qr of the affine grassmannian to z ff s the third algebra b is a combinatorially defined subalgebra of a0 which we call the affine fomin stanley

equivariant quantum schubert calculus abstract we study the t equivariant quantum cohomology of the grassmannian we prove the vanishing of a certain class of equivariant quantum littlewood richardson coefficients which implies an equivariant quantum pieri rule as in the equivariant case this implies an algorithm to compute the equivariant quantum littlewood richardson coefficients

bayesian optimal designs for estimating a set of symmetrical quantiles we propose multiple objective bayesian optimal designs for the logit model as an example we consider the design problem for estimating several percentiles with possibly unequal interest in each of the percentiles characteristics of these designs are studied and illustrated for the case when the interest lies in estimating the three quartiles we compare these optimal designs with the sequential designs generated via a generalized polya urn model and found the latter to be highly efficient in addition comparisons are made between locally optimal designs and bayesian optimal designs

a pieri type formula for h t sln c b the singular cohomology of the grassmann variety of k planes in cn has a basis s indexed by partitions the classical pieri formula is an explicit rule for determining the coefficients in the expansion of the cup products1m s c1m s where 1m is a column of length m and s1m is the mth chern class of the tautological bundle lascoux and schutzenberger c r acad sci paris294 1982 447 450 formulated a generalization of pieri s formula to the cohomology of the flag variety sln c b and briefly indicated an algebraic proof manivel cours specialises 3 1998 provides details of this proof a geometric proof was given by sottile ann inst fourier grenoble 46 1996 89 110 in this paper we state and prove a generalization of this pieri type formula for the t equivariant cohomology of the flag variety we use the algebraic description of the t equivariant cohomology of the flag variety due to kostant and kumar adv math 62 1986 187 237 and arabia bull soc math france117 1989 129 165 and our new formula exposes an equality of certain structure constants in this algebra our proof is an induction based on the original idea in lascoux and schutzenberger

quantum integrability and generalised quantum schubert calculus abstract we introduce and study a new mathematical structure in the generalised quantum cohomology theory for grassmannians namely we relate the schubert calculus to a quantum integrable system known in the physics literature as the asymmetric six vertex model our approach offers a new perspective on already established and well studied special cases for example equivariant k theory and in addition allows us to formulate a conjecture on the so far unknown case of quantum equivariant k theory

littlewood richardson coefficients for reflection groups abstract in this paper we explicitly compute all littlewood richardson coefficients for semisimple and kac moody groups g that is the structure constants also known as the schubert structure constants of the cohomology algebra h g p c where p is a parabolic subgroup of g these coefficients are of importance in enumerative geometry algebraic combinatorics and representation theory our formula for the littlewood richardson coefficients is purely combinatorial and is given in terms of the cartan matrix and the weyl group of g however if some off diagonal entries of the cartan matrix are 0 or 1 the formula may contain negative summands on the other hand if the cartan matrix satisfies a i j a j i 4 for all i j then each summand in our formula is nonnegative that implies nonnegativity of all littlewood richardson coefficients we extend this and other results to the structure coefficients of the t equivariant cohomology of flag varieties g p and bott samelson varieties i g

a positive monk formula in the s1 equivariant cohomology of type a peterson varieties peterson varieties are a special class of hessenberg varieties that have been extensively studied for example by peterson kostant and rietsch in connection with the quantum cohomology of the flag variety in this manuscript we develop a generalized schubert calculus and in particular a positive chevalley monk formula for the ordinary and borel equivariant cohomology of the peterson variety y in type a n 1 with respect to a natural s 1 action arising from the standard action of the maximal torus on flag varieties as far as we know this is the first example of positive schubert calculus beyond the realm of kac moody flag varieties g p our main results are as follows first we identify a computationally convenient basis of h s1 y which we call the basis of peterson schubert classes second we derive a manifestly positive integral chevalley monk formula for the product of a cohomology degree 2 peterson schubert class with an arbitrary peterson schubert class both h s1 y and h y are generated in degree 2 finally by using our chevalley monk formula we give explicit descriptions via generators and relations of both the s 1 equivariant cohomology ring h s1 y and the ordinary cohomology ring h y of the type a n 1 peterson variety our methods are both directly from and inspired by those of the gkm goresky kottwitz macpherson theory and classical schubert calculus we discuss several open questions and directions for future work

positivity in equivariant quantum schubert calculus a conjecture of d peterson proved by w graham states that the structure constants of the t equivariant cohomology of a homogeneous space g p satisfy a certain positivity property in this paper we show that this positivity property holds in the more general situation of equivariant quantum cohomology 1 introduction it is well known that the integral cohomology of the homogeneous space x g p for g a connected semisimple complex lie group and p a parabolic subgroup satisfies a positivity property its structure constants are nonnegative integers equal to the number of intersection points of three schubert varieties in general position whose codimensions add up to the dimension of x recently graham gr has proved a conjecture of peterson p asserting that h x the t equivariant cohomology of x where t c r is a maximal torus in g enjoys a more general positivity property

excited young diagrams and equivariant schubert calculus we describe the torus equivariant cohomology ring of isotropic grassmannians by using a localization map to the torus fixed points we present two types of formulas for equivariant schubert classes of these homogeneous spaces the first formula involves combinatorial objects which we call excited young diagrams and the second one is written in terms of factorial schur q or p functions as an application we give a giambelli type formula for the equivariant schubert classes we also give combinatorial and pfaffian formulas for the multiplicity of a singular point in a schubert variety

schubert classes in the equivariant cohomology of the lagrangian grassmannian abstract let lg n denote the lagrangian grassmannian parametrizing maximal isotropic lagrangian subspaces of a fixed symplectic vector space of dimension 2n for each strict partition 1 k with 1 n there is a schubert variety x let t denote a maximal torus of the symplectic group acting on lg n consider the t equivariant cohomology of lg n and the t equivariant fundamental class of x the main result of the present paper is an explicit formula for the restriction of the class to any torus fixed point the formula is written in terms of factorial analogue of the schur q function introduced by ivanov as a corollary to the restriction formula we obtain an equivariant version of the giambelli type formula for lg n as another consequence of the main result we obtained a presentation of the ring h t lg n

positivity in the grothendieck group of complex flag varieties abstract we prove a conjecture of a s buch concerning the structure constants of the grothendieck ring of a flag variety with respect to its basis of schubert structure sheaves for this we show that the coefficients in this basis of the structure sheaf of any subvariety with rational singularities have alternating signs equivalently the class of the dualizing sheaf of such a subvariety is a nonnegative combination of classes of dualizing sheaves of schubert varieties

adapting for the missing link we consider the fitting of generalized linear models in wich the link function is assumed to be unknown and propose the following computational method first estimate regression coefficients using the canonical link then estimate the link via a kernel smoother treating the direction in the predictor space determined by the regression coefficients as known then reestimate the direction using the estimated link and alternate between these two steps we show that under fairly general conditions n 1 2 consistent estimates of the direction are obtained a small monte carlo study is presented

equivariant quantum cohomology of the odd symplectic grassmannian the odd symplectic grassmannian mathrm ig mathrm ig k 2n 1 parametrizes k dimensional subspaces of mathbb c 2n 1 which are isotropic with respect to a general necessarily degenerate symplectic form the odd symplectic group acts on mathrm ig with two orbits and mathrm ig is itself a smooth schubert variety in the submaximal isotropic grassmannian mathrm ig k 2n 2 we use the technique of curve neighborhoods to prove a chevalley formula in the equivariant quantum cohomology of mathrm ig i e a formula to multiply a schubert class by the schubert divisor class this generalizes a formula of pech in the case k 2 and it gives an algorithm to calculate any multiplication in the equivariant quantum cohomology ring

a note on nonparametric estimation of the effective dose in quantal bioassay for the common binary response model we propose a direct method for the nonparametric estimation of the effective dose level the estimator is obtained by the composition of a nonparametric estimate of the quantile response curve and a classical density estimate the new method yields a simple and reliable monotone estimate of the effective dose level curve ed and is appealing to users of conventional smoothing methods as kernel estimators local polynomials series estimators or smoothing splines moreover it is computationally very efficient because it does not require a numerical inversion of a monotonized estimate of the quantile dose response curve we prove asymptotic normality of the new estimate and compare it with an available alternative estimate based on a monotonized nonparametric estimate of the dose response curve and calculation of the inverse function by means of a simulation study