impact of multicollinearity on small sample hydrologic regression models 1 often hydrologic regression models are developed with ordinary least squares ols procedures the use of ols with highly correlated explanatory variables produces multicollinearity which creates highly sensitive parameter estimators with inflated variances and improper model selection it is not clear how to best address multicollinearity in hydrologic regression models here a monte carlo simulation is developed to compare four techniques to address multicollinearity ols ols with variance inflation factor screening vif principal component regression pcr and partial least squares regression pls the performance of these four techniques was observed for varying sample sizes correlation coefficients between the explanatory variables and model error variances consistent with hydrologic regional regression models the negative effects of multicollinearity are magnified at smaller sample sizes higher correlations between the variables and larger model error variances smaller r2 the monte carlo simulation indicates that if the true model is known multicollinearity is present and the estimation and statistical testing of regression parameters are of interest then pcr or pls should be employed if the model is unknown or if the interest is solely on model predictions is it recommended that ols be employed since using more complicated techniques did not produce any improvement in model performance a leave one out cross validation case study was also performed using low streamflow data sets from the eastern united states results indicate that ols with stepwise selection generally produces models across study regions with varying levels of multicollinearity that are as good as biased regression techniques such as pcr and pls

spherical prediction variance properties of central composite and box behnken designs for quadratic regression on the hypercube a single number criterion such as a g efficiency that is based on the prediction variance is often included as one of the criteria when selecting a response surface design as an alternative to the single number criterion approach the variance dispersion graph presented by giovannitti jensen and myers is a graphical technique for evaluating prediction variance properties throughout the experimental region three properties of interest are the maximum minimum and average spherical prediction variances given the spherical radius as an alternative to the computer based approach requiring an optimization algorithm to evaluate these properties the maximum minimum and spherical prediction variances for central composite and box behnken designs can be determined analytically and are functions only of the radius and the design parameters these functions yield the exact values of the spherical prediction variance properties of central composite and box behnke

gromov witten classes quantum cohomology and enumerative geometry the paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry in particular it contains an axiomatic treatment of gromov witten classes and a discussion of their properties for fano varieties cohomological field theories are defined and it is proved that tree level theories are determined by their correlation functions application to counting rational curves on del pezzo surfaces and projective spaces are given

a measure of rotatability for response surface designs a measure that quantifies the amount of rotatability in a given response surface design is introduced in this article the measure which is expressible as a percentage takes the value 100 if and only if the design is rotatable one of the main advantages of this measure is that it can be used to repair a nonrotatable design by the addition of experimental runs that maximize the percent rotatability over a spherical region of interest four numerical examples are given to illustrate the applications of this measure

the effective cone of the space of parametrized rational curves in a grassmannian we determine the effective cone of the quot scheme parametrizing all rank r degree d quotient sheaves of the trivial bundle of rank n on mathbb p 1 more specifically we explicitly construct two effective divisors which span the effective cone and we also express their classes in the picard group in terms of a known basis

gromov invariants for holomorphic maps from riemann surfaces to grassmannians two compactifications of the space of holomorphic maps of fixed degree from a compact riemann surface to a grassmannian are studied it is shown that the uhlenbeck compactification has the structure of a projective variety and is dominated by the algebraic compactification coming from the grothendieck quot scheme the latter may be embedded into the moduli space of solutions to a generalized version of the vortex equations studied by bradlow this gives an effective way of computing certain intersection numbers known as gromov invariants on the space of holomorphic maps into grassmannians we carry out these computations in the case where the riemann surface has genus one department of mathematics university of utah salt lake city utah 84112 e mail address bertram math utah edu department of mathematics brown university providence rhode island 02912 e mail address daskal gauss math brown edu department of mathematics university of california irvine california 92717 e mail address raw math uci edu license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

two characteristic numbers for smooth plane curves of any degree we use a sequence of blow ups over the projective space parametrizing plane curves of degree d to obtain some enumerative results concerning smooth plane curves of arbitrary degree for d 4 this gives a first modern verification of results of h g zeuthen

the degree of the variety of rational ruled surfaces and gromov witten invariants we compute the degree of the variety parametrizing rational ruled surfaces of degree d in p 3 by relating the problem to gromov witten invariants and quantum cohomology

gromov s compactness of pseudo holomorphic curves and symplectic geometry on utilise certaines idees de schoerr et uhlenbeck sur les equations aux derivees partielles pour demontrer la compacite de gromov des courbes pseudoholomorphes

joint probability of precipitation and discharge deficits in the netherlands 1 the netherlands are situated at the downstream end of the rhine river a large part of the country can be supplied with water from the river in the case of precipitation deficits for drought assessment it is therefore necessary to consider the joint distribution of precipitation and discharge deficits a transformed bivariate normal distribution as well as a bivariate gumbel distribution are fitted to this data in addition nearest neighbor resampling is used to estimate joint probabilities of precipitation and discharge deficits both the reproduction of the marginal distributions and the dependence structure are explored it is found that the transformed bivariate normal distribution underestimates the probability that both the precipitation and discharge deficit are extreme due to its asymptotic independence nearest neighbor resampling also underestimates this probability mainly because the upper tails of the marginal distributions are not properly reproduced by the simulations from the two fitted bivariate distributions a novel bivariate distribution is constructed with transformed normal marginals and a logistic gumbel dependence structure which gives the best description of the upper tail of the joint distribution the use of a failure region based on economic damage rather than on joint exceedances considerably reduces the differences between the probabilities of drought from the various bivariate models

regional hydrologic analysis ordinary and generalized least squares revisited generalized least squares gls regional regression procedures have been developed for estimating river flow quantiles a widely used gls procedure employs a simplified model error structure and average covariances when constructing an approximate residual error covariance matrix this paper compares that gls estimator an idealized gls estimator based on the simplifying assumptions of with true underlying statistics in a region the best possible gls estimator obtained using the true residual error covariarice matrix and the ordinary least squares estimator useful analytic expressions are developed for the variance of and for previously examined cases the average sampling mean square error mses of was the same as the mses of and the mses of usually was larger than the mses of both and the loss in efficiency of was mostly due to estimating streamflow statistics employed in the construction of the residual error covariance matrix rather than the simplifying assumptions in presently employed gls estimators the new analytic expressions were used to compare the performance of the ols and gls estimators for new cases representing greater model variability across sites as well as the effect return period has on the estimators relative performance for a more heteroscedastic model error variance and larger return periods some increase in the mses of relative to the mses of was observed

on the cohomology of brill noether loci over quot schemes abstract let c be a smooth projective irreducible curve over an algebraic closed field k of characteristic 0 we consider brill noether loci over the moduli space of morphisms from c to a grassmannian g m n of m planes in k n and the corresponding quot schemes of quotients of a trivial vector bundle on c compactifying the spaces of morphisms we study in detail the case in which m 2 n 4 we prove results on the irreducibility and dimension of these brill noether loci and we address explicit formulas for their cohomology classes we study the existence problem of these spaces which is closely related with the problem of classification of vector bundles over curves

polar sets on metric spaces we show that if x is a proper metric measure space equipped with a doubling measure supporting a poincare inequality then subsets of x with zero p capacity are precisely the p polar sets that is a relatively compact subset of a domain in x is of zero p capacity if and only if there exists a p superharmonic function whose set of singularities contains the given set in addition we prove that if x is a p hyperbolic metric space then the p superharmonic function can be required to be p superharmonic on the entire space x we also study the the following question if a set is of zero p capacity does there exist a p superharmonic function whose set of singularities is precisely the given set

development of regional regression relationships with censored data when no discharge record is available for a site a regional regression relationship can be used to estimate low flow quantiles problems arise in the derivation of such models when some at site quantile estimates are reported as zero one concern is that quantile estimates reported as zero may be in the range from zero to the measurement threshold a second concern is that a logarithmic transformation cannot be used with zero quantile estimates so traditional log linear least squares estimators cannot be computed this paper uses visual examples and monte carlo simulation to compare the performance of techniques for estimating the parameters of a regional regression model when some at site quantile estimates are zero ordinary least squares ols techniques employed in practice include adding a small constant to all at site quantile estimates denoted olsc or neglecting all observation reported as zero denoted olsd olsc and olsd performed poorly compared to the use of a tobit model which is a maximum likelihood estimator mle procedure that represents the below threshold estimates as a range from zero to the threshold level for a small amount of censoring the olsd method can be acceptable a weighted tobit model that accounts for the heteroscedasticity of the residuals in the regression model provided relatively little gain over the ordinary tobit model

center points in second order response surface designs previous approaches for selecting the number of center points in a second order response surface design are explained and an integrated variance criterion is offered calculations extending results in the literature are made for various composite and box behnken designs for 2 k 8 the results are compared and discussed and indicate that fewer center points than initially recommended are generally appropriate

an index of rotatability an index of rotatability is suggested that will enable the experimenter to obtain an immediate appreciation of the overall shape of specified variance contours for symmetrical second order designs values of the index are tabulated for the central composite designs for two to eight factors three designs of roquemore 1976 are assessed via the index comparisons are made with an entirely different index suggested by khuri 1988 it is concluded that both indexes are useful and sensibly consistent

response surface designs in flexible regions abstract when both variance and bias errors are present the choice of an experimental design to fit a response surface depends on the choice of the region of interest box and draper 1959 1963 and draper and lawrence 1965a 1966 investigated spherical and cuboidal regions respectively both of these regions can however be regarded as special cases of a more general flexible region family depending on a single parameter in this manner the sensitivity of the design criterion can be studied and more general results can be obtained the problem is set out here for k predictor variables and specific designs are provided for k 2 predictor variables when variance and bias errors are of roughly the same order a broad rule of thumb that emerges from this work is that for practical situations involving both variance and bias errors and for designs of the composite type used for fitting a second order surface the design points should be near the boundary of the region of interest over which a uni

degree of the generalized plucker embedding of a quot scheme and quantum cohomology we compute the degree of the generalized plucker embedding kappa of a quot scheme x over pp 1 the space x can also be considered as a compactification of the space of algebraic maps of a fixed degree from pp 1 to the grassmanian rm grass m n then the degree of the embedded variety kappa x can be interpreted as an intersection product of pullbacks of cohomology classes from rm grass m n through the map psi that evaluates a map from pp 1 at a point x in pp 1 we show that our formula for the degree verifies the formula for these intersection products predicted by physicists through quantum cohomology cite va92 cite in91 cite wi94 we arrive at the degree by proving a version of the classical pieri s formula on the variety x using a cell decomposition of a space that lies in between x and kappa x

conformal field theory and the cohomology of the moduli space of stable bundles let g be a compact riemann surface of genus g 2 and let be a line bundle over g of degree 1 then the moduli space of rank 2 stable bundles v over g such that v was shown by seshadri 19 to be a nonsingular projective variety ng its rational cohomology ring h ng is exceedingly rich and despite years of study has never quite been computed in full the object of this paper is to give an essentially complete characterization of this ring or at least to reduce the problem to a matter of linear algebra in particular we find a proof of newstead s conjecture that pg1 ng 0 we also obtain a formula for the volume of ng which can be regarded as a twisted version of the formula for the degree 0 moduli space recently announced by witten the approach we shall take is not from algebraic geometry but from mathematical physics it relies on the su 2 wess zumino witten model which is a functor zk associating a finite dimensional vector space to each riemann surface with marked points the relationship with the moduli space is that when the level k of the functor is even the vector space associated to g with no marked points can be identified with h ng l where l is a fixed line bundle over ng now the work of verlinde provides us with a means of calculating the dimension of any vector space arising from our functor and in particular dimh ng l which we shall denote d g k on the other hand newstead 14 found explicit generators for h ng and we can also express d g k in terms of them using a riemann roch theorem equating the two formulas enables us to evaluate any monomial in the generators on the fundamental class of ng and by poincare duality this is sufficient at least in principle to determine the ring structure of h ng in the discussion above one important point has been skated over the su 2 wzw model is of course associated to bundles of degree 0 not degree 1 so in order to make use of verlinde s work it is necessary to formulate a twisted version of the field theory this is carried out in 2 but the crucial properties of the twisted theory analogous to those which make the ordinary theory a modular functor are not proved in this paper rather we will confine ourselves to exploring the consequences of these claims and hope to return to justify them in a later paper an outline of the remaining sections goes as follows in 3 we review those parts of verlinde s work we shall need show how they must be modified in the twisted case and work out some explicit formulas for d g k in 4 we study the cohomology of ng which we regard throughout as a space of representations via the theorem of narasimhan and seshadri we define newstead s generators and i of h ng