applications of the ashtekar gravity to four dimensional hyperkahler geometry and yang mills instantons the ashtekar jacobson smolin mason newman equations are used to construct the hyperkahler metrics on four dimensional manifolds these equations are closely related to anti self dual yang mills equations of the infinite dimensional gauge lie algebras of all volume preserving vector fields several examples of hyperkahler metrics are presented through the reductions of anti self dual connections for any gauge group anti self dual connections on hyperkahler manifolds are constructed using the solutions of both nahm and laplace equations

moment methods for decision analysis decision models involving continuous probability distributions almost always require some form of approximation the usual approach to evaluating these kinds of models is to construct a discrete approximation for each continuous distribution and compute value lotteries and certain equivalents using these discrete approximations although decision analysts are quite comfortable with this approach there has been relatively little consideration of how these discrete approximations affect the results of the analysis in the first part of this paper we review three common methods of constructing discrete approximations and compare their performance in a simple example the results of the example suggest a different approach that offers potential improvements in accuracy and efficiency over the usual approach the basic idea is that given discrete approximations that accurately represent the moments of assessed input distributions we may easily and accurately compute the moments of the output distribution or value lotteries these moments then summarize what we know about the value lottery and certain equivalent and provide the basis for computing approximate value lotteries and certain equivalents in this paper we discuss the methods supporting this moment approach and evaluate their performance in the context of the example

maximum likelihood estimates via duality for log convex models when cell probabilities are subject to convex constraints the purpose of this article is to derive and illustrate a method for fitting models involving both convex and log convex constraints on the probability vector s of a product multinomial distribution we give a two step algorithm to obtain maximum likelihood estimates of the probability vector s and show that it is guaranteed to converge to the true solution some examples are discussed which illustrate the procedure

comparing the areas under two or more correlated receiver operating characteristic curves a nonparametric approach methods of evaluating and comparing the performance of diagnostic tests are of increasing importance as new tests are developed and marketed when a test is based on an observed variable that lies on a continuous or graded scale an assessment of the overall value of the test can be made through the use of a receiver operating characteristic roc curve the curve is constructed by varying the cutpoint used to determine which values of the observed variable will be considered abnormal and then plotting the resulting sensitivities against the corresponding false positive rates when two or more empirical curves are constructed based on tests performed on the same individuals statistical analysis on differences between curves must take into account the correlated nature of the data this paper presents a nonparametric approach to the analysis of areas under correlated roc curves by using the theory on generalized u statistics to generate an estimated covariance matrix

on anova expansions and strategies for choosing the anchor point the classic lebesgue anova expansion offers an elegant way to represent functions that depend on a high dimensional set of parameters and it often enables a substantial reduction in the evaluation cost of such functions once the anova representation is constructed unfortunately the construction of the expansion itself is expensive due to the need to evaluate high dimensional integrals a way around this is to consider an alternative formulation known as the anchored anova expansion this formulation requires no integrals but has an accuracy that depends sensitively on the choice of a special parameter known as the anchor point we present a comparative study of several strategies for the choice of this anchor point and argue that the optimal choice of this anchor point is the center point of a sparse grid quadrature this choice induces no additional cost and as we shall show results in a natural truncation of the anova expansion the efficiency and accuracy is illustrated through several standard benchmarks and this choice is shown to outperform the alternatives over a range of applications c 2010 elsevier inc all rights reserved

three point approximations for continuous random variables this paper compares a number of approximations used to estimate means and variances of continuous random variables and or to serve as substitutes for the probability distributions of such variables with particular emphasis on three point approximations numerical results from estimating means and variances of a set of beta distributions indicate surprisingly large differences in accuracy among approximations in current use with some of the most popular ones such as the pert and triangular density function approximations faring poorly a simple new three point approximation which is a straightforward extension of earlier work by pearson and tukey outperforms the others significantly in these tests and also performs well in related multivariate tests involving the dirichlet family of distributions it offers an attractive alternative to currently used approximations in a variety of applications

profile likelihood inference for highly accurate diagnostic tests nous envisageons une inference sur la vraisemblance de profil basee sur la distribution multinomiale pour evaluer la precision d un test diagnostic la methode s applique a des donnees ordinales quand la precision est evaluee au moyen de l aire sous la courbe caracteristique du receveur courbe roc des resultats de simulation suggerent que les intervalles de confiance derives ont vies probabilites de couverture acceptables meme lorsque les tailles d echantillons sont petites et que le test diagnostic a une haute precision les methodes s etendent a des contextes stratifies et a des situations ou les evaluations sont correlees nous illustrons les methodes sur des donnees provenant d un essai clinique sur la detection du cancer de l ovaire

self dual manifolds with positive ricci curvature we prove that the connected sums cp2 cp2 and cp2 cp2 cp2 admit self dual metrics with positive ricci curvature moreover every self dual metric of positive scalar curvature on cp2 cp2 is conformal to a metric with positive ricci curvature supported in part by nsf grant dms 9204093

better estimation of pert activity time parameters this paper builds upon earlier work from the decision risk analysis area in presenting simple easy to use approximations for the mean and variance of pert activity times these approximations offer significant advantages over the pert formulas currently being taught and used as well as over recently proposed modifications for instance they are several orders of magnitude more accurate than their pert counterparts in estimating means and variances of beta distributions if the data required for all methods are obtained accurately moreover they utilize probability data that can be assessed more reliably than those required by the pert formulas while still requiring just three points from each activity time probability distribution using the proposed approximations can significantly improve the accuracy of probability statements about project completion time and their use complements ongoing efforts to improve pert analyses of networks involving multiple critical paths

on twistor spaces of anti self dual hermitian surfaces we consider a complex surface m with anti self dual hermitian metric h and study the holomorphic properties of its twistor space z we show that the naturally defined divisor line bundle x is isomorphic to the 1 2 power of the canonical bundle of z if and only if there is a kahler metric of zero scalar curvature in the conformal class of h this has strong consequences on the geometry of m which were also found by c boyer 3 using completely different methods we also prove the existence of a very close relation between holomorphic vector fields on m and z in the case that m is compact and kahler

a probabilistic interpretation of the final rankings in ahp abstract the analytic hierarchy process ahp has become a popular and practical tool for dealing with complex decision problems it provides a ranking for the decision alternatives this article recommends treating the pairwise comparison input data as random variables this will allow the determination of whether the differences between alternatives are statistically significant

modeling uncertainty in multi criteria decision analysis abstract his paper provides a review of multiple criteria decision analysis mcda for cases where attribute evaluations are uncertain the main aim is to identify different tools which can be used to represent uncertain evaluations and to broadly survey the available decision models that can be used to support uncertain decision making the review includes models using probabilities or probability like quantities explicit risk measures such as quantiles and variances fuzzy numbers and scenarios the practical assessment of uncertain outcomes and preferences associated with these outcomes is also discussed

singular hyperbolic monopoles we study the twistor theory of singular hyperbolic su 2 monopoles following the approach taken by kronheimer 9 in the euclidean case we use our results to show that the moduli space of charge 1 monopoles possesses a natural 2 sphere of scalar flat kahler metrics in the zero mass limit the metrics reduce to a class of metrics first studied by lebrun in 10

correlations and copulas for decision and risk analysis the construction of a probabilistic model is a key step in most decision and risk analyses typically this is done by defining a joint distribution in terms of marginal and conditional distributions for the model s random variables we describe an alternative approach that uses a copula to construct joint distributions and pairwise correlations to incorporate dependence among the variables the approach is designed specifically to permit the use of an expert s subjective judgments of marginal distributions and correlations the copula that underlies the multivariate normal distribution provides the basis for modeling dependence but arbitrary marginals are allowed we discuss how correlations can be assessed using techniques that are familiar to decision analysts and we report the results of an empirical study of the accuracy of the assessment methods the approach is demonstrated in the context of a simple example including a study of the sensitivity of the results to the assessed correlations

operationalizing approximate multiattribute utility functions for use in practice abstract decision analysis provides defensible methods for rank ordering alternatives based on the calculation of expected utility this technique can be quite time consuming and conceptually difficult for the decision maker as many input parameters are needed to be assessed the recognition of such difficulties prompted researchers to call for simplifying approximations which retain the integrity of the original problem but relieve the assessment burden on the decision maker winkler 1982 this call has been answered with the recent development by kirkwood of taylor series approximations to expected utility which have never been applied in practice this paper further addresses the need to simplify decision analysis through the development of a simulation based technique which allows the kirkwood approximations to be used in practice the technique generates confidence intervals for the true but unknown certain equivalents which are used to rank order alternatives

twistor cr manifolds and three dimensional conformal geometry a cr i e partially complex 5 manifold is contructed as a sphere bundle over an arbitrary 3 manifold with conformal metric this so called twistor cr manifold is show to capture completely the original geometry and necessary and sufficient conditions are given for an abstract cr manifold to arise via the construction the above correspondence is then used to prove that a twistor cr manifold is locally imbeddable as a real hypersurface in c3 only if it is real analytic with respect to a suitable atlas introduction there is a very beautiful interplay between the conformal geometry of riemannian manifolds and complex analysis examples of which are found both in the classical theorem asserting the existence of isothermal coordinates on a surface and in the theory of self dual riemannian 4 manifolds penrose 1976 atiyah et al 1978 intermediating between these two theories is a relationship between 3dimensional conformal geometry and cr partially complex manifolds of dimension 5 in this paper we will explain this relationship focusing on the imbedding problem when can these partially complex manifolds be realized as real hypersurfaces in complex 3 manifolds the idea of associating a cr manifold with a conformal riemannian 3 manifold originated with penrose 1975 1983 who starting with a real analytic space like hypersurface in a real analytic lorentzian 4 manifold constructed a real hypersurface in a complex 3 manifold if the space like hypersurface is totally geodesic or merely all umbilic penroses s cr manifold coincides in our terminology with the twistor cr manifold of the conformal structure induced on the space like hypersurface after establishing notation and conventions in 0 principal results on cr manifolds are reviewed and the notions of cr vector bundle and cr contact form are introduced in 1 a general machine for constructing involutive distributions is produced and in 2 this construction is used to associate to every conformal riemannian 3 manifold a cr 5 manifold the cr manifolds arising in this way are characterized abstractly in 4 in terms of properties presented in 3 then in 5 it is shown that these cr manifolds are isomorphic to real hypersurfaces of complex received by the editors march 15 1983 and in revised form july 8 1983 1980 mathematics subject classificcation primary 32f25 secondary 53a30 83c99 work supported in part by nsf grant mcs 812 0790 001984 american mathematical society 0002 9947 84 1 00 25 per page

compact self dual manifolds with positive scalar curvature on etudie l existence d une metrique auto duale a courbure scalaire positive sur une variete compacte simplement connexe qui n est pas conformement equivalente a ces metriques standard sur s 4 ou cp 2

double solid twistor spaces the case of arbitrary signature in a recent paper 9 we constructed a series of new moishezon twistor spaces which are a kind of variant of the famous lebrun twistor spaces in this paper we explicitly give projective models of another series of moishezon twistor spaces on ncp2 for arbitrary n 3 which can be regarded as a generalization of the twistor spaces of double solid type on 3cp2 studied by kreusler kurke poon and the author similarly to the twistor spaces of double solid type on 3cp2 projective models of the present twistor spaces have a natural structure of double covering of a cp2 bundle over cp1 we explicitly give a defining polynomial of the branch divisor of the double covering whose restriction to fibers is degree four if n 4 these are new twistor spaces to the best of the author s knowledge we also compute the dimension of the moduli space of these twistor spaces differently from 9 the present investigation is based on analysis of pluri half anticanonical systems of the twistor spaces

using bayesian networks for bankruptcy prediction some methodological issues abstract this study provides operational guidance for building naive bayes bayesian network bn models for bankruptcy prediction first we suggest a heuristic method that guides the selection of bankruptcy predictors based on the correlations and partial correlations among variables the method aims at eliminating redundant and less relevant variables a naive bayes model is developed using the proposed heuristic method and is found to perform well based on a 10 fold validation analysis the developed naive bayes model consists of eight first order variables six of which are continuous we also provide guidance on building a cascaded model by selecting second order variables to compensate for missing values of first order variables second we analyze whether the number of states into which the six continuous variables are discretized has an impact on the model s performance our results show that the model s performance is the best when the number of states for discretization is either two or three starting from four states the performance starts to deteriorate probably due to over fitting finally we experiment whether modeling continuous variables with continuous distributions instead of discretizing them can improve the model s performance our finding suggests that this is not true one possible reason is that continuous distributions tested by the study do not represent well the underlying distributions of empirical data finally the results of this study could also be applicable to business decision making contexts other than bankruptcy prediction

a copulas based approach to modeling dependence in decision trees this paper presents a general framework based on copulas for modeling dependent multivariate uncertainties through the use of a decision tree the proposed dependent decision tree model allows multiple dependent uncertainties with arbitrary marginal distributions to be represented in a decision tree with a sequence of conditional probability distributions this general framework could be naturally applied in decision analysis and real options valuations as well as in more general applications of dependent probability trees while this approach to modeling dependencies can be based on several popular copula families as we illustrate we focus on the use of the normal copula and present an efficient computational method for multivariate decision and risk analysis that can be standardized for convenient application

the most critical path in a pert network a heuristic approach abstract the classical pert approach uses a deterministic critical path to estimate the probability of completing a project by a given deadline in general such a path is not the most critical path and does not provide the lowest probability of completion time this paper studies the most critical path problem and formulates it as an optimal path problem through a deterministic network with a two attribute fractional objective function a heuristic approach is developed for the optimal path problem which gives a solution to the most critical path problem the illustrative examples and computational results demonstrate that the proposed procedure provides critical activities and estimates for the probabilities of completion time that are more accurate than those resulting from the classical pert approach further our computational experiments show that the proposed heuristic performs well in identifying the most critical paths

estimating the impact of uncertainty on a deterministic multiattribute evaluation a method is presented to estimate the impact of uncertainty on the results of a multiattribute evaluation prior to conducting a complete probabilistic multiattribute utility analysis the method assumes that a deterministic analysis has been completed using a weighted additive multiattribute value evaluation function and that either an additive or multiplicative utility function is appropriate for the multiattribute utility analysis an approximation procedure is developed to estimate whether uncertainty about attribute levels could change the deterministic evaluation results an illustrative application investigates the accuracy of the procedure and its use to simplify analysis of the effects of uncertainty

empirical likelihood inference for the area under the roc curve for a continuous scale diagnostic test the most commonly used summary index of the receiver operating characteristic curve roc is the area under the curve auc that measures the accuracy of the diagnostic test in this article we propose an empirical likelihood el approach for the inference on the auc first we define an el ratio for the auc and show that its limiting distribution is a scaled chi square distribution we then obtain an el based confidence interval for the auc using the scaled chi square distribution this el inference for the auc can be extended to stratified samples and the resulting limiting distribution is a weighted sum of independent chi square distributions additionally we conduct simulation studies to compare the relative performance of the proposed el based interval with the existing normal approximation based intervals and bootstrap intervals for the auc

a linear bayesian stochastic approximation to update project duration estimates by relaxing the unrealistic assumption of probabilistic independence on activity durations in a project this paper develops a hierarchical linear bayesian estimation model statistical dependence is established between activity duration and the amount of resource as well as between the amount of resource and the risk factor upon observation or assessment of the amount of resource required for an activity in near completion the posterior expectation and variance of the risk factor can be directly obtained in the bayesian scheme then the expected amount of resources required for and the expected duration of upcoming activities can be predicted we simulate an application project in which the proposed model tracks the varying critical path activities on a real time basis and updates the expected project duration throughout the entire project in the analysis the proposed model improves the prediction accuracy by 38 36 compared to the basic pert approach

risk taking in stochastic pert networks abstract this paper studies a pert problem which incorporates project managers risk taking behavior in stochastic environments in this stochastic pert network which we refer to as riskpert activity durations are nonnegative random variables and a project manager uses a disutility function to evaluate each path and chooses that path which maximizes his expected disutility as an optimal or critical path this path as opposed to the classical critical path is used as a basis to analyze the pert network and to estimate the probability of completion time the general riskpert is difficult to solve however special cases when activity durations are statistically independent and disutility functions are linear exponential quadratic or linear exponential are solvable exactly the illustrative examples demonstrate that riskpert captures managers risk taking behavior and provides results that are more realistic than those of the classical pert models furthermore this paper extends some of the riskpert models to include multi dimensional pert networks

the nonparametric behrens fisher problem with dependent replicates purely nonparametric methods are developed for general two sample problems in which each experimental unit may have an individual number of possibly correlated replicates in particular equality of the variances or higher moments of the distributions of the data is not assumed even under the null hypothesis of no treatment effect thus a solution for the so called nonparametric behrens fisher problem is proposed for such models the methods are valid for metric count ordered categorical and even dichotomous data in a unified way point estimators of the treatment effects as well as their asymptotic distributions will be studied in detail for small sample sizes the distributions of the proposed test statistics are approximated using satterthwaite welch type t approximations extensive simulation studies show favorable performance of the new methods in particular in small sample size situations a real data set illustrates the application of the proposed methods

twistors kahler manifolds and bimeromorphic geometry i by considering deformations of the moishezon twistor spaces of cp2 cp2 constructed in 20 we showthat theblowupof c2 at n points in general position admits an asymptotically flat scalar flat kahler metric in each kahler class at least provided that the given points are nearly collinear department of mathematics suny at stony brook stony brook new york 11794 e mail address claude math sunysb edu this content downloaded from 157 55 39 153 on mon 19 sep 2016 04 58 00 utc all use subject to http about jstor org terms

equivariant connected sums of compact self dual manifolds due to a recent theorem of taubes self dual conformal structures on compact four dimensional manifolds exist in abundance 43 therefore finding examples in general is no longer an issue in this subject to deepen our understanding of self dual conformal geometry we take the path of trying to understand the possible symmetry groups of this geometry i e the groups of orientation preserving conformal transformations the first systematic approach which constructs a large collection of compact self dual manifolds with a non trivial group of symmetries is lebrun s hyperbolic ansatz 22 the condition of this ansatz is that the symmetry group contains an s l u 1 which acts on the manifold semi freely meaning that the isotropy group at any point of the manifold is either trivial or the entire group s i with this condition on the group action and some favourable topological conditions on the underlying manifold the geometrical equations of self duality on the given four dimensional manifold are reduced to the s 1 monopole equations on hyperbolic 3 space with this setup lebrun explicitly constructed a large number of compact self dual manifolds with semi free s l symmetry including the fubinistudy metric on the complex projective plane ci 2 and a one parameter family of self dual metrics on c c first found in 38 we shall call the self dual metrics on nci 2 n 1 constructed by lebrun s hyperbolic ansatz the lebrun metrics after this very successful ansatz it is natural to try to find new examples which are beyond the scope of that method

score and profile likelihood confidence intervals for contingency table parameters a straightforward approach to computing score and profile likelihood confidence intervals for contingency table parameters is described the computational approach herein avoids two main limitations of existing methods 1 compared with existing methods this paper s approach is applicable to a much broader class of parameters 2 unlike existing methods this paper s approach is not case specific and hence lends itself to a general yet very simple computational algorithm this paper describes the sliding quadratic computational algorithm and illustrates its use on examples that have not been previously considered in the literature a small scale simulation study highlights the advantage of using score and profile likelihood intervals rather than wald intervals

discretizing distributions with exact moments error estimate and convergence analysis the maximum entropy principle is a powerful tool for solving underdetermined inverse problems this paper considers the problem of discretizing a continuous distribution which arises in various applied fields we obtain the approximating distribution by minimizing the kullback leibler information relative entropy of the unknown discrete distribution relative to an initial discretization based on a quadrature formula subject to some moment constraints we study the theoretical error bound and the convergence of this approximation method as the number of discrete points increases we prove that i the theoretical error bound of the approximate expectation of any bounded continuous function has at most the same order as the quadrature formula we start with and ii the approximate discrete distribution weakly converges to the given continuous distribution moreover we present some numerical examples that show the advantage of the method and apply to numerically solving an optimal portfolio problem

einstein metrics spinning top motions and monopoles on considere la construction de solutions pour l equation d einstein autoduale on applique la construction de lebrun a la sphere de berger la 3 sphere de metrique 1 2 2 2 i 3 3 2 ou 1 2 3 est une base de 1 formes invariante a gauche satisfaisant d i j k e ijk j k et i 3 est une constante

killing vectors in self dual euclidean einstein spaces using the formalism of complex h spaces we show that all real euclidean self dual spaces that admit at least one killing vector may be gauged so that only two distinct types of killing vectors appear in kahler coordinates these are the generators of a translational or a rotational symmetry we give explicit forms both for the killing vectors and for the constraint on the kahler potential function which allows for such a killing vector in the translational case we show how all such spaces are determined by the general solution of the three dimensional flat laplace s equation and how these are related to the multi taub nut metrics of gibbons and hawking in the rotational case we simplify the equation determining but this is not sufficient to obtain the general solution

the calabi flow on kahler surfaces with bounded sobolev constant i we consider the formation of singularities along the calabi flow by assuming the uniformly bounded sobolev constants on kahler surfaces we prove that if curvature tensor is not uniformly bounded then one can form a singular model called deepest bubble such deepest bubble has to be a scalar flat ale kahler metric in certain kahler classes on toric fano surfaces the sobolev constants are a priori bounded along the calabi flow with small calabi energy we can also show in certain cases no deepest bubble can form along the flow it follows that the curvature tensor is uniformly bounded and the flow exists for all time and converges to an extremal metric subsequently to illustrate our results more clearly we focus on an example on mathbb cp 2 blown up three points at generic position our result also implies existence of constant scalar curvature metrics on mathbb cp 2 blown up three points at generic position in the kahler classes where the exceptional divisors have the same area

generating scenario trees for multistage decision problems in models of decision making under uncertainty we often are faced with the problem of representing the uncertainties in a form suitable for quantitative models if the uncertainties are expressed in terms of multivariate continuous distributions or a discrete distribution with far too many outcomes we normally face two possibilities either creating a decision model with internal sampling or trying to find a simple discrete approximation of the given distribution that serves as input to the model this paper presents a method based on nonlinear programming that can be used to generate a limited number of discrete outcomes that satisfy specified statistical properties users are free to specify any statistical properties they find relevant and the method can handle inconsistencies in the specifications the basic idea is to minimize some measure of distance between the statistical properties of the generated outcomes and the specified properties we illustrate the method by single and multiple period problems the results are encouraging in that a limited number of generated outcomes indeed have statistical properties that are close to or equal to the specifications we discuss how to verify that the relevant statistical properties are captured in these specifications and argue that what are the relevant properties will be problem dependent

certainty equivalents for three point discrete distribution approximations three point discrete distribution approximations are often used in decision and risk analyses to represent probability distributions for continuous random variables e g as probability nodes in decision or probability trees performance evaluations of such approximations have generally been based on their accuracies in estimating moments for the underlying distributions however moment based comparisons for recently proposed approximations have been limited moreover very little research has addressed the accuracies of any such approximations in representing expected utilities or certainty equivalents of the underlying distributions this is of potential concern since recent research shows three point discrete distribution approximations exist that match the first several moments of an underlying distribution exactly while counterintuitively approximating its certainty equivalents poorly this paper compares the best two approximations for estimating means and variances identified in an earlier study with promising approximations proposed more recently specifically it examines how accurately six simple general purpose three point approximations represent certainty equivalents for continuous random variables as the level of risk aversion is varied as well as how accurately they estimate means and variances the results show that several of these are quite accurate over a variety of test distributions when the level of risk aversion and the characteristics of the distributions are within reasonable bounds their robust performance is significant for decision analysis practice

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

using experts noisy quantile judgments to quantify risks theory and application to agribusiness motivated by a unique agribusiness setting this paper develops an optimization based approach to estimate the mean and standard deviation of probability distributions from noisy quantile judgments provided by experts the approach estimates the mean and standard deviations as weighted linear combinations of quantile judgments where the weights are explicit functions of the expert s judgmental errors the approach is analytically tractable and provides flexibility to elicit any set of quantiles from an expert the approach also establishes that using an expert s quantile judgments to deduce the distribution parameters is equivalent to collecting data with a specific sample size and enables combining the expert s judgments with those of other experts it also shows analytically that the weights for the mean add up to one and the weights for the standard deviation add up to zero these properties have been observed numerically in the literature in the last 30 years but without a systematic explanation th

non abelian infinite algebra of generalized symmetries for the sdiff 2 toda equation we determine the non abelian algebra of generalized symmetries for the sdiff 2 toda equation a pde for a single function of three independent variables the solutions of which determine self dual vacuum solutions of the einstein field equations this algebra is a realization of two copies of the abstract algebra sdiff 2 along with an additional pair of elements that have derivation like properties on both of the copies it contains as a subalgebra the doubly infinite abelian algebra equivalent to the infinite hierarchy of higher flows found by takasaki and takebe an infinite prolongation of the jet bundle for the original pde to include all the variables allowed in their hierarchy is required for the presentation of this generalization because these symmetries have non zero commutators they generate a recursion relation allowing the generation and description of the entire algebra

index energy estimates for yang mills connections and einstein metrics we prove a conformally invariant estimate for the index of schrodinger operators acting on vector bundles over four manifolds related to the classical cwikel lieb rozenblum estimate applied to yang mills connections we obtain a bound for the index in terms of its energy which is conformally invariant and captures the sharp growth rate furthermore we derive an index estimate for einstein metrics in terms of the topology and the einstein hilbert energy lastly we derive conformally invariant estimates for the betti numbers of an oriented four manifold with positive scalar curvature