demazure modules and weyl modules the twisted current case we study finite dimensional respresentations of twisted current algebras and show that any graded twisted weyl module is isomorphic to level one demazure modules for the twisted affine kac moody algebra using the tensor product property of demazure modules we obtain by analyzing the fundamental weyl modules dimension and character formulas moreover we prove that graded twisted weyl modules can be obtained by taking the associated graded modules of weyl modules for the loop algebra which implies that its dimension and classical character are independent of the support and depend only on its classical highest weight these results were previously known for untwisted current algebras and are new for all twisted types

multi dimensional weyl modules and symmetric functions the weyl modules in the sense of v chari and a pressley cp over the current lie algebra on an affine variety are studied we show that local weyl modules are finite dimensional and generalize the tensor product decomposition theorem from cp more explicit results are stated for currents on a non singular affine variety of dimension d with coefficients in the lie algebra sl r the weyl modules with highest weights proportional to the vector representation one are related to the multi dimensional analogs of harmonic functions the dimensions of such local weyl modules are calculated in the following cases for d 1 we show that the dimensions are equal to powers of r for d 2 we show that the dimensions are given by products of the higher catalan numbers the usual catalan numbers for r 2

integrable representations of root graded lie algebras abstract in this paper we study the category of representations of a root graded lie algebra l which are integrable as representations of a finite dimensional semisimple subalgebra g and whose weights are bounded by some dominant weight of g we link this category to the module category of an associative algebra whose structure we determine for map algebras and sl n a our approach unifies recent work of chari and her collaborators on map algebras of fourier and savage and their collaborators on equivariant map algebras as well as the classical work of seligman on isotropic lie algebras

the winding number of pf 1 for polynomials p and meromorphic extendibility of f abstract let be the open unit disc in c the paper deals with the following conjecture if f is a continuous function on b such that the change of argument of p f 1 around b is nonnegative for every polynomial p such that p f 1 has no zero on b then f extends holomorphically through we prove a related result on meromorphic extendibility for smooth functions with finitely many zeros of finite order which in particular implies that the conjecture holds for real analytic functions

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

lie algebras graded by finite root systems and intersection matrix algebras this paper classifies the lie algebras graded by doubly laced finite root systems and applies this classification to identify the intersection matrix algebras arising from multiply affinized cartan matrices of types b c f and g this completes the determination of the lie algebras graded by finite root systems initiated by berman and moody who studied the simply laced finite root systems of rank 2

central extension of quadratic lie algebras and its relation to dihedral homology if a is a k algebra where k is a commutative ring then kassel and w x w x loday in 5 study sl a s gl a gl a where gl a is the lie n n n n algebra of n n matrices over a they define a steinberg lie algebra st a which they prove is a universal central lie algebra extension of n sl a furthermore the kernel of this extension is isomorphic to a cyclic n homology group which in our current notation is designated as hc a 1 in this paper we prove a similar result for a a k algebra with involution a a a which acts trivially on k given a form parameter l and l g center a such that ll s 1 we have a form algebra a l l see w x 1 the lie algebra adaptation of the general l quadratic group gq a l will be denoted as gq l and will be a sub lie algebra of 2 n 2 n gl a we will show that the commutator subalgebra of gq l does not 2 n 2 n depend on the choice of parameter l this subalgebra will be denoted as sq a and will be called the special quadratic lie algebra of a there is 2 n another subalgebra sq l which does depend on the choice of l we 2 n

the argument principle and holomorphic extendibility to finite riemann surfaces let m be a finite riemann surface and let a m be the algebra of all continuous functions on m bm which are holomorphic on m we prove that a continuous function on bm extends to a function in a m if and only if for every f g in a m such that f g 0 on bm the change of argument of f g is nonnegative

weyl demazure and fusion modules for the current algebra of sl r 1 we construct a poincare birkhoff witt type basis for the weyl modules v chari a pressley weyl modules for classical and quantum affine algebras represent theory 5 2001 191 223 math qa 0004174 of the current algebra of as a corollary we prove the conjecture made in v chari a pressley weyl modules for classical and quantum affine algebras represent theory 5 2001 191 223 math qa 0004174 v chari a pressley integrable and weyl modules for quantum affine in quantum groups and lie theory durham 1999 in london math soc lecture note ser vol 290 cambridge univ press cambridge 2001 pp 48 62 math qa 0007123 on the dimension of the weyl modules in this case further we relate the weyl modules to the fusion modules defined in b feigin s loktev on generalized kostka polynomials and the quantum verlinde rule in differential topology infinite dimensional lie algebras and applications in amer math soc transl ser 2 vol 194 1999 pp 61 79 math qa 9812093 of the current algebra and the demazure modules in level one representations of the corresponding affine algebra in particular this allows us to establish substantial cases of the conjectures in b feigin s loktev on generalized kostka polynomials and the quantum verlinde rule in differential topology infinite dimensional lie algebras and applications in amer math soc transl ser 2 vol 194 1999 pp 61 79 math qa 9812093 on the structure and graded character of the fusion modules

meromorphic extendibility and rigidity of interpolation abstract let t be the unit circle f be an holder continuous function on t 1 2 and a be the algebra of continuous function in the closed unit disk d that are holomorphic in d then f extends to a meromorphic function in d with at most m poles if and only if the winding number of f h on t is bigger or equal to m for any h a such that f h 0 on t

irreducible finite dimensional representations of equivariant map algebras suppose a finite group acts on a scheme x and a finite dimensional lie algebra g the corresponding equivariant map algebra is the lie algebra m of equivariant regular maps from x to g we classify the irreducible finite dimensional representations of these algebras in particu lar we show that all such representations are tensor products of evaluation representations and one dimensional representations and we establish conditions ensuring that they are all evaluation representations for example this is always the case if m is perfect our results can be applied to multiloop algebras current algebras the onsager algebra and the tetrahedron algebra doing so we easily recover the known classifications of irreducible finite dimensional representations of these algebras moreover we obtain previously unknown classifica tions of irreducible finite dimensional representations of other types of equivariant map algebras such as the generalized onsager algebra

improved moment estimation formulas using more than three subjective fractiles pert type subjective estimations are used in many stochastic decision models to estimate the random variables mean and standard deviation s d the approach is based on the beta distribution assumption also most pert type formulas use only three estimated fractiles we point out that i it is desirable to consider a substantially richer set of distributions than the beta in developing pert type formulas ii it may be beneficial to use more than three fractile estimates in pert type formulas we then develop formulas for estimating the mean and s d that are based on a substantially richer set of distributions than the beta and that use more than three estimated fractiles these formulas perform better than the best currently vailable formulas when the subjective distribution is not restricted to be beta

double and single bisection methods for subjective probability assessment in a location scale family a common problem is to determine by an elicitation process the parameters of a subjective distribution in a location scale family one method is to elicit assessments of the median and one other quartile equate the assessed median to the location parameter and estimate the scale from the difference in a second method all three quartiles are elicited and then the scale is estimated from the interquartile range with either method the location and scale estimates are not made independently these methods are here studied by introducing probability models for the elicitation process itself it is shown that the second full quartiles method has important advantages not held by the first

analysis of least absolute deviation summary we develop a unified l 1 based analysis of variance type method for testing linear hypotheses like the classical l2 based analysis of variance the method is coordinate free in the sense that it is invariant under any linear transformation of the covariates or regression parameters moreover it allows singular design matrices and heterogeneous error terms a simple approximation using stochastic perturbation is proposed to obtain cut offvalues for the resulting test statistics both test statistics and distributional approximations can be computed using standard linear programming an asymptotic theory is derived for the method special cases of one and multi way analysis of variance and analysis of covariance models are worked out in detail the main results of this paper can be extended to general quantile regression extensive simulations show that the method works well in practical settings the method is also applied to a dataset from general social surveys

central extensions of lie algebras graded by finite root systems lie algebras graded by finite irreducible reduced root systems have been classified up to central extensions by berman and moody benkart and zelmanov and neher in this paper we determine the central extensions of these lie algebras and hence describe them completely up to isomorphism

using expert advice on a skew judgmental distribution a decision maker is interested in a quantity i he consults an expert who provides three fractiles of her probability distribution for i the problem discussed in this paper is how the decision maker should use the expert s fractiles to produce his distribution for i a key ingredient has to be the decision maker s opinion of the expert our analysis shows how the bayesian approach clearly incorporates this feature with three fractiles provided information about the skewness of the distribution of i is available in addition to location and scale our development uses the skew logistic to accommodate this additional feature and much of the paper is concerned with skewness extending the methods of lindley for the symmetric case the ideas easily extend to several experts if the unrealistic assumption is made that the decision maker regards them as independent

holomorphic arcs on singularities the study of formal arcs was initiated by nash in a 1967 preprint arc spaces of smooth varieties have a rather transparent structure but difficult problems arise for arcs passing through singularities the nash conjecture on the irreducible components of such arc spaces was proved for surfaces by fernandez de bobadilla and pe pereira and for toric singularities by ishii and kollar but counter examples were found in higher dimensions by ishii and kollar de fernex and johnson and kollar here we start the study of holomorphic arcs these are holomorphic maps from the closed unit disk to a complex analytic space as one expects there is not much conceptual difference between the set of formal arcs and the set of holomorphic arcs since every formal arc can be approximated by holomorphic arcs however a formal deformation of an arc is a much more local object than a holomorphic deformation thus in many cases the space of holomorphic arcs has infinitely many connected components while the space of formal arcs always has only finitely many for a complex analytic space we define two variants the space of arcs and the space of short arcs and we study their connected components for short arcs we obtain complete answers for surface singularities and for isolated quotient singularities in all dimensions

generalized ahlfors functions let be a bordered riemann surface with genus g and m boundary components let z z be a smooth family of smooth jordan curves in c which all contain the point 0 in their interior let p and let f be the family of all bounded holomorphic functions f on s such that f p 0 and f z z for almost every z then there exists a smooth up to the boundary holomorphic function f 0 f with at most 2g m 1 zeros on s so that f 0 z z for every z and such that f 0 p f p for every f f if in addition all the curves z z are strictly convex then f 0 is unique among all the functions from the family f

on meromorphic extendibility let d be a bounded domain in the complex plane whose boundary consists of finitely many pairwise disjoint real analytic simple closed curves let f be an integrable function on bd in the paper we show how to compute the candidates for poles of a meromorphic extension of f through d and thus reduce the question of meromorphic extendibility to the question of holomorphic extendibility let a d be the algebra of all continuous functions on d which are holomorphic on d we prove that a continuous function f on bd extends meromorphically through d if and only if there is an n n 0 such that the change of argument of pf q along bd is bounded below by 2 n for all p q a d such that pf q 0 on bd if this is the case then the meromorphic extension of f has at most n poles in d counting multiplicity

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

lie algebras graded by finite root systems and the intersection matrix algebras of slodowy this paper is about toroidal lie algebras certain intersection matrix lie algebras defined by slodowy and their relationship to one another and to certain lie algebra analogues of steinberg groups the main result of the paper is the identification of the intersection matrix algebras arising from multiply affinized cartan matrices of types a d and e with certain steinberg lie algebras and toroidal lie algebras propositions 5 9 and 5 10 a major part of the paper studies and classifies lie algebras graded by finite root systems these become the princi pal tool in our analysis of intersection matrix algebras each lie algebra graded by a simply laced finite root system of rank 2 has attached to it an algebra which according to the type and rank is either commutative and associative only associative or alternative all these possibilities occur in our description of inter section matrix algebras let r be any associative algebra with identity not necessarily finite dimen sional over a field k of characteristic 0 for each positive integer n the associative algebra m r of n n matrices with coefficients in r forms a lie algebra over k under the commutator product we denote this lie algebra by ol r let eis be the i j matrix unit of m r and assume that n 2 the subalgebra e r of oi r generated by the elements reis r r i 4 j is an ideal of 91 r and is perfect i e it is its own derived algebra now any perfect lie algebra o has a universal central extension also perfect called a universal covering algebra u c a of o ga so in particular e r has a u c a that we will denote by t r we define 112 r by the exact sequence 0 1 0 f2 r t r e r 0 dedicated to our teacher maria j wonenburger both authors gratefully acknowledge the support of the natural sciences and engineering research council of canada

lie algebras graded by 3 graded root systems and jordan pairs covered by grids let r be an irreducible root system a lie algebra l is called graded by r if l is graded with grading group the root lattice of r such that the only nonzero homogeneous subspaces of l have degree 0 or a root in r the grading is induced by the adjoint action of a split cartan subalgebra of a finite dimensional simple lie subalgebra of l with root system r and l is generated by the homogeneous subspaces of nonzero degree this class of lie algebras was introduced and studied by s berman and r moody in invent math 108 1992 where in particular a classification up to central equivalence is given in the simply laced case the doubly laced cases have recently been classified by g benkart and e zelmanov let r be a 3 graded root system i e r is not of type e 8 f 4 or g 2 in this paper lie algebras graded by r are described in a unified way without case by case considerations namely it is shown that a lie algebra l is 3 graded if and only if l is a central extension of the tits kantor koecher algebra of a jordan pair covered by a grid whose associated 3 graded root system is isomorphic to r this result is then used to classify lie algebras graded by r we give the classification of jordan pairs covered by a grid and describe their tits kantor koecher algebras one of the advantages of this approach is that it works over rings containing 1 2 and 1 3 and also for infinite root systems another application is the description of slodowy s intersection matrix algebras arising from multiply affinized cartan matrices

linking numbers and boundaries of varieties called a cornerstone of algebraic geometry gh p 62 is a simple consequence of the fact that analytic varieties carry a natural orientation recast in terms of linking numbers it is our principal motivation it implies the following let m be a smooth oriented compact 3 manifold in c3 suppose that m bounds a bounded complex 2 variety v here bounds means in the sense of stokes theorem i e that b v m as currents let a be an algebraic curve in c3 which is disjoint from m consider the linking number link m a of m and a since this linking number is equal to the intersection number i e the sum of the intersection indices of v and a by the positivity