moments and fourier transforms of b splines abstract the problem of determining the moments and the fourier transforms of b splines with arbitrary knots is considered there exists a simple connection between the moments of such splines and the so called extended stirling numbers of the second kind which are defined in section 2 some recurrence relations for the moments of b splines with arbitrary knots are given in section 3 in the case of equidistant knots we have also further recurrences for the forward central and perfect b splines the explicit formulas for the moments are given in section 3 the fourier transforms of b splines is treated in section 4 the final section is devoted to so called stieltjes series connected with the nonnegative weight function w x and such that a b w x dx 0 in some closed interval a b it is proved that such series for the particular values of the independent variable may be expressed by the finite sums which contain the nodes and coefficients of the optimal in the davies sense quadrature formulas

n ary algebras a review with applications this paper reviews the properties and applications of certain n ary generalizations of lie algebras in a self contained and unified way these generalizations are algebraic structures in which the two entries lie bracket has been replaced by a bracket with n entries each type of n ary bracket satisfies a specific characteristic identity which plays the role of the jacobi identity for lie algebras particular attention will be paid to generalized lie algebras which are defined by even multibrackets obtained by antisymmetrizing the associative products of its n components and that satisfy the generalized jacobi identity and to filippov or n lie algebras which are defined by fully antisymmetric n brackets that satisfy the filippov identity three lie algebras have surfaced recently in multi brane theory in the context of the bagger lambert gustavsson model because of this filippov algebras will be discussed at length including the cohomology complexes that govern their central extensions and their deformations it turns out that whitehead s lemma extends to all semisimple n lie algebras when the skewsymmetry of the lie or n lie algebra bracket is relaxed one is led to a more general type of n algebras the n leibniz algebras these will be discussed as well since they underlie the cohomological properties of n lie algebras the standard poisson structure may also be extended to the n ary case we shall review here the even generalized poisson structures whose generalized jacobi identity reproduces the pattern of the generalized lie algebras and the nambu poisson structures which satisfy the filippov identity and determine filippov algebras finally the recent work of bagger lambert and gustavsson on superconformal chern simons theory will be briefly discussed emphasis will be made on the appearance of the 3 lie algebra structure and on why the a4 model may be formulated in terms of an ordinary lie algebra and on its nambu bracket generalization

on the linear independence of multivariate b splines i triangulations of simploids this paper is concerned with the construction of certain collections of linearly independent multivariate b splines whose linear span is appropriate for approximation by constructing linear projectors onto these spline spaces we are able to estimate local and global approximation errors as well as the condition number of the b spline basis our approach is based upon some combinatorial properties of certain triangulations of a class of n polytopes which we call simploids

integral error formulae for the scale of mean value interpolations which includes kergin and hakopian interpolation in this paper we provide an integral error formula for a certain scale of mean value interpolations which includes the multivariate polynomial interpolation schemes of kergin and hakopian this formula involves only derivatives of order one higher than the degree of the interpolating polynomial space and from it we can obtain sharp l infty estimates these l infty estimates are precisely those that numerical analysts want to guarantee that a scheme based on such an interpolation has the maximum possible order

graded multiplicities in the exterior algebra abstract this paper deals with the graded multiplicities of the smallest irreducible representations of a simple lie algebra in its exterior algebra an explicit formula for the graded multiplicity of the adjoint representation in terms of the weyl group exponents was conjectured by a joseph a proof of this conjecture based on the properties of macdonald polynomials is given in the present paper the same method allows us to calculate the multiplicity of the simple module with highest weight equal to the short dominant root

an algorithm for constructing configuration of knots for bivariate b splines this paper is concerned with an algorithm for constructing suitable configurations of knot sets for bivariate b splines on arbitrary planar domains the objective is to determine positions for the knots realizing a stable basis of b splines with optimal smoothness furthermore we wish to reduce the number of b splines not vanishing at any given point of the domain uniform knot configurations w dahmen and c a micchelli numerical algorithms for least squares approximation by multivariate b splines numerical methods of approximation theory birkhauser basel 1981 are covered as a special case the paper includes some numerical experiments that illustrate the performance of the algorithm

multivariate spline functions b spline basis and polynomial interpolations the main result of this paper is a formula relating a multivariate b spline and a certain directional derivative to a lower order b spline the structure of discontinuous b splines is given a formula relates the discontinuous k variate b spline to a k 1 variate b spline with the help of these formulas we have succeeded in determining the exact class of local and global smoothness of a multivariate b spline depending on the configuration of the knots

the adjoint representation inside the exterior algebra of a simple lie algebra abstract for a simple complex lie algebra g we study the space of invariants a g g g which describes the isotypic component of type g in g as a module over the algebra of invariants g g as main result we prove that a is a free module of rank twice the rank of g over the exterior algebra generated by all primitive invariants in g g with the exception of the one of highest degree

the interacting multiple model algorithm for systems with markovian switching coefficients an important problem in filtering for linear systems with markovian switching coefficients dynamic multiple model systems is the management of hypotheses which is necessary to limit the computational requirements a novel approach to hypotheses merging is presented for this problem the novelty lies in the timing of hypotheses merging when applied to the problem of filtering for a linear system with markovian coefficients the method is an elegant way to derive the interacting multiple model imm algorithm evaluation of the imm algorithm shows that it performs well at a relatively low computational load these results imply a significant change in the state of the art of approximate bayesian filtering for systems with markovian coefficients

on the linear independence of multivariate b splines ii complete configurations the first part of this paper is concerned with global characterizations of both the multivariate b spline and the multivariate truncated power function as smooth piecewise polynomials in the second part of the paper we establish combinatorial criteria for the linear independence of multivariate b splines corresponding to certain configurations of knot sets

homogeneous simplex splines homogeneous simplex splines also known as cone splines or multivariate truncated power functions are discussed from a perspective of homogeneous divided differences and polar forms this makes it possible to derive the basic properties of these splines in a simple and economic way in addition a construction of spaces of homogeneous simplex splines is considered which in the nonhomogeneous setting is due to dahmen micchelli and seidel a proof for this construction is presented based on knot insertion restricting the homogeneous splines to a sphere gives rise to spaces of spherical simplex splines

the hypoelliptic laplacian on a compact lie group abstract let g be a compact lie group and let g be its lie algebra in this paper we produce a hypoelliptic laplacian on g g which interpolates between the classical laplacian of g and the geodesic flow this deformation is obtained by producing a suitable deformation of the dirac operator of kostant we show that various poisson formulas for the heat kernel can be proved using this interpolation by methods of local index theory the paper was motivated by papers by atiyah and frenkel in connection with localization formulas in equivariant cohomology and with kac s character formulas for affine lie algebras in a companion paper we will use similar methods in the context of selberg s trace formula

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

general two stage extended kalman filters a general two stage extended kalman filter gtsekf which extends the linear general two stage kalman filter to nonlinear systems is further proposed a new nonlinear two stage transformation is introduced to facilitate achieving this extension as in the linear one the gtsekf is derived mainly by applying the nonlinear two stage transformation to the well known extended kalman filter ekf and is shown to be equivalent to the ekf with a decoupled computing structure a nonlinear filter for estimating constant parameters in dynamic systems is presented to illustrate one application of the proposed gtsekf a literature example is also given to demonstrate the correctness and usefulness of the proposed results

brief modified stochastic luenberger observers a modified stochastic luenberger observer mslo structure is proposed to recover the optimal performance of the coventional slo for obtaining full state estimates in linear discrete time stochastic systems the optimal mslo omslo which gives the mmse estimates is derived by using the general two stage kalman filter a reduced order form of the omslo is also proposed for systems having singular measurement noises the connection between the omslo and the optimal minimal order observer of leondes and novak is also shown

a lie algebraic approach to the local index theorem on compact homogeneous spaces abstract using a k theory point of view bott related the atiyah singer index theorem for elliptic operators on compact homogeneous spaces to the weyl character formula this article explains how to prove the local index theorem for compact homogenous spaces using lie algebra methods the method follows in outline the proof of the local index theorem due to berline and vergne but the use of kostant s cubic dirac operator in place of the riemannian dirac operator leads to substantial simplifications an important role is also played by the quantum weil algebra of alekseev and meinrenken

clifford algebra analogue of the hopf koszul samelson theorem theρ decompositionc g end vρ c p and the g module structure of g

equivariant cohomology and the maurer cartan equation let g be a compact connected lie group acting smoothly on a manifold m goresky kottwitz macpherson described a small cartan model for the equivariant cohomology of m quasi isomorphic to the standard cartan complex of equivariant differential forms in this paper we construct an explicit cochain map from the small cartan model into the standard cartan model inducing an isomorphism in cohomology the construction involves the solution of a remarkable inhomogeneous maurer cartan equation this solution has further applications to the theory of transgression in the weil algebra and to the chevalley koszul theory of the cohomology of principal bundles

the dimension and basis of spaces of multivariate splines abstract the purpose of this paper is to study the recent development of certain aspects of multivariate spline spaces more specifically we will survey the existence of bivariate spline the dimension and basis of the bivariate spline spaces s k with various partitions furthermore we will also introduce some results on the higher dimensional splines

powers of the euler product and commutative subalgebras of a complex simple lie algebra if mathfrak g is a complex simple lie algebra and k does not exceed the dual coxeter number of mathfrak g then the absolute value of the kth coefficient of the dim mathfrak g power of the euler product may be given by the dimension of a subspace of wedge k mathfrak g defined by all abelian subalgebras of mathfrak g of dimension k this has implications for all the coefficients of all the powers of the euler product involved in the main results are dale peterson s 2rank theorem on the number of abelian ideals in a borel subalgebra of mathfrak g an element of type and my heat kernel formulation of macdonald s function theorem a set dalcove of special highest weights parameterized by all the alcoves in a weyl chamber generalizing young diagrams of null m core when mathfrak g text lie mathit sl m mathbb c and the homology and cohomology of the nil radical of the standard maximal parabolic subalgebra of the affine kac moody lie algebra

generalization of harish chandra s basic theorem for riemannian symmetric spaces of non compact type abstract a basic exact sequence by harish chandra related to the invariant differential operators on a riemannian symmetric space g k is generalized for each k type in a certain class which we call single petaled the argument also includes a further generalization of broer s generalization of the chevalley restriction theorem

optimal solution of the two stage kalman estimator the two stage kalman estimator was originally proposed to reduce the computational complexity of the augmented state kalman filter it was also applied to the tracking of maneuvering targets by treating the target acceleration as a bias term except in certain restrictive conditions the conventional two stage estimators are suboptimal in the sense that they are not equivalent to the augmented state filter in this paper the authors propose a new two stage kalman estimator i e new structure which is an extension of friedland s estimator and is optimal in general conditions in addition we provide some analytic results to demonstrate the computational advantages of two stage estimators over augmented ones

exact multivariate bayesian bootstrap distributions of moments the common unknown probability law p of a random sample y 1 y n is assigned a dirichlet process prior with index it is shown that the posterior joint density of several moments of p converges as r 0 to a multivariate b spline which is therefore the bayesian bootstrap joint density of the moments the result provides the basis for possible default nonparametric bayesian inference on unknown moments

a parallel decomposition for kalman filters a decomposition is given for the implementation of the kalman filter as a collection of parallel processors this decomposition is based on the representation of the system as a direct sum of observability subspaces

immanant inequalities and 0 weight spaces let n e n and let a aij be an n x n complex matrix let sn be the group of permutations of x 1 n and let w n be the set of all partitions of n as one knows to each a e n we may associate an irreducible complex representation vm sn aut ya such that using standard notation x a is the character of va generalizing the determinant and permanent of a one defines the immanants of a by putting for any a e

multivariate spline spaces in principle all of the properties and applications of multivariate spline functions can be derived from the basic theories in chapter 1 the expressions of the multivariate spline functions in 2 of chapter 1 are inconvenient to be applied directly because the expressions also rely on the solutions of the global conformality conditions so as to determine all the smooth cofactors

on the optimality of two stage state estimation in the presence of random bias sufficient conditions for the optimality of a two stage state estimator in the presence of random bias are derived under an algebraic constraint on the correlation between the state and bias process noises the optimal estimate of the system state can be obtained as a linear combination of the output of the first stage a bias free filter and the second stage a bias filter because the algebraic constraint is restrictive in practice the results indirectly indicate that for most practical systems the proposed solution to the two stage estimation problem will be suboptimal

comments on optimal solution of the two stage kalman estimator the optimal two stage kalman estimator otske is analyzed c s hsieh and f c chen see ibid vol 44 p 194 9 1999 comparison with two estimators shows that the optimal two stage kalman estimator has no advantage especially no computational advantage over the augmented state kalman filter

casimir elements associated with levi subalgebras of simple lie algebras and their applications abstract let g be a simple lie algebra h a levi subalgebra and c h u h the casimir element defined via the restriction of the killing form on g to h we study c h eigenvalues in g h and related h modules without loss of generality one may assume that h is a maximal levi then g is equipped with the natural z grading g i z g i such that g 0 h and g i is a simple h module for i 0 we give explicit formulae for the c h eigenvalues in each g i i 0 and relate eigenvalues of c h in g 1 to the dimensions of abelian subspaces of g 1 then we prove that if a g 1 is abelian whereas g 1 is not then dim a dim g 1 2 moreover if dim dim g 1 2 then a has an abelian complement the z gradings of height 2 are closely related to involutions of g and we provide a connection of our theory to an extension of the strange formula of freudenthal de vries

the stable evaluation of multivariate simplex splines this paper gives a general method for the stable evaluation of multivariate simplex splines based on the well known recurrence relation of micchelli 12 this paper deals with two problems which arise in the implementation of the recurrence relation first the coefficients in the recurrence are shown to be efficiently computable via the dual simplex method of linear programminig secondly the problem of evaluation along mesh boundaries is discussed in detail

optimal minimal order least squares estimators via the general two stage kalman filter a direct derivation of the optimal minimal order least squares estimator is presented using the general two stage kalman filter using this new result the reduced order estimators of o reilly 1982 and fairman and luk 1985 are readily shown to be equivalent a practical implementation issue to consider these two estimators is also addressed

a generalization of cachazo douglas seiberg witten conjecture for symmetric spaces we extend the original cachazo douglas seiberg witten conjecture on the structure of the chiral ring of classical supersymmetric yang mills theory to symmetric spaces

the non commutative weil algebra for any compact lie group g together with an invariant inner product on its lie algebra g we define the non commutative well algebra w g as a tensor product of the universal enveloping algebra u g and the clifford algebra cl 8 just like the usual we

tensor factorization and spin construction for kac moody algebras abstract in this paper we discuss the factorization phenomenon which occurs when a representation of a lie algebra is restricted to a subalgebra and the result factors into a tensor product of smaller representations of the subalgebra we analyze this phenomenon for symmetrizable kac moody algebras including finite dimensional semi simple lie algebras we present a few factorization results for a general embedding of a symmetrizable kac moody algebra into another and provide an algebraic explanation for such a phenomenon using spin construction we also give some application of these results for semi simple finite dimensional lie algebras we extend the notion of spin functor from finite dimensional to symmetrizable kac moody algebras which requires a very delicate treatment we introduce a certain category of orthogonal g representations for which surprisingly the spin functor gives a g representation in bernstein gelfand gelfand category o also for an integrable representation spin produces an integrable representation we give the formula for the character of spin representation for the above category and work out the factorization results for an embedding of a finite dimensional semi simple lie algebra into its untwisted affine lie algebra finally we discuss the classification of those representations for which spin is irreducible

invariant theory in exterior algebras and amitsur levitzki type theorems abstract this article discusses invariant theories in some exterior algebras which are closely related to amitsur levitzki type theorems first we consider the exterior algebra on the vector space of square matrices of size n and look at the invariants under conjugations we see that the algebra of these invariants is isomorphic to the exterior algebra on an n dimensional vector space moreover we give a cayley hamilton type theorem for these invariants the anticommutative version of the cayley hamilton theorem this cayley hamilton type theorem can also be regarded as a refinement of the amitsur levitzki theorem we discuss two more amitsur levitzki type theorems related to invariant theories in exterior algebras one is a famous amitsur levitzki type theorem due to kostant and rowen and this is related to o v invariants in 2 v the other is a new amitsur levitzki type theorem and this is related to gl v invariants in 2 v s 2 v

bivariate spline interpolation at grid points we describe algorithms for constructing point sets at which interpolation by spaces of bivariate splines of arbitrary degree and smoothness is possible the splines are defined on rectangular partitions adding one or two diagonals to each rectangle the interpolation sets are selected in such a way that the grid points of the partition are contained in these sets and no large linear systems have to be solved our method is to generate a net of line segments and to choose point sets in these segments which satisfy the schoenberg whitney condition for certain univariate spline spaces such that a principle of degree reduction can be applied in order to include the grid points in the interpolation sets we give a sufficient schoenberg whitney type condition for interpolation by bivariate splines supported in certain cones this approach is completely different from the known interpolation methods for bivariate splines of degree at most three our method is illustrated by some numerical examples

quasi identities on matrices and the cayley hamilton polynomial abstract we consider certain functional identities on the matrix algebra m n that are defined similarly as the trace identities except that the coefficients are arbitrary polynomials not necessarily those expressible by the traces the main issue is the question of whether such an identity is a consequence of the cayley hamilton identity we show that the answer is affirmative in several special cases and moreover for every such an identity p and every central polynomial c with zero constant term there exists m n such that the affirmative answer holds for c m p in general however the answer is negative we prove that there exist antisymmetric identities that do not follow from the cayley hamilton identity and give a complete description of a certain family of such identities

on the cachazo douglas seiberg witten conjecture for simple lie algebras we prove a part of the cachazo douglas seiberg witten conjecture uniformly for any simple lie algebra the main ingredients in the proof are garland s result on the lie algebra cohomology of kostant s result on the diagonal cohomolgy of and its connection with abelian ideals in a borel subalgebra of and a certain deformation of the singular cohomology of the infinite grassmannian introduced by belkale kumar

general two stage kalman filters a general two stage kalman filter that is equivalent to but numerically more efficient than the standard single stage kalman filter is developed for general time varying linear discrete time systems analytical results defining the reduction in computational burdens are presented simulation results that validate the predicted efficiency improvements are shown as well

on covariants in exterior algebras for the even special orthogonal group abstract let g s o 2 n c be the even special orthogonal group and let m 2 n resp m 2 n be the space of symmetric resp skew symmetric complex matrices with respect to the usual transposition we study the structure of b m 2 n m 2 n g the space of g equivariant skew symmetric matrix valued alternating multilinear maps on the space of symmetric n tuples of matrices with g acting by conjugation further we decompose b as the direct sum b b b where b m 2 n m 2 n g we prove that b is a free module over a certain subalgebra of invariants a m 2 n g of rank 2n we give an explicit description for the basis of this module furthermore we prove new trace polynomial identities for symmetric matrices finally we show using a computer assisted computation made with the lie software that b m 2 n m 2 n g doesn t satisfy a similar property

adjoint affine fusion and tadpoles we study affine fusion with the adjoint representation for simple lie algebras elementary and universal formulas determine the decomposition of a tensor product of an integrable highest weight representation with the adjoint representation using the refined affine depth rule we prove that equally striking results apply to adjoint affine fusion for diagonal fusion a coefficient equals the number of nonzero dynkin labels of the relevant affine highest weight minus 1 a nice lattice polytope interpretation follows and allows the straightforward calculation of the genus 1 1 point adjoint verlinde dimension the adjoint affine fusion tadpole explicit formulas piecewise polynomial in the level are written for the adjoint tadpoles of all classical lie algebras we show that off diagonal adjoint affine fusion is obtained from the corresponding tensor product by simply dropping non dominant representations

the optimal reduced order estimator for systems with singular measurement noise the optimal reduced order estimator is completely characterized by necessary conditions resulting from the optimal projection equations the solution consists of one riccati equation and two lyapunov equations coupled by two projections explicit expressions for all of the estimator parameters are given the relation between the reduced order singular estimator and the full order optimal singular estimator which is of reduced order itself is investigated it is shown that under certain conditions the optimal estimator is recovered from the reduced order estimator

moments of dirichlet splines and their applications to hypergeometric functions abstract dirichlet averages of multivariate functions are employed for a derivation of basic recurrence formulas for the moments of multivariate dirichlet splines an algorithm for computing the moments of multivariate simplex splines is presented applications to hypergeometric functions of several variables are discussed

abelian ideals in a borel subalgebra of a complex simple lie algebra let mathfrak g be a complex simple lie algebra and mathfrak b a fixed borel subalgebra of mathfrak g we describe the abelian ideals in mathfrak b in a uniform way that is independent of the classification of complex simple lie algebras as an application we derive a formula for the maximal dimension of a commutative lie subalgebra of mathfrak g

group valued equivariant localization we prove a localization formula for group valued equivariant de rham cohomology of a compact g manifold this formula is a non trivial generalization of the localization formula of berline vergne and atiyah bott for the usual equivariant de rham cohomology we derive from this result a duistermaat heckman formula for group valued moment maps as an application we prove part of witten s conjectures about intersection pairings on moduli spaces of flat connections on 2 manifolds

quasiharmonic polynomials for coxeter groups and representations of cherednik algebras we introduce and study deformations of finite dimensional modules over rational cherednik algebras our main tool is a generalization of usual harmonic polynomials for each coxeter group the so called quasiharmonic polynomials a surprising application of this approach is the construction of canonical elementary symmetric polynomials and their deformations for all coxeter groups

the dirac operator on compact quantum groups for the q deformation gq 0 q 1 of any simply connected simple compact lie group g we construct an equivariant spectral triple which is an isospectral deformation of that defined by the dirac operator d on g our quantum dirac operator dq is a unitary twist of d considered as an element of ug cl g the commutator of dq with a regular function on gq consists of two parts one is a twist of a classical commutator and so is automatically bounded the second is expressed in terms of the commutator of the associator with an extension of d we show that in the case of the drinfeld associator the latter commutator is also bounded