exit problems associated with affine reflection groups we obtain a formula for the distribution of the first exit time of brownian motion from the alcove of an affine weyl group in most cases the formula is expressed compactly in terms of pfaffians expected exit times are derived in the type widetilde a case the results extend to other markov processes we also give formulas for the real eigenfunctions of the dirichlet and neumann laplacians on alcoves observing that the hot spots conjecture of j rauch is true for alcoves

cortical surface based analysis ii inflation flattening and a surface based coordinate system the surface of the human cerebral cortex is a highly folded sheet with the majority of its surface area buried within folds as such it is a difficult domain for computational as well as visualization purposes we have therefore designed a set of procedures for modifying the representation of the cortical surface to i inflate it so that activity buried inside sulci may be visualized ii cut and flatten an entire hemisphere and iii transform a hemisphere into a simple parameterizable surface such as a sphere for the purpose of establishing a surface based coordinate system r 1999 academic press

geometrically accurate topology correction of cortical surfaces using nonseparating loops in this paper we focus on the retrospective topology correction of surfaces we propose a technique to accurately correct the spherical topology of cortical surfaces specifically we construct a mapping from the original surface onto the sphere to detect topological defects as minimal nonhomeomorphic regions the topology of each defect is then corrected by opening and sealing the surface along a set of nonseparating loops that are selected in a bayesian framework the proposed method is a wholly self contained topology correction algorithm which determines geometrically accurate topologically correct solutions based on the magnetic resonance imaging mri intensity profile and the expected local curvature applied to real data our method provides topological corrections similar to those made by a trained operator

suboptimal minimum cluster volume cover based method for measuring fractal dimension a new method for calculating fractal dimension is developed in this paper the method is based on the box dimension concept however it involves direct estimation of a suboptimal covering of the data set of interest by finding a suboptimal cover this method is better able to estimate the required number of covering elements for a given cover size than is the standard box counting algorithm moreover any decrease in the error of the covering element count directly increases the accuracy of the fractal dimension estimation in general our method represents a mathematical dual to the standard box counting algorithm by not solving for the number of boxes used to cover a data set given the size of the box instead the method chooses the number of covering elements and then proceeds to find the placement of smallest hyperellipsoids that fully covers the data set this method involves a variant of the fuzzy c means clustering algorithm as well as the use of the minimum cluster volume clustering algorithm a variety of fractal dimension estimators using this suboptimal covering method are discussed finally these methods are compared to the standard box counting algorithm and wavelet decomposition methods for calculating fractal dimension by using one dimensional cantor dust sets and a set of standard brownian random fractal images

models of fractal river basins two distinct models for self similar and self affine river basins are numerically investigated they yield fractal aggregation patterns following nontrivial power laws in experimentally relevant distributions previous numerical estimates on the critical exponents when existing are confirmed and superseded a physical motivation for both models in the present framework is also discussed

robust estimation of fractal measures for characterizing the structural complexity of the human brain optimization and reproducibility high resolution isotropic three dimensional reconstructions of human brain gray and white matter structures can be characterized to quantify aspects of their shape volume and topological complexity in particular methods based on fractal analysis have been applied in neuroimaging studies to quantify the structural complexity of the brain in both healthy and impaired conditions the usefulness of such measures for characterizing individual differences in brain structure critically depends on their within subject reproducibility in order to allow the robust detection of between subject differences this study analyzes key analytic parameters of three fractal based methods that rely on the box counting algorithm with the aim to maximize within subject reproducibility of the fractal characterizations of different brain objects including the pial surface the cortical ribbon volume the white matter volume and the gray matter white matter boundary two separate datasets originating from different imaging centers were analyzed comprising 50 subjects with three and 24 subjects with four successive scanning sessions per subject respectively the reproducibility of fractal measures was statistically assessed by computing their intra class correlations results reveal differences between different fractal estimators and allow the identification of several parameters that are critical for high reproducibility highest reproducibility with intra class correlations in the range of 0 9 0 95 is achieved with the correlation dimension further analyses of the fractal dimensions of parcellated cortical and subcortical gray matter regions suggest robustly estimated and region specific patterns of individual variability these results are valuable for defining appropriate parameter configurations when studying changes in fractal descriptors of human brain structure for instance in studies of neurological diseases that do not allow repeated measurements or for disease course longitudinal studies

determinantal forms for composite schur and q functions via the boson fermion correspondence the boson fermion correspondence is used to represent a composite schur function as a fermionic expectation value wick s theorem then enables it to be written as a determinant generalizing the giambelli formula for ordinary schur functions as single hook determinants the schur q function analogue recently introduced in the context of the bkp hierarchy is also given

dkp and mdkp hierarchy of soliton equations abstract in a series of papers date jimbo kashiwara and miwa djkm studied soliton equations using representation of lie algebras in particular they found a remarkable new hierarchy of kp type namely bkp one can also study the same problem using vertex operators for the infinite rank affine lie algebras a b combining the ideas of kac and djkm we found the realization of the basic representation and the half spin representations for d as well in particular we show that dkp and bkp are essentially the same moreover one gets a modified dkp hierarchy

symmetric function products and plethysms and the boson fermion correspondence we use the boson fermion correspondence for s and q functions to establish some interesting properties concerning outer products and plethysms of s functions or q functions by power sum symmetric functions the techniques which are developed are also applied to computing the inverse kostka foulkes matrix which is the transition matrix between hall littlewood symmetric functions and s functions in some simple cases

binomial determinants paths and hook length formulae abstract we give a combinatorial interpretation for any minor or binomial determinant of the matrix of binomial coefficients this interpretation involves configurations of nonintersecting paths and is related to young tableaux and hook length formulae

schur functions and affine lie algebras abstract we make use of the representation theory of the infinite dimensional lie algebrasa b and formula to derive explicit formulas relating schur sp functions to schur ss functions

clustering algorithms based on volume criteria clustering algorithms such as the k means algorithm and the fuzzy c means algorithm are based on the minimization of the trace of the fuzzy within fluster scatter matrix in this paper we explore the use of determinant volume criteria for clustering we derive an algorithm called the minimum scatter volume msv algorithm that minimizes the scatter volume and another algorithm called the minimum cluster volume mcv that minimizes the sum of the volumes of the individual clusters the behavior of msv is shown to be similar to that of k means whereas mcv is more versatile

young diagrammatic methods for the representation theory of the classical groups of type bn cn dn since the appearance of the celebrated article the classical groups of weyl 7 numerous influences have occurred in the representation theory of the classical groups especially it has become folklore that the facts in this field should be expressed only in terms of young diagrams the aim of this paper is one of the realizations of this principle a classical model of our approach is the results for gl n which are known as littlewood richardson coefficients and the kostka coefficients these two coefficients can be stated in the combinatorial manner using only young diagrams independently of the rank n of gl n we generalize these results to the other classical groups over c namely we give explicit formulas to decompose the tensor product of two irreducible representations into irreducible constituents also we have obtained an interesting relationship linking the irreducible characters of sp and so which we call duality relation between the representations of sp and so as for explicit formulas for the multiplicities of weights in the form of polynomials in the rank n we refer to 3

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

existence and partial regularity results for the heat flow for harmonic maps on considere le probleme d evolution t u m u n u 0 dans m r u 0 u 0 sur m ou m et n sont des varietes de riemann lisses compactes et u 0 l application harmonique de m dans n on demontre l existence d une solution globale faible

vertex operator realization of symplectic and orthogonal s functions it is shown that symplectic and orthogonal schur functions can be realized by the action of the modes of certain vertex operators on the function the properties of these vertex operators allow one to reproduce the basic propreties of these types of symmetric functions the vertex operator modes are shown to obey free fermionic relations which provide applications such as calculating products and plethysms as well as generating hirota type partial differential equations pdes which have symplectic s functions as functions

a combinatorial proof of the giambelli identity for schur functions the giambelli identity provides a formula for expressing an arbitrary schur function of shape as a determinant of schur functions of hook shapes l 1k we give the first complete combinatorial proof of the giambelli identity we show how to derive various hook formulas from the giambelli identity and show how to extend our methods to derive extensions of the giambelli identity and the hook formula for the number of standard tableaux to certain skew shapes

lacunarity definition for ramified data sets based on optimal cover lacunarity is a measure of how data fills space it complements fractal dimension which measures how much space is filled this paper discusses the limitations of the standard gliding box algorithm for calculating lacunarity which leads to a re examination of what lacunarity is meant to describe two new lacunarity measures for ramified data sets are then presented that more directly measure the gaps in a ramified data set these measures are rigorously defined an algorithm for estimating the new lacunarity measure using fuzzy c means clustering algorithm is developed the lacunarity estimation algorithm is used to analyze two and three dimensional cantor dusts applications for these measures include biological modeling and target detection within ramified data sets

partial regularity for weak heat flows into spheres in this paper we show that a weak heat flow of harmonic maps from a compact riemannian manifold possibly with boundary into a sphere satisfying the monotonicity inequality and the energy inequality is regular off a closed set of m dimensional hausdorff measure zero

on some identities of schur q functions abstract we give a proof of the jozefiak pragacz identity using the representation theory of affine lie algebras new results of this type are also obtained

advances in the implementation of the box counting method of fractal dimension estimation the box counting analysis is an appropriate method of fractal dimension estimation for images with or without self similarity however this technique including processing of the image and definition of the range of box sizes requires a proper implementation to be effective in practice the objectives of this study were thus 1 to determine how to prepare an image for box counting analysis 2 to define reasonable preferences for using the fractal dimension calculator software and 3 to develop a routine procedure for defining the most appropriate range of box sizes for any one piece image four fractal images were chosen for this study the koch curve koch coastline koch boxes and cross tree our results show that the skeletons provide better material for the box counting method since only lines and or curves are responsible for the fractal dimension value in the procedure of box counting for fractal dimension estimation the image must be surrounded by a four square frame with the least possible area and the condition of linear relationship must be satisfied in a log log plot fractal dimension is to be estimated over the minimum number of boxes covering the image for each box size after superimposing a reasonable number of grid offsets in many cases 25 of the shorter image side may provide an appropriate value for largest box size however for noisy or dispersed patterns a smaller box size than this is needed in the log log plot with 12 box sizes some points corresponding to smaller box sizes deviate from the straight line from a certain point on the box size corresponding to this breakpoint will provide an appropriate smallest box size the exercise of determining the most appropriate range of box sizes must be performed repeatedly for every individual image

nonintersecting paths pfaffians and plane partitions gessel and viennot have developed a powerful technique for enumerating various classes of plane partitions the purpose of this article is to show by similar means that one may use pfaffiants to enumerate configurations of nonintersecting paths in which the initial and or terminal vertices of the paths are allowed to vary over specified regions of the digraph this leads to the possibility of enumerating classes of plane partitions in which the shape is allowed to vary whereas the previous applications of gessel and viennot were largely confined to plane partitions of a given shape

an efficient implementation of the gliding box lacunarity algorithm lacunarity is a measure of how data fills space it complements fractal dimension which measures how much space is filled many researchers use the gliding box algorithm to calculate lacunarity this paper introduces a fast algorithm for making this calculation this algorithm is akin to fast box counting algorithms used by some researchers to estimate fractal dimension a simplified gliding box measure equation along with key pseudo code implementations for the algorithm are presented applications for the gliding box lacunarity measure have ranged from biological community modelling to target detection

a dual of macmahon s theorem on plane partitions a classical theorem of macmahon states that the number of lozenge tilings of any centrally symmetric hexagon drawn on the triangular lattice is given by a beautifully simple product formula in this paper we present a counterpart of this formula corresponding to the exterior of a concave hexagon obtained by turning 120 after drawing each side macmahon s hexagon is obtained by turning 60 after each step

data dimensionality estimation methods a survey in this paper data dimensionality estimation methods are reviewed the estimation of the dimensionality of a data set is a classical problem of pattern recognition there are some good reviews algorithms for clustering data prentice hall englewood cliffs nj 1988 in literature but they do not include more recent developments based on fractal techniques and neural autoassociators the aim of this paper is to provide an up to date survey of the dimensionality estimation methods of a data set paying special attention to the fractal based methods

initial boundary value problems for anisotropic landau lifshitz equations in three or four space dimensions abstract in this paper we will show the existence of partially regular solutions to the initial boundary value problem for landau lifshitz equations with nonpositive anisotropy constants in three or four space dimensions the partial regularity is proved up to the boundary both for the dirichlet problem and for the neumann problem in addition for the neumann case a generalized stability condition which ensures the partial regularity is given for equations with positive or negative anisotropy coefficients we will give two results of existence and uniqueness for the solutions corresponding to ground states

the littlewood richardson rule and the boson fermion correspondence the boson fermion correspondence is applied to derive explicit formulae for expressing the product of s functions in terms of sums of s functions associated to non standard partitions

design and implementation of an estimator of fractal dimension using fuzzy techniques abstract this paper presents a new method for estimating the fractal dimension of one dimensional profiles in this approach the real fractal dimension d is considered as an implicit continuous function of the estimated fractal dimension de the resolution and several other elements by approximating this function from a number of experimental data we can obtain more precise estimates of the fractal dimension d this approximation is done using a fuzzy logic controller and an averaging procedure permitting to respectively decrease two kinds of estimation errors 1 systematic errors which are associated with values of d resolution trends of profiles and etc 2 stochastic errors which are mainly caused by the choice of the sequence ek representing the sizes of structuring elements corresponding to different scales the effectiveness of this method is shown by estimating fractal dimensions for two sample functions and a number of natural and synthetic fibers

error bounds on the estimation of fractal dimension the use of fractal dimension as an index of complexity is widely used in natural science studies related to the significance of the estimates obtained have rarely been presented in the past this paper is intended to present an error analysis in the estimation of the fractal dimension of a particular class of objects namely graphs of knopp functions the algorithm used to estimate the fractal dimension is the variation method various numerical approximations of the variation will be presented using functional properties of the chosen parametric family of sets we derive error bounds useful in the assessment of the accuracy that can be achieved in the numerical approximation of the dimension at a given scale and resolution

new schur function series this paper presents some new product identities for certain summations of schur functions these identities are generalizations of some famous identities known to littlewood and appearing in macdonald s book we refer to these identities as the littlewood type formulas in addition analogues for summations of characters of the other classical groups are also given the littlewood type formulas in this paper are separated into two classes the rational schur function series and the generalized schur function series an application of a rational schur function series to the infinite product representation of the elliptic theta functions is also given we prove these littlewood type formulas using the cauchy binet formula the cauchy binet formula is a basic but powerful tool applicable in the present context which can be derived from our pfaffian formula as we explain q 1998 academic press

shifted tableaux and the projective representations of symmetric groups les groupes de representation de s n classes de conjugaison algebres de clifford et la representation spin de base produits induits pour des representations spin une application caracteristique pour des caracteres de spin tableaux decales et q fonctions de schur les caracteres de spin irreductibles de s n et a n une analogue decalee de la regle lr produits tensoriels interieurs

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed