bayesian image restoration an application to edge preserving surface recovery bayesian methods for recovering a 2 d surface are discussed it is assumed that there is a textural image that can be modeled by a markov random field and that the original surface is composed of different surfaces each of which is associated with one textural state both parametric and nonparametric methods are used to enforce smoothness of these surfaces iterative procedures are examined for simultaneous restoration of the textural image and estimation of underlying parameters from the estimated textural image and the estimated parameters an estimate for the original surface is obtained two illustrative examples are presented

characterization of some fuzzy relation equations provided with one solution on a finite set abstract in this paper some fuzzy relation equations provided with one solution on a finite set are characterized we consider fuzzy relation equations h q t with q f x x y and card x card y after recalling the definition of equivalent fuzzy relation equations we introduce the definition of equivalent ones which allows us to constrain our research without loss of generality to fuzzy relation equations where t does not have zero components

binary restoration of thin objects in multidimensional imagery we present a method for restoration of noisy tomographic images for detecting thin objects such as explosives use of a weighted mean square estimate optimizes the solution to place emphasis on the infrequent but significant local structure associated with thin objects experimental results show successful restoration at very high noise levels

numerical analysis on fuzzy relation equations with various operators abstract in this study we investigate the properties of the approximate solutions of fuzzy relation equations for composite operators defined by max min hamacher and its special case disjunctive sum conjunctive product a quasi newton method is adopted to find the solutions apart from particular issues relevant to individual operators comparisons among the various operators are made

jump and sharp cusp detection by wavelets summary a method is proposed to detect jumps and sharp cusps in a function which is observed with noise by checking if the wavelet transformation of the data has significantly large absolute values across fine scale levels asymptotic theory is established and practical implementation is discussed the method is tested on simulated examples and applied to stock market return data the analysis of change points which describe sudden localised changes has recently found increasing interest change points can be used to model practical problems arising in fields such as quality control economics medicine signal and image processing and physical sciences for example in electroencephalogram signals sharp cusps exhibit the accelerations and decelerations in the beating of the hearts many practical problems like this involve functions which have jumps and sharp cusps the recently developed theory of wavelets has drawn much attention from both math ematicians statisticians and engineers in the seminal work of donoho 1993 donoho johnstone 1994 1995a b and donoho johnstone et al 1995 orthonormal bases of compactly supported wavelets have been used to estimate functions the theory of wavelets permits decomposition of functions into localised oscillating components this is an ideal tool to study localised changes such as jumps and sharp cusps in one dimension as well as several dimensions unlike traditional smoothing methods based on a fixed spatial scale the wavelet method is a multiresolution approach and has local adaptivity in this

fuzzy linear systems abstract fuzzy linear systems of equations and inequalities over a bounded chain are studied both from theoretical and computational points of view a unified approach is presented for solving such systems completed with polynomial time algorithms the main results are concerned with establishing the consistency of the system computing all kinds of solutions or marking the contradictory equations respectively inequalities if the system is inconsistent applications in fuzzy linear programming multivalued logic fuzzy matrix equations or inequalities fuzzy relation equations or inclusions fuzzy automata and medicine are presented

fuzzy relational equations and the inverse problem abstract the inverse problem concerned with fuzzy relations is investigated the conditions for the existence of a solution are shown and an analytical solution is given a method for the improvement of the solution is proposed

the fixed point property in jb triples and preduals of jbw triples abstract we prove that for every member x in the class of real or complex jb triples or preduals of jbw triples the following assertions are equivalent 1 x has the fixed point property 2 x has the super fixed point property 3 x has normal structure 4 x has uniform normal structure 5 the banach space of x is reflexive as a consequence a real or complex c algebra or the predual of a real or complex w algebra having the fixed point property must be finite dimensional

artistic edge and corner enhancing smoothing two important visual properties of paintings and painting like images are the absence of texture details and the increased sharpness of edges as compared to photographic images painting like artistic effects can be achieved from photographic images by filters that smooth out texture details while preserving or enhancing edges and corners however not all edge preserving smoothers are suitable for this purpose we present a simple nonlinear local operator that generalizes both the well known kuwahara filter and the more general class of filters known in the literature as ldquocriterion and value filter structure rdquo this class of operators suffers from intrinsic theoretical limitations which give rise to a dramatic instability in presence of noise especially on shadowed areas such limitations are discussed in the paper and overcome by the proposed operator a large variety of experimental results shows that the output of the proposed operator is visually similar to a painting comparisons with existing techniques on a large set of natural images highlight conditions on which traditional edge preserving smoothers fail whereas our approach produces good results in particular unlike many other well established approaches the proposed operator is robust to degradations of the input image such as blurring and noise contamination

change point estimation using nonparametric regression we consider a regression model in which the mean function may have a discontinuity at an unknown point we propose an estimate of the location of the discontinuity based on one side nonparametric regression estimates of the mean function the change point estimate is shown to converge in probability at rate 0 n 1 and to have the same asymptotic distribution as maximum likelihood estimates considered by other authors under parametric regression models confidence regions for the location and size of the change are also discussed

edge preserving and peak preserving smoothing an alternative procedure is developed to the smoothed linear fitting method of mcdonald and owen the procedure is based on the detection of discontinuities by comparing at any given position three smooth fits diagnostics are used to detect discontinuities in the regression function itself edge detection or in its first derivative peak detection an application in electron microscopy is discussed

resolution of composite interval valued fuzzy relation equations abstract in this study we investigate the properties and the solution procedure of composite interval valued fuzzy relation equations with max min operator this is a generalization of existing constant valued fuzzy relation problems the study includes both theoretical development and numerical analyses it is concluded that our proposed methodology not only solves the existing fuzzy relation equations but also reveals the necessary and sufficient conditions for more general problems therefore this study facilitates the understanding of solution structure and wider applications

discontinuous versus smooth regression given measurements x i y i i 1 n we discuss methods to assess whether an underlying regression function is smooth continuous or differentiable or whether it has discontinuities the variance of the measurements is assumed to be unknown and is estimated simultaneously by regressing squared differences of the data formed with various span sizes on the span size itself we obtain an asymptotic linear model with dependent errors the parameters of this asymptotic linear model include the sum of the squared jump sizes as well as the variance of the measurements both parameters can be consistently estimated with mean squared error rates of convergence of n 2 3 for the sum of squared jump sizes and n 1 for the error variance we derive the asymptotic constants of the mean squared error mse and discuss the dependence of mse on the maximum span size l the test for the existence of jumps is formulated for the null hypothesis that the sum of squared jump sizes is 0 the asymptotic distribution of the test statistic is obtained essentially via a central limit theorem for u statistics we motivate and illustrate the methods with data surrounded by a scientific controversy concerning the question whether the growth of children occurs smoothly or rather in jumps

eigen fuzzy sets and fuzzy relations abstract when r is a fuzzy relation between the elements of a finite set x the fuzzy sets a of x such that r a a max min composition are called eigen fuzzy sets the main result of this paper is the determination of the greatest eigen fuzzy set associated with a given fuzzy relation and we give three methods illustrated by an example we then state that the greatest eigen fuzzy set associated with r the transitive closure of r is exactly the one associated with r finally we describe how to obtain all fuzzy relations keeping invariant a given fuzzy set

discontinuous regression surfaces fitting we suggest a three stage procedure to recover discontinuous regression surfaces when noisy data are present in the first stage jump candidate points are detected using a jump detection criterion a local principal component line is then fitted through these points in a neighborhood of a design point this line provides a first order approximation to the true jump location curve in that neighborhood in the third stage observations on the same side of the line as the given point are combined using a weighted average procedure to fit the surface at that point if there are no jump candidate points in the neighborhood then all observations in that neighborhood are used in the surface fitting if however the center of the neighborhood is on a jump location curve only those observations on one side of the line are used thus blurring is automatically avoided around the jump locations this methodology requires o n k 2 computation where n is the sample size and k is the window width its assumptions on the model are flexible some numerical results are presented to evaluate the surface fit and to discuss the selection of the window widths

infinite fuzzy relational equations on a complete brouwerian lattice abstract let i be a finite and j an infinite index sets a a ij i j and b b i i i a given matrix and a column vector with elements from a complete brouwerian lattice in this paper the problem of a minimal solution of a fuzzy relational equation a x b where is the sup inf composite operation is investigated

bandwidth selection for changepoint estimation in nonparametric regression nonparametric estimation of abrupt changes in a regression function involves choosing smoothing bandwidth parameters the performance of estimation procedures depends heavily on this choice so far little attention has been paid to the crucial issue of choosing appropriate bandwidth parameters in practice in this article we propose a bootstrap procedure for selecting the bandwidth parameters in a nonparametric two step estimation method this method results in a fully data driven procedure for estimating a finite but possibly unknown number of changepoints in a regression function we evaluate the performance of the data driven procedure via a simulation study which reveals that the fully automatic procedure performs quite well as an illustration we apply the procedure to some real data

uniform smoothness implies super normal structure property we prove that a uniformly smooth banach space x has supernormal structure property more precisely we prove that if the modulus of smoothness of x satisfies lim 0 x 1 2 then x and x are superreflexives and have super normal structure property

solution of fuzzy equations with extended operations abstract assuming that is any operation defined on a product set x y and taking values on a set z it can be extended to fuzzy sets by means of zadeh s extension principle given a fuzzy subset c of z it is here shown how to solve the equation a b c or a b c when a fuzzy subset a of x or a fuzzy subset b of y is given the methodology we provide includes as a special case the resolution of fuzzy arithmetical operations i e when stands for or extended to fuzzy numbers fuzzy subsets of the real line the paper is illustrated with several examples in fuzzy arithmetic

relational equations in totally ordered lattices and their complete resolution abstract the set of all solutions of a composite fuzzy relation equation of sanchez inform and control 30 1976 defined on finite spaces is studied by determining and characterizing all the lower solutions of such an equation

some results on fuzzy relation equations provided with one solution in a previous paper 3 we characterized the fuzzy relation equations h q t where q f x y provided with one solution on a finite set considering card x card y or card x card y in this paper we settle the same question for card x card y and s q t o moreover we show that every fuzzy relation equation provided with one solution is equivalent to a fuzzy relation sub equation of it

an alternative approach to the resolution of fuzzy relation equations abstract in this paper we propose a solution procedure by means of a quasi characteristic matrix the matrix is so defined that not only the solution set can be determined but also the type of solution set such as the non empty set full set unique minimal solution and unique solution can be identified before the solution set is determined therefore it provides an insight into the structure of fuzzy relation equations

solving interval valued fuzzy relation equations solving systems of fuzzy relation equations is an important topic in fuzzy set theory this paper studies the composite interval valued fuzzy relation equations after analyzing the properties of its solution set we convert the fuzzy relation equations into a fuzzy relation inequality system and propose an efficient computational procedure to generate the whole solution set examples are included to illustrate the idea and algorithm

coping with discontinuities in computer vision their detection classification and measurement the general principles of detection classification and measurement of discontinuities are studied the following issues are discussed detecting the location of discontinuities classifying discontinuities by their degrees measuring the size of discontinuities and coping with the random noise and designing optimal discontinuity detectors an algorithm is proposed for discontinuity detection from an input signal s for degree k discontinuity detection and measurement a detector p phi is used where p is the pattern and phi is the corresponding filter if there is a degree k discontinuity at location t sub 0 then in the filter response there is a scaled pattern alpha p at t sub 0 where alpha is the size of the discontinuity this reduces the problem to searching for the scaled pattern in the filter response a statistical method is proposed for the approximate pattern matching to cope with the random noise a study is made of optimal detectors which minimize the effects of noise

consistencies and rates of convergence of jump penalized least squares estimators we study the asymptotics for jump penalized least squares regression aiming at approximating a regression function by piecewise constant functions besides conventional consistency and convergence rates of the estimates in l 2 0 1 our results cover other metrics like skorokhod metric on the space of cadlag functions and uniform metrics on c 0 1 we will show that these estimators are in an adaptive sense rate optimal over certain classes of approximation spaces special cases are the class of functions of bounded variation piecewise holder continuous functions of order 0 1 and the class of step functions with a finite but arbitrary number of jumps in the latter setting we will also deduce the rates known from change point analysis for detecting the jumps finally the issue of fully automatic selection of the smoothing parameter is addressed

local likelihood tracking of fault lines and boundaries summary we suggest locally parametric methods for estimating curves such as boundaries of density supports or fault lines in response surfaces in a variety of spatial problems the methods are based on spatial approximations to the local likelihood that the curve passes through a given point in the plane as a function of that point the local likelihood might be a regular likelihood computed locally with kernel weights e g in the case of support boundary estimation or a local version of a likelihood ratio statistic e g in fault line estimation in either case the local likelihood surface represents a function which is relatively large near the target curve and relatively small elsewhere therefore the curve may be estimated as a ridge line of the surface we require only a numerical algorithm for tracking the projection of a ridge into the plane this approach offers several potential advantages over alternative methods first the local log likelihood surface can be graphed and the degree of ridginess assessed visually to determine how the level of local smoothing should be varied in different spatial locations in order to emphasize the ridge and hence the curve adequately secondly the local likelihood surface does not need to be computed in anything like its entirety once we have a reasonable approximation to a point on the curve we may track it by numerically walking along the ridge line thirdly the method is appropriate without change for many different types of spatial explanatory vanables gridded stochastic or otherwise three examples are explored in detail fault lines in response surfaces and in intensity or density surfaces and boundaries of supports of probability densities

a local polynomial jump detection algorithm in nonparametric regression we suggest a one dimensional jump detection algorithm based on local polynomial fitting for jumps in regression functions zero order jumps or jumps in derivatives first order or higherorder jumps if jumps exist in the mth order derivative of the underlying regression function then an m 1 order polynomial is fitted in a neighborhood of each design point we then characterize the jump information in the coefficients of the highest order terms of the fitted polynomials and suggest an algorithm for jump detection this method is introduced briefly for the general setup and then presented in detail for zero order and first order jumps several simulation examples are discussed we apply this method to the bombay india sea level pressure data

lower solutions of systems of fuzzy equations lower solutions of a system of fuzzy equations are determined by lower solutions of each equation the result is applied to fuzzy relation equations so that they are decomposed to simple fuzzy relation equations

when is a fuzzy relation decomposable in two fuzzy sets two necessary and sufficient conditions are given in order to decompose an assigned fuzzy relation in two fuzzy sets

on the convergence exponent of decomposable relations this paper deals with an decomposable fuzzy relation on 0 1 the decomposition problem of a fuzzy relation can be stated as follows given a fuzzy relation r determine whether there exist two fuzzy sets a and b such that r a b where the operator is nothing but godel implication at first two necessary and sufficient conditions that a fuzzy relation r is decomposable are given then the solution set of a and b with a b r is obtained in the end the convergence of the decomposable fuzzy relation is investigated and an algorithm is shown to calculate the convergence exponent for a given decomposable fuzzy relation

on a new geometric constant related to the von neumann jordan constant abstract the von neumann jordan constant c nj x is computed for x being l 2 l 1 and l l 1 space by introducing a new geometric constant x t these partly give an answer to an open question posed by kato et al some basic properties of this new coefficient are investigated moreover we obtain a new class of banach spaces with uniform normal structure

finitary solvability conditions for systems of fuzzy relation equations in this paper we consider the problem of solving systems of fuzzy relation equations two types of these systems with different compositions are considered and processed simultaneously a new sufficient condition and new solvability criteria are proposed for systems of both types all these conditions are finitary and thus can be easily verified the sufficient condition and criteria of solvability characterize a relationship between a skeleton nxn matrix n is a number of equations and a vector of the right hand side

solution sets of finite fuzzy relation equations with sup inf composition over bounded brouwerian lattices this paper considers the resolution of finite fuzzy relation equations with sup inf composition over a bounded brouwerian lattice the solution sets of finite fuzzy relation equations on a bounded brouwerian lattice are described in a similar way as those of linear spaces of n dimensional vectors in linear algebra

inference on segmented polynomial models segmented polynomial regression models with unknown change points are used in a wide variety of biological settings the application that stimulated this work uses a segmented polynomial model to examine the optimal hematocrit hypothesis we discuss problems in the fitting of these models and compare by simulation two methods of inference in these models that based on the chi squared approximation to the distribution of the likelihood ratio statistic and that based on the asymptotic normality of the least squares estimates the results show that of the two only the likelihood ratio statistic produces reliable inference concerning the change point

finite fuzzy relation equations with unique solution in complete brouwerian lattices abstract we characterize finite fuzzy relation equations having a unique solution for sake of completeness we first recall some results already established in foregoing papers in the context of the linear lattices new results are established in the general setting of the complete brouwerian lattices suitable examples illustrate the theorems obtained

jump information criterion for statistical inference in estimating discontinuous curves nonparametric regression analysis when the regression function is discontinuous has many applications existing methods for estimating a discontinuous regression curve usually assume that the number of jumps in the regression curve is known beforehand which is unrealistic in some situations although there has been research on estimation of a discontinuous regression curve when the number of jumps is unknown the problem remains mostly open because such research often requires assumptions on other related quantities such as a known minimum jump size in this paper we propose a jump information criterion which consists of a term measuring the fidelity of the estimated regression curve to the observed data and a penalty related to the number of jumps and the jump sizes the number of jumps can then be determined by minimizing our criterion theoretical and numerical studies show that our method works well

kernel type estimators of jump points and values of a regression function in the fixed design nonparametric regression model kernel type estimators of the locations of jump points and the corresponding sizes of jump values of the regression function are proposed these kernel type estimators are analyzed with almost sure results and limiting distributions using the limiting distributions we are able to test the number of jump points and give asymptotic confidence intervals for the sizes of jump values of the regression function simulation studies demonstrate that the asymptotic results hold for reasonable sample sizes

normal structure and the generalized james and zbăganu constants abstract we present several sufficient conditions for normal structure of a banach space in terms of the generalized james von neumann jordan zb ganu constants and the coefficient of weak orthogonality of sims our results significantly improve many known results in the literatures a negative answer is given to the question of llorens fuster e llorens fuster zbganu constant and normal structure fixed point theory 9 2008 159 172

on james and von neumann jordan constants and sufficient conditions for the fixed point property abstract in this paper we prove that a banach space x and its dual space x have uniform normal structure if c nj x 1 3 2 the garcia falset coefficient r x is estimated by the c nj x constant and the weak orthogonality coefficient introduced by b sims finally we present an affirmative answer to a conjecture by l maligranda concerning the relation between the james and c nj x constants for a banach space

nonparametric change point estimation consider a sequence of independent random variables x i 1 leq i leq n having cdf f for i leq theta n and cdf g otherwise a class of strongly consistent estimators for the change point theta in 0 1 is proposed the estimators require no knowledge of the functional forms or parametric families of f and g furthermore f and g need not differ in their means or other measure of location the only requirement is that f and g differ on a set of positive probability the proof of consistency provides rates of convergence and bounds on the error probability for the estimators the estimators are applied to two well known data sets in both cases yielding results in close agreement with previous parametric analyses a simulation study is conducted showing that the estimators perform well even when f and g share the same mean variance and skewness

jump detection in a regression curve and its derivative curve estimation from observed noisy data has broad applications in certain applications the underlying regression curve may have singularities including jumps and roofs valleys i e jumps in the first order derivative of the regression curve at some unknown positions representing structural changes of the related process detection of such singularities is important for understanding the structural changes in the literature a number of jump detection procedures have been proposed most of which are based on estimation of the one sided first order derivatives of the true regression curve in this paper motivated by certain related research in image processing we propose an alternative jump detection procedure besides the first order derivatives we suggest using helpful information about jumps in the second order derivatives as well theoretical justifications and numerical studies show that this jump detector works well in applications this procedure is then extended for detecting roofs v

resolution of matrix equations over arbitrary brouwerian lattices this paper studies the problem of solving a matrix equation over an arbitrary not necessarily complete brouwerian lattice a criterion for the solvability and a method for finding all solutions of the equation are obtained over an arbitrary self dual brouwerian lattice an equivalent condition for the unique solution a necessary and sufficient condition for the minimal solution and a procedure for constructing minimal solutions less than or equal to any given solution of the equation are presented it is shown that the results of the paper can be applied to the determination of generalized inverses of a matrix over an arbitrary brouwerian lattice

conditions under which the solution sets of fuzzy relational equations over complete brouwerian lattices form lattices it is well known that the solution set of a fuzzy relational equation with sup inf composition is a join semilattice in general not a meet semilattice this paper investigates the conditions under which the solution sets of fuzzy relational equations with sup inf composition over complete brouwerian lattices form lattices we first give some properties of the decompositions of elements in complete lattices then present a necessary and sufficient condition that the meet of a solution with any other solution is again a solution finally we show some necessary and sufficient conditions that the solution sets are lattices

the half half plot the half half hh plot is a new graphical method to investigate qualitatively the shape of a regression curve the empirical hh plot counts observations in the lower and upper quarter of a strip that moves horizontally over the scatterplot the plot displays jumps clearly and reveals further features of the regression curve we prove a functional central limit theorem for the empirical hh plot with rate of convergence in a simulation study the good performance of the plot is demonstrated the method is also applied to two case studies the proofs and one more case study are deferred to a supplement which is available online

resolution of finite fuzzy relation equations abstract general schemes for solving fuzzy relation equations with finite sets are derived in this paper the relationship between the results derived here and those of previous publications regarding this subject is also discussed

edge preserving smoothers for image processing abstract classical smoothers have limited usefulness in image processing because sharp edges tend to be blurred there is a literature on edge preserving smoothers but these all require moderately large smooth stretches here we discuss an approach to this problem called sigma filtering and propose an improvement based on running m estimation both computational and theoretical aspects are developed for image processing the methods have a niche between standard filtering approaches and bayes markov random field analysis

convexity conditions and normal structure of banach spaces we prove that f convexity the property dual to p convexity of kottman implies uniform normal structure moreover in the presence of the worth property normal structure follows from a weaker convexity condition than f convexity the latter result improves the known fact that every uniformly nonsquare space with the worth property has normal structure

finite fuzzy relation equations with a unique solution in linear lattices abstract we characterize fuzzy relation equations possessing a unique solution the equations presented here involve fuzzy sets and fuzzy relations assuming values in a linear lattice and defined on finite sets a suitable example also shows that our main result generally does not hold if the fuzzy equations are assigned on complete brouwerian lattices not necessarily linear

on solving relational equations in brouwerian lattices abstract the problems of finding generic and minimal solutions of a fuzzy relational equations in brouwerian lattices are considered

some results in the setting of fuzzy relation equations theory abstract in this paper we point out certain theoretical aspects of fuzzy relation equations connecting some known results between composite and o fuzzy equations several theorems are obtained and some examples are provided too

smoothing with split linear fits we introduce a family of smoothing algorithms that can produce discontinuous output unlike most commonly used smoothers that tend to blur discontinuities in the data this smoother can be used for smoothing with edge detection we cite examples of other approaches to two dimensional smoothing with edge detection in image processing and apply our one dimensional smoother to sea surface temperature data where the discontinuities arise from changes in ocean currents

solving nonlinear optimization problems with fuzzy relation equation constraints abstract an optimization model with a nonlinear objective function subject to a system of fuzzy relation equations is presented since the solution set of the fuzzy relation equations is in general a non convex set when it is not empty conventional nonlinear programming methods are not ideal for solving such a problem in this paper a genetic algorithm ga is proposed this ga is designed to be domain specific by taking advantage of the structure of the solution set of fuzzy relation equations the individuals from the initial population are chosen from the feasible solution set and are kept within the feasible region during the mutation and crossover operations the construction of test problems is also developed to evaluate the performance of the proposed algorithm

the characteristic of convexity of a banach space and normal structure we present some sufficient conditions for normal structure of banach spaces and their dual spaces in terms of the characteristic of convexity the james constant and the coefficient of weak orthogonality many known results are improved and strengthened we also show that some of our results are sharp

sufficient conditions for uniform normal structure of banach spaces and their duals in this paper we present some sufficient conditions for the uniform normal structure of banach spaces and their dual spaces more precisely many known conditions implying the uniform normal structure of banach spaces are shown to imply uniform normal structure of their dual spaces as well

jump surface estimation edge detection and image restoration surface estimation is important in many applications when conventional smoothing procedures e g running averages local polynomial kernel smoothing procedures smoothing spline procedures are used for estimating jump surfaces from noisy data jumps are blurred at the same time when noise is removed in recent years new smoothing methodologies have been proposed in the statistical literature for detecting jumps in surfaces and for estimating jump surfaces with jumps preserved we provide a review of these methodologies because a monochrome image can be considered a jump surface of the image intensity function with jumps at the outlines of objects edge detection and image restoration problems in image processing are closely related to the jump surface estimation problem in statistics we also review major methodologies on edge detection and image restoration and discuss connections and differences among these methods and related methods in the statistical literature

fuzzy relation equations on a finite set abstract the family of all the solutions of a fuzzy relation equation on a finite set is considered it is characterized by the set of all the lower solutions which can be obtained by a combinatorial algorithm

some geometric measures of spheres in banach spaces in this paper we first give relations between pythagorean parameters and other well known parameters the coefficient of weak orthogonality james and von neumann jordan constants consequently some known results in j gao on the generalized pythagorean parameters and the applications in banach spaces discrete contin dyn syst ser b 8 3 2007 557 567 j gao on some geometric parameters in banach spaces j math anal appl 344 2007 114 122 a jimenez melado e llorens fuster s saejung the von neumann jordan constant weak orthogonality and normal structure in banach spaces proc am math soc 134 2006 355 364 are deduced and strengthened secondly we present several sufficient conditions for a banach space and its dual to have normal structure finally some open questions posed at the end of gao 2007 are answered in the negative