existence of solutions for variational inequalities on riemannian manifolds abstract we establish the existence and uniqueness results for variational inequality problems on riemannian manifolds and solve completely the open problem proposed in s z nemeth variational inequalities on hadamard manifolds nonlinear anal 52 2003 1491 1498 also the relationships between the constrained optimization problem and the variational inequality problems as well as the projections on riemannian manifolds are studied

equilibrium problems on riemannian manifolds with applications abstract we study equilibrium problem on general riemannian manifolds focusing on existence of solutions and the convexity properties of the solution set our approach consists in relating the equilibrium problem to a suitable variational inequality problem on riemannian manifolds and is completely different from previous ones on this topic in the literature we apply our results to the mixed variational inequality and the nash equilibrium problem moreover we formulate and analyze the convergence of the proximal point algorithm for the equilibrium problem in particular correct proofs are provided for the results claimed in j math anal appl 388 2012 61 77 i e theorems 3 5 and 4 9 there regarding the existence of the mixed variational inequality and the domain of the resolvent for the equilibrium problem on hadamard manifolds

general ranked set sampling with cost considerations nahhas wolfe and chen 2002 biometrics 58 964 971 considered optimal set size for ranked set sampling rss with fixed operational costs this framework can be very useful in practice to determine whether rss is beneficial and to obtain the optimal set size that minimizes the variance of the population estimator for a fixed total cost in this article we propose a scheme of general rss in which more than one observation can be taken from each ranked set this is shown to be more cost effective in some cases when the cost of ranking is not so small we demonstrate using the example in nahhas wolfe and chen 2002 biometrics 58 964 971 by taking two or more observations from one set even with the optimal set size from the rss design can be more beneficial

the modified box counting method analysis of some characteristic parameters abstract one of the most popular ways of estimating fractal dimension df is the box counting method bcm the major problem with this method is that it leads to results with a very high percentage of error the modified box counting method mbcm was developed as a methodic procedure to set sequence and range the procedure eliminates two problems of the computerized bcm the border effect and noninteger values of e the mbcm is a new powerful tool very simple to use allowing accurate estimation of df

influence of image resolution and evaluation algorithm on estimates of the lacunarity of porous media in recent years experience has demonstrated that the classical fractal dimensions are not sufficient to describe uniquely the interstitial geometry of porous media at least one additional index or dimension is necessary lacunarity a measure of the degree to which a data set is translationally invariant is a possible candidate unfortunately several approaches exist to evaluate it on the basis of binary images of the object under study and it is unclear to what extent the lacunarity estimates that these methods produce are dependent on the resolution of the images used in the present work the gliding box algorithm of allain and cloitre phys rev a 44 3552 1991 and two variants of the sandbox algorithm of chappard et al j pathol 195 515 2001 along with three additional algorithms are used to evaluate the lacunarity of images of a textbook fractal the sierpinski carpet of scanning electron micrographs of a thin section of a european soil and of light transmission photographs of a togolese soil the results suggest that lacunarity estimates as well as the ranking of the three tested systems according to their lacunarity are affected strongly by the algorithm used by the resolution of the images to which these algorithms are applied and at least for three of the algorithms producing scale dependent lacunarity estimates by the scale at which the images are observed depending on the conditions under which the estimation of the lacunarity is carried out lacunarity values range from 1 02 to 2 14 for the three systems tested and all three of the systems used can be viewed alternatively as the most or the least lacunar some of this indeterminacy and dependence on image resolution is alleviated in the averaged lacunarity estimates yielded by chappard et al s algorithm further research will be needed to determine if these lacunarity estimates allow an improved unique characterization of porous media

estimating the dimension of a fractal summary we suggest refinements of the box counting method which address the obvious problems caused by the incomplete information and inaccessibility of the limit a method for the statistical analysis of these corrected data is developed and tested on simulated and real data

nonsmooth analysis and hamilton jacobi equations on riemannian manifolds we establish some perturbed minimization principles and we develop a theory of subdifferential calculus for functions defined on riemannian manifolds then we apply these results to show existence and uniqueness of viscosity solutions to hamilton jacobi equations defined on riemannian manifolds

efficient regression analysis with ranked set sampling this article is motivated by a lung cancer study where a regression model is involved and the response variable is too expensive to measure but the predictor variable can be measured easily with relatively negligible cost this situation occurs quite often in medical studies quantitative genetics and ecological and environmental studies in this article by using the idea of ranked set sampling rss we develop sampling strategies that can reduce cost and increase efficiency of the regression analysis for the above mentioned situation the developed method is applied retrospectively to a lung cancer study in the lung cancer study the interest is to investigate the association between smoking status and three biomarkers polyphenol dna adducts micronuclei and sister chromatic exchanges optimal sampling schemes with different optimality criteria such as a d and integrated mean square error imse optimality are considered in the application with set size 10 in rss the improvement of the optimal schemes over simple random sampling srs is great for instance by using the optimal scheme with imse optimality the imses of the estimated regression functions for the three biomarkers are reduced to about half of those incurred by using srs

can the fractal dimension of images be measured abstract fractal dimension is a popular parameter for explaining certain phenomena and for describing natural textures the problem of estimating the fractal dimension of a profile or an image is more difficult and devious than theory suggests this paper studies the accuracy and robustness of two common estimators of fractal dimension box counting and the variation method using two types of data brownian and takagi poor results are demonstrated from applying theory directly called naive estimation data is then interpreted in the most optimistic way possible by matching the estimator to the known fractal dimension experiments quantify the effects of resolution or fineness of sampling and quantization or rounding of sampled values increasing resolution enhances the estimators when true dimension d is large but may possibly due to quantization effect degrade estimators when d is small quantization simply causes shifts in estimates the results suggest that one should not place much reliance in the absolute value of a fractal estimate but that the estimates do vary monotonically with d and might be useful descriptors in tasks such as image segmentation and description

nonparametric two sample procedures for ranked set samples data abstract ranked set samples have been shown to lead to improved methods of estimation in parametric settings under specific distributional forms when actual measurement of the sample observations is difficult but ranking them is relatively easy the earliest work with ranked set data concentrated on estimating a population mean or variance more recently a ranked set sample estimator of a cumulative distribution function was developed and used to obtain a simultaneous confidence interval for the function in this article we take the next logical step and use this ranked set empirical distribution function to construct distribution free competitors to the standard mann whitney wilcoxon estimation and testing procedures the appropriate null distribution tables for the associated test are presented for the case of perfect ranking asymptotic relative efficiency comparisons between the simple random sample mann whitney wilcoxon procedures and their ranked set analogues are discussed and the results of a s

an iterative method for finding common solutions of equilibrium and fixed point problems we introduce an iterative method for finding a common element of the set of solutions of an equilibrium problem and of the set of fixed points of a finite family of nonexpansive mappings in a hilbert space we prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality which is the optimality condition for a minimization problem

estimating fractal dimension with fractal interpolation function models fractal dimension fd is a feature which is widely used to characterize medical images previously researchers have shown that fd separates important classes of images and provides distinctive information about texture the authors analyze limitations of two principal methods of estimating fd box counting bc and power spectrum ps bc is ineffective when applied to data limited low resolution images ps is based on a fractional brownian motion fbm model a model which is not universally applicable the authors also present background information on the use of fractal interpolation function fif models to estimate fd of data which can be represented in the form of a function they present a new method of estimating fd in which multiple fif models are constructed the mean of the fd s of the fif models is taken as the estimate of the fd of the original data the standard deviation of the fd s of the fif models is used as a confidence measure of the estimate the authors demonstrate how the new method can be used to characterize fractal texture of medical images in a pilot study they generated plots of curvature values around the perimeters of images of red blood cells from normal and sickle cell subjects the new method showed improved separation of the image classes when compared to bc and ps methods

new existence results for equilibrium problems we consider equilibrium problems in the framework of the formulation proposed by blum and oettli which includes variational inequalities nash equilibria in noncooperative games and vector optimization problems for instance as particular cases we establish new sufficient and or necessary conditions for existence of solutions of such problems our results are based upon the relation between equilibrium problems and certain auxiliary convex feasibility problems together with extensions to equilibrium problems of gap functions for variational inequalities then we apply our results to some particular instances of equilibrium problems obtaining results which include among others a new lemma of the alternative for convex optimization problems

equilibrium programming using proximal like algorithms we compute constrained equilibria satisfying an optimality condition important examples include convex programming saddle problems noncooperative games and variational inequalities under a monotonicity hypothesis we show that equilibrium solutions can be found via iterative convex minimization in the main algorithm each stage of computation requires two proximal steps possibly using bregman functions one step serves to predict the next point the other helps to correct the new prediction to enhance practical applicability we tolerate numerical errors

variational inequalities on hadamard manifolds the notion of variational inequalities is extended to hadamard manifolds and related to geodesic convex optimization problems existence and uniqueness theorems for variational inequalities on hadamard manifolds are proved a convexity property of the solution set of a variational inequality on a hadamard manifold is presented

monotonicity resolvents and yosida approximations of operators on hilbert manifolds abstract monotonicity resolvents and yosida approximations of operators on manifolds of class c 3 are considered strong convergence of difference approximations to differential equations is shown by using resolvents our motivation is to give an elementary step toward constructing nonlinear semigroup theory on infinite dimensional manifolds

fractal feature analysis and classification in medical imaging following b b mandelbrot s fractal theory 1982 it was found that the fractal dimension could be obtained in medical images by the concept of fractional brownian motion an estimation concept for determination of the fractal dimension based upon the concept of fractional brownian motion is discussed two applications are found 1 classification 2 edge enhancement and detection for the purpose of classification a normalized fractional brownian motion feature vector is defined from this estimation concept it represented the normalized average absolute intensity difference of pixel pairs on a surface of different scales the feature vector uses relatively few data items to represent the statistical characteristics of the medial image surface and is invariant to linear intensity transformation for edge enhancement and detection application a transformed image is obtained by calculating the fractal dimension of each pixel over the whole medical image the fractal dimension value of each pixel is obtained by calculating the fractal dimension of 7 7 pixel block centered on this pixel

convergence of numerical box counting and correlation integral multifractal analysis techniques abstract a systematic study of the rate of convergence for a numerical box counting and a numerical correlation integral algorithm for determining the generalized fractal dimension d q are described the algorithms are applied to euclidean point sets koch constructions and a symmetric chaotic mapping the results provide a basis for estimating the size of a fractal subset needed for measurement of the generalized dimension d q in particular the number of points n 5 required to assure 5 convergence of the algorithms is given within a factor of 4 by log 10 n 5 2 54 d q 0 11 for the fractal sets studied here approximately 25 times as many points are needed for 1 convergence approximately 0 1 times as many points are needed for 25 convergence the box based correlation integral algorithm employed in the present studies which is well suited to the analysis of large data sets is also described

multiple resolution texture analysis and classification textures are classified based on the change in their properties with changing resolution the area of the gray level surface is measured at serveral resolutions this area decreases at coarser resolutions since fine details that contribute to the area disappear fractal properties of the picture are computed from the rate of this decrease in area and are used for texture comparison and classification the relation of a texture picture to its negative and directional properties are also discussed

evaluating the fractal dimension of surfaces fractal objects derive from many interface phenomena as they arise in for example materials science chemistry and geology hence the problem of estimating fractal dimension becomes of both theoretical and practical importance existing algorithms implement the standard definitions of fractal dimension directly but as we show often give unreliable results when applied to digitized and quantized data we present a new algorithm for estimating the fractal dimension of surfaces the variation method that is more reliable and robust than the standard ones it is based on a new definition of fractal dimension particularly suited for graphs of functions the variation method is validated with both fractional brownian surfaces and takagi surfaces two classes of mathematical objects with known fractal dimension and is shown to give more accurate results than the classical algorithms finally our new algorithm is applied to data from sand blasted metal surfaces

characteristics of natural scenes related to the fractal dimension many objects in images of natural scenes are so complex and erratic that describing them by the familiar models of classical geometry is inadequate in this paper we exploit the power of fractal geometry to generate global characteristics of natural scenes in particular we are concerned with the following two questions 1 can we develop a measure which can distinguish between different global backgrounds e g mountains and trees and 2 can we develop a measure that is sensitive to change in distance or scale we present a model based on fractional brownian motion which will allow us to recover two characteristics related to the fractal dimension from silhouettes the first characteristic is an estimate of the fractal dimension based on a least squares linear fit we show that this feature is stable under a variety of real image conditions and use it to distinguish silhouettes of trees from silhouettes of mountains next we introduce a new theoretical concept called the average holder constant and relate it mathematically to the fractal dimension it is shown that this measurement is sensitive to scale in a predictable manner and hence provides the potential for use as a range indicator corroborating experimental results are presented

maximum likelihood estimates of the fractal dimension for random spatial patterns summary two approximate maximum likelihood estimates of the fractal dimension are suggested for random point patterns and planar curves based on the distributions of the palm probability and the spectrum for estimation of the fractal dimension of self similar patterns in space we usually measure the slope of a linear log log relationship between pairs of points the box counting method plots the number of pixels which intersects the pattern under consider ation versus length of the pixel unit and the walking dividers method plots the total length of polygons which approximate the considered curves versus length of the polygon s side and so on see mandelbrot 1982 p 33 for example using these methods there has been a large number of papers reporting the dimensions of fractal sets especially since mandelbrot 1977 second order properties of a stochastic process such as the auto covariance and the spectrum are also available for examining self similarity and measuring its indices this is especially easy for stochastic processes on a real line for instance ogata 1988 and ogata abe 1991 applied one dimensional point processes to the occurrence times of major earthquakes in the world and japan for a time span of about one century investigating the self similarity by the auto covariance spectrum dispersion time curves and so called r s statistic in particular fractal dimension and the hurst number or the self similarity index are estimated objectively by maximizing a spectral log likelihood the maximum likelihood method is generally expected to provide an efficient estimate together with its standard error therefore in this paper we develop maximum likelihood methods for planar point and curve patterns two types of approximate likelihood functions are considered one is equivalent to an isotropic poisson likelihood by modelling an intensity function of points under the palm probability and the other is the so called spectral likelihood based on the distribution of the periodogram which is an estimate of the spectrum these two independent estimation methods are compared by using a certain artificially generated clustering point pattern then a map of epicentres of earthquakes and a topographic contour line are analyzed to compare the two methods

estimating the intrinsic dimension of data with a fractal based method in this paper the problem of estimating the intrinsic dimension of a data set is investigated a fractal based approach using the grassberger procaccia algorithm is proposed since the grassberger procaccia algorithm 1983 performs badly on sets of high dimensionality an empirical procedure that improves the original algorithm has been developed the procedure has been tested on data sets of known dimensionality and on time series of santa fe competition

on the imaging of fractal surfaces an analysis is presented of the imaging of surfaces modeled by fractal brownian elevation functions of the sort used in computer graphics it is shown that if lambertian reflectance modest surface slopes and the absence of occlusions and self shadowing are assumed a fractal surface with fourier power spectrum proportional to f sup beta produces an image with power spectrum proportional to f sup 2 beta here f is the spatial frequency and beta is related to the fractional dimension value this allows one to use the spectral falloff of the images to predict the fractal dimension of the surface

regression estimator in ranked set sampling summary situations in this paper we assume that the ranking is done on the basis of a concomitant variable x regression type rss estimators of the population mean of y will be proposed by utilizing this concomitant variable x in both the ranking process of the units and the estimation process when the population mean of x is known when x has unknown mean double sampling will be used to obtain an estimate for the population mean of x it is found that when x and y jointly follow a bivariate normal distribution our proposed rss regression estimator is more efficient than rss and simple random sampling srs naive estimators unless the correlation between x and y is low ipi 0 4 moreover it is always superior to the regression estimator under srs for all p when normality does not hold this approach could still perform reasonably well as long as the shape of the distribution of the concomitant variable x is only slightly departed from symmetry for heavily skewed distributions a remedial measure will be suggested an example of estimating the mean plutonium concentration in surface soil on the nevada test site nevada u s a will be considered

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

equilibrium problems in hadamard manifolds abstract an equilibrium theory is developed in hadamard manifolds the existence of equilibrium points for a bifunction is proved under suitable conditions and applications to variational inequality fixed point and nash equilibrium problems are provided the convergence of picard iteration for firmly nonexpansive mappings along with the definition of resolvents for bifunctions in this setting is used to devise an algorithm to approximate equilibrium points

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