Hausdorff Methods Research Articles

Nakagami-fuzzy imaging for grading brain tumors by analyzing fractal complexity

Gliomas are the brain tumors in glial cells, which are categorized into four numerical grades, I-II-III-IV, to quantize the aggressiveness and severity of the tumors; while divided into two major groups, high-grade (HG) and low-grade (LG), in general. Among many differences between these groups, one of the most distinct and characteristic features could be seen in the shape of the tumor boundaries by magnetic resonance imaging (MRI). Due to aggressive nature of the HG tumors in proliferation phase, the boundaries of HG tumors become more shape-wise complex compared to the LG tumors, which could be differentiated by analyzing the fractal complexity of the cell membranes. However, the complexity cannot be either manually calculated or estimated by eye inspection without a reference point with one single image or sometimes even with an image set. Therefore, we present an automated glioma grading framework to provide an insight on the grades with a novel contouring and fractal dimension analysis system. The primary component of the proposed system is an automated Nakagami imaging module with a specialized fuzzy c-means algorithm to contour the boundaries of the whole tumors. The contoured images, afterwards, are analyzed by the Minkowski–Bouligand and Hausdorff methods for two panning options to generate the fractal dimensions and to estimate the fractal complexities for classifying the gliomas The results are greatly encouraging that the overall classification accuracy is computed as 88.31 % using the basic support vector machines (SVM) classifier; while as 91.96 % with the arbitrary thresholding appended. The outcomes of this paper with implementable mathematical infrastructure would be very useful and beneficial as an expert system in intelligent and automatic glioma grading, for researchers and medical experts.

An assessment of the efficiency of spatial distances in linear object matching on multi-scale, multi-source maps

ABSTRACTDistance in geospatial sciences has many applications, including the calculation of spatial similarity degree in object-matching problems. Various distances have so far been utilised for this purpose. However, no study has examined the efficiency of methods used for finding solutions for linear object matching in data sets with different or the same scales and sources. The present study investigated the efficiency of the most important and applicable spatial distances (13 types of distance methods) in vector data sets with different scales and sources. To this end, we employed three data sets of urban roads network of different sources with the scales of 1:2000, 1:5000 and 1:25,000. In the considered approach, the data sets are initially pre-processed to unify the format and coordinate system as well as removing topological errors. The corresponding objects in the data sets are then identified, and one-to-null, null-to-one, one-to-one, one-to-many, many-to-one and many-to-many relations are extracted. Ultimately, the method with the minimum dispersion in calculation of the distances between corresponding objects is selected as the efficient method. The results indicated that the short-line median and mean Hausdorff methods achieved higher efficiencies compared to the other employed methods. In addition to achieving a smaller variance compared to other introduced methods, these two methods are well capable of identifying one-to-many (many-to-one) and many-to-many relations.

Hausdorff methods for approximating the convex Edgeworth–Pareto hull in integer problems with monotone objectives

Adaptive methods for the polyhedral approximation of the convex Edgeworth–Pareto hull in multiobjective monotone integer optimization problems are proposed and studied. For these methods, theoretical convergence rate estimates with respect to the number of vertices are obtained. The estimates coincide in order with those for filling and augmentation H-methods intended for the approximation of nonsmooth convex compact bodies.

Convergence of hausdorff approximation methods for the Edgeworth–Pareto hull of a compact set

The Hausdorff methods comprise an important class of polyhedral approximation methods for convex compact bodies, since they have an optimal convergence rate and possess other useful properties. The concept of Hausdorff methods is extended to a problem arising in multicriteria optimization, namely, to the polyhedral approximation of the Edgeworth–Pareto hull (EPH) of a convex compact set. It is shown that the sequences of polyhedral sets generated by Hausdorff methods converge to the EPH to be approximated. It is shown that the Estimate Refinement method, which is most frequently used to approximate the EPH of convex compact sets, is a Hausdorff method and, hence, generates sequences of sets converging to the EPH.

Strongly nonatomic densities defined by certain matrices

Abstract Drewnowski and Paúl proved about ten years ago that for any strongly nonatomic submeasure η on the power set P(ℕ) of the set ℕ of all natural numbers the ideal of all null sets of η has the Nikodym property (NP). They stated the problem whether the converse is true in general. By presenting an example, Alon, Drewnowski and Łuczak proved recently that the answer is negative. Nevertheless, it is of mathematical interest to identify classes of submeasures η such that η is strongly nonatomic if and only if the set of all null sets of η has the Nikodym property. In this context, the authors proved some years ago that this equivalence holds, for instance, if one restricts the attention to the case of densities defined by regular Riesz matrices or by nonnegative regular Hausdorff methods. Also sufficient and necessary conditions in terms of the matrix coefficients are given, that the defined density is strongly nonatomic. In this paper we extend these investigations to the class of generalized Riesz matrices, introduced by Drewnowski, Florencio and Paúl in 1994.

Optimal growth order of the number of vertices and facets in the class of Hausdorff methods for polyhedral approximation of convex bodies

The internal polyhedral approximation of convex compact bodies with twice continuously differentiable boundaries and positive principal curvatures is considered. The growth of the number of facets in the class of Hausdorff adaptive methods of internal polyhedral approximation that are asymptotically optimal in the growth order of the number of vertices in approximating polytopes is studied. It is shown that the growth order of the number of facets is optimal together with the order growth of the number of vertices. Explicit expressions for the constants in the corresponding bounds are obtained.

Statistical gap Tauberian theorems in metric spaces

By using the concept of statistical convergence we present statistical Tauberian theorems of gap type for the Cesàro, Euler–Borel family and the Hausdorff families applicable in arbitrary metric spaces. In contrast to the classical gap Tauberian theorems, we show that such theorems exist in the statistical sense for the convolution methods which include the Taylor and the Borel matrix methods. We further provide statistical analogs of the gap Tauberian theorems for the Hausdorff methods and provide an explanation as to how the Tauberian rates over the gaps may differ from those of the classical Tauberian theorems.

Tauberian Theorems via Statistical Convergence

The concept of statistical convergence, which is related to the usual concept of convergence in probability, provides a regular summability method for abstract metric spaces. By using probabilistic tools, we provide some Tauberian theorems which have best possible order Tauberian conditions. Furthermore, these methods can be used to unify and improve the classical pointwise Tauberian theorems of summability theory for the random walk type methods as proved by Bingham, and Hausdorff methods as proved by Lorentz.

A Tauberian theorem for Hausdorff methods

Let H = ( H , χ ) H = (H,\chi ) be a regular Hausdorff summability method defined by the function χ ∈ BV [ 0 , 1 ] \chi \in {\text {BV}}\left [ {0,1} \right ] . It is shown that if χ \chi is absolutely continuous on [ 0 , 1 ] \left [ {0,1} \right ] , then the methods H H and V ⋅ H V \cdot H are equivalent for bounded sequences, where V V belongs to a certain class of summability methods which includes the Cesàro methods C α ( α > 0 ) {C_\alpha }(\alpha {\text { > }}0) , the Abel method A A , and the methods A ⋅ C α ( α > − 1 ) A \cdot {C_\alpha }(\alpha {\text { > }} - 1) .

On Integral Abel-Type and Logarithmic Methods of Summability

In this paper, we define an integral logarithmic method of summability, extending the integral Abel-type methods defined by Jakimovski [6]. We examine the behaviour of the product of this method with integral Hausdorff methods. A full scale of strict inclusions for integral Abel-type methods is obtained and the integral logarithmic method is placed in this scale.

A Note on Some Prime Hausdorff Methods of Summability

Given a matrix A = (ank) (n, k = 0, 1, 2, …), let (A) denote the set of all sequences x = {xk} such that {An(x)} ∊ c where An(x) = Σk=0∞ankxk (n≥0) and c denotes the set of all convergent sequences. It is well known (see e.g. Zeller [6] or Zeller and Beekmann [7], p. 48) that given an unbounded sequence x, there exists a regular (=permanent) matrix A with ank = 0 for k > n (and indeed with ann ≠ 0) such that (A) = c ⊕x, the linear space spanned by c and x. We call A an Einfolgenverfahren. (See [7].) In [4] Rhoades considered, inconclusively, the question whether there exists a Hausdorff matrix H such that (H)= c ⊕ x (for arbitrary unbounded sequence x).

Absolute summability of orthogonal series by Hausdorff methods

Sufficient conditions are obtained for the absolute summability of orthogonal series by the Hausdorff methods.

On the nonequivalence of conservative Hausdorff methods and Hausdorff moment sequences

On the nonequivalence of conservative Hausdorff methods and Hausdorff moment sequences

Inclusion theorems for generalized Hausdorff summability methods

The purpose of this note is to demonstrate how some of the classical Hausdorff inclusion theorems extend to the case where the sequences have their domains and ranges in topological vector spaces. We follow the terminology in [3] and [1] for the topological vector spaces and Hausdorff summability methods respectively. Suppose (X, Tl) and (Y, T2) are locally convex separated topological vector spaces. We denote by L(X, Y) the linear functions from X to Y which are continuous with respect to the topologies T, and T2. DEFINITION 1. If fmn,L(X, Y) (m, n=O, 1, 2, * ), then the matrix M = (fVn) is called a summability method from X to Y. Suppose now that (Z, Ts) is also a locally convex separated topological vector space. DEFINITION 2. If M1 is a summability method from X to Y and M2 is a summability method from X to Z with the property that for each sequence {xn4 of points in X for which {Ym} =M({xn}) is T2-convergent, the sequence {Zm } =M2({xn}) is T3-convergent, then we say M2 includes M1. We indicate this by M2DM1. We wish to consider Hausdorff methods H(j.A) = 6bt6, where ,u =diag (,o, Al, ), a is the differencing matrix, and MiEL(X, Y) (for example). We denote by N(Mi) the null space of ,u,. Suppose we have two such Hausdorff methods Hi = Hi(A) = 6ui6, where Ai =diag(uio, Ail, 1i2, * (i 2), lk EL(X, Y) and M2kEL(X, Z).

A characterization of totally regular [𝐽,𝑓(𝑥)] transforms

0. Introduction. Our main object is to prove a necessary and sufficient condition for [J, f (x) ] summation methods to be totally regular (?1). But we take this opportunity to establish also an inclusion theorem for regular [J, f (x) ] means (?2). A totally regular summation method is one which sums a sequence s} n=O, 1, ***, to s, whenever s--s, both for s finite and for s infinite. Necessary and sufficient conditions for a triangular method to be totally regular were discovered by W. A. Hurwitz [5] who also characterized the totally regular Hausdorff methods [H, g] as those generated by nondecreasing g(t), 0<t <1, with g(O+) =g(0) =0 and g(1) = 1. Conditions for total regularity of general methods of summation are more complicated than those for a triangular method and were provided by H. Hurwitz [4]. Here we supply a corresponding result for [J, f(x) ] methods. The [J, f(x)] transform was introduced by Jakimovski [6] in the following way. Let f(x) be a function differentiable infinitely often in (0, oo ). With a sequence { sn }, n = 0, 1, * * *, associate the transform

The Lebesgue Constants for Regular Hausdorff Methods

The unboundedness of the sequence of Lebesgue constants (norms), at a point, of certain transforms implies, as is well known, that there exist (i) a continuous function whose transform fails to converge to the function at the point in question (the du Bois-Reymond singularity), and (ii) another such function whose transform, while converging everywhere to the function, does not do so uniformly in any neighbourhood of the stipulated point (the Lebesgue singularity). The converses also hold in our case.

The Consistency of Norlund and Hausdorff Methods (Solution of a Problem of E. Ullrich)

The Consistency of Norlund and Hausdorff Methods (Solution of a Problem of E. Ullrich)

On the Total Relative Strength of Some Hausdorff Methods Equivalent to Identity

On the Total Relative Strength of Some Hausdorff Methods Equivalent to Identity

On the “collective Hausdorff method.”

Hausdorff methods have been extensively investigated by F. Hausdorif and others [1, 3, 5].1 A of summation A is said to be stronger than another B if every sequence summed by B is also summed by A. Two methods of summation are consistent if any sequence which can be summed by both methods has the same limit assigned to it by both of them. I shall say that A contains B if A is stronger than B and also consistent with B. A containing ordinary convergence is called regular. F. Hausdorff proved that any two regular Hausdorff methods are consistent [3]. This enabled R. P. Agnew to introduce the collective Hausdorff method I by the definition: { Sn } is summable & to the sum S if { Sn } is summable to S by any regular Hausdorff [1]. He raised at the same time the question whether there is a matrix of summation containing ,. I shall now show that the answer is in the negative. More precisely I prove the following:

On Hausdorff's methods of summability. II

A general theory of ‘strength’ for Hausdorff methods T of summability was given in the first part of this paper, to which we shall refer in the following as I. Those methods T which are regular (that is, at least as strong as ordinary convergence C), are defined by the linear transformsof a given sequence (sk). Here ø(t) (that is, each of its real and imaginary parts) is of bounded variation (b.v.) in the closed interval 〈0, 1〉 [see I, (4)]. The additional condition for consistency (with C) is that ø(t) is continuous at t = 0, and that ø(1) − ø(0) = 1 [I, H. 3]. We write also T ˜ ø(t).