Soft clustering – Fuzzy and rough approaches and their extensions and derivatives

Clustering is one of the most widely used approaches in data mining with real life applications in virtually any domain. The huge interest in clustering has led to a possibly three-digit number of algorithms with the k-means family probably the most widely used group of methods. Besides classic bivalent approaches, clustering algorithms belonging to the domain of soft computing have been proposed and successfully applied in the past four decades. Bezdek’s fuzzy c-means is a prominent example for such soft computing cluster algorithms with many effective real life applications. More recently, Lingras and West enriched this area by introducing rough k-means. In this article we compare k-means to fuzzy c-means and rough k-means as important representatives of soft clustering. On the basis of this comparison, we then survey important extensions and derivatives of these algorithms; our particular interest here is on hybrid clustering, merging fuzzy and rough concepts. We also give some examples where k-means, rough k-means, and fuzzy c-means have been used in studies.     

δ - and δ ′-shocks and the transportation and concentration processes

Using the definitions of δ - and δ ′-shocks for some systems of conservation laws, the corresponding Rankine–Hugoniot conditions are derived. We also derive the balance laws describing area, volume, mass and momentum transportation between the area outside the wave front and the wave front. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quasipower and hypergeometric series — construction and evaluation

It is proved that some power series converging very slowly in a neighbourhood of the point 1 can be transformed intoquasipower series. The latter converge faster but are more complicated because they contain some hypergeometric series₂ F ₁. Standard methods of the values evaluation for needed hypergeometric series with the aid of recurrence relations are not sufficiently efficient for some variable values. Therefore a new method, formally similar to Levin's transforms, is proposed. More generally, this is a method of approximative evaluating of such a solution of an inhomogeneous recurrence relation of order one which has some particular asymptotic properties.The efficacity of the proposed methods is analyzed in detail for Euler's dilogarithm. This is a typical function whose power series is approached with difficulties ifz1. In particular, its Padé approximants are sufficiently accurate only for, sayx[–1, 1/2]. Hermite-Padé approximation is more effective. Resulting irrational approximants generalize in some sense partial sums of the quasipower series introduced here.     

Trial and Error

Trial and error is a problem-solving strategy everyone uses at one time or another.In trial and error,you try ananswer.(The word trial comes from try.)If the answer is in error,you try something else.You keep trying untilyou get a correct answer.Trial and error is a particularly good strategy if a question has only a few possible answers.     

Problem-Solving and Sequence

In 1971,the famous mathematician George Polya,introduced four basic steps or phases for solving problems: Step1 Understand the Problem Step2 Decide on a Plan Step 3 Carry out the Plan Step4 Look Back     

E—matrices and Several Necessary and Sufficient Conditions

AER-x.,Aiscalledanonnegative(nonPositive)matrixifau>O(a,jO(aiirtO),markedA>O(AO(i#j),auO,wecallp(A)=max{IA,l,lA,l'.-.,lAnl},(whereA,,A2,..',A"areeigenva1ueofA)asspectralradiusofA.LemmalIfAe7'.,thenareanonnegativesquarematrxBandapositivenumberksuchthatA=B-kI(whereIistheidentitymatrixofordern).ProofLetbti=…     

Structure and Application of Algebraic Spline Curve and Surface

Algebraic-geometry foundations for non-linear spline functions are estab-lished.The structure of algebraic spline curves and surfaces represented by implicitforms are investigated.The necessary and sufficient conditions for smooth connection ofalgebraic curves(or surfaces)are presented.     

Transitivity and Chaos

Several different definitions of transitivity and their relationship are carefullydiscussed for general spaces, and it is proved that a continuous map on a metric space ischaotic in the sense of Devaney if and only if it is periodic orbit transitive or periodic orbitstrongly transitive.     

Ramanujan and extensions and contractions of continued fractions

If a continued fraction K _n=1^∞ a _n /b _n is known to converge but its limit is not easy to determine, it may be easier to use an extension of K _n=1^∞ a _n /b _n to find the limit. By an extension of K _n=1^∞ a _n /b _n we mean a continued fraction K _n=1^∞ c _n /d _n whose odd or even part is K _n=1^∞ a _n /b _n . One can then possibly find the limit in one of three ways:

Orthogonal and Round

Civilized man is surrounded on all sides, indoors and out, by a subtle, seldom-noticed conflict between two ancient ways of shaping things: the orthogonal and the round. Cars on circular wheels, guided by hand on circular steering wheels, move on streets that intersect like the lines of a rectangular lattice. Buildings and houses are made up mostly of right angles, relieved occasionally by circular domes and windows. At rectangular or circular tables, with rectangular napkins on our laps, we eat from circular plates and drink from glasses with circular cross sections. We light cylindrical cigarettes with matches torn from rectangular packs, and we pay the rectangular bill with rectangular bank notes and circular coins.     

The locally uniformly non‐square points of Orlicz–Bochner sequence spaces

In this paper, we investigate the locally uniformly non‐square point of Orlicz–Bochner sequence spaces endowed with Luxemburg norm. Analysing and combining the generating function M and properties of the real Banach space X , we get sufficient and necessary conditions of locally uniformly non‐square point, which contributes to criteria for locally uniform non‐squareness in Orlicz–Bochner sequence spaces. The results generalize the corresponding results in the classical Orlicz sequence spaces.     

Criteria for complex strongly extreme points of Musielak–Orlicz function spaces

The concepts of complex strongly extreme point and complex midpoint local uniform rotundity in complex Banach spaces are introduced. It is proved that every complex strongly extreme point is a complex extreme point and every complex locally uniformly rotund point is a complex strongly extreme point. Moreover, criteria for complex strongly extreme points of the unit ball and, as a corollary, criteria for complex midpoint local uniform rotundity in Musielak–Orlicz function spaces are given.     

Local rotundity structure of generalized Orlicz–Lorentz sequence spaces

We give some criteria for extreme points and strong U-points in generalized Orlicz–Lorentz sequence spaces, which were introduced in [P. Foralewski, H. Hudzik, L. Szymaszkiewicz, On some geometric and topological properties of generalized Orlicz–Lorentz sequence spaces, Math. Nachr. (in press)] (cf. [G.G. Lorentz, An inequality for rearrangements, Amer. Math. Monthly 60 (1953) 176–179; M. Nawrocki, The Mackey topology of some F-spaces, Ph.D. Dissertation, Adam Mickiewicz University, Poznań, 1984 (in Polish)]). Some examples show that in these spaces the notion of the strong U-point is essentially stronger than the notion of the extreme point. This paper is related to the results from [A. Kamińska, Extreme points in Orlicz–Lorentz spaces, Arch. Math. 55 (1990) 173–180] (see Remark 1).     

Uniform Gateaux differentiability and weak uniform rotundity in Musielak-Orlicz function spaces of Bochner type equipped with the Luxemburg norm

Criteria for weak uniform rotundity and reflexivity of Musielak-Orlicz function spaces of Bochner type equipped with the Luxemburg norm are given. Although, criteria for uniform Gateaux differentiability and weak uniform rotundity of Musielak-Orlicz function spaces of real functions equipped with the Luxemburg norm were known, they can be also easily deduced from our main results.     

Noncommutative Orlicz–Hardy spaces associated with growth functions

In this paper, we first present several basic properties of growth functions, and then prove a Hölder type inequality on noncommutative Orlicz spaces associated with a growth function. Moreover, we prove Riesz and Szegö type factorization theorems and the contractivity of the conditional expectation on noncommutative Orlicz–Hardy spaces associated with a growth function.     

Dual Orlicz–Brunn–Minkowski theory

In this paper, a dual Orlicz–Brunn–Minkowski theory is presented. An Orlicz radial sum and dual Orlicz mixed volumes are introduced. The dual Orlicz–Minkowski inequality and the dual Orlicz–Brunn–Minkowski inequality are established. The variational formula for the volume with respect to the Orlicz radial sum is proved. The equivalence between the dual Orlicz–Minkowski inequality and the dual Orlicz–Brunn–Minkowski inequality is demonstrated. Orlicz intersection bodies are defined and the Orlicz–Busemann–Petty problem is posed.     

Orlicz-Sobolev空间的弱局部一致凸性

本文研究了Orlicz-Sobolev空间的弱局部一致凸性.通过运用Orlicz空间和Sobolev空间的技巧,得到了赋Luxemburg范数的Orlicz-Sobolev空间具有弱局部一致凸性的充要条件和赋Orlicz范数的Orlicz-Sobolev空间具有弱局部一致凸性的充分条件.     

Geometric Properties of Symmetric Spaces with Applications to Orlicz–Lorentz Spaces

We deal with the basic convexity properties –rotundity, and uniform, local uniform and full rotundity –- for symmetric spaces. A characterization of Orlicz–Lorentz spaces with the Kadec–Klee property for pointwise convergence is given. These results are applied to obtain criteria of convexity properties for Orlicz–Lorentz sequence spaces, and some new proofs of the sufficiency part of criteria for rotundity and uniform rotundity for Orlicz–Lorentz function spaces.     

Musielak–Orlicz–Bochner function spaces which are uniformly noncreasy

In this paper, the criteria for uniform noncreasiness of Musielak–Orlicz–Bochner function spaces are given. Moreover authors also prove that the space (resp ) is uniformly noncreasy if and only if the space (resp ) is uniformly convex or uniformly smooth. As a corollary, the criteria for uniform noncreasiness of Musielak–Orlicz function spaces are given.