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.     

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:

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.     

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.     

Four functionals fixed point theorem

The Four Functionals Fixed Point Theorem is a generalization of the original, as well as the functional generalizations, of the Leggett–Williams Fixed Point Theorem. In the Four Functionals Fixed Point Theorem, neither the upper nor the lower boundary of the underlying set is required to map below or above the boundary in the functional sense. As an application, the existence of a positive solution to a second-order right focal boundary value problem is considered by applying both standard and nonstandard choices of functionals. An extension to multivalued maps is provided for completeness.     

抽象凸空间中集值映射的不动点定理及其应用

A new fixed point theorem and the selection property for upper semi-continuous set-valued mappings in abstract convexity space are established. As their applications the existence of Nash equilibrium for n-person non-cooperative generalized games is proved.     

A critical remark on “Fixed point theorems for occasionally weakly compatible mappings”

In the present paper, we show that under contractive conditions, the existence of a common fixed point and occasional weak compatibility are equivalent conditions. We also show that contractive conditions employed by Jungck and Rhoades [Fixed point theorems for occasionally weakly compatible mappings, Fixed Point Theory 7(2) (2006) 287–296; Fixed Point Theory 9 (2008) 383–384 (erratum)] do not provide a nontrivial setting for the application of occasional weak compatible mappings. Finally, we improve the results of Jungck and Rhoades by employing a proper setting.     

A renorming of some nonseparable Banach spaces with the Fixed Point Property

We prove that for every Banach space which can be embedded in c₀(Γ) (for instance, reflexive spaces or more generally spaces with M-basis) there exists an equivalent renorming which enjoys the (weak) Fixed Point Property for non-expansive mappings. As a consequence, we solve a longtime open question in Metric Fixed Point Theory: Every reflexive Banach can be renormed to satisfy the Fixed Point Property. Furthermore, this norm can be chosen arbitrarily closed to the original norm.     

Fixed point theorems for random pseudo-contractive mappings

Very recently in Fierro et al. (2009) [6], we obtained a general principle to prove the existence of Random Fixed Point Theorems. As a consequence of this, we have been able to obtain various generalizations for pseudo-contractive mappings with rather simple proofs. In addition, while we were deriving these extensions for random operators, some deterministic results arose, which also appear to be new.     

First-order three-point boundary value problems at resonance

We consider three-point boundary value problems for a system of first-order equations in perturbed systems of ordinary differential equations at resonance. We obtain new results for the above boundary value problems with nonlinear boundary conditions. The existence of solutions is established by applying a version of Brouwer’s Fixed Point Theorem which is due to Miranda.     

Iterative processes related to Riordan arrays: The reciprocation and the inversion of power series

We point out how the Banach Fixed Point Theorem, together with the Picard successive approximation methods yielded by it, allows us to treat some mathematical methods in combinatorics. In particular we get, in this way, a proof of and an iterative algorithm for deriving the Lagrange Inversion Formula.     

Edelstein's method and fixed point theorems for some generalized nonexpansive mappings

A new condition for mappings, called condition (C), which is more general than nonexpansiveness, was recently introduced by Suzuki [T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008) 1088-1095]. Following the idea of Kirk and Massa Theorem in [W.A. Kirk, S. Massa, Remarks on asymptotic and Chebyshev centers, Houston J. Math. 16 (1990) 364-375], we prove a fixed point theorem for mappings with condition (C) on a Banach space such that its asymptotic center in a bounded closed and convex subset of each bounded sequence is nonempty and compact. This covers a result obtained by Suzuki [T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008) 1088-1095]. We also present fixed point theorems for this class of mappings defined on weakly compact convex subsets of Banach spaces satisfying property (D). Consequently, we extend the results in [T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008) 1088-1095] to many other Banach spaces.     

Strong convergence of the iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems of an infinite family of nonexpansive mappings

In this paper, we introduce an iterative scheme based on the extragradient approximation method for finding a common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions of a mixed equilibrium problem, and the set of solutions of the variational inequality problem for a monotone L-Lipschitz continuous mapping in a real Hilbert space. Then, the strong convergence theorem is proved under some parameters controlling conditions. Applications to optimization problems are given. The results obtained in this paper improve and extend the recent ones announced by Wangkeeree [R. Wangkeeree, An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings, Fixed Point Theory and Applications (2008) 17. doi:10.1155/2008/134148. Article ID 134148], Kumam and Katchang [P. Kumam, P. Katchang, A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings, Nonlinear Anal. Hybrid Syst. (2009) doi:10.1016/j.nahs.2009.03.006] and many others.     

Inequalities and Nash equilibria

This paper concentrates on the problem of the existence of equilibrium points for non-cooperative generalized N-person games, N-person games of normal form and their related inequalities. We utilize the K-K-M lemma to obtain a theorem and then use it to obtain a new Fan-type inequality and minimax theorems. Various new equilibrium point theorems are derived, with the necessary and sufficient conditions and with strategy spaces with no fixed point property. Examples are given to demonstrate that these existence theorems cover areas where other existence theorems break down.