reasonable properties for the ordering of fuzzy quantities ii as a continuation of the first part related to the first and second class of ordering approaches this paper deals with the fulfilment of reasonable properties in the third class of ordering approaches to do so we briefly introduce fuzzy relations on which the third class of approaches is based then we recall some transitivity related concepts and an ordering procedure based on a acyclic fuzzy relation acyclicity is a very weak restriction on a fuzzy relation we prove that many fuzzy relations used for the comparison of fuzzy quantities satisfy some conditions stronger than acyclicity so we give a widely applicable formulation to derive a total ranking order from a fuzzy relation with our formulation we examine all the ordering indices in the third class with respect to the proposed axioms in part i

dimension of the boundary of quasiconformal siegel disks we study quasiconformal siegel disks with critical points in their boundaries the main result asserts that every subarc of the boundary of the siegel disk has the hausdorff dimension strictly larger than 1 and that the boundary does not have a tangent at any point

cones and decomposition of sub and supermartingales abstract we study sub and supermartingales with values in the class f c of all convex fuzzy subsets of a banach space having compact support the terms sub and super are understood with respect to a partial order not necessarily inclusion in this general framework orders with respect to which a doob type decomposition can hold are naturally associated to convex subcones with a special property termed f c properness decomposition theorems into the sum or difference of a martingale and a compensator are proven in the sub case the martingale need exist only in a superspace of f c

rational rotation numbers for maps of the circle we consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms that is maps like x to x t frac c 2 pi sin 2 pi x bmod 1 withc 1 we prove that the set of parameter values corresponding to irrational rotation numbers has lebesgue measure 0 in other words the intervals on which frequency locking occurs fill up the set of full measure

convergence theorem for fuzzy martingales abstract we study fuzzy set valued measures in a banach space and their relationships to fuzzy random variables our main result is a convergence theorem for fuzzy martingales our tool is a radon nikodym type theorem for fuzzy measures which are absolutely continuous with respect to probability measures

an embedding theorem for fuzzy numbers on banach spaces and its applications in this work we give an embedding theorem for fuzzy number space n x on a banach space x as its applications we show that some measurability concepts for fuzzy number valued mappings are equivalent which partially solves an open problem given by butnariu fuzzy sets and systems 31 1989 77 additionally we generalize some convergence theorems and lusin theorem to fuzzy random variables

on the numerical computation of diophantine rotation numbers of analytic circle maps in this paper we present a numerical method to compute diophantine rotation numbers of circle maps with high accuracy we mainly focus on analytic circle diffeomorphisms but the method also works in the case of enough finite differentiability the keystone of the method is that under these conditions the map is conjugate to a rigid rotation of the circle moreover although it is not fully justified by our construction the method turns out to be quite efficient for computing rational rotation numbers we discuss the method through several numerical examples

siegel discs herman rings and the arnold family we show that the rotation number of an analytically linearizable element of the arnold family fa b x x a bsin 27rx mod 1 a b c ir o b 1 27r satisfies the brjuno condition conversely for every brjuno rotation number there exists an analytically linearizable element of the arnold family along the way we prove the necessity of the brjuno condition for linearizability of p x d z az 1 z d d and e z azez a e27ia at 0 we also investigate the complex arnold family and classify its possible fatou components finally we show that the siegel discs of pa d and e are quasidiscs with a critical point on the boundary if the rotation number is of constant type

non computable impressions of computable external rays of quadratic polynomials we discuss computability of impressions of prime ends of compact sets in particular we construct quadratic julia sets which possess explicitly described non computable impressions

on the convergence of fuzzy martingales the purpose of this paper is to prove the convergence theorem for fuzzy martingales without assuming that their values are lipschitz or continuous fuzzy numbers on rn this approach allows us to deduce many results on convergence of fuzzy numbers and fuzzy random variables from the relevant results on real numbers and real valued random variables that appear as their support functions based on the concept of fuzzy downward martingale and applications of convergence theorem of fuzzy martingales the strong law of large numbers for fuzzy random variables is proved here the convergence is in the uniform metric but not the separable metric on fuzzy number space

measurablility concepts for fuzzy mappings abstract in a series of recent works alternative measurability concepts for fuzzy mappings were introduced and studied in this paper we show that some of these concepts are equivalent or strongly inter related

siegel disks with smooth boundaries we show the existence of angles r z such that the quadratic polynomial p z e2i z z2 has a siegel disk with c smooth boundary this result was first announced by r perez marco in 1993

area and hausdorff dimension of sierpinski carpet julia sets we prove the existence of rational maps whose julia sets are sierpinski carpets having positive area such rational maps can be constructed such that they either contain a cremer fixed point a siegel disk or are infinitely renormalizable we also construct some sierpinski carpet julia sets with zero area but with hausdorff dimension two moreover for any given number s in 1 2 we prove the existence of sierpinski carpet julia sets having hausdorff dimension exactly s

one hundred years of complex dynamics holomorphic dynamics has been a key area of study for 100 years in this survey we focus on the dynamics of rational functions

self similarity of siegel disks and hausdorff dimension of julia sets let f z ez z where is an irrational number of bounded type according to siegel f is linearizable on a disk containing the origin in this paper we show the hausdorff dimension of the julia set j f is strictly less than two and if is a quadratic irrational such as the golden mean then the siegel disk for f is self similar about the critical point in the latter case we also show the rescaled first return maps converge exponentially fast to a system of commuting branched coverings of the complex plane

on strong laws of large numbers for random upper semicontinuous functions a short proof of the strong law of large numbers for random upper semicontinuous functions fuzzy random variables with compact support is given this approach allows us to deduce many results on convergence of fuzzy random variables from the relevant results on random sets that appear as their level sets

a transcendental julia set of dimension 1 we construct a non polynomial entire function whose julia set has finite 1 dimensional spherical measure and hence hausdorff dimension 1 in 1975 baker proved the dimension of such a julia set must be at least 1 but whether this minimum could be attained has remained open until now our example also has packing dimension 1 and is the first transcendental julia set known to have packing dimension strictly less than 2 it is also the first example with a multiply connected wandering domain where the dynamics can be completely described

semi order fuzzy supermartingales and submartingales with continuous time a new suitable semi order in the fuzzy number space en d the semi order fuzzy supermartingales and submartingales with continuous time and their characteristics are introduced the famous riesz decomposition and doob meyer decomposition theorems in standard martingale theory are generalized to semi order fuzzy supermartingales the proof of doob meyer decomposition is rather technical and so the theory of fuzzy valued measures generated by increasing fuzzy processes is developed

a decomposition theorem for herman maps abstract in 1980s thurston established a topological characterization theorem for postcritically finite rational maps in this paper a decomposition theorem for a class of postcritically infinite branched covering termed herman map is developed it s shown that every herman map can be decomposed along a stable multicurve into finitely many siegel maps and thurston maps such that the combinations and rational realizations of these resulting maps essentially dominate the original one this result is motivated by a non expanding version of mcmullen s problem and thurston s theory on characterization of rational maps it enables us to prove a thurston type theorem for rational maps with herman rings

on the julia set of a typical quadratic polynomial with a siegel disk let 0 1 be an irrational number with continued fraction expan sion a1 a 2 a 3 and consider the quadratic polynomial p z 7 e 2i z z 2 by performing a trans quasiconformal surgery on an associated blaschke product model we prove that if log an o p n a sn 1 then the julia set of p is locally connected and has lebesgue measure zero in particular it follows that for almost every 0 1 the quadratic p has a siegel disk whose boundary is a jordan curve passing through the critical point of p b y standard renormalization theory these results generalize to the quadratics which have siegel disks of higher periods

decomposition theorems for fuzzy supermartingales and submartingales doob decomposition and riesz decomposition theorems in standard martingale theory are generalized to fuzzy martingales the concepts of two types of riesz decomposition for fuzzy supermartingales are given and the necessary and sufficient conditions of that fuzzy supermartingale or submartingale resp supermartingale has doob decomposition resp riesz decomposition are discussed in detail

are there critical points on the boundaries of singular domains generalizing a result of e ghys we prove a general theorem that implies that if a rational functionf of the riemann sphere of degree 2 leaves invariant a singular domainc a disk or a ring on which the rotation number off satisfies a diophantine condition provided that on bar c f is injective then each boundary component ofc contains critical point off the injectivity condition is always satisfied for singular disks associated to linearizable periodic elliptic points off z z n a withne n 2 andae we also show that the singular disks associated to periodic elliptic points off z e az that satisfy a diophantine condition are unbounded in in the end of the paper we give a survey of the theory of iteration of entire functions of

quadratic julia sets with positive area we recently proved the existence of quadratic polynomials having a julia set with positive lebesgue measure we present the ideas of the proof and the techniques involved

a convergence theorem of fuzzy valued martingales in the extended hausdorff metric h in this paper we shall give a new embedding method to prove a convergence theorem for fuzzy valued random variables in the sense of the extended hausdorff metric h without the restriction of fuzzy sets satisfying the lipschitz condition

a pivoted lll algorithm abstract luk and tracy 2008 7 developed a matrix interpretation of the lll algorithm building on their work 7 we propose to add pivoting to the algorithm we prove that our new algorithm always terminates and we construct a class of ill conditioned reduced matrices to illustrate the advantages of pivoting

non computable julia sets polynomial julia sets have emerged as the most studied examples of fractal sets generated by a dynamical system apart from the beautiful mathematics one of the reasons for their popularity is the beauty of the computer generated images of such sets the algorithms used to draw these pictures vary the most naive work by iterating the center of a pixel to determine if it lies in the julia set milnor s distance estimator algorithm mil uses classical complex analysis to give a one pixel estimate of the julia set this algorithm and its modifications work quite well for many examples but it is well known that in some particular cases computation time will grow very rapidly with increase of the resolution moreover there are examples even in the family of quadratic polynomials when no satisfactory pictures of the julia set exist in this paper we study computability properties of julia sets of quadratic polynomials under the definition we use a set is computable if roughly speaking its image can be generated by a computer with an arbitrary precision under this notion of computability we show

upper bound for the size of quadratic siegel disks if is an irrational number we let p n q n n 0 be the approximants given by its continued fraction expansion the bruno series b is defined as b alpha sum n geq 0 frac log q n 1 q n the quadratic polynomial p z e 2i z z 2 has an indifferent fixed point at the origin if p is linearizable we let r be the conformal radius of the siegel disk and we set r 0 otherwise yoccoz proved that if b then r 0 and p is not linearizable in this article we present a different proof and we show that there exists a constant c such that for all irrational number with b we have b alpha log r alpha c together with former results of yoccoz see y this proves the conjectured boundedness of b logr

doob s stopping theorem for fuzzy super sub martingales with discrete time fuzzy random variables are introduced as random variables whose values are not reals but fuzzy sets the concepts of fuzzy conditional expectation and fuzzy super sub martingales are given in this paper we study doob s stopping theorem on fuzzy super sub martingales with discrete parameters

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

convergence of set valued and fuzzy valued martingales the purpose of this paper is to prove the convergence theorems of set valued and fuzzy valued martingales in kuratowski mosco sense without assuming that their values are compact or of compact level sets

limit theorems for fuzzy random variables a strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables whose values are fuzzy sets with compact levels the proofs are based on embedding theorems as well as on probability techniques in banach space