on additive transformations preserving a multiplicative matrix function abstract let m n k be the ring of all n n matrices over a division ring k and f be a multiplicative matrix function from m n k to a multiplicative abelian group with zero g 0 f ab f a f b a b m n k we call an additive transformation on m n k preserves a multiplicative matrix function f if f a f a a m n k in this paper we characterize all additive surjective transformations on m n k over any division ring k chk 2 that leave a non trivial multiplicative matrix function invariant applications to several related preservers are considered

the stein stromberg covering theorem in metric spaces abstract in 8 naor and tao extended to the metric setting the o d log d bounds given by stein and stromberg for lebesgue measure in r d deriving these bounds first from a localization result and second from a random vitali lemma here we show that the stein stromberg original argument can also be adapted to the metric setting giving a third proof we also weaken the hypotheses and additionally we sharpen the estimates for lebesgue measure

variations on the boman covering lemma abstract we explore some variants of the boman covering lemma and their relationship to the boundedness properties of the maximal operator let 1 p and let q be its conjugate exponent we prove that the strong type q q of the uncentered maximal operator by itself implies certain generalizations of the boman covering lemma for the exponent p and in turn these generalizations entail the weak type q q of the centered maximal operator we show by example that it is possible for the uncentered maximal operator to be unbounded for all 1 s while the conclusion of the lemma holds for every 1 p thus the latter condition is much weaker also the boundedness of the centered maximal operator entails weak versions of the lemma

random martingales and localization of maximal inequalities abstract let x d be a metric measure space for r 0 consider the hardy littlewood maximal operator m r f x def sup r r 1 b x r b x r f d we show that if there is an n 1 such that one has the microdoubling condition b x 1 1 n r b x r for all x x and r 0 then the weak 1 1 norm of m r has the following localization property m r l 1 x l 1 x sup r 0 m r r n r l 1 x l 1 x an immediate consequence is that if x d is ahlfors david n regular then the weak 1 1 norm of m r is n log n generalizing a result of stein and stromberg 1983 47 we show that this bound is sharp by constructing a metric measure space x d that is ahlfors david n regular for which the weak 1 1 norm of m 0 is n log n the localization property of m r is proved by assigning to each f l 1 x a distribution over random martingales for which the associated random doob maximal inequality controls the weak 1 1 inequality for m r

the quaternion formalism for mobius groups in four or fewer dimensions abstract the mobius group of r n defines n dimensional inversive geometry this geometry can serve as an alternative to projective geometry in providing a common foundation for spherical euclidean and hyperbolic geometry accordingly the mobius group plays an important role in geometry and topology the modern emphasis on low dimensional topology makes it timely to discuss a useful quaternion formalism for the mobius groups in four or fewer dimensions the present account is self contained it begins with the representation of quaternions by 2 x 2 matrices of complex numbers it discusses 2 x 2 matrices of quaternions and how a suitably normalized subgroup of these matrices extended by a certain involution related to sense reversal is 2 1 homomorphic to the mobius group acting on r 4 it provides details of this action and the relation of this action to various models of the classical geometries in higher dimensions n 5 the best description of the mobius group is probably by means of n 2 n 2 lorentz matrices in the lower dimensions covered by the quaternion formalism this alternative lorentz formalism is a source of interesting homomorphisms a sampling of these homomorphisms is computed explicitly both for intrinsic interest and for an illustration of the ease with which one can handle the quaternion formalism

pointwise theorems for amenable groups in this paper we prove the pointwise ergodic theorem for general locally compact amenable groups along folner sequences that obey some restrictions these restrictions are mild enough so that such sequences exist for all amenable groups we also prove a generalization of the shannon mcmillan breiman theorem to all discrete amenable groups

applications of axiomatic fuzzy set clustering method on management strategic analysis abstract in this paper first the axiomatic fuzzy set afs clustering method x d liu w wang t y chai the fuzzy clustering analysis based on afs theory ieee transactions on systems man and cybernetics part b 35 5 2005 1013 1027 is investigated further by improving the algorithm to be more applicable then it is used to analyze the evaluation results of 30 companies which have been studied and analyzed by g s liang et al g s liang t y chou t c han cluster analysis based on fuzzy equivalence relation european journal of operational research 166 2005 160 171 compared with liang s algorithm the proposed method is more transparent understandable and interpretable this method can be applied to the management strategic analysis based on the datasets described by mixed features such as real numbers boolean logical values linguistic descriptions the illustrative examples show that the interpretations of the clustering results of the 30 companies are almost consistent with the expert s intuitions

the fuzzy theory based on afs algebras and afs structure abstract in this paper we define a more general algebra class called afs algebras which includes ei algebra and eii algebra published by the author which are completely distributive molecular lattices we establish the fuzzy theory using afs algebras and afs structure published by the author and propose the concept of fuzzy variable and a simple controlling model based on afs theory

quaternionic modular groups abstract matrices whose entries belong to certain rings of algebraic integers are known to be associated with discrete groups of transformations of inversive n space or hyperbolic n 1 space h n 1 in particular groups operating in the hyperbolic plane or hyperbolic 3 space may be represented by 2 2 matrices whose entries are rational integers or real or imaginary quadratic integers the theory is extended here to groups operating in h 4 or h 5 and matrices over one of the three basic systems of quaternionic integers quaternionic modular groups are shown to be subgroups of the rotation groups of regular honeycombs of h 4 and h 5 for four dimensional groups the division ring of quaternions is treated as a clifford algebra results in hyperbolic 5 space derive from the homeomorphism of inversive 4 space and the quaternionic projective line

orientation preserving mobius transformations in r 4 and quaternionic determinants abstract this paper explores m r 4 the group of orientation preserving mobius transformations acting in r 4 on the one hand m r 4 is given by the group of 2 2 matrices over the quaternions h with determinant d derived from the corresponding 4 4 matrices over the complex numbers c on the other hand we know that in general m r n may be given in terms of 2 2 matrices over the clifford algebra c n with n 1 generators thus when n 4 m r 4 is given in terms of 2 2 matrices with entries drawn from c 4 and determinant defined in terms of the entries of the given matrix we note that the skew field h may be considered as a clifford algebra c 3 based on two generators i and j or more generally i 1 and i 2 where i j k or i 1 i 2 k while the set of elements 1 i j k form a basis of h regarded as a 4 dimensional real vector space thus h is embedded in c 4 in the paper we reconcile the two representations of m r 4 by comparing the generating sets of the underlying groups of matrices a relationship between a determinants d and is also exposed

theory of topological molecular lattices abstract the aim of this paper is to bring the theories of point set topology fuzzy topology and l fuzzy topology into a unified theory the theory of topological molecular lattices for this purpose we first introduce the concepts of molecule for completely distributive complete lattice molecular lattice for short the cotopology on it and the concept of remote neighborhoods of molecules therefrom we develop the theory of pointwise topology on molecular lattices

on afs algebra part ii this paper addresses the properties and structures of ei algebra and ein algebra in the framework of liu fuzzy sets and systems 95 1998 179 the main contribution of the paper is that the expressions of both irreducible elements or irreducible elements in molecular lattice em and ex1 xnm are developed and that the standard minimal family of elements in molecular lattice em and ex1 xnm is formulated it is proved that neither molecular lattice em nor molecular lattice ex1 xnm is a fuzzy lattice

quaternions and matrices of quaternions abstract we give a brief survey on quaternions and matrices of quaternions present new proofs for certain known results and discuss the quaternionic analogues of complex matrices the methods of converting a quaternion matrix to a pair of complex matrices and homotopy theory are emphasized

conjugacy invariants of sl 2 h abstract in this paper we use the quaternionic formalism of mobius transformations on r 4 to derive conjugacy invariants on sl 2 h and hence on psl 2 h we then use these invariants to distinguish between the various conjugacy classes of psl 2 h

the weak type 1 1 bounds for the maximal function associated to cubes grow to infinity with the dimension let md be the centered hardy littlewood maximal function associated to cubes in r d with lebesgue measure and let cd denote the lowest constant appearing in the weak type 1 1 inequality satised by md we show that cd 1 as d 1 thus answering for the case of cubes a longstanding

delambre gauss formulas for augmented right angled hexagons in hyperbolic 4 space abstract we study the geometry of right angled hexagons in the hyperbolic 4 space h 4 via clifford numbers or quaternions we show how to augment alternate sides of such a hexagon and arbitrarily orient each line and plane involved so that for the non augmented sides we can define quaternion half side lengths whose angular parts are obtained from half the euler angles associated to a certain orientation preserving isometry of the euclidean 3 space we also define appropriate complex half side lengths for the augmented sides of the augmented hexagon we further explain how to geometrically read off the quaternion half side lengths for a given oriented augmented right angled hexagon in h 4 our main result is a set of generalized delambre gauss formulas for oriented augmented right angled hexagons in h 4 involving the quaternion half side lengths and the complex half side lengths we also show in the appendix how the same method gives delambre gauss formulas for oriented right angled hexagons in h 3 from which the well known laws of sines and of cosines can be deduced these formulas generalize the classical delambre gauss formulas for spherical hyperbolic triangles

approaches to the representations and logic operations of fuzzy concepts in the framework of axiomatic fuzzy set theory ii in this paper the representations of fuzzy concepts based on raw data have been investigated within the framework of afs axiomatic fuzzy set theory first a brief review of afs theory is presented and a completely distributive lattice the e i algebra is proposed secondly two kinds of e i algebra representations of fuzzy concepts are derived in detail in order to represent the membership functions of fuzzy concepts in the interval 0 1 the norm of afs algebra is defined and studied finally the relationships of various representations with their advantages and drawbacks are analyzed

a 3d ginibre point field we introduce a family of three dimensional random point fields using the concept of the quaternion determinant the kernel of each field is an n dimensional orthogonal projection on a linear space of quaternionic polynomials we find explicit formulas for the basis of the orthogonal quaternion polynomials and for the kernel of the projection for number of particles n rightarrow infty we calculate the scaling limits of the point field in the bulk and at the center of coordinates we compare our construction with the previously introduced fermi sphere point field process

a quaternion qr algorithm this paper extends the francis qr algorithm to quaternion and antiquaternion matrices it calculates a quaternion version of the schur decomposition using quaternion unitary similarity transformations following a finite step reduction to a hessenberg like condensed form a sequence of implicit qr steps reduces the matrix to triangular form eigenvalues may be read off the diagonal eigenvectors may be obtained from simple back substitutions for serial computation the algorithm uses only half the work and storage of the unstructured francis qr iteration by preserving quaternion structure the algorithm calculates the eigenvalues of a nearby quaternion matrix despite rounding errors

concept analysis via rough set and afs algebra formal concept analysis fca was originally proposed by wille 1982 which is an important theory for data analysis and knowledge discovery afs axiomatic fuzzy set algebra was proposed by liu x liu the fuzzy theory based on afs algebras and afs structure journal of mathematical analysis and applications 217 1998 459 478 x liu the topology on afs algebra and afs structure journal of mathematical analysis and applications 217 1998 479 489 which is a semantic methodology relating to the fuzzy theory combining above two theories we propose afs formal concept which can be viewed as the generalization and development of monotone concept proposed by deogun and saquer 2003 moreover we show that the set of all afs formal concepts forms a complete lattice afs formal concept can be applied to represent the logic operations of queries in information retrieval furthermore we give an approach to find the afs formal concepts whose intents extents approximate any element of afs algebra by virtue of rough set theory

quaternions further contributions to a matrix oriented approach abstract by representing quaternions as four dimensional vectors and the multiplication of quaternions as matrix by vector product we employ a matrix oriented approach to the topic we investigate properties of the fundamental real matrix associated with a quaternion and examine a frequently considered quaternion equation from which the n th power of a quaternion can be determined