a common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type abstract in this paper we proved a common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type and a property e a introduced in m aamri d el moutawakil some new common fixed point theorems under strict contractive conditions j math anal appl 270 2002 181 188 our theorem generalizes theorem 2 2 of m aamri d el moutawakil common fixed points under contractive conditions in symmetric spaces appl math e notes 3 2003 156 162 and theorem 2 of m aamri d el moutawakil some new common fixed point theorems under strict contractive conditions j math anal appl 270 2002 181 188

missing modules the gnome lie algebra and e10 we study the embedding of kac moody algebras into borcherds or gener alized kac moody algebras which can be explicitly realized as lie algebras of physical states of some completely compactified bosonic string the extra missing states can be decomposed into irreducible highest or lowest weight missing modules w r t the relevant kac moody subalgebra the corresponding lowest weights are associated with imaginary simple roots whose multiplicities can be simply understood in terms of cer tain polarization states of the associated string we analyse in detail two examples where the momentum lattice of the string is given by the unique even unimodular lorentzian latticeii 1 1 orii 9 1 respectively the former leads to the borcherds algebra gii 1 1 which we call gnome lie algebra with maximal kac moody subalgebraa1 by the use of the denominator formula a complete set of imaginary simple roots can be exhibited while the ddf construction provides an explicit lie algebra basis in terms of purely longitudinal states of the compactified string in two dimensions the second example is the borcherds algebra gii 9 1 whose maximal kac moody subalgebra is the hyperbolic algebra e10 the imaginary simple roots at level 1 which give rise to irreducible low est weight modules fore10 can be completely characterized furthermore our explicit analysis of two non trivial level 2 root spaces leads us to conjecture that these are in fact the only imaginary simple roots for gii 9 1

a note on a paper of hicks and rhoades abstract our aim is to point out an error in the proof of lemma 2 in the paper fixed point theory in symmetric spaces with applications to probabilistic spaces by t l hicks b e rhoades nonlinear analysis 36 1999 331 344 and to indicate a way to repair it

symmetric spaces approach to some fixed point results abstract in this paper we present a simple and unified approach to the fixed point results on cone symmetric spaces and metric type spaces based on symmetric spaces fixed point theory we also give a new characterization of semi metric spaces with open balls

vertex operators for physical states of bosonic string the commutator of the virasoro operator lm and general vertex operator v for the mass level l is explicitly calculated by demanding that a physical vertex operator must be of conformal dimension j 1 a set of algebraic equations for determining the vertex operators corresponding to all physical states of bosonic strings is obtained explicit expressions for the first three mass levels are given

common fixed point theorems for some new generalized contractive type mappings in this paper some new generalized contractive type conditions for a pair of mappings in metric space are defined some common fixed point results for these mappings are presented

involutive automorphisms and iwasawa decomposition of some hyperbolic kac moody algebras the involutive automorphisms of hyperbolic kac moody algebras are computed from satake diagrams of these algebras which are then used to furnish a general procedure of their iwasawa decomposition in particular 1 1 and 2 1 are considered as representative examples to illustrate the efficacy of the underlying treatment

hyperbolic kac moody superalgebras we present a classification of the hyperbolic kac moody hkm superalgebras the hkm superalgebras of rank larger or equal than 3 are finite in number 213 and limited in rank 6 the dynkin kac diagrams and the corresponding simple root systems are determined we also discuss a class of singular sub super algebras obtained by a folding procedure

equivalent contractive conditions in symmetric spaces in this paper we introduce 22 contractive conditions in symmetric spaces we prove that all these contractive conditions are equivalent then we use these conditions to get some common fixed point theorems for noncompatible maps

dynkin diagrams for hyperbolic kac moody algebras hyperbolic algebras such as e10 are based on minkowskian root spaces that revert to finite or affine root systems upon the removal of any simple root in string theory terms finite lie affine kac moody and hyperbolic kac moody algebras are generated by tachyon vertex operators trachyon plus photon vertex operators and vertex operators for all mass levels respectively the 136 possible hyperbolic dynkin diagrams between the ranks 3 and 10 are classified and exhibited completing an earlier enumeration by kac of the 18 rank 7 10 cases the rank 2 hyperbolic algebras infinite in number have been classified by lepowsky and moody 1979

binary tree versus single level tree classification of white blood cells abstract for the problem of white blood cell recognition the use of various binary tree classification schemes is compared with the application of single tree classifiers in principle in a multi class problem binary tree classifiers have the advantage that only a restricted number of features per branch point is needed enabling an economical design of the classification process taking into account prior probabilities for all classes while these reasons remain valid the results presented here show that binary tree classifiers do not necessarily improve the correct recognition rate

fixed point theorems for non self mappings in symmetric spaces under φ weak contractive conditions and an application to functional equations in dynamic programming in this paper we prove some common fixed point theorems for two pairs of non self weakly compatible mappings enjoying common limit range property besides satisfying a generalized f weak contractive condition in symmetric spaces we furnish some illustrative examples to highlight the realized improvements in our results over the corresponding relevant results of the existing literature we extend our main result to four finite families of mappings in symmetric spaces using the notion of pairwise commuting mappings finally we utilize our results to discuss the existence and uniqueness of solutions of certain system of functional equations arising in dynamic programming

cone uniform cone locally convex and cone metric spaces endpoints set valued dynamic systems and quasi asymptotic contractions abstract the concepts of cone pseudometrics cone uniform spaces generated by these pseudometrics cone pseudodistances set valued quasi asymptotic contractions with respect to these pseudodistances and cone closed maps are introduced and studied conditions guaranteeing the existence and uniqueness of endpoints stationary points of these contractions and conditions ensuring that for each starting point each generalized sequence of iterations of these contractions in particular each dynamic process converges and then the limit is an endpoint are all established also the concept of the cone locally convex space as a special case of the cone uniform space is introduced and examples of quasi asymptotic contractions in cone metric spaces are constructed the definitions results ideas and methods are new for set valued dynamic systems in cone uniform cone locally convex and cone metric spaces and even for single valued maps in these spaces

dynkin diagrams of hyperbolic kac moody superalgebras hyperbolic kac moody superalgebras are classified in terms of their dynkin diagrams these types of kac moody superalgebras are those whose diagrams revert to either that of simple or affine superalgebras upon deletion of one of the vertices it is found that the maximum rank of this type of algebra is 6

on cone metric spaces a survey abstract using an old m krein s result and a result concerning symmetric spaces from s radenovic z kadelburg quasi contractions on symmetric and cone symmetric spaces banach j math anal 5 1 2011 38 50 we show in a very short way that all fixed point results in cone metric spaces obtained recently in which the assumption that the underlying cone is normal and solid is present can be reduced to the corresponding results in metric spaces on the other hand when we deal with non normal solid cones this is not possible in the recent paper m a khamsi remarks on cone metric spaces and fixed point theorems of contractive mappings fixed point theory appl 2010 7 pages article id 315398 doi 10 1115 2010 315398 the author claims that most of the cone fixed point results are merely copies of the classical ones and that any extension of known fixed point results to cone metric spaces is redundant also that underlying banach space and the associated cone subset are not necessary in fact khamsi s approach includes a small class of results and is very limited since it requires only normal cones so that all results with non normal cones which are proper extensions of the corresponding results for metric spaces cannot be dealt with by his approach

weyl group orbits on kac moody root systems let be a dynkin diagram and let be the simple roots of the corresponding kac moody root system let denote the cartan subalgebra let w denote the weyl group and let denote the set of all roots the action of w on and hence on is the discretization of the action of the kac moody algebra understanding the orbit structure of w on is crucial for many physical applications we show that for the simple roots i and j are in the same w orbit if and only if vertices i and j in the dynkin diagram corresponding to i and j are connected by a path consisting only of single edges we introduce the notion of the cayley graph of the weyl group action on real roots whose connected components are in one to one correspondence with the disjoint orbits of w for a symmetric hyperbolic generalized cartan matrix a of rank we prove that any two real roots of the same length lie in the same w orbit we show that if the generalized cartan matrix a contains zeros then there are simple roots that are stabilized by simple root reflections in w that is w does not act simply transitively on real roots we give sufficient conditions in terms of the generalized cartan matrix a equivalently for w to stabilize a real root using symmetry properties of the imaginary light cone in the hyperbolic case we deduce that the number of w orbits on imaginary roots on a hyperboloid of fixed radius is bounded above by the number of root lattice points on the hyperboloid that intersect the closure of the fundamental region for w

coincidence and fixed points in symmetric spaces under strict contractions abstract some common fixed point theorems due to aamri and el moutawakil m aamri d el moutawakil some new common fixed point theorems under strict contractive conditions j math anal appl 270 2002 181 188 and pant and pant r p pant v pant common fixed points under strict contractive conditions j math anal appl 248 2000 327 332 proved for strict contractive mappings in metric spaces are extended to symmetric semi metric spaces under tight conditions some related results are derived besides discussing illustrative examples which establish the utility of results proved in this note

from dynkin diagram symmetries to fixed point structures any automorphism of the dynkin diagram of a symmetrizable kac moody algebra g induces an automorphism of g and a mapping between highest weight modules of g for a large class of such dynkin diagram automorphisms we can describe various aspects of these maps in terms of another kac moody algebra the orbit lie algebra g in particular the generating function for the trace of over weight spaces which we call the twining character of g with respect to the automorphism is equal to a character of g the orbit lie algebras of untwisted affine lie algebras turn out to be closely related to the fixed point theories that have been introduced in conformal field theory orbit lie algebras and twining characters constitute a crucial step towards solving the fixed point resolution problem in conformal field theory

endpoints of set valued dynamical systems of asymptotic contractions of meir keeler type and strict contractions in uniform spaces abstract in this paper we introduce the concepts of the set valued dynamical systems of asymptotic contractions of meir keeler type and set valued dynamical systems of strict contractions in uniform spaces and we present a method which is useful for establishing conditions guaranteeing the existence and uniqueness of endpoints of these contractions and the convergence to these endpoints of all generalized sequences of iterations of these contractions the result concerning the investigations of problems of the set valued asymptotic fixed point theory include some well known results of meir and keeler kirk and suzuki concerning the asymptotic fixed point theory of single valued maps in metric spaces the result concerning set valued strict contractions in which the contractive coefficient is not constant is different from the result of yuan concerning the existence of endpoints of tarafdar vyborny generalized contractions in which the contractive coefficient is constant in bounded metric spaces and provides some examples of tarafdar yuan topological contractions in compact uniform spaces definitions and results presented here are new for set valued dynamical systems in uniform locally convex and metric spaces and even for single valued maps examples show a fundamental difference between our results and the well known ones

common fixed point theorems of gregus type for weakly compatible mappings satisfying generalized contractive conditions abstract we prove a common fixed point theorem of gregus type for four mappings satisfying a generalized contractive condition in metric spaces using the concept of weak compatibility which generalizes theorems of i altun d turkoglu b e rhoades fixed points of weakly compatible mappings satisfying a general contractive condition of integral type fixed point theory appl 2007 2007 article id 17301 a djoudi l nisse gregus type fixed points for weakly compatible mappings bull belg math soc 10 2003 369 378 a djoudi a aliouche common fixed point theorems of gregus type for weakly compatible mappings satisfying contractive conditions of integral type j math anal appl 329 1 2007 31 45 p vijayaraju b e rhoades r mohanraj a fixed point theorem for a pair of maps satisfying a general contractive condition of integral type int j math math sci 15 2005 2359 2364 x zhang common fixed point theorems for some new generalized contractive type mappings j math anal appl 333 2 2007 780 786 we prove also a common fixed point theorem which generalizes theorem 3 5 of h k pathak m s khan t rakesh a common fixed point theorem and its application to nonlinear integral equations comput math appl 53 2007 961 971 and common fixed point theorems of gregus type using a strict generalized contractive condition a property e a and a common property e a