super lie groups global topology and local structure a general mathematical framework for the super lie groups of supersymmetric theories is presented the definition of super lie group is given in terms of supermanifolds and two theorems analogous to theorems in classical lie group theory are proved the relationship of the super lie groups defined here to the formal groups of berezin and kac and the graded lie groups of kostant is analyzed

even sectors of lie superalgebras as locally convex lie algebras let g be a finite dimensional lie superalgebra over a graded commutative local augmented arens michael ground algebra a lie group is associated with a number of pleasant properties to the even sector g0 viewed as an ordinary complete locally convex lie algebra which is in general infinite dimensional

fix points of meromorphic functions and iterated entire functions let f be a rational function of degree k and let g be a transcendental meromorphic function gross and osgood 13 proved that the composite function fog has infinitely many fix points if k 3 if k 2 then fog may have only a finite number of fix points and all such f and g have been characterized by gross and osgood there is a corresponding result for the case that f is a transcendental meromorphic function and that g is a polynomial in fact the two cases are closely connected since as pointed out by gross and yang 14 p 214 proof of theorem 23 fog has infinitely many fix points if and only if gof does this paper deals with the fix points offog wherefand g are both transcendental it has been proved in 6 that f 0 g has infinitely many fix points if f and g are entire this confirmed a conjecture of gross 111 there are a number of papers s 12 14 22 27 283 besides 6 dealing with fix points of fog for entire f and g but their methods do not seem to extend to the case of meromorphic f and in fact very little seems to be known about this case gross 1 l has asked whether fog has infinitely many fix points if f is transcendental and meromorphic and if g is transcendental and entire using some ideas from 6 we shall prove that this is indeed the case iffsatisfies a certain condition to be discussed afterward

normal families new perspectives this paper surveys some surprising applications of a lemma characterizing normal families of meromorphic functions on plane domains these include short and efficient proofs of generalizations of i the picard theorems ii gol dberg s theorem a meromorphic function on c which is the solution of a first order algebraic differential equation has finite order and iii the fatou julia theorem the julia set of a rational function of degree d 2 is the closure of the repelling periodic points we also discuss bloch s principle and provide simple solutions to some problems of hayman connected with this principle over twenty years ago on the way to a partial explication of the phenomenon known as bloch s principle i proved a little lemma characterizing normal families of holomorphic and meromorphic functions on plane domains 68 over the years the lemma has grown and in dextrous hands proved amazingly versatile with applications to a wide variety of topics in function theory and related areas with the renewed interest in normal families arising largely from the important role they play in complex dynamics it seems sensible to survey some of the most striking of these applications to the one variable theory with the aim of making this technique available to as broad an audience as possible that is the purpose of this report one pleasant aspect of the theory is that judicious application of the lemma often leads to proofs which seem almost magical in their brevity in such cases we have made no effort to resist the temptation to write out complete proofs hardly anything beyond a basic knowledge of function theory is required to understand what follows so the reader is urged to take courage and plough on through and now we turn to our tale 1 let d be a domain in the complex plane c we shall be concerned with analytic maps i e meromorphic functions f d r2 received by the editors october 15 1997 and in revised form may 26 1998 1991 mathematics subject classification primary 30d45 secondary 30d35 34a20 58f23

baker campbell hausdorff relations for the orthosymplectic group osp 1 2 baker campbell hausdorff relations are presented for the connected supergroup associated with the orthosymplectic algebra osp 1 2

superspaces and supersymmetries a theory of graded banach modules over a banach grassmann algebra is developed and applied to differential geometry of super manifolds the explicit structure of superspaces carrying poincare supersymmetry and extended supersymmetry including central charges is described

analysis on generalized superspaces the analysis over commutative algebras generalized supercommutative algebras that is differentiation and integration for functions defined on superspace over a commutative algebra is studied

on factorization of certain entire functions abstract let p and q be polynomials and let be an entire function suppose that q and are nonconstant we show that the function h z p z e z q z has a factorization h z g z with nonlinear meromorphic left and right factors and g if and only if p q and have such a factorization with a common right factor this confirms conjectures of f gross g d song and c c yang

variational calculus on supermanifolds and invariance properties of superspace field theories the foundations of a variational calculus on fibered supermanifolds are outlined giving applications to the formulation of superspace field theories utiyama theorems and conservation laws related to local gauge and general invariance of the theory are proved

analysis on the p adic superspace i generalized functions gaussian distribution the theory of generalized functions on the p adic superspace qn mp over the super commutative banach superalgebra 0 1 with the trivial 1 annihilator is proposed the p adic gaussian supermeasure is defined as a generalized function on the p adic superspace the connection between the p adic gaussian integration and the p adic gamma function of morita is considered

differential equations frobenius theorem and local flows on supermanifolds the classical frobenius theorem both in its local and global formulations is generalised to superanalytic supermanifolds as an application it is proved that a coset space g h where both g and h are super lie groups is a supermanifold existence and uniqueness of local flows of tangent vector fields is proved

generalized complex superspace involutions of superfields distinguished involutions called transposition and conjugation in the algebra of complex supernumbers are introduced on the base of these involutions an analysis is developed over complex superspaces consistency of integration under supertransformations between two complex superspaces is shown

stability of semilinear stochastic evolution equations stability of moments of the mild solution of a semilinear stochastic evolution equation is studied and sufficient conditions are given for the exponential stability of the pth moment in terms of liapunov function sufficient conditions for sample continuity of the solution are also obtained and the exponential stability of sample paths is proved three examples are given to illustrate the theory

feedback stabilization of some nonlinear stochastic systems on hilbert space has been considered by ichikawa lo where a generates a strongly continuous semigroup t t t 2 0 on a real separable hilbert space x f and g are nonlinear and satisfy lipschitz condition together with f 0 0 and g 0 0 that is f and g are bounded perturbations but in many practical situations f and g might be bounded or unbounded even uncertain that is f g e 6 where 6 is a family of nonlinear unbounded operators the problem of stabilization is to design a state feedback control law that assures exponential stability of the zero state uniformly with respect to perturbations f g e 6 this stabilization problem for perturbed uncertain linear systems in finite dimensions has received considerable attention in current literature 3 13 and references thereof we have considered similar problems for linear evolution equations in infinite dimensional spaces in our earlier work 14 151 in this paper we show that our approach can be extended to both deterministic and stochastic semilinear evolution equations based on perturbation theory of semigroups and some results of ichikawa furthermore in addition to exponential stability in the mean square sense for stochastic systems stability of sample paths is also obtained to our knowledge this problem involving infinite dimensional systems with unbounded and nonlinear perturbations has not been considered in the literature before

characteristic functions and invariants of supermatrices the characteristic functions and the invariants of supermatrices are studied it is shown that the euclidean algorithm is useful in obtaining a system of invariants

on the fixpoints of composite entire functions of finite order suppose that f and g are transcendental entire functions such that the composition f f g has finite order and suppose that q is a nonconstant rational function we show that n r 1 f q o t r f the theorem is related to results of bergweiler goldstein and others

invariant supersymmetric multilinear forms and the casimir elements of p type lie superalgebras the casimir elements of the p type lie superalgebras are investigated depending on the class of algebras under consideration either there do not exist any nontrivial casimir elements at all or else the casimir elements are highly degenerate basic to the investigation is a lemma about invariant supersymmetric multilinear forms on a finite dimensional module over a lie superalgebra some comments on the cartan subalgebras of a lie superalgebra are also included an appendix provides some information on multilinear algebra with e commuting scalars

on the fix points of composite transcendental entire functions a transcendental entire function f z is called composite iff f z f g z for some nonlinear entire functionsf and g a point z0 in the complex plane is called a fix point of f iff f z zo gross 3 conjectured that every composite transcendental entire function must have infinitely many fix points it has been shown thus far that the conjecture is valid for finite order entire functions by several authors e g z 4 71 via different approaches in a previous paper 111 we investigated the fixpoints of a certain type of infinite order composite entire function there it is required that g be an entire function of positive finite order with a finite nevanlinna exceptional value and f be an arbitrary entire function such that the hyperorder off g is less than the order of g in this note we shall treat some other class of composite entire functions the tools are as usual based on nevanlinna s theory of meromorphic functions and an elegant result due to steinmetz s corollary i our result is the following theorem

composite meromorphic functions and growth of the spherical derivative abstract let f and h be transcendental meromorphic and g a transcendental entire function it is shown that if h grows slower than g in a suitable sense then there exists an unbounded sequence zn such that f g zn h zn

composite bank laine functions and a question of rubel a bank laine function is an entire function e satisfying e z 1 at every zero of e we determine all bank laine functions of form e fog with f g entire further we prove that if h is a transcendental entire function of finite order then there exists a path tending to infinity on which h and all its derivatives tend to infinity thus establishing for finite order a conjecture of rubel

a new class of stabilizing controllers for uncertain dynamical systems this paper is concerned with the problem of designing a stabilizing controller for a class of uncertain dynamical systems the vector of uncertain parameters q cdot is time varying and its values q t lie within a prespecified bounding set q in r p furthermore no statistical description of q cdot is assumed and the controller is shown to render the closed loop system practically stable in a so called guaranteed sense that is the desired stability properties are assured no matter what admissible uncertainty q cdot is realized within the perspective of previous research in this area this paper contains one salient feature the class of stabilizing controllers which we characterize is shown to include linear controllers when the nominal system happens to be linear and time invariant in contrast in much of the previous literature see for example 1 2 7 and 9 a linear system is stabilized via nonlinear control another feature of this paper is the fact that the

fixed points of composite meromorphic functions and normal families we show that there exists a function f meromorphic in the plane c such that the family of all functions g holomorphic in the unit disc d for which f g has no fixed point in d is not normal this answers a question of hinchliffe who had shown that this family is normal if f c does not consist of exactly one point in d we also investigate the normality of the family of all holomorphic functions g such that f g z h z for some non constant meromorphic function h

supergroups have no extra topology the global topology of supergroups is studied and it is found that they are equivalent as far as homotopy is concerned to the product of the underlying ordinary lie groups

linear canonical supertransformation on a fock superspace a linear canonical supertransformation is implemented by an inner product preserving operator which is necessarily not continuous

lie groups and lie algebras with generalized supersymmetric parameters matrices with symmetric parameters the most general extension of supersymmetric parameters are investigated the superdeterminants of such matrices are defined lie groups consisting of these matrices and their lie algebras are studied

color analysis theory of γ graded integrable evolution equations and super nijenhuis operators using the generalized grassmann variables or color variables a theory of graded integrable evolution equations is presented by elevating the treatments of magri gel fand dorfman and fuchssteiner of nonlinear integrable bosonic evolution equations to the graded case as an example it is shown that kupershmidt s super kdv is characterized by a z2 graded nijenhuis operator compatible with the underlying z2 graded hamiltonian structure

graded algebras and geometry based on yang baxter operators the graded algebras such as the e symmetric algebra generalized lie algebras and the concept of generalized lie cartan pairs are given in terms of graded yang baxter operators