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

parasupercoherent states and color supergroups parabosonic coherent states are illustrated from the displacement operator point of view such a construction requires the use of the green cusson ansatze a specific representation of parastatistics leading to color supergroups a comparison with a previous result is shown through the annihilation operator and the minimum uncertainty contexts the extension to the parasupersymmetric case is also considered

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

tensor products are projective geometries abstract given a finite dimensional vector space v we construct a family of projective geometries whose flats are certain subspaces of v and show that there is a one to one correspondence between this family of projective geometries and the set of equivalence classes of tensor decompositions of v this provides a practical method for finding a tensor decomposition of a finite dimensional kg module or proving that no nontrivial tensor decomposition exists

short presentations for three dimensional unitary groups abstract we give a presentation of length o log2 g for the groups g psu3 q this result has applications in recent algorithms to compute the structure of permutation groups and matrix groups

baker campbell hausdorff relations for supergroups one parameter subgroups the exponential mapping and canonical coordinates have previously been defined for connected supergroups in this paper a technique for obtaining analytical expressions linking different coordinate schemes for supergroups is presented these are the baker campbell hausdorff relations the method is illustrated in detail with the examples of supergroups based on the supersymmetric quantum mechanical superalgebra sqm 2 and on the inonu wigner contraction isop 1 2 of the simple superalgebra osp 1 2

invariance of the approximately reachable set under nonlinear perturbations this paper considers nonlinear perturbations of control systems with linear dynamics and seeks to analyze whether the approximately reachable set may be left unchanged by this perturbation under suitable conditions it is shown that this analysis may be reduced to the presumably simpler analysis of such invariance for a family of affine perturbations interest centers on the context of infinite dimensional state spaces so the system may for example correspond to a hyperbolic or parabolic partial differential equation

invariance of the reachable set under nonlinear perturbations for the abstract controlled volterra equation qquad x t bar x t int 0 t boldsymbol psi t s phi s x s bf b s v s ds on 0 t we consider the reachable set mathcal k phi x t text for some v in mathfrak b viewing phi cdot cdot as a nonlinear perturbation of an otherwise linear control problem conditions are obtained under which mathcal k phi mathcal k 0 for a suitable class mathfrak f of such nonlinear perturbations phi whenever the linear problem is known to have a reachable set invariant under affine perturbations phi cdot cdot g in mathfrak z the results generalize those obtained by naito 7 for control of the heat equation

admissible controllability of vibrating systems with constrained controls we study the control of a vibrating string when the controls f x t are applied only on a limited subset of the string and constrained by continuous functions alpha x beta x as alpha x leqq f x t leqq beta x we find that it is possible to transfer almost all states of vibration into rest by a constrained control the higher space dimensional vibrating systems are also considered

effective black box constructive recognition of classical groups abstract we describe a black box las vegas algorithm to construct standard generators for classical groups defined over finite fields we assume that the field has size at least 4 and that oracles to solve certain problems are available subject to these assumptions the algorithm runs in polynomial time a practical implementation of our algorithm is distributed with the computer algebra system magma

tensor products of q superalgebra representations and q series identities we work out examples of tensor products for distinct q generalizations of euclidean oscillator and type superalgebras in cases where the method of highest weight vectors will not apply in particular we use the three term recurrence relations for askey wilson polynomials to decompose the tensor product of representations from the positive discrete series and representations from the negative discrete series we show that various q analogues of the exponential function can be used to mimic the exponential mapping from a lie algebra to its lie group and we compute the corresponding matrix elements of the group operators on these representation spaces we show that the matrix elements themselves transform irreducibly under the action of the quantum superalgebra the most important q series identities derived here are interpreted as the expansion of the matrix elements of a group operator via the exponential mapping in a tensor product basis in terms of the matrix elements in a reduced basis they involve q hypergeometric series with base

on the theory of one parameter subgroups of supergroups the one parameter subgroups of supergroups are defined and their primary properties are delineated parametrization schemes analogous to those used in lie group theory are investigated in particular exponentiation is introduced and related to the one parameter subgroups canonical coordinates of three kinds are investigated the third of which has no analog in conventional lie theory we illustrate our results with the examples of the supersymmetric quantum mechanical superalgebra sqm 2 and the simple superalgebra osp 1 2

constructive recognition of sl3 q abstract we present a constructive recognition algorithm for groups of lie type sl 3 q this is a necessary component for constructive recognition algorithms of quasisimple groups of lie type

supersymmetric quantum mechanics defined as sesquilinear forms a formulation of supersymmetric quantum mechanics in terms of sesquilinear forms is presented otherwise occurring rather complicated domain questions are solved in a simple and natural way the authors formulate the axioms and describe a number of models supersymmetry transformations are implemented as superunitary transformations in a space defined over a grassmann algebra generalised bogoliubov transformations acting in that space are also formulated

constructive recognition of classical groups in even characteristic abstract let g x gl d f be a classical group in its natural representation defined over a finite field f of even characteristic we present las vegas algorithms to construct standard generators for g which permit us to write an element of g as a straight line program in x and to construct an involution as a straight line program in x if f 4 then the algorithms run in time polynomial in the size of the input subject to the existence of a discrete logarithm oracle for f

controllability of second order semilinear neutral functional differential systems in banach spaces abstract sufficient conditions for controllability of semilinear second order neutral functional differential systems in banach spaces are established using the theory of strongly continuous cosine families the results are obtained by using the leray schauder alternative

approximate controllability of semilinear functional equations in hilbert spaces in this paper approximate and complete controllability for semilinear functional differential systems is studied in hilbert spaces sufficient conditions are established for each of these types of controllability the results address the limitation that linear systems in infinite dimensional spaces with compact semigroup cannot be completely controllable the conditions are obtained by using the schauder fixed point theorem when the semigroup is compact and the banach fixed point theorem when the semigroup is not compact

fast constructive recognition of black box orthogonal groups abstract we present an algorithm that constructively recognises when a given black box group is a nontrivial homomorphic image of the orthogonal group e d q for known e d and q the algorithm runs in polynomial time assuming oracles for handling sl 2 q subgroups and discrete logarithms in f q

subgroups of exceptional groups of lie type generated by long root elements ii characteristic two let g be an exceptional group of lie type over a finite field f q p and assume the weyl rank of g is at least four let x be the collection of long root subgroups of g and r the collection of long root elements of g in 7 8 j we determined those subgroups l of g generated by long root subgroups with o l 1 this was done by studying the geometry of g acting on s with classification theorems only used in 7 to identify the possible isomorphism classes for l in part i 9 of this series we determined those subgroups l of g generated by long root elements with o l 1 where p 2 in that paper we used aschbacher s powerful result classifying the groups in chev p for p odd 21 in order to identify l in this paper we complete the determination in the case p 2 using the theorems of aschbacher 11 fischer 131 and timmesfeld 181 to identify l the determination of the subgroups of classical groups generated by elements central in a long root subgroup was carried out by kantor 151 in all characteristics using the aforementioned classification theorems for references on earlier attempts at these determinations for the classical groups see the introduction of 15 in another paper 111 we make use of our results to determine the minimal degree of a permutation representation of an exceptional group we also foresee these results having application to the more general problem of determining the maximal subgroups of an exceptional group

controllability of neutral functional integrodifferential systems in banach spaces abstract sufficient conditions for controllability of neutral functional integrodifferential systems in a banach space are established the results are obtained by using the schaefer fixed point theorem an example is provided to illustrate the theory

topological theory of non autonomous parabolic evolution inclusions on a noncompact interval and applications in this paper we consider the topological structure of the solution set of non autonomous parabolic evolution inclusions with time delay defined on non compact intervals the result restricted to compact intervals is then extended to non autonomous parabolic control problems with time delay moreover as the applications of the information about the structure we establish the existence result of global integral solutions for non autonomous cauchy problems subject to nonlocal condition and prove the invariance of a reachability set for non autonomous control problems under single valued nonlinear perturbations finally some illustrating examples are supplied

complete controllability of one dimensional vibrating systems with bang bang controls this paper deals with the problem of controllability for one dimensional vibrating systems with controls which are scalar valued functions of time only we give some constraint sets of controls and prove exact controllability of vibrations in those constraint sets by studying the moment problem attached to this control problem precisely using these results and lyapunov s convexity theorem for vector measures we obtain complete controllability only with bang bang controls

supergroups factorizations through matrix realization using a matrix realization generic elements of the supergroups u m n and osp m 2n are obtained through the exponentiation of the corresponding super lie algebra elements the emphasis is put on the contribution of the odd part the application to the factorization problem is given and the relationship with nonlinear differential superequations is clarified

short presentations for finite groups we conjecture that every finite groupghas a short presentation in terms of generators and relations in the sense that the totallengthof the relations is log g o 1 we show that it suffices to prove this conjecture for simple groups motivated by applications in computational complexity theory we conjecture that for finite simple groups such a short presentation is computable in polynomial time from the standard name ofg assuming in the case of lie type simple groups overgf pm that an irreducible polynomialfof degreemovergf p and a primitive root ofgf pm are given we verify this stronger conjecture for all finite simple groups except for the three families of rank 1 twisted groups we do not handle the unitary groupspsu 3 q 2a2 q the suzuki groupssz q 2b2 q and the ree groupsr q 2g2 q in particular all finite groups g without composition factors of these types have presentations of length o log g 3 for groups of lie type normal or twisted of rank 2 we use a reduced version of the curtis steinberg tits presentation

controllability of sobolev type integrodifferential systems in banach spaces sufficient conditions for controllability of sobolev type integrodifferential systems in banach spaces are established the results are obtained using compact semigroups and the schauder fixed point theorem as an example is provided to illustrate the results

constructive recognition of mathrm psl 2 q existing black box and other algorithms for explicitly recognising groups of lie type over gf q have asymptotic running times which are polynomial in q whereas the input size involves only log q this has represented a serious obstruction to the efficient recognition of such groups recently brooksbank and kantor devised new explicit recognition algorithms for classical groups these run in time that is polynomial in the size of the input given an oracle that recognises psl 2 q explicitly the present paper in conjunction with an earlier paper by the first two authors provides such an oracle the earlier paper produced an algorithm for explicitly recognising sl 2 q in its natural representation in polynomial time given a discrete logarithm oracle for gf q the algorithm presented here takes as input a generating set for a subgroup g of gl d f that is isomorphic modulo scalars to psl 2 q where f is a finite field of the same characteristic as gf q it returns the natural representation of g modulo scalars since a faithful projective representation of psl 2 q in cross characteristic or a faithful permutation representation of this group is necessarily of size that is polynomial in q rather than in log q elementary algorithms will recognise psl 2 q explicitly in polynomial time in these cases given a discrete logarithm oracle for gf q our algorithm thus provides the required polynomial time oracle for recognising psl 2 q explicitly in the remaining case namely for representations in the natural characteristic this leads to a partial solution of a question posed by babai and shalev if g is a matrix group in characteristic p determine in polynomial time whether or not op g is trivial

black box exceptional groups of lie type ii abstract if a black box group g is known to be isomorphic to an exceptional simple group g of lie type of twisted rank 1 other than any f 4 2 2 e or d 4 3 2 e over a field of known size q and if suitable sl 2 and discrete log oracles are available when q is odd then a polynomial time las vegas algorithm is given that produces a constructive isomorphism between g and g

some examples of compact supermanifolds with non abelian fundamental group a detailed construction is given of a supermanifold which is compact and has far from trivial topology higher dimensional examples are also described the grassmann algebra from which the supermanifolds are constructed may have finite or infinite dimension

controllability of semilinear control systems dominated by the linear part while various equivalent conditions for controllability have been obtained in the case of linear control systems controllability problems of semilinear control systems usually require some complicated and limited assumptions in this paper we show the approximate controllability of an abstract semilinear control system under the assumption which has a simple form and can be easily checked in many examples

baker campbell hausdorff relations for the super poincare group baker campbell hausdorff relations are obtained for the connected supergroup associated with the super poincare algebra iosp 1 4

recognition of finite exceptional groups of lie type let q be a prime power and let g be an absolutely irreducible subgroup of gld f where f is a finite field of the same characteristic as fq the field of q elements assume that g g q a quasisimple group of exceptional lie type over fq which is neither a suzuki nor a ree group we present a las vegas algorithm that constructs an isomorphism from g to the standard copy of g q if g 6 3d4 q with q even then the algorithm runs in polynomial time subject to the existence of a discrete log oracle

black box exceptional groups of lie type if a black box group is known to be isomorphic to an exceptional simple group of lie type of twisted rank 1 other than any f4 q over a field of known size a las vegas algorithm is given to produce a constructive isomorphism in view of its timing this algorithm yields an upgrade of all known nearly linear time monte carlo permutation group algorithms to las vegas algorithms when the input group has no composition factor isomorphic to any group f4 q or g2 q

groups of automorphisms of the canonical commutation and anticommutation relations observables of supersymmetric quantum mechanics are coded by taking the antisymmetric tensor product with anticommuting parameters next the authors define superunitary transformations which mix bosonic and fermionic degrees of freedom in order to construct automorphisms of the canonical anti commutation relations c a cr conversely every automorphism of the c a cr is implemented by an essentially unique superunitary transformation