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

multi hamiltonian structure for the finite defocusing ablowitz ladik equation we study the poisson structure associated to the defocusing ablowitz ladik equation from a functional analytical point of view by reexpressing the poisson bracket in terms of the associated caratheodory function using this expression we are able to introduce a family of compatible poisson brackets that form a multi hamiltonian structure for the ablowitz ladik equation furthermore we show using some of these new poisson brackets that the geronimus relations between orthogonal polynomials on the unit circle and those on the interval define an algebraic and symplectic mapping between the ablowitz ladik and toda hierarchies 2008 wiley periodicals inc

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

invariant solutions of the supersymmetric sine gordon equation a comprehensive symmetry analysis of the supersymmetric sine gordon equation is performed two different forms of the supersymmetric system are considered we begin by studying a system of partial differential equations corresponding to the coefficients of the various powers of the anticommuting independent variables next we consider the super sine gordon equation expressed in terms of a bosonic superfield involving anticommuting independent variables in each case a lie super algebra of symmetries is determined and a classification of all subgroups having generic orbits of codimension 1 in the space of independent variables is performed the method of symmetry reduction is systematically applied in order to derive invariant solutions of the supersymmetric model several types of algebraic hyperbolic and doubly periodic solutions are obtained in explicit form

poisson geometry of the analog of the miura maps and backlund darboux transformations for equations of toda type and periodic toda flows we show that the analog of the miura maps and backlund darboux transformations for a general class of equations of toda type and for a generalized class of periodic toda flows are isomorphisms of poisson lie groups

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

on schur flows for the finite schur dmkdv flows a non local poisson structure is introduced and shown to be linked via backlund darboux transformations to linear and quadratic poisson structures for the toda lattice two different lax representations for the schur flows are used one to construct backlund darboux transformations and the other to solve the cauchy problem via the trigonometric moment problem

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

a representation formula for maps on supermanifolds we analyze the notion of morphisms of rings of superfunctions which is the basic concept underlying the definition of supermanifolds as ringed spaces i e following berezin leites manin etc we establish a representation formula for all pull back morphisms from the algebra of functions on an ordinary manifolds to the superalgebra of functions on an open subset of a superspace we then derive two consequences of this result the first one is that we can integrate the data associated with a morphism in order to get a nonunique map defined on an ordinary space and uniqueness can be achieved by restriction to a scheme the second one is a simple and intuitive recipe to compute pull back images of a function on a manifold m by a map from a superspace to m

supermanifolds and automorphisms of super lie groups the geometrical theory of supermanifolds is developed and applied to the super lie groups the resulting structural equations and the adjoint representation are studied and used to find the automorphisms of super lie groups the n 1 supergravity symmetry group the so called graded poincare group is explicitly studied and possible applications are outlined

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

attitude synchronization of multiple spacecraft with cone avoidance constraints abstract in this paper we consider a team of spacecraft which requires changing its orientation to a common attitude using a decentralized control scheme under a connected communication topology while satisfying cone avoidance constraints due to blind celestial objects plume impingement and so on for this purpose we first combine consensus theory and optimization theory to develop a quaternion based attitude consensus protocol based on the communication graph at each time step each spacecraft generates a guidance command or reference attitude trajectory by synthesizing a series of laplacian like matrix p t using semidefinite programming sdp which involves linear matrix inequalities lmis it is analytically shown that this series of matrices p t is capable of collectively driving the initial attitudes to a common consensus attitude for satisfying cone avoidance constraints exclusion zones are then identified and expressed as lmis this identification of the exclusion zones gives rise to selecting safe waypoints from the reference attitude trajectory and then to passing through the selected waypoints while avoiding the exclusion zones via proper control inputs this solution procedure is demonstrated via numerical simulations of coordinated attitude rendezvous and attitude formation acquisition of multiple spacecraft with cone avoidance manoeuvres

global properties of supermanifolds we construct new examples of supermanifolds and determine the vector bundle structure of the supermanifolds commonly used in physics we show that any supermanifold admits a foliation whose leaves are locally tangent to the soul directions in the coordinate charts and which is one of a nested sequence of foliations we point out that the existence of these foliations implies restrictions on the possible topologies of supermanifolds for example a compact supermanifold with a single even dimension must have vanishing euler characteristic we also show that a globally defined superfield on a nice compact supermanifold must be constant along the leaves of the foliations by this mechanism the global topology of a supermanifold can be used to impose physically interesting constraints on superfields as an example we exhibit a supermanifold which has the local geometry of flat superspace but is such that all globally defined superfields are chiral

quadratically constrained attitude control via semidefinite programming we consider the attitude control problem augmented with an arbitrary number of nonconvex quadratic constraints by utilizing the implicit magnitude constraint of the state vector i e quaternion we are able to reduce this problem to a convex quadratically constrained quadratic program or a semidefinite program this result leads to an efficient solution strategy for general quadratically constrained attitude control problems an example demonstrates the proposed methodology

factorization of random jacobi operators and backlund transformations we show that a positive definite random jacobi operatorl over an abstract dynamical systemt x x can be factorized asl d2 whered is again a random jacobi operator but defined over a new dynamical systems y y which is an integral extension oft an isospectral random toda deformation ofl corresponds to an isospectral random volterra deformation ofd the factorization leads to commuting backlund transformations which can be written explicitly in terms of titchmarsh weyl functions in the periodic case the backlund transformations are time 1 maps of a toda flow with a time dependent hamiltonian

structure of supermanifolds and supersymmetry transformations after giving a global constraint free lagrangian formulation of then 1 superspace supergravity in terms of super fibre bundles and differential forms over a supermanifold we show that the concept of body manifold of a supermanifold provides a natural manner to reduce the theory to spacetime this reduction however is not canonical and the various ways in which it can be done give rise to transformations of the field variables which generalise the known invariances of then 1 spacetime supergravity under supersymmetry transformations and spacetime diffeomorphisms

grassmann analogs of classical matrix groups in general if the parameters of a real or complex algebraic matrix group are replaced by grassmann parameters without changing the algebraic constraints the resulting set fails to form a group it is shown how to remedy this defect of naive grassmannification by generalizing the constraint relations in particular it is shown how to define grassmann analogs of the orthogonal unitary and symplectic groups

existence and uniqueness for mean field equations on multiply connected domains at the critical parameter we consider the mean field equation begin aligned left begin array l quad l vardelta u rho frac e u int varomega e u 0 hbox in varomega u 0 hbox on partial varomega end array right end aligned 1 where varomega subset mathbb r 2 is an open and bounded domain of class c 1 in his 1992 paper suzuki proved that if varomega is a simply connected domain then eq 1 admits a unique solution for rho in 0 8 pi this result for varomega a simply connected domain has been extended to the case rho 8 pi by chang chen and the second author however the uniqueness result for varomega a multiply connected domain has remained a long standing open problem which we solve positively here for rho in 0 8 pi to obtain this result we need a new version of the classical bol s inequality suitable to be applied on multiply connected domains our second main concern is the existence of solutions for 1 when rho 8 pi we obtain a necessary and sufficient condition for the solvability of the mean field equation at rho 8 pi which is expressed in terms of the robin s function gamma for varomega for example if eq 1 has no solution at rho 8 pi then gamma has a unique nondegenerate maximum point as a by product of our results we solve the long standing open problem of the equivalence of canonical and microcanonical ensembles in the onsager s statistical description of two dimensional turbulence on multiply connected domains

graded gauge theory the mathematical background for a graded extension of gauge theories is investigated after discussing the general properties of graded lie algebras and what may serve as a model for a graded lie group the graded fiber bundle is constructed its basis manifold is supposed to be the so called superspace i e the product of the minkowskian space time with the grassmann algebra spanned by the anticommuting lorentz spinors the vertical subspaces tangent to the fibers are isomorphic with the graded extension of the su n lie algebra the connection and curvature are defined then on this bundle the two different gradings are either independent of each other or may be unified in one common grading which is equivalent to the choice of the spin statistics dependence the yang mills lagrangian is investigated in the simplified case the conformal symmetry breaking is discussed as well as some other physical consequences of the model

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

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

chern simons forms on principal superfiber bundles a graded weil homomorphism is defined for principal superfiber bundles and the related transgression or chern simons forms are introduced as an example of the application of these concepts a superextension of the dirac monopole is discussed

on the bihamiltonian structure of the supersymmetric kdv hierarchies a lie superalgebraic approach we give a lie superalgebraic interpretation of the bihamiltonian structure of known supersymmetric kdv equations we show that the loop algebra of a lie superalgebra carries a natural poisson pencil and we subsequently deduce the bihamiltonian structure of the supersymmetric kdv hierarchies by applying to loop superalgebras an appropriate reduction technique this construction can be regarded as a superextension of the drinfeld sokolov method for building a kdv type hierarchy from a simple lie algebra

a differential equation approach to the singular value decomposition of bidiagonal matrices we consider the problem of approximating the singular value decomposition of a bidiagonal matrix by a one parameter family of differentiable matrix flows it is shown that this approach can be fully expressed as an autonomous homogeneous and cubic dynamical system asymptotic behavior is justified by the established theory of the toda lattice

the haar integral for lie supergroups each supermanifold can be considered as equivalent to a certain family of real manifolds the haar integral of a lie supergroup is defined using this equivalence a useful general formula is derived together with explicit construction methods of the simple lie supergroups of types spl n m el osp n 2r el and b n el and the extended poincare group

non degeneracy and uniqueness of solutions to singular mean field equations on bounded domains abstract the aim of this paper is to complete the program initiated in 51 23 and then carried out by several authors concerning non degeneracy and uniqueness of solutions to mean field equations in particular we consider mean field equations with general singular data on non smooth domains the argument is based on the alexandrov bol inequality and on the eigenvalues analysis of linearized singular liouville type problems

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

superfiber bundle structure of gauge theories with faddeev popov fields in this work the mathematical structure of principal superfiber bundle ps is used to give a geometrical description of gauge theories the base space of ps is an s4 2 supermanifold xs four commuting and two anticommuting variables and the structure group an sn 0 supergroup gs where n is the dimension of the gauge group for the classical theory the body of ps is the usual principal fiber bundle p of gauge theories gauge and faddeev popov fields arise as superfields components of the connections in ps in a local coordinate system brs becchi rouet and stora and anti brs transformations are gauge transformations in ps of parameters the ghost and antighost superfields respectively in the case of soul flat connections which are connections in ps coming from connections in p the brs and anti brs transformations are finite translations along the anticommuting directions of xs and generate an s0 2 supergroup

on the backlund transformation for the gel fand dickey equations we study the backlund transformation of the gel fand dickey equations and in particular how the factorization ofnth order differential operators leads to lax type equations for first order operators generalizing work of adler and moser 1 in a similar fashion we study the toda equations

super twistor realization of unitary super representations of u p q n this paper is concerned with demonstrating that the twistor realization of the ladder representation of u p q can be generalized to super representations of u p q n this is achieved by considering the generalization of twistor elementary states to a super algebraic category unitarity of these super representations follows from the positive definiteness of a super twistor scalar product constructed in this paper although generalizations of the ladder representations have been well studied by other means it will be shown that the super twistor generalization is especially natural and merits special investigation

integration on supermanifolds and a generalized cartan calculus a suggestion by berezin for a method of integration on supermanifolds is given a precise differential geometric meaning by assuming that a supermanifold is the total space of a fibre bundle with connection the relevant objects for integration are identified as suitable horizontal vertical projections of hyperforms the latter are generalizations of differential forms having both covariant and contravariant indices the exterior calculus of these projected hyperforms is developed analogously to the cartan calculus by introducing appropriate derivations and determining their commutators respectively anticommutators

supersymmetric distributions hilbert spaces of supersymmetric functions and quantum fields the recently investigated hilbert krein and other positivity structures of the superspace are considered in the framework of superdistributions these tools are applied to problems raised by rigorous supersymmetric quantum field theory

on the existence of global integral forms on supermanifolds it is shown that the berezin approach to integration on supermanifolds can be applied to cases where the supermanifold is a twisted extension of a real manifold this is done by showing that supermanifolds admit a subatlas of coordinate charts with transition functions of a quite restricted kind

the axioms of supermanifolds and a new structure arising from them on presente une analyse des supervarietes sur un algebre arbitraire commutative graduee en partant d un ensemble d axiomes dont le premier est que les derivations de la structure faisceau d une supervariete sont localement libres

how different are the supermanifolds of rogers and dewitt a dewitt supermanifold always has the structure of a vector bundle over an ordinary spacetime manifold whereas a rogers supermanifold is not so restricted corresponding to the vector space fibers of the dewitt supermanifold a rogers supermanifold has a foliation by submanifolds or leaves parametrized by soul coordinates only we show that the universal covering space of any leaf always admits a flat metric if the covering space is complete in this metric it must in fact be a vector space we combine this result with known theorems about foliations to give conditions under which a compact rogers supermanifold with a single even dimension is necessarily a quotient space of flat superspace we also show that a supermanifold defined by a polynomial equation in flat superspace is always of the dewitt type finally we exhibit new supermanifold structures forr2 and the 2 torus which show that the foliation of a rogers supermanifold can be quite exotic

a formulation of supergravity based on cartan s connection the supergravity is formulated as a gauge theory with the yang mills lagrangian quadratic in curvature to arrive at such a description the super fiber bundle based on a superspace with the osp n 4 symmetry group is introduced the connection in this bundle becomes the cartan connection when osp n 4 is broken to sl 2 c so n in the reduced bundle this leads automatically to curvature and torsion explicit calculations of the lagrangian are carried out in terms of the cartan killing metrics and the curvature form for n 1 and n 8 the final theory describes the coupling of geometry to fermion fields these fields emerge as solder forms of the cartan connection

characteristic classes of super vector bundles a theory of characteristic classes for complex super vector bundles over supermanifolds is developed such characteristic classes are proved to fulfill the usual properties and it is shown that under suitable conditions they can be represented in terms of supersmooth curvature forms

consensus seeking in multiagent systems under dynamically changing interaction topologies this note considers the problem of information consensus among multiple agents in the presence of limited and unreliable information exchange with dynamically changing interaction topologies both discrete and continuous update schemes are proposed for information consensus this note shows that information consensus under dynamically changing interaction topologies can be achieved asymptotically if the union of the directed interaction graphs have a spanning tree frequently enough as the system evolves

superspace first order partial differential equations through the cartan kahler integration theorem in the paper first order partial differential equations are studied on superspace the key point is the use of a suitable generalization of the cartan kahler integration theorem on superspace by means of this result it is possible to investigate the characteristic fields and the structure of the solutions

factorization properties of birational mappings abstract we analyse birational mappings generated by transformations on q q matrices which correspond respectively to two kinds of transformations the matrix inversion and a permutation of the entries of the q q matrix remarkable factorization properties emerge for quite general involutive permutations it is shown that factorization properties do exist even for birational transformations associated with noninvolutive permutations of entries of q q matrices and even for more general transformation which are rational transformations but no longer birational the existence of factorization relations independent of q the size of the matrices is underlined the relations between the polynomial growth of the complexity of the iterations the existence of recursions in a single variable and the integrability of the mappings are sketched for the permutations yielding these properties all these results show that permutations of the entries of the matrix yielding factorization properties are not so rare in contrast the occurrence of recursions in a single variable or of the polynomial growth of the complexity are of course less frequent but not completely exceptional

nonlocal quadratic poisson algebras monodromy map and bogoyavlensky lattices a new lax representation for the bogoyavlensky lattice is found and its r matrix interpretation is elaborated the r matrix structure turns out to be related to a highly nonlocal quadratic poisson structure on a direct sum of associative algebras the theory of such nonlocal structures is developed and the poisson property of the monodromy map is worked out in the most general situation some problems concerning the duality of lax representations are raised

elementary toda orbits and integrable lattices we show that key features of several important integrable lattices appear naturally in a framework of the full toda flows using special symplectic leaves for these flows we construct a family of bi hamiltonian integrable lattices that interpolates between the nonrelativistic and relativistic toda lattices

supermanifolds and brs transformations we use a supermanifold formalism to support a geometrical interpretation of a recently proposed superfield formulation of brs becchi ronet stora and anti brs transformations

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

the attitude control problem a general framework for the analysis of the attitude tracking control problem for a rigid body is presented a large family of globally stable control laws is obtained by using the globally nonsingular unit quaternion representation in a lyapunov function candidate whose form is motivated by the consideration of the total energy of the rigid body the controllers share the common structure of a proportional derivative feedback plus some feedforward which can be zero the model independent case the coriolis torque compensation or an adaptive compensation these controller structures are compared in terms of the requirement on the a priori model information guaranteed transient performance and robustness the global stability of the luh walker paul robot end effector controller is also analyzed in this framework

on exact discretization of cubic quintic duffing oscillator application of the intersection theory to the construction of n point finite difference equations associated with classical integrable systems is discussed as an example we present a few exact discretizations of one dimensional cubic and quintic duffing oscillators sharing the form of the hamiltonian and canonical poisson bracket up to the integer scaling factor application of the intersection theory to the construction of n point finite difference equations associated with classical integrable systems is discussed as an example we present a few exact discretizations of one dimensional cubic and quintic duffing oscillators sharing the form of the hamiltonian and canonical poisson bracket up to the integer scaling factor

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

lattices supersymmetry and kahler fermions it is shown that a graded extension of the space group of a generalised simple cubic lattice exists in any space dimension d the fermionic variables which arise admit a kahlerian interpretation each graded space group is a subgroup of a graded extension of the appropriate euclidean group e d the relevance of this to the construction of lattice theories is discussed

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

variational calculus and poincare cartan formalism on supermanifolds in this work we show how the poincare cartan form can be used to describe the symmetries of lagrangian supersymmetric possibly p adic field theories we show that the existence of the poincare cartan form in supermanifold theory is ensured only in a relevant class of lagrangian densities moreover we give an abstract characterization of supersymmetric invariance based on the poincare cartan form

enlargeable graded lie algebras of supersymmetry a criterion to enlarge infinite dimensional lie algebras to analytic lie groups is used here for the extension of graded lie algebras of physical interest to super lie groups