braided vector fields on a quantum hyperboloid via the quantum group uq sl 2 we introduce a new method of defining a space of braided vector fields on a quantum hyperboloid our method is based on the lyubashenko sudbery construction see lyubashenko and sudbery 1998 j math phys 39 3487 and references therein and it consists in realizing its generators braided analogs of hyperbolic infinitesimal rotations via those of the quantum group uq sl 2 our main result consists in showing that the space of braided vector fields is a projective module over the coordinate algebra of the quantum hyperboloid

quantum spinors and spin groups from quantum clifford algebras a general construction of multiparametric quantum spinors and corresponding quantum groups associated to pseudo euclidean spaces is presented and their homomorphism to the respective groups is discussed this construction is based on a quantum clifford algebra and is described in detail for involutive pure twists intertwining braids for general braid operators that admit abstract volume elements a procedure is also given for deriving quantum analogues of these groups

q epsilon tensor for quantum and braided spaces the machinery of braided geometry introduced previously is used now to construct the e totally antisymmetric tensor on a general braided vector space determined by r matrices this includes natural q euclidean and q minkowski spaces the formalism is completely covariant under the corresponding quantum group such as soq 4 or soq 1 3 the hodge operator and differentials are also constructed in this approach

relativistic and newtonian κ space times the deformations of the galilei algebra and their associated noncommutative newtonian space times are investigated this is done by analyzing the possible nonrelativistic limits of an eleven generator pseudo extended poincare algebra p and their implications for the existence of a first order differential calculus the additional one form needed to achieve a consistent calculus on minkowski space is shown to be related to the additional central generator entering in the p hopf algebra in the process deformations of the extended galilei and galilei algebras are introduced which have respectively a cocycle and a bicrossproduct structure

wave equations on q minkowski space we give a systematic account of a component approach to the algebra of forms onq minkowski space introducing the corresponding exterior derivative hodge star operator coderivative laplace beltrami operator and lie derivative using this braided differential geometry we then give a detailed exposition of theq d alembert andq maxwell equation and discuss some of their non trivial properties such as for instance plane wave solutions for theq maxwell field we also give aq spinor analysis of theq field strength tensor

quantum deformation of lorentz group a one parameter quantum deformations l 2 ofsl 2 is introduced and investigated an analog of the iwasawa decomposition is proved the compact part of this decomposition coincides withs u 2 whereas the solvable part is identified as a pontryagin dual ofs u 2 it shows thats l 2 is the result of the dual version of drinfeld s double group construction applied tos u 2 the same construction applied to any compact quantum groupgc is discussed in detail in particular the explicit formulae for the haar measures on the pontryagin dualgd ofgc and on the double groupg are given we show that there exists remarkable 1 1 correspondence between representations ofg and bicovariant bimodules tensor bundles overgc the theory of smooth representations ofs l 2 is the same as that ofsl 2 clebsh gordon coefficients are however modified the corresponding tame bicovariant bimodules ons u 2 are classified an application to 4d differential calculus is presented the nonsmooth case is also discussed

conservation laws for linear equations on quantum minkowski spaces the general linear equations with constant coefficients on quantum minkowski spaces are considered and the explicit formulae for their conserved currents are given the proposed procedure can be simplified for invariant equations the derived method is then applied to klein gordon dirac and wave equations on different classes of minkowski spaces in the examples also symmetry operators for these equations are obtained they include quantum deformations of classical symmetry operators as well as an additional operator connected with deformation of the leibnitz rule in non commutative differential calculus

noncommutative fields and actions of twisted poincare algebra within the context of the twisted poincare algebra there exists no noncommutative analog of the minkowski space interpreted as the homogeneous space of the poincare group quotiented by the lorentz group the usual definition of commutative classical fields as sections of associated vector bundles on the homogeneous space does not generalize to the noncommutative setting and the twisted poincare algebra does not act on noncommutative fields in a canonical way we make a tentative proposal for the definition of noncommutative classical fields of any spin over the moyal space which has the desired representation theoretical properties we also suggest a way to search for noncommutative minkowski spaces suitable for studying noncommutative field theory with deformed poincare symmetries

on a distributed control problem arising in dynamic optimization of a fixed size population the practical motivation for this paper is provided by the recruitment problem faced by many organizations of fixed size to keep the average age young and thus keep innovation and productivity high while at the same time keeping levels of recruitment high a typical example is an academy of sciences the problem is formalized by an infinite horizon optimal control model for a first order pde with nonlocal dynamics a mckendrick type equation based on the nonstandard necessary optimality condition proved in the paper the following results are established i stationarity of the optimal recruitment density ii strong ergodicity of the optimal solution iii principle of bipolar recruitment in the case where the productivity of the organization is measured by the average age of the members the analysis involves a new type of transversality condition for the costate system the stability with respect to perturbations of the optimal solution of a noncoercive bang bang type problem boundedness and stability of the solution of a specific volterra integral equation of the second kind

quantum instantons with classical moduli spaces we introduce a quantum minkowski space time based on the quantum group su 2 q extended by a degree operator and formulate a quantum version of the anti self dual yang mills equation we construct solutions of the quantum equations using the classical adhm linear data and conjecture that up to gauge transformations our construction yields all the solutions we also find a deformation of penrose s twistor diagram giving a correspondence between the quantum minkowski space time and the classical projective space p 3

the classical n body problem within a generalized statistical mechanics the modifications that must be introduced within the framework of standard classical n body statistical mechanics techniques in order to deal with the recently proposed generalized scenario of non extensive statistical mechanics are discussed a generalized version of the equipartition theorem is shown to hold the ideal gas is revisited as a simple application

which deformations of the poincare group we analyse the present status of poincare group in considering it as a fundamental object independent of the minkowski space we examine the observables associated with it after having introduced the notion of kinematical observable we derive the class of deformed poincare groups which are compatible with physics

bifurcation diagram and stability for a one parameter family of planar vector fields we consider the 1 parameter family of planar quintic systems x y3 x3 y x my5 introduced by a bacciotti in 1985 it is known that it has at most one limit cycle and that it can exist only when the parameter m is in 0 36 0 6 in this paper using the bendixon dulac theorem we give a new unified proof of all the previous results we shrink this to 0 547 0 6 and we prove the hyperbolicity of the limit cycle we also consider the question of the existence of polycycles the main interest and difficulty for studying this family is that it is not a semi complete family of rotated vector fields when the system has a limit cycle we also determine explicit lower bounds of the basin of attraction of the origin finally we answer an open question about the change of stability of the origin for an extension of the above systems

free braided differential calculus braided binomial theorem and the braided exponential map braided differential operators i are obtained by differentiating the addition law on the braided covector spaces introduced previously such as the braided addition law on the quantum plane these are affiliated to a yang baxter matrix r the quantum eigenfunctions expr x v of the i braided plane waves are introduced in the free case where the position components xi are totally noncommuting a braided r binomial theorem and a braided taylor theorem expr a f x f a x are proven these various results precisely generalize to a generic r matrix and hence to n dimensions the well known properties of the usual one dimensional q differential and q exponential as a related application it is shown that the q heisenberg algebra px qxp 1 is a braided semidirect product c x c p of the braided line acting on itself a braided weyl algebra and similarly for its generalization to an arbitrary r matrix

q lorentz group and braided coaddition on q minkowski space we present a new version ofq minkowski space which has both a coaddition law and anslq 2 decomposition the additive structure forms a braided group rather than a quantum one in the process we obtain aq lorentz group which coacts covariantly on thisq minkowski space

on the deformability of heisenberg algebras based on the vanishing of the second hochschild cohomology group of the weyl algebra it is shown that differential algebras coming from quantum groups do not provide a non trivial deformation of quantum mechanics for the case of aq oscillator there exists a deforming map to the classical algebra it is shown that the differential calculus on quantum planes with involution i e if one works in position momentum realization can be mapped on aq difference calculus on a commutative real space although this calculus leads to an interesting discretization it is proved that it can be realized by generators of the undeformed algebra and does not possess a proper group of global transformations

braided affine geometry and q analogs of wave operators the main goal of this review is to compare different approaches to constructing the geometry associated with a hecke type braiding in particular with that related to the quantum group uq sl n we place emphasis on the affine braided geometry related to the so called reflection equation algebra rea all objects of such a type of geometry are defined in the spirit of affine algebraic geometry via polynomial relations on generators we begin by comparing the poisson counterparts of quantum varieties and describe different approaches to their quantization also we exhibit two approaches to introducing q analogs of vector bundles and defining the chern connes index for them on quantum spheres in accordance with the serre swan approach the q vector bundles are treated as finitely generated projective modules over the corresponding quantum algebras besides we describe the basic properties of the rea used in this construction and compare different ways of defining q analogs of partial derivatives and differentials on the rea and algebras close to them in particular we present a way of introducing a q differential calculus via koszul type complexes the elements of the q calculus are applied to defining q analogs of some relativistic wave operators

spin in the q deformed poincare algebra we investigate spin as algebraic structure within the q deformed poincare algebra proceeding in the same manner as in the undeformed case the q pauli lubanski vector the q spin casimir and the q little algebras for the massless and the massive case are constructed explicitly

on stability of bang bang type controls from the theory of nonlinear optimal control problems it is known that the solution stability w r t data perturbations and conditions for strict local optimality are closely related facts for important classes of control problems sufficient optimality conditions can be formulated as a combination of the independence of active constraints gradients and certain coercivity criteria in the case of discontinuous controls however common pointwise coercivity approaches may fail in the paper we consider sufficient optimality conditions for strong local minimizers which make use of an integrated hamilton jacobi inequality in the case of linear system dynamics we show that the solution stability including the switching points localization is ensured under relatively mild regularity assumptions on the switching function zeros for the objective functional local quadratic growth estimates in l1 sense are provided an example illustrates stability as well as instability effects in case the regularity condition is violated

quantum group covariant noncommutative geometry an algebraic formulation of the quantum group covariant noncommutative geometry in the framework of the r matrix approach to the theory of quantum groups is given structure groups are considered taking values in the quantum groups the notion of noncommutative connections and curvatures is introduced transforming as comodules under the local coaction of the structure group that is an exterior extension of glq n these noncommutative connections and curvatures generate glq n covariant quantum algebras for these algebras combinations of generators are found which are coinvariants under the local coaction of the exterior extension of the quantum group and one can formally consider these coinvariants as noncommutative images of the lagrangians for the topological chern simons models non abelian gauge theories and the einstein gravity also an explicit realization of such covariant quantum algebras is presented via the investigation of the coset construction glq n 1 glq n glq 1

separation of noncommutative differential calculus on quantum minkowski space noncommutative differential calculus on quantum minkowski space is not separated with respect to the standard generators in the sense that partial derivatives of functions of a single generator can depend on all other generators it is shown that this problem can be overcome by a separation of variables we study the action of the universal l matrix appearing in the coproduct of partial derivatives on generators powers of the resulting quantum minkowski algebra valued matrices are calculated this leads to a nonlinear coordinate transformation which essentially separates the calculus a compact formula for general derivatives is obtained in form of a chain rule with partial jackson derivatives it is applied to the massive quantum klein gordon equation by reducing it to an ordinary q difference equation the rest state solution can be expressed in terms of a product of q exponential functions in the separated variables

the quantized universal enveloping algebras uq iso n uq e 3 1 and uq e n and the representation theory of uq e 3 we construct the hopf algebras of uq iso n and uq iso 3 1 as regular functionals on their function algebras examine the possible involutions on these objects derive the quantized euclidean algebras uq e n and finally give the irreducible representations of uq e 3 depending on two parameters which fix the value of the casimir operators

on the structure of inhomogeneous quantum groups we investigate inhomogeneous quantum groups g built from a quantum group h and translations the corresponding commutation relations contain inhomogeneous terms under certain conditions which are satisfied in our study of quantum poincare groups 12 we prove that our construction has correct size find the r matrices and the analogues of minkowski space for g

on representations of the q deformed lorentz and poincare algebras we construct explicitly all finite dimensional representations of the quantum lorentz group slq 2 c based on this we prove that the q deformed lorentz algebra which was recently introduced can be considered as a hopf algebra dual to slq 2 c the generators of the q deformed lorentz algebra act via a differential representation on a hilbert space built of corepresentation spaces of the quantum lorentz group a chiral decomposition of the q deformed lorentz algebra is proposed it is shown how spinor bases for the q deformed poincare algebra can be constructed

d 3 real quantum conformal algebra using the equivalence of o 3 2 with d 3 conformal algebras we describe the d 3 quantum conformal algebra by the q deformed cartan weyl basis of a real hopf algebra uq o 3 2 with mod q mod 1 the universal r matrix written in terms of d 3 conformal generators is investigated the hopf subalgebras of uq o 3 2 and their quasitriangularity are discussed the analogies with d 4 quantum conformal algebra are pointed out

quantum poincare group without dilatation and twisted classical algebra a quantum deformation of a poincare group was constructed using as a starting point su 2 2 conformal group and twistorlike definition of the minkowski space this leads to a pure twisted quantum poincare group the relative simplicity of the q deformation permits one to find out the important properties of the q groups of the space time transformations in particular it is shown that the geometrical meaning of the twisting is the transition to the another space time coordinates frame so that one can freely choose the most convenient deformation among the ones related to each other by twists the small subgroup used in the induced representation construction is proven to be twisted with respect to initial q group this last fact means that commutation relations for creation and destruction operators are not strongly defined by the q group of the space time transformations

solutions of klein gordon and dirac equations on quantum minkowski spaces covariant differential calculi and exterior algebras on quantum homogeneous spaces endowed with the action of inhomogeneous quantum groups are classified in the case of quantum minkowski spaces they have the same dimensions as in the classical case formal solutions of the corresponding klein gordon and dirac equations are found the fock space construction is sketched

geometrical tools for quantum euclidean spaces we apply one of the formalisms of noncommutative geometry to n q the quantum space covariant under the quantum group so q n over n q there are two so q n covariant differential calculi for each we find a frame a metric and two torsion free covariant derivatives which are metric compatible up to a conformal factor and which have a vanishing linear curvature this generalizes results found in a previous article for the case of 3 q as in the case n 3 one has to slightly enlarge the algebra n q for n odd one needs only one new generator whereas for n even one needs two as in the particular case n 3 there is a conformal ambiguity in the natural metrics on the differential calculi over n q while in our previous article the frame was found by hand here we disclose the crucial role of the quantum group covariance and exploit it in the construction as an intermediate step we find a homomorphism from the cross product of n q with u q so n into n q an interesting result in itself

generalized singular point quantity and integrability of degenerate resonant singular point in this paper integrability and generalized complex resonant center condition of degenerate resonant singular point for a class of complex polynomial differential system were studied the concept of generalized singular point quantity of degenerate resonant singular point was proposed and the construction of that was studied two methods of computing generalized singular point quantities were given furthermore the sufficient and necessary condition of integrability of degenerate resonant singular point was discussed for the first time

on the classification of quantum poincare groups using the general theory of 10 quantum poincare groups without dilatations are described and investigated the description contains a set of numerical parameters which satisfy certain polynomial equations for most cases we solve them and give the classification of quantum poincare groups each of them corresponds to exactly one quantum minkowski space the poincare series of these objects are the same as in the classical case we also classify possibler matrices for the fundamental representation of the group

braided momentum in the q poincare group the q poincare group of m schlieker et al z phys c 53 79 1992 is shown to have the structure of a semidirect product and coproduct b soq 1 3 where b is a braided quantum group structure on the q minkowski space of four momentum with braided coproduct p p 1 1 p here the necessary b is not a usual kind of quantum group but one with braid statistics similar braided vectors and covectors v r v r exist for a general r matrix the abstract structure of the q lorentz group is also studied

deformed minkowski spaces classification and properties using general but simple covariance arguments we classify the quantum minkowski spaces for dimensionless deformation parameters this requires a previous analysis of the associated lorentz groups which reproduces a previous classification by woronowicz and zakrzewski as a consequence of the unified analysis presented we give the commutation properties the deformed and central length element and the metric tensor for the different spacetime algebras

sufficient second order conditions for bang bang control problems we provide sufficient optimality conditions for optimal control problems with bang bang controls building on a structural assumption on the adjoint state we additionally need a weak second order condition this second order condition is formulated with functions from an extended critical cone and it is equivalent to a formulation posed on measures supported on the set where the adjoint state vanishes if our sufficient optimality condition is satisfied we obtain a local quadratic growth condition in l 1 omega

metric regularity and stability of optimal control problems for linear systems this paper studies stability properties of the solutions of optimal control problems for linear systems the analysis is based on an adapted concept of metric regularity the strong bi metric regularity which is introduced and investigated in the paper it allows one to give a more precise description of the effect of perturbations on the optimal solutions in terms of a holder type estimate and to investigate the robustness of this estimate the holder exponent depends on a natural number k which is known as the controllability index of the reference solution an inverse function theorem for strongly bi metrically regular mappings is obtained which is used in the case k 1 for proving stability of the solution of the considered optimal control problem under small nonlinear perturbations moreover a new stability result with respect to perturbations in the matrices of the system is proved in the general case k geq 1

regularization and implicit euler discretization of linear quadratic optimal control problems with bang bang solutions we analyze the implicit euler discretization for a class of convex linear quadratic optimal control problems with control appearing linearly constraints are defined by lower and upper bounds for the controls and the cost functional may depend on a regularization parameter without any structural assumption on the optimal control we prove convergence of order 1 w r t the mesh size for the discrete optimal values under the additional assumption that the optimal control is of bang bang type and the switching function satisfies a growth condition around their zeros we show that the solutions are calm functions of perturbation and regularization parameters by applying this result to the implicit euler discretization we improve existing error estimates for discretizations based on the explicit euler method numerical experiments confirm the theoretical findings and demonstrate the usefulness of implicit methods and regularization in case of bang bang controls

quantum deformation of the poincare supergroup and kappa deformed superspace the classical r matrix for the n 1 super poincare algebra given by lukierski et al 1993 is used to describe the graded poisson structure on the n 1 poincare supergroup the standard correspondence principle between the even odd poisson brackets and anti commutators leads to the consistent quantum deformation of the super poincare group with the deformation parameter q described by the fundamental mass parameter k k 1 in q the k deformation of n 1 superspace as dual to the k deformed supersymmetry algebra is discussed

bifurcation at the equator for a class of quintic polynomial differential system abstract the center problem and bifurcation of limit cycles for degenerate singular point and infinity are far from being solved in general in this paper we studied center conditions and bifurcation of limit cycles at the degenerate singular point and infinity the equator in a class of quintic polynomial differential system with two small parameters and eight normal parameters the method is based on two homeomorphic transformations of the infinity and degenerate singular point into linear singular point which allows us to compute the generalized lyapunov constants the singular point quantities for the origin and infinity the center conditions for the degenerate singular point and infinity are derived respectively the limit cycle configurations of 7 2 and 2 6 are obtained under simultaneous perturbation at the origin and infinity as far as we know it is the first time that the bifurcation of limit cycles from a degenerate singular point and infinity simultaneously is studied in planar differential system

algebraic q integration and fourier theory on quantum and braided spaces an algebraic theory of integration on quantum planes and other braided spaces is introduced in the one dimensional case a novel picture of the jackson q integral as indefinite integration on the braided group of functions in one variable x is obtained here x is treated with braid statistics q rather than the usual bosonic or grassmann ones it is shown that the definite integral x x can also be evaluated algebraically as multiples of the integral of a q gaussian with x remaining as a bosonic scaling variable associated with the q deformation further composing the algebraic integration with a representation then leads to ordinary numbers for the integral integration is also used to develop a full theory of q fourier transformation f the braided addition x x 1 1 x and braided antipode s is used to define a convolution product and prove a convolution theorem it is also proven that f2 s the analogous results are proven on any braided group including integration and fourier transformation on quan

the euclidean hopf algebra u q e n and its fundamental hilbert space representations the euclidean hopf algebra uq en dual of fun rnq soq 1 n is constructed by realizing it as a subalgebra of the differential algebra diff rnq on the quantum euclidean space rnq in fact the previous realization g fiore commun math phys 169 475 500 1995 of uq 1 so n is extended within diff rnq through the introduction of q derivatives as generators of q translations the fundamental hilbert space representations of uq en turn out to be of highest weight type and rather simple lattice regularized versions of the classical ones the vectors of a basis of the singlet i e zero spin irrep can be realized as normalizable functions on rnq going to distributions in the limit q 1

generalized q exponentials related to orthogonal quantum groups and fourier transformations of noncommutative spaces essential prerequisites for the study of q deformed physics are particle states in position and momentum representation in order to relate x and p space by fourier transformations the appropriate q exponential series related to orthogonal quantum symmetries is constructed it turns out to be a new q special function giving rise to q plane wave solutions that transform with a noncommutative phase under translations

q euclidean space and quantum wick rotation by twisting the quantum matrix algebra r21x1x2 x2x1r is studied and proposed for the standard 2 2 case as the coordinates of q deformed euclidean space the algebra in this simplest case is isomorphic to the usual quantum matrices mq 2 but in a form which is naturally covariant under the euclidean rotations suq 2 suq 2 a quantum wick rotation is introduced that twists this system precisely into the approach to q minkowski space based on braided matrices and their associated spinorial q lorentz group

realization of u q so n within the differential algebra on r q n we realize the hopf algebrauq 1 so n as an algebra of differential operators on the quantum euclidean spacer q n the generators are suitableq deformed analogs of the angular momentum components on ordinaryr n the algebra fun r q n of functions onr q n splits into a direct sum of irreducible vector representations ofuq 1 so n the latter are explicitly constructed as highest weight representations

the inhomogeneous quantum groups from differential calculi with classical dimension from the bicovariant first order differential calculus on inhomogeneous hopf algebra b we construct the set of right invariant maurer cartan one forms considered as a right invariant basis of a bicovariant b bimodule over which we develop the woronowicz general theory of differential calculus on quantum groups in this formalism we introduce suitable functionals on b which control the inhomogeneous commutation rules in particular we find that the homogeneous part of commutation rules between the translations and those between the generators of the homogeneous part of b and translations are controlled by different r matrices satisfying characteristic equations

structures on braided spaces structures on quantum and braided spaces of the type defined via an r matrix are studied these include q minkowski and q euclidean spaces as additive braided groups the duality between the braided groups of vectors and covectors is proved and some first applications to braided geometry are made

bifurcation of a piecewise smooth cubic system via expansion of melnikov function abstract asymptotic expansion formulae of the first order melnikov function near the perturbed center of a cubic system are investigated studied through a suitable transformation by expanding the formulae we obtain that the given cubic system can have at least ten limit cycles

numerical approximations in optimal control of a class of heterogeneous systems the paper presents a numerical procedure for solving a class of optimal control problems for heterogeneous systems the latter are described by parameterized systems of ordinary differential equations coupled by integrals along the parameter space such problems arise in economics demography epidemiology management of biological resources etc the numerical procedure includes discretization and a gradient projection method for solving the resulting discrete problem a main point of the paper is the performed error analysis which is based on the property of metric regularity of the system of necessary optimality conditions associated with the considered problem

q deformed poincare algebra theq differential calculus for theq minkowski space is developed the algebra of theq derivatives with theq lorentz generators is found giving theq deformation of the poincare algebra the reality structure of theq poincare algebra is given the reality structure of theq differentials is also found the real laplacian is constructed finally the comultiplication counit and antipode for theq poincare algebra are obtained making it a hopf algebra

kappa deformation of poincare superalgebra with classical lorentz subalgebra and its graded bicrossproduct structure the kappa deformed d 4 poincare superalgebra written in hopf superalgebra form is transformed to the basis with classical lorentz subalgebra generators we show that in such a basis the kappa deformed d 4 poincare superalgebra can be written as a graded bicrossproduct we show that the kappa deformed d 4 superalgebra acts covariantly on a kappa deformed chiral superspace

quantum deformations of nonsemisimple algebras the example of d 4 inhomogeneous rotations a general class of deformations of the complexified d 4 poincare algebra o 3 1 c t4 c is considered with a classical undeformed subalgebra o 3 c t4 c and deformed relations preserving the o 3 c tensor structure we distinguish the class of quantum deformations the complex noncocommutative hopf algebras which depend on one complex mass parameter further we consider the real hopf algebras obtained by imposing the reality conditions for any choice of real metric o 4 o 3 1 or o 2 2 the parameter becomes real all e g minkowski as well as euclidean real quantum algebras with standard reality condition contain as nonlinearities the hyperbolic functions of the energy operator and can be interpreted as introducing an imaginary time lattice the symmetries of the models with real time lattice are described by a real quantum algebra with nonstandard reality conditions and trigonometric nonlinearities

the center problem for a family of systems of differential equations having a nilpotent singular point abstract we study the analytic system of differential equations in the plane x y t i 0 f q p 2 i s where p q n p q s n 1 p q 0 n n and f i p i q i t are quasi homogeneous vector fields of type t p q and degree i with f q p y 0 t and q q p 2 s 1 0 0 the origin of this system is a nilpotent and monodromic isolated singular point we prove for this system the existence of a lyapunov function and we solve theoretically the center problem for such system finally as an application of the theoretical procedure we characterize the centers of several subfamilies

quantum 2 1 kinematical algebras a global approach in this paper we give an approach to quantum deformations of the 2 1 kinematical lie algebras within a scheme that simultaneously describes all groups of motions of classical geometries in n 3 dimensions we cover at once all the kinematical geometries including the quantum versions of inonu wigner contractions which are defined in a natural way and relate q deformations as expected we thus obtain some q deformations previously known for the three dimensional euclidean and 2 1 poincare algebras and also some new q deformations for these and other kinematical algebras such as the 2 1 de sitter galilei and newton hooke algebras

q deformed relativistic wave equations based on the representation theory of the q deformed lorentz and poincare symmetries q deformed relativistic wave equation are constructed the most important cases of the dirac proca rarita schwinger and maxwell equations are treated explicitly the q deformed wave operators look structurally like the undeformed ones but they consist of the generators of a noncommutative minkowski space the existence of the q deformed wave equations together with previous results on the representation theory of the q deformed poincare symmetry solve the q deformed relativistic one particle problem

second order analysis for bang bang control problems of pdes in this paper we derive some sufficient second order optimality conditions for control problems of partial differential equations pdes when the cost functional does not involve the usual quadratic term for the control or higher nonlinearities for it though not always in this situation the optimal control is typically bang bang two different control problems are studied the second differs from the first in the presence of the l 1 norm of the control this term leads to optimal controls that are sparse and usually take only three different values we call them bang bang bang controls though the proofs are detailed in the case of a semilinear elliptic state equation the approach can be extended to parabolic control problems some hints are provided in the last section to extend the results

coadditive differential complexes on quantum groups and quantum spaces a regular way to define an additive coproduct or coaddition on q deformed differential complexes is proposed for quantum groups and quantum spaces related to the hecke type r matrices several examples of braided coadditive differential bialgebras hopf algebras are presented

local bifurcation of limit cycles and integrability of a class of nilpotent systems of differential equations we study the analytic system of differential equations in the plane which can be written in a suitable coordinates system as email protected email protected email protected i 0 f q p 2 i s where p email protected n p 0 email protected n and f i p i q i t are quasi homogeneous vector fields of type t p q and degree i with f q p y 0 t and q q p 2 s 1 0 0 the origin of this system is a nilpotent and monodromic isolated singular point we show the taylor expansion of the return map near the origin for this system which allow us to generate small amplitude limit cycles bifurcating from the critical point also as an application of the theoretical procedure we characterize the centers and we generate limit cycles of small amplitude from the origin of several families finally we give a new family integrable analytically which includes the centers of the systems studied

on the decoupling of the homogeneous and inhomogeneous parts in inhomogeneous quantum groups we show that if there exists a realization of a hopf algebra h in a h module algebra then one can split their cross product into the tensor product algebra of itself with a subalgebra isomorphic to h and commuting with this result applies in particular to the algebra underlying inhomogeneous quantum groups like the euclidean groups which are obtained as cross products of the quantum euclidean spaces qn with the quantum groups of rotation uq so n of qn for which it has no classical analogue

high order discrete approximations to mayer s problems for linear systems this paper presents a discretization scheme for mayer s type optimal control problems of linear systems the scheme is based on second order volterra fliess approximations and on an augmentation of the control variable in a control set of higher dimension compared with the existing results it has the advantage of providing a higher order accuracy which may make it more efficient when aiming for a certain precision error estimations depending on the controllability index of the system at the solution are proved by using a recent result about stability of the optimal solution with respect to disturbances numerical results are provided which show the sharpness of the error estimations

quantum d 4 poincare superalgebra the kappa deformation of d 4 poincare algebra is extended to the n 1 d 4 poincare superalgebra by the contraction of real hopf superalgebra uq osp 1 mod 4 q real we obtain real hopf algebra ukappa p4 1 p4 1 is written in the form which in the limit of vanishing fermionic generators supercharges reduces to the kappa poincare algebra ukappa p4 p4 describes d 4poincare algebra proposed by lukierski nowicki and ruegg 1991