an example of a generalized brownian motion we present an example of a generalized brownian motion it is given by creation and annihilation operators on a twisted fock space ofl2 these operators fulfill for a fixed 1 1 the relationsc f c g c g c f f g 1 f g l2 we show that the distribution of these operators with respect to the vacuum expectation is a generalized gaussian distribution in the sense that all moments can be calculated from the second moments with the help of a combinatorial formula we also indicate that our brownian motion is one component of ann dimensional brownian motion which is invariant under the quantum groups u n of woronowicz with v2

stochastic integral representation theorem for quantum semimartingales abstract the quantum stochastic integral of ito type formulated by hudson and parthasarathy is extended to a wider class of adapted quantum stochastic processes on boson fock space an ito formula is established and a quantum stochastic integral representation theorem is proved for a class of unbounded semimartingales which includes polynomials and wick exponentials of the basic martingales in quantum stochastic calculus

a transformation system for deductive database modules with perfect model semantics abstract we present a transformation system for deductive database ddb modules we show that it preserves several data dependency properties of a ddb and is correct for the perfect model semantics of ddbs perfect models are not directly amenable to logical reasoning since logically equivalent ddbs may have different perfect models we develop an approach which involves using a condition on data dependencies in ddbs stratification compatibility to pass from a logical equivalence to equivalence under perfect model semantics this is readily applicable to the transformation system

linearly implicit lienard systems abstract the presence of nonlinearities in the capacitance and the inductance in van der pol type electrical circuits defines a linearly implicit or quasilinear counterpart of the classical lienard systems when the reactances remain positive the existence of a unique attracting periodic solution follows with minor modifications as in the classical setting novel results are obtained when the values of reactances may vanish at certain points of the state space these points yield singularities of the model and the existence of an attracting periodic solution can be characterized in terms of the behavior of certain smooth solutions crossing the singular manifold through so called i singularities

bifurcations and chaos of the bonhoeffer van der pol model periodic and chaotic behaviour of the bonhoeffer van der pol model of a nerve membrane driven by a periodic stimulating current a1 cos omega t is investigated results show that there exist ordinary and reversed period doubling cascades and a mode locking state at low driving amplitudes a1 there are period doubling and chaotic states but no impulse solutions when a1 is larger than a0 0 749 there are chaotic reversed period doubling and mode locking states and there also exist impulse trains a mode locking state with period 4 over a very large range of amplitudes is also found at a1 1 7059 the system goes back to a one period state

fermion ito s formula and stochastic evolutions an ito product formula is proved for stochastic integrals against fermion brownian motion and used to construct unitary processes satisfying stochastic differential equations as in the corresponding boson theory 10 11 these give rise to stochastic dilations of completely positive semigroups

relaxation oscillation and canard explosion abstract we give a geometric analysis of relaxation oscillations and canard cycles in singularly perturbed planar vector fields the transition from small hopf type cycles to large relaxation cycles which occurs in an exponentially thin parameter interval is described as a perturbation of a family of singular cycles the results are obtained by means of two blow up transformations combined with standard tools of dynamical systems theory the efficient use of various charts is emphasized the results are applied to the van der pol equation

a fock space representation for the quantum lorentz gas a fock space representation is given for the quantum lorentz gas i e for random schrodinger operators of the form h p2 v p2 x xj acting in h l2 rd with poisson distributed xjs an operator h is defined in k h p h l2 p d l2 p d h by the action of h on its fibers in a direct integral decomposition the stationarity of the poisson process allows a unitarily equivalent description in terms of a new family h k k rd where each h k acts in p a tip j math phys 35 113 1994 the space p is then unitarily mapped upon the symmetric fock space over l2 rd dx with the intensity of the poisson process the average number of points xj per unit volume the scatterer density and the equivalent of h k is determined averages now become vacuum expectation values and a further unitary transformation removing in dx is made which leaves the former invariant the resulting operator hf k has an interesting structure on the nth fock layer we encounter a single particle mo

bifurcation phenomena and attractive periodic solutions in the saturating inductor circuit in this paper we investigate bifurcation phenomena such as those of the periodic solutions for the unperturbed nonlinear system g x cbxz 0 with g x z ac x 2 1c x 2 and ao1 bo0 when we add the two competing termskf t cg x with f t a time periodic analytic forcing function and go0 the dissipative parameter the resulting differential equation g x xcbxcg xkf t z 0 describes approximately an electronic system known as the saturating inductor circuit for any periodic orbit of the unperturbed system we provide conditions which give rise to the appearance of subharmonic solutions furthermore we show that other bifurcation phenomena arise as there is a periodic solution with the same period as the forcing function f t which branches off the origin when the perturbation is switched on we also show that such a solution which encircles the origin attracts the entire phase space when the dissipative parameter becomes large enough we then compute numerically the basins of attraction of the attractive periodic solutions by choosing specific examples of the forcing function f t which are dictated by experience we provide evidence showing that all the dynamics of the saturating inductor circuit is organized by the persistent subharmonic solutions and by the periodic solution around the origin

a quantum nonadapted ito formula and stochastic analysis in fock scale abstract a generalized definition of quantum stochastic qs integrals and differentials is given in the free of adaptiveness and dimensionality form in terms of malliavin derivative on a projective fock space and their uniform continuity with respect to the inductive limite convergence is proved a new form of qs calculus based on an inductive algebraic structure in an indefinite space is developed and a nonadaptive generalization of the qs ito formula for its representation in fock space is derived 1 the problem of solution of general qs evolution equations in a hilbert space is solved in terms of the constructed operator representation of chronological products defined in the indefinite space and the isometry and homomorphism property respectively for operators and maps of these solutions corresponding to the peseudounitary and homomorphism property of the qs integrable generators is proved

annihilation derivative creation derivative and representation of quantum martingales on the basis of the quantum white noise theory we introduce the notion of creation and annihilation derivatives of fock space operators and study the differentiability of white noise operators we define the hitsuda skorohod quantum stochastic integrals by the adjoint actions of quantum stochastic gradients and show explicit formulas for their creation and annihilation derivatives as an application we derive direct formulas for the integrands in the quantum stochastic integral representation of a regular quantum martingale

cohomology of power sets with applications in quantum probability square integrable wiener functionals may be represented as sums of multiple ito integrals this leads to an identification of such functionals with square integrable functions on the symmetric measure space of the lebesgue spacer when the pointwise product of wiener functionals is thus carried over the product takes a pleasing form cf wick s theorem and various non commutative perturbations of this wiener product have been considered here we employ cohomological arguments to analyse deformations of an abstract wiener product this leads to the construction of levy fields which are neither bosonic nor fermionic and also gives rise to homotopies between quasi free boson and fermion fields finally we unify existence and uniqueness results for quantum stochastic differential equations by treating mixed noise differential equations

quantum q white noise and a q central limit theorem we establish a connection between the azema martingales and certain quantum stochastic processes with increments satisfyingq commutation relations this leads to a theory ofq white noise onq bialgebras and to a generalization of the fock space representation theorem for white noise on bialgebras in particular quantum azema noise q interpolations between fermion and boson quantum brownian motion and unitary evolutions withq independent multiplicative increments are studied it follows from our results that the azema martingales and theq interpolations are central limits of sums ofq independent identically distributed quantum random variables

gaussian hypercontractivity revisited abstract recent work on convolution type products on fock space allow an elementary treatment of hypercontractivity for second quantised contractions in the fermi case new results are obtained lending further support to a conjecture of l gross

quantum ito s formula and stochastic evolutions using only the boson canonical commutation relations and the riemann lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes this leads to a noncommutative ito product formula a realisation of the classical poisson process in fock space which gives a noncommutative central limit theorem the construction of solutions of certain noncommutative stochastic differential equations and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps the classical ito product formula for stochastic differentials with respect to brownian motion and the poisson process is a special case

a martingale characterization of canonical commutation and anticommutation relations using a martingale condition and some restrictions on moments up to fourth order the characterisation problem of boson fermion and classical brownian motions is studied from a unified point of view entirely within the framework of elementary operator theory global commutation and anticommutation rules turn out to be consequences of corresponding commutation and anticommutation rules between past and future observables

quantum stochastic calculus with maximal operator domains quantum stochastic calculus is extended in a new formulation in which its stochastic integrals achieve their natural and maximal domains operator adaptedness conditional expectations and stochastic integrals are all defined simply in terms of the orthogonal projections of the time filtration of fock space together with sections of the adapted gradient operator free from exponential vector domains our stochastic integrals may be satisfactorily composed yielding quantum ito formulas for operator products as sums of stochastic integrals the calculus has seen two reformulations since its discovery one closely related to classical ito calculus the other to noncausal stochastic analysis and malliavin calculus our theory extends both of these approaches and may be viewed as a synthesis of the two the main application given here is existence and uniqueness for the attal meyer equations for implicit definition of quantum stochastic integrals

quantum random walk for uq su 2 and a new example of quantum noise we derive the quantum version of the central limit theorem for sample sums of q independent q identically distributed quantum variables for q r 1 in particular we consider uq su 2 for which we obtain in the limit a state on the algebra generated by q commuting q oscillators indexed by pairs of real numbers r s where 0 r s giving a new example of non additive quantum noise

quantum and non causal stochastic calculus the quantum stochastic calculus initiated by hudson and parthasarathy and the non causal stochastic calculus originating with the papers of hitsuda and skorohod are two potent extensions of the ito calculus currently enjoying intensive development the former provides a quantum probabilistic extension of schrodinger s equation enabling the construction of a markov process for a quantum dynamical semigroup the latter allows the treatment of stochastic differential equations which involve terms which anticipate the future in this paper the close relationship between these theories is displayed and a noncausal quantum stochastic calculus already in demand from physics is described