mean square stability of second order runge kutta methods for stochastic differential equations in a previous paper we proposed the stochastic generalization of classical second order two stage explicit runge kutta rk methods the obtained stochastic schemes have second order in the weak sense in this paper the numerical stability of these rk schemes is studied the study focuses on stability with respect to the second moment ms stability figures of the stability domains of the numerical schemes are shown numerical examples that confirm the theoretical results are also presented

uv divergence free qft on noncommutative plane we formulate noncommutative qauntum field theory in terms of fields defined as mean value over coherent states of the noncommutative plane no product is needed in this formulation and noncommutativity is carried by a modified fourier transform of fields as a result the theory is uv finite and the cutoff is provided by the noncommutative parameter

persistence and extinction in stochastic delay logistic equation by incorporating ornstein uhlenbeck process abstract the persistence and extinction pe are interesting topics in mathematics this research analyzed pe of stochastic logistic equations sle by incorporating the ornstein uhlenbeck process slop and stochastic delay logistic equation sdle by incorporating the ornstein uhlenbeck process sldop firstly we proved that slop and sldop have positive solutions likewise for stochastic permanence sp weak persistence in the mean wpm non persistence in the mean npm and extinction the sufficient conditions are established for slop and sldop subsequently for numerical simulation we used 4 stage stochastic runge kutta srk4 to illustrate the effectiveness of the results

the composite euler method for stiff stochastic differential equations in this paper we present the composite euler method for the strong solution of stochastic differential equations driven by d dimensional wiener processes this method is a combination of the semi implicit euler method and the implicit euler method at each step either the semi implicit euler method or the implicit euler method is used in order to obtain better stability properties we give criteria for selecting the semi implicit euler method or the implicit euler method for the linear test equation the convergence properties of the composite euler method depend on the criteria for selecting the methods numerical results suggest that the convergence properties of the composite euler method applied to nonlinear sdes is the same as those applied to linear equations the stability properties of the composite euler method are shown to be far superior to those of the euler methods and numerical results show that the composite euler method is a very promising method

a statistical method of estimation and simulation for systems of stochastic differential equations summary this paper proposes a new statistical method of estimation and simulation for systems of stochastic differential equations by using this method we can easily derive a discretised process from a system of stochastic differential equations the discretised process is tractable enough to simulate the system and to construct the likelihood which is used to estimate the coefficients of the stochastic differential equations from discrete observation

step size control in the numerical solution of stochastic differential equations abstract we introduce a variable step size algorithm for the pathwise numerical approximation of solutions to stochastic ordinary differential equations the algorithm is based on a new pair of embedded explicit runge kutta methods of strong order 1 5 1 0 where the method of strong order 1 5 advances the numerical computation and the difference between approximations defined by the two methods is used for control of the local error we show that convergence of our method is preserved though the discretization times are not stopping times any more and further we present numerical results which demonstrate the effectiveness of the variable step size implementation compared to a fixed step size implementation

stabilizing and destabilizing heegaard splittings of sufficiently complicated 3 manifolds let m1 and m2 be compact orientable 3 manifolds with incompressible boundary and m the manifold obtained by gluing with a homeomorphism phi partial m 1 to partial m 2 we analyze the relationship between the sets of low genus heegaard splittings of m1 m2 and m assuming the map phi is sufficiently complicated this analysis yields counter examples to the stabilization conjecture a resolution of the higher genus analogue of a conjecture of gordon and a result about the uniqueness of expressions of heegaard splittings as amalgamations

asymptotically efficient runge kutta methods for a class of itoˆ and stratonovich equations in certain applications of stochastic differential equations approximate solutions must be found that depend only on samples of the driving process it is known that the order of convergence of such approximations is limited and that some are asymptotically efficient in the sense that they minimize the leading coefficient in the expansion of mean square errors as power series in the sample step size this article develops asymptotically efficient runge kt tta methods that involve evaluations either of the coefficients of an ito equation or of the coefficients of the corresponding stratonovich equation simpler approximations which converge with the maximum possible order but which are not asymptotically efficient are also defined the runge kutta methods for ito equations differ from those designed for ordinary differential equations in that they involve terms in the square root of the sample step size the approximations are tested along with the classical euler method on five examples the simulatio

asymptotic expansions and lie algebras for some nonlinear filtering problems state estimation problems for systems involving small parameters are treated by both analytical and lie algebraic approximation techniques an asymptotic expansion for the unnormalized conditional density corresponding to the case of observations of a gauss markov process through a weak polynomial nonlinearity is computed and a convergence result is derived the expansion is related to certain approximations of the associated estimation lie algebra the convergence result is based on arguments used recently to prove existence and uniqueness and to estimate the tail behavior of solutions to nonlinear filtering problems with unbounded coefficients these arguments are in turn adapted from the analytical theory of parabolic pde s simulation results are presented to further assess the performance of the resulting approximate filters

lagrangian quadrature schemes for computing weak solutions of quantum stochastic differential equations lagrangian quadrature schemes for computing weak solutions of lipschitzian quantum stochastic differential equations are introduced and studied this is accomplished within the framework of the hudson parthasarathy formulation of quantum stochastic calculus and subject to matrix elements of solution being sufficiently differentiable results concerning convergence of these schemes in the topology of the locally convex space of solution are presented numerical examples are given

a newton cotes method for quantum stochastic differential equations abstract we present an implicit simpson s rule for the numerical computation of quantum stochastic differential equations we derive the method present convergence results and a numerical example our method avoid the high cost of high order methods as well the cost of obtaining starting values for multi step methods

the three dimensional noncommutative gross neveu model this work is dedicated to the study of the noncommutative gross neveu model as is known in the canonical weyl moyal approach the model is inconsistent basically due to the separation of the amplitudes into planar and nonplanar parts we prove that if instead a coherent basis representation is used the model becomes renormalizable and free of the aforementioned difficulty we also show that although the coherent state procedure breaks lorentz symmetry in odd dimensions in the gross neveu model this breaking can be kept under control by assuming the noncommutativity parameters to be small enough we also make some remarks on ordering prescriptions used in the literature

the liouville arnold nekhoroshev theorem for non compact invariant manifolds under certain conditions generalized action angle coordinates can be introduced near non compact invariant manifolds of completely and partially integrable hamiltonian systems

numerical treatment of stochastic differential equations we define general runge kutta approximations for the solution of stochastic differential equations sde these approximations are proved to converge in quadratic mean to the solution of an sde with a corrected drift the explicit form of the correction term is given concerning the order of convergence we show that in general it is impossible for the quadratic mean of the one step error to be of an order greater than o h 3 this order is attained e g by the stochastic analogue of heun s method in the n dimensional case the highest order of convergence is in general only o h 2 attained by euler s method the order o h 3 can only be reached if nabla x sigma r sigma s nabla x sigma s sigma r for the diffusion matrix

noncommutative 3d harmonic oscillator we find transformation matrices allowing us to express a noncommutative three dimensional harmonic oscillator in terms of an isotropic commutative oscillator following the philosophy of simplicity approach noncommutative parameters have a physical interpretation in terms of an external magnetic field furthermore we show that for a particular choice of noncommutative parameters there is an equivalent anisotropic representation whose transformation matrices are far more complicated we indicate a way to obtain the more complex solutions from the simple ones

critical heegaard surfaces in this paper we introduce critical surfaces which are described via a 1 complex whose definition is reminiscent of the curve complex our main result is that if the minimal genus common stabilization of a pair of strongly irreducible heegaard splittings of a 3 manifold is not critical then the manifold contains an incompressible surface conversely we also show that if a non haken 3 manifold admits at most one heegaard splitting of each genus then it does not contain a critical heegaard surface in the final section we discuss how this work leads to a natural metric on the space of strongly irreducible heegaard splittings as well as many new and interesting open questions

feynman path integral on the non commutative plane we formulate the feynman path integral on a non commutative plane using coherent states the propagator for a free particle exhibits uv cutoff induced by the parameter of non commutativity

representations of the lie superalgebra gl 1 n in a gel fand zetlin basis and wigner quantum oscillators an explicit construction of all finite dimensional irreducible representations of the lie superalgebra in a gel fand zetlin basis is given particular attention is paid to the so called star type i representations unitary representations and to a simple class of representations v p with p any positive integer then the notion of wigner quantum oscillators wqos is recalled in these quantum oscillator models the unitary representations of are physical state spaces of the n particle d dimensional oscillator so far physical properties of wqos were described only in the so called fock spaces w p leading to interesting concepts such as non commutative coordinates and a discrete spatial structure here we describe physical properties of wqos for other unitary representations including certain representations v p of gl 1 dn these new solutions again have remarkable properties following from the spectrum of the hamiltonian and of the position momentum and angular momentum operators formulae are obtained that give the angular momentum content of all the representations v p of associated with the n particle three dimensional wqo for these representations v p we also consider in more detail the spectrum of the position operators and their squares leading to interesting consequences in particular a classical limit of these solutions is obtained that is in agreement with the correspondence principle

least area incompressible surfaces in 3 manifolds let m be a riemannian manifold and let f be a closed surface a map f f m is called least area if the area of f is less than the area of any homotopic map from f to m note that least area maps are always minimal surfaces but that in general minimal surfaces are not least area as they represent only local stationary points for the area function the existence of least area immersions in a homotopy class of maps has been established when the homotopy class satisfies certain injectivity conditions on the fundamental group 18 17 in this paper we shall consider the possible singularities of such immersions our results show that the general philosophy is that least area surfaces intersect least meaning that the intersections and self intersections of least area immersions are as small as their homotopy classes allow when measured correctly one should note that evidence supporting this view had been found by meeks yau in their embedding theorems for minimal disks and 2 spheres 13 143 our main result asserts that if a least area immersion is homotopic to an embedding then it has no self intersections which clearly exemplifies the above philosophy the precise result is the following

a high order discontinuous galerkin method for ito stochastic ordinary differential equations in this paper we develop a high order discontinuous galerkin dg method for strong solution of ito stochastic ordinary differential equations sdes driven by one dimensional wiener processes motivated by the dg method for deterministic ordinary differential equations odes we first construct an approximate deterministic ode with a random coefficient on each element using the well known wong zakai approximation theorem since the resulting ode converges to the solution of the corresponding stratonovich sde we apply a transformation to the drift term to obtain a deterministic ode which converges to the solution of the original sde the corrected equation is then discretized using the standard dg method for deterministic odes we prove that the proposed stochastic dg sdg method is equivalent to an implicit stochastic runge kutta method then we study the numerical stability of the sdg scheme applied to linear sdes with an additive noise term the method is shown to be numerically stable in the mean sense and also a stable as a result it is suitable for solving stiff sdes moreover the method is proved to be convergent in the mean square sense numerical evidence demonstrates that our proposed dg scheme for sdes with additive noise has a strong convergence order of 2 p 1 when p degree piecewise polynomials are used when applied to sdes with multiplicative noise the sdg method is strongly convergent with order p several linear and nonlinear test problems are presented to show the accuracy and effectiveness of the proposed method in particular we demonstrate that our proposed scheme is suitable for stiff stochastic differential systems