on long time evolution in general relativity and geometrization of 3 manifolds this paper introduces relations between the long time asymptotic behavior of einstein vacuum space times and the geometrization of 3 manifolds envisioned by thurston the relations are obtained by analysing the asymptotic behavior of a cmc foliation by compact cauchy surfaces and the induced curve of 3 manifold geometries the cheeger gromov theory is introduced in this context and a number of open problems are considered from this viewpoint

global controllability for smooth nonlinear systems a geometric approach the paper consists of two parts in part i a global controllability theory is constructed local controllability criteria are integrated by means of the qualitative behavior of the dynamics involved leading to a global result the relation with global observability is discussed part ii is concerned with controllability by means of two vectorfields it is well known that each connected paracompact manifold of class c k 2 leqq k leqq infty or k omega carries a globally controllable set of two c k 1 vectorfields as an application of part i we construct control systems of the form dot x x uy defined on a compact manifold with the property that for x a morse smale vectorfield with no periodic orbits there exists a large class of vectorfields y such that the system is globally controllable by means of a bang bang two valued control it is stressed that the techniques applied imply the existence of a multitude of basically different globally controllable systems with two

lie brackets and local controllability a sufficient condition for scalar input systems we develop a general formalism based on exponential lie series which can be used to study control variations we apply the formalism to prove a result conjectured by h hermes that gives a sufficient condition for small time local controllability for single input systems

on the breakdown criterion in general relativity we give a geometric criterion for the breakdown of an einstein vacuum space time foliated by a constant mean curvature or maximal foli ation more precisely we show that the foliated space time can be extended as long as the the second fundamental form and the first derivatives of the logarithm of the lapse of the foliation remain uniformly bounded we make no restrictions on the size of the initial data

the detection and estimation of long memory in stochastic volatility abstract we propose a new time series representation of persistence in conditional variance called a long memory stochastic volatility lmsv model the lmsv model is constructed by incorporating an arfima process in a standard stochastic volatility scheme strongly consistent estimators of the parameters of the model are obtained by maximizing the spectral approximation to the gaussian likelihood the finite sample properties of the spectral likelihood estimator are analyzed by means of a monte carlo study an empirical example with a long time series of stock prices demonstrates the superiority of the lmsv model over existing short memory volatility models

arbitrage with fractional gaussian processes abstract while the arbitrage opportunity in the black scholes model driven by fractional brownian motion has a long history the arbitrage strategy in the black scholes model driven by general fractional gaussian processes is in its infancy the development of stochastic calculus with respect to fractional gaussian processes allowed us to study such models in this paper following the idea of shiryaev 1998 an arbitrage strategy is constructed for the black scholes model driven by fractional gaussian processes when the stochastic integral is interpreted in the riemann stieltjes sense arbitrage opportunities in some fractional gaussian processes including fractional brownian motion sub fractional brownian motion bi fractional brownian motion weighted fractional brownian motion and tempered fractional brownian motion are also investigated

controllability in codimension one abstract local and global controlability of analytic affine control systems with an arbitrary number of controls are studied assuming strong accessibility of and codimension one of the lie algebra t generated by the input vector fields i e dim t p n 1 at every p m controls are assumed to have no a priori bound from the study of the set h where a necessary condition for local controllability at a point is verified and assuming some transversality relations an easy to verify geometric condition is proved h is a submanifold and is locally controllable at every point of h outside a codimension one submanifold a geometric sufficient condition for global controllability on simply connected manifolds is then obtained if every leaf of t intersects the manifold h and some transversality relations including those involved in the local condition are verified is globally controllable

local controllability of scalar input systems on 3 manifolds abstract local controllability of generic c k pairs of vector fields in a three dimensional connected manifold m is studied showing that in general a necessary condition is also sufficient on an open dense subset of the set of points where it is verified the results obtained are used to determine local controllability for scalar input affine control systems

local controllability of nonlinear systems abstract the local controllability of control systems with an arbitrary number of controls is considered first on an open set and then at a given point necessary conditions are derived concerning the lie algebra t generated by the input vector fields and applied to gradient systems the main results are a geometric sufficient condition for local controllability at a point and an equivalent condition based on the computation of lie brackets assuming t to be n 1 dimensional

proof of the positive mass theorem ii the positive mass theorem states that for a nontrivial isolated physical system the total energy which includes contributions from both matter and gravitation is positive this assertion was demonstrated in our previous paper in the important case when the space time admits a maximal slice here this assumption is removed and the general theorem is demonstrated abstracts of the results of this paper appeared in 11 and 13

n dimensional controllability with n 1 controls let m be a connected real analytic n dimensional manifold f g 1 g n 1 be complete real analytic vector fields on m which are linearly independent at some point of m and u 1 u n 1 be real valued controls consider the controllability of the system dot x t f x t sum min i 1 max n 1 u i t g i x t x 0 x 0 in m necessary and sufficient conditions are given so that this system is controllable on any simply connected domain d contained in m on which g 1 g n 1 are linearly independent these conditions depend on the computation of lie brackets at those points where f g 1 g n 1 are linearly dependent

geometric conditions for local controllability abstract small time local controllability of a c1 affine control system with a generic drift vector field and n 1 linearly independent input vector fields on an n dimensional connected c1 manifold m is studied showing that a necessary condition namely the drift vector field belonging to the convex hull of the control vector fields at the given point is also sufficient on an open dense subset h1 of the set of points where it is verified local controllability at the points in the interior of h 1 is also considered when the input vector fields generate an involutive distribution and a criterion based on the type of contract of h with the hyperplane spanned by the input vector fields is proved an example shows a situation where the sufficient condition for small time local controllability of h sussmann siam j control option 25 1987 158 194 cannot be applied but in which the method now presented distinguishes the possible cases

local controllability along a reference trajectory abstract local controllability along a reference trajectory of analytic and smooth affine single input control systems without any a priori bounds on the control is considered and a geometric proof of a version of the hermes conjecture is given

on the boundedness of effective potentials arising from string compactifications we study effective potentials coming from compactifications of string theory we show that under mild assumptions such potentials are bounded from below in four dimensions giving an affirmative answer to a conjecture proposed by the second author in douglas jhep 3 71 2010 we also derive some sufficient conditions for the existence of critical points all proofs and mathematical hypotheses are discussed in the context of their relevance to the physics of the problem

the mass of asymptotically hyperbolic manifolds motivated by certain problems in general relativity and riemannian geometry we study manifolds which are asymptotic to the hyperbolic space in a certain sense it is shown that an invariant the so called total mass can be defined unambiguously a positive mass theorem is established by using the spinor method

the hodge cohomology of a conformally compact metric on etudie le laplacien de hogde agissant sur des k formes differentielles pour une classe de varietes de riemann completes a courbure sectionnelle negative proche de l infini ces varietes ont des compactifications c sur lesquelles la metrique est conforme a une metrique lisse jusqu au bord avec un facteur conforme 2 une fonction definissante pour le bord

local controllability in 3 manifolds small time local controllability of a two input ck affine control system on a 3 dimesional connected ck manifold m is studied showing that in general a necessary condition is also sufficient on an open dense subset of the set of points where it is verified an example shows a situation where the sufficient condition for small time local controllability of sussmann 8 cannot be applied but in which the method now presented distinguishes the possible cases

no arbitrage leverage and completeness in a fractional volatility model abstract when the volatility process is driven by fractional noise one obtains a model which is consistent with the empirical market data depending on whether the stochasticity generators of log price and volatility are independent or are the same two versions of the model are obtained with different leverage behaviors here the no arbitrage and completeness properties of the models are rigorously studied

on local controllability let x y be real analytic vector fields on a real analytic n dimensional manifold m consider a control system on m with dynamics described by 1 qquad dx dt x x t u t y x t quad x 0 p where an admissible control is lebesgue measurable with values u t leqq 1 the relationship between the map 2 qquad s 1 cdots s n to exp s 1 y circ cdots circ exp s n text ad n 1 x y circ exp tx p being one one on a neighborhood of 0 in mathbb r n for each 0 t varepsilon and other known necessary conditions for local controllability are studied in dimension n 2 many necessary conditions are equivalent and also sufficient for n geqq 3 the map 2 being locally one one implies many necessary conditions are satisfied but these need not be sufficient examples which illustrate what occurs geometrically are given

control systems which generate decomposable lie algebras let x y be real analytic vector fields on an n manifold m ad x y x y denote the lie product inductively ad k 1 x y x ad k x y and 1 ad v x y v 0 the attainable set at time t denoted download full size image t for the control system 1 x x x u t y x x 0 p u t 1 is contained in the integral manifold through p of the lie algebra l 1 generated by 1 l 1 is said to be decomposable at p if essentially when dim l 1 p n the solution of 1 can be written as exp tx exp f 1 t u v 1 exp f n t u v n p with v 1 v n l 1 this is true for example at any p if l 1 is nilpotent the f i t u are shown to satisfy a system on n of the form 2 x i u t g i t x 1 x i 1 x i 0 0 i 1 n where if l 1 is nilpotent each g i is a polynomial in x 1 x i 1 in this case reachability properties of 2 for controls being countably additive measures impluses reduce to the study of algebraic equations homogeneity allows these to be considered on projective space n giving local results by bezout s theorem similar results hold for controls in the unit ball of krener s results on approximating systems of the form 1 by systems which generate nilpotent lie algebras can then be used to remove the condition that l 1 be nilpotent the method applies to show that if m denotes the set of all products of k elements of 1 with k m and dim span m 1 p dim span m for all odd m then exp tx p int download full size image t for all t 0