gravimagnetic anomaly formulas for polyhedra of spatially linear media formulas for the gravity potential field and field gradient tensor are derived for a polyhedral target body of a spatially linear density medium the formulas also define the magnetic potential and field in the case of a medium of spatially linear magnetization this work generalizes existing solutions for the gravity field of a polyhedral target of linearly varying density the formulas are analyzed for singularities and for numerical error growth error growth with increasing target distance is found to be higher than in the corresponding uniform polyhedral case examination of the error sources reveals that some error reduction is possible on this basis new algorithms of improved error control over existing algorithms are proposed for the linear medium computational results confirm the expected improvement

geometric formulation of mechanical systems with one sided constraints a geometric formulation of analytical mechanics of systems with holonomic and holonomic one sided constraints is proposed this is an extension of work by tulczyjew 1986 in which non holonomic constraints were also considered but not the one sided case the introduction of lagrange multipliers to give explicit equations of motion was not considered by tulczyjew either this allows the authors to study problems of motion of particles hitting moving walls they verify that the conditions of collision are satisfied

a multiclass station with markovian feedback in heavy traffic this paper proves a heavy traffic limit theorem for a multiclass service station with markovian feedback this result generalizes the one proved by reiman reiman m i 1988 a multiclass feedback queue in heavy traffic adv appl probab 20 179 207 our approach also significantly simplifies reiman s original proof numerical examples are presented to illustrate the effectiveness of the qnet method which is rooted in the theorem

stability of constrained markov modulated diffusions a family of constrained diffusions in a random environment is considered constraint set is a polyhedral cone and coefficients of the diffusion are governed by in addition to the system state a finite state markov process that is independent of the driving noise such models arise as limit objects in the heavy traffic analysis of generalized jackson networks gjn with markov modulated arrival and processing rates we give sufficient conditions which in particular includes a requirement on the regularity of the underlying skorohod map for positive recurrence and geometric ergodicity when the coefficients only depend on the modulating markov process i e they are independent of the system state a complete characterization for stability and transience is provided the case where the pathwise skorohod problem is not well posed but the underlying reflection matrix is completely s is treated as well as consequences of geometric ergodicity various results such as exponential integrability of invariant measures and central limit results clt for fluctuations of long time averages of process functionals about their stationary values are obtained conditions for stability are formulated in terms of the averaged drift where the average is taken with respect to the stationary distribution of the modulating markov process finally steady state distributions of the underlying gjn are considered and it is shown that under suitable conditions such distributions converge to the unique stationary distribution of the constrained random environment diffusion

stochastic differential equations with reflecting boundary conditions on resout des equations differentielles stochastiques a conditions aux limites reflechissantes par une approche directe basee sur le probleme de skorokhod

reflected brownian motion in a cone with radially homogeneous reflection field this work is concerned with the existence and uniqueness of a strong markov process that has continuous sample paths and the following ad ditional properties i the state space is a cone in dimensions d 3 and the process behaves in the interior of the cone like ordinary brownian motion ii the process reflects instantaneously at the boundary of the cone the direction of reflection being fixed on each radial line emanating from the vertex of the cone iii the amount of time that the process spends at the vertex of the cone is zero i e the set of times for which the process is at the vertex has zero lebesgue measure the question of existence and uniqueness is cast in precise mathematical terms as a submartingale problem in the style used by stroock and varadhan for diffusions on smooth domains with smooth boundary conditions the question is resolved in terms of a real parameter a which in general depends in a rather complicated way on the geometric data of the problem i e on the cone and the directions of reflection however a criterion is given for determining whether a 0 it is shown that there is a unique continuous strong markov process satisfying i iii above if and only if a 2 there is a unique continuous strong markov process satisfying i and ii above it reaches the vertex of the cone almost surely and remains there these results are illustrated in concrete terms for some special cases the process considered here serves as a model for comparison with a re flected brownian motion in a cone having a nonradially homogeneous reflection field this is discussed in a subsequent work by kwon

stability of queueing networks and scheduling policies we develop a programmatic procedure for establishing the stability of queueing networks and scheduling policies the method uses linear or nonlinear programming to determine what is an appropriate quadratic functional to use as a lyapunov function if the underlying system is markovian our method establishes not only positive recurrence and the existence of a steady state probability distribution but also the geometric convergence of an exponential moment we illustrate this method on several example problems

multidimensional reflected brownian motions having exponential stationary distributions on considere la distribution stationnaire d un mouvement brownien reflechi dans un domaine d dimensionnel g on etudie deux classes de processus de ce type

uniqueness for diffusions with piecewise constant coefficients letl be a second order partial differential operator inrd letrd be the finite union of disjoint polyhedra suppose that the diffusion matrix is everywhere non singular and constant on each polyhedron and that the drift coefficient is bounded and measurable we show that the martingale problem associated withl is well posed

stochastic differential games in a non markovian setting stochastic differential games are considered in a non markovian setting typically in stochastic differential games the modulating process of the diffusion equation describing the state flow is taken to be markovian then nash equilibria or other types of solutions such as pareto equilibria are constructed using hamilton jacobi bellman hjb equations but in a non markovian setting the hjb method is not applicable to examine the non markovian case this paper considers the situation in which the modulating process is a fractional brownian motion fractional noise calculus is used for such models to find the nash equilibria explicitly although fractional brownian motion is taken as the modulating process because of its versatility in modeling in the fields of finance and networks the approach in this paper has the merit of being applicable to more general gaussian stochastic differential games with only slight conceptual modifications this work has applications in finance to stock price modeling which incorporates the effect of institutional investors and to stochastic differential portfolio games in markets in which the stock prices follow diffusions modulated with fractional brownian motion

functional limit theorems for a simple auction we consider a continuous transparent auction with one asset buyers and sellers arrive at the auction according to independent renewal processes willing to transact a random number of units shares of the asset for one ofn possible prices upon arrival they buy sell as many units of the asset as they can for the best price available at that time however if the prices offered for sale purchase are worse than their initial preferences they do not transact but instead they stay at the auction posting buying selling orders and waiting for a suitable counterparty we model the numbers of outstanding orders of each type together with the numbers of assets exchanged up to date by customers of different types by stochastic processes and find the limiting distributions of these suitably scaled processes

mechanical systems subjected to generalized non holonomic constraints we study mechanical systems subject to constraint functions that can be dependent at some points and independent at the rest such systems are modelled by means of generalized codistributions we discuss how the constraint force can transmit an impulse to the motion at the points of dependence and derive an explicit formula to obtain the post impact momentum in terms of the pre impact momentum

directional derivatives of oblique reflection maps given an oblique reflection map and functions documentclass aastex usepackage amsbsy usepackage amsfonts usepackage amssymb usepackage bm usepackage mathrsfs usepackage pifont usepackage stmaryrd usepackage textcomp usepackage portland xspace usepackage amsmath amsxtra pagestyle empty declaremathsizes 10 9 7 6 begin document psi chi in mathcal d lim end document the space of k valued functions that have finite left and right limits at every point the directional derivative documentclass aastex usepackage amsbsy usepackage amsfonts usepackage amssymb usepackage bm usepackage mathrsfs usepackage pifont usepackage stmaryrd usepackage textcomp usepackage portland xspace usepackage amsmath amsxtra pagestyle empty declaremathsizes 10 9 7 6 begin document nabla chi gamma psi end document of along evaluated at is defined to be the pointwise limit as documentclass aastex usepackage amsbsy usepackage amsfonts u

least controls for a class of constrained linear stochastic systems this paper introduces a class of constrained linear stochastic systems the main objective is to investigate whether there exists a least control in a sense that it exerts least effort in controlling the system to within a predetermined region our approach of finding a least control is a to characterize a class of functional polyhedral sets of functions which have least elements and b to construct a least control process through these least elements existence of a least control is established and the connection between a least control and a solution of the dynamic complementarity problem dcp is also discussed

ideality criterion for unilateral constraints in time dependent impulsive mechanics we construct a new geometric framework based on the concepts of left and right jet bundles of a classical space time v in order to analyze the impulsive behavior of a unilateral constraint s the setup allows deep insights into how one can choose an ideality criterion for the constraint s when the hypothesis of conservation of kinetic energy is assumed we show that the conservation of kinetic energy alone univocally determines the impulsive reaction when the codimension of s is 1 and that it leaves the impulsive reaction partially undetermined when the codimension of s is greater than 1 if the codimension of s is greater than 1 we prove that an additional minimality requirement determines a physically meaningful constitutive characterization of s we show that both the newton like and the poisson like approaches to the description of the reactive impulse are equivalent in the sense that both give the same results about the ideality criterion moreover we prove that the same results hold using the cl

numerical approximations for stochastic differential games the markov chain approximation method is a widely used robust relatively easy to use and efficient family of methods for the bulk of stochastic control problems in continuous time for reflected jump diffusion type models it has been shown to converge under broad conditions and there are good algorithms for solving the numerical problems if the dimension is not too high versions of these methods have been used in applications to various two player differential and stochastic dynamic games for a long time and proofs of convergence are available for some cases mainly using pde type techniques in this paper purely probabilistic proofs of convergence are given for a broad class of such problems where the controls for the two players are separated in the dynamics and cost function and which cover a substantial class not dealt with in previous works discounted and stopping time cost functions are considered finite horizon problems and problems where the process is stopped on first hitting an a priori given boundary can be dealt with by adapting the methods of h j kushner and p dupuis numerical methods for stochastic control problems in continuous time 2nd ed springer verlag berlin new york 2001 as done in this paper for the treated problems the essential conditions are the weak sense existence and uniqueness of solutions an almost everywhere continuity condition and that a weak local consistency condition holds almost everywhere for the numerical approximations just as for the control problem there are extensions to problems with controlled variance and jumps

on the singularities of the gravity field of a homogeneous polyhedral body the study of the gravity field of arbitrary polyhedral bodies of homogeneous density has provoked a series of publications over the last decades some of the researchers represented an arbitrary three dimensional body in terms of contours obtained by the intersection of horizontal planes with the body

the dynamics of discrete mechanical systems with perfect unilateral constraints the dynamics of discrete mechanical systems with perfect unilateral constraints is formulated in a very general setting the well posedness of the resulting evolution problem is studied it is proved that existence and uniqueness of a maximal solution is ensured provided strong assumptions are made on the regularity of the data they are supposed to be analytic simple examples show that this regularity assumption may not be relaxed sufficient conditions to ensure that the maximal solution is defined for all time are supplied the continuous dependence of the solution on initial conditions is also studied and the numerical computation of the solution is discussed

diffusion approximations for some multiclass queueing networks with fifo service disciplines the diffusion approximation is proved for a class of multiclass queueing networks under fifo service disciplines in addition to the usual assumptions for a heavy traffic limit theorem a key condition that characterizes this class is that aj x j matrixg known as the workload contents matrix has a spectral radius less than unity wherej represents the number of service stations the j l th component of matrixg can be interpreted as the long run average amount of future work for stationj that is embodied in a unit of immediate work at stationl this class includes as a special case the feedforward multiclass queueing network and the rybko stolyar network under fifo service discipline a new approach is taken in establishing the diffusion limit theorem the traditional approach is based on an oblique reflection mapping but such a mapping is not well defined for the network under consideration our approach takes two steps arst establishing thec tightness of the scaled queueing processes and then completing the proof for the convergence of the scaled queueing processes by invoking the weak uniqueness for the limiting processes which are semimartingale reflecting brownian motions

subharmonic solutions with prescribed minimal period for some second order impulsive differential equations abstract this paper uses critical point theory and variational methods to investigate the subharmonic solutions with prescribed minimal period for a class of second order impulsive differential equations the conditions for the existence of subharmonic solutions are established our results rectify some known results in the literature and will allow us in the future to deal with the subharmonic solutions for more extensive impulsive problems

a limit theorem for financial markets with inert investors we study the effect of investor inertia on stock price fluctuations with a market microstructure model comprising many small investors who are inactive most of the time it turns out that semi markov processes are tailor made for modelling inert investors with a suitable scaling we show that when the price is driven by the market imbalance the log price process is approximated by a process with long range dependence and non gaussian returns distributions driven by a fractional brownian motion consequently investor inertia may lead to arbitrage opportunities for sophisticated market participants the mathematical contributions are a functional central limit theorem for stationary semi markov processes and approximation results for stochastic integrals of continuous semimartingales with respect to fractional brownian motion

lagrangian submanifolds and higher order mechanical systems higher order lagrangian and hamiltonian systems time dependent or independent are interpreted in terms of lagrangian submanifolds of sympletic higher order tangent bundles the relation between both formalisms is given

strong solutions of stochastic equations with rank based coefficients we study finite and countably infinite systems of stochastic differential equations in which the drift and diffusion coefficients of each component particle are determined by its rank in the vector of all components of the solution we show that strong existence and uniqueness hold until the first time three particles collide motivated by this result we improve significantly the existing conditions for the absence of such triple collisions in the case of finite dimensional systems and provide the first condition of this type for systems with a countable infinity of particles

brownian approximations of multiclass open queueing networks we study a multiclass open queueing network with a set of single server stations that operate under a combination of fifo first in first out and priority service disciplines and are subject to random breakdowns assuming that the primitive processes in particular external arrivals service requirements service capacities up and down times and the routing mechanism follow two moment approximations based on functional central limit theorems we develop a semi martingale reflected brownian motion srbm approximation for the performance processes such as workload queue lengths and sojourn times we illustrate through numerical examples in comparison against simulation that the srbm approximation while not always supported by a limit theorem exhibits good accuracy in most cases through analyzing special networks we also discuss the existence of the srbm approximation in relation to the stability and the heavy traffic limits of the networks

a class of polynomially solvable linear complementarity problems although the general linear complementarity problem lcp is np complete there are special classes that can be solved in polynomial time one example is the type where the defining matrix is nondegenerate and for which the n step property holds in this paper we consider an extension of the property to the degenerate case by introducing the concept of an extended n step vector and matrix it is shown that the lcp defined by such a matrix is polynomially solvable as well

numerical approximations for stochastic differential games the ergodic case the markov chain approximation method is a widely used relatively easy to use and efficient family of methods for the bulk of stochastic control problems in continuous time for reflected jump diffusion type models it has been shown to converge under broad conditions and there are good algorithms for solving the numerical problems if the dimension is not too high we consider a class of stochastic differential games with a reflected diffusion system model and ergodic cost criterion and where the controls for the two players are separated in the dynamics and cost function it is shown that the value of the game exists and that the numerical method converges to this value as the discretization parameter goes to zero the actual numerical method solves a stochastic game for a finite state markov chain and ergodic cost criterion the essential conditions are nondegeneracy and that a weak local consistency condition hold almost everywhere for the numerical approximations just as for the control problem

a heavy traffic limit theorem for networks of queues with multiple customer types the central result of this paper is a heavy traffic limit theorem for the vector of total station workloads in an open network of queues with multiple customer types under first come first served and priority disciplines the limit process is a regulated brownian motion on the nonnegative orthant with parameters specified from the first two moments of the interarrival and service time distributions and a matrix of reduced routing information through the phenomenon of state space collapse associated limit results for queue length workload and sojourn time processes by customer type are obtained jointly as simple transformations of the total workload limit process diffusion approximations based on the theorem are discussed

validity of heavy traffic steady state approximations in multiclass queueing networks the case of queue ratio disciplines a class of stochastic processes known as semi martingale reflecting brownian motions srbms is often used to approximate the dynamics of heavily loaded queueing networks in two influential papers bramson bramson m 1998 state space collapse with applications to heavy traffic limits for multiclass queueing networks queueing systems 30 89 148 and williams williams rj 1998b diffusion approximations for open multiclass queueuing networks sufficient conditions involving state space collapse queueing systems 30 27 88 laid out a general and structured approach for proving the validity of such heavy traffic approximations in which an srbm is obtained as a diffusion limit from a sequence of suitably normalized workload processes however for multiclass networks it is still not known in general whether the steady state distribution of the srbm provides a valid approximation for the steady state distribution of the original network in this paper we study the case of queue ratio disciplines and provide a set of sufficient conditions under which the above question can be answered in the affirmative in addition to standard assumptions made in the literature towards the stability of the pre and post limit processes and the existence of diffusion limits we add a requirement that solutions to the fluid model are attracted to the invariant manifold at a linear rate for the special case of static priority networks such linear attraction is known to hold under certain conditions on the network primitives the analysis elucidates interesting connections between stability of the pre and post limit processes their respective fluid models and state space collapse and identifies the respective roles played by all of the above in establishing validity of heavy traffic steady state approximations

gravimetric analysis of uniform polyhedra analytical formulas for the gravity anomaly of a uniform polyhedral body are subject to numerical error that increases with distance from the target while the anomaly decreases this leads to a limited range of target distances in which the formulas are operational beyond which the calculations are dominated by rounding error we analyze the sources of error and propose a combination of numerical and analytical procedures that exhibit advantages over existing methods namely 1 errors that diminish with distance 2 enhanced operating range and 3 algorithmic simplicity the latter is achieved by avoiding the need to transform coordinates and the need to discriminate between projected observation points that lie inside on or outside a target facet boundary our error analysis is verified in computations based on a published code and on a code implementing our methods the former requires a numerical precision of one part in 1016 double precision in problems of geophysical interest whereas our c

gravity anomalies of arbitrary 3d polyhedral bodies with horizontal and vertical mass contrasts up to cubic order abstracta new singularity free analytical formula has been developed for the gravity field of arbitrary 3d polyhedral mass bodies with horizontally and vertically varying density contrast using third order polynomial functions first the observation sites are moved to the origin of the coordinate system then the volume and surface integral theorems are invoked successively to transform the volume integrals into surface integrals over polygonal faces and into line integrals over the edges of the polyhedral mass bodies furthermore singularity free closed form solutions are derived for these line integrals over the edges thus the observation sites can be located inside on or outside the 3d distributions a synthetic prismatic mass body is adopted to verify the accuracy and singularity free property of our newly developed analytical expressions excellent agreements are obtained between our solutions and other published closed form solutions with relative errors in the order of 10 12 to 10 13 in a

on the geometry of non holonomic lagrangian systems we present a geometric framework for non holonomic lagrangian systems in terms of distributions on the configuration manifold if the constrained system is regular an almost product structure on the phase space of velocities is constructed such that the constrained dynamics is obtained by projecting the free dynamics if the constrained system is singular we develop a constraint algorithm which is very similar to that developed by dirac and bergmann and later globalized by gotay and nester special attention to the case of constrained systems given by connections is paid in particular we extend the results of koiller for caplygin systems an application to the so called non holonomic geometry is given

transformation techniques and fixed point theories to establish the positive solutions of second order impulsive differential equations abstract using a new method for dealing with the impulse term of second impulsive differential equations we are concerned with determining values of for which there exist positive solutions of the following boundary value problems in the form of x t f t x t 0 t j t t k x t k x t k c k x t k k 1 2 n a x 0 b x 0 a x 1 b x 1 0 1 h s x t d t where 0 is a positive parameter c k is a real sequence with c k 1 k 1 2 n the dependence of positive solution x t on the parameter is also studied i e lim x or lim x 0 the proof of our main result is based upon transformation techniques and fixed point theories this is probably the first time the existence of positive solutions of second order impulsive differential equations has been studied by applying this technique

boundary value problems for higher order linear impulsive differential equations in this paper higher order linear impulsive differential equations with fixed moments of impulses subject to linear boundary conditions are studied green s formula is defined for piecewise differentiable functions properties of green s functions for higher order impulsive boundary value problems are introduced an appropriate example of the green s function for a boundary value problem is provided furthermore eigenvalue problems and basic properties of eigensolutions are considered

on semimonotone matrices with nonnegative principal minors in this paper we examine two well known classes of matrices in linear complementarity theory semimonotone matrices and matrices possessing nonnegative principal minors we derive two sets of conditions under which the two classes are identical

on the simultaneous presence of unilateral and kinetic constraints in time dependent impulsive mechanics the simultaneous presence of unilateral and kinetic constraints acting on a mechanical system with a finite number of degrees of freedom is framed in the geometric context of left and right jet bundles of the classical space time bundle of the system the survey gives three main cases and several subcases some of which are mathematically correct but physically meaningless the existence of at least one frame of reference for which the whole set of constraints can be at rest is the criterion selecting the physically relevant systems for these systems the conservation of kinetic energy possibly together with a standard gauss s requirement on the impulsive reaction is shown to give a well posed criterion of ideality of the constraints the application of the criterion to several examples is presented and the corresponding results are critically analyzed

mechanical systems subjected to impulsive constraints a geometrical description of mechanical systems subjected to impulsive non holonomic constraints is given their motions are determined by means of suitable projection operators which allow us to evaluate the instantaneous jumps of momenta due to the action of impulsive reaction forces several paradigmatic examples are investigated from this viewpoint a disk falling into a plane a rolling ball hitting a wall and the collision of two balls

an ergodic control problem for constrained diffusion processes existence of optimal markov control an ergodic control problem for a class of constrained diffusion processes is considered the goal is the almost sure minimization of long term cost per unit time the main result of the paper is that there exists an optimal markov control for the considered problem it is shown that under the assumption of regularity of the skorohod map and the assumption that the drift vector field takes values in a certain cone of stability the class of controlled diffusion processes considered have strong uniform in control stability properties the role of the boundary is critical in obtaining the stability and ergodic control results for the class of controlled constrained diffusion processes considered in this work since the domains are unbounded and the corresponding unconstrained diffusions are typically transient these stability properties are key in obtaining appropriate tightness estimates once these estimates are available the remaining work lies in identifying weak limits of a certain family of occupation measures in this regard an extension to the echeverria weiss kurtz characterization of invariant measures of markov processes to the case of constrained controlled processes considered in this paper is proved this characterization result is also crucially used in proving the compactness of the family of invariant measures of markov processes corresponding to all possible markov controls

explicit solution to a robust queueing control problem we consider the robust optimal control of a law of large numbers approximation of a stochastic network the robust control problem is formulated as a differential game with one player choosing the policies that determine service and routing assignments and the other choosing quantities such as the arrival and service rates subject to constraints the cost to be minimized by the first player and maximized by the second is the time until the origin is reached an explicit formula is given for the value function and some of its basic properties are studied

a numerical scheme for invariant distributions of constrained diffusions reflected diffusions in polyhedral domains are commonly used as approximate models for stochastic processing networks in heavy traffic stationary distributions of such models give useful information on the steady state performance of the corresponding stochastic networks and thus it is important to develop reliable and efficient algorithms for numerical computation of such distributions in this work we propose and analyze a monte carlo scheme based on an euler type discretization of the reflected stochastic differential equation using a single sequence of time discretization steps which decrease to zero as time approaches infinity appropriately weighted empirical measures constructed from the simulated discretized reflected diffusion are proposed as approximations for the invariant probability measure of the true diffusion model almost sure consistency results are established that in particular show that weighted averages of polynomially growing continuous functionals evaluated on the discretized simulated system converge a s to the corresponding integrals with respect to the invariant measure proofs rely on constructing suitable lyapunov functions for tightness and uniform integrability and characterizing almost sure limit points through an extension of echeverria s criteria for reflected diffusions regularity properties of the underlying skorohod problems play a key role in the proofs rates of convergence for suitable families of test functions are also obtained a key advantage of monte carlo methods is the ease of implementation particularly for high dimensional problems a numerical example

hamiltonian theory of constrained impulsive motion this paper considers systems subject to nonholonomic constraints which are not uniform on the whole configuration manifold when the constraints change the system undergoes a transition in order to comply with the new imposed conditions building on previous work on the hamiltonian theory of impact we tackle the problem of mathematically describing the classes of transitions that can occur we propose a comprehensive formulation of the transition principle that encompasses the various impulsive regimes of hamiltonian systems our formulation is based on the partial symplectic formalism which provides a suitable framework for the dynamics of nonholonomic systems we pay special attention to mechanical systems and illustrate the results with several examples

convergence rates for rank based models with applications to portfolio theory we determine rates of convergence of rank based interacting diffusions and semimartingale reflecting brownian motions to equilibrium bounds on fluctuations of additive functionals are obtained using transportation cost information inequalities for markov processes we work out various applications to the rank based abstract equity markets used in stochastic portfolio theory for example we produce quantitative bounds including constants for fluctuations of market weights and occupation times of various ranks for individual coordinates another important application is the comparison of performance between symmetric functionally generated portfolios and the market portfolio this produces estimates of probabilities of beating the market

analytical computation of the full gravity tensor of a homogeneous arbitrarily shaped polyhedral source using line integrals the analytical computation of the full gravity tensor from a polyhedral source of homogeneous density is presented with emphasis on its algorithmic implementation the theoretical development is based on the subsequent transition of the general expressions from volume to surface and from surface to line integrals defined along the closed polygons building each polyhedral face however the accurate numerical computation of the obtained transcendental expressions is linked with the relative position of the computation point and its corresponding projections on the plane of each face and on the line of each segment with respect to the polygons defining each face depending on this geometric setup the application of the divergence theorem of gauss leads to the appearance of additional correction terms valid only for these boundary conditions and crucial for the correct numerical evaluation of the polyhedral related gravity quantities at those locations of the computation point a program in fortran is su

open queueing networks in heavy traffic this paper presents heavy traffic limit theorems for the queue length and sojourn time processes associated with open queueing networks these limit theorems state that properly normalized sequences of queue length and sojourn time processes converge weakly to a certain diffusion as the network traffic intensity converges to unity the limit diffusion is reflected brownian motion on the nonnegative orthant this process behaves like brownian motion on the interior of its state space and reflects instantaneously on the boundaries the reflection direction is a constant for each boundary hyperplane

dissipative systems with constraints throughout the paper m will be a c compact m dimensional connected riemannian manifold with a c riemannian metric aa4 q4 called the configuration space the dimension m of a4 is called the degree of freedom the tangent bundle 7 44 of m the velocity phase space and the cotangent bundle t m of a4 the momentum phase space the vector bundle isomorphism p tm t m defined by p u w v w is a c diffeomorphism and will be referred to as the mass operator otherwise called the legendre transformation the function k tm r defined by k v v v is called the kinetic energy of the system the constraint z is a c mapping qhzq which assigns to each qe m an n dimensional subspace c c t m the subvector bundle xm of tm defined by the vectors of all z q e m is a regular submanifold of tm and if c is the orthogonal complement of c in t m the same is true for the subvector bundle zim defined by the distribution q h zi the bundles cm elm have dimensions m n 2m n and there are canonically defined c bundle mappings p tm cm p tm 2 m defined by pv va p v v where vi v are the orthogonal projections of v on c zi by means of the mass operator one can translate em z m p

power law polya s urn and fractional brownian motion we introduce a natural family of random walks s n on mathbb z that scale to fractional brownian motion the increments x n s n s n 1 in pm 1 have the property that given x k k n the conditional law of x n is that of x n k n where k n is sampled independently from a fixed law mu on the positive integers when mu has a roughly power law decay precisely when mu lies in the domain of attraction of an alpha stable subordinator for 0 alpha 1 2 the walks scale to fractional brownian motion with hurst parameter alpha 1 2 the walks are easy to simulate and their increments satisfy an fkg inequality in a sense we describe they are the natural fractional analogues of simple random walk on mathbb z

heavy traffic convergence of a controlled multiclass queueing system this paper provides a rigorous proof of the connection between the optimal sequencing problem for a two station two customer class queueing network and the problem of control of a multidimensional diffusion process obtained as a heavy traffic limit of the queueing problem in particular the diffusion problem which is one of singular control of a brownian motion is used to develop policies which are shown to be asymptotically nearly optimal as the traffic intensity approaches one in the queueing network the results are proved by a viscosity solution analysis of the related hamilton jacobi bellman equations

a law of large numbers for limit order books we define a stochastic model of a two sided limit order book in terms of its key quantities best bid ask price and the standing buy sell volume density for a simple scaling of the discreteness parameters that keeps the expected volume rate over the considered price interval invariant we prove a limit theorem the limit theorem states that given regularity conditions on the random order flow the key quantities converge in probability to a tractable continuous limiting model in the limit model the buy and sell volume densities are given as the unique solution to first order linear hyperbolic pdes specified by the expected order flow parameters we calibrate order flow dynamics to market data for selected stocks and show how our model can be used to derive endogenous shape functions for models of optimal portfolio liquidation under market impact

comparison of gravimetric formulas for uniform polyhedra the gravitational anomaly of a polyhedral target of uniform density is amenable to analytical solution and has received wide attention we analyze a number of apparently dissimilar solutions for consistency and comment on their suitability for numeric computation consistency is determined by evaluating a method over a special contour for which all methods must agree numerical performance is assessed through operation counts and algorithm classification into vertex line or surface type each type having a known accuracy bound for a given floating point precision we verify performance predictions through model calculations in particular we demonstrate the effectiveness of revised vertex and line algorithms and of a new analytical surface method our scheme of analysis and classification has general application it provides criteria by which existing and novel solutions may be examined for consistency and numerical performance

a rational route to randomness the concept of adaptively rational equilibrium a r e is introduced agents adapt their beliefs over time by choosing from a finite set of different predictor or expectations functions each predictor is a function of past observations and has a performance or fitness measure which is publicly available agents make a rational choice concerning the predictors based upon their past performance this results in a dynamics across predictor choice which is coupled to the equilibrium dynamics of the endogenous variables as a simple but typical example we consider a cobweb type demand supply model where agents can choose between rational and naive expectations in an unstable market with small positive information costs for rational expectations a high intensity of choice to switch predictors leads to highly irregular equilibrium prices converging to a strange attractor the irregularity of the equilibrium time paths is explained by the existence of a so called homoclinic orbit and its associated complicated dynamical phenomena thus local instability and global complicated dynamics may be a feature of a fully rational notion of equilibrium

heavy tails and long range dependence in on off processes and associated fluid models on off models are common inputs for a variety of communication network models as well as storage and inventory models a stationary renewal process alternating periods of activity active transmission fluid buildup etc with periods of inactivity silence no transmission inputs off etc induces a stationary indicator process which indicates if the system is active or not heavy tails for the on periods induces a covariance function for the indicator process which decreases slowly at a rate characteristic of long range dependence this has dramatic consequences for fluid models where fluid flows in at constant rate and there is a constant rate of release

an analytical expression for the gravity field of a polyhedral body with linearly varying density over the past 24 years the gravity fields of homogeneous polyhedra have been studied extensively paul 1974 barnett 1976 okabe 1979 golizdra 1981 strakhov et al 1986a 1986b gotze and lahmeyer 1988 pohanka 1988 ivan 1990 pohanka 1990 holstein and ketteridge 1996 this intensive scrutiny is well deserved because polyhedra provide a framework in which complex body shapes can be modeled parsimoniously and because the analytic expressions for the gravity fields of uniform polyhedra involve only elementary transcendental functions available as part of the run time library for most computational programming languages

a boundary property of semimartingale reflecting brownian motions we consider a class of reflecting brownian motions on the non negative orthant inr k in the interior of the orthant such a process behaves like brownian motion with a constant covariance matrix and drift vector at each of the k 1 dimensional faces that form the boundary of the orthant the process reflects instantaneously in a direction that is constant over the face we give a necessary condition for the process to have a certain semimartingale decomposition and then show that the boundary processes appearing in this decomposition do not charge the set of times that the process is at the intersection of two or more faces this boundary property plays an essential role in the derivation performed in a separate work of an analytical characterization of the stationary distributions of such semimartingale reflecting brownian motions

nonideal unilateral constraints in impulsive mechanics a geometric approach we use the differential geometric framework of the first jet bundle of the classical space time bundle to study the impulsive behavior of a mechanical system with a finite number of degrees of freedom subject to nonideal unilateral constraints we show that this framework allows deeper insights on the concepts of nonideal constitutive characterization and of coefficient of restitution of the constraints we study the relations among newton poisson and stronge definitions of coefficient of restitution we reveal the inconsistency of the criticisms based on the energy balance of the impact for the newton definition we show the equivalence of the three definitions in the nonideal smooth case we prove the equivalence of newton and poisson ones and the insufficiency of the stronge one in the nonideal rough case we analyze the relation between coefficient of restitution and coulomb s friction coefficient in the rough case we present also several physically meaningful examples

generalizations of the hidden minkowski property abstract the linear complementarity problem whose defining feature is the extended n step property is known to be solvable in polynomial time this paper focuses on the investigation of the subclass of p 0 matrices that possess this particular property we present a necessary condition and a separate sufficient condition for membership in the class it is shown that each of the conditions defines a set of matrices that properly contains the transposed hidden minkowski set thus generalizing previous work developed in ref j s pang r chandrasekaran linear complementarity problems solvable by a polynomially bounded pivoting algorithm math program study 25 1985 13 27 and in ref w d morris jr j lawrence geometric properties of hidden minkowski matrices siam j matrix anal appl 10 1989 229 232 to prove the necessity result we use a matrix analytical approach instead of a geometric argument as in the previous case to prove the sufficiency result we first establish two characterizations of singular irreducible k 0 matrices

gravimagnetic similarity in anomaly formulas for uniform polyhedra gravitational and magnetic anomalies of an arbitrary target body are linked through poisson s differential relation for a uniform polyhedral target poisson s relation reduces to an algebraic link between gravity and magnetic anomaly formulas the derivation is given in tensor form it identifies for each target facet edge a vector function in terms of which the gravitational and magnetic potential and field anomaly formulas are similarly expressed as appropriately weighted linear combinations this similarity unifies the theory of uniform polyhedral anomalies it benefits analysis and construction of software that naturally embraces all anomalies in a single code the analysis is exemplified by a discussion of singularities and by the adaptation of three gravity field algorithms to the remaining gravitational and magnetic cases while retaining the respective computational advantages of the former gravity field algorithms

on positive recurrence of constrained diffusion processes let g c r k be a convex polyhedral cone with vertex at the origin given as the intersection of half spaces g i i 1 n where n i and d i denote the inward normal and direction of constraint associated with g i respectively stability properties of a class of diffusion processes constrained to take values in g are studied under the assumption that the skorokhod problem defined by the data n i d i i 1 n is well posed and the skorokhod map is lipschitz continuous explicit conditions on the drift coefficient b of the diffusion process are given under which the constrained process is positive recurrent and has a unique invariant measure define c i d i i 0 i 1 n then the key condition for stability is that there exists 0 and a bounded subset a of g such that for all x g a b x c and dist b x c where c denotes the boundary of c

a versatile solution for the gravity anomaly of 3d prism meshed bodies with depth dependent density contrast abstractwe have developed a generalized solution for computing the gravity anomalies of 3d irregular mass bodies with complicated density contrast variation the 3d irregular shaped bodies can be approximated flexibly by a collection of finite juxtaposed right rectangular prisms the complicated density contrast variation of each prism can be well represented by a depth dependent polynomial function a novel analytic solution of gravity anomalies due to a right rectangular prism with an arbitrary order of polynomial density contrast function of depth is then derived the solution is singularity free in the upper half space over the prism and its singularity in the lower half space containing the prism is resolved by assigning their limit values to the singular terms the numerical stability of the solution is also evaluated through numerical tests hence the solution can be used to compute the gravity anomalies of 3d irregular bodies with variable density contrasts without singularities when computation

geometric formulation of carnot s theorem a geometrical approach for one sided constraints is given the riemannian metric is used in order to define convenient projectors which give the post impulses in terms of the pre impulses a formulation of carnot s theorem within this geometric framework is exhibited

workload models for stochastic networks value functions and performance evaluation this paper concerns control and performance evaluation for stochastic network models structural properties of value functions are developed for controlled brownian motion cbm and deterministic fluid workload models leading to the following conclusions outside of a null set of network parameters the following hold i the fluid value function is a smooth function of the initial state under further minor conditions the fluid value function satisfies the derivative boundary conditions that are required to ensure it is in the domain of the extended generator for the cbm model exponential ergodicity of the cbm model is demonstrated as one consequence ii the fluid value function provides a shadow function for use in simulation variance reduction for the stochastic model the resulting simulator satisfies an exact large deviation principle while a standard simulation algorithm does not satisfy any such bound iii the fluid value function provides upper and lower bounds on performance for the cbm model this follows from an extension of recent linear programming approaches to performance evaluation

nonnegative volume matching by cubic c 1 splines on clough tocher splits we describe a numerical method to construct volume matching c1 splines satisfying the frequently occurring requirement of nonnegativity the approximated histogram is assumed to be defined on a partition of a polygonal domain into polygonal subdomains our idea consists of applying cubic c1 splines on clough tocher refinements of a triangulation that is compatible with the given partition the proposed algorithm always works successfully for nonnegative histograms

large fluctuations in a deterministic multiclass network of queues in this paper we investigate a relatively simple deterministic four class two queue multiclass open network of single server fifo queues with traffic intensity one at each queue our purpose is to better understand the effect of feedback with class dependent service times at the queues the example is sufficiently tractable that we are able to describe its transient behavior in great detail the transient behavior depends strongly on the initial conditions and for some initial conditions the sample paths of the queue length processes at individual stations have sudden large fluctuations a large jump up followed immediately by a large jump down these large fluctuations occur because batches of customers with short service times build up in the queues consistent with recent work by dai and wang 1993 on brownian network models these fluctuations rule out conventional heavy traffic limit theorems we show how to obtain proper heavy traffic limits for this example by weakening the topology or enlarging the space of prospective limits and changing the topology this example also dramatically demonstrates a disadvantage of the fifo discipline compared to other disciplines like head of the line processor sharing hol ps among the classes at each queue under which the large fluctuations do not occur finally the critical arrival rate for stability in our example actually depends on the service discipline being even lower if the classes with longer service times are given high priority at each queue this phenomenon can occur in the network setting because individual queues can be empty when there is work in the network

3d vector gravity potential and line integrals for the gravity anomaly of a rectangular prism with 3d variable density contrast three dimensional rectangular prisms are building blocks for calculating gravity anomalies from irregular 3d mass bodies with spatially variable density contrasts a 3d vector gravity potential is defined for a 3d rectangular prism with density contrast varying in depth and horizontally the vertical component of the gravity anomaly equals the flux of the 3d vector gravity potential through the enclosed surface of the prism thus the 3d integral for the gravity anomaly is reduced to a 2d surface integral in turn a 2d vector gravity potential is defined the vertical component of the gravity anomaly equals the net circulation of the 2d vector gravity potential along the enclosed contour bounding the surfaces of the prism the 3d integral for the gravity anomaly is reduced to 1d line integrals further analytical or numerical solutions can then be obtained from the line integrals depending on the forms of the density contrast functions if an analytical solution cannot be obtained the line integral method is semianalytical requiring numerical quadratures to be carried out at the final stages singularity and discontinuity exist in the algorithm and the method of exclusive infinitesimal sphere or circle is effective to remove them then the vector potential line integral method can calculate the gravity anomaly resulting from a rectangular prism with density contrast varying simply in one direction and sophisticatedly in three directions the advantage of the method is that the constraint to the form of the density contrast is greatly reduced and the numerical calculation for the gravity anomaly is fast

reflected brownian motion on an orthant we consider a k dimensional diffusion process z whose state space is the nonnegative orthant on the interior of the orthant z behaves like a kdimensional brownian motion with arbitrary covariance matrix and drift vector at each of the k 1 dimensional hyperplanes that form the boundary of the orthant z reflects instantaneously in a direction that is constant over that hyperplane there is no extant theory of multidimensional diffusion that applies to this process because the boundary of its state space is not smooth we adopt an approach that requires a restriction on the directions of reflection but reiman has shown that this restriction is met by all diffusions z arising as heavy traffic limits in open k station queuing networks our process z is defined as a path to path mapping of k dimensional brownian motion from this construction it follows that z is a continuous markov process and a semimartingale using the latter property we obtain a change of variable formula from which one can develop a complete analytical theory for the process z

simple necessary and sufficient conditions for the stability of constrained processes in recent years a new approach has emerged for analyzing the stability properties of constrained stochastic processes in this approach one associates with the stochastic model a deterministic model or a family of deterministic models and under appropriate conditions stability of the stochastic model follows if all solutions of the deterministic model are attracted to the origin in the present work we show that a rather sharpcharacterization for the stability of the deterministic model is possible when it can be represented in terms of what we call a regular skorokhod map let g be a convex polyhedral cone with vertex at the origin given as the intersection of half spaces gi i 1 n where ni and di denote the inward normal and direction of constraint associated with gi suppose that the skorokhod problem defined by the data ni di i 1 n is regular under these conditions the deterministic model mentioned above will correspond to a law of large numbers limit of the original stochas

reflected brownian motion in a cone semimartingale property in this paper the object of study is reflected brownian motion in a cone ind dimensions d 3 with nonconstant oblique reflection on each radial line emanating from the vertex of the cone the basic question considered here is when is this process a semimartingale conditions for the existence and uniqueness of the process for which the vertex is an instantaneous state were given by kwon which is resolved in terms of a real parameter depending on the cone and the direction of reflection it is shown that starting from any point of the cone the process is a semimartingale if 1 0 and not a semimartingale if 2

existence and uniqueness of semimartingale reflecting brownian motions in an orthant this work is concerned with the existence and uniqueness of a class of semimartingale reflecting brownian motions which live in the non negative orthant of d loosely speaking such a process has a semimartingale decomposition such that in the interior of the orthant the process behaves like a brownian motion with a constant drift and covariance matrix and at each of the d 1 dimensional faces that form the boundary of the orthant the bounded variation part of the process increases in a given direction constant for any particular face so as to confine the process to the orthant for historical reasons this pushing at the boundary is called instantaneous reflection in 1988 reiman and williams proved that a necessary condition for the existence of such a semimartingale reflecting brownian motion srbm is that the reflection matrix formed by the directions of reflection be completely l in this work we prove that condition is sufficient for the existence of an srbm and that the srbm is unique in law it follows from the uniqueness that an srbm defines a strong markov process our results have potential application to the study of diffusions arising as approximations tomulti class queueing networks

brownian motion in a wedge with variable skew reflection does planar brownian motion confined to a wedge by skew reflection on the sides approach the vertex of the wedge this question has been answered by varadhan and williams in the case where the direction of reflection is constant on each of the sides but here we address the question when the direction reflected is allowed to vary a necessary condition and a sufficient condition are obtained for the vertex to be reached the conditions are of a geometric nature and the gap between them is quite small

an escape time criterion for queueing networks asymptotic risk sensitive control via differential games we consider the problem of risk sensitive control of a stochastic network in controlling such a network an escape time criterion can be useful if one wishes to regulate the occurrence of large buffers and buffer overflow in this paper a risk sensitive escape time criterion is formulated which in comparison to the ordinary escape time criteria penalizes exits that occur on short time intervals more heavily the properties of the risk sensitive problem are studied in the large buffer limit and related to the value of a deterministic differential game with constrained dynamics we prove that the game has value and that the value is the viscosity solution of a pde for a simple network the value is computed demonstrating the applicability of the approach

on the distribution of multidimensional reflected brownian motion let z z t t 0 be a reflected brownian motion on the k dimensional nonnegative orthant with the direction of reflection constant over each boundary surface such processes arise in heavy traffic theory for k station networks of queues this paper continues our study of the diffusion process z with attention restricted to the case k 2 for simplicity most of the results extend directly to higher dimensions our notation and style of argument are designed to suggest appropriate generalizations for arbitrary k wherever possible the backward equation for the transition density function with boundary and initial conditions the corresponding forward equation and the equation for the steady state distribution are all derived informally also presented are various calculations relating to steady state distributions including a moment formula and the derivation of a condition involving the drifts and directions of reflection that we conjecture to be necessary and sufficient for existence

reduction of some classical non holonomic systems with symmetry two types of nonholonomic systems with symmetry are treated i the configuration space is a total space of a g principal bundle and the constraints are given by a connection ii the configuration space is g itself and the constraints are given by left invariant forms the proofs are based on the method of quasicoordinates in passing a derivation of the maurer cartan equations for lie groups is obtained simple examples are given to illustrate the algorithmical character of the main results

reduction of constrained mechanical systems and stability of relative equilibria a mechanical system with perfect constraints can be described under some mild assumptions as a constrained hamiltonian system m h d w m thephase space is a symplectic manifold h thehamiltonian a smooth function onm d theconstraint submanifold a submanifold ofm andw theprojection bundle a vector sub bundle oft d m the reduced tangent bundle alongd we prove that when these data satisfy some suitable conditions the time evolution of the system is governed by a well defined differential equation ond we define constrained hamiltonian systems with symmetry and prove a reduction theorem application of that theorem is illustrated on the example of a convex heavy body rolling without slipping on a horizontal plane two other simple examples show that constrained mechanical systems with symmetry may have an attractive or repulsive set of relative equilibria

nonholonomic mechanical systems with symmetry this work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of lagrangian mechanics and with a view to control theoretical applications the basic methodology is that of geometric mechanics applied to the formulation of lagrange d alembert generalizing the use of connections and momentum maps associated with a given symmetry group to this case we begin by formulating the mechanics of nonholonomic systems using an ehresmann connection to model the constraints and show how the curvature of this connection enters into lagrange s equations unlike the situation with standard configuration space constraints the presence of symmetries in the nonholonomic case may or may not lead to conservation laws however the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables this momentum equation which plays an important role in control problems involves parallel transport operators and is computed explicitly in coordinates an alternative description using a body reference frame relates part of the momentum equation to the components of the euler poincar e equations along those symmetry directions consistent with the constraints one of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not an example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard a modified version of the skateboard which uses momentum coupling for locomotion generation we construct a synthesis of the mechanical connection and the ehresmann connection defining the constraints obtaining an important new object we call the nonholonomic connection when the nonholonomic connection is a principal connection for the given symmetry group we show how to perform lagrangian reduction in the presence of nonholonomic constraints generalizing previous results which only held in special cases several detailed examples are given to illustrate the theory september 1994 revised march 1995 revised june 1995

geometric formulation of mechanical systems subjected to time dependent one sided constraints a geometrical setting for time dependent impulsive constraints is given this description permits us to treat in a systematic way a wide family of mechanical systems subjected to time dependent constraints like moving walls etc if the impulsive constraints remain then an affine projector can be defined giving the new initial data in terms of the old ones several examples are discussed

non degeneracy of wiener functionals arising from rough differential equations malliavin calculus is about sobolev type regularity of functionals on wiener space the main example being the ito map obtained by solving stochastic differential equations rough path analysis is about strong regularity of the solution to possibly stochastic differential equations we combine arguments of both theories and discuss the existence of a density for solutions to stochastic differential equations driven by a general class of non degenerate gaussian processes including processes with sample path regularity worse than brownian motion

reduction of constrained systems with symmetries a general model is proposed for constrained dynamical systems on a symplectic manifold which covers among others the description of lagrangian and hamiltonian systems with nonholonomic constraints and the canonical description of mechanical systems with a singular lagrangian the reduction properties of these systems in the presence of symmetry are investigated within this general framework

non holonomic lagrangian systems in jet manifolds a geometrical setting in terms of jet manifolds is developed for time dependent non holonomic lagrangian systems an almost product structure on the evolution space is constructed in such a way that the constrained dynamics is obtained by projection of the free dynamics a constrained poincare cartan 2 form is defined if the non holonomic system is singular a constraint algorithm is constructed special attention is devoted to caplygin systems and a reduction theorem is proved