strong mixed integer programming formulations for trained neural networks we present strong mixed integer programming mip formulations for high dimensional piecewise linear functions that correspond to trained neural networks these formulations can be used for a number of important tasks such as verifying that an image classification network is robust to adversarial inputs or solving decision problems where the objective function is a machine learning model we present a generic framework which may be of independent interest that provides a way to construct sharp or ideal formulations for the maximum of d affine functions over arbitrary polyhedral input domains we apply this result to derive mip formulations for a number of the most popular nonlinear operations e g relu and max pooling that are strictly stronger than other approaches from the literature we corroborate this computationally showing that our formulations are able to offer substantial improvements in solve time on verification tasks for image classification networks

small and strong formulations for unions of convex sets from the cayley embedding there is often a significant trade off between formulation strength and size in mixed integer programming when modeling convex disjunctive constraints e g unions of convex sets adding auxiliary continuous variables can sometimes help resolve this trade off however standard formulations that use such auxiliary continuous variables can have a worse than expected computational effectiveness which is often attributed precisely to these auxiliary continuous variables for this reason there has been considerable interest in constructing strong formulations that do not use continuous auxiliary variables we introduce a technique to construct formulations without these detrimental continuous auxiliary variables to develop this technique we introduce a natural non polyhedral generalization of the cayley embedding of a family of polytopes and show it inherits many geometric properties of the original embedding we then show how the associated formulation technique can be used to construct small and strong formulation for a wide range of disjunctive constraints in particular we show it can recover and generalize all known strong formulations without continuous auxiliary variables

quasipotential and exit time for 2d stochastic navier stokes equations driven by space time white noise we are dealing with the navier stokes equation in a bounded regular domain mathcal o of mathbb r 2 perturbed by an additive gaussian noise partial w q delta partial t which is white in time and colored in space we assume that the correlation radius of the noise gets smaller and smaller as delta searrow 0 so that the noise converges to the white noise in space and time for every delta 0 we introduce the large deviation action functional s delta t and the corresponding quasi potential u delta and by using arguments from relaxation and gamma convergence we show that u delta converges to u u 0 in spite of the fact that the navier stokes equation has no meaning in the space of square integrable functions when perturbed by space time white noise moreover in the case of periodic boundary conditions the limiting functional u is explicitly computed finally we apply these results to estimate of the asymptotics of the expected exit time of the solution of the stochastic navier stokes equation from a basin of attraction of an asymptotically stable point for the unperturbed system

on mathematical programming with indicator constraints in this paper we review the relevant literature on mathematical optimization with logical implications i e where constraints can be either active or disabled depending on logical conditions to hold in the case of convex functions the theory of disjunctive programming allows one to formulate these logical implications as convex nonlinear programming problems in a space of variables lifted with respect to its original dimension we concentrate on the attempt of avoiding the issue of dealing with large nlps in particular we review some existing results that allow to work in the original space of variables for two relevant special cases where the disjunctions corresponding to the logical implications have two terms then we significantly extend these special cases in two different directions one involving more general convex sets and the other with disjunctions involving three terms computational experiments comparing disjunctive programming formulations in the original space of variables with straightforward bigm ones show that the former are computationally viable and promising

weak solutions of the landau lifshitz bloch equation abstract the landau lifshitz bloch llb equation is a formulation of dynamic micromagnetics valid at all temperatures treating both the transverse and longitudinal relaxation components important for high temperature applications we study llb equation in case the temperature raised higher than the curie temperature the existence of weak solution is showed and its regularity properties are also discussed in this way we lay foundations for the rigorous theory of llb equation that is currently not available

on the separation of disjunctive cuts disjunctive cuts for mixed integer linear programs mips were introduced by egon balas in the late 1970s and have been successfully exploited in practice since the late 1990s in this paper we investigate the main ingredients of a disjunctive cut separation procedure and analyze their impact on the quality of the root node bound for a set of instances taken from miplib library we compare alternative normalization conditions and try to better understand their role in particular we point out that constraints that become redundant because of the disjunction used can produce over weak cuts and analyze this property with respect to the normalization used finally we introduce a new normalization condition and analyze its theoretical properties and computational behavior along the way we make use of a number of small numerical examples to illustrate some basic and often misinterpreted disjunctive programming features

random dynamical systems generated by stochastic navier stokes equations on a rotating sphere abstract in this paper we first prove the existence and uniqueness of the solution to the stochastic navier stokes equations on the rotating 2 dimensional unit sphere then we show the existence of an asymptotically compact random dynamical system associated with the equations

analysis of time fractional and space nonlocal stochastic incompressible navier stokes equation driven by white noise abstract in this paper we consider the stochastic time space fractional incompressible navier stokes equation driven by white noise we firstly derive the regularity of the nonlocal stochastic convolution then we prove the global existence and uniqueness of the mild solution finally the holder continuity for the stochastic time space fractional incompressible navier stokes equation driven by white noise is established the optimal holder continuity accords with the result of integer order time incompressible navier stokes equation

on stochastic evolution equations for nonlinear bipolar fluids well posedness and some properties of the solution abstract we investigate the stochastic evolution equations describing the motion of a non newtonian fluids excited by multiplicative noise of levy type we show that the system we consider has a unique global strong solution we also give some results concerning the properties of the solution we mainly prove that the unique solution satisfies the markov feller property this enables us to prove by means of some results from ergodic theory that the semigroup associated to the unique solution admits at least an invariant measure which is ergodic and tight on a subspace of the lebesgue space l 2

integration of brownian vector fields summary using the wiener chaos decomposition we show that strong solutions of non lipschitzian s d e s are given by random markovian kernels the example of sobolev flows is studied in some detail exhibiting interesting phase transitions

stochastic navier stokes equations with caputo derivative driven by fractional noises abstract in this paper we consider the extended stochastic navier stokes equations with caputo derivative driven by fractional brownian motion we firstly derive the pathwise spatial and temporal regularity of the generalized ornstein uhlenbeck process then we discuss the existence uniqueness and holder regularity of mild solutions to the given problem under certain sufficient conditions which depend on the fractional order and hurst parameter h the results obtained in this study improve some results in existing literature

global flows with invariant gibbs measures for euler and navier stokes two dimensional fluids we construct a family of probability spaces p 0 associated with the euler equation for a two dimensional inviscid incompressible fluid which carries a pointwise flow t time evolution leavingp globally invariant t is obtained as the limit of galerkin approximations associated with euler equations p is also in invariant measure for a stochastic process associated with a navier stokes equation with viscosity stochastically perturbed by a white noise force

local and global strong solutions to the stochastic incompressible navier stokes equations in critical besov space abstract considering the stochastic navier stokes system in r d forced by a multiplicative white noise we establish the local existence and uniqueness of the strong solution when the initial data take values in the critical space b p r d p 1 r d the proof is based on the contraction mapping principle stopping time and stochastic estimates then we prove the global existence of strong solutions in probability if the initial data are sufficiently small which contain a class of highly oscillating large data

global strong solution to the three dimensional stochastic incompressible magnetohydrodynamic equations the three dimensional incompressible magnetohydrodynamic equations with stochastic external forces are considered first the existence and uniqueness of local strong solution to the stochastic magnetohydrodynamic equations are proved when the external forces satisfy some conditions the proof is based on the contraction mapping principle stopping time and stochastic estimates the strong solution is a weak solution for the fluid variables with a given complete probability space and a given brownian motion then the global existence of strong solutions in probability is established if the initial data are sufficiently small and the noise is multiplicative and non degenerate

strong formulations for quadratic optimization with m matrices and indicator variables we study quadratic optimization with indicator variables and an m matrix i e a psd matrix with non positive off diagonal entries which arises directly in image segmentation and portfolio optimization with transaction costs as well as a substructure of general quadratic optimization problems we prove under mild assumptions that the minimization problem is solvable in polynomial time by showing its equivalence to a submodular minimization problem to strengthen the formulation we decompose the quadratic function into a sum of simple quadratic functions with at most two indicator variables each and provide the convex hull descriptions of these sets we also describe strong conic quadratic valid inequalities preliminary computational experiments indicate that the proposed inequalities can substantially improve the strength of the continuous relaxations with respect to the standard perspective reformulation

strong solution for a stochastic model of two dimensional second grade fluids existence uniqueness and asymptotic behavior abstract we investigate a stochastic evolution equation for the motion of a second grade fluid filling a bounded domain of r 2 global existence and uniqueness of strong probabilistic solution is established in contrast to previous results on this model we show that the sequence of galerkin approximation converges in mean square to the exact strong probabilistic solution of the problem we also give two results on the long time behavior of the solution mainly we prove that the strong solution of our stochastic model converges exponentially in mean square to the stationary solution of the time independent second grade fluids equations if the deterministic part of the external force does not depend on time if the deterministic forcing term explicitly depends on time then the strong probabilistic solution decays exponentially in mean square

fiber dynamics in turbulent flows general modeling framework the paper at hand deals with the modeling of turbulence effects on the dynamics of a long slender elastic fiber independent of the choice of the drag model a general aerodynamic force concept is derived on the basis of the velocity field for the randomly fluctuating component of the flow its construction as a centered differentiable gaussian field complies thereby with the requirements of the stochastic k epsilon turbulence model and kolmogorov s universal equilibrium theory on local isotropy

critical strong feller regularity for markov solutions to the navier stokes equations abstract the main purpose of this paper is to show that markov solutions to the 3d navier stokes equations driven by gaussian noise have the strong feller property up to the critical topology given by the domain of the stokes operator to the power one fourth

existence of densities for the 3d navier stokes equations driven by gaussian noise we prove three results on the existence of densities for the laws of finite dimensional functionals of the solutions of the stochastic navier stokes equations in dimension 3 in particular under very mild assumptions on the noise we prove that finite dimensional projections of the solutions have densities with respect to the lebesgue measure which have some smoothness when measured in a besov space this is proved thanks to a new argument inspired by an idea introduced in fournier and printems 2010

large deviation principles for 2 d stochastic navier stokes equations driven by levy processes abstract in this paper we establish a large deviation principle for the two dimensional stochastic navier stokes equations driven by levy processes which involves the study of the levy noise and the investigation of the effect of the highly nonlinear unbounded drifts

support vector machines with the ramp loss and the hard margin loss in the interest of deriving classifiers that are robust to outlier observations we present integer programming formulations of vapnik s support vector machine svm with the ramp loss and hard margin loss the ramp loss allows a maximum error of 2 for each training observation while the hard margin loss calculates error by counting the number of training observations that are in the margin or misclassified outside of the margin svm with these loss functions is shown to be a consistent estimator when used with certain kernel functions in computational studies with simulated and real world data svm with the robust loss functions ignores outlier observations effectively providing an advantage over svm with the traditional hinge loss when using the linear kernel despite the fact that training svm with the robust loss functions requires the solution of a quadratic mixed integer program qmip and is np hard while traditional svm requires only the solution of a continuous quadratic program qp we are able to find good solutions and prove optimality for instances with up to 500 observations solution methods are presented for the new formulations that improve computational performance over industry standard integer programming solvers alone

martingale solutions for the three dimensional stochastic nonhomogeneous incompressible navier stokes equations driven by levy processes abstract in this paper the three dimensional stochastic nonhomogeneous incompressible navier stokes equations driven by levy processes consisting of the brownian motion the compensated poisson random measure and the poisson random measure are considered in a bounded domain we obtain the existence of martingale solutions the construction of the solution is based on the classical galerkin approximation method the stopping times the stochastic compactness method and the jakubowski skorokhod theorem

on the existence and uniqueness of solutions to stochastic three dimensional lagrangian averaged navier stokes equations we prove the existence and uniqueness of solutions for a stochastic version of the threedimensional lagrangian averaged navier stokes equation in a bounded domain to this end we previously prove some existence and uniqueness results for an abstract stochastic equation and justify that our model falls within this framework

the disjunctive graph machine representation of the job shop scheduling problem abstract the disjunctive graph is one of the most popular models used for describing instances of the job shop scheduling problem which has been very intensively explored in this paper a new time and memory efficient representation of the disjunctive graph is proposed it has the form of a graph matrix and combines advantages of a few classical graph representations enabling easy operating on the problem data computational experiments have proved higher efficiency of the proposed approach over the classical ones

the local structure of turbulence in incompressible viscous fluid for very large reynolds numbers 1 we shall denote by u p u x 1 x 2 x 3 t 1 2 3 the components of velocity at the moment t at the point with rectangular cartesian coordinates x 1 x 2 x 3 in considering the turbulence it is natural to assume the components of the velocity u p at every point p x 1 x 2 x 3 t of the considered domain g of the four dimensional space x 1 x 2 x 3 t are random variables in the sense of the theory of probabilities cf for this approach to the problem millionshtchikov 1939 denoting by the mathematical expectation of the random variable a we suppose that 2 and d u d x 2 are finite and bounded in every bounded subdomain of the domain g

a lift and project cutting plane algorithm for mixed 0 1 programs we propose a cutting plane algorithm for mixed 0 1 programs based on a family of polyhedra which strengthen the usual lp relaxation we show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point this cut is found through the solution of a linear program that has about twice the size of the usual lp relaxation a lifting step is used to reduce the size of the lp s needed to generate the cuts an additional strengthening step suggested by balas and jeroslow is then applied we report our computational experience with a preliminary version of the algorithm this approach is related to the work of balas on disjunctive programming the matrix cone relaxations of lovasz and schrijver and the hierarchy of relaxations of sherali and adams

global solutions to random 3d vorticity equations for small initial data abstract one proves the existence and uniqueness in l p r 3 3 3 2 p 2 of a global mild solution to random vorticity equations associated to stochastic 3 d navier stokes equations with linear multiplicative gaussian noise of convolution type for sufficiently small initial vorticity this resembles some earlier deterministic results of t kato 16 and are obtained by treating the equation in vorticity form and reducing the latter to a random nonlinear parabolic equation the solution has maximal regularity in the spatial variables and is weakly continuous in l 3 l 3 p 4 p 6 3 with respect to the time variable furthermore we obtain the pathwise continuous dependence of solutions with respect to the initial data in particular one gets a locally unique solution of 3 d stochastic navier stokes equation in vorticity form up to some explosion stopping time adapted to the brownian motion

a hierarchy of maximum entropy closures for galerkin systems of incompressible flows abstract we propose a maximum entropy closure strategy for dissipative dynamical systems building on and generalizing earlier examples noack niven 2012 11 focus is placed on galerkin systems arising from a projection of the incompressible navier stokes equation onto orthonormal expansion modes the maximum entropy closure is motivated by a simple analytical example and elaborated to a hierarchical framework with sufficient conditions for the existence of solutions

some results concerning invertible morishima and anti morishima matrices whose associated digraphs are trees abstract this paper addresses the following questions given that a is an invertible morishima or anti morishima matrix under what conditions will a 1 be a morishima or anti morishima matrix also given the associated digraph of a and the fact that a a 1 are morishima or anti morishima matrices what can be said about the digraph of a 1 a class of matrices is introduced where there questions are investigated

stochastic tamed 3d navier stokes equations existence uniqueness and ergodicity in this paper we prove the existence of a unique strong solution to a stochastic tamed 3d navier stokes equation in the whole space as well as in the periodic boundary case then we also study the feller property of solutions and prove the existence of invariant measures for the corresponding feller semigroup in the case of periodic conditions moreover in the case of periodic boundary and degenerated additive noise using the notion of asymptotic strong feller property proposed by hairer and mattingly ann math 164 993 1032 2006 we prove the uniqueness of invariant measures for the corresponding transition semigroup

exponential mixing for stochastic pdes the non additive case we establish a general criterion which ensures exponential mixing of parabolic stochastic partial differential equations spde driven by a non additive noise which is white in time and smooth in space we apply this criterion on two representative examples 2d navier stokes ns equations and complex ginzburg landau cgl equation with a locally lipschitz noise due to the possible degeneracy of the noise doob theorem cannot be applied hence a coupling method is used in the spirit of kuksin and shirikyan j math pures appl 1 567 602 2002 and mattingly commun math phys 230 421 462 2002

mixed 0 1 programming by lift and project in a branch and cut framework we investigate the computational issues that need to be addressed when incorporating general cutting planes for mixed 0 1 programs into a branch and cut framework the cuts we use are of the lift and project variety some of the issues addressed have a theoretical answer but others are of an experimental nature and are settled by comparing alternatives on a set of test problems the resulting code is a robust solver for mixed 0 1 programs we compare it with several existing codes on a wide range of test problems it performs as well as or better than some of the best currently available mixed integer programming codes

a new method to predict the evolution of the power spectral density for a finite amplitude sound wave a method to predict the effect of nonlinearity on the power spectral density of a plane wave traveling in a thermoviscous fluid is presented as opposed to time domain methods the method presented here is based directly on the power spectral density of the signal not the signal itself the burgers equation is employed for the mathematical description of the combined effects of nonlinearity and dissipation the burgers equation is transformed into an infinite set of linear equations that describe the evolution of the joint moments of the signal a method for solving this system of equations is presented only a finite number of equations is appropriately selected and solved by numerical means for the method to be applied all appropriate joint moments must be known at the source if the source condition has gaussian characteristics it is a gaussian noise signal or a gaussian stationary and ergodic stochastic process then all the joint moments can be computed from the power spectral density of the signal at the source numerical results from the presented method are shown to be in good agreement with known analytical solutions in the preshock region for two benchmark cases i sinusoidal source signal and ii a gaussian stochastic process as the source condition

on the theory of statistical and isotropic turbulence the statistical theory of turbulence initiated by taylor 1935 and v karman howarth 1938 has recently been developed so far that a satisfactory explanation of the spectral distribution of energy among the turbulent eddies can be given in fact kolmogoroff 1941 a b and independently onsager 1945 and v weizsaecker 1948 have introduced a similarity hypothesis which allows a determination of the spectrum for eddies with large reynolds numbers and the author heisenberg 1948 has extended these calculations to include those frequency components which have small reynolds numbers since the distribution of energy among the largest eddies must be a geometrical and not a statistical problem one may say that the statistical part of the spectrum is now well understood recently batchelor townsend 1947 1948 a b have studied the decay of turbulent motion caused by a mesh grating in a wind tunnel and the following discussions will apply the statistical theory to this problem

mild solutions of the stochastic mhd equations driven by fractional brownian motions abstract the incompressible magnetohydrodynamic equations driven by additive fractional brownian motions are considered we firstly establish the local existence and uniqueness of the mild solution in l p space on a smooth bounded domain in r d d 2 3 the proof is based on the semigroup theory fixed point theorem and the results of stochastic pdes of linear parabolic type in the proof the eigenvalue problem with perfectly conducting wall condition is considered to weaken the requirements of noise terms finally the global existence of mild solutions is also established by energy estimate

2d stochastic navier stokes equations driven by jump noise abstract in the paper we are studying the existence and uniqueness of the solution of an abstract nonlinear equation driven by a multiplicative noise of levy type our result is formulated in an abstract setting this type of equation covers the stochastic 2d navier stokes equations the 2d stochastic magneto hydrodynamic equations the 2d stochastic boussinesq model for the benard convection the 2d stochastic magnetic bernard problem the 3d stochastic leray model for the navier stokes equations and several stochastic shell models of turbulence

analysis of a stochastic implicit interface model for an immersed elastic surface in a fluctuating fluid we present some mathematical analyses of a recently proposed stochastic implicit interface model for an elastic surface immersed in an incompressible viscous fluid subject to fluctuation forces we derive suitable a priori estimates and establish the well posedness of pathwise solutions and provide uniform control on the solutions in probability

well posedness of stochastic second grade fluids abstract the theory of turbulent newtonian fluids shows that the choice of the boundary condition is a relevant issue because it can modify the behavior of a fluid by creating or avoiding a strong boundary layer in this study we consider stochastic second grade fluids filling a two dimensional bounded domain with the navier slip boundary condition with friction we prove the well posedness of this problem and establish a stability result our stochastic model involves a multiplicative white noise and a convective term with third order derivatives which significantly complicate the analysis

martingale and stationary solutions for stochastic navier stokes equations we prove the existence of martingale solutions and of stationary solutions of stochastic navier stokes equations under very general hypotheses on the diffusion term the stationary martingale solutions yield the existence of invariant measures when the transition semigroup is well defined the results are obtained by a new method of compactness

on two classes of matrices with positive diagonal solutions to the lyapunov equation abstract we characterize real indecomposable quasi jacobi matrices of class d i e those which satisfy the lyapunov equation pa a p q with p diagonal and both p and q positive definite the subclass d 2 of class d when also q is diagonal is also characterized in the case of general indecomposable real matrices

global l2 solutions of stochastic navier stokes equations this paper concerns the cauchy problem in r d for the stochastic navier stokes equation 1 u u u u p f u u p g u o w u 0 u 0 div u 0 driven by white noise w under minimal assumptions on regularity of the coefficients and random forces the existence of a global weak martingale solution of the stochastic navier stokes equation is proved in the two dimensional case the existence and pathwise uniqueness of a global strong solution is shown a wiener chaos based criterion for the existence and uniqueness of a strong global solution of the navier stokes equations is established

local martingale and pathwise solutions for an abstract fluids model we establish the existence and uniqueness of both local martingale and local pathwise solutions of an abstract nonlinear stochastic evolution system the primary application of this abstract framework is to infer the local existence of strong pathwise solutions to the 3d primitive equations of the oceans and atmosphere forced by a nonlinear multiplicative white noise instead of developing our results specifically for the 3d primitive equations we choose to develop them in a slightly abstract framework which covers many related forms of these equations atmosphere oceans coupled atmosphere ocean on the sphere on the beta plane approximation etc and the incompressible navier stokes equations in applications all of the details are given for the beta plane approximation of the oceans equations

stochastic modelling in fluid dynamics ito versus stratonovich suppose the observations of lagrangian trajectories for fluid flow in some physical situation can be modelled sufficiently accurately by a spatially correlated it stochastic process with zero mean

existence of a martingale solution of the stochastic navier stokes equations in unbounded 2d and 3d domains abstract stochastic navier stokes equations in 2d and 3d possibly unbounded domains driven by a multiplicative gaussian noise are considered the noise term depends on the unknown velocity and its spatial derivatives the existence of a martingale solution is proved the construction of the solution is based on the classical faedo galerkin approximation the compactness method and the jakubowski version of the skorokhod theorem for nonmetric spaces moreover some compactness and tightness criteria in nonmetric spaces are proved compactness results are based on a certain generalization of the classical dubinsky theorem

a theorem concerning certain sign symmetric matrices whose inverses are morishima abstract suppose a is an invertible sign symmetric matrix whose associated digraph d a is a tree then a 1 will be morishima iff a 0 for all interior points in d a a 1 will be anti morishima iff a 0 for all interior points in d a

the stochastic korteweg de vries equation on a bounded domain this paper considers the stochastic korteweg de vries equation on a bounded domain the existence of a weak martingale solution is discussed by the galerkin s approximation and compactness method based on this result we establish the unique continuation property for the stochastic korteweg de vries equation

local and global well posedness of spde with generalized coercivity conditions abstract in this paper we establish the local and global existence and uniqueness of solutions for general nonlinear evolution equations with coefficients satisfying some local monotonicity and generalized coercivity conditions an analogous result is obtained for stochastic evolution equations in hilbert space with additive noise as applications the main results are applied to obtain simpler proofs in known cases as the stochastic 3d navier stokes equation the tamed 3d navier stokes equation and the cahn hilliard equation but also to get new results for stochastic surface growth pde and stochastic power law fluids

existence of strong solutions for ito s stochastic equations via approximations given strong uniqueness for an ito s stochastic equation with discontinuous coefficients we prove that its solution can be constructed on any probability space by using for example euler s polygonal approximations stochastic equations in d and in domains in d are considered

convex programming for disjunctive convex optimization given a finite number of closed convex sets whose algebraic representation is known we study the problem of finding the minimum of a convex function on the closure of the convex hull of the union of those sets we derive an algebraic characterization of the feasible region in a higher dimensional space and propose a solution procedure akin to the interior point approach for convex programming

stochastic hall magneto hydrodynamics system in three and two and a half dimensions we introduce the stochastic hall magneto hydrodynamics hall mhd system in three and two and a half dimensions with infinite dimensional multiplicative noise white in time and prove the global existence of a martingale solution via a stochastic galerkin approximation and applications of prokhorov s skorokhod s and martingale representation theorems as well as the pressure term through de rham s theorem adapted to processes the hall term represents mathematically a very singular nonlinear term unprecedented in the previous work the results extend many others on the deterministic hall mhd and stochastic mhd systems and navier stokes equations in contrast to the stochastic mhd system the path wise uniqueness in the two and a half dimensional case is an open problem

local and global existence of smooth solutions for the stochastic euler equations with multiplicative noise we establish the local existence of pathwise solutions for the stochastic euler equations in a three dimensional bounded domain with slip boundary conditions and a suitable nonlinear multiplicative noise in the two dimensional case we obtain the global existence of these solutions with additive or linear multiplicative noise finally we show that in the three dimensional case the addition of linear multiplicative noise provides a regularizing effect the global existence of solutions occurs with high probability if the initial data is sufficiently small or if the noise coefficient is sufficiently large

on inviscid limits for the stochastic navier stokes equations and related models we study inviscid limits of invariant measures for the 2d stochastic navier stokes equations as shown by kuksin j stat phys 115 1 2 469 492 2004 the noise scaling sqrt nu is the only one which leads to non trivial limiting measures which are invariant for the 2d euler equations using a moser type iteration for stochastic drift diffusion equations we show that any limiting measure mu 0 is in fact supported on bounded vorticities relationships of mu 0 to the long term dynamics of 2d euler in l infty with the weak topology are discussed we also obtain a drift independent modulus of continuity for a stationary deterministic model problem which leads us to conjecture that in fact mu 0 is supported on c 0 moreover in view of the batchelor krainchnan 2d turbulence theory we consider inviscid limits for a weakly damped stochastic navier stokes equation in this setting we show that only an order zero noise scaling with respect to leads to a nontrivial limiting measure in the inviscid limit

nonlinear chance constrained problems with applications to hydro scheduling we present a branch and cut algorithm for a class of nonlinear chance constrained mathematical optimization problems with a finite number of scenarios unsatisfied scenarios can enter a recovery mode this class corresponds to problems that can be reformulated as deterministic convex mixed integer nonlinear programming problems with indicator variables and continuous scenario variables but the size of the reformulation is large and quickly becomes impractical as the number of scenarios grows the branch and cut algorithm is based on an implicit benders decomposition scheme where we generate cutting planes as outer approximation cuts from the projection of the feasible region on suitable subspaces the size of the master problem in our scheme is much smaller than the deterministic reformulation of the chance constrained problem we apply the branch and cut algorithm to the mid term hydro scheduling problem for which we propose a chance constrained formulation a computational study using data from ten hydroplants in greece shows that the proposed methodology solves instances faster than applying a general purpose solver for convex mixed integer nonlinear programming problems to the deterministic reformulation and scales much better with the number of scenarios

almost sure existence of global weak solutions for incompressible mhd equations in negative order sobolev space abstract in this paper we consider the cauchy problem of the incompressible magnetohydrodynamics mhd equations in the periodic space domain t n where n 2 after a suitable randomization to the initial data we obtain the almost sure existence of global weak solutions with the initial data in h s t n s 1 0 specially the global weak solution is unique when n 2

3d tamed navier stokes equations driven by multiplicative levy noise existence uniqueness and large deviations abstract in this paper we show the existence and uniqueness of a strong solution to stochastic 3d tamed navier stokes equations driven by multiplicative levy noise with periodic boundary conditions then we establish the large deviation principles of the strong solution on the state space d 0 t h 1 where the weak convergence approach plays a key role

a branch and cut method for 0 1 mixed convex programming we generalize the disjunctive approach of balas ceria and cornuejols 2 and devevlop a branch and cut method for solving 0 1 convex programming problems we show that cuts can be generated by solving a single convex program we show how to construct regions similar to those of sherali and adams 20 and lovasz and schrijver 12 for the convex case finally we give some preliminary computational results for our method

perspective cuts for a class of convex 0 1 mixed integer programs we show that the convex envelope of the objective function of mixed integer programming problems with a specific structure is the perspective function of the continuous part of the objective function using a characterization of the subdifferential of the perspective function we derive perspective cuts a family of valid inequalities for the problem perspective cuts can be shown to belong to the general family of disjunctive cuts but they do not require the solution of a potentially costly nonlinear programming problem to be separated using perspective cuts substantially improves the performance of branch cut approaches for at least two models that either naturally or after a proper reformulation have the required structure the unit commitment problem in electrical power production and the mean variance problem in portfolio optimization

bi perspective functions for mixed integer fractional programs with indicator variables perspective functions have long been used to convert fractional programs into convex programs more recently they have been used to form tight relaxations of mixed integer nonlinear programs with so called indicator variables motivated by a practical application maximising energy efficiency in an ofdma system we consider problems that have a fractional objective and indicator variables simultaneously to obtain a tight relaxation of such problems one must consider what we call a bi perspective bi p function an analysis of bi p functions leads to the derivation of a new kind of cutting planes which we call bi p cuts computational results indicate that bi p cuts typically close a substantial proportion of the integrality gap

ergodicity of the 2 d navier stokes equation under random perturbations a 2 dimensional navier stokes equation perturbed by a sufficiently distributed white noise is considered existence of invariant measures is known from previous works the aim is to prove uniqueness of the invariant measures strong law of large numbers and convergence to equilibrium

magnetohydrodynamic turbulent flows existence results abstract in this paper we provide a detailed investigation of the problem of existence of solutions to the stochastic magnetohydrodynamic equations which describe the motion of turbulent particles of fluids in a magnetic field the results are proved through a delicate combination of galerkin approximation scheme with some deep compactness results of analytic and probabilistic nature

simultaneous convexification of bilinear functions over polytopes with application to network interdiction we study the simultaneous convexification of graphs of bilinear functions g k x y y intercal a k x over x in ensuremath xi x in 0 1 n e x geq f and y in delta m y in r m 1 intercal y leq 1 we propose a constructive procedure to obtain a linear description of the convex hull of the resulting set this procedure can be applied to derive convex and concave envelopes of certain bilinear functions to study unary expansions of integer variables in mixed integer bilinear sets and to obtain convex hulls of sets with complementarity constraints exploiting the structure of xi the procedure naturally yields stronger linearizations for bilinear terms in a variety of practical settings in particular we demonstrate the effectiveness of the approach by strengthening the traditional dual formulation of network interdiction problems and report encouraging preliminary numerical results

stochastic equations in hilbert space with application to navier stokes equations in any dimension abstract we give an existence theorem for an abstract nonlinear stochastic evolution equation in a hilbert space the result is applicable to the stochastic navier stokes equation in any dimension with a nonlinear noise term

on unbiased stochastic navier stokes equations a random perturbation of a deterministic navier stokes equation is considered in the form of an spde with wick type nonlinearity the nonlinear term of the perturbation can be characterized as the highest stochastic order approximation of the original nonlinear term u nabla u this perturbation is unbiased in that the expectation of a solution of the perturbed equation solves the deterministic navier stokes equation the perturbed equation is solved in the space of generalized stochastic processes using the cameron martin version of the wiener chaos expansion it is shown that the generalized solution is a markov process and scales effectively by catalan numbers

two dimensional navier stokes equations driven by a space time white noise abstract we study the two dimensional navier stokes equations with periodic boundary conditions perturbed by a space time white noise it is shown that although the solution is not expected to be smooth the nonlinear term can be defined without changing the equation we first construct a stationary martingale solution then we prove that for almost every initial data with respect to a measure supported by negative spaces there exists a unique global solution in the strong probabilistic sense

weak solutions to stochastic 3d navier stokes α model of turbulence α asymptotic behavior abstract we study the asymptotic behavior of weak solutions to the stochastic 3d navier stokes model as approaches zero the main result provides a new construction of the weak solutions of stochastic 3d navier stokes equations as approximations by sequences of solutions of the stochastic 3d navier stokes model

nonlinear stochastic partial differential equations with singular diffusivity and gradient stratonovich noise abstract we study existence and uniqueness of a variational solution in terms of stochastic variational inequalities svi to stochastic nonlinear diffusion equations with a highly singular diffusivity term and multiplicative stratonovich gradient type noise we derive a commutator relation for the unbounded noise coefficients in terms of a geometric killing vector condition the drift term is given by the total variation flow respectively by a singular p laplace type operator we impose nonlinear zero neumann boundary conditions and precisely investigate their connection with the coefficient fields of the noise this solves an open problem posed in barbu et al 2013 7 and barbu and rockner 2015 10

kolmogorov constants for the second order structure function and the energy spectrum we examine the behavior of the kolmogorov constants c 2 c k and c k1 which are respectively the prefactors of the second order longitudinal structure function and the three dimensional and one dimensional longitudinal energy spectrum in the inertial range we show that their ratios c 2 c k1 and c k c k1 exhibit clear dependence on the microscale reynolds number r implying that they cannot all be independent of r in particular it is found that c k1 c 2 0 25 1 95r 0 68 the study further reveals that the widely used relation c 2 4 02c k1 holds only asymptotically when r 10 5 it is also found that c 2 has much stronger r dependence than either c k or c k1 if the latter indeed has a systematic dependence on r we further show that the varying dependence on r of these three numbers can be attributed to the difference of the inertial range in real and wave number space with the inertial range in real space known to be much shorter than that in wave number space

stochastic 3d tamed navier stokes equations existence uniqueness and small time large deviation principles abstract in this paper we first give a direct approach to the existence and uniqueness of strong solutions to stochastic 3d tamed navier stokes equations in case of dirichlet boundary conditions for the stokes laplacian then we prove a small time large deviation principle for the solutions

stochastic generalized magnetohydrodynamics equations with not regular multiplicative noise well posedness and invariant measure abstract in this paper we study a stochastic magnetohydrodynamics mhd system with fractional diffusion and resistivity 0 in r d d 2 3 our main goal is to identify the conditions on under which we can prove the existence of a martingale solution the pathwise uniqueness of solution and the existence of invariant measure when the noises are multiplicative and take values in functional space bigger than the space of square integrable functions roughly speaking we prove that if 1 0 and the driving noises take values in h then the stochastic system has at least a weak martingale solution we also establish the pathwise uniqueness of solution whenever d 2 finally under the latter condition and under the addition of linear damping to the equations we are able to establish the existence of an invariant measure

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references

layer methods for stochastic navier stokes equations using simplest characteristics we propose and study a layer method for stochastic navier stokes equations snse with spatial periodic boundary conditions and additive noise the method is constructed using conditional probabilistic representations of solutions to snse and exploiting ideas of the weak sense numerical integration of stochastic differential equations we prove some convergence results for the proposed method including its first mean square order results of numerical experiments on two model problems are presented

some possible new directions for combinatorial matrix analysis using the facts that 1 much of the combinatorial matrix analysis to date has been concerned with properties of trees and 2 trees as digraphs have many important special properties we suggest several classes of matrices which can probably be successfully studied with the help of combinatorial methods these classes include the following a unipethic matrices a class derived by retaining the property that in the irreducible case the digraph defined in the usual way is strong and unipathic if it is a tree b bipartite matrices a class obtained by retaining the property that a tree is a bipartite digraph these matrices are closely related to 2 cyclic matrices but they are not the same class c matrices such that their digraph has at least one cycle of length p and no longer cycle these matrices are an obvious generalization of trees they are primarily of interest for small values of p relative to the size of the matrix d hessenberg matrices because their successive principal minors can be shown to always satisfy a difference equation and therefore a are one sort of generalization of the tridiagonal matrices outside of the class of trees e matrices which are such that the digraph is a k tree for k 1 in particular the 2 trees it is easy to show that k trees are a natural generalization of trees and they include the band matrices five diagonal matrices are for example 2 trees

some structural results concerning certain classes of qualitative matrices and their inverses abstract this paper presents some precise structural results concerning combinatorially symmetric sign symmetric and sign antisymmetric invertible matrices whose associated diagraphs are trees in particular given an invertible sign antisymmetric matrix a whose associated digraph is a tree and the fact that a 1 is sign antisymmetric we are able to completely determine the associated digraph of a 1

homogenization of a stochastic nonlinear reaction diffusion equation with a large reaction term the almost periodic framework abstract homogenization of a stochastic nonlinear reaction diffusion equation with a large nonlinear term is considered under a general besicovitch almost periodicity assumption on the coefficients of the equation we prove that the sequence of solutions of the said problem converges in probability towards the solution of a rather different type of equation namely the stochastic nonlinear convection diffusion equation which we explicitly derive in terms of appropriate functionals we study some particular cases such as the periodic framework and many others this is achieved under a suitable generalized concept of convergence for stochastic processes