liouvillian solutions of n th order homogeneous linear differential equations be a linear differential operator with coefficients in f a finite algebraic extension of q x we shall show that one can find in a finite number of steps a basis for the vector space of liouvillian solutions of l y 0 i e those solutions which can be built up from the rational functions by algebraic operations taking exponentials and by integration see section 2 for a precise definition in particular we show how to decide if all solution of l y 0 are liouvillian or if there are any solutions which are liouvillian our algorithm in conjunction with the algorithm in 8 also allows one to determine the algebraic relationships among the liouvillian solutions of l y 0 and in particular determine if all solutions are algebraic c f 111 for second order linear homogeneous equations over q x the problem of determining if all solutions are algebraic functions was considered by fuchs klein and schwarz but none of these mathematicians seems to have presented a complete decision procedure building on the work of klein baldassarri and dwork 21 have given such a procedure baldassarri 11 has extended this to consider linear homogeneous equations whose coefficients are algebraic functions for third order equations painleve and boulanger gave a procedure which in effect reduced the problem of finding algebraic solutions of l y 0 to the problem of effectively bounding the torsion of the jacobian variety of a given curve a complete procedure for deciding if all solutions of an n th order homo

second order necessary conditions for nonlinear optimization problems in banach spaces and their application to an optimal control problem we study second order necessary conditions for nonlinear optimization problems with equality and set constraints in banach spaces the necessary conditions are given in the primal form and the dual form by the use of second order neustadt derivative moreover we apply the necessary conditions to an optimal control problem without state constraints

generalized hypergeometric functions and rational curves on calabi yau complete intersections in toric varieties we formulate general conjectures about the relationship between the a model connection on the cohomology of ad dimensional calabi yau complete intersectionv ofr hypersurfacesv1 vr in a toric varietyp and the system of differential operators annihilating the special generalized hypergeometric series 0 constructed from the fan using this generalized hypergeometric series we propose conjectural mirrorsv ofv and the canonicalq coordinates on the moduli spaces of calabi yau manifolds

finite surface discretization for incompressible navier stokes equations and coupled conservation laws abstract we present a new finite surface discretization fsd aiming at the incompressible navier stokes equations nse and other coupled conservation laws this discretization defines the velocities as surfaced averaged values living on the faces of the pressure volumes in which the mass is set to be conserved consequently the calculation of the mass balance on these control volumes is exact which allows more accurate information to be kept in the velocity field and produces a very accurate prediction of the pressure in the next time step the proposed discretization reduces the stencil size of the poisson equation in the projection method compared to the finite volume and finite difference discretizations due to highly accurate mass conservation the compact sixth order approximation of fsd can be used with an explicit fourth order pressure treatment this property greatly reduces the cost and complexity of the implementation we present the discrete evolution equation of the surface averaged velocities together with the enforcement of mass conservation and the solution procedure for the pressure the approximation of the nse under this new discretization uses a combination of finite difference and finite volume methods the proposed method is validated using standard laminar test cases we identify the conditions under which a fourth order pressure treatment can support the sixth order and eighth order approximations of the convection term using fourier analysis the performance of the method is evaluated on turbulent channel flows up to friction reynolds number of 950 the quantitative relationships between the accuracy of the solution and grid size are identified we present two performance indices for comparison with other methods at the error level of 1 per mille the proposed method is 28 times faster than the classic second order scheme

velocity vorticity helicity formulation and a solver for the navier stokes equations for the three dimensional incompressible navier stokes equations we present a formulation featuring velocity vorticity and helical density as independent variables we find the helical density can be observed as a lagrange multiplier corresponding to the divergence free constraint on the vorticity variable similar to the pressure in the case of the incompressibility condition for velocity as one possible practical application of this new formulation we consider a time splitting numerical scheme based on an alternating procedure between vorticity helical density and velocity bernoulli pressure systems of equations results of numerical experiments include a comparison with some well known schemes based on pressure velocity formulation and illustrate the competitiveness on the new scheme as well as the soundness of the new formulation

numerical solution for an aggregation equation with degenerate diffusion abstract a numerical method for approximating weak solutions of an aggregation equation with degenerate diffusion is introduced the numerical method consists of a finite element method together with a mass lumping technique and an extra stabilizing term plus a semi implicit euler time integration then we carry out a rigorous passage to the limit as the spatial and temporal discretization parameters tend to zero and show that the sequence of finite element approximations converges toward the unique weak solution of the model at hands in doing so nonnegativity is attained due to the stabilizing term and the acuteness on partitions of the computational domain and hence a priori energy estimates of finite element approximations are established as we deal with a nonlinear problem some form of strong convergence is required the key compactness result is obtained via an adaptation of a riesz frechet kolmogorov criterion by perturbation a numerical example is also presented

the long time l2 and h1 stability of linearly extrapolated second order time stepping schemes for the 2d incompressible navier stokes equations abstract herein we present a study on the long time stability of finite element discretizations of a generalized class of semi implicit second order time stepping schemes for the 2d incompressible navier stokes equations these remarkably efficient schemes require only a single linear solve per time step through the use of a linearly extrapolated advective term our result develops a class of sufficient conditions such that if external forcing is uniformly bounded in time velocity solutions are uniformly bounded in time in both the l2 and h1 norms we provide numerical verification of these results we also demonstrate that divergence free finite elements are critical for long time h1 stability

unconditional long time stability of a velocity vorticity method for the 2d navier stokes equations we prove unconditional long time stability for a particular velocity vorticity discretization of the 2d navier stokes equations the scheme begins with a formulation that uses the lamb vector to couple the usual velocity pressure system to the vorticity dynamics equation and then discretizes with the finite element method in space and implicit explicit bdf2 in time with the vorticity equation decoupling at each time step we prove the method s vorticity and velocity are both long time stable in the l 2 l2 and h 1 h1 norms without any timestep restriction moreover our analysis avoids the use of gronwall type estimates which leads us to stability bounds with only polynomial instead of exponential dependence on the reynolds number numerical experiments are given that demonstrate the effectiveness of the method

unconditional stability and convergence of fully discrete schemes for 2d viscous fluids models with mass diffusion in this work we develop fully discrete in time and space numerical schemes for two dimensional incompressible fluids with mass diffusion also so called kazhikhov smagulov models we propose at most h 1 conformed finite elements only globally continuous functions to approximate all unknowns velocity pressure and density although the limit density solution of continuous problem will have h 2 regularity a backward euler in time scheme is considered decoupling the computation of the density from the velocity and pressure unconditional stability of the schemes and convergence towards the unique global in time weak solution of the models is proved since a discrete maximum principle cannot be ensured we must use a different interpolation inequality to obtain the strong estimates for the discrete density from the used one in the continuous case this inequality is a discrete version of the gagliardo nirenberg interpolation inequality in 2d domains moreover the discrete density is truncated in some adequate terms of the velocity pressure problem

error estimates of a linear decoupled euler fem scheme for a mass diffusion model we present error estimates of a linear fully discrete scheme for a three dimensional mass diffusion model for incompressible fluids also called kazhikhov smagulov model all unknowns of the model velocity pressure and density are approximated in space by c 0 finite elements and in time an euler type scheme is used decoupling the density from the velocity pressure pair if we assume that the velocity and pressure finite element spaces satisfy the inf sup condition and the density finite element space contains the products of any two discrete velocities we first obtain point wise stability estimates for the density under the constraint lim h k 0 h k 0 h and k being the space and time discrete parameters respectively and error estimates for the velocity and density in energy type norms at the same time afterwards error estimates for the density in stronger norms are deduced all these error estimates will be optimal of order mathcal o h k for regular enough solutions without imposing nonlocal compatibility conditions at the initial time finally we also study two convergent iterative methods for the two problems to solve at each time step which hold constant matrices independent of iterations

necessary and sufficient optimality conditions for a class of nonsmooth minimization problems the purpose of this paper is to derive in a unified way second order necessary and sufficient optimality criteria for four types of nonsmooth minimization problems thediscrete minimax problem thediscrete l1 approximation the minimization of theexact penalty function and the minimization of theclassical exterior penalty function our results correct and supplement conditions obtained by various authors in recent papers

second order optimality conditions for problems with c1 data abstract in this paper we obtain second order optimality conditions of karush kuhn tucker type and fritz john one for a problem with inequality constraints and a set constraint in nonsmooth settings using second order directional derivatives in the necessary conditions we suppose that the objective function and the active constraints are continuously differentiable but their gradients are not necessarily locally lipschitz in the sufficient conditions for a global minimum x we assume that the objective function is differentiable at x and second order pseudoconvex at x a notion introduced by the authors i ginchev v i ivanov higher order pseudoconvex functions in i v konnov d t luc a m rubinov eds generalized convexity and related topics in lecture notes in econom and math systems vol 583 springer 2007 pp 247 264 the constraints are both differentiable and quasiconvex at x in the sufficient conditions for an isolated local minimum of order two we suppose that the problem belongs to the class c 1 1 we show that they do not hold for c 1 problems which are not c 1 1 ones at last a new notion parabolic local minimum is defined and it is applied to extend the sufficient conditions for an isolated local minimum from problems with c 1 1 data to problems with c 1 one

maximal degeneracy points of gkz systems motivated by mirror symmetry we study certain integral representations of solutions to the gel fand kapranov zelevinsky gkz hypergeometric system some of these solutions arise as period integrals for calabi yau manifolds in mirror symmetry we prove that for a suitable compactification of the parameter space there exists certain special boundary points which we called maximal degeneracy points at which all but one solutions become singular 3 2 96 email hosono sci toyama u ac jp email lian max math brandeis edu email yau math harvard edu

second order conditions in c 1 1 constrained vector optimization we consider the constrained vector optimization problem minc f x g x k where f n m and g n p are c1 1 functions and c m and k p are closed convex cones with nonempty interiors two type of solutions are important for our considerations namely w minimizers weakly efficient points and i minimizers isolated minimizers we formulate and prove in terms of the dini directional derivative second order necessary conditions for a point x0 to be a w minimizer and second order sufficient conditions for x0 to be an i minimizer of order two we discuss the reversal of the sufficient conditions under suitable constraint qualifications of kuhn tucker type the obtained results improve the ones in liu neittaanmaki k i ek 21

singularities of the hypergeometric system associated with a monomial curve we compute using d module restrictions the slopes of the irregular hypergeometric system associated with a monomial curve we also study rational solutions and reducibility of such systems

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

on error analysis for the 3d navier stokes equations in velocity vorticity helicity form we present a rigorous numerical analysis and computational tests for the galerkin finite element discretization of the velocity vorticity helicity formulation of the equilibrium navier stokes equations nses this formulation recently derived by the authors is the first nse formulation that directly solves for helicity and the first velocity vorticity formulation to naturally enforce incompressibility of the vorticity and preliminary computations confirm its potential we present a numerical scheme prove stability existence of solutions uniqueness under a small data condition and convergence and provide numerical experiments to confirm the theory and illustrate the effectiveness of the scheme on a benchmark problem

on the long time stability of the implicit euler scheme for the two dimensional navier stokes equations in this paper we study the stability for all positive time of the fully implicit euler scheme for the two dimensional navier stokes equations more precisely we consider the time discretization scheme and with the aid of the discrete gronwall lemma and the discrete uniform gronwall lemma we prove that the numerical scheme is stable

on the long time stability of a temporal discretization scheme for the three dimensional viscous primitive equations in this article a semi discretized euler scheme to solve the three dimensional viscous primitive equations is studied based on suitable assumptions on the initial data and forcing terms the long time stability of the proposed scheme is proven by showing that the h 1 norm in space variables of the solutions is bounded at each time step when the time step satisfies certain smallness condition

conditional stability and convergence of a fully discrete scheme for three dimensional navier stokes equations with mass diffusion we construct a fully discrete numerical scheme for three dimensional incompressible fluids with mass diffusion in density velocity pressure formulation also called the kazhikhov smagulov model we will prove conditional stability and convergence by using at most c 0 finite elements although the density of the limit problem will have h 2 regularity the key idea of our argument is first to obtain pointwise estimates for the discrete density by imposing the constraint lim h k to 0 h k 0 on the time and space parameters k h afterwards under the same constraint on the parameters strong estimates for the discrete density in l infty h 1 and for the discrete laplacian of the density in l 2 l 2 are obtained from here the compactness and convergence of the scheme can be concluded with similar arguments as we used in math comp to appear where a different scheme is studied for two dimensional domains which is unconditionally stable and convergent moreover we study the asymptotic behavior of the numerical scheme as the diffusion parameter lambda goes to zero obtaining convergence as k h lambda to 0 towards a weak solution of the density dependent navier stokes system provided that the constraint lim lambda h k to 0 h lambda 2 k 0 on h k lambda is satisfied

second order optimality conditions based on parabolic second order tangent sets in this paper we discuss second order optimality conditions in optimization problems subject to abstract constraints our analysis is based on various concepts of second order tangent sets and parametric duality we introduce a condition called second order regularity under which there is no gap between the corresponding second order necessary and second order sufficient conditions we show that the second order regularity condition always holds in the case of semidefinite programming

first and second order optimality conditions for nonsmooth vector optimization using set valued directional derivatives we investigate a nonsmooth vector optimization problem with a feasible set defined by a generalized inequality constraint an equality constraint and a set constraint both necessary and sufficient optimality conditions of first and second order for weak solutions and firm solutions are established in terms of fritz john lagrange multiplier rules using set valued directional derivatives and tangent cones and second order tangent sets we impose steadiness and strict differentiability for first and second order necessary conditions respectively stability and l stability for first and second order sufficient conditions respectively the obtained results improve or include some recent known ones several illustrative examples are also provided

on second order fritz john type optimality conditions in nonsmooth multiobjective programming we study a multiobjective optimization program with a feasible set defined by equality constraints and a generalized inequality constraint we suppose that the functions involved are frechet differentiable and their frechet derivatives are continuous or stable at the point considered we provide necessary second order optimality conditions and also sufficient conditions via a fritz john type lagrange multiplier rule and a set valued second order directional derivative in such a way that our sufficient conditions are close to the necessary conditions some consequences are obtained for parabolic directionally differentiable functions and c1 1 functions in this last case expressed by means of the second order clarke subdifferential some illustrative examples are also given

strict efficiency in vector optimization the notion of strict minimum of order m for real optimization problems is extended to vector optimization its properties and characterization are studied in the case of finite dimensional spaces multiobjective problems also the notion of super strict efficiency is introduced for multiobjective problems and it is proved that in the scalar case all of them coincide necessary conditions for strict minimality and for super strict minimality of order m are provided for multiobjective problems with an arbitrary feasible set when the objective function is frechet differentiable necessary and sufficient conditions are established for the case m 1 resulting in the situation that the strict efficiency and super strict efficiency notions coincide

on second order conditions in unconstrained optimization the main purpose of this paper is to establish the second order nonsmooth sufficient unconstrained optimality condition for so called l stable at some point functions and in this way to generalize some previous results in this direction we provide the comparisons with other results by examples

inclusion of no slip boundary conditions in the meevc scheme abstract this work presents three methods for enforcing tangential velocity boundary conditions for the meevc scheme which was shown to be mass enstrophy energy and vorticity conserving scheme in the case of inviscid flow 1 while the normal velocity component can be strongly imposed in a div conforming formulation for the velocity field inclusion of the tangential velocity needs to be set through an appropriate choice of vorticity boundary conditions three methods to impose the tangential velocity boundary condition will be discussed the kinematic dirichlet formulation the kinematic neumann formulation and the dynamic neumann formulation the conservation properties of each of the resulting schemes are analyzed and numerical results are shown for the taylor green vortex and for the dipole collision test cases these confirm that kinematic neumann vorticity boundary conditions perform best

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

gevrey solutions of irregular hypergeometric systems in two variables we describe the gevrey series solutions at singular points of the irregular hypergeometric system gkz system associated with an affine plane monomial curve we also describe the irregularity complex of such a system with respect to its singular support and in particular we prove using elementary methods that this irregularity complex is a perverse sheaf as assured by a theorem of z mebkhout

the a hypergeometric system associated with a monomial curve introduction inthis paper we make a detailed analysis of the hypergeometric system or gkz system associated with a monomial curve and integral hence resonant exponents we describe all rational solutions and show in theorem 1 10 that they are in fact laurent polynomials we also show that for any exponent there are at most two linearly independent laurent solutions and that the upper bound is reached if and only if the curve is not arithmetically cohen macaulay we then construct for all integral parameters a basis of local solutions in terms of the roots of the generic univariate polynomial 0 5 associated with we also determine in theorem 3 7 the holonomic rank r for all z 2 and show that d r d 1 where d is the degree of the curve moreover the value d 1 is attained only for those exponents for which there are two linearly independent rational solutions and therefore r d for all if and only if the curve is arithmetically cohen macaulay inorder to place these results intheir appropriate con text we recall the defin ition

gkz generalized hypergeometric systems in mirror symmetry of calabi yau hypersurfaces we present a detailed study of the generalized hypergeometric system introduced by gel fand kapranov and zelevinski gkz hypergeometric system in the context of toric geometry gkz systems arise naturally in the moduli theory of calabi yau toric varieties and play an important role in applications of the mirror symmetry we find that the grobner basis for the so called toric ideal determines a finite set of differential operators for the local solutions of the gkz system at the special point called the large radius limit we find a close relationship between the principal parts of the operators in the gkz system and the intersection ring of a toric variety as applications we analyze general three dimensional hypersurfaces of fermat and non fermat types with hodge numbers up toh 1 1 3 we also find and analyze several non landau ginzburg models which are related to singular models

second order optimality conditions with the envelope like effect for nonsmooth vector optimization in infinite dimensions abstract second order necessary conditions and sufficient conditions with the envelope like effect for optimality in nonsmooth vector optimization are established we use approximations as generalized derivatives imposing strict differentiability for necessary conditions and differentiability for sufficient conditions and avoiding continuous differentiability convexity conditions are not imposed explicitly the results make it clear when the envelope like effect occurs and improve or include several recent existing ones examples are provided to show advantages of our theorems over some known ones in the literature

long time stability of a classical efficient scheme for two dimensional navier stokes equations this paper considers the long time stability property of a popular semi implicit scheme for the two dimensional incompressible navier stokes equations in a periodic box that treats the viscous term implicitly and the nonlinear advection term explicitly we consider both the semidiscrete discrete in time but continuous in space and fully discrete schemes with either fourier galerkin spectral or fourier pseudospectral collocation methods we prove that in all cases the scheme is long time stable provided that the timestep is sufficiently small the long time stability in the l 2 and h 1 norms further leads to the convergence of the global attractors and invariant measures of the scheme to those of the navier stokes equations at vanishing timestep

a novel solution method for unsteady incompressible euler flow using the vorticity bernoulli pressure formulation abstract a novel numerical formulation and time integration method for the incompressible euler flow equations is presented using the vorticity bernoulli pressure system lagrangian advection of vorticity carrying particles is adopted from classical vortex methods that are known to give accurate results for vortex dominated flows and allow larger time steps than explicit eulerian schemes solving only one scalar poisson equation per time step instead of a vector valued equation in vortex methods makes the new scheme theoretically more efficient on large computational grids the equation of motion in rotation form is integrated in time under certain approximation assumptions and solved explicitly for the flow velocity the velocity is eliminated as an unknown in the continuity equation which is efficiently solved for the bernoulli pressure using the predicted vorticity field obtained by lagrangian advection the bernoulli pressure is efficiently obtained from the poisson equation in finite volume formulation an estimated velocity field is calculated from the equation of motion to effectively counterbalance the discretization errors made by the prediction in a corrector step the kinematic relation between vorticity and estimated velocity is reformulated and solved in a lagrangian framework as well the velocity field is corrected at the end of the time step two dimensional cases on cartesian grids are considered numerical results for three flow cases demonstrate the locally smooth and accurate behavior and the second order spatial accuracy of the solution the new scheme produces significantly less artificial vortex distortion than the classical vortex method the favorable properties of the solutions with respect to accuracy and long time stability of the overall kinetic energy and enstrophy are also emphasized

second order optimality conditions with the envelope like effect in nonsmooth multiobjective mathematical programming ii optimality conditions second order necessary conditions and sufficient conditions with the envelope like effect for optimality in nonsmooth multiobjective mathematical programming are established we use set valued second order directional derivatives and impose strict differentiability for necessary conditions and l stability for sufficient conditions avoiding continuous differentiability the results improve and sharpen several recent existing ones examples are provided to show advantages of our theorems over some known ones in the literature in part i we consider l stability and second order set valued directional derivatives of vector functions part ii is devoted to second order necessary optimality conditions and sufficient ones

duality and monodromy reducibility of a hypergeometric systems we study hypergeometric systems ha in the sense of gelfand kapranov and zelevinsky under two aspects the structure of their holonomically dual system and reducibility of their rank module we prove in the first part that rank jumping parameters always correspond to reducible systems we show further that the property of being reducible is invariant modulo the lattice and obtain as a special instance a theorem of alicia dickenstein and timur sadykov on reducibility of mellin systems in the second part we study a conjecture of nobuki takayama which states that the holonomic dual of ha is of the form ha for suitable we prove the conjecture for all matrices a and generic parameter exhibit an example that shows that in general the conjecture cannot hold and present a refined version of the conjecture questions on both duality and reducibility have been quite difficult to answer with classical methods this paper may be seen as an example of the usefulness and scope of applications of the homological tools for a hypergeometric systems developed in matusevich et al j amer math soc 18 919 941 2005

linear relations among algebraic solutions of differential equations abstract for any univariate polynomial with coefficients in a differential field of characteristic zero and any integer q there exists an associated nonzero linear ordinary differential equation lode with the following two properties each term of the lode lies in the differential field generated by the rational numbers and the coefficients of the polynomial and the q th power of each root of the polynomial is a solution of this lode this lode is called a q th power resolvent of the polynomial we will show how one can get a resolvent for the logarithmic derivative of the roots of a polynomial from the th power resolvent of the polynomial where is an indeterminate that takes the place of q we will demonstrate some simple relations among the algebraic and differential equations for the roots and their logarithmic derivatives we will also prove several theorems regarding linear relations of roots of a polynomial over constants or the coefficient field of the polynomial depending upon the nondifferential galois group finally we will use a differential resolvent to solve the riccati equation

an efficient second order in time scheme for approximating long time statistical properties of the two dimensional navier stokes equations we investigate the long time behavior of the following efficient second order in time scheme for the 2d navier stokes equations in a periodic box begin array ll frac 3 omega n 1 4 omega n omega n 1 2k nabla perp 2 psi n psi n 1 cdot nabla 2 omega n omega n 1 nu delta omega n 1 f n 1 quad delta psi n omega n end array the scheme is a combination of a 2nd order in time backward differentiation and a particular explicit adams bashforth treatment of the advection term therefore only a linear constant coefficient poisson solver is needed at each time step we prove uniform in time bounds on this scheme in dot l 2 dot h 1 per and dot h 2 per provided that the time step is sufficiently small these time uniform estimates further lead to the convergence of long time statistics stationary statistical properties of the scheme to that of the nse itself at vanishing time step

global well posedness of the velocity vorticity voigt model of the 3d navier stokes equations abstract the velocity vorticity formulation of the 3d navier stokes equations was recently found to give excellent numerical results for flows with strong rotation in this work we propose a new regularization of the 3d navier stokes equations which we call the 3d velocity vorticity voigt vvv model with a voigt regularization term added to momentum equation in velocity vorticity form but with no regularizing term in the vorticity equation we prove global well posedness and regularity of this model under periodic boundary conditions we prove convergence of the model s velocity and vorticity to their counterparts in the 3d navier stokes equations as the voigt modeling parameter tends to zero we prove that the curl of the model s velocity converges to the model vorticity which is solved for directly as the voigt modeling parameter tends to zero finally we provide a criterion for finite time blow up of the 3d navier stokes equations based on this inviscid regularization

second order optimality conditions with the envelope like effect in nonsmooth multiobjective mathematical programming i l stability and set valued directional derivatives abstract second order necessary conditions and sufficient conditions with the envelope like effect for optimality in nonsmooth multiobjective mathematical programming are established we use set valued second order directional derivatives and impose strict differentiability for necessary conditions and l stability for sufficient conditions avoiding continuous differentiability the results improve and sharpen several recent existing ones examples are provided to show advantages of our theorems over some known ones in the literature in part i we consider l stability and second order set valued directional derivatives of vector functions part ii is devoted to second order necessary optimality conditions and sufficient ones

residual based vms subgrid modeling for vortex flows this paper presents a residual based subgrid modeling approach for large eddy simulations les based on the variational multiscale method as a cure for the problem of preservation of vortices in numerical flow simulation this approach combines a splitting of the non linear term in the navier stokes equations into strain and vorticity with a residual based modeling of the subgrid problems the benefit is that certain driving phenomena normally not present in subgrid modeling e g vortex stretching can be seen in the equations here we focus on two of the subgrid terms arising from the subgrid scale problem the effect of the two terms are illustrated in an les of a three dimensional flow around a wing where the main feature is the formation and preservation of a tip vortex an important phenomenon in many aerodynamic and hydrodynamical applications we see that the addition of the new subgrid terms correctly counteracts the dissipative effect arising from numerics and turbulence modeling on the vortex and thus strongly improves prediction of the tip vortex c 2009 elsevier b v all rights reserved

efficient and long time accurate second order methods for the stokes darcy system we propose and study two second order in time implicit explicit methods for the coupled stokes darcy system that governs flows in karst aquifers and other subsurface flow systems the first method is a combination of a second order backward differentiation formula and the second order gear s extrapolation approach the second is a combination of the second order adams moulton and second order adams bashforth methods both algorithms only require the solution of decoupled stokes and darcy problems at each time step hence these schemes are very efficient and can be easily implemented using legacy codes we establish the unconditional and uniform in time stability for both schemes the uniform in time stability leads to uniform in time control of the error which is highly desirable for modeling physical processes e g contaminant sequestration and release that occur over very long time scales error estimates for fully discretized schemes using finite element spatial discretizations are derived nume

second order conditions for optimization problems with constraints using a projective approach new necessary conditions and new sufficient conditions for optimization problems with explicit or implicit constraints are examined they are compared to previous ones a particular emphasis is given to mathematical programming problems with non polyhedral constraints this case occurs in particular when the constraints are defined in functional spaces

optimality conditions in mathematical programming and composite optimization new second order optimality conditions for mathematical programming problems and for the minimization of composite functions are presented they are derived from a general second order fermat s rule for the minimization of a function over an arbitrary subset of a banach space the necessary conditions are more accurate than the recent results of kawasaki 1988 and cominetti 1989 but more importantly in the finite dimensional case they are twinned with sufficient conditions which differ by the replacement of an inequality by a strict inequality we point out the equivalence of the mathematical programming problem with the problem of minimizing a composite function our conditions are especially important when one deals with functional constraints when the cone defining the constraints is polyhedral we recover the classical conditions of ben tal zowe 1982 and cominetti 1990

unconditional stability and long term behavior of transient algorithms for the incompressible navier stokes and euler equations abstract this paper examines the long term behavior dissipativity and unconditional nonlinear stability properties of time integration algorithms for the incompressible navier stokes equations including both direct schemes and fractional step projection methods the algorithms are termed nonlinear unconditionally stable if in the absence of a forcing term but for arbitrary initial conditions the computed kinetic energy decreases for arbitrary step sizes i e l 2 stability such an a priori stability estimate is a characteristic feature of the navier stokes system similarly the algorithms are said to exhibit asymptotic t independent long term dissipative behavior if as the continuum flow generated by the navier stokes equations the algorithmic flow also exhibits an absorbing set and a maximal attractor both direct and fractional step methods are described which inherit these fundamental properties of the navier stokes system for any time step size moreover a subclass of algorithms which retain these strong notions of nonlinear stability and long term dissipative behavior is identified which in addition has the remarkable property of being linear within the time step the key implication is that no iterations are required to compute the solution within a time step a specific member of this class is shown to possess the same absorbing set as the continuum navier stokes system a second order accurate unconditionally stable method is also identified which retains the property of linearity within a time step numerical analysis and computational aspects involved in the implementation of these methods are addressed in detail and illustrated in representative numerical simulations

approximation by an iterative method for regular solutions for incompressible fluids with mass diffusion we study the approximation by means of an iterative method towards strong and more regular solutions for incompressible navier stokes equations with mass diffusion in addition some convergence rates for the error between the approximation and the exact solution will be given for weak strong and more regular norms

natural vorticity boundary conditions on solid walls abstract we derive a new kind of boundary conditions for the vorticity equation with solid wall boundaries for fluid flow problems the formulation uses a dirichlet condition for the normal component of vorticity and neumann type conditions for the tangential components in a galerkin integral formulation the tangential condition is natural i e it is enforced by a right hand side functional and does not impose a boundary constraint on trial and test spaces the functional involves the pressure variable and we discuss several velocity vorticity formulations where the proposed condition is appropriate several numerical experiments are given that illustrate the validity of the new approach

swarming patterns in a two dimensional kinematic model for biological groups we construct a continuum model for the motion of biological organisms experiencing social interactions and study its pattern forming behavior the model takes the form of a conservation law in two spatial dimensions the social interactions are modeled in the velocity term which is nonlocal in the population density and includes a parameter that controls the interaction length scale the dynamics of the resulting partial integrodifferential equation may be uniquely decomposed into incompressible motion and potential motion for the purely incompressible case the model resembles one for fluid dynamical vortex patches there exist solutions which have constant population density and compact support for all time numerical simulations produce rotating structures which have circular cores and spiral arms and are reminiscent of naturally observed phenomena such as ant mills the sign of the social interaction term determines the direction of the rotation and the interaction length scale affects the degree o

first and second order sufficient conditions for strict minimality in nonsmooth vector optimization abstract in this paper we present first and second order sufficient conditions for strict local minima of orders 1 and 2 to vector optimization problems with an arbitrary feasible set and a twice directionally differentiable objective function with this aim the notion of support function to a vector problem is introduced in such a way that the scalar case and the multiobjective case in particular are contained the obtained results extend the multiobjective ones to this case moreover specializing to a feasible set defined by equality inequality and set constraints first and second order sufficient conditions by means of lagrange multiplier rules are established

numerics and hydrodynamic stability toward error control in computational fluid dynamics we critically review the available error analysis in computational fluid dynamics cfd and come to the conclusion that the existing error estimates are meaningless in most cases of interest we propose a new approach to error analysis in cfd aiming at reliable and efficient adaptive quantitative error control this is based on a precise analysis of hydrodynamic stability coupled with galerkin orthogonality we prove a priori and a posteriori type error estimates in a model case for pipe flow formulate corresponding adaptive algorithms and discuss the potential of this approach for adaptive error control in cfd

hypergeometric functions for projective toric curves abstract we produce a decomposition of the parameter space of the a hypergeometric system associated to a projective monomial curve as a union of an arrangement of lines and its complement in such a way that the analytic behavior of the solutions of the system is explicitly controlled within each term of the union

long term stability estimates and existence of a global attractor in a finite element approximation of the navier stokes equations with numerical subgrid scale modeling variational multiscale methods lead to stable finite element approximations of the navier stokes equations dealing with both the indefinite nature of the system pressure stability and the velocity stability loss for high reynolds numbers these methods enrich the galerkin formulation with a subgrid component that is modeled in fact the effect of the subgrid scale on the captured scales has been proved to dissipate the proper amount of energy needed to approximate the correct energy spectrum thus they also act as effective large eddy simulation turbulence models and allow one to compute flows without the need to capture all the scales in the system in this article we consider a dynamic subgrid model that enforces the subgrid component to be orthogonal to the finite element space in the l 2 sense we analyze the long term behavior of the algorithm proving the existence of appropriate absorbing sets and a compact global attractor the improvements with respect to a finite element galerkin approximation are the long term estimates for the subgrid component which are translated to effective pressure and velocity stability thus the stabilization introduced by the subgrid model into the finite element problem does not deteriorate for infinite time intervals of computation

finite element approximation of the nonstationary navier stokes problem part ii stability of solutions and error estimates uniform in time in this paper assumptions about the stability of a solution are introduced in the numerical analysis of the nonstationary navier stokes problem for the purpose of extending local a priori error estimates and local a posteriori error estimates globally in time