triangulations of subanalytic sets and locally subanalytic manifolds if two polyhedrons are locally subanalytically homeomorphic that is the graph is locally subanalytic they are pl homeomorphic a lo cally subanalytic manifold is one whose coordinate transformations are locally subanalytic it is proved that a locally subanalytic manifold has a unique pl manifold structure a semialgebraic manifold also is considered

long time numerical solution of the navier stokes equations based on a sequential regularization formulation the sequential regularization method is a reformulation of the unsteady navier stokes equations from the viewpoint of constrained dynamical systems or the approximate helmholtz hodge projection in this paper we study the long time behavior of the sequential regularization formulation we give a uniform in time estimate between the solution of the reformulated system and that of the navier stokes equations we also conduct an error analysis for the temporal discrete system and show that the error bound is independent of time a couple of long time flow examples are computed to demonstrate this method

high order algebraic splitting for magnetohydrodynamics simulation this paper proposes analyzes and tests high order algebraic splitting methods for magnetohydrodynamic mhd flows the main idea is to apply at each time step yosida type algebraic splitting to a block saddle point problem that arises from a particular incremental formulation of mhd by doing so we dramatically reduce the complexity of the nonsymmetric block schur complement by decoupling it into two stokes type schur complements each of which is symmetric positive definite and also is the same at each time step we prove the splitting is o t3 accurate and if used together with block pressure correction is fourth order a full analysis of the solver is given both as a linear algebraic approximation but also in a finite element context that uses the natural spatial norms numerical tests are given to illustrate the theory and show the effectiveness of the method

note on helicity balance of the galerkin method for the 3d navier stokes equations we study the helicity balance of the galerkin method for the 3d navier stokes equations and show that although it does not appear to correctly balance helicity in the usual sense it instead admits a slightly altered helicity balance that matches that of the underlying physics up to boundary conditions we consider helicity treatment in the commonly used galerkin discretization for the 3d navier stokes equations nse it is well known that the nse conserves helicity 13 in the absence of viscous or body forces and admits a precise helicity balance when these forces are present this balance is believed critical in flow structure development 12 and is the basis for the cascade of helicity joint with energy through the inertial range 4 therefore it is very desirable that the helicity balance be preserved in a numerical scheme however in contrast to the energy balance the helicity balance does not hold in the usual way for the standard galerkin approximations see 2 7 and thus creative and naturally more expensive discretizations have been developed that do so first in 10 for axisymmetric flow and later in 14 for the full 3d nse we show herein that the standard galerkin method instead balances a slightly altered discrete helicity type quantity

global stability of a delayed sir epidemic model with density dependent birth and death rates an sir epidemic model with density dependent birth and death rates is formulated in our model it is assumed that the total number of the population is governed by logistic equation the transmission of infection is assumed to be of the standard form namely proportional to i t h n t h where n t is the total variable population size i t is the size of the infective population and a time delay h is a fixed time during which the infectious agents develop in the vector we consider transmission dynamics for the model stability of an endemic equilibrium is investigated the stability result is stated in terms of a threshold parameter that is a basic reproduction number r 0

a nonlinear aids epidemic model with screening and time delay abstract a nonlinear mathematical model to study the effect of time delay in the recruitment of infected persons on the transmission dynamics of hiv aids is proposed and analyzed in modeling the dynamics the population is divided into four subclasses the susceptibles the hiv positives or infectives that do not know they are infected the hiv positives that know they are infected and the aids patients susceptibles are assumed to become infected via sexual contacts with both types of infectives the model is analyzed using stability theory of delay differential equations both the disease free and the endemic equilibria are found and their stability is investigated it is shown that the introduction of time delay in the model has a destabilizing effect on the system and periodic solutions can arise by hopf bifurcation numerical simulations are also carried out to investigate the influence of key parameters on the spread of the disease to support the analytical conclusion and to illustrate possible behavioral scenario of the model

on conservation laws of navier stokes galerkin discretizations abstract we study conservation properties of galerkin methods for the incompressible navier stokes equations without the divergence constraint strongly enforced in typical discretizations such as the mixed finite element method the conservation of mass is enforced only weakly and this leads to discrete solutions which may not conserve energy momentum angular momentum helicity or vorticity even though the physics of the navier stokes equations dictate that they should we aim in this work to construct discrete formulations that conserve as many physical laws as possible without utilizing a strong enforcement of the divergence constraint and doing so leads us to a new formulation that conserves each of energy momentum angular momentum enstrophy in 2d helicity and vorticity for reference the usual convective formulation does not conserve most of these quantities several numerical experiments are performed which verify the theory and test the new formulation

on the leray deconvolution model for the incompressible magnetohydrodynamics equations abstract we extend leray deconvolution modeling to the incompressible magnetohydrodynamics mhd the resulting model is shown to be well posed and have attractive limiting behavior both in its filtering radius and order of deconvolution additionally we present and study a numerical scheme for the model based on an extrapolated crank nicolson finite element method we show the numerical scheme is unconditionally stable preserves energy and cross helicity and optimally converges to the mhd solution numerical experiments are provided that verify convergence rates and test the scheme on benchmark problems of channel flow over a step and the orszag tang vortex problem

effectiveness non effectiveness in semialgebraic and pl geometry f acquistapace r benedetti and f broglia dipartimento di matematica universit i di pisa via f buonarroti 2 1 56127 pisa italy oblatum 14 iv 1989 25 i 1990 0 introduction and statement of the main results working in real semialgebraic geometry one could easily have the feeling that the main facts can be realized by algorithms e g path connecting points triangul ations good stratifications and the main trouble is to construct reasonably fast algorithms in fact it is this in principle constructive nature of semialgebraic geometry and the existence of powerful computers which make it interesting in view of concrete applications on the other hand nabutovsky s examples of non recursive functions associ ated to natural semialgebraic constructions see i n tell us that there are some conceptual limitations to the above attitude in this paper we consider some further examples of effectiveness non effect iveness results in semialgebraic and pl geometry our starting point was the study of shiota yokoi solution of the subanalytic hence semialgebraic hauptvermutung sy and the attempt to make it effective before starting we need to introduce some notations we shall use systematically the notation

asymptotic properties of an hiv aids model with a time delay a mathematical model for hiv aids with explicit incubation period is presented as a system of discrete time delay differential equations and its important mathematical features are analysed the disease free and endemic equilibria are found and their local stability investigated we use the lyapunov functional approach to show the global stability of the endemic equilibrium qualitative analysis of the model including positivity and boundedness of solutions and persistence are also presented the hiv aids model is numerically analysed to asses the effects of incubation period on the dynamics of hiv aids and the demographic impact of the epidemic using the demographic and epidemiological parameters for zimbabwe

enforcing energy helicity and strong mass conservation in finite element computations for incompressible navier stokes simulations abstract we study a finite element scheme for the 3d navier stokes equations nse that globally conserves energy and helicity and through the use of scott vogelius elements enforces pointwise the solenoidal constraints for velocity and vorticity a complete numerical analysis is given including proofs for conservation laws unconditional stability and optimal convergence we also show the method can be efficiently computed by exploiting a connection between this method its associated penalty method and the method arising from using grad div stabilized taylor hood elements finally we give numerical examples which verify the theory and demonstrate the effectiveness of the scheme

the field of reals with multisummable series and the exponential function we show that the field of real numbers with multisummable real power series is model complete o minimal and polynomially bounded further expansion by the exponential function yields again a model complete and o minimal structure which is exponentially bounded and in which the gamma function on the positive real line is definable 2000 mathematics subject classification primary 03c10 32b05 32b20 secondary 26e05

convergence analysis of the energy and helicity preserving scheme for axisymmetric flows we give an error estimate for the energy and helicity preserving scheme ehps in second order finite difference setting on axisymmetric incompressible flows with swirling velocity this is accomplished by a weighted energy estimate along with careful and nonstandard local truncation error analysis near the geometric singularity and a far field decay estimate for the stream function a key ingredient in our a priori estimate is the permutation identities associated with the jacobians which are also a unique feature that distinguishes ehps from standard finite difference schemes

a high order discontinuous galerkin method for 2d incompressible flows in this paper we introduce a high order discontinuous galerkin method for two dimensional incompressible flow in the vorticity stream function formulation the momentum equation is treated explicitly utilizing the efficiency of the discontinuous galerkin method the stream function is obtained by a standard poisson solver using continuous finite elements there is a natural matching between these two finite element spaces since the normal component of the velocity field is continuous across element boundaries this allows for a correct upwinding gluing in the discontinuous galerkin framework while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability the method is efficient for inviscid or high reynolds number flows optimal error estimates are proved and verified by numerical experiments

energy and helicity preserving schemes for hydro and magnetohydro dynamics flows with symmetry we propose a class of simple and efficient numerical scheme for incompressible fluid equations with coordinate symmetry by introducing a generalized vorticity stream formulation the divergence free constraints are automatically satisfied in addition with explicit treatment of the nonlinear terms and local vorticity boundary condition the navier stokes mhd respectively equation essentially decouples into 2 4 respectively scalar equation and thus the scheme is very efficient moreover with proper discretization of the nonlinear terms the scheme preserves both energy and helicity identities numerically this is achieved by recasting the nonlinear terms convection vorticity stretching geometric source lorentz force and electro motive force in terms of jacobians this conservative property is valid even in the presence of the pole singularity for axisymmetric flows the exact conservation of energy and helicity has effectively eliminated excessive numerical viscosity numerical examples have demonstrated both accuracy and efficiency of the scheme finally local mesh refinement near the boundary can also be easily incorporated into the scheme without extra cost

an energy and helicity conserving finite element scheme for the navier stokes equations we present a new finite element scheme for solving the navier stokes equations that exactly conserves both energy int omega u 2 and helicity int omega u cdot nabla times u in the absence of viscosity and external force we prove stability exact conservation and convergence for the scheme energy and helicity are exactly conserved by using a combination of the usual convective form with the rotational form of the nonlinearity and solving for both velocity and a projected vorticity in a trapezoidal time discretization numerical results are presented that compare the scheme to the usual trapezoidal schemes

large oscillations arising in a dispersive numerical scheme abstract we study the oscillatory behavior that arises in solutions of a dispersive numerical scheme for the hopf equation whenever the classical solution of the equation develops a singularity modulation equations are derived that describe period two oscillations so long as the solution of those equations takes values for which the equations are hyperbolic however those equations have an elliptic region that may be entered by its solutions in a finite time after which the corresponding period two oscillations are seen to break down this kind of phenomenon has not been observed for integrable schemes the generation and propagation of period two oscillations are asymptotically analyzed and a matching formula is found for the transition between oscillatory and nonoscillatory regions modulation equations are also presented for period three oscillations numerical experiments are carried out that illustrate our analysis

smooth functions in o minimal structures abstract fix an o minimal expansion of the real exponential field that admits smooth cell decomposition we study the density of definable smooth functions in the definable continuously differentiable functions with respect to the definable version of the whitney topology this implies that abstract definable smooth manifolds are affine moreover abstract definable smooth manifolds are definably c diffeomorphic if and only if they are definably c 1 diffeomorphic

o minimal hauptvermutung for polyhedra i milnor discovered two compact polyhedra which are homeomorphic but not pl homeomorphic a counterexample to the hauptvermutung he constructed the homeomorphism by a finite procedure repeated infinitely often informally we call a procedure constructive if it consists of an explicit procedure that is repeated only finitely many times in this sense milnor did not give a constructive procedure to define the homeomorphism between the two polyhedra in the case where the homeomorphism is semialgebraic the author and yokoi proved that the polyhedra in r n are pl homeomorphic in that article the required pl homeomorphism was not constructively defined from the given homeomorphism in the present paper we obtain the pl homeomorphism by a constructive procedure starting from the homeomorphism we prove in fact that for any ordered field r equipped with any o minimal structure two definably homeomorphic compact polyhedra in r n are pl homeomorphic the o minimal hauptvermutung theorem 1 1 together with the fact that any compact definable set is definably homeomorphic to a compact polyhedron we can say that o minimal topology is tame

energy balance and mass conservation in reduced order models of fluid flows abstract in this paper we investigate theoretically and computationally the conservation properties of reduced order models roms for fluid flows specifically we investigate whether the roms satisfy the same or similar energy balance and mass conservation as those satisfied by the navier stokes equations all of our theoretical findings are illustrated and tested in numerical simulations of a 2d flow past a circular cylinder at a reynolds number r e 100 first we investigate the rom energy balance we show that using the snapshot average for the centering trajectory which is a popular treatment of nonhomogeneous boundary conditions in roms yields an incorrect energy balance then we propose a new approach in which we replace the snapshot average with the stokes extension theoretically the stokes extension produces an accurate energy balance numerically the stokes extension yields more accurate results than the standard snapshot average especially for longer time intervals our second contribution centers around rom mass conservation we consider roms created using two types of finite elements the standard taylor hood th element which satisfies the mass conservation weakly and the scott vogelius sv element which satisfies the mass conservation pointwise theoretically the error estimates for the sv rom are sharper than those for the th rom numerically the sv rom yields significantly more accurate results especially for coarser meshes and longer time intervals

the real field with convergent generalized power series we construct a model complete and o minimal expansion of the field of real numbers in which each real function given on 0 1 by a series cnxn with 0 n and cn r n 1 is definable this expansion is polynomially bounded

remarks on singularities dimension and energy dissipation for ideal hydrodynamics and mhd for weak solutions of the incompressible euler equations there is energy conservation if the velocity is in the besov space b 3 s with s greater than 1 3 b 3 s consists of functions that are lip s i e holder continuous with exponent s measured in the l p norm here this result is applied to a velocity field that is lip 0 except on a set of co dimension on which it is lip agr 1 with uniformity that will be made precise below we show that the frisch parisi multifractal formalism is valid at least in one direction for such a function and that there is energy conservation if analogous conservation results are derived for the equations of incompressible ideal mhd i e zero viscosity and resistivity for both energy and helicity in addition a necessary condition is derived for singularity development in ideal mhd generalizing the beale kato majda condition for ideal hydrodynamics

computational design for long term numerical integration of the equations of fluid motion the integral constraints on quadratic quantities of physical importance such as conservation of mean kinetic energy and mean square vorticity will not be maintained in finite difference analogues of the equation of motion for two dimensional incompressible flow unless the finite difference jacobian expression for the advection term is restricted to a form which properly represents the interaction between grid points as derived in this paper it is shown that the derived form of the finite difference jacobian prevents nonlinear computational instability and thereby permits long term numerical integrations

longer time accuracy for incompressible navier stokes simulations with the emac formulation abstract in this paper we consider the recently introduced emac formulation for the incompressible navier stokes ns equations which is the only known ns formulation that conserves energy momentum and angular momentum when the divergence constraint is only weakly enforced since its introduction the emac formulation has been successfully used for a wide variety of fluid dynamics problems we prove that discretizations using the emac formulation are potentially better than those built on the commonly used skew symmetric formulation by deriving a better longer time error estimate for emac while the classical results for schemes using the skew symmetric formulation have gronwall constants dependent on exp c r e t with r e the reynolds number it turns out that the emac error estimate is free from this explicit exponential dependence on the reynolds number additionally it is demonstrated how emac admits smaller lower bounds on its velocity error since incorrect treatment of linear momentum angular momentum and energy induces lower bounds for l 2 velocity error and emac treats these quantities more accurately results of numerical tests for channel flow past a cylinder and 2d kelvin helmholtz instability are also given both of which show that the advantages of emac over the skew symmetric formulation increase as the reynolds number gets larger and for longer simulation times

oscillation analysis of advertising capital model analytical and numerical studies abstract this paper mainly deals with the oscillation of nonlinear delay differential equation which is used to describe advertising capital model analytically and numerically firstly the condition of oscillation of the analytic solution is presented by the technique of the theory of characteristic secondly the asymptotic behavior of non oscillatory analytic solution is verified thirdly the methods are applied to the mentioned equation some conditions under which the numerical solution oscillates are obtained moreover it is proved that every non oscillatory numerical solution tends to the steady state of the model finally some numerical simulations for verifying the theoretical findings are provided

vorticity boundary condition and related issues for finite difference schemes this paper discusses three basic issues related to the design of finite difference schemes forunsteadyviscous incompressible flows using vorticity formulations the boundary condition for vorticity an efficient time stepping procedure and the relation between these schemes and the ones based on velocity pressure formulation we show that many of the newly developed global vorticity boundary conditions can actually be written as some local formulas derived earlier we also show that if we couple a standard centered difference scheme with third or fourth order explicit runge kutta methods the resulting schemes havenocell reynolds number constraints for high reynolds number flows these schemes are stable under the cfl condition given by the convective terms finally we show that the classical mac scheme is the same as thom s formula coupled with second order centered differences in the interior in the sense that one can define discrete vorticity in a natural way for the mac scheme and get the same values as the ones computed from thom s formula we use this to derive an efficient fourth order runge kutta time discretization for the mac scheme from the one for thom s formula we present numerical results for driven cavity flow at high reynolds number 105

an energy preserving mac yee scheme for the incompressible mhd equation we propose a simple and efficient finite difference method for the incompressible mhd equation the numerical method combines the advantage of the mac scheme for the navier stokes equation and yee s scheme for the maxwell equation in particular the semi discrete version of our scheme introduces no numerical dissipation and preserves the energy identity exactly c

summation by parts projections and stability ii we have derived stability results for high order finite difference approximations of mixed hyperbolic parabolic initial boundary value problems ibvp the results are obtained using summation by parts and a new way of representing general linear boundary conditions as an orthogonal projection by slightly rearranging the analytic equations we can prove strict stability for hyperbolic parabolic ibvp furthermore we generalize our technique so as to yield strict stability on curvilinear non smooth domains in two space dimensions finally we show how to incorporate inhomogeneous boundary data while retaining strict stability using the same procedure one can prove strict stability in higher dimensions as well

global stability of multigroup epidemic model with group mixing and nonlinear incidence rates in this paper we introduce a basic reproduction number for a multigroup seir model with nonlinear incidence of infection and nonlinear removal functions between compartments then we establish that global dynamics are completely determined by the basic reproduction number r0 it shows that the basic reproduction number r0 is a global threshold parameter in the sense that if it is less than or equal to one the disease free equilibrium is globally stable and the disease dies out whereas if it is larger than one there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population finally two numerical examples are also included to illustrate the effectiveness of the proposed result

an efficient dynamically adaptive mesh for potentially singular solutions abstract we develop an efficient dynamically adaptive mesh generator for time dependent problems in two or more dimensions the mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic pdes for the mesh map when coupled to a physical problem the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior the overall mesh strategy is simple to implement avoids interpolation and can be easily incorporated into a broad range of applications the efficacy of the mesh is first demonstrated by two examples of blowing up solutions to the 2 d semilinear heat equation these examples show that the mesh can follow with high adaptivity a finite time singularity process the focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2 d boussinesq fluid a challenging problem the moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically the numerical results show the fast collapse to small scales and an exponential vorticity growth

mass momentum and energy conserving mamec discretizations on general grids for the compressible euler and shallow water equations the paper explains a method by which discretizations of the continuity and momentum equations can be designed such that they can be combined with an equation of state into a discrete energy equation the resulting mamec discretizations conserve mass momentum as well as energy although no explicit conservation law for the total energy is present essential ingredients are i discrete convection that leaves the discrete energy invariant and ii discrete consistency between the thermodynamic terms of particular relevance is the way in which finite volume fluxes are related to nodal values the method is an extension of existing methods based on skew symmetry of discrete operators because it allows arbitrary equations of state and a larger class of grids than earlier methods the method is first illustrated with a one dimensional example on a highly stretched staggered grid in which the mamec method calculates qualitatively correct results and a non skew symmetric finite volume method becomes unstable a further example is a two dimensional shallow water calculation on a rectilinear grid as well as on an unstructured grid the conservation of mass momentum and energy is checked and losses are found negligible up to machine accuracy

the elementary theory of restricted analytic fields with exponentiation numbers with exponentiation is model complete when we combine this with hovanskii s finiteness theorem 9 it follows that the real exponential field is o minimal in o minimal expansions of the real field the definable subsets of r share many of the nice structural properties of semialgebraic sets for example definable subsets have only finitely many connected components definable sets can be stratified and triangulated and continuous definable maps are piecewise trivial see 5 in this paper we will prove a quantifier elimination result for the real field augmented by exponentiation and all restricted analytic functions and use this result to obtain o minimality we were led to this while studying work of ressayre 13 and several of his ideas emerge here in simplified form however our treatment is formally independent of the results of 16 17 9 and 13

a note on helicity conservation in leray models of incompressible flow abstract we prove conservation of a regularized helicity h l u w d x for the leray model and its variants of turbulent flow where w is the solution of a leray regularized vorticity equation the usual definition of helicity is h u u d x which is considered by navier stokes flows but is not a conserved quantity of the leray model however if u is a leray solution then the difference between h and h l is that h l uses a regularized vorticity and h uses the curl of a regularized velocity the results are extended to show that the standard crank nicolson finite element method for leray models conserves both discrete energy and discrete regularized helicity

graph mappings and poincare duality we give a new geometric description for a compact oriented pseduo manifold x of the poincare duality map from the integral cohomology of x to the integral homology of x our construction takes a multi valued lipschitz map on x with values in a sphere sn to its geometric measure theoretic graph in x sn and then to the slice of this graph as an integral cycle on x this construction is compatible with analogous constructions on algebraic cocycles on projective varieties employed by the authors and others

convergence of a galerkin method for 2 d discontinuous euler flows we prove the convergence of a discontinuous galerkin method approximating the 2 d incompressible euler equations with discontinuous initial vorticity 0 2 l 2 furthermore when 0 2 l the whole sequence is shown to be strongly convergent this is the first convergence result in numerical approximations of this general class of discontinuous flows some important flows such as vortex patches belong to this class c 2000 john wiley sons inc

stiefel whitney classes and the conormal cycle of a singular variety a geometric construction of sullivan s stiefel whitney homology classes of a real analytic variety x is given by means of the conormal cycle of an embedding of x in a smooth variety we prove that the stiefel whitney classes define additive natural transformations from certain constructible functions to homology we also show that for a complex analytic variety these classes are the mod 2 reductions of the chern macpherson classes we present a new definition of the stiefel whitney homology classes of a possibly singular real analytic variety x the original definition due to sullivan s involves a triangulation of x its geometric meaning is unclear our definition uses the conormal cycle of an embedding of x in a smooth variety the conormal cycle of a subanalytic subset x of an analytic manifold m was defined by the first author f4 using geometric measure theory the conormal cycle is an integral current representing up to sign kashiwara s characteristic cycle of the sheaf dmrx f4 4 7 our construction of the stiefel whitney classes is based on the fundamental observation that the conormal cycle of a real analytic variety is antipodally symmetric mod 2 we use this observation to give a definition of stiefel whitney classes which is parallel to the first author s definition of the chern schwartz macpherson classes of a complex analytic variety f5 this definition of chern classes is related to earlier work of brylinski dubson kashiwara bdk and sabbah sa in fact we show that the stiefel whitney homology classes of a complex analytic variety are the mod 2 reductions of the chern classes we prove that our stiefel whitney classes satisfy axioms similar to the deligne grothendieck axioms for chern classes and we prove a specialization formula for the stiefel whitney classes of a family of varieties we show that the stiefel whitney classes of an affine real analytic variety x are represented by the polar cycles of x introduced for simplicial spaces by banchoff b and mccrory mc and we give a new proof of the combinatorial formula for stiefel whitney classes of manifolds our central result is a specialization formula for the conormal cycle theorem 3 7 we use this formula to prove the basic results of kashiwara and schapira s calculus of subanalytically constructible functions sc ks 9 7 as well as the pushforward and specialization formulas for stiefel whitney classes received by the editors october 2 1995 1991 mathematics subject classification primary 14p25 57r20 secondary 14p15 49q15

stability analysis of a time delayed sir epidemic model with nonlinear incidence rate stability of sir models has been studied extensively within the framework of disease epidemiology we formulate a nonlinear mathematical model to study the role of nonlinear incidence rates and the effect of time delay in a nonlinear logistically growing time delayed sir model with variable population size the existence and stability of the possible equilibria are examined in terms of a certain threshold condition 0 the basic reproduction number for mathematical tractability the stability of the endemic equilibrium is shown using lyapunov functional approach with linear incidence rate numerical simulations are carried out to investigate the influence of the key parameters on the spread of the disease to support the analytical conclusion and illustrate possible behavioral scenarios of the model

quasianalytic denjoy carleman classes and o minimality we show that the expansion of the real field generated by the functions of a quasianalytic denjoy carleman class is model complete and o minimal provided that the class satisfies certain closure conditions some of these structures do not admit analytic cell decomposition and they show that there is no largest o minimal expansion of the real field

numerical approximation of the voigt regularization for incompressible navier stokes and magnetohydrodynamic flows we study voigt regularizations for the navier stokes equations nses and magnetohydrodynamic mhd equations in the presence of physical boundary conditions in particular we develop the first finite element numerical algorithms for these systems prove stability and convergence of the algorithms and test them computationally on problems of practical interest it is found that unconditionally stable implementations of the voigt regularization can be made from a simple change to existing nse and mhd codes and moreover optimal convergence of the developed algorithms solutions to physical solutions can be obtained if lower order mixed finite elements are used finally we show that for several benchmark problems the voigt regularization on a coarse mesh produces good approximations to nse and mhd systems that is the voigt regularization provides accurate reduced order models for nse and mhd flows