adaptive unstructured volume remeshing ii application to two and three dimensional level set simulations of multiphase flow in part i adaptive unstructured volume remeshing i the method j comput phys in press we presented an adaptive remeshing algorithm that automatically adjusts the size of the elements of meshes of unstructured triangles 2d and unstructured tetrahedra 3d with time and position in the computational domain in order to efficiently resolve the relevant physical scales here we illustrate the performance of an implementation of the algorithm in finite element level set simulations of deformable droplet and fluid fluid interface interactions breakup and coalescence in multiphase flows the wide range of length scales characterizing the dynamics are accurately resolved as demonstrated by comparison to experiments and to theoretical and sharp interface boundary integral numerical results the computational cost is found to be competitive even with respect to boundary integral methods for the first time using an interface capturing level set method we successfully simulate the inertia driven impact and rebound of a liquid droplet from a liquid interface and find agreement with recent experimental results

three dimensional front tracking we describe a three dimensional front tracking algorithm discuss its numerical implementation and present studies to validate the correctness of this approach based on the results of the two dimensional computations we expect three dimensional front tracking to significantly improve computational efficiencies for problems dominated by discontinuities in some cases for which the interface computations display considerable numerical sensitivity we expect a greatly enhanced capability

when is a regular local ring a locality let a be a regular local ring and let r be a polynomial extension of a in a finite number of variables several attempts have been made to study the structure of projective modules over r and to find the minimal numbers of generators of ideals of r to wit z v 4 3 61 the methods employed in these references depend very much on the structure of a the cases dealt with are formal power series rings over fields and local rings at smooth points of algebraic varieties cohen s structure theorem tells us precisely what regular local rings are formal power series rings we present here the following characterization for a regular local ring to be a locality over a field

a 1 connectedness in reductive algebraic groups using sheaves of a 1 connected components we prove that the morel voevodsky singular construction on a reductive algebraic group fails to be a 1 local if the group does not satisfy suitable isotropy hypotheses as a consequence we show the failure of a 1 invariance of torsors for such groups on smooth affine schemes over infinite perfect fields we also characterize a 1 connected reductive algebraic groups over a field of characteristic 0

a hybrid particle level set method for improved interface capturing in this paper we propose a new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field our method uses lagrangian marker particles to rebuild the level set in regions which are underresolved this is often the case for flows undergoing stretching and tearing the overall method maintains a smooth geometrical description of the interface and the implementation simplicity characteristic of the level set method our method compares favorably with volume of fluid methods in the conservation of mass and purely lagrangian schemes for interface resolution the method is presented in three spatial dimensions

the point set method front tracking without connectivity abstract we describe a point set method for extracting the normal curvature and surface area from unordered data points residing on a surface this capability relaxes front tracking s reliance on connectivity between interfacial points and allows front tracking to model topological changes at an interface naturally we use this capability in 2d and 3d front tracking calculations to model coalescence we also describe a simple projection scheme which allows us to suppress parasitic currents in front tracking models

two integrals and some modified versions critical remarks the aim of this paper is to discuss different constructions of integrals sections 3 and 4 based on decomposable measures section 1 according to the classification of the continuous t conorms in essentially two types namely v and archimedean t conorms there exist mainly two types of integrals namely the constructions of sugeno section 3 and of weber section 4 further constructions corresponding to the archimedean case result to be special cases or not well defined section 4 in all cases a crucial property is some restricted distribution law for the pair with an appropriate operation section 2 some applications shall illustrate the use of the two integrals section 5

the range of null additive fuzzy and non fuzzy measures abstract in this paper a wide class of non additive set functions is investigated for finite null additive fuzzy and non fuzzy measures saks decomposition type theorem theorem on darboux property and lyapunov s type theorems are proved

implicit tracking for multi fluid simulations in this work we introduce a new method for tracking the interfaces among several immiscible fluids to advance the solution in time we use a muscl type scheme which couples a partial volume representation with a level set one to build the numerical fluxes in particular we show the positiveness and conservation properties of this method some numerical tests are given to demonstrate the conservativeness and the performances of our method

on the number of generators of ideals in polynomial rings for an ideal i in a noetherian ring r let i be the minimal number of generators of i it is well known that there is a sequence of inequalities i i i i i 1 that are strict in general however murthy conjectured in 1975 that i i i for ideals in polynomial rings whose height equals i i the purpose of this article is to prove a stronger form of the conjecture in case the base field is infinite of characteristic different from 2 namely the equality i i i holds for any ideal i irrespective of its height

lebesgue and saks decompositions of σ finite fuzzy measures two kinds of absolute continuity of fuzzy measures are distinguished and relations between them are established the scope of fuzzy measures under our consideration is limited in a family of those that are finite and absolute continuous with respect to a finite autocontinuous fuzzy measure under this restriction lebesgue decomposition and saks decomposition are proved

a note on the range of null additive fuzzy and non fuzzy measure abstract in this paper we point out that lemma 1 in pap fuzzy sets and systems 65 1994 105 115 is incorrect and give a correct proof of theorem 1 saks decomposition in pap

physical formulation and numerical algorithm for simulating n immiscible incompressible fluids involving general order parameters we present a family of physical formulations and a numerical algorithm based on a class of general order parameters for simulating the motion of a mixture of n n 2 immiscible incompressible fluids with given densities dynamic viscosities and pairwise surface tensions the n phase formulations stem from a phase field model we developed in a recent work based on the conservations of mass momentum and the second law of thermodynamics the introduction of general order parameters leads to an extremely strongly coupled system of n 1 phase field equations on the other hand the general form enables one to compute the n phase mixing energy density coefficients in an explicit fashion in terms of the pairwise surface tensions we show that the increased complexity in the form of the phase field equations associated with general order parameters in actuality does not cause essential computational difficulties our numerical algorithm reformulates the n 1 strongly coupled phase field equations for general order parameters into 2 n 1 helmholtz type equations that are completely de coupled from one another this leads to a computational complexity comparable to that for the simplified phase field equations associated with certain special choice of the order parameters we demonstrate the capabilities of the method developed herein using several test problems involving multiple fluid phases and large contrasts in densities and viscosities among the multitude of fluids in particular by comparing simulation results with the langmuir de gennes theory of floating liquid lenses we show that the method using general order parameters produces physically accurate results for multiple fluid phases

the autocontinuity of set function and the fuzzy integral abstract the autocontinuity and some other concepts of a set function are introduced and sugeno s fuzzy measure with some annexed conditions is studied on a fuzzy measure space egoroffs theorem is proved it is also proved that f mean convergence is equivalent to convergence in measure for a sequence of measurable functions finally some convergence theorems of sequence of fuzzy integrals are proved especially using the concept of autocontinuity a necessary and sufficient condition for the convergence in measure theorem is given

injective stability for k 1 of classical modules in rao 1994 14 the second author and w van der kallen showed that the injective stabilization bound for k1 of the general linear group is d 1 over a regular affine algebra over a perfect c1 field where d is the krull dimension of the base ring which is finite and at least 2 in this article we prove that the injective stabilization bound for k1 of the symplectic group is d 1 over a geometrically regular ring containing a field where d is the dimension of the base ring which is finite and at least 2 using the local global principle for the transvection subgroup of the automorphism group of projective and symplectic modules we show that the injective stabilization bound is d 1 for k1 of projective and symplectic modules of global rank at least 1 and local rank at least 3 respectively in each of the two cases above

on a question of quillen let r be a regular local ring and f a regular parameter of r quillen asked whether every projective r f module is free we settle this question when r is a regular local ring of an affine algebra over a field k further if r has infinite residue field we show that projective modules over laurent polynomial extensions of r f are also free

affine grassmannians of group schemes and exotic principal bundles over bbb a 1 let bf g be a simple simply connected group scheme over a regular local scheme u let mathcal e be a principal bf g bundle over bbb a 1 u trivial away from a subscheme finite over u we show that mathcal e is not necessarily trivial and give some criteria of triviality to this end we define affine grassmannians for group schemes and study their bruhat decompositions for semi simple group schemes we also give examples of principal bf g bundles over bbb a 1 u with split bf g such that the bundles are not isomorphic to pullbacks from u

a cohomological classification of vector bundles on smooth affine threefolds we give a cohomological classification of vector bundles of rank 2 on a smooth affine threefold over an algebraically closed field having characteristic unequal to 2 as a consequence we deduce that cancellation holds for rank 2 vector bundles on such varieties the proofs of these results involve three main ingredients first we give a description of the first nonstable a1 homotopy sheaf of the symplectic group second these computations can be used in concert with f morel s a1 homotopy classification of vector bundles on smooth affine schemes and obstruction theoretic techniques stemming from a version of the postnikov tower in a1 homotopy theory to reduce the classification results to cohomology vanishing statements third we prove the required vanishing statements

stability results for projective modules over blowup rings let r be a normal affine domain of dimension n 3 over an algebraically closed field k suppose chark 0 or chark p n let g f1 fr be a r regular sequence and a r f1 g fr g let p be a stably free a module of rank n 1 then murthy proved that there exists a projective r module q such that q r a p and n 1 q r 9 theorem 2 10 as a consequence of murthy s result if f g c x1 xn with g 0 then all stably free modules over c x1 xn f g of rank n 1 are free 9 corollary 2 11 in this paper we prove the following result 3 5 which generalizes the above result of murthy

affine representability results in mathbb a 1 homotopy theory i vector bundles we establish a general affine representability result in a1 homotopy theory over a general base we apply this result to obtain representability results for vector bundles in a1 homotopy theory our results simplify and significantly generalize morel s a1 representability theorem for vector bundles

the nilpotence conjecture in k theory of toric varieties it is shown that all nontrivial elements in higher k groups of toric varieties are annihilated by iterations of the natural frobenius type endomorphisms this is a higher analog of the triviality of vector bundles on affine toric varieties

correspondence yet another note on the range of null additive fuzzy and non fuzzy measures in this paper we point out that the only if part of lemma 1 in a previous paper by pap the range of null additive fuzzy and non fuzzy measures fuzzy sets and systems 65 1994 105 115 is incorrect we give a correct necessary and sufficient condition and its proof

on triviality of the euler class group of a deleted neighbourhood of a smooth local scheme let k be an infinite perfect field and let x specr where r is the localization of a k algebra b of finite type with respect to a regular prime ideal p specb letd be a divisor onx with a decompositiond i 1di into its irreducible components di such that each di is nonsingular and they intersect transversally consider y x d a deleted neighbourhood of the closed point x0 of x let e be a vector bundle on y in this set up e is trivial in the following cases i dim x 3 gabber ga1 ii t 1 bhatwadekar rao br iii ranke min dimr 2 t rao ra1 the above exposition is borrowed from a paper of nisnevich ni which contains some beautiful applications of these purity theorems in the same set up now let y y be a closed point and let my be the corresponding maximal ideal of oy y it is natural to ask whether my is a complete intersection observing that the conormal bundle my my is trivial we consider a more general situation let i be the defining ideal of a closed subscheme z of y such that the conormal bundle i i2 defined on z is trivial is i a complete intersection the questions we are addressing in this paper are a little stronger in nature can a given basis of i i2 be lifted to a minimal set of generators of i we remark that with the hypothesis on x each component di is defined by a single equation fi which vanishes at the closed point x0 of x let m be the maximal ideal at x0 the condition of transversality of intersection of di s is equivalent to saying that the images of f1 ft in the a m vector space m m2 are linearly independent keeping this discussion in mind we now give an excerpt of our main results in commutative algebraic terms see 4 2 4 3 4 4

a surface marker algorithm coupled to an area preserving marker redistribution method for three dimensional interface tracking in this work we present a new interface tracking algorithm based on surface markers to reconstruct and advect interfaces in three dimensional space the algorithm is coupled to a marker redistribution method which is area preserving along the interface the interface is described by a set of closed lines which define a coarse lagrangian quadrangular mesh in the fixed computational grid fixed markers are located where the lines cross and are maintained during the whole simulation at each time step the interface is first reconstructed and then all markers are advected by following streamlines in the reconstruction step new markers are determined by computing the intersections of the interface lines with the grid cell faces and by adding area conservation markers between fixed and grid intersection markers intersection and conservation markers defined at the previous time step are discarded the method maintains a smooth geometrical description of the interface for both two dimensional and three dimensional tests with accurate volume conservation even in very challenging situations where the fluid bodies progressively deform and stretch developing localized regions with very high curvature and thin fluid filaments the method compares favorably with front capturing methods such as volume of fluid and level set and other hybrid techniques such as the particle level set method

a sharp interface method for incompressible two phase flows we present a sharp interface method for computing incompressible immiscible two phase flows it couples the level set and volume of fluid techniques and retains their advantages while overcoming their weaknesses it is stable and robust even for large density and viscosity ratios on the order of 1000 to 1 the numerical method is an extension of the second order method presented by sussman m sussman a second order coupled levelset and volume of fluid method for computing growth and collapse of vapor bubbles journal of computational physics 187 2003 110 136 in which the previous method treated the gas pressure as spatially constant and the present method treats the gas as a second incompressible fluid the new method yields solutions in the zero gas density limit which are comparable in accuracy to the method in which the gas pressure was treated as spatially constant this improvement in accuracy allows one to compute accurate solutions on relatively coarse grids thereby providing a speed up over continuum or ghost fluid methods

lebesgue and saks decompositions of b decomposable measures abstract the properties of decomposable measures are investigated for t conorms which are more general than archimedean t conorms the continuity of such set functions with respect to suitably chosen submeasures is proved as an important consequence a lebesgue type decomposition theorem is obtained a version of saks decomposition for order continuous decomposable measures with respect to arbitrary t conorms is proved

pseudo atoms and darboux property for set multifunctions in this paper we introduce and study the concepts of pseudo atom and darboux property for set multifunctions we establish relationships between the darboux property and non atomicity in the multivalued case we also prove that the range of a multimeasure having the darboux property is a convex set

on the null additivity and the autocontinuity of a fuzzy measure abstract in this paper further research on the null additivity and autocontinuity of fuzzy measures is made we prove that autocontinuity from below and autocontinuity from above of a fuzzy measure are equivalent when this fuzzy measure is finite furthermore if the space x is countable then the null additivity is equivalent to the autocontinuity

on the complete intersection conjecture of murthy abstract suppose a k x 1 x 2 x n is a polynomial ring over a field k and i is an ideal in a m p murthy conjectured that i i i 2 where denotes the minimal number of generators recently fasel 3 settled this conjecture affirmatively when k is an infinite perfect field with 1 2 k always we are able to do the same when k is an infinite field in fact we prove similar results for ideals i in a polynomial ring a r x that contains a monic polynomial and r is essentially smooth algebra over an infinite field k or r is a regular ring over a perfect field k

a mixed markers and volume of fluid method for the reconstruction and advection of interfaces in two phase and free boundary flows in this work we present a new mixed markers and volume of fluid vof algorithm for the reconstruction and advection of interfaces in the two dimensional space the interface is described by using both the volume fraction function c as in vof methods and surface markers which locate the interface within the computational cells the c field and the markers are advected by following the streamlines new markers are determined by computing the intersections of the advected interface with the grid lines then other markers are added inside each cut cell to conserve the volume fraction c a smooth motion of the interface is obtained typical of the marker approach with a good volume conservation as in standard vof methods in this article we consider a few typical two dimensional tests and compare the results of the mixed algorithm with those obtained with vof methods translations rotations and vortex tests are performed showing that many problems of the vof technique can be solved and a good accuracy in the geometrical motion and mass conservation can be achieved

pseudo additive measures and integrals abstract we suggest pseudo additive measures based on a pseudo addition and discuss integrals with respect to pseudo additive measures a pseudo additive measure is a special type of fuzzy measures to define an integral a multiplication corresponding to a pseudo addition is introduced the resulting integral is an extension of the lebesgue integral in this context radon nikodym like theorems are shown

an efficient algorithm for incompressible n phase flows abstract we present an efficient algorithm within the phase field framework for simulating the motion of a mixture of n n 2 immiscible incompressible fluids with possibly very different physical properties such as densities viscosities and pairwise surface tensions the algorithm employs a physical formulation for the n phase system that honors the conservations of mass and momentum and the second law of thermodynamics we present a method for uniquely determining the mixing energy density coefficients involved in the n phase model based on the pairwise surface tensions among the n fluids our numerical algorithm has several attractive properties that make it computationally very efficient i it has completely de coupled the computations for different flow variables and has also completely de coupled the computations for the n 1 phase field functions ii the algorithm only requires the solution of linear algebraic systems after discretization and no nonlinear algebraic solve is needed iii for each flow variable the linear algebraic system involves only constant and time independent coefficient matrices which can be pre computed during pre processing despite the variable density and variable viscosity of the n phase mixture iv within a time step the semi discretized system involves only individual de coupled helmholtz type including poisson equations despite the strongly coupled phase field system of fourth spatial order at the continuum level v the algorithm is suitable for large density contrasts and large viscosity contrasts among the n fluids extensive numerical experiments have been presented for several problems involving multiple fluid phases large density contrasts and large viscosity contrasts in particular we compare our simulations with the de gennes theory and demonstrate that our method produces physically accurate results for multiple fluid phases we also demonstrate the significant and sometimes dramatic effects of the gravity density ratios pairwise surface tensions and drop sizes on the n phase configurations and dynamics the numerical results show that the method developed herein is capable of dealing with n phase systems with large density ratios large viscosity ratios and pairwise surface tensions and that it can be a powerful tool for studying the interactions among multiple types of fluid interfaces

the serre problem for discrete hodge algebras all rings in this paper will be commutative and noetherian let a b be a split injection of rings a finitely generated projective b module is called extended from a if there exists a finitely generated projective a module q such that p q b furthermore we call an c eglr b resp slr b extended from a if can be written as f i with fi glr a resp slr a and 7 e b the group of elementary matrices now let a be a ring in this paper we will consider a algebras of the following type a x1 xn i where i is an ideal generated by monomials such an a algebra is called a discrete hodge algebra over a and vice versa every discrete hodge algebra over a is of this type 5 for the proofs of most of our results we can restrict ourselves to the special class of square free discrete hodge algebras over a by which is meant that i can be generated by square free monomials the varieties over a corresponding to these square flee discrete hodge algebras are unions of intersections of coordinate hyperplanes h i x 1 xn anixi 0 i 1 n typical examples are a x 1 xn and a x 1 x 2 x i x 2 the main result of this paper is the following

the euler class group of a polynomial algebra with coefficients in a line bundle let a be a commutative noetherian ring and p be a projective a module of rank text dim a 1 an intriguing open question is to find the precise obstruction for p to split as p simeq q oplus a for some a module q in this paper we settle this question when a r t for some ring r containing the field of rationals and p is a projective a module of rank text dim r

atoms of fuzzy measures and fuzzy integrals abstract some analytical properties of sugeno s fuzzy measures especially of atoms of fuzzy measures are investigated a fuzzy integral of riemann type is defined through atoms and it is proved that a continuous function is fuzzy integrable in the sense of riemann and has the same integral as that of sugeno

euler class groups and the homology of elementary and special linear groups abstract we improve homology stability ranges for elementary and special linear groups over rings with many units our result implies stability for unstable quillen k groups and proves a conjecture of bass for commutative local rings with infinite residue fields we show that the obstruction to further stability is given by milnor witt k theory as an application we construct euler classes of projective modules with values in the cohomology of the milnor witt k theory sheaf for d dimensional commutative noetherian rings with infinite residue fields we show that the vanishing of the euler class is necessary and sufficient for an oriented projective module p of rank d to split off a rank 1 free direct summand along the way we obtain a new presentation of milnor witt k theory and of symplectic k 2 simplifying the classical matsumoto moore presentation

on fuzzy measures defined by fuzzy integrals on definit une mesure floue comme une fonction ensemble sur 7b f qui est une algebre de sous ensembles de l ensemble x

fuzzy integrals based on pseudo additions and multiplications on definit un semi groupe ou un groupe qui est isomorphe au semi groupe ou du groupe ou la mesure de lebesque est definie on definit une integrale par pseudo addition et pseudo multiplication qui sont des operations de semi anneaux distributives et associatives

pseudo atoms atoms and a jordan type decomposition in effect algebras abstract the aim of this paper is to introduce and investigate the concept of pseudo atoms of a real valued function m defined on an effect algebra l a few examples of pseudo atoms and atoms are given in the context of null additive null null additive and pseudo null additive functions and also some fundamental results for pseudo atoms under the assumption of null null additivity are established the notions of total variation m positive variation m and negative variation m of a real valued function m on l are studied elaborately and it is proved for a modular measure m which is of bounded total variation defined on a d lattice l that m is pseudo atomic or atomic if and only if its total variation m is pseudo atomic or atomic finally a jordan type decomposition theorem for an extended real valued function m of bounded total variation defined on an effect algebra l is proved and some properties on decomposed parts of m such as continuity from below pseudo atomicity or atomicity and being measure are discussed a characterization for the function m to be of bounded total variation is established here and used in proving above mentioned jordan type decomposition theorem

the monoid structure on homotopy obstructions abstract let a be a commutative noetherian ring containing a field k with 1 2 k dim a d and let p be a projective a module with r a n k p n let l o p denote the set of all pairs i where i is an ideal of a and p i i 2 is a surjective map the homotopy relations on l o p induced by l o p t leads to a set 0 l o p of equivalence classes in l o p there are two distinguished elements e 0 e 1 0 l o p respectively the images of 0 0 and a 0 define the obstruction class e p e 0 0 l o p to be called the nori homotopy class of p the following results are under suitable smoothness or regularity hypotheses we prove if 2 n d 2 then 0 l o p has a natural structure of a monoid which is a group if p q a when 2 n d 3 we prove p q a e p e 1 the additive zero further we give a definition of a euler class group e p under suitable smoothness hypotheses we prove if p q a and 2 n d 3 then there is natural isomorphism e p 0 l o p of groups

hierarchical x fem for n phase flow n 2 the extended finite element method x fem has been successfully used in two phase flow problems involving a moving interface in order to simulate problems involving more than two phases the x fem has to be further extended the proposed approach is presented in the case of a quasi static stokes n phase flow and it is based on using an ordered collection of level set functions to describe the location of the phases a level set hierarchy allows describing triple junctions avoiding overlapping or voids between materials moreover an enriched solution accounting for several simultaneous phases inside one element is proposed the interpolation functions corresponding to the enriched degrees of freedom require redefining the associated ridge function accounting for all the level sets the computational implementation of this scheme involves calculating integrals in elements having several materials inside an adaptive quadrature accounting for the interfaces locations is proposed to accurately compute these integrals examples of the hierarchical x fem approach are given for a n phase stokes problem in 2 and 3 dimensions

a variational level set approach to multiphase motion a coupled level set method for the motion of multiple junctions of e g solid liquid and grain boundaries which follows the gradient flow for an energy functional consisting of surface tension proportional to length and bulk energies proportional to area is developed the approach combines the level set method of s osher and j a sethian with a theoretical variational formulation of the motion by f reitich and h m soner the resulting method uses as many level set functions as there are regions and the energy functional is evaluated entirely in terms of level set functions the gradient projection method leads to a coupled system of perturbed by curvature terms hamilton jacobi equations the coupling is enforced using a single lagrange multiplier associated with a constraint which essentially prevents a regions from overlapping and b the development of a vacuum the numerical implementation is relatively simple and the results agree with and go beyond the theory as given in 12 other applications of this methodology including the decomposition of a domain into subregions with minimal interface length are discussed finally some new techniques and results in level set methodology are presented

a coupled level set and volume of fluid method for computing 3d and axisymmetric incompressible two phase flows we present a coupled level set volume of fluid clsvof method for computing 3d and axisymmetric incompressible two phase flows this method combines some of the advantages of the volume of fluid method with the level set method to obtain a method which is generally superior to either method alone we present direct comparisons between computations made with the clsvof method and computations made with the level set method the volume of fluid method and the boundary integral method we also compare our computations to the exact solution for an oscillating ellipse due to lamb and experimental results obtained for a rising gas bubble in liquid obtained by hnat and buckmaster our computational examples focus on flows in which surface tension forces and changes in topology are dominant features of the flow

a finite element technique for multifluid incompressible flow using eulerian grids the paper presents a finite element method for 3d incompressible fluid flows with capillary free boundaries it uses a fixed eulerian grid of 10 nodes p2 p1 tetrahedra and tracks the free boundary using a six nodes p2 triangular surface grid in order to improve the mass conservation properties of the method a local enrichment of the finite element basis in the elements intersected by the free boundaries is employed in addition to the surface tracking it also advects a smooth indicator function for an easy identification of the fluid properties in the different parts of the domain the advective part of the navier stokes equations is split and integrated with a characteristic method the remaining generalized stokes problem is resolved by means of an inexact outer inner uzawa iteration with a properly chosen preconditioner the performance of this technique is evaluated on several problems involving droplets in viscous liquids

a front tracking method for the computations of multiphase flow direct numerical simulations of multiphase flows using a front tracking method are presented the method is based on writing one set of governing equations for the whole computational domain and treating the different phases as one fluid with variable material properties interfacial terms are accounted for by adding the appropriate sources as functions at the boundary separating the phases the unsteady navier stokes equations are solved by a conventional finite volume method on a fixed structured grid and the interface or front is tracked explicitly by connected marker points interfacial source terms such as surface tension are computed on the front and transferred to the fixed grid advection of fluid properties such as density is done by following the motion of the front the method has been implemented for fully three dimensional flows as well as for two dimensional and axisymmetric ones first the method is described for the flow of two or more isothermal phases the representation of the moving interface and its dynamic restructuring as well as the transfer of information between the moving front and the fixed grid are discussed applications and extensions of the method to homogeneous bubbly flows atomization flows with variable surface tension solidification and boiling are then presented

on the number of generators of modules over polynomial affine rings preprint enviat per a la seva publicacio en una revista cientifica mathematische zeitschrift 1991 vol 208 p 11 21 https doi org 10 1007 bf02571506

vector bundles on contractible smooth schemes we discuss algebraic vector bundles on smooth k schemes x contractible from the standpoint of a homotopy theory when k c the smooth manifolds x c are contractible as topological spaces the integral algebraic k theory and integral motivic cohomology of such schemes are that of spec k one might hope that furthermore and in analogy with the classification of topological vector bundles on manifolds algebraic vector bundles on such schemes are all isomorphic to trivial bundles this is almost certainly true when the scheme is affine however in the non affine case this is false we show that essentially every smooth a contractible strictly quasi affine scheme that admits a u torsor whose total space is affine for u a unipotent group possesses a non trivial vector bundle indeed we produce explicit arbitrary dimensional families of non isomorphic such schemes with each scheme in the family equipped with as many i e arbitrary dimensional moduli of non isomorphic vector bundles of every sufficiently large rank n as one desires neither the schemes nor the vector bundles on them are distinguishable by algebraic k theory we also discuss the triviality of vector bundles for certain smooth complex affine varieties whose underlying complex manifolds are contractible but that are not necessarily a contractible

pseudo atoms of fuzzy and non fuzzy measures in this paper we introduce the concept of pseudo atoms of set functions and establish some fundamental conclusions about pseudo atoms on the assumption of null null additivity we give the necessary and sufficient conditions which insure that the set of non null measure contains no pseudo atoms further we obtain several decomposition theorems about pseudo atoms and atoms of set functions

an accurate adaptive solver for surface tension driven interfacial flows a method combining an adaptive quad octree spatial discretisation geometrical volume of fluid interface representation balanced force continuum surface force surface tension formulation and height function curvature estimation is presented the extension of these methods to the quad octree discretisation allows adaptive variable resolution along the interface and is described in detail the method is shown to recover exact equilibrium to machine accuracy between surface tension and pressure gradient in the case of a stationary droplet irrespective of viscosity and spatial resolution accurate solutions are obtained for the classical test case of capillary wave oscillations an application to the capillary breakup of a jet of water in air further illustrates the accuracy and efficiency of the method the source code of the implementation is freely available as part of the gerris flow solver