rational singularities and almost split sequences the main aim of this paper is to relate almost split sequences to singularity theory by showing that the mckay quiver built from the finite dimen sional representations of a finite subgroup g of gl 2 c where c is the complex numbers is isomorphic to the ar quiver of the reflexive modules of the quotient singularity associated with g over the past decade almost split sequences have been playing an increasingly important role in the representation theories of finite dimensional algebras and classical orders see 5 and 3 8 for basic existence theorems in these contexts while they have been known for some time to exist in higher dimensional situations 3 it has not been at all clear how they related to singularity theory if at all the main aim of this paper is to relate almost split sequences to singularity theory by showing that the mckay quiver built from the fimte dimensional complex represen tations of a finite subgroup g of gl 2 c where c is the complex numbers is isomorphic to the ar quiver of the reflexive modules over the quotient singularity r associated with g as in the case of finite dimensional algebras the ar quiver of reflexive r modules is defined in terms of the almost split sequences of reflexive r modules in the case g c sl 2 c mckay observed that the underlying graph of the mckay quiver with the trivial module removed is isomorphic to the desingulariza tion graph of the associated singularity various explanations of this phenomenon have been given by knorrer gonzalez sprinberg verdier 6 and artin verdier 1 which along the way have established most explicitly in 1 a natural one to one correspondence between the indecomposable reflexive r modules and the nodes of the desingularization graph but why the almost split sequences describe the edge of the desingularization graph still remains to be explained an effort has been made to make this paper as self contained as possible in particular no prior knowledge of almost split sequences is required before describing the contents of the six sections of this paper we fix some notation throughout this paper g is a finite group k an algebraically closed field of characteristic not dividing the order of g and v a two dimensional k representation of g setting s k x y the k algebra of formal power series the two dimen sional representation v gives a linear action of g on s as a group of k algebra automorphisms we denote by s g the skew group ring given by this action

poisson cohomology of plane poisson structures with isolated singularities revisited continuing the work of monnier israel j math 129 189 207 2002 we determine the gerstenhaber algebra structure over the poisson cohomology groups for a large class of poisson structures with isolated singularities over the plane it reveals that there exists a gaga type phenomenon we give an explicit description via generators and relations for the case of simple singularities

maximal cohen macaulay modules over the cone of an elliptic curve let r k y1 y2 y3 f f y13 y23 y33 where k is an algebraically closed field with chark 3 using atiyah bundle classification over elliptic curves we describe the matrix factorizations of the graded indecomposable reflexive r modules equivalently we describe explicitly the indecomposable bundles over the projective curve v f pk2 using the fact that over the completion r of r every reflexive module is gradable we obtain a description of the maximal cohen macaulay modules over r k y1 y2 y3 f

optimal control problems for the navier stokes system coupled with the k ω turbulence model optimal control of fluid dynamics systems has gained attention in the last several years from the scientific community because of its potential use in design of new engineering devices and optimization of existing ones many research works have extensively studied the optimal control for the system of navier stokes but the problem of turbulence in these works is usually not taken into account because of the many difficulties arising from the numerical implementation and solution of the optimality system in this work turbulence is considered by coupling the k two equation turbulence model with the averaged navier stokes system the complete optimality system is derived and the existence of a weak solution proven some numerical examples are reported

an improved k ω φ α turbulence model applied to near wall separated and impinging jet flows and heat transfer abstract a turbulence model based on elliptic blending concept referred to as improved k model compared against the original k model developed previously is developed and verified this model consists of four governing equations among them the k and equations are based on the wilcox s k model with some modifications and improvements according to the original k model and the and equations are extracted from the original k model directly without any change the improved k model is applied to near wall separated and impinging jet flows and convective heat transfer i e the 2d fully developed channel flow the 2d backward facing step flow the 2d impinging jet flow and the convective heat transfer in the 2d fully developed channel flow and the 2d impinging jet flow the computational results are compared with available dns and experimental data and also to those computed using the original k model and the popular menter s sst k model it is shown that the improved k model has better numerical stability higher computational efficiency and more concise form than the original k model in addition compared with the original k model the improved k model can yield similar velocity profiles in the fully developed channel flow and step flow and friction and pressure coefficients in the step flow and very close temperature profiles in the fully developed channel flow moreover it shows significant improvements on the predictions for the fluid flow and heat transfer in the impinging jet flow as a whole the improved k model predicts better results than both of the original k model and the sst k model

crossed modules for lie rinehart algebras abstract we introduce the notion of crossed module for lie rinehart algebras and prove that they are classified by the third cohomology of lie rinehart algebras developed in j huebschmann j reine angew math 408 1990 57 113 g s rinehart trans amer math soc 108 1963 195 222

integration of holomorphic lie algebroids we prove that a holomorphic lie algebroid is integrable if and only if its underlying real lie algebroid is integrable thus the integrability criteria of crainic fernandes theorem 4 1 in crainic fernandes in ann math 2 157 2003 do also apply in the holomorphic context without any modification as a consequence we prove that a holomorphic poisson manifold is integrable if and only if its real part or imaginary part is integrable as a real poisson manifold

on the large order behavior of φ44 we continue the rigorous study of the large order behavior of the perturbation series for the 4 model in 4 dimensions started in 1 in this paper we prove a result announced in 1 we show that the exact radius of convergence of the borel transform of the renormalized perturbation series for 44 is greater than or equal to the expected value given by the position of the first renormalon 2 this result holds for any vector 2 2 model withn components and makes use of the lipatov bound of 1 this result is based on a partial resummation of counterterms similar to the one of 3 but in a phase space analysis of the renormalized series

a duality approach to the symmetry of bernstein sato polynomials of free divisors abstract in this paper we prove that the bernstein sato polynomial of any free divisor for which the d s module d s h s admits a spencer logarithmic resolution satisfies the symmetry property b s 2 b s this applies in particular to locally quasi homogeneous free divisors for instance to free hyperplane arrangements or more generally to free divisors of linear jacobian type we also prove that the bernstein sato polynomial of an integrable logarithmic connection e and of its dual e with respect to a free divisor of linear jacobian type are related by the equality b e s b e s 2 our results are based on the behaviour of the modules d s h s and d s e s h s under duality

further experiments on supersonic turbulent flow development in a square duct the mean flow structure of supersonic turbulent adiabatic wall flow in a square duct is investigated experimentally over a development length x d 0 50 for a uniform flow mach 3 9 condition at the duct inlet the results show that a secondary flow cell structure develops which is similar to that for the incompressible case development of the primary flow is influenced by the combined effects of the secondary flow and the streamwise adverse pressure gradient total pressure axial mean velocity and mach number profiles are presented which show that the outer flow is sensitive primarily to the streamwise pressure gradient while flow in the near wall region is dominated by the secondary flow axial mean velocity profiles plotted in terms of van driest scaled variables show that a well defined log law region exists in the near wall layer this region exists in the presence of a secondary flow which continuously modifies spanwise wall shear stress behavior along the length of the duct

critical exponents for the percolation problem and the yang lee edge singularity gives details of a calculation of critical exponents for a class of field theory models that have an interaction cubic in the fields the results have already been reported give the exponents to third order in epsilon where epsilon 6 d and d is the dimensionality of space the class of models includes the percolation problem and the yang lee edge singularity so the authors give explicit results for the exponents to order epsilon 3 in these cases by resummation methods based on the symptotic behaviour of the epsilon expansion they obtain numerical estimates for these exponents for a number of interesting values of d

cohomology of restricted lie rinehart algebras and the brauer group abstract we give an interpretation of the brauer group of a purely inseparable extension of exponent 1 in terms of restricted lie rinehart cohomology in particular we define and study the category p lr a of restricted lie rinehart algebras over a commutative algebra a we define cotriple cohomology groups h p l r l m for l p lr a and m a beck l module we classify restricted lie rinehart extensions thus we obtain a classification theorem for regular extensions considered by hochschild

the poincare series of the module of derivations of affine monomial curves abstract let a be a graded k algebra and m be a finitely generated a module the poincare series p m a z is the formal power series i dim k tor i a k m z i we study the poincare series of der k k s the module of derivations of a numerical semigroup ring k s and we relate it to the poincare series of k over k s and to the type of s we then use this in order to determine the poincare series of der k k s or at least its rationality for some classes of examples we finally give an example of a non rational p der k k s k s z

determinantal representations of singular hypersurfaces in pn abstract a global determinantal representation of projective hypersurface x p n is a matrix whose entries are linear forms in homogeneous coordinates and whose determinant defines the hypersurface we study the properties of such representations for singular possibly reducible or non reduced hypersurfaces in particular we obtain the decomposability criteria for determinantal representations of globally reducible hypersurfaces further we classify the determinantal representations in terms of the corresponding kernel sheaves on x finally we extend the results to the case of symmetric self adjoint representations with implications to hyperbolic polynomials and the generalized lax conjecture

non perturbative renormalization using dimensional regularization applications to the epsilon expansion the authors give a prescription for the one loop renormalisation of the imaginary parts of vertex functions in g phi 4 which are generated when g 0 within the dimensional regularisation scheme this involves another non perturbative renormalisation over and above the usual one loop perturbative renormalisation the authors use an extended minimal subtraction scheme so that the resulting renormalisation group functions have a simple dependence on epsilon 4 d in contrast to the usual case however both the beta function and the renormalised coupling gr are in general complex for real g and the fixed point for epsilon gr 0 is non perturbative in epsilon this fixed point determines the imaginary parts of the critical exponents which are generated when epsilon 0 and allows the determination of the high order behaviour of the perturbation series in epsilon for these exponents the generalisation of these ideas to the o n symmetric g phi 2 2 model is also given

twisted poincare duality between poisson homology and poisson cohomology abstract a version of the twisted poincare duality is proved between the poisson homology and cohomology of a polynomial poisson algebra with values in an arbitrary poisson module the duality is achieved by twisting the poisson module structure in a canonical way which is constructed from the modular derivation in the case that the poisson structure is unimodular the twisted poincare duality reduces to the poincare duality in usual sense the main result generalizes the work of launois and richard 8 for the quadratic poisson structures and zhu 25 for the linear poisson structures

hilbert series of modules over lie algebroids we consider modules m over lie algebroids ga which are of finite type over a local noetherian ring a using ideals j a such that ga j j and the length lga m jm we can define in a natu

differential operators on monomial curves abstract let k be an algebraically closed field of characteristic 0 let n 0 be a numerical semigroup and let a k be the corresponding semigroup algebra we give an explicit description of the ring d d a of k linear differential operators on a the associated graded ring grd a and the module of derivations derk a we also classify all graded d modules which are finitely generated torsion free a modules of rank 1 considered as modules over the sub ring a d a

formulation of the k w turbulence model revisited this paper presents a reformulated version of the author sk model of turbulence revisions include the addition of just one new closure coefficient and an adjustment to the dependence of eddy viscosity on turbulence properties the result is a significantly improved model that applies to both boundary layers and free shear flows and that has very little sensitivity to finite freestream boundary conditions on turbulence properties the improvements to the k model facilitate a significant expansion of its range of applicability the new model like preceding versions provides accurate solutions for mildly separated flows and simple geometries such as that of a backward facing step the model s improvement over earlier versions lies in its accuracy for even more complicated separated flows this paper demonstrates the enhanced capability for supersonic flow into compression corners and a hypersonic shock wave boundary layer interaction the excellent agreement is achieved without introducing any compressibility modifications to the turbulence model

lie rinehart cohomology and integrable connections on modules of rank one abstract let k be an algebraically closed field of characteristic 0 let r be a commutative k algebra and let m be a torsion free r module of rank one with a connection we consider the lie rinehart cohomology with values in end r m with its induced connection and give an interpretation of this cohomology in terms of the integrable connections on m when r is an isolated singularity of dimension d 2 we relate the lie rinehart cohomology to the topological cohomology of the link of the singularity and when r is a quasi homogenous hypersurface of dimension two we give a complete computation of the cohomology

renormalization in quantum field theory and the riemann hilbert problem i the hopf algebra structure of graphs and the main theorem this paper gives a complete selfcontained proof of our result announced in 6 showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the riemann hilbert problem we shall first show that for any quantum field theory the combinatorics of feynman graphs gives rise to a hopf algebra which is commutative as an algebra it is the dual hopf algebra of the enveloping algebra of a lie algebra whose basis is labelled by the one particle irreducible feynman graphs the lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa the corresponding lie group g is the group of characters of we shall then show that using dimensional regularization the bare unrenormalized theory gives rise to a loop where c is a small circle of complex dimensions around the integer dimension d of space time our main result is that the renormalized theory is just the evaluation at z d of the holomorphic part of the birkhoff decomposition of we begin to analyse the group g and show that it is a semi direct product of an easily understood abelian group by a highly non trivial group closely tied up with groups of diffeomorphisms the analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title

next to k leading log expansions by chord diagrams green functions in a quantum field theory can be expanded as bivariate series in the coupling and a scale parameter the leading logs are given by the main diagonal of this expansion i e the subseries where the coupling and the scale parameter appear to the same power then the next to leading logs are listed by the next diagonal of the expansion where the power of the coupling is incremented by one and so on we give a general method for deriving explicit formulas and asymptotic estimates for any hbox next to k leading log expansion for a large class of single scale green functions these green functions are solutions to dyson schwinger equations that are known by previous work to be expressible in terms of chord diagrams we look in detail at the green function for the fermion propagator in massless yukawa theory as one example and the green function of the photon propagator in quantum electrodynamics as a second example as well as giving general theorems our methods are combinatorial but the consequences are physical giving information on which terms dominate and on the dichotomy between gauge theories and other quantum field theories

renormalization of gauge fields a hopf algebra approach we study the connes kreimer hopf algebra of renormalization in the case of gauge theories we show that the ward identities and the slavnov taylor identities in the abelian and non abelian case respectively are compatible with the hopf algebra structure in that they generate a hopf ideal consequently the quotient hopf algebra is well defined and has those identities built in this provides a purely combinatorial and rigorous proof of compatibility of the slavnov taylor identities with renormalization

differential operators on an affine curve let x denote an irreducible affine algebraic curve over an algebraically closed field k of characteristic zero denote by 3 x the sheaf of differential operators on x and 2 x f x 3 x the ring of global differential operators on x the following is established theorem 3 x is a finitely generated k algebra and a noetherian ring furthermore 2 x has a unique minimal non zero ideal j and 3 x j is a finite dimensional k algebra let x denoted the normalisation of x and n x x the projection map the main technique is to compare 2 x with 3 x

improvement of corner flow prediction using the quadratic constitutive relation this paper presents results of the implementation of spalart s quadratic constitutive relation into the reynolds averaged navier stokes equations on two applications encountered in applied aerodynamics these applications one in external flow and one in internal flow are intended to be representative of the problem of corner flow simulation with the corner separation size being generally overestimated by classical turbulence models the first application is the dlr f6 wing body configuration used in the drag prediction workshop where a separation occurs at the wing body junction in the case without fairing the second application is the shock induced separation in a rectangular duct it is shown that the quadratic constitutive relation greatly improves the accuracy of the junction flow simulation without worsening the quality of the solution elsewhere

supersonic corner flow predictions using the quadratic constitutive relation a series of reynolds averaged navier stokes simulations is performed for a supersonic wall bounded turbulent corner flow these simulations are compared to a high order implicit large eddy simulation for the same geometry and flow conditions inclusion of the quadratic constitutive relation in the reynolds averaged navier stokes simulation results in significant improvement in qualitative agreement with the large eddy simulation specifically the presence of secondary flow a counter rotating vortex pair the range of valid values for the constant in the quadratic constitutive relation formulation is explored additionally the effects on this range from different turbulence models and momentum thickness reynolds numbers for the corner are also examined these effects of the quadratic constitutive relation are explored using both the finite difference fluid solver overflow and the finite volume fluid solver us3d the results indicate that the quadratic constitutive relation term directly affects the str

comparison of explicit and traditional algebraic stress models of turbulence a critical comparison of explicit vs traditional algebraic stress models of turbulence is made in an effort to clear up the confusion that appears to have been generated by the recently published literature on the subject in which disparate approaches are adopted the only way that general second order closures can formally lead to fully explicit algebraic stress models in a global sense is in the limit of equilibrium homogeneous turbulence when these fully explicit models are then applled to turbulent flows that are far from equilibrium a singularity can arise which can be removed by a systematic regularization when solved explicitly either analytically or numerically the traditional implicit algebraic stress models have either multiple solutions or singularities which tends to explain why they have had problems in applications to complex flows thus it is argued that traditional algebraic stress models are intrinsically ill behaved and should be abandoned in future applications in favor of regularized explicit algebraic stress models it is furthermore argued that these should be based on the homogeneous equilibrium hypothesis which allows for more general second order closures to be used to obtain single valued models

the structure of renormalization hopf algebras for gauge theories i representing feynman graphs on bv algebras we study the structure of renormalization hopf algebras of gauge theories we identify certain hopf subalgebras in them whose character groups are semidirect products of invertible formal power series with formal diffeomorphisms this can be understood physically as wave function renormalization and renormalization of the coupling constants respectively after taking into account the slavnov taylor identities for the couplings as generators of a hopf ideal we find hopf subalgebras in the corresponding quotient as well in the second part of the paper we explain the origin of these hopf ideals by considering a coaction of the renormalization hopf algebras on the batalin vilkovisky bv algebras generated by the fields and couplings constants the so called classical master equation satisfied by the action in the bv algebra implies the existence of the above hopf ideals in the renormalization hopf algebra finally we exemplify our construction by applying it to yang mills gauge theory

gerstenhaber algebras and bv algebras in poisson geometry the purpose of this paper is to establish an explicit correspondence between various geometric structures on a vector bundle with some well known algebraic structures such as gerstenhaber algebras and bv algebras some applications are discussed in particular we find an explicit connection between the koszul brylinski operator and the modular class of a poisson manifold as a consequence we prove that poisson homology is isomorphic to poisson cohomology for unimodular poisson structures

on co morphisms of lie pseudoalgebras and groupoids we give a unified description of morphisms and comorphisms of lie pseudoalgebras showing that the both types of morphisms can be regarded as subalgebras of a lie pseudoalgebra called the sum we also provide similar descriptions for morphisms and comorphisms of lie algebroids and groupoids

six loop renormalization group functions of o n symmetric phi6 theory and epsilon expansions of tricritical exponents up to epsilon3 we present a six loop calculation for wilson s function and for the crossover exponent of the tricritical o n symmetric 6 theory in d 3 dimensions the counterterms of all but one of the 29 diagrams can be calculated analytically while for one diagram high precision numerical calculations together with the pslq integer relation search algorithm was used to come up with an analytical result the divergences of the dimensionally regularized diagrams are removed by minimal subtraction

lie algebroid cohomology and lie algebroid extensions abstract we consider the extension problem for lie algebroids over schemes over a field given a locally free lie algebroid q over a scheme x and a coherent sheaf of lie o x algebras l we determine the obstruction to the existence of extensions 0 l e q 0 and classify the extensions in terms of a suitable lie algebroid hypercohomology group in the preliminary sections we study free lie algebroids and recall some basic facts about lie algebroid hypercohomology

groupoids and faa di bruno formulae for green functions in bialgebras of trees we prove a faa di bruno formula for the green function in the bialgebra of p trees for any polynomial endofunctor p the formula appears as relative homotopy cardinality of an equivalence of groupoids

duality and products in algebraic co homology theories abstract the origin and interplay of products and dualities in algebraic co homology theories is ascribed to a a hopf algebra structure on the relevant universal enveloping algebra this provides a unified treatment for example of results by van den bergh about hochschild co homology and by huebschmann about lie rinehart co homology

the fundamental module of a normal local domain of dimension 2 the fundamental module e of a normal local domain r m of dimension 2 is defined by the nonsplit exact sequence 0 a k a e a m a 0 where k is the canonical module of r we prove that if r is complete with r m c then e is decomposable if and only if r is a cyclic quotient singularity various other properties of fundamental modules will be discussed 0 introduction let r m be a normal local domain of dimension 2 which possesses the canonical module k let c r denote the category of finitely generated reflexive r modules note that a finitely generated r module is an object in c r if and only if it is a maximal cohen macaulay module over r by definition k is a reflexive module of rank 1 and it satisfies extr r m k r m see herzog and kunz 8 for the details we denote the duality with respect to r resp k by resp that is homr r and homr k remark that a finitely generated r module m lies in c r if and only if m m or equivalently m m we make the following definition 0 1 since ext m k ext2r r m k r m there uniquely exists the nonsplit exact sequence 0 k e m 0 which is called the fundamental sequence in 3 in particular the module e appearing in the middle term of this sequence is also unique up to isomorphism we call e the fundamental module of r e is said to be decomposable if there is an isomorphism e a e b where a and b are nontrivial ideals of r otherwise e is indecomposable in this paper we are interested in the properties of the fundamental modules in particular we are mostly concerned with its decomposability in fact we can prove that if r is complete with the residue field c then r has the decomposable fundamental module if and only if r is a cyclic quotient singularity see theorem 2 1 and if this is the case then the ar quiver of r will be easily described in 1 we summarize some elementary properties of fundamental modules in particular we can prove that if r is a hypersurface then the fundamental module of r is isomorphic to the third syzygy of k in 2 our main theorem 2 1 stated above will be proved the reader will notice that the keys of the proof are the first author s theorem 11 theorem 1 1 and the theorem of herzog auslander auslander 3 theorem 4 9 herzog 71 received by the editors june 26 1986 and in revised form july 29 1987 1980 mathematics subject classification 1985 revision primary 13h10 13c13 secondary 14b05 14j17

validation of the steady state hover formulation for accurate performance predictions this paper shows accurate predictions for hover performance regardless of planform geometry blade tip mach number or disk loading to prove this statement sensitivity analyses were performed along with performance predictions for four rotor designs planform effects were also studied such as the blade anhedral showing the strong sensitivity of the rotor blade performance due to geometric features the steady state solution methodology with imposed froude boundary conditions is shown to give accurate results for relatively coarse grid sizes this approach leads to reduced computational costs as compared to time dependent simulations it is also recognized that given the current accuracy of the available experimental data the use of more advanced computational fluid dynamics methods may not be fully justified to advance the accuracy of modern computational fluid dynamics methods a comprehensive experimental dataset is required

the signed permutation group on feynman graphs the feynman rules assign to every graph an integral which can be written as a function of a scaling parameter l assuming l for the process under consideration is very small so that contributions to the renormalization group are small we can expand the integral and only consider the lowest orders in the scaling the aim of this article is to determine specific combinations of graphs in a scalar quantum field theory that lead to a remarkable simplification of the first non trivial term in the perturbation series it will be seen that the result is independent of the renormalization scheme and the scattering angles to achieve that goal we will utilize the parametric representation of scalar feynman integrals as well as the hopf algebraic structure of the feynman graphs under consideration moreover we will present a formula which reduces the effort of determining the first order term in the perturbation series for the specific combination of graphs to a minimum

triple cohomology of lie rinehart algebras and the canonical class of associative algebras abstract we introduce a bicomplex which computes the triple cohomology of lie rinehart algebras we prove that the triple cohomology is isomorphic to the rinehart cohomology provided the lie rinehart algebra is projective over the corresponding commutative algebra as an application we construct a canonical class in the third dimensional cohomology corresponding to an associative algebra and extend sridharan s result on almost commutative algebras

numerical evidence of multiple solutions for the reynolds averaged navier stokes equations in this paper evidence is presented for the existence of multiple solutions of reynolds averaged navier stokes equations with the one equation spalart allmaras and two equation wilcox k turbulence models on fixed grids in three dimensions and how they were obtained is described the two major configurations considered are an academic extruded two dimensional airfoil geometry and the trap wing test case the observed appearance of the multiple solutions seems to be closely related to smooth body separation sometimes massive routinely observed in flows over high lift configurations especially near stall angles of attack the results are obtained and cross verified with two stabilized finite element codes streamwise upwind petrov galerkin which provide residual converged results for complex flows with second order discretizations in the paper the ways multiple solutions have been obtained are described including such obvious ones as providing a different initial guess to the steady state solver

on the application of isogeometric finite volume method in numerical analysis of wet steam flow through turbine cascades abstract the isogeometric finite volume analysis is utilized in this research to numerically simulate the two dimensional viscous wet steam flow between stationary cascades of a steam turbine for the first time in this approach the analysis suitable computational mesh with curved boundaries is generated for the fluid flow by employing a non uniform rational b spline nurbs surface that describes the cascade geometry and the governing equations are then discretized by the nurbs representation thanks to smooth and accurate geometry representation of the nurbs formulation the employed isogeometric framework not only resolves issues concerning the conventional mesh generation techniques of the finite volume method in steam turbine problems but also as validated against well established experiments significantly improves the accuracy of the numerical solution in addition the shock location in the cascade is predicted and tracked with a sufficient accuracy

rings of differential operators on invariant rings of tori let k be an algebraically closed field of characteristic zero and g a torus acting diagonally on k for a subset 8 of s 1 2 s set up u e k u o if j e 8 then g acts on d up the ring of regular functions on up and we study the ring d d up g of all differential operators on the invariantvring more generally suppose that is a set of subsets of s such that each invariant ring d u g b e has the same quotient field we prove thatn g d d up g is noetherian and finitely generated as a k algebra now g acts on each d d up and there is a natural map 0 n d d ub n d d up g d ya g obtained by restriction of the differential operators we find necessary and sufficient conditions for 0 to be suuective and describe the kernel of the algebras n g d up g and n g a d up g carry a natural filtration given by the order of the differential operators we show that the associated graded rings are finitely generated commutative algebras and are gorensetin rings we also determine the centers of n ga d up5 g and n ga d d u g introduction throughout this paper k will be an algebraically closed field of characteristic zero if k is a commutative k algebra we denote by do k the set of k linear maps k k and if 1 a k linear map f k k belongs to dp k provided the map f r defined by f r s f rs rf s for s e k belongs to dp l k for all r e k the set d k upaodp k forms a subring of endk k called the ring of differential operators on k we consider k as a left d k module where f r f r for all f e d k r e k the ring of differential operators on the invariant ring of a finite group acting on a polynomial ring has been studied in ka l1 and l2 we carry out a similar study for the invariant ring of a torus we assume that the torus g acts on ks as a group of diagonal matrices if jb is a subset of s 1 2 s then g acts on ua u e ksluj o if j e jb also g acts on p u via gf u f g 1u for g e g f e g u 3 u e u 3 more generally suppose that for some set of subsets a of s we have y ue ua where each invariant ring p u c has the same quotient field f then each d p u 6 is a received by the editors june 18 1986 and in revised form december 11 1986 1980 mclthemcltiss subject clclssificcltion 1985 revision primary 16a33 secondary 13n05 14l30 p 1987 american mathematical s clety 0002 9947 87 1 00 25 per page

improvements to the quadratic constitutive relation based on nasa juncture flow data modifications to the quadratic constitutive relations qcrs of spalart strategies for turbulence modelling and simulation international journal of heat and fluid flow vol 21 2000 pp 252 2

grid influence on turbulence model coefficient uncertainties in transonic wall bounded flows the goal of this work is to determine the effect of grid topology and flow solver discretization on the uncertainty quantification and sensitivity analysis of commonly used turbulence models in rey

rank 2 cohen macaulay modules over singularities of type x13 x23 x33 x43 abstract we describe by matrix factorizations all graded rank 2 maximal cohen macaulay modules over singularities of type x 1 3 x 2 3 x 3 3 x 4 3

deformations of algebras defined by tilting bundles abstract in this paper we produce noncommutative algebras derived equivalent to deformations of schemes with tilting bundles we do this in two settings first proving that a tilting bundle on a scheme lifts to a tilting bundle on an infinitesimal deformation of that scheme and then expanding this result to c equivariant deformations over schemes with a good c action in both these situations the endomorphism algebra of the lifted tilting bundle produces a deformation of the original endomorphism algebra and this is a graded deformation in the c equivariant case we apply our results to rational surface singularities generalising the deformed preprojective algebras of crawley boevey and holland and also to symplectic situations where the deformations produced are related to symplectic reflection algebras

investigation of a nonlinear reynolds averaged navier stokes closure for corner flows this study gives an insight into the corner flow around a wing mounted on a flat plate reynolds averaged navier stokes simulations and experimental data are compared for a reynolds number based on the wing chord rec equal to 3 105 and an angle of attack of 12 deg the flat plate and wing boundary layers are fully turbulent and the reynolds number based on the momentum thickness of the incoming boundary layer is equal to re 2200 it is shown that the reynolds averaged navier stokes simulation using the spalart allmaras model with the boussinesq closure predicts a massive corner flow separation whereas on the experimental side the separation is confined to a small area better predictions are obtained when resorting to the spalart allmaras model with the quadratic constitutive relation closure a comprehensive investigation of the numerical results using the experimental data is performed to get a better understanding of the behavior of the spalart allmaras quadratic constitutive relation model for t

turbulence modeling in rotating and curved channels assessing the spalart shur correction aunie edapproachtosystem rotationandstreamline curvatureeffectsintheframeworkofsimpleeddy viscosity turbulence models is exercised in a range of rotating and curved channel e ows the spalart allmaras sa oneequation turbulence model spalart p r and allmaras s r a one equation turbulence model for aerodynamic flows aiaapaper 92 0439 1992 modie ed in thismanner is shown to bequitecompetitivewith advanced nonlinear and reynolds stress models and to be much more accurate than the original sa model and other eddyviscosity models that are widely used for industrial e ow computations the new term adds about 20 to the computing cost but does not degrade convergence

stability of landau ginzburg branes we evaluate the ideas of stability at the landau ginzburg lg point in moduli space of compact calabi yau manifolds using matrix factorizations to b model the topological d brane category the standard requirement of unitarity at the ir fixed point is argued to lead to a notion of r stability for matrix factorizations of quasihomogeneous lg potentials the d0 brane on the quintic at the landau ginzburg point is not obviously unstable aiming to relate r stability to a moduli space problem we then study the action of the gauge group of similarity transformations on matrix factorizations we define a naive moment maplike flow on the gauge orbits and use it to study boundary flows in several examples gauge transformations of nonzero degree play an interesting role for brane antibrane annihilation we also give a careful exposition of the grading of the landau ginzburg category of b branes and prove an index theorem for matrix factorizations

lie algebroids holonomy and characteristic classes abstract we extend the notion of connection in order to study singular geometric structures namely we consider a notion of connection on a lie algebroid which is a natural extension of the usual concept of a covariant connection it allows us to define holonomy of the orbit foliation of a lie algebroid and prove a stability theorem we also introduce secondary or exotic characteristic classes thus providing invariants which generalize the modular class of a lie algebroid

data parallel line relaxation method for the navier stokes equations the gauss seidel line relaxation method is modie ed for the simulation of viscous e ows on massively parallel computers the resulting data parallel line relaxation method is shown to have good convergence properties for a seriesoftestcases thenewmethodrequiressignie cantlymorememorythanthepreviouslydevelopeddata parallel relaxation methods but it reaches a steady state solution in much less time for all cases tested to date in addition the data parallel line relaxation method shows good convergence properties even on the high cell aspect ratio grids required to simulate high reynolds number e ows the new method is implemented using message passing on the cray t3e and the parallel performance of the method on this machine is discussed the data parallel line relaxation method combines the fast convergence of the gauss seidel line relaxation method with a high parallel efe ciency and thus shows promise for large scale simulation of viscous e ows

a new fast hybrid adaptive grid generation technique for arbitrary two dimensional domains this paper describes a new fast hybrid adaptive grid generation technique for arbitrary two dimensional domains this technique is based on a cartesian background grid with square elements and quadtree decomposition a new algorithm is introduced for the distribution of boundary points based on the curvature of the domain boundaries the quadtree decomposition is governed either by the distribution of the boundary points or by a size function when a solution based adaptive grid is desired the resulting grid is quaddominant and ready for the application of finite element multi grid or line relaxation methods all the internal angles in the final grid have a lower bound of 45 and an upper bound of 135 although our main interest is in grid generation for unsteady flow simulations the technique presented in this paper can be employed in many other fields several application examples are provided to illustrate the main features of this new approach copyright 2010 john wiley sons ltd