vector invariants of sylp gl n fq and their hilbert ideals abstract we describe the hilbert ideal of the vector invariants of a p sylow subgroup of the general linear group

on quandle homology groups of alexander quandles of prime order in this paper we determine the integral quandle homology groups of alexander quandles of prime order as a special case this settles the delayed f ibonacci conjecture by m niebrzydowski and j h przytycki in 7 moreover we determine the cohomology group of the alexander quandle and obtain relatively simple presentations of all higher degree cocycles which generate the cohomology group furthermore we prove that the integral quandle homology of a finite connected alexander quandle is annihilated by the order of the quandle

rings of invariants for modular representations of the klein four group we study the rings of invariants for the indecomposable modular representations of the klein four group for each such representation we compute the noether number and give minimal generating sets for the hilbert ideal and the field of fractions we observe that with the exception of the regular representation the hilbert ideal for each of these representations is a complete intersection

a coupled finite volume solver for the solution of incompressible flows on unstructured grids this paper reports on a newly developed fully coupled pressure based algorithm for the solution of laminar incompressible flow problems on collocated unstructured grids the implicit pressure velocity coupling is accomplished by deriving a pressure equation in a procedure similar to a segregated simple algorithm using the rhie chow interpolation technique and assembling the coefficients of the momentum and continuity equations into one diagonally dominant matrix the extended systems of continuity and momentum equations are solved simultaneously and their convergence is accelerated by using an algebraic multigrid solver the performance of the coupled approach as compared to the segregated approach exemplified by simple is tested by solving five laminar flow problems using both methodologies and comparing their computational costs results indicate that the number of iterations needed by the coupled solver for the solution to converge to a desired level on both structured and unstructured meshes is grid independent for relatively coarse meshes the cpu time required by the coupled solver on structured grid is lower than the cpu time required on unstructured grid on dense meshes however this is no longer true for low and moderate values of the grid aspect ratio the number of iterations required by the coupled solver remains unchanged while the computational cost slightly increases for structured and unstructured grid systems the required number of iterations is almost independent of the grid size at any value of the grid expansion ratio recorded cpu time values show that the coupled approach substantially reduces the computational cost as compared to the segregated approach with the reduction rate increasing as the grid size increases

explicit generators of the invariants of finite groups abstract let r denote a commutative and associative ring with 1 and let a denote a finitely generated commutative r algebra let g denote a finite group of r algebra automorphisms of a in the case that r is a field of characteristic 0 noether constructed a finite set of r algebra generators of the invariants of g this paper proves that the same construction produces a set of generators of the invariants of g when g is invertible in r generators of the invariants of g are also explicitly described in the case that g is solvable and g is invertible in r

computation of invariants for reductive groups abstract we will give an algorithm for computing generators of the invariant ring for a given representation of a linearly reductive group the algorithm basically consists of a single grobner basis computation we will also show a connection between some open conjectures in commutative algebra and finding good degree bounds for generating invariants

numerical study of the turbulent flow past an airfoil with trailing edge separation presentation d une methode numerique par volume fini pour la resolution des equations de navier stokes bidimensionnelles incompressibles et stationnaires en coordonnees generales curvilignes application de la methode aux ecoulements turbulents sur des profils avec et sans separation au bord de sortie posterieur comparaison des calculs avec des donnees experimentales

castelnuovo mumford regularity by approximation abstract the castelnuovo mumford regularity of a module gives a rough measure of its complexity we bound the regularity of a module given a system of approximating modules whose regularities are known such approximations can arise naturally for modules constructed by inductive combinatorial means we apply these methods to bound the regularity of ideals constructed as combinations of linear ideals and the module of derivations of a hyperplane arrangement as well as to give degree bounds for invariants of finite groups

an asymptotic preserving all speed scheme for the euler and navier stokes equations we present an asymptotic preserving all speed scheme for the simulation of compressible flows valid at all mach numbers ranging from very small to order unity the scheme is based on a semi implicit discretization which treats the acoustic part implicitly and the convective and diffusive parts explicitly this discretization which is the key to the asymptotic preserving property provides a consistent approximation of both the hyperbolic compressible regime and the elliptic incompressible regime the divergence free condition on the velocity in the incompressible regime is respected and an the pressure is computed via an elliptic equation resulting from a suitable combination of the momentum and energy equations the implicit treatment of the acoustic part allows the time step to be independent of the mach number the scheme is conservative and applies to steady or unsteady flows and to general equations of state one and two dimensional numerical results provide a validation of the asymptotic preserving all speed properties

vector invariant fields of finite classical groups abstract let w be an n dimensional vector space over a finite field f q of any characteristic and mw denote the direct sum of m copies of w let f q m w gl w and f q m w gl w denote the vector invariant ring and vector invariant field respectively where gl w acts on w in the standard way and acts on mw diagonally we prove that there exists a set of homogeneous invariant polynomials f 1 f 2 f m n f q m w gl w such that f q m w gl w f q f 1 f 2 f m n we also prove the same assertions for the special linear groups in any characteristic and the orthogonal unitary and symplectic groups in odd characteristic

on the castelnuovo mumford regularity of rings of polynomial invariants we show that when a group acts on a polynomial ring over a eld the ring of invariants has castelnuovo mumford regularity at most zero as a consequence we prove a well known conjecture that the invariants are always generated in degrees at most n jgj 1 where n 1 is the number of polynomial generators andjgj 1 is the order of the group we also prove some other related conjectures in invariant theory

sharpening the generalized noether bound in the invariant theory of finite groups abstract we consider linear representations of a finite group g on a finite dimensional vector space over a field f in which g is invertible by a theorem due to e noether in char 0 and to fleischmann and fogarty in general the ring of invariants is generated by homogeneous elements of degree at most g schmid domokos and hegedus sharpened noether s bound when g is not cyclic and char f 0 we prove that the sharpened bound holds over general fields if g is not cyclic and g is invertible in f then the ring of invariants is generated by elements of degree at most 3 4 g if g is even and at most 5 8 g if g is odd

fixed fields of triangular matrix groups let k be any field and g be a finite subgroup of gln k then g acts on the rational function field k x1 x2 xn by k automorphisms defined by xj 1 i naijxi if aij g let k x1 xn g f k x1 xn f ffor any g be the fixed field miyata shows that k x1 xn g is rational i e purely transcendental over k provided that g consists of upper triangular matrices we will show that in this situation a transcendence basis f1 fn for k x1 xn g can be choosen with each fi being a polynomial in k x1 xn in fact this theorem follows from a more general result

the depth of invariant rings and cohomology abstract let g be a finite group acting linearly on a vector space v over a field k of positive characteristic p and let p g be a sylow p subgroup ellingsrud and skjelbred compositio math 41 1980 233 244 proved the lower bound depth k v g min dim v p 2 dim v for the depth of the invariant ring with equality if g is a cyclic p group let us call the pair g v flat if equality holds in the above in this paper we use cohomological methods to obtain information about the depth of invariant rings and in particular to study the question of flatness for g of order not divisible by p2 it ensues that g v is flat if and only if h1 g k v 0 or dim vp 2 dim v we obtain a formul a for the depth of the invariant ring in the case that g permutes a basis of v and has order not divisible by p2 in this situation g v is usually not flat moreover we introduce the notion of visible flatness of pairs g v and prove that this implies flatness for example the groups sl2 q so3 q su3 q sz q and r q with many interesting representations in defining characteristic are visibly flat in particular if g sl2 q and v is the space of binary forms of degree n or a direct sum of such spaces then g v is flat for all q pr with the exception of a finite number of primes p along the way we obtain results about the buchsbaum property of invariant rings and about the depth of the cohomology modules hi g k v we also determine the support of the positive cohomology h g k v as a module over k v g in the appendix the visibly flat pairs g v of a group with bn pair and an irreducible representation are classified

invariants of the diagonal cp action on v3 abstract let c p denote the cyclic group of order p where p 3 is prime we denote by v 3 the indecomposable three dimensional representation of c p over a field f of characteristic p we compute a set of generators in fact a sagbi basis for the ring of invariants f v 3 v 3 c p our main result confirms the conjecture of shank r j shank classical covariants and modular invariants in h e a campbell d l wehlau eds invariant theory in all characteristics crm proc lecture notes vol 35 amer math soc 2004 pp 241 249 for this example that all modular rings of invariants of c p are generated by rational invariants norms and transfers

two dimensional vector invariant rings of abelian p groups abstract for any faithful representation v of a non trivial p group over a field of characteristic p 0 it is known that the ring of vector invariants of m copies of v is not cohen macaulay if m 3 however much less is known about the case m 2 in this paper we show that if m 2 and the group is an abelian p group then the ring of invariants of 2v is a complete intersection in some cases and is not cohen macaulay in most cases as a corollary we obtain that if the field is f p and the ring of invariants of the representation v is a polynomial ring then the ring of invariants of 2v is either a complete intersection or not cohen macaulay

the limiting absorption and amplitude principles for the diffraction problem with two unbounded media we consider the classical diffraction problem for the wave propagation in the case where the propagation speed is piecewise constant and the surface separating two media is unbounded the validity of the limiting absorption and amplitude principles is proved

a numerical method for large eddy simulation in complex geometries we discuss the development of a numerical algorithm and solver capable of performing large eddy simulation in the very complex geometries often encountered in industrial applications the algorithm is developed for unstructured hybrid grids is non dissipative yet robust at high reynolds numbers on highly skewed grids simulation results for a variety of flows are presented

degree bounds for syzygies of invariants abstract suppose that g is a linearly reductive group good degree bounds for generators of invariant rings were given in proc amer math soc 129 4 2001 955 here we study minimal free resolutions of invariant rings for finite linearly reductive groups g it was recently shown in adv math 156 1 2000 23 electron res announc amer math soc 7 2001 5 adv math 172 2002 151 that rings of invariants are generated in degree at most the group order g in characteristic 0 this degree bound is a classical result by emmy noether see math ann 77 1916 89 given an invariant ring of a finite linearly reductive group g we prove that the ideal of relations of a minimal set of generators is generated in degree at most 2 g

homomorphisms localizations and a new algorithm to construct invariant rings of finite groups abstract let g be a finite group acting on a polynomial ring a over the field k and let a g denote the corresponding ring of invariants let b be the subalgebra of a g generated by all homogeneous elements of degree less than or equal to the group order g then in general b is not equal to a g if the characteristic of k divides g however we prove that the field of fractions quot b coincides with the field of invariants quot a g quot a g we also study various localizations and homomorphisms of modular invariant rings as tools to construct generators for a g we prove that there is always a nonzero transfer c a g of degree g such that the localization a g c can be generated by fractions of homogeneous invariants of degrees less than 2 g 1 if a sym v f g with finite dimensional f g module v then c can be chosen in degree one and 2 g 1 can be replaced by g let n denote the image of the classical noether homomorphism see the definition in the paper we prove that n contains the transfer ideal and thus can be used to calculate generators for a g by standard elimination techniques using grobner bases this provides a new construction algorithm for a g

the momentum interpolation method based on the time marching algorithm for all speed flows the time marching approach has clear physical meaning and strict mathematical nature and has been applied in computation of compressible flows widely and extended to many uniform algorithms for all speed flows remedy for its weakness in the problem of checkerboard decoupling of pressure field for incompressible flows is envisaged with the time marching momentum interpolation method mim taken into account in this paper existing preconditioning methods for suppressing decoupling and time marching mim are analyzed for this purpose and algorithms of time marching mim are proposed for steady and unsteady flows and for all speed flows asymptotic analysis shows that the supposed time marching mim has at least a third order accuracy better than the existing time marching coupling methods which only have an accuracy of the same order as the adopted scheme has effects of the time step sizes on the ability of the time marching mim to suppress the checkerboard pressure decoupling are particularly discussed in terms of the dual time stepping approach and it is revealed how the decreased sizes of either the pseudo or physical time step increases the possibility of decoupling and how choi s modification in which the history of the interface velocity is decided by itself instead of the arithmetic average of the velocities on its adjacent nodes removes the unphysical pressure oscillation with small size of the physical time step but leads to divergence with the pseudo time step as well as a remedy for the pseudo time step such methods are recommended as implicit methods and the local time step method with a proposed modification of the time marching mim preventing accuracy loss due to very large time step size numerical experiments support the theoretical analyses and show the validity of the time marching mim proposed

vector invariants for the two dimensional modular representation of a cyclic group of prime order in this paper we study the vector invariants of the 2 dimensional indecomposable representation v2 of the cyclic group cp of order p over a field f of characteristic p f mv2 cp this ring of invariants was first studied by david richman 1990 20 who showed that the ring required a generator of degree m p 1 thus demonstrating that the result of noether in characteristic 0 that the ring of invariants of a finite group is always generated in degrees less than or equal to the order of the group does not extend to the modular case he also conjectured that a certain set of invariants was a generating set with a proof in the case p 2 this conjecture was proved by campbell and hughes 1997 in 3 later shank and wehlau 2002 in 24 determined which elements in richman s generating set were redundant thereby producing a minimal generating set we give a new proof of the result of campbell and hughes shank and wehlau giving a minimal algebra generating set for the ring of invariants f mv2 cp in fact our proof does much more we show that our minimal generating set is also a sagbi basis for f mv2 cp further our results provide a procedure for finding an explicit decomposition of f mv2 into a direct sum of indecomposable cp modules finally noting that our representation of cp on v2 is as the p sylow subgroup of sl2 fp we describe a generating set for the ring of invariants f mv2 sl2 fp and show that p m 2 p 1 is an upper bound for the noether number for m 2

the module structure of a group action on a polynomial ring a finiteness theorem consider a group acting on a polynomial ring over a finite field we study the polynomial ring as a module for the group and prove a structure theorem with several striking corollaries for example any indecomposable module that appears as a summand must also appear in low degree and so the number of isomorphism types of such summands is finite there are also applications to invariant theory giving a priori bounds on the degrees of the generators

numerical time step restrictions as a result of capillary waves the propagation of capillary waves on material interfaces between two fluids imposes a strict constraint on the numerical time step applied to solve the equations governing this problem and is directly associated with the stability of interfacial flow simulations the explicit implementation of surface tension is the generally accepted reason for the restrictions on the temporal resolution caused by capillary waves in this article a fully coupled numerical framework with an implicit treatment of surface tension is proposed and applied demonstrating that the capillary time step constraint is in fact a constraint imposed by the temporal sampling of capillary waves irrespective of the type of implementation the presented results show that the capillary time step constraint can be exceeded by several orders of magnitude with the explicit as well as the implicit treatment of surface tension if capillary waves are absent furthermore a revised capillary time step constraint is derived by studying the temporal resolution of capillary waves based on numerical stability and signal processing theory including the doppler shift caused by an underlying fluid motion the revised capillary time step constraint assures a robust aliasing free result as demonstrated by representative numerical experiments and is in the static case less restrictive than previously proposed time step limits associated with capillary waves

pressure velocity coupling allowing acoustic calculation in low mach number flow low mach number flow computation in co located grid arrangement requires pressure velocity coupling in order to prevent the checkerboard phenomenon two broad categories of pressure velocity coupling methods for unsteady flows can be distinguished based on the time step dependency of the coupling coefficient in the definition of the transporting velocity on a face of a control volume as an example of the time step independent category the ausm up scheme is studied as an example of the second category rhie chow momentum interpolation methods are studied within the momentum interpolation techniques again two broad categories can be distinguished based on the time step dependency of the coupling coefficient used for unsteady flow computations but when a steady state is reached variants of rhie chow interpolation methods in each subcategory are studied on critical test cases the result of the study is that for a good representation of unsteady flows containing acoustic information the pressure velocity coupling coefficient must explicitly depend on the time step but that the transporting velocity must become independent of the time step when a steady state is reached

zero separating invariants for finite groups abstract we fix a field k of characteristic p for a finite group g denote by g and g respectively the minimal number d such that for every finite dimensional representation v of g over k and every v v g 0 or v v 0 respectively there exists a homogeneous invariant f k v g of positive degree at most d such that f v 0 let p be a sylow p subgroup of g which we take to be trivial if the group order is not divisible by p we show that g p if n g p p is cyclic we show g n g p if g is p nilpotent and p is not normal in g we show g g l where l is the smallest prime divisor of g these results extend known results in the non modular case to the modular case

the rack space the main result of this paper is a new classification theorem for links smooth embeddings in codimension 2 the classifying space is the rack space and the classifying bundle is the first james bundle we investigate the algebraic topology of this classifying space and report on calculations given elsewhere apart from de ning many new knot and link invariants including generalised james hopf invariants the classification theorem has some unexpected applications we give a combinatorial interpretation for pi 2 of a complex which can be used for calculations and some new interpretations of the higher homotopy groups of the 3 sphere we also give a cobordism classification of virtual links

on the finite volume discretization of discontinuous body force field on collocated grid application to vof method discontinuities in the body force field typically appear at the interface of two fluid systems modeled with the volume of fluid vof and discretized with the finite volume method the discontinuous body force fields are represented as abruptly variable in the present study gravity and continuum surface force csf are considered such strongly variable body forces can produce unphysical spikes in the velocity field when collocated variable arrangement is used the spikes can be eliminated following a force field discretization rule which is deduced by imposing a constraint requiring a zero velocity solution when the forces applied to the system are equilibrated with the gradient of the pressure field it is shown as a byproduct of the present work that a zero velocity solution can only be obtained if the force field is conservative on the discrete level which applies also for the studied case of a stationary bubble finally the case of a rising bubble demonstrates that the proposed rule should be obeyed generally although it is obtained for a quiescent fluid

unified formulation of the momentum weighted interpolation for collocated variable arrangements abstract momentum weighted interpolation mwi is a widely used discretisation method to prevent pressure velocity decoupling in simulations of incompressible and low mach number flows on meshes with a collocated variable arrangement despite its popularity a unified and consistent formulation of the mwi is not available at present in this work a discretisation procedure is devised following an in depth analysis of the individual terms of the mwi derived from physically consistent arguments based on which a unified formulation of the mwi for flows on structured and unstructured meshes is proposed including extensions for discontinuous source terms in the momentum equations as well as discontinuous changes of density as shown by the presented analysis and numerical results the mwi enforces a low pass filter on the pressure field which suppresses oscillatory solutions furthermore the numerical dissipation of kinetic energy introduced by the mwi is shown to converge with third order in space and is independent of the time step if the mwi is derived consistently from the momentum equations in the presence of source terms the low pass filter on the pressure field can be shaped by a careful choice of the interpolation coefficients to ensure the filter only acts on the driving pressure gradient that is associated with the fluid motion which is shown to be vitally important for the accuracy of the numerical solution to this end a force balanced discretisation of the source terms is proposed that precisely matches the discretisation of the pressure gradients and preserves the force applied to the flow using representative test cases of incompressible and low mach number flows including flows with discontinuous source terms and two phase flows with large density ratios the newly proposed formulation of the mwi is favourably compared against existing formulations and is shown to significantly reduce or even eliminate solution errors with an increased stability for flows with large density ratios

moderately ramified actions in positive characteristic in characteristic 2 and dimension 2 wild mathbb z 2 mathbb z actions on k u v ramified precisely at the origin were classified by artin who showed in particular that they induce hypersurface singularities we introduce in this article a new class of wild quotient singularities in any characteristic p 0 and dimension n ge 2 arising from certain non linear actions of mathbb z p mathbb z on the formal power series ring k u 1 dots u n these actions are ramified precisely at the origin and their rings of invariants in dimension 2 are hypersurface singularities with an equation of a form similar to the form found by artin when p 2 in higher dimension the rings of invariants are not local complete intersection in general but remain quasi gorenstein we establish several structure results for such actions and their corresponding rings of invariants

quandle cocycles from invariant theory abstract let g be a group any g module m has an algebraic structure called a g family of alexander quandles given a 2 cocycle of a cohomology associated with this g family topological invariants of handlebody knots in the 3 sphere are defined we develop a simple algorithm to algebraically construct n cocycles of this g family from g invariant group n cocycles of the abelian group m we present many examples of 2 cocycles of these g families using facts from modular invariant theory

quandle cohomology and state sum invariants of knotted curves and surfaces the 2 twist spun trefoil is an example of a sphere that is knotted in 4 dimensional space a proof is given in this paper that this sphere is distinct from the same sphere with its orientation reversed our proof is based on a state sum invariant for knotted surfaces developed via a cohomology theory of racks and quandles also known as distributive groupoids a quandle is a set with a binary operation the axioms of which model the reidemeister moves in classical knot theory colorings of diagrams of knotted curves and surfaces by quandle elements together with cocycles of quandles are used to define state sum invariants for knotted circles in 3 space and knotted surfaces in 4 space cohomology groups of various quandles are computed herein and applied to the study of the state sum invariants non triviality of the invariants is proved for a variety of knots and links and conversely knot invariants are used to prove non triviality of cohomology for a variety of quandles

the noether bound in invariant theory of finite groups abstract let r be a commutative ring v a finitely generated free r module and g glr v a finite group acting naturally on the graded symmetric algebra a sym v let ag denote the minimal number m such that the ring ag of invariants can be generated by finitely many elements of degree at most m furthermore let h g be a normal subgroup such that the index g h is invertible in r in this paper we prove the inequality a g a h g h for h 1 and g invertible in r we obtain noether s bound ag g which so far had been shown for arbitrary groups only under the assumption that the factorial of the group order g is invertible in r

conservation properties of unstructured staggered mesh schemes classic cartesian staggered mesh schemes have a number of attractive properties they do not display spurious pressure modes and they have been shown to locally conserve mass momentum kinetic energy and circulation to machine precision recently a number of generalizations of the staggered mesh approach have been proposed for unstructured triangular or tetrahedral meshes these unstructured staggered mesh methods have been created to retain the attractive pressure aspects and mass conservation properties of the classic cartesian mesh method this work addresses the momentum kinetic energy and circulation conservation properties of unstructured staggered mesh methods it is shown that with certain choices of the velocity interpolation unstructured staggered mesh discretizations of the divergence form of the navier stokes equations can conserve kinetic energy and momentum both locally and globally in addition it is shown that unstructured staggered mesh discretizations of the rotational form of the navier stokes equations can conserve kinetic energy and circulation both locally and globally the analysis includes viscous terms and a generalization of the concept of conservation in the presence of viscosity to include a negative definite dissipation term in the kinetic energy equation these novel conserving unstructured staggered mesh schemes have not been previously analyzed it is shown that they are first order accurate on nonuniform two dimensional unstructured meshes and second order accurate on uniform unstructured meshes numerical confirmation of the conservation properties and the order of accuracy of these unstructured staggered mesh methods is presented

coinvariants and the regular representation of a cyclic p group we consider an indecomposable representation of a cyclic p group z p r over a field of characteristic p we show that the top degree of the corresponding ring of coinvariants is less than frac r 2 3r p r 2 this bound also applies to the degrees of the generators for the invariant ring of the regular representation

decomposing modular coinvariants we consider the ring of coinvariants for a modular representation of a cyclic group of prime order p we show that the classes of the terminal variables in the coinvariants have nilpotency degree p and that the coinvariants are a free module over the subalgebra generated by these classes an incidental result we have is a description of a grobner basis for the hilbert ideal and a decomposition of the corresponding monomial basis for the coinvariants with respect to the monomials in the terminal variables

compressive vof method with skewness correction to capture sharp interfaces on arbitrary meshes the accurate and efficient modelling of two phase flows is at present mostly limited to structured unskewed meshes due to the additional topological and numerical complexity of arbitrary unstructured meshes compressive vof methods which discretize the interface advection with algebraic differencing schemes are computationally efficient and inherently applicable to arbitrary meshes however compressive vof methods evidently suffer severely from numerical diffusion on meshes with topological skewness in this paper we present a compressive vof method using a state of the art donor acceptor advection scheme which includes novel modifications to substantially reduce numerical diffusion on arbitrary meshes without adding computational complexity the new methodology accurately captures evolving interfaces on any arbitrary non overlapping mesh and conserves mass within the limits of the applied solver tolerance a thorough validation of the presented methods is conducted examining the pure advection of the interface indicator function as well as the application to evolving interfaces with surface tension crucially the results on equidistant cartesian and arbitrary tetrahedral meshes are shown to be comparable and accurate

the second order projection method for the backward facing step flow abstract in an effort to demonstrate the validity of the second order projection method for incompressible and viscous flows inside irregular domains we use the method to study the flow in the two dimensional channel with a backward facing step the emphasis of our study is on the flow separation and reattachment phenomena calculations are performed over a wide range of reynolds numbers including the laminar transitional and turbulent regimes defined by experiments in the laminar regime we find excellent agreement with experiments in terms of the dependence of the reattachment length on the reynolds number and streamwise velocity profiles for reynolds numbers beyond the laminar regime where there are no two dimensional experimental data available for comparison we find characteristic flow structures similar to those observed in three dimensional experiments for such physically unstable flows we are able to show certain numerical convergence in the statistical behavior of the solution comparison with the random vortex method is also made and similar flow behavior is found in both solutions

the fundamental theorems of vector invariants abstract new proofs are given of the first fundamental theorems of vector invariants for the special linear and special orthogonal groups the novelty of the proofs is that they rely on reducing the theorems to the case of 2 2 matrices there is a close connection between the vector invariants of the special linear group or special orthogonal group of n n matrices for all n 2 and those of the analogous group of 2 2 matrices this connection is described using the notion of leading monomials properties of leading monomials are also used to give new proofs of the second fundamental theorems of vector invariants for the general linear and orthogonal groups

a new degree bound for vector invariants of symmetric groups let r be a commutative ring v a finitely generated free rmodule and g glr v a finite group acting naturally on the graded symmetric algebra a s v let g v g denote the minirnal number m such lxhat the ring ag of invariants can be generated by finitely many elements of degree at most m for g en and v n kn the k fold direct sum of the natural perrnutation module one knows that p v n kn en n provided that n is invertible in r this was used by e noether to prove p v g igi if igl g a in this paper we prove p v n 1z en masc n kn n 1 for arbitrary commutative rings r and show equality for n ps a prime power and a z or any ring with n 1r o ou r results imply g v g maz lgl rank v lgi 1 for any ring with igi g r

a sharp bound for the castelnuovo mumford regularity of subspace arrangements abstract we show that the ideal of an arrangement of d linear subspaces of projective space is d regular in the sense of castelnuovo and mumford answering a question of b sturmfels in particular this implies that the ideal of an arrangement of d subspaces is generated in degrees less than or equal to d

spectral analysis for optical fibres and stratified fluids i the limiting absorption principle the wave equation t2 h 0 withh c2 x y x y z x y 1 z z x y x rk y rm describes the propagation of acoustical waves in stratified fluids m 1 k 2 as well as electromagnetic waves in layered dielectric media m 1 k 2 and in optical fibres m 2 k 1 and p x y 1 the density p x y and signal speed c x y are possibly long range perturbations of a density p0 y and speed c0 y i e p x y p0 y c x y c0 y 0 as z positive commutator methods are used to show that the spectrum of h is absolutely continuous except possibly for a sequence of isolated eigenvalues of finite multiplicity that can accumulate only at zero and away from those eigenvalues a limiting absorption principle for h is established it is proven that the resolvent h k 1 of h has a norm limit as an operator between l2 rk m 1 z2 2 c2 1 dnz and l2 rk m 1 z2 2 c2 1 dnz for 12 as k c r approaches the real axis either from above or from below

non linear group actions with polynomial invariant rings and a structure theorem for modular galois extensions let g be a finite p group and k be a field of characteristic p 0 we show that g has a non linear faithful action on a polynomial ring u of dimension n log p g such that the invariant ring ug is also polynomial this contrasts with the case of linear and graded group actions with polynomial rings of invariants where the classical theorem of chevalley shephard todd and serre requires g to be generated by pseudo reflections our result is part of a general theory of trace surjective g algebras which in the case of p groups coincide with the galois ring extensions in the sense of chase harrison and rosenberg galois theory and galois cohomology of commutative rings mem amer math soc 52 1965 15 33 we consider the dehomogenized symmetric algebra dk a polynomial ring with non linear g action containing u as a retract and we show that dgk is a polynomial ring thus u turns out to be universal in the sense that every trace surjective g algebra can be constructed from u by forming quotients and extending invariants as a consequence we obtain a general structure theorem for galois extensions with given p group as galois group and any prescribed commutative k algebra r as invariant ring this is a generalization of the artin schreier witt theory of modular galois field extensions of degree ps

numerical method for coupled fluid flow heat transfer and stress analysis using unstructured moving meshes with cells of arbitrary topology abstract a numerical method is presented that can be used for both solid body stress analysis and fluid flow predictions independently as well as in a coupled manner the method uses an integral form of equations governing mass energy and momentum balance for an arbitrary control volume a detailed description is provided of a novel second order accurate spatial discretisation technique which can accommodate unstructured moving meshes with cells of arbitrary topology this is accompanied by a fully implicit temporal discretisation which makes the method stable for any time step size the resulting set of coupled non linear algebraic equations is solved by employing a segregated approach leading to a decoupled set of linear algebraic equations for each dependent variable with a sparse diagonally dominant coefficient matrix these equations are solved by an iterative conjugate gradient solver which retains the sparsity of the coefficient matrix thus achieving a very efficient use of computer resources the method has been tested on a number of fluid flow and stress analysis problems and numerical calculations show favourable agreement with analytical and or experimental results in order to demonstrate the full capabilities of the present method an example of numerically complex coupled fluid flow stress analysis and heat transfer calculations is presented

the invariants of modular indecomposable representations of mathbb z _ p 2 we consider the invariant ring for an indecomposable representation of a cyclic group of order p 2 over a field mathbb f of characteristic p we describe a set of mathbb f algebra generators of this ring of invariants and thus derive an upper bound for the largest degree of an element in a minimal generating set for the ring of invariants this bound as a polynomial in p is of degree two

an all speed roe type scheme and its asymptotic analysis of low mach number behaviour a new scheme all speed roe scheme was proposed for all speed flows compared with traditional preconditioned roe scheme all speed roe scheme changes non linear eigenvalues in the numerical dissipation terms of roe type schemes with an asymptotic analysis the low mach number behaviour of the scheme is studied theoretically in two ways in one way all speed roe scheme is regarded as finite magnification of low speed roe scheme in the low mach number limit in the other way a general form of all speed roe scheme is analyzed both ways demonstrate that all speed roe scheme has the same low mach number behaviour as the original governing equation in the continuous case which includes three important features pressure variation scales with the square of the mach number the zero order velocity field is subject to a divergence constraint and the second order pressure satisfies a poisson type equation in the case of constant entropy the analysis also leads to an unexpected conclusion that the velocity filed computed by traditional preconditioned roe scheme does not satisfy the divergence constraint as the mach number vanishes moreover the analysis explains the reason of checkerboard decoupling and shows that momentum interpolation method provides a similar mechanism as traditional preconditioned roe scheme inherently possesses to suppress checkerboard decoupling in the end general rulers for modifying non linear eigenvalues are obtained finally several numerical experiments are provided to support the theoretical analysis all speed roe scheme has a sound foundation and is expected to be widely studied and applied to all speed flow calculations

the second main theorem vector for the modular regular representation of c2 we study the ring of invariants for a finite dimensional representation v of the group c2 of order 2 in characteristic 2 let denote a generator of c2 and x1 y1 xm ym a basis of v then xi xi and yi yi xi to our knowledge this ring for any prime p was first studied by david richman 12 in 1990 he gave a first main theorem for v2 c2 that is he proved that the ring of invariants when p 2 is generated by xi ni yi2 xiyi tr a 2 a m where a 0 1 m ya y1a1y2a2 ymam and tr a ya y1 x1 a1 y2 x2 a2 ym xm am in this paper we prove the second main theorem for v2 c2 that is we show that all relations between these generators are generated by relations of type i i axitr a i 0 and of type ii tr a tr b l ixi lnltr i l j k ni l jxj ltr l k for all m we also derive relations of type iii which are simpler and can be used in place of the relations of type ii

global degree bounds and the transfer principle for invariants abstract the main purpose of this paper is to verify a conjecture of derksen and kemper concerned with the boundedness of rings of invariants we shall make use of some results connected with the transfer principle in invariant theory these results are dealt with in some generality because of their independent interest and potential for further applications

radiation condition and limiting amplitude principle for acoustic propagators with two unbounded media consider the diffraction problem for perturbed acoustic propagators with perturbations decreasing slowly at infinity the propagation speed is discontinuous at the interface of two unbounded media and the interface may be an arbitrary and smooth surface locally a sommerfeld radiation condition is introduced for the acoustic propagator and is then used to establish the limiting absorption principle and the resolvent estimate at low frequencies for such an operator furthermore we prove the existence of a unique solution to the diffraction problem and the validity of the limiting amplitude principles for the acoustic propagator

generalized formulations for the rhie chow interpolation in this paper generalized formulations for the rhie chow interpolation for co located grid discretizations are derived these generalized formulations eliminate the major known defects in the standard rhie chow interpolation including the following dependence of the converged solution on the value of the under relaxation factor saw tooth pressure oscillations in transient problems with small time steps and incorrect or non converged solutions for problems with discontinuities the generalized formulations are also shown to be applicable to a wider range of flow conditions than the standard rhie chow interpolation the derivation of the rhie chow interpolation is first recalled and its numerical errors are analyzed then the generalized formulations are presented and explained and the way in which they eliminate or counter some of the known defects of the standard rhie chow interpolation are outlined the generalized formulations are then verified and validated with numerical experiments including experiments with flow in porous media and in a packed bed it is then concluded that the generalized formulations presented in this work represent an advance over the standard rhie chow interpolation with a negligible increase in computational cost

a coupled pressure based computational method for incompressible compressible flows pressure based flow solvers couple continuity and linearized truncated momentum equations to derive a poisson type pressure correction equation and use the well known simple algorithm momentum equations and the pressure correction equation are typically solved sequentially in many cases this method results in slow and often difficult convergence the current paper proposes a novel computational algorithm solving for pressure and velocity simultaneously within a pressure correction coupled solution approach using finite volume method on structured and unstructured meshes the method can be applied to both incompressible and subsonic compressible flows for subsonic compressible flows the energy equation is also coupled with flow field and the density of fluid is obtained by equation of state the procedure eliminates the pressure correction step the most expensive component of the simple like algorithms the proposed coupled continuity momentum energy equation method can be used to simulate steady state or transient flow problems the method has been tested on several cfd benchmark cases with excellent results showing dramatically improved numerical convergence and significant reduction in computational time

the noether numbers and the davenport constants of the groups of order less than 32 abstract the computation of the noether numbers of all groups of order less than thirty two is completed it turns out that for these groups in non modular characteristic the noether number is attained on a multiplicity free representation it is strictly monotone on subgroups and factor groups and it does not depend on the characteristic algorithms are developed and used to determine the small and large davenport constants of these groups for each of these groups the noether number is greater than the small davenport constant whereas the first example of a group whose noether number exceeds the large davenport constant is found answering partially a question posed by geroldinger and grynkiewicz

the noether number for the groups with a cyclic subgroup of index two abstract for the finite groups with a cyclic subgroup of index two the exact degree bound for the generators of rings of polynomial invariants is determined

a continuum method for modeling surface tension abstract a new method for modeling surface tension effects on fluid motion has been developed interfaces between fluids of different properties or colors are represented as transition regions of finite thickness across which the color variable varies continuously at each point in the transition region a force density is defined which is proportional to the curvature of the surface of constant color at that point it is normalized so that the conventional description of surface tension on an interface is recovered when the ratio of local transition region thickness to local radius of curvature approaches zero the continuum method eliminates the need for interface reconstruction simplifies the calculation of surface tension enables accurate modeling of two and three dimensional fluid flows driven by surface forces and does not impose any modeling restrictions on the number complexity or dynamic evolution of fluid interfaces having surface tension computational results for two dimensional flows are given to illustrate the properties of the method

the noether numbers for cyclic groups of prime order abstract the noether number of a representation is the largest degree of an element in a minimal homogeneous generating set for the corresponding ring of invariants we compute the noether number for an arbitrary representation of a cyclic group of prime order and as a consequence prove the 2 p 3 conjecture

morrey campanato estimates for helmholtz equations with two unbounded media we prove uniform morrey campanato estimates for helmholtz equations in the case of two unbounded inhomogeneous media separated by an interface they imply weighted l 2 estimates for the solution we prove also a uniform l 2 estimate em without weight for the trace of the solution on the interface

on generators of the hilbert ideal for cyclic groups in modular invariant theory abstract let g be a cyclic group of order p 2 and v be a faithful indecomposable representation of g over a field f of characteristic p we show that the hilbert ideal of the invariant ring is generated by polynomials of degree at most g whenever dim v 4 p or dim v p 2 2 p proving a conjecture of derksen and kemper in this particular case

invariants of finite groups over fields of characteristicp abstract let v denote a finite dimensional k vector space and let g denote a finite group of k linear automorphisms of v let v m denote the direct sum of m copies of v and let g act on the symmetric algebra k v m of v m by the diagonal action on v m a result of noether implies that if char k 0 then k v m g can be generated as a k algebra by polynomials whose degrees are g no matter how large m is this paper proves that this result no longer holds when the characteristic of k divides g more precisely it is proved in this case that there is a positive number depending only on g and char k such that every set of k algebra generators of k v m g contains a generator whose degree is m

an unconditionally stable convergent finite difference method for navier stokes problems on curved domains a new finite difference scheme is presented for solving two dimensional transient incompressible navier stokes problems on a bounded simply connected region for which there exists a c 2 invertible mapping onto the unit square the method is proven to be unconditionally stable and convergent and reduces to the well known krzhivitski and ladyzhenskaya scheme for rectangular domains

a note on the hilbert ideals of a cyclic group of prime order the hilbert ideal is the ideal generated by positive degree invariant polynomials of a finite group for a cyclic group of prime order p we show that the image of the transfer lie in the ideal generated by invariants of degree at most p 1 consequently we show that the hilbert ideal corresponding to an indecomposable representation is generated by polynomials of degree at most p confirming a conjecture of harm derksen and gregor kemper for this case

improved piso algorithms for modeling density varying flow in conjugate fluid porous domains two modified segregated piso algorithms are proposed which are constructed to avoid the development of spurious oscillations in the computed flow near sharp interfaces of conjugate fluid porous domains the new collocated finite volume algorithms modify the rhie chow interpolation to maintain a correct pressure velocity coupling when large discontinuous momentum sources associated with jumps in the local permeability and porosity are present the re distributed resistivity rdr algorithm is based on spreading flow resistivity over the grid cells neighboring a discontinuity in material properties of the porous medium the face consistent pressure fcp approach derives an auxiliary pressure value at the fluid porous interface using momentum balance around the interface such derived pressure correction is designed to avoid spurious oscillations as would otherwise arise with a strictly central discretization the proposed algorithms are successfully compared against published data for the velocity and pressure for two reference cases of viscous flow the robustness of the proposed algorithms is additionally demonstrated for strongly reduced viscosity i e higher reynolds number flows and low darcy numbers i e low permeability of the porous regions in the domain for which solutions without unphysical oscillations are computed both rdr and fcp are found to accurately represent porous media flow near discontinuities in material properties on structured grids two modified segregated piso algorithms are proposed the rhie chow interpolation is modified to maintain a correct pressure velocity coupling when large discontinuous momentum sources are present the proposed algorithms are successfully compared against published data for the velocity and pressure for two reference cases of viscous flow the robustness of the proposed algorithms is additionally demonstrated for high reynolds number flows and low darcy numbers

on vector invariants over finite fields abstract let f denote a finite field and let s m n f denote a set of generators of the invariants of sl n f acting on mn component vectors this paper proves that if m n 1 then s m n f must contain a generator whose degree is greater than or equal to m n 2 f 1 similar results are obtained for the vector invariants of other groups of matrices with entries in f

symmetric powers and modular invariants of elementary abelian p groups abstract let e be an elementary abelian p group of order q p n let w be a faithful indecomposable representation of e with dimension 2 over a field k of characteristic p and let v s m w with m q we prove that the rings of invariants k v e are generated by elements of degree q and relative transfers this extends recent work of wehlau 18 on modular invariants of cyclic groups of order p if m p we prove that k v e is generated by invariants of degree 2 q 3 extending a result of fleischmann sezer shank and woodcock 6 for cyclic groups of order p our methods are primarily representation theoretic and along the way we prove that for any d q with d m q s d v is projective relative to the set of subgroups of e with order m and that the sequence s d v d 0 is periodic with period q modulo summands which are projective relative to the same set of subgroups these results extend results of almkvist and fossum 1 on cyclic groups of prime order

vector invariants ofu2 fp a proof of a conjecture of richman abstract in this paper we prove a conjecture of d r richman concerning the vector invariants of the group u 2 f p let v be a vector space of dimension 2 with basis x y over the field f p and let f p x y be the symmetric algebra of v over f p if denotes a generator of u 2 f p then we may assume x x and y x y let a n be the symmetric algebra of v n we obtain an automorphism of a n of order p by using the diagonal action of extended to the whole of a n the subalgebra of polynomials left invariant by this action is called the ring of vector invariants of u 2 f p richman conjectured that these rings of invariants have certain sets of generators and gave a proof in the case p 2 we prove his conjecture for all primes