the riemann problem for a class of nonlinear degenerate equations in the present paper we study the riemann problem for degenerate equations in which the pressure is constant over some range of stains by the liu entropy condition the riemann solutions of twelve regions in the v u plane are completely constructed moreover we obtain the invariant regions for the riemann solutions c 2012 elsevier inc all rights reserved

the cohomological dimension of a resultant variety in prime characteristicp 2 2 let f s t s x s q 2 x st q x t 1 f i f 3 be a collection of bii i0 i1 i2 nary quadratic forms with indeterminate coefficients over a field k and let w x r s k x the aim of this paper is to compute the cohomoi j 1f if 3 0 f jf 2 logical dimension of the ideal i r generated by the following sylvester resultants res f f res f f res f f and res f q f f 1 2 1 3 2 3 1 2 3 the closed points of the affine variety defined by i correspond to coefficients of the three forms having a common zero thus in the w x terminology of l1 i is a resultant system for f f and f compare 1 2 3 w x with theorem 1 b in l1 in general if f f are binary forms of 1 s degrees d g d g g d one can show that a resultant system is 1 2 s s w x generated by at least 1 q 1 q d elements theorem 1 in l1 and ts3 t one would like to know if this lower bound is sharp in our case where s s 3 and d s d s d s 2 the question is whether 1 2 3 4 i can be generated by three elements this would imply that h r s 0 i and this is the motivation behind the main result of this paper 4 it turns out that it is possible to show that h r s 0 working in the i subring of joint invariants of g s gl 2 k acting on r via change of variables in s t this calculation is based on some recent results by lyubeznik on f modules which include an algorithm for deciding whether local cohomology modules over regular rings of prime characteristic p vanish to the best of the author s knowledge theorem 9 in this paper is the first application of this algorithm yan has recently produced a 4 w x topological argument showing that h r vanishes in characteristic 0 y i

generators and defining relations for the ring of differential operators on a smooth affine algebraic variety in prime characteristic abstract for the ring of differential operators d o x on a smooth affine algebraic variety x over a perfect field of characteristic p 0 explicit algebra generators and defining relations are found a finite set of generators and a finite set of defining relations are given explicitly for the module der k o x of derivations on the algebra o x of regular functions on the variety x

convergence of the fractional step lax friedrichs scheme and godunov scheme for the isentropic system of gas dynamics a convergence theorem of the fractional step lax friedrichs scheme and godunov scheme for an inhomogeneous system of isentropic gas dynamics 1 5 3 is established by using the framework of compensated compactness meanwhile a corresponding existence theorem of global solutions with large data containing the vacuum is obtained

finiteness properties of local cohomology modules an application of d modules to commutative algebra the main goal of this paper is to establish finiteness properties of local cohomology modules in characteristic 0 that would be analogous to those proven by c huneke and r sharp in characteristicp 0 our method based on the theory of algebraicd modules seems to be the first application ofd modules to commutative algebra

on invariants and endomorphism rings of certain local cohomology modules abstract let r m denote an n dimensional gorenstein ring for an ideal i r with grade i c we define new numerical invariants i j i as the socle dimensions of h m i h i n j r in case of a regular local ring containing a field these numbers coincide with the lyubeznik numbers i j r i we use d d i d dim r i to characterize the surjectivity of the natural homomorphism f r hom r h i r c r h i r c r as a technical tool we study several natural homomorphisms moreover we prove a few results on i j i

numerical calabi yau metrics we develop numerical methods for approximating ricci flat metrics on calabi yau hypersurfaces in projective spaces our approach is based on finding balanced metrics and builds on recent theoretical work by donaldson we illustrate our methods in detail for a one parameter family of quintics we also suggest several ways to extend our results

local cohomology modules of a smooth z algebra have finitely many associated primes let r be a commutative noetherian ring that is a smooth mathbb z algebra for each ideal mathfrak a of r and integer k we prove that the local cohomology module h k mathfrak a r has finitely many associated prime ideals this settles a crucial outstanding case of a conjecture of lyubeznik asserting this finiteness for local cohomology modules of all regular rings

associated primes of local cohomology of flat extensions with regular fibers and σ finite d modules abstract in this manuscript we study the following question raised by mel hochster let r m k be a local ring and s be a flat extension with regular closed fiber is v m s ass s h i i s finite for every ideal i s and i n we prove that the answer is positive when s is either a polynomial or a power series ring over r and dim r i r 1 in addition we analyze when this question can be reduced to the case where s is a power series ring over r an important tool for our proof is the use of finite d modules which are not necessarily finitely generated as d modules but whose associated primes are finite we give examples of this class of d modules and applications to local cohomology

some results on local cohomology of polynomial and formal power series rings the one dimensional case abstract in this paper we prove several results on the finiteness of local cohomology of polynomial and formal power series rings in particular we give a partial affirmative answer for a question of l nunez betancourt 2014 14

lyubeznik numbers of monomial ideals let r k x1 xn be the polynomial ring in n independent variables where k is a field in this work we will study bass numbers of local cohomology modules h r i r supported on a squarefree monomial ideal i r among them we are mainly interested in lyubeznik numbers we build a dictionary between the modules h r i r and the minimal free resolution of the alexander dual ideal i that allow us to interpret lyubeznik numbers as the obstruction to the acyclicity of the linear strands of i the methods we develop also help us to give a bound for the injective dimension of the local cohomology modules in terms of the dimension of the small support

on the support of local cohomology we examine the question of whether the support of an arbitrary local coho mology module of a finitely generated module over a noetherian ring with support in a given ideal must be closed in the zariski topology several results giving an armative answer to this question are given in particular we show that the support is closed when the given ideal has cohomological dimension at most two or the ambient ring is local of dimension at most four

bass numbers of local cohomology modules let a be a regular local ring of positive characteristic this paper is concerned with the local cohomology modules of a itself but with respect to an arbitrary ideal of a the results include that all the bass numbers of all such local cohomology modules are finite that each such local cohomology module has finite set of associated prime ideals and that whenever such a local cohomology module is artinian then it must be injective this last result had been proved earlier by hartshorne and speiser under the additional assumptions that a is complete and contains its residue field which is perfect the paper ends with some low dimensional evidence related to questions about whether the analogous statements for regular local rings of characteristic 0 are true

convergence of the viscosity solutions for the system of nonlinear elasticity abstract in this paper some special entropy pairs of lax type are constructed for the system of nonlinear elasticity in which the progressive terms are functions of a single variable the necessity estimates for the major terms are obtained by using the theory of singular perturbation of the ordinary differential equations the special entropy pairs are used to prove that the family of young measures uniquely determined by the viscosity solution sequence is a family of dirac measures hence a convergence theorem on the viscosity solutions is established by applying the method of compensated compactness

cohomological dimension lyubeznik numbers and connectedness in mixed characteristic abstract we establish a second vanishing theorem for local cohomology modules over regular rings of unramified mixed characteristic which relates the connectedness of the spectrum of a ring with the vanishing of local cohomology applying this and new results on the mixed characteristic lyubeznik numbers we further study connectedness properties of the spectra of a certain class of rings

d module structure of local cohomology modules of toric algebras let s be a toric algebra over a field k of characteristic 0 and let i be a monomial ideal of s we show that the local cohomology modules h i i s are of finite length over the ring of differential operators d s k generalizing the classical case of a polynomial algebra s as an application we compute the characteristic cycles of some local cohomology modules

existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in eulerian and lagrangian coordinates we prove the existence and compactness stability of entropy solutions for the hyperbolic systems of conservation laws corresponding to the isentropic gas dynamics where the pressure and density are related by a law for any 1 our results considerably extend and simplify the program initiated by diperna and provide a complete existence proof our methods are based on the compensated compactness and the kinetic formulation of systems of conservation laws 1996 john wiley sons inc

an example of an infinite set of associated primes of a local cohomology module let r m be a local noetherian ring let i subset r be any ideal and let m be a finitely generated r module in 1990 craig huneke conjectured that the local cohomology modules h i i m have finitely many associated primes for all i in this paper i settle this conjecture by constructing a local cohomology module of a local k algebra with an infinite set of associated primes and i do this for any field k

associated primes of local cohomology modules and matlis duality abstract let r m be a commutative noetherian local ring of dimension d and i an ideal of r we show that the set of associated primes of the local cohomology module h i 2 r is finite whenever r is regular also it is shown that if x 1 x d is a system of parameters for r then d h x 1 x i i r has infinitely many associated prime ideals for all i d 1 where d hom r e denotes the matlis dual functor and e e r r m is the injective hull of the residue field r m finally we explore a counterexample of grothendieck s conjecture by showing that if d 3 then the r module hom r r i h i d 1 r is not finitely generated where i x 1 x 2 x d

the complete matter sector in a three generation compactification we consider a calabi yau compactification paradigm with three light generations and anr symmetry from a special form of the tian yau manifold we also construct a new three generation model with markedly different phenomenology thecomplete spectrum of all light matter fields is obtained in a universal way and moreover in a physically suitable basis allowing a straightforward analysis of all their couplings here we discuss all the renormalizable yukawa couplings this computation can equally well be repeated for all compactification models based on calabi yau complete intersections in products of homogeneous spaces

yukawa couplings in heterotic compactification we present a practical algebraic method for efficiently calculating the yukawa couplings of a large class of heterotic compactifications on calabi yau three folds with non standard embeddings our methodology covers all of though is not restricted to the recently classified positive monads over favourable complete intersection calabi yau three folds since the algorithm is based on manipulating polynomials it can be easily implemented on a computer this makes the automated investigation of yukawa couplings for large classes of smooth heterotic compactifications a viable possibility

toward an efficient algorithm for deciding the vanishing of local cohomology modules in prime characteristic let r k x 1 dots x n be a ring of polynomials over a field k of characteristic p 0 there is an algorithm due to lyubeznik for deciding the vanishing of local cohomology modules h i i r where i subset r is an ideal this algorithm has not been implemented because its complexity grows very rapidly with the growth of p which makes it impractical in this paper we produce a modification of this algorithm that consumes a modest amount of memory

on the structure of the endomorphism ring of a certain local cohomology module let r m denote an n dimensional gorenstein ring for an ideal i r of height c we are interested in the endomorphism ring b homr hic r hic r it turns out that b is a commutative ring in the case of r m a regular local ring containing a field b is a cohen macaulay ring its properties are related to the highest lyubeznik number l dimkextrd k hic r d dimr i in particular r b if and only if l 1 moreover we show that the natural homomorphism extrd k hic r k is non zero

the d module structure of r f modules let r be a regular ring essentially of finite type over a perfect field k an r module m is called a unit r f module if it comes equipped with an isomorphism f e m m where f denotes the frobenius map on spec r and f e is the associated pullback functor it is well known that m then carries a natural d r module structure in this paper we investigate the relation between the unit r f structure and the induced d r structure on m in particular it is shown that if k is algebraically closed and m is a simple finitely generated unit r f module then it is also simple as a d r module an example showing the necessity of k being algebraically closed is also given

positively invariant regions for a problem in phase transitions positively invariant regions for the system v t p w x e v xx w t v x e w x are constructed where p p w 0 w e 0 such a choice of p is motivated by the maxwell construction for a van der waals fluid the method of an analysis is a modification of earlier ideas of chueh conley smoller 1 the results given here provide e independent l bounds on the solution w v

kinetic formulation of the isentropic gas dynamics and p systems we consider the 2 x 2 hyperbolic system of isentropic gas dynamics in both eulerian or lagrangian variables also called the p system we show that they can be reformulated as a kinetic equation using an additional kinetic variable such a formulation was first obtained by the authors in the case of multidimensio nal scalar conservation laws a new phenomenon occurs here namely that the advection velocity is now a combination of the macroscopic and kinetic velocities various applications are given we recover the invariant regions deduce new l estimates using moments lemma and prove l w stability for 7 3

bockstein homomorphisms in local cohomology let r be a polynomial ring in finitely many variables over the integers fix an ideal a of r we prove that for all but finitely prime integers p the bockstein homomorphisms on local cohomology h a r pr h k 1 a r pr are zero this vanishing of bockstein homomorphisms is predicted by lyubeznik s conjecture which states that when r is a regular ring the modules h a r have finitely many associated prime ideals we further show that when r is replaced by a hypersurface ring the vanishing of bockstein homomorphisms as well as the analogue of lyubeznik s conjecture need not hold bockstein homomorphisms have their origins in algebraic topology we establish a connection between algebraic and topological bockstein homomorphisms through stanley reisner theory

on the set of associated primes of a local cohomology module abstract assume r is a local cohen macaulay ring it is shown that ass r h l i r is finite for any ideal i and any integer l provided ass r h 2 x y r is finite for any x y r and ass r h 3 x 1 x 2 y r is finite for any y r and any regular sequence x 1 x 2 r furthermore it is shown that ass r h l i r is always finite if dim r 3 the same statement is even true for dim r 4 if r is almost factorial

constructing arbitrary torsion elements for a local cohomology module local cohomology modules are in general not finitely generated over the ambient ring towards a better understanding of the structure of these modules huneke 1992 hu asked if they have finitely many associated prime ideals the answer is negative several authors have constructed a ring whose local cohomology module has infinitely many associated primes in certain situations one can prove that a local cohomology module has infinitely many associated primes by showing that z pz embeds into the local cohomology module for each prime integer p we show that in fact each finitely generated abelian group embeds into a local cohomology module