positive soliton solutions for generalized quasilinear schrodinger equations with critical growth this paper is concerned with the positive solutions for generalized quasilinear schrodinger equations in rn with critical growth which have appeared from plasma physics as well as high power ultrashort laser in matter by using a change of variables and variational argument we obtain the existence of positive solutions for the given problem

soliton solutions for generalized quasilinear schrodinger equations abstract by introducing a new variable replacement we study the existence of nontrivial solutions for generalized quasilinear schrodinger equations which appear from plasma physics as well as high power ultrashort laser in matter

test ideals in diagonal hypersurface rings ii abstract let r k x 1 x n x 1 d x n d where k is a field of characteristic p p does not divide d and n 3 we describe a method for computing the test ideal for these diagonal hypersurface rings this method involves using a characterization of test ideals in gorenstein rings as well as developing a way to compute tight closures of certain ideals despite the lack of a general algorithm in addition we compute examples of test ideals in diagonal hypersurface rings of small characteristic relative to d including several that are not integrally closed these examples provide a negative answer to smith s question k e smith the multiplier ideal is universal test ideal comm algebra 28 12 2000 5912 5929 of whether the test ideal in general is always integrally closed

a characterization of rational singularities in terms of injectivity of frobenius maps the notions of f rational and f regular rings are defined via tight closure which is a closure operation for ideals in a commutative ring of positive characteristic the geometric significance of these notions has persisted and k e smith proved that f rational rings have rational singularities we now ask about the converse implication the answer to this question is yes and no for a fixed positive characteristic there is a rational singularity which is not f rational so the answer is no in this paper however we aim to show that the answer is yes in the following sense if a ring of characteristic zero has rational singularity then its modulo p reduction is f rational for almost all characteristic p this result leads us to the correspondence of f regular rings and log terminal singularities

strong and weak f regularity are equivalent for graded rings it is shown that the tight closure of a submodule in a artinian module is the same as its finitistic tight closure when the modules are graded over a finitely generated inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i graded ring over a perfect field as a corollary it is deduced that for such a graded ring strong and weak f regularity are equivalent as another application the following conjecture of hochster and huneke is proved let r m be a finitely generated inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i graded ring over a field with unique homogeneous maximal ideal m then r is weakly f regular if and only if r m is weakly f regular

on instability of excited states of the nonlinear schrodinger equation we introduce a new notion of linear stability for standing waves of the nonlinear schrodinger equation nls which requires not only that the spectrum of the linearization be real but also that the generalized kernel be not degenerate and that the signature of all the positive eigenvalues be positive we prove that excited states of the nls are not linearly stable in this more restrictive sense we then give a partial proof that this more restrictive notion of linear stability is a necessary condition to have orbital stability

f singularities under generic linkage abstract let r k x 1 x n be a polynomial ring over a prefect field of positive characteristic let i be an equi dimensional ideal in r and let j be a generic link of i in s r u i j c r we describe the parameter test submodule of s j in terms of the test ideal of the pair r i when a reduction of i is a complete intersection or almost complete intersection as an application we deduce a criterion for when s j has f rational singularities in these cases we also compare the f pure threshold of r i and s j

existence multiplicity and concentration of solutions for generalized quasilinear schrodinger equations with critical growth in this paper we study the following generalized quasilinear schrodinger equations div e2g2 u u e2g u g u u 2 v x u f u g u g u 2 2g u x n where e 0 n 3 g is a c1 even function g 0 1 and g s 0 for all s 0 where g u 0ug t dt under some suitable conditions we study the existence multiplicity and concentration of solutions by the variational methods

f injective singularities are du bois in this paper we prove that singularities of f injective type are du bois this extends the correspondence between singularities associated to the minimal model program and singularities defined by the action of frobenius in positive characteristic

cone orderings group majorizations and similarly separable vectors abstract we generalize some results on majorization in papers by wu and debnath s wu l debnath inequalities for convex sequences and their applications comput math appl 54 2007 525 534 and marshall et al a w marshall i olkin f proschan monotonicity of ratios of means and other applications of majorization in o shisha ed inequalities academic press new york 1967 pp 177 190 we present sufficient conditions for some vector inequalities to hold in the case of cone orderings and group induced cone orderings the framework used is based on results for similarly separable vectors given in paper m niezgoda bifractional inequalities and convex cones discrete math 306 2006 231 243

multiple solutions for a quasilinear schrodinger equation abstract in this paper we consider the quasilinear schrodinger equation u v x u u 2 u g x u x r n where g and v are periodic in x 1 x n and g is odd in u subcritical and satisfies a monotonicity condition we employ the approach developed in szulkin and weth 2009 2010 15 16 and obtain infinitely many geometrically distinct solutions

test ideals in local rings it is shown that certain aspects of the theory of tight closure are well behaved under localization let j be the parameter test ideal for r a complete local cohen macaulay ring of positive prime characteristic for any multiplicative system u c r it is shown that ju 1r is the parameter test ideal for u 1 r this is proved by proving more general localization results for the here introduced classes of f ideals of r and f submodules of the canonical module of r which are annihilators of r modules with an action of frobenius it also follows that the parameter test ideal cannot be contained in any parameter ideal of r tight closure has produced surprising new results for instance that the absolute integral closure of a complete local domain r of prime characteristic is a cohen macaulay algebra for r as well as tremendously simple proofs for otherwise difficult theorems such as the fact that the ring of invariants of a linearly reductive group acting on a regular ring is cohen macaulay the definition of tight closure is recalled in section 1 but we refer the reader to the papers of hochster and huneke listed in the bibliography for more about its applications while primarily a prime characteristic notion tight closure offers insight into arbitrary commutative rings containing q by fairly standard reduction to characteristic p techniques despite its successes the tight closure operation which in its principal setting is a closure operation performed on ideals in a commutative noetherian ring of prime characteristic remains poorly understood for example tight closure is not known to behave well under localization given an ideal i of a ring r letting i denote the tight closure of i it is not known whether or not

ground state solutions for generalized quasilinear schrodinger equations with variable potentials and berestycki lions nonlinearities by introducing some new tricks we prove that the following generalized quasilinear schrodinger equation div g2 u u g u g u u 2 v x u f u x rn admits two classes of ground state solutions under the general berestycki lions assumptions on the nonlinearity f which are almost necessary conditions as well as some weak assumptions on the potential v moreover we also give a minimax characterization of the ground state energy our results improve and complement the previous ones in the literature by introducing some new tricks we prove that the following generalized quasilinear schrodinger equation div g2 u u g u g u u 2 v x u f u x rn admits two classes of ground state solutions under the general berestycki lions assumptions on the nonlinearity f which are almost necessary conditions as well as some weak assumptions on the potential v moreover we also give a minimax characterization of the ground state energy our results improve and complement the previous ones in the literature

an elementary computation of the f pure threshold of an elliptic curve abstract we compute the f pure threshold of a degree three homogeneous polynomial in three variables with an isolated singularity the computation uses elementary methods to prove a known result of bhatt and singh from 2

ground states for a class of critical quasilinear coupled superlinear elliptic systems abstract in this work we consider the following class of quasilinear coupled systems u a x u u 2 u g u x u 2 u v x r n v b x v v 2 v h v x v 2 v u x r n where n 3 and a b r n r are positive potentials r n r is a nonnegative continuous function 0 and 2 satisfying 2 2 on the nonlinear terms we assume that g h are in c 1 class which are superlinear functions at infinity and at the origin we deal with nonlinearities g and h being subcritical or critical the coupling term is a subcritical function which is superlinear at infinity our main theorem is stated without the well known ambrosetti rabinowitz condition at infinity using a change of variable we turn the quasilinear coupled system into a nonlinear coupled system where we establish a variational approach based on nehari method

discreteness and rationality of f jumping numbers on singular varieties we prove that the f jumping numbers of the test ideal tau x delta mathfrak a t are discrete and rational under the assumptions that x is a normal and f finite scheme over a field of positive characteristic p kx is mathbb q cartier of index not divisible p and either x is essentially of finite type over a field or the sheaf of ideals mathfrak a is locally principal this is the largest generality for which discreteness and rationality are known for the jumping numbers of multiplier ideals in characteristic zero

centers of f purity in this paper we study a positive characteristic analogue of the centers of log canonicity of a pair r we call these analogues centers of f purity we prove positive characteristic analogues of subadjunction like results prove new stronger subadjunction like results and in some cases lift these new results to characteristic zero using a generalization of centers of f purity which we call uniformly f compatible ideals we give a characterization of the test ideal which unifies several previous characterizations finally in the case that 0 we show that uniformly f compatible ideals coincide with the annihilators of the mathcal f e r k submodules of e r k as defined by lyubeznik and smith

formulas for multiplier ideals on singular varieties we prove a generalization of demailly ein lazarsfeld s subadditivity formula and musta s summation formula for multiplier ideals to the case of singular varieties using characteristic p methods as an application of our formula we improve hochster huneke s result on the growth of symbolic powers of ideals in singular affine algebras

f singularities of pairs and inversion of adjunction of arbitrary codimension we generalize the notions of f regular and f pure rings to pairs r mathfrak a t of rings r and ideals mathfrak a subset r with real exponent t 0 and investigate these properties these f singularities of pairs correspond to singularities of pairs of arbitrary codimension in birational geometry via this correspondence we prove a sort of inversion of adjunction of arbitrary codimension which states that for a pair x y of a smooth variety x and a closed subscheme y subsetneq x if the restriction z y z to a normal gorenstein closed subvariety z subsetneq x is klt resp lc then the pair x y z is plt resp lc near z

the intersection homology d module in finite characteristic for y a closed normal subvariety of codimension c of a smooth variety x brylinski and kashiwara showed that the local cohomology module c y x x contains a unique simple x submodule denoted by y x in this paper the analogous result is shown for x and y defined over a perfect field of finite characteristic moreover a local construction of y x is given relating it to the theory of tight closure from the construction one obtains a criterion for the x simplicity of c y x x

global existence of small solutions to a relativistic nonlinear schrodinger equation on etudie le probleme de cauchy associea une equation de schrodinger non lineaire modelisant une impulsion laser ultra intense et ultra courte dans un plasma les termes non lineaires nouveaux sont dus aux effets relativistes et a la force ponderomotrice nous prouvons l existence locale et l unicite de solutions petites lorsque la dimension de l espace transverse est egale a 2 ou 3 ainsi que l existence locale pour des donnees initiales de taille arbitraire en dimension un d espace transverse

the jumping coefficients of non q gorenstein multiplier ideals abstract let a o x be a coherent ideal sheaf on a normal complex variety x and let c 0 be a real number de fernex and hacon associated a multiplier ideal sheaf to the pair x a c which coincides with the usual notion whenever the canonical divisor k x is q cartier we investigate the properties of the jumping numbers associated to these multiplier ideals we show that the set of jumping numbers of a pair is unbounded countable and satisfies a certain periodicity property we then prove that the jumping numbers form a discrete set of real numbers if the locus where k x fails to be q cartier is zero dimensional it follows that discreteness holds whenever x is a threefold with rational singularities furthermore we show that the jumping numbers are rational and discrete if one removes from x a closed subset w x of codimension at least three which does not depend on a we also obtain that outside of w the multiplier ideal reduces to the test ideal modulo sufficiently large primes p 0

periodic and asymptotically periodic quasilinear elliptic systems in this work we are concerned with the existence and nonexistence of ground state solutions for the following class of quasilinear schrodinger coupled systems taking into account periodic or asymptotically periodic potentials the nonlinear terms are superlinear at infinity and at the origin by using a change of variable we turn the quasilinear system into a nonlinear system where we can establish a variational approach with a fine analysis on the nehari method for the nonexistence result we compare the potentials with periodic potentials proving the nonexistence of ground state solutions

on the arithmetic of tight closure we provide a negative answer to an old question in tight closure theory by showing that the containment x 3y 3 in x 4 y 4 z 4 in k x y z x 7 y 7 z 7 holds for infinitely many but not for almost all prime characteristics of the field k this proves that tight closure exhibits a strong dependence on the arithmetic of the prime characteristic the ideal x y z subset k x y z u v w x 7 y 7 z 7 ux 4 vy 4 wz 4 x 3y 3 has then the property that the cohomological dimension fluctuates arithmetically between 0 and 1

multiplier ideal sheaves and existence of kahler einstein metrics of positive scalar curvature abstract to study c0a priori estimates for solutions to certain complex monge ampere equations i introduce a coherent sheaf of ideals and show that it satisfies various global algebrogeometric conditions including a cohomology vanishing theorem this technique is used to establish the existence of kahler einstein metrics of positive scalar curvature on a very large class of compact complex manifolds

a frobenius characterization of rational singularity in 2 dimensional graded rings a ring r is said to be f rational if for every prime p in r the local ring r p has the property that every system of parameters ideal is tightly closed as defined by hochster huneke it is proved that if r is a 2 dimensional graded ring with an isolated singularity at the irrelevant maximal ideal m then the following are equivalent 1 r has a rational singularity at m 2 r is f rational 3 a r 0 here a r as defined by goto watanabe denotes the least nonvanishing graded piece of the local cohomology module h m r the proof of this result relies heavily on the properties of derivations of r and suggests further questions in that direction paradigmatically if one knows that d a satisfies a certain property for every derivation d what can one conclude about the original ring element a

f rational rings have rational singularities it is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudorational a key point in the proof is a characterization of f rational local rings as those cohen macaulay local rings r m in which the local cohomology module h d m r where d is the dimension of r have no submodules stable under the natural action of the frobenius map an analog for finitely generated algebras over a field of characteristic zero is developed which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i without reference to a desingularization