existence and multiplicity of nontrivial solutions for a class of modified nonlinear fourth order elliptic equations on r n in this paper we study the following fourth order elliptic equation of the form 2 u u v x u 2 u 2 u h x u x r n where 2 is the biharmonic operator 0 n 6 v c r n r and h c r n r r under appropriate assumptions on v x and h x u some existence and multiplicity results for nontrivial solutions are obtained via variational methods

on model complete differential fields we develop a geometric approach to definable sets in differentially closed fields with emphasis on the question of orthogonality to a given strongly minimal set equivalently within a family of ordinary differential equations we consider those equations that can be transformed by differential algebraic transformations so as to yield solutions of a given fixed first order ode x we show that this sub family is usually definable in particular if x lives on a curve of positive genus as a corollary we show the existence of many model complete superstable theories of differential fields

high energy solutions and nontrivial solutions for fourth order elliptic equations abstract in this paper we study the existence of nontrivial solutions and infinitely many high energy solutions for a class of nonlinear fourth order elliptic equations in r n via variational methods three main theorems are obtained

a chevalley theorem for difference equations by a theorem of chevalley the image of a morphism of varieties is a constructible set the algebraic version of this fact is usually stated as a result on extension of specializations or lifting of prime ideals we present a difference analog of this theorem the approach is based on the philosophy that occasionally one needs to pass to higher powers of where is the endomorphism defining the difference structure in other words we consider difference pseudo fields which are finite direct products of fields rather than difference fields we also prove a result on compatibility of pseudo fields and present some applications of the main theorem e g constrained extension and uniqueness of differential picard vessiot rings with a difference parameter

sign changing solutions for some fourth order nonlinear elliptic problems in this paper we consider the existence and multiplicity of sign changing solutions for some fourth order nonlinear elliptic problems and some existence and multiple are obtained the weak solutions are sought by means of sign changing critical theorems

on von neumann regular rings with an automorphism abstract we study the first order theory of bezout difference rings in particular we show that rings of sequences very rarely have decidable theories as difference rings or even decidable model completions

non integrability by discrete quadratures we give a necessary condition for integrability by discrete quadratures of systems of difference equations the discrete variational equations along algebraic solutions must have virtually solvable galois groups this necessary condition a la morales and ramis is used in order to prove that q analogues of painleve i and painleve iii equations are not integrable by discrete quadratures

local andre oort conjecture for the universal abelian variety we prove a p adic version of the andre oort conjecture for subvarieties of the universal abelian varieties let g and n be integers with n 3 and p a prime number not dividing n let r be a finite extension of w mathbb f p alg the ring of witt vectors of the algebraic closure of the field of p elements the moduli space mathcal a mathcal a g 1 n of g dimensional principally polarized abelian varieties with full level n structure as well as the universal abelian variety pi mathcal x to mathcal a over mathcal a may be defined over r we call a point xi in mathcal x r r special if mathcal x pi xi is a canonical lift and is a torsion point of its fibre employing the model theory of difference fields and work of moonen on special subvarieties of mathcal a we show that an irreducible subvariety of mathcal x r containing a dense set of r special points must be a special subvariety in the sense of mixed shimura varieties

effective uniform bounding in partial differential fields abstract motivated by the effective bounds found in 12 for ordinary differential equations we prove an effective version of uniform bounding for fields with several commuting derivations more precisely we provide an upper bound for the size of finite solution sets of partial differential polynomial equations in terms of data explicitly given in the equations and independent of parameters our methods also produce an upper bound for the degree of the zariski closure of solution sets whether they are finite or not

positive solutions for n 1 1 three point boundary value problems with coefficient that changes sign abstract in this paper we establish existence results for positive solutions for the n 1 1 three point boundary value problems consisting of the equation u n a t f u t 0 t 0 1 with one of the following boundary value conditions u 0 u u 1 u u i 0 0 for i 1 2 n 2 and u n 2 0 u n 2 u n 2 1 u n 2 u i 0 0 for i 0 1 n 3 where 0 1 0 0 and a 0 1 r may change sign and r f 0 0 0 is a parameter our approach is based on the leray schauder degree theory this paper is motivated by eloe and henderson nonlinear anal 28 1997 1669 1680

existence of triple positive solutions of two point right focal boundary value problems on time scales we consider the following boundary value problem 1 n 1y d n t 1 p 1f t y s n 1 t email protected a b t y d n a 0 email protected email protected p 1 y d n s b 0 email protected email protected n 1 where n 2 1 p n 1 is fixed and t is a time scale by applying fixed point theorems for operators on a cone existence criteria are developed for triple positive solutions of the boundary value problem we also include examples to illustrate the usefulness of the results obtained

invariant varieties for polynomial dynamical systems we study algebraic dynamical systems and more generally s varieties f a n c a n c given by coordinatewise univariate polynomials by refining an old theorem of ritt on compositional identities amongst polynomials more precisely we find a nearly canonical way to write a polynomial as a composition of clusters from which one may easily read off possible compositional identities our main result is an explicit description of the weakly skew invariant varieties that is for a fixed field automorphism s c c those algebraic varieties x a n c for which f x x s as a special case we show that if f x c x is a polynomial of degree at least two that is not conjugate to a monomial chebyshev polynomial or a negative chebyshev polynomial and x a 2 c is an irreducible curve that is invariant under the action of x y f x f y and projects dominantly in both directions then x must be the graph of a polynomial that commutes with f under composition as consequences we deduce a variant of a conjecture of zhang on the existence of rational points with zariski dense forward orbits and a strong form of the dynamical manin mumford conjecture for liftings of the frobenius we also show that in models of acfa 0 a disintegrated set defined by s x f x for a polynomial f has morley rank one and is usually strongly minimal that model theoretic algebraic closure is a locally finite closure operator on the nonalgebraic points of this set unless the skew conjugacy class of f is defined over a fixed field of a power of s and that nonorthogonality between two such sets is definable in families if the skew conjugacy class of f is defined over a fixed field of a power of s

multiplicity of solutions for a class of fourth elliptic equations we show that there exist at least three nontrivial solutions for a class of fourth elliptic equations under navier boundary conditions by linking approaches

independence in computable algebra abstract we give a sufficient condition for an algebraic structure to have a computable presentation with a computable basis and a computable presentation with no computable basis we apply the condition to differentially closed real closed and difference closed fields with the relevant notions of independence to cover these classes of structures we introduce a new technique of safe extensions that was not necessary for the previously known results of this kind we will then apply our techniques to derive new corollaries on the number of computable presentations of these structures the condition also implies classical and new results on vector spaces algebraically closed fields torsion free abelian groups and archimedean ordered abelian groups

solvability for a nonlinear second order three point boundary value problem this paper investigates the existence of nontrivial solution for the three point boundary value problem u f t u 0 0 t 1 u 0 0 u 1 u where 0 1 r 1 f c 0 1 r r under certain growth conditions on the nonlinearity f several sufficient conditions for the existence of nontrivial solution are obtained by using leray schauder nonlinear alternative as an application some examples to demonstrate our results are given

the dynamical mordell lang problem for etale maps we prove a dynamical version of the mordell lang conjecture for etale endomorphisms of quasiprojective varieties we use p adic methods inspired by the work of skolem mahler and lech combined with methods from algebraic geometry as special cases of our result we obtain a new proof of the classical mordell lang conjecture for cyclic subgroups of a semiabelian variety and we also answer positively a question of keeler rogalski stafford for critically dense sequences of closed points of a noetherian integral scheme

geometric axioms for differentially closed fields with several commuting derivations abstract a geometric first order axiomatization of differentially closed fields of characteristic zero with several commuting derivations in the spirit of pierce and pillay 1998 13 is formulated in terms of a relative notion of prolongation for kolchin closed sets

on algebraic sigma groups we introduce the categories of algebraic varieties and groups over a difference field k under a linearly closed assumption on k a we prove an isotriviality theorem for groups this theorem immediately yields the key lemma in a proof of the manin mumford conjecture the present paper crucially uses ideas of pilay and ziegler 2003 but in a model theory free manner the applications to manin mumford are inspired by hrushovski s work 2001 and are also closely related to papers of pink and roessler 2002 and 2004

strongly minimal expansions of ℂ definable in o minimal fields we characterize those functions f definable in o minimal expansions of the reals for which the structure f is strongly minimal such functions must be complex constructible possibly after conjugating by a real matrix in particular we prove a special case of the zilber dichotomy an algebraically closed field is definable in certain strongly minimal structures which are definable in an o minimal field

embedding problems over large fields in this paper we study galois theoretic properties of a large class of fields a class which includes all fields satisfying a universal local global principle for the existence of rational points on varieties as applications we give a positive answer to a long standing conjecture which originates in an unpublished note of roquette and asserts that the absolute galois group of a countable hilbertian pac field is profinite free see f j problem 24 41 secondly we give new evidence for the conjecture of shafarevich which asserts that the absolute galois group of qab is profinite free finally one of the most interesting applications of the theory we develop here is the insight in the galois structure of the totally eg adic numbers

a critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations abstract for an even functional on a banach space the symmetric mountain pass lemma gives a sequence of critical values which converges to zero under the same assumptions on the functional this paper establishes a new critical point theorem which provides a sequence of critical points converging to zero the theorem is applied to sublinear elliptic equations then a sequence of solutions converging to zero is obtained

applications of local linking to critical point theory we prove the existence of nontrivial critical points of functionals with a local linking at 0 applications to elliptic boundary value problems and to hamiltonian systems are given c 1995 academic press inc

several sufficient conditions of solvability for a nonlinear higher order three point boundary value problem with all derivatives in this paper firstly some errors in the proof of our paper several sufficient conditions of solvability for a nonlinear higher order three point boundary value problem on time scales appl math comput 190 2007 566 575 are pointed and we make the corresponding correction when t r then the more general problem with all derivatives is considered under certain growth conditions on the nonlinearity several sufficient conditions for the existence and uniqueness of nontrivial solution are obtained by using leray schauder nonlinear alternative and banach fixed point theorem

infinitely many solutions for fourth order elliptic equations abstract in this paper we study a class of fourth order elliptic equations setting on r n based on the minimax methods in critical point theory we obtain the existence of infinitely many large energy and small energy solutions which unifies and improves the recent results of yin and wu y yin x wu high energy solutions and nontrivial solutions for fourth order elliptic equations j math anal appl 375 2011 699 705

on the complexity of the integral closure the computation of the integral closure of an affine ring has been the focus of several modern algorithms we will treat here one related problem the number of generators the integral closure of an affine ring may require this number and the degrees of the generators in the graded case are major measures of cost of the computation we prove several polynomial type bounds for various kinds of algebras and establish in characteristic zero an exponential type bound for homogeneous algebras with a small singular locus

triple positive solutions of boundary value problems for p laplacian dynamic equations on time scales a new triple fixed point theorem is applied to investigate the existence of at least three positive solutions of boundary value problems for p laplacian dynamic equations on time scales

the existence of countably many positive solutions for a system of nonlinear singular boundary value problems with the p laplacian operator abstract in this paper we study the existence of countable many positive solutions for a class of nonlinear singular boundary value systems with p laplacian operator p 1 u a 1 t f u v 0 0 t 1 p 2 v a 2 t g u v 0 0 t 1 1 p 1 u 0 1 p 1 u 0 0 1 p 1 u 1 1 p 1 u 1 0 2 p 2 v 0 2 p 2 v 0 0 2 p 2 v 1 2 p 2 v 1 0 where p i s s p i 2 s p i 1 f g are lower semi continuous functions a i t has countable many singularities on 0 1 2 i 1 2 by using the fixed point theorem of cone expansion and compression of norm type the existence of countable many positive solutions for nonlinear singular boundary value system with p laplacian operator are obtained

remarks on multiple nontrivial solutions for quasi linear resonant problems in this paper morse theory and local linking are used to study the existence of multiple nontrivial solutions for a class of dirichlet boundary value problems with double resonance at infinity and at 0

generalized basic hypergeometric equations this paper deals with regular singular generalized q hypergeometric equations with either large or small galois groups in particular we consider the fundamental problem of finding appropriate galoisian substitutes for the usual notion of local monodromy

soliton solutions for quasilinear schrodinger equations ii abstract for a class of quasilinear schrodinger equations we establish the existence of ground states of soliton type solutions by a variational method

high energy solutions of modified quasilinear fourth order elliptic equations with sign changing potential abstract in the present paper we consider the following modified quasilinear fourth order elliptic equation 1 1 2 u u v x u 1 2 u 2 u f x u for x r n u x h 2 r n where 2 is the biharmonic operator v c r n r and f c r n r r n 5 are allowed to be sign changing two main theorems on the existence of nontrivial solutions and infinitely many high energy solutions for eq 1 1 are obtained via variational methods

positive solutions for nonlinear three point boundary value problems on time scales abstract let t be a time scale such that 0 t t consider the following three point boundary value problem on time scales u t a t f t u t 0 t 0 t u 0 u 0 0 u u t where 0 0 0 t 0 t and d t 1 0 by using fixed point theorems in cones some new and general results are obtained for the existence of single and multiple positive solutions of the above problem in particular our criteria generalize and improve some known results

multiple nontrivial solutions for some fourth order semilinear elliptic problems abstract in this paper we consider the existence of multiple nontrivial solutions for some fourth order semilinear elliptic boundary value problems the weak solutions are sought by means of morse theory and local linking

nonstandard methods for bounds in differential polynomial rings abstract motivated by the problem of the existence of bounds on degrees and orders in checking primality of radical partial differential ideals the nonstandard methods of van den dries and schmidt l van den dries k schmidt bounds in the theory of polynomial rings over fields a nonstandard approach invent math 76 1984 77 91 are here extended to differential polynomial rings over differential fields among the standard consequences of this work are a partial answer to the primality problem the equivalence of this problem with several others related to the ritt problem and the existence of bounds for characteristic sets of minimal prime differential ideals and for the differential nullstellensatz

existence results for singular three point boundary value problems on time scales abstract we prove the existence of a positive solution for the three point boundary value problem on time scale t given by y f x y 0 x 0 1 t y 0 0 y p y 2 1 where p 0 1 t is fixed and f x y is singular at y 0 and possibly at x 0 y we do so by applying a fixed point theorem due to gatica oliker and waltman j differential equations 79 1989 62 for mappings that are decreasing with respect to a cone we also prove the analogous existence results for the related dynamic equations y f x y 0 y f x y 0 and y f x y 0 satisfying similar three point boundary conditions

on some fourth order semilinear elliptic problems in r n the existence of two solutions for a fourth order semilinear elliptic problem involving critical growth from the viewpoint of sobolev embedding was established the basic tools used in the analysis were the mountain pass theorem constrained minimization and concentration compactness principle the existence of positive principle eigenvalues for the corresponding linear elliptic problem was also investigated

primitive permutation groups of bounded orbital diameter we give a description of infinite families of finite primitive permutation groups for which there is a uniform finite upper bound on the diameter of all orbital graphs this is equivalent to describing families of finite permutation groups such that every ultraproduct of the family is primitive a key result is that in the almost simple case with socle of fixed lie rank apart from very specific cases there is such a diameter bound this is proved using recent results on the model theory of pseudofinite fields and difference fields

existence results for an even order boundary value problem on time scales abstract let t be a time scale with t 1 t 2 t 3 t we investigate the existence of solutions to the nonlinear even order three point boundary value problem 1 n y 2 n t f t y t t t 1 t 3 t y 2 i 1 t 1 0 y 2 i t 3 y 2 i 1 t 3 y 2 i 1 t 2 0 i n 1 where n n t 2 t 1 t 3 0 and 1 are given constants

existence of solution and positive solution of bvp for nonlinear third order dynamic equation abstract in this paper we are concerned with the following third order two point boundary value problem on time scale t u t f t u t u t 0 t a b u a a u 2 b b u a c some existence criteria of solution and positive solution are established by using leray schauder fixed point theorem and our main conditions are local an example is also included to illustrate the importance of the results obtained

the uniform companion for large differential fields of characteristic 0 we show that there is a theory uc of differential fields in several commuting derivatives of characteristic 0 which serves as a model companion for every theory of large and differential fields extending a model complete theory of pure fields as an application we introduce differentially closed ordered fields differentially closed p adic fields and differentially closed pseudo finite fields

on green s functions and positive solutions for boundary value problems on time scales in this paper we offer a form of self adjoint differential equations on time scales so that the associated green s function is found symmetric in the usual sense for this purpose together with the delta derivative we employ the nabla derivative as well we introduce the concepts of lebesgue delta and nabla integrals on time scales next sign properties of the green s function are investigated and existence results for positive solutions of nonlinear boundary value problems are established upper and lower bounds for these positive solutions also are given

definable sets in algebraically closed valued fields elimination of imaginaries it is shown that if k is an algebraically closed valued field with valuation ring r then th k has elimination of imaginaries if sorts are added whose elements are certain cosets in k n of certain definable r submodules of k n for all n 1 the proof involves the development of a theory of independence for unary types which play the role of 1 types followed by an analysis of germs of definable functions from unary sets to the sorts contents

the existence of positive solutions for nonlinear singular boundary value system with p laplacian in this paper we study the existence of positive solutions for the following nonlinear singular boundary value problem with p laplacian f p u a t f u t 0 01 f is lower semi continuous functions by using the fixed point theorem of cone expansion and compression of norm type the existence of positive solution and infinitely many positive solutions for nonlinear singular boundary value problem p laplacian are obtained

difference galois theory of linear differential equations abstract we develop a galois theory for linear differential equations equipped with the action of an endomorphism this theory is aimed at studying the difference algebraic relations among the solutions of a linear differential equation the galois groups here are linear difference algebraic groups i e matrix groups defined by algebraic difference equations

solutions for a quasilinear schrodinger equation a dual approach abstract we consider quasilinear stationary schrodinger equations of the form 1 u u 2 u g x u x r n introducing a change of unknown we transform the search of solutions u x of 1 into the search of solutions v x of the semilinear equation 2 v 1 1 2f 2 v g x f v x r n where f is suitably chosen if v is a classical solution of 2 then u f v is a classical solution of 1 variational methods are then used to obtain various existence results

proof mining and effective bounds in differential polynomial rings abstract using the functional interpretation from proof theory we analyze nonconstructive proofs of several central theorems about polynomial and differential polynomial rings we extract effective bounds some of which are new to the literature from the resulting proofs in the process we discuss the constructive content of noetherian rings and the nullstellensatz in both the classical and differential settings sufficient background is given to understand the proof theoretic and differential algebraic framework of the main results

standing waves for quasilinear schrodinger equations with indefinite potentials abstract we consider quasilinear schrodinger equations in r n of the form u v x u u u 2 g u where g u is 4 superlinear unlike all known results in the literature the schrodinger operator v is allowed to be indefinite hence the variational functional does not satisfy the mountain pass geometry by a local linking argument and morse theory we obtain a nontrivial solution for the problem in case that g is odd we get an unbounded sequence of solutions

solutions for fourth order elliptic equations on rn involving uδ u2 and sign changing potentials abstract we obtain existence and multiplicity results for fourth order elliptic equations on r n involving u u 2 and sign changing potentials our results generalize some recent results on this kind of problems to study this kind of problems we first consider the case that the potential v is coercive so that the working space can be compactly embedded into lebesgue spaces then we studied the case that the potential v is bounded so that the working space is exactly h 2 r n which can not be compactly embedded into lebesgue spaces anymore to deal with this more difficult case we study the weak continuity of the term in the energy functional corresponding to the term u u 2 in the equation

generalized hasse schmidt varieties and their jet spaces building on the abstract notion of prolongation developed in 10 the theory of iterative hasse schmidt rings and schemes is introduced simultaneously generalising difference and hasse schmidt differential rings and schemes this work provides a unified formalism for studying difference and differential algebraic geometry as well as other related geometries as an application hasse schmidt jet spaces are constructed generally allowing the development of the theory for arbitrary systems of algebraic partial difference differential equations where constructions by earlier authors applied only to the finite dimensional case in particular it is shown that under appropriate separability assumptions a hasse schmidt variety is determined by its jet spaces at a point

infinitely many solutions for fourth order elliptic equations with general potentials abstract this paper is concerned with the following fourth order elliptic equations 2 u u v x u f x u in r n u h 2 r n where u r n r under general assumption on potential v x and f x u is indefinite sign and sublinear as u we establish the existence of infinitely many solutions by using the genus properties in critical point theory recent results from the literature are extended

several sufficient conditions of solvability for a nonlinear higher order three point boundary value problem on time scales abstract in this paper a nonlinear higher order three point boundary value problem on time scales is considered under certain growth conditions on the nonlinearity several sufficient conditions for the existence of nontrivial solution are obtained by using leray schauder nonlinear alternative as an application some examples to demonstrate our results are given

positive solutions of nonlinear three point boundary value problems abstract let a c 0 1 b c 0 1 0 let 1 t be the unique solution of the linear boundary value problem u t a t u t b t u t 0 t 0 1 u 0 0 u 1 1 we study the existence of positive solutions to the nonlinear boundary value problem u t a t u t b t u t h t f u 0 t 0 1 u 0 0 u u 1 where 0 0 and f c 0 0 we show the existence of at least one positive solution if f is either superlinear or sublinear by applying the fixed point theorem in cones

existence of solutions for some fourth order nonlinear elliptic problems in this paper we consider the existence of positive negative and sign changing solutions for some fourth order semilinear elliptic boundary value problems we present new results on invariant sets of the gradient flows of the corresponding variational functionals the structure of the invariant sets will be built into minimax procedures to construct the sign changing solutions

double positive solutions of three point boundary value problems for p laplacian dynamic equations on time scales a new double fixed point theorem is applied to investigate the existence of at least double positive solutions of three point boundary value problems for p laplacian dynamic equations on a time scale