fixed points and approximate solutions for nonlinear operator equations we study a nonlinear operator equation and obtain a sequence of approximate solutions by extending the monotone iterative technique to new situations some examples and applications are presented

irreducible affine space curves and the uniform artin rees property on the prime spectrum abstract we give a negative answer to the problem open for twenty years as to whether the full uniform artin rees property holds on the prime spectrum of an excellent ring it was known to hold locally on the prime spectrum of such a ring

on the existence and uniqueness for higher order periodic boundary value problems abstract this paper discusses the existence and uniqueness for the n th order periodic boundary value problem l n u t f t u t 0 t 2 u i 0 u i 2 i 0 1 n 1 where l n u t u n t i 0 n 1 a i u i t is an n th order linear differential operator n 2 and f 0 2 r r is continuous in the case that l n has an even order derivative we present some new spectral conditions for the nonlinearity f t u to guarantee the existence and uniqueness these spectral conditions allow f t u to be a superlinear growth and are the extension for the spectral separation condition presented recently in y li existence and uniqueness for higher order periodic boundary value problems under spectral separation conditions j math anal appl 322 2 2006 530 539

positivity and lower and upper solutions for fourth order boundary value problems abstract this paper is devoted to the study the boundary value problem u 4 t f t u t for all t i 0 1 u 0 u 1 u 0 u 1 0 we prove the existence of at least one two or three solutions in the presence of a pair of not necessarily ordered lower and upper solutions the proof follows from maximum principles related to the operator u 4 m u and amann s three solutions theorem

the method of lower and upper solutions for third order periodic boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem l n u t t u t u i 0 u i 2 i r i 0 n 1 with ln an nth order linear operator and a caratheodory function we obtain sufficient conditions to guarantee the validity of the monotone method for this problem for this we study the sign of the green function of the operator t 1n with tnu lnu mu m 0 furthermore existence results are obtained we show that the results obtained are optimal given a lower solution and an upper solution we obtain the existence of solution between and when either or this general study is applied to the third order problem u t pu t qu t t u t u i 0 u i 2 i r i 0 1 2 with p q r for this problem we obtain the best value on the constant m to guarantee that the green function of the operator t 13 is positive

periodic solutions of some fourth order nonlinear differential equations abstract in this paper using mawhin s continuation theorem we obtain existence results concerning t periodic solutions of fourth order nonlinear differential equations of the type u p u g t u e t where p r g r 2 r is continuous t periodic in t and e r r is continuous t periodic with mean value zero applications are given for the extended fisher kolmogorov and swift hohenberg equations with forcing term

existence multiplicity and infinite solvability of positive solutions to a nonlinear fourth order periodic boundary value problem abstract the existence of n and infinitely many positive solutions is proved for the nonlinear fourth order periodic boundary value problem u 4 t u t u t f t u t 0 t 1 u i 0 u i 1 i 0 1 2 3 where n is an arbitrary natural number and 2 2 0 1 2 2 2 2 4 2 1 0 this kind of fourth order boundary value problems usually describes the equilibrium state of an elastic beam with periodic boundary condition the main results show that the problem may have n or infinitely many positive solutions provided the growth rates of nonlinear term f t l are appropriate on some bounded subsets of its domain

boundary value problems for higher order linear impulsive differential equations in this paper higher order linear impulsive differential equations with fixed moments of impulses subject to linear boundary conditions are studied green s formula is defined for piecewise differentiable functions properties of green s functions for higher order impulsive boundary value problems are introduced an appropriate example of the green s function for a boundary value problem is provided furthermore eigenvalue problems and basic properties of eigensolutions are considered

the theory of d sequences and powers of ideals in this paper we study the question of finding depth r i given an ideal i in a commutative ring this problem is difficult in general even for simple examples depth r z can be difficult to compute brodmann has shown that for any commutative noetherian ring depth r z become stable for large n finding this value however is quite hard this problem originally arose through consideration of the following example suppose k is a field and x xij is an n

positive solutions to nonlinear first order pbvps with parameter on time scales abstract in this paper we consider the following nonlinear first order periodic boundary value problem with parameter on a time scale t x t p t x t f x t t 0 t t x 0 x t where 0 is a parameter for suitable 0 some existence multiplicity and nonexistence criteria of positive solutions are established by using well known results from the fixed point index

integral degree of a ring reduction numbers and uniform artin rees numbers abstract the supremum of reduction numbers of ideals having principal reductions is expressed in terms of the integral degree a new invariant of the ring which is finite provided the ring has finite integral closure as a consequence one obtains bounds for the castelnuovo mumford regularity of the rees algebra and for the artin rees numbers

twin positive solutions of nonlinear first order boundary value problems on time scales abstract in this paper we consider the following nonlinear first order boundary value problem on time scale t x t f x t t 0 t t x 0 x t where 0 1 some existence criteria for at least two positive solutions are established by using the well known twin fixed point theorem due to avery and henderson

existence of multiple positive periodic solutions for delay differential equation whose order is a multiple of 4 abstract this article shows the existence of positive periodic solutions for retarded advanced neutral and ordinary differential equations whose order is a multiple of 4 first we find a positive green s function explicitly then assuming that the coefficient of the unknown in the linear part of the equation is bounded above and below by positive constants we find one and then three solutions by applying the krasnoselskii and legget william fixed point theorems

existence and uniqueness for higher order periodic boundary value problems under spectral separation conditions abstract this paper deals with the existence and uniqueness for the nth order periodic boundary value problem l n u t f t u t 0 t 2 u i 0 u i 2 i 0 1 n 1 where l n u t u n t i 0 n 1 a i u i t is an nth order linear differential operator and f 0 2 r r is continuous we present some spectral conditions for the nonlinearity f t u to guarantee the existence and uniqueness these spectral conditions are the generalization for nonresonance condition of duffing equation

more on squaring and multiplying large integers methods of squaring and multiplying large integers are discussed the obvious o n sup 2 methods turn out to be best for small numbers existing o n sup log sup 3 log sup 2 spl ap o n sup 1 585 methods become better as the numbers get bigger new methods that are o sup log5 log sup 3 spl ap 0 n sup 1 465 o n sup log sup 7 log sup 4 spl ap o n sup 1 404 and o n sup log sup 9 log sup 5 spl ap o n sup 1 365 presented in actual experiments all of these methods turn out to be faster than fft multipliers for numbers that can be quite large 37 000 000 bits squaring seems to be fundamentally faster than multiplying but it is shown that t sub multiplyspl les 2t sub square o n

periodic boundary value problems for nonlinear higher order ordinary differential equations we study a class of boundary value problems for linear and nonlinear ordinary differential equations of higher order we study the solution set and present a maximum principle as a consequence we show the validity of the monotone method and obtain the extremal solutions between the upper and lower solutions

uniform artin rees bounds for syzygies abstract let r m be a local noetherian ring let m be a finitely generated r module and let f be a free resolution of m we find a uniform bound h such that the artin rees containment i n f i im i 1 i n h im i 1 holds for all integers i d for all integers n h and for all ideals i of r in fact we show that a considerably stronger statement holds the uniform bound h holds for all ideals and all resolutions of dth syzygy modules in order to prove our statements we introduce the concept of koszul annihilating sequences

remarks on the lower and upper solutions method for second and third order periodic boundary value problems we study the solvability and the approximation of the solutions by monotone iteration of the second and third order periodic boundary value problems bvps u cu t u u 0 u email protected u 0 u email protected and email protected au bu t u u 0 u email protected u 0 u email protected u 0 u email protected in the presence of lower and upper solutions which may not satisfy the usual ordering condition to this end some maximum and antimaximum principles are stated and proved for certain linear differential operators naturally associated with the previously stated problems

uniform bounds in generalized cohen macaulay rings abstract we establish a uniform bound for the castelnuovo mumford regularity of associated graded rings of parameter ideals in a generalized cohen macaulay ring as consequences we obtain uniform bounds for the relation type and the postulation number moreover we show that generalized cohen macaulay rings can be characterized by the existence of such uniform bounds

the method of lower and upper solutions for second third fourth and higher order boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem u n t t u t u i a u i b i r i 0 n 1 where is a caratheodory function we obtain necessary and sufficient conditions in to guarantee the existence of solutions between a lower solution and an upper solution for n 2 if n 3 either or and n 4 if furthermore we obtain sufficient conditions in for n 2 k 6 when