analytic torsion and holomorphic determinant bundles i bott chern forms and analytic torsion we attach secondary invariants to any acyclic complex of holomorphic hermitian vector bundles on a complex manifold these were first introduced by bott and chern bot c our new definition uses quillen s superconnections we also give an axiomatic characterization of these classes these results will be used in bgs2 and bgs3 to study the determinant of the cohomology of a holomorphic vector bundle

existence of positive solutions for fourth order m point boundary value problem with variable parameters in this paper we investigate the existence of positive solutions for a class of nonlinear fourth order m point boundary value problem with two variable parameters by using the fixed point index theory we give some new existence results

on nilpotency of matrices over antirings abstract an antiring is a semiring which is zerosumfree i e a b 0 implies a b 0 for any a b in this semiring in this paper we study the nilpotency of matrices over commutative antirings we first provide some properties and characterizations of the nilpotent matrices in terms of principal permanental minors main diagonals and permanental adjoint matrices when a family of matrices are simultaneously considered we establish some characterizations of the simultaneous nilpotence for a family of matrices

multiple sign changing solutions for fourth order m point boundary value problems abstract in this paper we study the existence and multiplicity of nontrivial solutions for the fourth order m point boundary value problems making use of the theory of the fixed point index in a cone and the leray schauder degree under general conditions on nonlinearity we prove that there exist at least six different nontrivial solutions for the fourth order m point boundary value problems furthermore if the nonlinearity is odd we obtain that there exist at least eight different nontrivial solutions

bases in semimodules over commutative semirings abstract in this paper the bases of a semimodule over a commutative semiring r are investigated some properties and characterizations of the bases are discussed and some equivalent conditions for a basis to be a free basis in a finitely generated free semimodule over r are given the different possible cardinalities for a basis in a finitely generated free semimodule over r are considered and some equivalent descriptions are obtained for a commutative semiring r satisfying the property that any two bases for a finitely generated free r semimodule have the same cardinality partial results obtained in the paper develop and generalize the corresponding results for commutative join semirings and for commutative zerosumfree semirings

matrices over semirings introducing the concept of semi invertibility of square matrices over semirings some sufficient conditions for semi invertibility of square matrices over various kinds of semirings are obtained also a necessary and sufficient condition for semi invertibility of square matrices over semirings is obtained

about a numerical method of successive interpolations for functional hammerstein integral equations a new and robust numerical method for hammerstein functional integral equations is proposed the convergence and the numerical stability of the method are mathematically proved and tested on some examples

on simultaneously nilpotent fuzzy matrices abstract nilpotent fuzzy matrices play a crucial role in the study of fuzzy matrices in this paper we shall extend the nilpotence to the notion of simultaneous nilpotence for a finite set of fuzzy matrices the notion of simultaneous nilpotence relates to the infinite products of a finite number of fuzzy matrices which converge to the zero matrix properties of the simultaneous nilpotence will be established in the study of consecutive powers of a fuzzy matrix a controllable fuzzy matrix can be characterized by an associated nilpotent fuzzy matrix in this paper we propose the notion of simultaneously controllable fuzzy matrices which can be thought of as a generalization of the notion of controllable fuzzy matrices similar to the nilpotent characterization for a controllable fuzzy matrix the simultaneously controllable fuzzy matrices can be characterized by a finite set of associated simultaneously nilpotent fuzzy matrices

eigenvalues of fourth order singular sturm liouville boundary value problems abstract in this paper by establishing a new comparison theorem and constructing upper and lower solutions some sufficient conditions of existence of positive solutions for the following nonlinear fourth order singular sturm liouville eigenvalue problem 1 p t p t u t f t u t 0 1 u 0 u 1 0 u 0 lim t 0 p t u t 0 u 1 lim t 1 p t u t 0 are established due to the schauder s fixed point theorem for large enough where 0 0 f and p can be singular at t 0 and or 1 moreover f can also be singular at u 0 in addition some peculiar cases are discussed and some further results are obtained

positive solutions of the three point boundary value problem for second order differential equations with an advanced argument abstract in this paper by using the fixed point index method we establish the existence of at least one or at least two positive solutions to the three point boundary value problem for a second order differential equation with an advanced argument u t a t f u h t 0 t 0 1 u 0 0 u u 1 where 0 1 0 and 1 0 as an application we also give an example to demonstrate our result

invertible and nilpotent matrices over antirings abstract in this paper we characterize invertible matrices over an arbitrary commutative antiring s with 1 and find the structure of gl n s we find the number of nilpotent matrices over an entire commutative finite antiring we prove that every nilpotent n n matrix over an entire antiring can be written as a sum of log 2 n square zero matrices and also find the necessary number of square zero summands for an arbitrary trace zero matrix to be expressible as their sum

existence and multiplicity of solutions of a kind of fourth order boundary value problem abstract in this paper the existence and multiplicity results of the solutions are obtained for the fourth order boundary value problem u 4 t f t u t for all t 0 1 subject to u 0 u 1 u 0 u 1 0 where f is continuous the monotone operator theory and critical point theory are employed to discuss this problem respectively

multiple solutions for fourth order boundary value problem abstract in this paper we study the existence and multiplicity of nontrivial solutions for the fourth order two point boundary value problems making use of the theory of fixed point index in cone and leray schauder degree under general conditions on nonlinearity we prove that there exist at least six different nontrivial solutions for the fourth order two point boundary value problems furthermore if the nonlinearity is odd we obtain that there exist at least eight different nontrivial solutions

generalized fuzzy h bi ideals and h quasi ideals of hemirings a new kind of generalized fuzzy h ideals of a hemiring namely the set membership variant set membership variantlogical orq fuzzy h bi ideal resp h quasi ideal is studied and the relationships between these generalized fuzzy h ideals are described some characterization theorems of prime and semiprime set membership variant set membership variantlogical orq fuzzy h bi ideals resp h quasi ideals of a hemiring are also given in particular we show that the h hemiregular hemirings and h intra hemiregular hemirings can be described by using some of their generalized fuzzy h ideals finally the implication based fuzzy h bi ideals resp h quasi ideals of a hemiring are considered

positive solutions of the nonlinear fourth order beam equation with three parameters abstract this paper is concerned with the existence and nonexistence of positive solutions of the nonlinear fourth order beam equation u 4 t u t u t f t u t 0 t 1 u 0 u 1 u 0 u 1 0 where f t u 0 1 0 0 is continuous and and are parameters we show that there exists a 0 such that the above boundary value problem bvp has at least two one and no positive solutions for 0 and respectively furthermore by using the semiorder method on cones of banach space we establish a uniqueness criterion for positive solution of the bvp in particular such a positive solution u t of the bvp depends continuously on the parameter i e u t is nondecreasing in lim 0 u t 0 and lim u t for any t 0 1

inversion of matrices over a commutative semiring it is a well known consequence of the elementary theory of vector spaces that if a and b are n by n matrices over a field or even a skew field such that ab 1 then ba 1 this result remains true for matrices over a commutative ring however it is not in general true for matrices over noncommutatives rings in this paper we show that if a and b are n by n matrices over a commutative semiring then the equation ab 1 implies ba 1 we give two proofs one algebraic in nature the other more combinatorial both proofs use a generalization of the familiar product law for determinants over a commutative semiring

positive solutions of three point boundary value problems for second order impulsive differential equations with advanced arguments in this paper we discuss three point boundary value problems for second order impulsive differential equations with advanced arguments we establish sufficient conditions under which such problems have positive solutions to obtain the results we use the fixed point index

positive solutions to a multi point boundary value problem with delay abstract the paper is concerned with the existence of positive solutions for the nonlinear multi point boundary value problems of a class of second order delay differential equations our results improve some recent results obtained in d q jiang multiple positive solutions for boundary value problems of second order delay differential equations appl math lett 15 2002 575 583 dingyong bai yuantong xu existence of positive solutions for boundary value problems of second order delay differential equations appl math lett 18 2005 621 630

on simultaneously nilpotent antiring matrices antirings are an important type of semirings which generalize boolean algebra fuzzy algebra distributive lattice and incline in this paper we study the issue of nilpotent antiring matrices provide some properties and characterizations of the simultaneous nilpotence for a finite number of antiring matrices present some methods for calculating the simultaneously nilpotent index of a finite number of antiring matrices partial results obtained in this paper generalize and develop the corresponding ones on nilpotent antiring matrices and on simultaneously nilpotent fuzzy matrices lattice matrices

linear operators strongly preserving r potent matrices over semirings abstract we consider the problem of characterizing those linear operators l on the matrices over a semiring such that l x is idempotent if and only if x is complete characterizations are obtained for many semirings including the nonnegative reals the nonnegative integers the two element boolean algebra and the fuzzy scalars

necessary and sufficient conditions for the existence of quasi symmetric positive solutions of singular boundary value problem abstract this paper investigates the existence of quasi symmetric positive solutions for fourth order singular three point boundary value problem we first give definitions of quasi symmetric lower and upper solutions by constructing quasi symmetric lower and upper solutions and using the schauder fixed point theorem necessary and sufficient conditions for existence results are obtained

the existence and the uniqueness of symmetric positive solutions for a fourth order boundary value problem the purpose of this paper is to investigate the existence and the uniqueness of symmetric positive solutions for a class of fourth order boundary value problem y 4 t f t y t t 0 1 y 0 y 1 y 0 y 1 0 by using the fixed point index method we establish the existence of at least one or at least two symmetric positive solutions for the above boundary value problem further by using a fixed point theorem of general a concave operators we also present criteria which guarantee the existence and uniqueness of symmetric positive solutions for the above boundary value problem

symmetric positive solutions for nonlocal boundary value problems of fourth order abstract the singular fourth order nonlocal boundary value problem u t h t f t u 0 t 1 u 0 u 1 0 1 p s u s d s u 0 u 1 0 1 q s u s d s is considered under some suitable conditions concerning the first eigenvalue of the corresponding linear operator where p q l 1 0 1 h 0 1 0 is continuous symmetric on 0 1 and may be singular at t 0 and t 1 f 0 1 0 0 is continuous and f x is symmetric on 0 1 for all x 0 the existence of at least one symmetric positive solution is obtained by the application of the fixed point index in cones

inner products on semimodules over a commutative semiring abstract in this paper the inner products on the semimodules over a commutative semiring are investigated some characterizations for inner products and for orthogonal sets in the semimodules are given and standard orthogonal basis and orthogonal subsemimodule are discussed in particular an equivalent description is obtained for a semifield s satisfying the property that every standard orthogonal set in a finitely generated semimodule m over s can be extended to a standard orthogonal basis for m also the analogue of the kronecker capelli theorem is proved to be valid for a matrix equation over a commutative semiring partial results obtained in this paper generalize and develop corresponding results for semilinear spaces of n dimensional vectors over commutative zerosumfree semirings

upper and lower solution method for fourth order four point boundary value problems this paper is concerned with the fourth order ordinary differential equation u 4 t f t u t u t 0 0 some new existence results are obtained by developing the upper and lower solution method and the monotone iterative technique

noncommuting graphs of matrices over semirings we calculate diameters and girths of commuting graphs of the set of all nilpotent matrices over a semiring the group of all invertible matrices over a semiring and the full matrix semiring

positive solutions for a class of boundary value problem with integral boundary conditions in banach spaces this paper investigates the existence and multiplicity of positive solutions for a class of nonlinear boundary value problem of second order differential equations with integral boundary conditions in ordered banach spaces the arguments are based upon a specially constructed cone and the fixed point theory in a cone for strict set contraction operators the nonexistence of a positive solution is also studied

on nilpotent fuzzy matrices abstract in this paper we study the issue of nilpotent fuzzy matrices we first provide some properties of nilpotent fuzzy matrices in terms of eigenvalues when a finite number of fuzzy matrices are simultaneously considered we establish some characterizations of the simultaneous nilpotence for a finite number of fuzzy matrices on the other hand it is well known that the max min algebra is one kind of lattice we shall extend the results on simultaneous nilpotence to matrices on a bounded distributive lattice

positive solutions of second order two delay differential systems with twin parameter abstract the paper is concerned with the existence of positive solutions for a second order two delay differential systems with twin parameter firstly we present some sufficient conditions with the twin parameter belonging to an open rectangular region eigenvalue region to ensure the existence of positive solutions for the differential systems with singularity secondly we deal with the superlinear semipositone problem of the differential systems and establish the existence of positive solutions for the systems the proofs are based on a fixed point theorem in cones

a rank theorem for vandermonde matrices we show that certain matrices built from vandermonde matrices are of full rank this result plays a key role in the construction of the limit theory of generic polynomials

existence and multiplicity of solutions for fourth order boundary value problems with parameters abstract in this paper we study the existence and multiplicity of the solutions for the fourth order boundary value problem bvp u 4 t u t u t f t u t 0 t 1 u 0 u 1 u 0 u 1 0 where f 0 1 r r is continuous r and r are parameters by means of the idea of the decomposition of operators shown by chen w y chen a decomposition problem for operators xuebao of dongbei renmin university 1 1957 95 98 see also m krasnosel skii topological methods in the theory of nonlinear integral equations gostehizdat moscow 1956 and the critical point theory we obtain that if the pair is on the curve 2 4 satisfying 2 2 then the above bvp has at least one two three and infinitely many solutions for being in different interval respectively

second order boundary value problems with integral boundary conditions abstract in this paper we investigate the existence of positive solutions of nonlocal second order boundary value problems with integral boundary conditions

positive solutions of fourth order singular three point eigenvalue problems abstract in this paper by establishing a new comparison theorem and constructing upper and lower solutions some sufficient conditions of existence of positive solutions for the following singular fourth order three point eigenvalue problem u 4 t f t u t 0 1 u 0 u u 1 0 u 0 u u 1 0 are established due to schauder s fixed point theorem for large enough where 0 1 are constants f can be singular at t 0 and or 1 u 0 moreover some peculiar cases are discussed and some further results are obtained

positive solutions to a type of multi point boundary value problem with delay and one dimensional p laplacian abstract this paper deals with a type of multi point boundary value problem with delay and one dimensional p laplacian by using guo krasnosel skii fixed point theorem and generalization of the leggett williams fixed point theorem due to avery and peterson we obtain that bvp 1 1 has at least one positive solution or three positive solutions

positive solutions for fourth order differential equations with deviating arguments and integral boundary conditions abstract in this paper we investigate integral boundary value problems for fourth order differential equations with deviating arguments we discuss our problem both for advanced or delayed arguments we establish sufficient conditions under which such problems have positive solutions to obtain the existence of multiple at least three positive solutions we use a fixed point theorem due to avery and peterson an example is also included to illustrate that corresponding assumptions are satisfied the results are new

on nilpotent incline matrices inclines are additively idempotent semirings in which products are less than or equal to either factor thus they generalize boolean algebra fuzzy algebra and distributive lattice this paper studies the nilpotent incline matrices in detail it is proved that an incline matrix is nilpotent if and only if it has index and the zero vector is its unique standard eigenvector the nilpotent matrices over an incline without nilpotent elements are characterized in terms of principal minors main diagonals nilpotent indices and adjoint matrices also some properties of the reduction of nilpotent matrices over an additively residuated incline without nilpotent elements are established the results obtained here generalize the corresponding ones on fuzzy matrices and lattice matrices shown in the references

positive solutions of fourth order four point boundary value problems with p laplacian operator in this paper we consider the existence of positive solutions for the singular fourth order p laplacian equation p u t f t u t 0 t 1 with the four point boundary conditions u 0 0 u 1 au u 0 0 u 1 bu where p t t p 2t p 1 0 1 f c 0 1 0 0 may be singular at t 0 and or 1 and u 0 by using the upper and lower solution method and fixed point theorems the existence of positive solutions to the above the boundary value problem is obtained

symmetric positive solutions of boundary value problems with integral boundary conditions under conditions weaker than those used by ma we establish various results on the existence and nonexistence of symmetric positive solutions to fourth order boundary value problems with integral boundary conditions the arguments are based upon a specially constructed cone and the fixed point theory in a cone our results improve and extend that obtained in h ma symmetric positive solutions for nonlocal boundary value problems of fourth order nonlinear anal 68 2008 645 651 we illustrate our results by one example which cannot be handled using the existing results

calculus on arithmetic surfaces in a2 and a3 arakelov introduces an intersection calculus for arithmetic surfaces that is for stable models of curves over a number field in this paper we intend to show that his intersection product has a lot of useful properties more precisely we show that the following properties from the theory of algebraic surfaces have an analogue in our situation

positive solutions for a class of boundary value problems with integral boundary conditions this paper investigates the existence and multiplicity of positive solutions for a class of nonlinear boundary value problems of fourth order differential equations with integral boundary conditions the arguments are based upon a specially constructed cone and the fixed point theory in cone the nonexistence of positive solutions is also studied

searching the least value method for solving fourth order nonlinear boundary value problems this paper obtains a searching least value slv method for a class of fourth order nonlinear boundary value problems is investigated the argument is based on the reproducing kernel space w 5 0 1 the approximate solutions u n x and u n k x are uniformly convergent to the exact solution u x and u k x k 1 2 3 4 respectively numerical results are verified that the method is quite accurate and efficient for this kind of problem

upper and lower solution method for a fourth order four point boundary value problem on time scales in this work we consider a fourth order four point boundary value problem on time scales we establish criteria for the existence of a solution by developing the upper and lower solution method and the monotone iterative technique

three positive fixed points of nonlinear operators on ordered banach spaces we generalize the fixed point theorem of leggett williams which is a theorem giving conditions that imply the existence of three fixed points of an operator defined on a cone in a banach space we then show how to apply our theorem to prove the existence of three positive solutions to a second order discrete boundary value problem 2001 elsevier science ltd all rights reserved

on the nilpotent matrices over d01 lattice abstract in this paper we study nilpotent matrices over d 01 lattice a distributive lattice l is said to be d 01 lattice if 0 1 l and 0 x 1 for all x l a necessary and sufficient condition for a d 01 lattice matrix a to be a nilpotent matrix will be given we also establish some important properties of nilpotent matrices

multiple solutions of some boundary value problems with parameters abstract in this paper we study the existence and multiplicity of nontrivial solutions for the following second order dirichlet nonlinear boundary value problem with odd order derivative u t a u t b u t f t u t for all t 0 1 with u 0 u 1 0 where a b r 1 f c 1 0 1 r 1 r 1 by using the morse theory we impose certain conditions on f which are able to guarantee that the problem has at least one nontrivial solution two nontrivial solutions and infinitely many solutions separately

heights and arakelov s intersection theory introduction this paper relates the neron tate height on jacobian varieties to an intersection theory on arithmetic surfaces constructed by s arakelov a1 here i extend and prove a statement of arakelov to the effect that his intersection pairing is a neron pairing a2i i use this to prove an analogue of the hodge index theorem for arithmetic surfaces a result due independently to g faltings the results of this paper have been extracted from my thesis written under the direction of michael artin it differs from the thesis in three ways firstly it employs the language of local height functions as defined by a neron and j tate and expounded by s lang li in the thesis i used neron s language of local intersection pairings n1 the second major difference is in the proof of theorem 1 3 in my thesis i proved this result by using work of raynaud and picard functors r 1 this method was suggested by m artin in this paper i provide a simpler proof observed by s lang and independently by b gross finally the development of arakalov s theory in the context of neron functions in section 2 is due to s lang who has kindly given me leave to publish it this exposition of my results is due in considerable part to s lang it is essentially an edited transcript of a series of conversations between us which he provided it is my great pleasure to thank both him and michael artin for their help and encouragement 1 the neron pairing neron ne has given a pairing between divisors and points on abelian varieties and also on arbitrary varieties under suitable conditions we shall give a complement to the neron theory on curves let c be a curve by which we mean a complete regular curve geometrically irreducible over a field k with an absolute value v let

existence of positive solutions of lidstone boundary value problems abstract we show that there exists at least one positive solution for lidstone boundary value problem 1 n u 2 n t f t u t u t u 2 n 1 t 0 t 1 u 2 i 0 u 2 i 1 0 i 0 1 n 1 under some suitable conditions our proof is based upon global bifurcation techniques

positive solutions of fourth order nonlinear singular sturm liouville eigenvalue problems abstract we consider the existence of positive solutions for the following fourth order singular sturm liouville eigenvalue problems 1 p t p t u t g t f t u u 0 0 t 1 1 u 0 1 u 0 0 1 u 1 1 u 1 0 2 u 0 2 lim t 0 p t u t 0 2 u 1 2 lim t 1 p t u t 0 where 0 g p may be singular at t 0 and or 1 moreover f t x y may also have singularity at x 0 and or y 0 by using fixed point theory in cones an explicit interval for is derived such that for any in this interval the existence of at least one positive solution to the boundary value problem is guaranteed our results extend and improve many known results including singular and nonsingular cases

global structure of nodal solutions for a fourth order two point boundary value problem we consider the fourth order two point boundary value problem x kx lx a t f x 0 t 1 x 0 x 1 x 0 x 1 0 where a c 0 1 0 with a t 0 on any subinterval of 0 1 f c 1 r 0 r c r r satisfies f x x 0 for all x 0 we give conditions on the constants k l and the function f x that guarantee the existence and multiplicity results of nodal solutions the proofs of our main results are based upon disconjugate operator theory and the global bifurcation techniques

determinantal identities over commutative semirings abstract we present a development of determinantal identities over commutative semirings this includes a generalization of the cauchy binet and laplace theorems as well as results on compound matrices and adjoints it is further shown that laplace s theorem is a special case of the grade s adjoint identity

multiply sign changing solutions for fourth order nonlinear boundary value problems abstract this paper deals with the existence of sign changing solutions for the fourth order nonlinear boundary value problem u 4 f u u 0 t 1 u 0 u 1 u 0 u 1 0 where f r 2 r is a continuously odd function some conditions on f guaranteeing the existence and multiplicity of sign changing solutions are presented the existence and multiplicity conditions concern the eigenline of the associated two parameter linear eigenvalue problem the discussion is based on the fixed point index theory in cones and an anti symmetrical extension method of solution

multiple positive solutions for boundary value problems of second order delay differential equations with one dimensional p laplacian abstract we consider the boundary value problems p x t q t f t x t x t 1 x t 0 p s s p 2 s p 1 t 0 1 subject to some boundary conditions by using a generalization of the leggett williams fixed point theorem due to avery and peterson we provide sufficient conditions for the existence of at least three positive solutions to the above problems

on nilpotent matrices over distributive lattices abstract in this paper nilpotent matrices over distributive lattices are considered some properties and characterizations for nilpotent matrices are established and in particular a necessary and sufficient condition for an n n nilpotent matrix to have the nilpotent index n is given also some properties of the reduction of a nilpotent matrix are obtained partial results obtained in this paper are results on nilpotent fuzzy matrices by h hashimoto

constructive proof of existence for a class of fourth order nonlinear bvps a new existence proof of solutions for a class of fourth order nonlinear boundary value problems is proposed the proof of the main results is based on the reproducing kernel theorem it is worthwhile to point out that the method presented in this paper can be applied for the existence proof of diverse kinds of boundary conditions

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

on the powers of matrices over a distributive lattice abstract in this paper the index and the period for a lattice matrix are estimated some necessary and sufficent conditions for the convergence of the powers of a lattice matrix are obtained also some equivalent conditions for a nilpotent lattice matrix are discussed

on positive solutions of some nonlinear fourth order beam equations abstract the existence uniqueness and multiplicity of positive solutions of the following boundary value problem is considered u 4 t f t u t 0 for 0 where 0 is a constant f 0 1 0 0 is continuous

positive solutions of fourth order nonlinear singular boundary value problems abstract we consider the existence of positive solutions for the following fourth order singular sturm liouville boundary value problem 1 p t p t u t g t f t u 0 0 t 1 1 u 0 1 u 0 0 1 u 1 1 u 1 0 2 u 0 2 lim t 0 p t u t 0 2 u 1 2 lim t 1 p t u t 0 where g p may be singular at t 0 and or 1 moreover f t x may also have singularity at x 0 the existence and multiplicity theorems of positive solutions for the fourth order singular sturm liouville boundary value problem are obtained by using the first eigenvalue of the corresponding linear problems our results significantly extend and improve many known results including singular and nonsingular cases

multiple solutions of some fourth order boundary value problems abstract for some fourth order boundary value problems several new existence theorems on multiple positive negative and sign changing solutions are obtained the critical point theory and the supersolution and subsolution method are employed to discuss this problem

positive solutions of fourth order two point boundary value problems in this paper by using the krasnoselskii fixed point theorem we study the existence of one or multiple positive solution of the fourth order two point boundary value problem y 4 t f t y t y t y 0 y 1 y 0 y 1 0 we also give some examples to illustrate our results

triple positive solutions of three point boundary value problems for fourth order differential equations using the avery and peterson fixed point theorem we investigate the existence of three positive solutions for the fourth order three point boundary value problem u 4 t h t f t u t u t 00 0 x 1 and h t is sign changing on 0 1 the interesting point is that the nonlinear term is allowed to depend on u

on nilpotent matrices over antirings in this paper the nilpotent matrices over commutative antirings are characterized in terms of principal permanental minors main diagonals and permanental adjoint matrices and a necessary and sufficient condition for a nilpotent matrix over a commutative antiring which has a given nilpotent index is obtained also a method for calculating the nilpotent index of any nilpotent matrix over a commutative entire antiring is given