approximation of biholomorphic mappings by automorphisms of cn in this paper we prove several results on approximation of biholomorphic mappings between domains in c n 2 by holomorphic automorphisms of c recall that the group of holomorphic automorphisms of c denoted aut c consists of those holomorphic mappings c c that have a holomorphic inverse 4 1 c c with the topology of uniform convergence on compact sets aut c is a topological group while the group aut c 1 of automorphisms of the complex plane consists only of the linear mappings az b a b e c a 4 0 the group aut c is very large and complicated when n 2 let us choose a linear coordinate system on c and write the coordinates as z z w where z z1 zn i c c n 1 and w z c then autc n contains mappings of the form

positive solutions of sub superlinear sturm liouville problems in this paper we introduce the notion of a strict lower upper solution to nonlinear sturm liouville boundary value problems based on the maximum principles we establish a result of leray schauder degree on the ordered intervals induced by the pairs of strict lower and upper solutions applying the result and the fixed point index theory in cones we obtain the global existence results of positive solutions for sub superlinear sturm liouville problems

eigenvalue problem for p laplacian three point boundary value problems on time scales abstract let t be a time scale such that 0 t t 0 and 0 t we consider the following p laplacian three point boundary problem on time scales p u t h t f u t 0 t 0 t u 0 u 0 u u t 0 where p 1 0 h c ld 0 t 0 and f c 0 0 some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem are established in doing so the usual restriction that f 0 lim u 0 f u p u and f lim u f u p u exist is removed an example is also given to illustrate the main results

nearest neighbor spacing distribution of prime numbers and quantum chaos we give heuristic arguments and computer results to support the hypothesis that after appropriate rescaling the statistics of spacings between adjacent prime numbers follows the poisson distribution the scaling transformation removes the oscillations in the nearest neighbor spacing distribution of primes these oscillations have the very profound period of length six we also calculate the spectral rigidity 3 for prime numbers by two methods after suitable averaging one of these methods gives the poisson dependence 3 l l 15

multiple positive solutions for nonlinear periodic problems abstract we consider a nonlinear periodic equation driven by the scalar p laplacian and with a caratheodory asymptotically p 1 linear nonlinearity using variational methods coupled with suitable truncation techniques we show that the problem has at least two positive solutions for the semilinear case p 2 using morse theory we show that the problem has three distinct positive solutions

a singular boundary value problem for the one dimensionalp laplacian abstract the singular boundary value problem formula is studied in this paper whereg s s p 2s p 1 the singularity may appear atu 0 and att 0 ort 1 and the functionfmay be discontinuous the authors prove that for anyp 1 and for any positive nonincreasing functionfand nonnegative measurable functionkwith some integrability conditions the above mentioned problem has a unique solution also the properties of the solution are discussed in the paper

the fredholm alternative at the first eigenvalue for the one dimensionalp laplacian abstract in this work we study the range of the operatoru u p 2 u 1 u p 2 u u 0 u t 0 p 1 we prove that all functionsh c1 0 t satisfying t0 h t sinp pt t dt 0 lie in the range but that ifp 2 andh 0 the solution set is bounded here sin pt t is a first eigenfunction associated to 1 we also show that in this case the associated energy functionalu 1 p t0 u p 1 p t0 u p t0 huis unbounded from below if 1 2 onw1 p0 0 t 1corresponds precisely to the best constant in thelp poincare inequality moreover we show that unlike the linear casep 2 forp 2 the range contains a nonempty open set inl 0 t

strongly pseudoconvex domains as subvarieties of complex manifolds in this paper we obtain existence and approximation results for closed complex subvarieties that are normalized by strongly pseudoconvex stein domains our sufficient condition for the existence of such subvarieties in a complex manifold x is expressed in terms of the morse indices and the number of positive levi eigenvalues of an exhaustion function on x examples show that our conditions cannot be weakened in general we obtain optimal results for subvarieties of this type in complements of compact complex submanifolds with griffiths positive normal bundle in the projective case these generalize classical theorems of remmert bishop and narasimhan concerning proper holomorphic maps and embeddings to bbb c n bbb p n backslash bbb p n 1

closed holomorphic 1 forms without zeros on stein manifolds we prove that a stein mainfold admits a closed holomorphic 1 form without zeros in every class of the first cohomology group we also prove an approximation result for closed holomorphic 1 forms without zeros defined in a neighborhood of a compact subset

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

the method of lower and upper solutions for fourth order two point boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem u iv f x u x u x u 0 u 1 u 0 u 1 0 where f 0 1 r r r is continuous

existence of positive solutions for singular boundary value problems involving the one dimensional p laplacian this paper studies the existence of positive solutions for singular dirichlet boundary value problems these results are obtained by using the global continuation theorem fixed point index theory and approximate method

existence of the second positive radial solution for a p laplacian problem we investigate the existence nonexistence and multiplicity of positive radial solutions for the p laplacian problem with boundary parameters for proofs we mainly use a combination of a fixed point theorem the method of upper and lower solutions and fixed point index theory in the frame of the ordinary differential equation ode technique

integrable embeddings and foliations a k submanifold l of an open n manifold m is called it weakly integrable wi resp it strongly integrable si if there exists a submersion phi m rightarrow bbb e n k such that l subset phi 1 0 resp l phi 1 0 in this work we study the following problem first stated in a particular case by costa et al it invent math 1988 which submanifolds l of an open manifold m are wi or si for general m we explicitly solve the case k n 1 and provide necessary and sufficient conditions for submanifolds to be wi and si in higher codimension as particular cases we recover the theorem of bouma and hector it indagationes math 1983 asserting that any open orientable surface is si in bbb e 3 and watanabe s and miyoshi s theorems it topology 1993 and 1995 claiming that any link is wi in an open 3 manifold in the case m bbb e n we fully characterize wi and si submanifolds we provide examples of 3 and 7 manifolds which are not wi and we show that a theorem by miyoshi it topology 1995 which states that any link in bbb e 3 is si does not hold in general the right analogue to miyoshi s theorem is also proved implying in particular the surprising result that no knot in bbb e 3 is si our results applied to the theory of foliations of euclidean spaces give rise to some striking corollaries using some topological invariants we classify all the submanifolds of bbb e n which can be realized as proper leaves of foliations we prove that bbb s 3 can be realized as a leaf of a foliation of bbb e n n geq 7 but not in bbb e 5 or bbb e 6 which partially answers a question by vogt it math ann 1993 we construct open 3 manifolds which cannot be leaves of a foliation of any compact 4 manifold but are proper leaves in bbb e 4 the theory of wi and si submanifolds is a framework where many classical tools of differential and algebraic topology play a prominent role phillips gromov h principle hirsh smale theory complete intersections seifert manifolds the theory of immersions and embeddings obstruction theory etc

generalized ambrosetti prodi conditions for nonlinear two point boundary value problems 1 under the assumptions that g is of class c2 strictly convex and such that o lim g u l lim g u l 2 u ro it 0 where 1 a2 are the first two eigenvalues of the dirichlet problem on 52 for a they proved that the holder space c a was split into two open sets 0 and o2 by a c manifold m such that i if k e oo 1 has no solution ii if 12 e m 1 has exactly one solution iii if h e 02 1 has exactly two solutions 422

multiple periodic solutions of a class of p laplacian by using the recent generalization of coincidence degree method the existence of multiple periodic solutions for a class of p laplacian is obtained under the existence of strict upper and lower solutions

lidstone polynomials and boundary value problems abstract we define lidstone polynomials provide their explicit representations give their relations with bernoulli polynomials obtain best possible error estimates in lidstone interpolation and establish several best possible sharp inequalities which compare favourably with the known results next we use these results to study even order differential equations together with lidstone boundary conditions

sharp conditions for the existence of sign changing solutions to equations involving the one dimensional p laplacian abstract we consider the boundary value problem involving the one dimensional p laplacian u p 2 u a x f u 0 0 x 1 u 0 u 1 0 where p 1 we establish sharp conditions for the existence of solutions with prescribed numbers of zeros in terms of the ratio f s s p 1 at infinity and zero our argument is based on the shooting method together with the qualitative theory for half linear differential equations

eigenvalues and the one dimensional p laplacian we consider the boundary value problem p u f t u 0 with p 1 t 0 1 u 0 u 1 0 and with 0 the value of is chosen so that the boundary value problem has a positive solution in addition we derive an explicit interval for such that for any in this interval the existence of a positive solution to the boundary value problem is guaranteed in addition the existence of two positive solutions for in an appropriate interval is also discussed

singular second order boundary value problems many other boundary conditions could be considered however since the analysis is similar we choose to omit the details for example if b denoted y 0 y 1 0 the results of this case follow from the results when b is denoted by i by making a change of variable problems of the form 1 l have been studied by a variety of authors see l 3 5 7 lo 141 for example the most advanced results to date seem to be those in 3 7 and 111 however in these papers f x y is decreasing in y for each x e 0 1 here we will not require this assumption for example we will show that y y o y 0 0 t 1 y 0 y 1 0 with 0 i 2 0 5 y 1 has a solution furthermore in fact in the case where f x y is decreasing in y our results improve those of 3 71 and 111 to show existence of a solution to 1 1 we will use the technique initiated in lo and ll the methods in those papers rely on the topological transversality theorem 8 91 the arzela ascoli theorem and apriori bounds on solutions

multiple nontrivial solutions for nonlinear periodic problems with the p laplacian abstract we consider a nonlinear periodic problem driven by the scalar p laplacian with a nonsmooth potential hemivariational inequality using the degree theory for multivalued perturbations of s operators and the spectrum of a class of weighted eigenvalue problems for the scalar p laplacian we prove the existence of at least three distinct nontrivial solutions two of which have constant sign

periodic solutions for a rayleigh type p laplacian equation with sign variable coefficient of nonlinear term as p laplacian equations have been widely applied in the field of fluid mechanics and nonlinear elastic mechanics it is necessary to investigate the periodic solutions of functional differential equations involving the scalar p laplacian by using lu s continuation theorem which is an extension of manasevich mawhin we study the existence of periodic solutions for a rayleigh type p laplacian equation f p x t f x t g 1 x email protected 1 t x email protected t g 2 x email protected 2 t x e t it is significant that the growth degree with respect to the variable u in g 1 u is allowed to be greater than p 1 and the coefficient b t of g 2 x email protected 2 t x can change sign in this paper which could be achieved rarely in the previous literature

fixed point theory and nonlinear problems introduction among the most original and far reaching of the contributions made by henri poincare to mathematics was his introduction of the use of topological or qualitative methods in the study of nonlinear problems in analysis his starting point was the study of the differential equations of celestial mechanics and in particular of their periodic solutions his work on this topic began with his thesis in 1879 and was developed in detail in his great three volume work methodes nouvelles de la mechanique celeste which appeared in the early 1890s and summarized his many memoirs of the intervening period it continued until his memoir shortly before his death in 1912 in which he put forward the unproved fixed point result usually referred to as poincare s last geometric theorem the ideas introduced by poincare include the use of fixed point theorems the continuation method and the general concept of global analysis the writer s acquaintance with poincare s influence came through contact with solomon lefschetz and marston morse both of whom were very explicit as to the role of poincare as an initiator in this direction of mathematical development in 1934 in the foreword to his colloquium volume on the calculus of variations in the large morse had put this forward very forcefully in the first paragraph

a foliation of the ball by complete holomorphic discs we show that the open unit ball mathbb b n of mathbb c n n 1 admits a nonsingular holomorphic foliation by complete properly embedded holomorphic discs

a homotopic deformation along p of a leray schauder degree result and existence for u p 2u ƒ t u 0 u 0 u t 0 p 1 soit f 0 1 r r une fonction continue et soit o p r r definie par o p s s p 2 s pour p fixe 1 on considere le probleme de dirichlet suivant o p u f t u 0 u 0 u t 0 ou t est un nombre reel positif

random walk on the prime numbers the one dimensional random walk rw where steps up and down are performed according to the occurrence of special primes is defined some quantities characterizing rw are investigated the mean fluctuation function f l displays perfect power law dependence f l l1 2 indicating that the defined rw is not correlated the number of returns of this special rw to the origin is investigated it turns out that this single very special realization of rw is a typical one in the sense that the usual characteristics used to measure rw take values close to the ones averaged over all random walks this fact suggests that random numbers of good quality could be obtained by means of rw on prime numbers the fractal structure on the subset of primes is also found

extending holomorphic sections from complex subvarieties in this paper we treat the classical problem of extending holomorphic mappings and sections from closed complex subvarieties of a complexmanifold our main results theorems 1 1 and 1 4 extend those of grauert gr2 and cartan car results of this type are commonly referred to as the okagrauert principle on stein manifolds our methods are similar to those developed in the papers of gromov gro henkin and leiterer hl2 and the authors fp1 fp2 the main addition here is the interpolation of a given holomorphic section on a complex subvariety of a stein manifold our globalization scheme follows very closely the one developed in fp2 to state our first result we recall the notion of a dominating spray introduced by m gromov gro sect 0 5 given a holomorphic vector bundle p e y over a complex manifold y we denote by 0y ey p 1 y the zero element in the fiberey and we observe thatey is ac linear subspace of the tangent space t0ye definition 1 a spray on a complex manifold y is a holomorphic vector bundle p e y together with a holomorphic map s e y such that for each y y s 0y y and the derivative ds t0ye tyy maps ey surjectively onto tyy

ramanujan fourier series the wiener khintchine formula and the distribution of prime pairs the wiener khintchine formula plays a central role in statistical mechanics it is shown here that the problem of prime pairs is related to autocorrelation and hence to a wiener khintchine formula experimental evidence is given for this c 1999 elsevier science b v all rights reserved

periodic solutions for second order differential inclusions with the scalar p laplacian we study periodic problems driven by the scalar p laplacian with a multivalued right hand side nonlinearity we prove two existence theorems in the first we assume nonuniform nonresonance conditions between two successive eigenvalues of the negative p laplacian with periodic boundary conditions in the second we employ certain landesman lazer type conditions our approach is based on degree theory

numerical methods for unilateral problems abstract the theory of variational inequalities besides being elegant and synthetic also provides a unified general framework for applying the numerical methods for solving many unrelated free boundary value problems in this paper we describe numerical experience on the use of variational inequalities and pade approximants to obtain approximate solutions to a class of unilateral boundary value problems of elasticity like those describing the equilibrium configuration of an elastic beam stretched over an elastic obstacle in the case of a known obstacle these problems can be alternatively formulated as nonlinear boundary value problems without constraints for which the technique of pade approximants can be successfully employed the variational inequality formulation is used to discuss the problem of uniqueness and existence of the solution

a class of nonlinear sturm liouville problems with infinitely many solutions on etudie l existence des solutions des problemes de sturm liouville non lineaires dont la partie lineaire est a la resonance on montre que de tels problemes peuvent avoir une infinite de solutions si les perturbations non lineaires sont periodiques

stein structures and holomorphic mappings we prove that every continuous map from a stein manifold x to a complex manifold y can be made holomorphic by a homotopic deformation of both the map and the stein structure on x in the absence of topological obstructions the holomorphic map may be chosen to have pointwise maximal rank the analogous result holds for any compact hausdorff family of maps but it fails in general for a noncompact family our main results are actually proved for smooth almost complex source manifolds x j with the correct handlebody structure the paper contains another proof of eliashberg s int j math 1 29 46 1990 homotopy characterization of stein manifolds and a slightly different explanation of the construction of exotic stein surfaces due to gompf ann math 148 2 619 693 1998 j symplectic geom 3 565 587 2005

multiple positive solutions for some p laplacian boundary value problems this paper deals with the existence of multiple positive solutions for the one dimensional p laplacian f p x t q t f t x t x t 0 t 0 1 subject to the following boundary value conditions x 0 i 1n a ix x i x 1 i 1n b ix x i where f p s s p 2 s p 1 by means of a fixed point theorem due to avery and peterson sufficient conditions are obtained that guarantee the existence of at least three positive solutions to the above boundary value problem

largest known twin primes the numbers 697053813 216352 1 are twin primes a search for large prime pairs of the form 3 30h 2 1 was performed at the university of paderborn germany in the framework of a project for parallel computing in computational number theory supported by the heinz nixdorf institute paderborn we used the arithmetic routines developed in this project for fast parallel and sequential computations with very large numbers the search was done until the first large prime pair 697053813 2 1 was found these numbers have 4932 digits starting with 1930880535 to our best knowledge the earlier records were 1692923232 10 1 4030 digit 4655478828 10 1 3439 digit found by dubner 1 and 1706595 2 1 3389 digit found by parady smith zarantonello 3 in 1990 the search consisted of the following steps 1 we started with the 2 nonnegative values of h below 2 the exponent 16352 was fixed during the calculations 2 both the case 1 and the case 1 were sieved with factors from 7 up to 2 after sieving 209571 candidates remained 3 using the probabilistic miller rabin primality test see knuth 2 pp 379 380 the candidates were tested first the 1 case and then the 1 case until a probable twin prime pair was found 4 this probable twin prime pair was tested with exact tests the 1 case using a lucasian test see riesel 5 and the 1 case using the test of brillhart lehmer and selfridge and the test of proth see ribenboim 4 pp 37 40 the number 2 x of twin primes below x is conjectured to be 2 x 2c2x ln x where c2 p 2 1 1 p 1 0 66016 computer experiments support this conjecture see ribenboim 4 p 202 here we test a sequence of numbers n 1 n h0 30h 2 where n is a large but fixed natural number h runs through the positive integers smaller than 2 30 and where h0 3 mod 30 n 0 mod 4 or h0 9 mod 30 n 1 mod 4 or h0 21 mod 30 n 3 mod 4 or h0 27 mod 30 n 2 mod 4 is satisfied because only in these cases is it true that n 1 and n 1 both are not divisible by 2 3 and 5 and a simple exact lucas test works for n 1 our experiments with a simple mapler program show that 11 lnn is a good lower estimate for the probability that n 1 and n 1 both are primes for the range n 2 where n is fixed and h runs up to 2 30 if we assume that this approximation holds up to n 2 we get 1 11 000 000 for received by the editor december 16 1994 1991 mathematics subject classification primary 11 04 secondary 11a41 rmaple is a registered trademark of waterloo maple software c 1996 american mathematical society

proper j holomorphic discs in stein domains of dimension 2 we prove the existence of global bishop discs in a strictly pseudoconvex stein domain in an almost complex manifold of complex dimension 2

the gaps between the gaps some patterns in the prime number sequence abstract it has long been known that the gaps between consecutive prime numbers cluster on multiples of 6 recently it was shown that the frequency of the gaps between the gaps is lower for multiples of 6 than for other values p kumar et al information entropy and correlation in prime numbers arxiv cond mat 0303110 this paper investigates higher moments of the prime number series and shows that they exhibit certain peculiarities in order to remove doubts as to whether these peculiarities are related to the known clustering of the gaps on multiples of 6 the results are compared to a benchmark series of simulated gaps

positive solutions for nonlinear periodic problems with concave terms abstract we consider a nonlinear periodic problem driven by the scalar p laplacian with a parametric concave term and a caratheodory perturbation whose potential primitive exhibits a p superlinear growth near without satisfying the usual in such cases ambrosetti rabinowitz condition using critical point theory and truncation techniques we prove a bifurcation type theorem describing the nonexistence existence and multiplicity of positive solutions as the parameter varies

generalizations of browder s degree theory the starting point of this paper is the recent important work of f e browder who extended degree theory to operators of monotone type the degree function of browder is generalized to maps of the form t f g where t is maximal monotone f is of class s bounded and g is an u s c compact multifunction it is also generalized to maps of the form f ng with f of class s and ng the nemitsky operator of a multifunction g x r satisfying various types of sign conditions some examples are also included to illustrate the abstract results

applications of statistical mechanics in number theory abstract the links between statistical physics and number theory are discussed first the attempts to prove the riemann hypothesis by means of the suitable spin model and the lee yang theorem about zeros of the partition function are shortly reviewed next the analogies between random walks and prime numbers are mentioned in the last section the partition function of the system whose energies are defined by the distances between consecutive primes is calculated the arguments are given that such a prime numbers gas behaves like a set of noninteracting harmonic oscillators

testing for scaling in natural forms and observables the general procedure of calculating fractal dimensions or other exponents is based on estimating some quantity as a function of scale and on assessing whether or not this function is a power law this power law manifests itself in a log quantity versus log scale plot as a linear region scaling it has thus become the practice to estimate dimensions by the slope of some linear region in those log log plots when we are dealing with exact fractals the koch curve for example there are no problems when however we are working with natural forms or observables problems begin to emerge in such cases the scaling region is subjectively estimated and often is only the result of the generic property of the quantity to increase monotonically or decrease monotonically as the scale goes to zero irrespective of the geometry of the object here we discuss these issues and suggest a procedure to deal with them

positive solutions and eigenvalue regions of two delay singular systems with a twin parameter abstract the paper is concerned with a two delay singular differential system with a twin parameter applying fixed point index theory we show the relationship between the asymptotic behaviors of nonlinearities at zero and infinity and the open regions eigenvalue regions of parameters which are correlated with delays such that the system has zero one and two positive solution s

continuity in weak topology and extremal problems of eigenvalues of the p laplacian we will study the dependence of eigenvalues of the one dimensional p laplacian on potentials or weights two results are obtained one is the continuity of eigenvalues in potentials with respect to the weak topologies of l spaces 1 and the other is the continuous differentiability of eigenvalues in potentials with respect to l norms as applications we will study some extremal problems of eigenvalues by developing some analytical methods

physical tests for random numbers in simulations we propose three physical tests to measure correlations in random numbers used in monte carlo simulations the first test uses autocorrelation times of certain physical quantities when the ising model is simulated with the wolff algorithm the second test is based on random walks and the third on blocks of ital n successive numbers we apply the tests to show that recent errors in high precision ising simulations using generalized feedback shift register algorithms are due to short range correlations in random number sequences

oka s principle for holomorphic sections of elliptic bundles 2 the h principle for locally trivial fibrations 2 1 the h principle over small subsets 2 2 the h principle over totally real submanifolds 2 3 local extension of the h principle 2 4 localizable extensions 2 5 the h principle for cm bundles 2 6 manifolds with totally real souls 2 7 totally real extensions 2 8 nicely localizable extensions 2 9 the basic h principle for z x 2 10 homomorphisms and holomorphic maps of rank r

existence and uniqueness results for some nonlinear boundary value problems u a s u b s0 dirichlet 2 u9 a s u9 b s 0 neumann 3 u a s u b and u9 a s u9 b periodic 4 w x here f r a r is an odd increasing homeomorphism k a b a r and w x 2 py2 f a b r a r equation 1 with f s s s s where p 1 and k t s n y 1 rt arises in the study of radial solutions for the p laplacian equation on an annular domain in n dimensions py2 n div u u q f x u u s 0 a x b x g r 5 w x and has been studied extensively see for instance 1 4 6 in this paper we shall establish existence results for 1 with each of the above boundary conditions 2 3 and 4 under various growth conditions on f in particular our results give the existence of radial solutions

multiple positive solutions for a superlinear problem a topological approach we study the multiplicity of positive solutions for a two point boundary value problem associated to the nonlinear second order equation u f x u 0 we allow x f x s to change its sign in order to cover the case of scalar equations with indefinite weight roughly speaking our main assumptions require that f x s s is below 1 as s 0 and above 1 as s in particular we can deal with the situation in which f x s has a superlinear growth at zero and at infinity we propose a new approach based on topological degree which provides the multiplicity of solutions applications are given for u a x g u 0 where we prove the existence of 2n 1 positive solutions when a x has n positive humps and a x is sufficiently large

boundary value problems occurring in plate deflection theory abstract the paper deals with several iterative methods for solving some boundary value problems occurring in plate deflection theory making it possible to avoid the lengthy problem preparation which is required to solve these problems by finite difference methods

holomorphic flexibility properties of complex manifolds we obtain results on approximation of holomorphic maps by algebraic maps the jet transversality theorem for holomorphic and algebraic maps between certain classes of manifolds and the homotopy principle for holomorphic submersions of stein manifolds to certain algebraic manifolds

asymptotics of invariant metrics in the normal direction and a new characterisation of the unit disk we give improvements of estimates of invariant metrics in the normal direction on strictly pseudoconvex domains specifically we will give the second term in the expansion of the metrics this depends on an improved localisation result and estimates in the one variable case finally we will give a new characterisation of the unit disk in mathbb c in terms of the asymptotic behaviour of quotients of invariant metrics

nonlinear resonant periodic problems with concave terms abstract we consider a nonlinear periodic problem driven by the scalar p laplacian with a concave term and a caratheodory perturbation we assume that this perturbation f t x is p 1 linear at and resonance can occur with respect to an eigenvalue m 1 m 2 of the negative periodic scalar p laplacian using a combination of variational techniques based on the critical point theory with morse theory we establish the existence of at least three nontrivial solutions useful in our considerations is an alternative minimax characterization of 1 0 the first nonzero eigenvalue that we prove in this work

oka s principle for holomorphic submersions with sprays the oka principle is a fundamental principle of complex analysis which says that on stein manifolds closed complex submanifolds of affine spaces analytic problems of cohomological or even homotopical nature have only topological obstructions the story began in 1939 when k oka proved that on a domain of holomorphy a secondcousin problem is solvable by holomorphic functions if it is solvable by continuous functions oka oka s result has the following equivalent formulation ife x is a holomorphicc bundle over adomainof holomorphy then every continuous section of e is homotopic to a holomorphic section in 1957 h grauert gra extended oka s result to sections of holomorphic fiber bundles with complex homogeneous fibers over stein manifolds see also the papers car fr hl in 1989 m gromov gr1 announced a major generalization of grauert s theorem replacing complex lie groups and homogeneous spaces by a larger class of complex manifolds with a spray a spray on a complex manifold y is a holomorphic vector bundle p e y with a holomorphic map s e y the spray map which is the identity on the zero section and whose restriction to each fibre of e is a submersion at zero such sprays exist on complex homogeneous spaces on complements of affine algebraic subvarieties of codimension at least two and more generally on manifolds whose holomorphic tangent bundle is spanned by finitely many c complete holomorphic vector fields the notion of spray extends in a natural way to submersions the following is theorem 4 5 in gr1 if h z x is a holomorphic submersion onto a stein manifold x such that every point x x has an open neighborhood u x such that z u h 1 u u admits a fiber dominating spray then the inclusion of the

existence and multiplicity results for nonlinear boundary value problems this paper studies the existence of positive solutions for a class of boundary value problems of elliptic degenerate equations by using bifurcation and fixed point index theories in the frame of approximation arguments the criteria of the existence multiplicity and nonexistence of positive solutions are established

the oka grauert principle without induction over the base dimension we give a new proof of grauert s theorem on oka s principle gra1 gra2 gra3 see also c in the case of smooth stein manifolds which does not use induction over the base dimension instead we use induction over the levels of a strictly plurisubharmonic exhausting function grauert s bump method the present paper is an edited version of our preprint h l2 from 1986 which is difficult to find and of bad printing quality we did not publish the paper in a journal at that time because we planned to write a book containing it but the book has not been written until now on the other hand in the meantime some interest to this proof appeared see e g the work of gromov gro therefore we think a publication of our proof could be useful even with a delay of 10 years this proof gives also some oka principle for arbitrary pseudoconvex manifolds see theorem 1 3 below for vector bundles vector bundles without further mention will refer to topological vector bundles with fibre cr this oka principle contains the following

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