singular kahler einstein metrics we study degenerate complex monge ampere equations of the form omega dd c f n e t f mu where omega is a big semi kahler form on a compact kahler manifold x of dimension n t in r and mu f omega n is a positive measure with density f in l p x omega n p 1 we prove the existence and unicity of continuous o plurisubharmonic solutions in case x is projective and omega psi omega where psi x to v is a proper birational morphism to a normal projective variety omega in ns r v is an ample class and mu has only algebraic singularities we prove that the solution is smooth in the regular locus of the equation we use these results to construct singular kahler einstein metrics of non positive curvature on projective klt pairs in particular on canonical models of algebraic varieties of general type

comparison of two capacities in ℂ n in the past few years several capacitary set functions have been introduced in connection with the study of analytic and plurisubharmonic functions of several variables 1 1 5 9 13 there are many different such functions and the relationship between them is not at all clear we shall consider here the relative capacity of bedford and taylor and the capacity defined in terms of certain tchebycheff constants as studied by zaharjata 13 and alexander 1 see sect 2 for the definitions we show that these capacities are essentially the same our main result theorem 2 1 gives the quantitative relationship between them it happens that the relationship between the two capacities is closely connected with a result of josefson 8 on the equivalence of locally and globally pluripolar sets the quantitative estimates of theorem 2 1 allow us to give in sect 4 a new proof of josefson s lemma about normalized polynomials which are very small on the sets where a given plurisubharmonic function is nearly o e josefson s own proof is a direct construction our original motivation was to give a proof based on capacity notions it turns out that the tchebycheff polynomials themselves already do the job e1 mir 7 has obtained an extension of josefson s theorem given a function v psh on an open set he obtains a global psh function u of small growth at infinity such that u is dominated by a function h v of v his choice of h is essentially h x l o g xl thus he gets u l o g iv and so the value of the global function u is o e whenever v oe our methods give a short proof of this in fact they apply to functions h satisfying

on stability and continuity of bounded solutions of degenerate complex monge ampere equations over compact kahler manifolds abstract we obtain a stability estimate for the degenerate complex monge ampere operator which generalizes a result of kolodziej 2003 12 in particular we obtain the optimal stability exponent and also treat the case when the right hand side is a general borel measure satisfying certain regularity conditions moreover our result holds for functions plurisubharmonic with respect to a big form thus generalizing the kahler form setting in kolodziej 2003 12 independently we also provide more detail for the proof in zhang 2006 18 on continuity of the solution with respect to a special big form when the right hand side is l p measure with p 1

power law periodicity in the tangent bifurcations of the logistic map abstract numerical studies were carried out for the average of the logistic map on the tangent bifurcations from chaos into periodic windows a critical exponent of 1 2 is found on the average amplitude as one approaches the transition additionally the averages oscillate with a period that decreases with the same exponent this power law periodicity is related to the reinjection mechanism of the map the undulations appear at control parameter values much earlier than the values where the critical exponent of the bifurcation shows significant changes in the average amplitude

reinjection probability density in type iii intermittency a method to obtain the reinjection probability density rpd for systems showing type iii intermittency phenomenon is presented the method emerges as a natural extension of a recent procedure we established to derive the rpd for type ii intermittency a new two parameter class of reinjection probability densities is presented to describe a broad class of type iii intermittency maps our rpd expression also provides information about the lower bound of the reinjection lbr the new characteristic relation e depends on the lbr and it has a critical exponent such that 1 0 5 the corresponding analytical duration probability densities are also derived in agreement with the numerical computation the main classical results can be extracted from our rpd as particular cases

smooth approximations of the conical kahler ricci flows in this note we show that the conical kahler ricci flows introduced in chen and wang bessel functions heat kernel and the conical kahler ricci flow j funct anal 269 2 2013 exist for all time t in 0 infty in the weak sense as in definition 1 2 as a key ingredient of the proof we show that a conical kahler ricci flow is actually the limit of a sequence of smooth kahler ricci flows

a new type of intermittent transition to chaos an intermittent transition to chaos in the presence of symmetry is investigated statistical properties of intermittency are found including the case of external noise

viscosity solutions to degenerate complex monge ampere equations we develop an alternative approach to degenerate complex monge ampere equations on compact kahler manifolds based on the concept of viscosity solutions and compare systematically viscosity concepts with pluripotential theoretic ones we generalize to the kahler case a theorem due to dinew and zhang in the projective case to the effect that their pluripotential solutions constructed previously by the authors are continuous

random walks kleinian groups and bifurcation currents let be a holomorphic family of representations of a finitely generated group g into psl 2 parameterized by a complex manifold we define a notion of bifurcation current in this context that is a positive closed current on describing the bifurcations of this family of representations in a quantitative sense it is the analogue of the bifurcation current introduced by demarco for holomorphic families of rational mappings on 1 our definition relies on the theory of random products of matrices so it depends on the choice of a probability measure on g

on the dirichlet problem for the complex monge ampere operator on considere le probleme de dirichlet pour l equation de monge ampere complexe sur un ouvert borne de c n

experimental evidence of characteristic relations of type i intermittency in an electronic circuit we experimentally observe the characteristic relations of type i intermittency c m kim et al phys rev lett 73 525 1994 in an inductance resistance diode circuit near a bifurcation point the reinjection probability distribution is of the form x ensuremath 1 2 and the characteristic relations are of the form l ensuremath propto ensuremath epsilon ensuremath 1 4 and constant when the lower bounds of the reinjection are at and above the tangent point respectively the results agree well with theoretical predictions

a thermodynamical formalism for monge ampere equations moser trudinger inequalities and kahler einstein metrics we develop a variational calculus for a certain free energy functional on the space of all probability measures on a kahler manifold x this functional can be seen as a generalization of mabuchi s k energy functional and its twisted versions to more singular situations applications to monge ampere equations of mean field type twisted kahler einstein metrics and moser trudinger type inequalities on miller manifolds are given tian s alpha invariant is generalized to singular measures allowing in particular a proof of the existence of kahler einstein metrics with positive ricci curvature that are singular along a given anti canonical divisor which combined with very recent developments concerning miller metrics with conical singularities confirms a recent conjecture of donaldson as another application we show that if the calabi flow in the anti canonical class exists for all times then it converges to a kahler einstein metric when a unique one exists which is in line with a well known conjecture c 2013 elsevier inc all rights reserved

equidistribution of zeros of random holomorphic sections for moderate measures we establish an equidistribution theorem for the zeros of random holomorphic sections of high powers of a positive holomorphic line bundle the equidistribution is associated with a family of singular moderate measures we also give a convergence speed for the equidistribution

noise effect on statistical properties of type i intermittency abstract in this work we analyze the noise effect on type i intermittency for which we develop a methodology based on a recently proposed technique used to model the reinjection probability density rpd this new methodology allows us to study the noise effect on the intermittency statistical properties for relatively large noise strengths in a quadratic map with different reinjection mechanisms we show that this procedure allows to predict the behavior of the noisy and noiseless system using the results of the m x function which is implemented to obtain the rpd function we also derive an analytical approximation for the probability density of the laminar lengths and we obtain results for the average laminar length all analytical approaches show a good agreement with the numerical results even though the statistical properties are calculated using either the noisy or noiseless data however in some cases the description of the noiseless system using the noisy data can be inaccurate in addition we show that occasionally the presence of noise could be not detected due to the results behave as they would be corresponding to a noiseless system this aspect may have important consequences especially when working with experimental data

experimental observation of characteristic relations of type iii intermittency in the presence of noise in a simple electronic circuit we investigate the characteristic relations of type ii and iii intermittencies in the presence of noise the theoretically predicted characteristic relation is that mathcal l ensuremath sim mathrm exp ensuremath epsilon 2 for a negative regime of ensuremath epsilon and mathcal l ensuremath sim ensuremath epsilon ensuremath ensuremath nu for the positive regime of ensuremath epsilon 1 2l ensuremath nu l1 where mathcal l is the average laminar length and 1 ensuremath epsilon is the slope of the local poincar e map around the tangent point we experimentally confirm these relations in a simple electronic circuit

on the moser trudinger inequality in complex space abstract in this paper we prove the pluricomplex counterpart of the moser trudinger and sobolev inequalities in complex space we consider these inequalities for plurisubharmonic functions with finite pluricomplex energy and we estimate the concerned constants

monge ampere measures on subvarieties in this article we address the question whether the complex monge ampere equation is solvable for measures with large singular part we prove that under some conditions there is no solution when th

hypersymmetry a z3 graded generalization of supersymmetry we propose a generalization of non commutative geometry and gauge theories based on ternary z3 graded structures in the new algebraic structures we define all products of two entities are left free the only constraining relations being imposed on ternary products these relations reflect the action of the z3 group which may be either trivial i e abc bca cab generalizing the usual commutativity or non trivial i e abc jbca with j e 2 i 3 the usual z2 graded structures such as grassmann lie and clifford algebras are generalized to the z3 graded case certain suggestions concerning the eventual use of these new structures in physics of elementary particles and fields are exposed

convergence of calabi yau manifolds abstract in this paper we study the convergence of calabi yau manifolds under kahler degeneration to orbifold singularities and complex degeneration to canonical singularities including the conifold singularities and the collapsing of a family of calabi yau manifolds

ricci flow and the metric completion of the space of kahler metrics we consider the space of kahler metrics as a riemannian submanifold of the space of riemannian metrics and study the associated submanifold geometry in particular we show that the intrinsic and extrinsic distance functions are equivalent we also determine the metric completion of the space of kahler metrics making contact with recent generalizations of the calabi yau theorem due to dinew and guedj zeriahi as an application we obtain a new analytic stability criterion for the existence of a kahler einstein metric on a fano manifold in terms of the ricci flow and the distance function we also prove that the kahler ricci flow converges as soon as it converges in the metric sense

averages and critical exponents in type iii intermittent chaos the natural measure in a map with type iii intermittent chaos is used to define critical exponents for the average of a variable from a dynamical system near bifurcation numerical experiments were done with maps and verify the analytical predictions physical experiments to test the usefulness of such exponents to characterize the nonlinearity at bifurcations were done in a driven electronic circuit with diode as a nonlinear element two critical exponents were determined at the same bifurcation one from the fitting of the average voltage across the diode and the other one from the average length of the laminar phase events both values are consistent with the predictions of a type iii intermittency of cubic nonlinearity the averages of variables in intermittent chaotic systems is a technique complementary to the measurements of laminar phase histograms to identify the nonlinear mechanisms the average exponents may have a broad application in ultrafast chaotic phenomena

d3 0 d2 0 differential calculi on certain noncommutative super spaces in this paper we construct a covariant differential calculus on a quantum plane with two parametric quantum group as a symmetry group the two cases d2 0 and d3 0 are completely established we also construct differential calculi n 2 and n 3 nilpotent on super quantum spaces with one and two parametric symmetry quantum supergroup

convergence of kahler ricci flow in this paper we prove a theorem on convergence of kahler ricci flow on a compact kahler manifold which admits a kahler ricci soliton

remarks on gagliardo nirenberg type inequality with critical sobolev space and bmo we consider the generalized gagliardo nirenberg inequality in mathbb r n in the homogeneous sobolev space dot h s r mathbb r n with the critical differential order s n r which describes the embedding such as l p mathbb r n cap dot h n r r mathbb r n subset l q mathbb r n for all q with p q n r which may be regarded as a generalization of the brezis gallouet wainger inequality

characterization of monge ampere measures with holder continuous potentials abstract we show that the complex monge ampere equation on a compact kahler manifold x of dimension n admits a holder continuous psh solution if and only if its right hand side is a positive measure with holder continuous super potential this property is true in particular when the measure has locally holder continuous potentials or when it belongs to the sobolev space w 2 n p 2 p x or to the besov space b 2 x for some 0 and p 1

the complex monge ampere type equation on compact hermitian manifolds and applications abstract we prove the existence and uniqueness of continuous solutions to the complex monge ampere type equation with the right hand side in l p p 1 on compact hermitian manifolds next we generalise results of eyssidieux guedj and zeriahi 17 18 to compact hermitian manifolds which a priori are not in the fujiki class these generalisations lead to a number of applications we obtain partial results on a conjecture of tosatti and weinkove 40 and on a weak form of a conjecture of demailly and paun 11

twisting of quantum differentials and the planck scale hopf algebra we show that the crossed modules and bicovariant differential calculi on two hopf algebras related by a cocycle twist are in 1 1 correspondence in particular for quantum groups which are cocycle deformation quantisations of classical groups the calculi are obtained as deformation quantisations of the classical ones as an application we classify all bicovariant differential calculi on the planck scale hopf algebra this is a quantum group which has an limit as the functions on a classical but non abelian group and a limit as flat space quantum mechanics we further study the noncommutative differential geometry and fourier theory for this hopf algebra as a toy model for planck scale physics the fourier theory implements a t duality like self duality the noncommutative geometry turns out to be singular when and is therefore not visible in flat space quantum mechanics alone

characteristic relations of type i intermittency in the presence of noise near the point of tangent bifurcation the scaling properties of the laminar length of type i intermittency are investigated in the presence of noise based on analytic and numerical studies we show that the scaling relation of the laminar length is dramatically deformed from 1 sqrt ensuremath epsilon for ensuremath epsilon g0 to mathrm exp 1 d ensuremath epsilon 3 2 for ensuremath epsilon l0 as ensuremath epsilon passes the bifurcation point ensuremath epsilon 0 the results explain why two coupled r ossler oscillators exhibit deformation of the scaling relation of the synchronous length in the nearly synchronous regime

holder continuity of solutions to the complex monge ampere equation with the right hand side in l p the case of compact kahler manifolds we prove that on compact kahler manifolds solutions to the complex monge ampere equation with the right hand side in lp p 1 are holder continuous

the kahler ricci flow through singularities we prove the existence and uniqueness of the weak kahler ricci flow on projective varieties with log terminal singularities we also show that the weak kahler ricci flow can be uniquely continued through divisorial contractions and flips if they exist finally we propose an analytic version of the minimal model program with ricci flow

stability and regularity of solutions of the monge ampere equation on hermitian manifolds abstract we prove stability of solutions of the complex monge ampere equation on compact hermitian manifolds when the right hand side varies in a bounded set in l p p 1 and it is bounded away from zero such solutions are shown to be holder continuous as an application we extend a recent result of szekelyhidi and tosatti on kahler einstein equation from kahler to hermitian manifolds

boundary invariants of pseudoconvex domains the purpose of this paper is to describe an invariant ax z which is associated with each point z in the boundary of a smoothly bounded pseudoconvex domain in cn the motivation for defining the invariant ax z referred to as the multitype of z comes from the study of the boundary regularity properties of solutions of the d neumann problem on domains of finite type by a domain of finite type we mean one that satisfies the definition given by d angelo 5 namely a domain such that at any boundary point the maximum order of contact of one dimensional complex analytic varieties with the boundary is bounded in 4 the author showed how one can give a proof of global regularity of the a neumann problem for pseudoconvex domains of finite type if there is an invariant such as the one described in this paper in a forthcoming article this invariant will be used to obtain a proof of subelliptic estimates for the same class of domains before giving the definition of the multitype ax z we must introduce some

complex monge ampere equation for measures supported on real submanifolds let x omega be a compact n dimensional kahler manifold on which the integral of omega n is 1 let k be an immersed real mathcal c 3 submanifold of x such that the tangent space at any point of k is not contained in any complex hyperplane of the real tangent space at that point of x let mu be a probability measure compactly supported on k with l p density for some p 1 we prove that the complex monge ampere equation dd c varphi omega n mu has a holder continuous solution

kahler ricci flow with degenerate initial class in the earlier joint work 3 we introduced the weak kahler ricci flow for various geometric motivations in this current work we take further consideration on setting up the weak flow by allowing the initial class to be not necessary kahler it s shown that the construction is compatible with the earlier construction in kahler case we also discuss the convergence as t 0 which is of great interest in this topic and provide related motivation 1 motivation and set up of weak flow the kahler ricci flow the complex version of ricci flow has been under intensive study over the last twenty some years in 18 and more recently 17 g tian proposed the intriguing program of constructing globally existing weak kahlerricci flow with canonical singular limit at time infinity and applying it to the study of algebraic manifolds and even kahler manifolds in general it can be viewed as the analytic version of the famous minimal model program in algebraic geometry in general people should expect the classic smooth kahler ricci flow to encounter singularity at some finite time which is completely decided by cohomology information according to the optimal existence result in 20 just as what people have been doing and had successes in cases for ricci flow surgeries for the underlying manifold should be expected for the kahler ricci flow we naturally expect the surgery to have flavours from algebraic geometry a simple example would be surfaces of general type one only needs the blowingdown of 1 curves when applying the general construction at the end of 3 to push the flow through the finite time of singularities where the measure restriction is actually not so involved as explained later in example 4 4 the degenerate class at the singularity time would become kahler for the new manifold because those 1 curves causing the cohomology degeneration would eventually been crushed to points things can get significantly more complicated for higher dimension for example a standard procedure called flip is introduced in the algebraic geometry context which is of great importance for the business about the algebraic minimal model program simply speaking one needs to blow up the manifold and then perform a different blowing down process naturally we should expect the transformation of the degenerate class is still not kahler in this note we confirm that this is not a problem under the assumption that formally the kahler ricci flow is instantly date received on january 6 2012 and accepted on august 22 2012 2010 mathematics subject classification primary 53c44 secondary 14e30 58j35

an inequality for mixed monge ampere measures we generalize an inequality for mixed monge ampere measures from kolodziej indiana univ math j 43 1321 1338 1994 we also give an example that shows that our assumptions are sharp the corresponding result in the setting of compact kahler manifold is also discussed

n 3 differential calculus and gauge theory on a reduced quantum plane we discuss the algebra of n n matrices as a reduced quantum plane a n 3 nilpotent deformed differential calculus involving a complex parameter q is constructed the two cases q 3rd and nth root of unity are completely treated as an application we establish a gauge field theory for the particular cases n 2 and n 3

necessary conditions for subellipticity of the overline partial neumann problem let a be a a closed form of type p q with l2 coefficients on a smoothly bounded domain q in c the problem of determining both the existence and regularity properties of the solution u of au a where u is orthogonal to the null space of a on p q l forms is known as the a neumann problem one of the principal methods used in the investigation of this problem is the proof of certain a priori subelliptic estimates let u be a neighborhood of a point z0 in the boundary of q a subelliptic estimate is said to hold in u if the estimate

determinantal point processes and fermions on complex manifolds large deviations and bosonization we study determinantal random point processes on a compact complex manifold x associated to a hermitian metric on a line bundle over x and a probability measure on x physically this setup describes a gas of free fermions on x subject to a u 1 gauge field and when x is the riemann sphere it specializes to various random matrix ensembles our general setup will also include the setting of weighted orthogonal polynomials in mathbb c n as well as in mathbb r n it is shown that in the many particle limit the empirical random measures on x converge exponentially towards the deterministic pluripotential equilibrium measure defined in terms of the monge ampere operator of complex pluripotential theory more precisely a large deviation principle ldp is established with a good rate functional which coincides with the normalized pluricomplex energy of a measure recently introduced in berman et al publ math de l ihes 117 179 245 2013 we also express the ldp in terms of the ray singer analytic torsion this can be seen as an effective bosonization formula generalizing the previously known formula in the riemann surface case to higher dimensions and the paper is concluded with a heuristic quantum field theory interpretation of the resulting effective boson fermion correspondence

real hypersurfaces orders of contact and applications a fundamental problem in the modem theory of several complex variables concerns the boundary behavior of the cauchy riemann equations suppose that d is an open domain in cn and that its boundary m is a smooth real submanifold of cn how does the geometry of m influence the function theory on d one approach to this question is the a neumann problem 6 there one studies the inhomogeneous cauchy riemann equations au a where a is a 0 q form with distribution coefficients on the closed domain d u m satisfying the necessary compatibility condition a a 0 the a neumann problem constructs a particular solution u which is orthogonal to the null space of the complex laplacian kohn 10a has solved this problem and has shown that good local regularity properties for u in terms of a follow from so called subelliptic estimates it is natural to seek geometric conditions on m that are necessary and sufficient for these estimates in this paper we will be concerned with a certain geometric condition on m that we feel is the right one for subellipticity although the motivation for the questions discussed here comes from partial differential equations the techniques come from algebraic geometry to see why we state theorems of kohn greiner and catlin concerning subellipticity kohn 1oa has discovered a sufficient condition for a subelliptic estimate near a fixed point p in m in case d is pseudoconvex and is contained in c2 this condition is equivalent to the finiteness of order of tangency of every complex analytic one dimensional manifold with m at p greiner 13 established the necessity of this condition in

universality results for zeros of random holomorphic sections in this work we prove an universality result regarding the equidistribution of zeros of random holomorphic sections associated to a sequence of singular hermitian holomorphic line bundles on a compact kahler complex space x namely under mild moment assumptions we show that the asymptotic distribution of zeros of random holomorphic sections is independent of the choice of the probability measure on the space of holomorphic sections in the case when x is a compact kahler manifold we also prove an off diagonal exponential decay estimate for the bergman kernels of a sequence of positive line bundles on x

asymptotic number of scattering resonances for generic schrodinger operators let v be the schrodinger operator acting on l 2 mathbb r d mathbb c with d geq 3 odd here v is a bounded real or complex function vanishing outside the closed ball of center 0 and of radius a let n v r denote the number of resonances of v with modulus r we show that if the potential v is generic in a sense of pluripotential theory then n v r c d a dr d o r d 3 over 16 epsilon quad mbox as r to infty for any e 0 where c d is a dimensional constant

exterior differentials of higher order and their covariant generalization we investigate a particular realization of generalized q differential calculus of exterior forms on a smooth manifold based on the assumption that dn 0 while dk 0 for k n it implies the existence of cyclic commutation relations for the differentials of first order and their generalization for the differentials of higher order special attention is paid to the cases n 3 and n 4 a covariant basis of the algebra of such q grade forms is introduced and the analogs of torsion and curvature of higher order are considered we also study a zn graded exterior calculus on a generalized clifford algebra

super potentials of positive closed currents intersection theory and dynamics we introduce a notion of super potential for positive closed currents of bidegree p p on projective spaces this gives a calculus on positive closed currents of arbitrary bidegree we define in particular the intersection of such currents and the pull back operator by meromorphic maps one of the main tools is the introduction of structural discs in the space of positive closed currents which gives a geometry on that space we apply the theory of super potentials to construct green currents for rational maps and to study equidistribution problems for holomorphic endomorphisms and for polynomial automorphisms

boundary regularity of proper holomorphic mappings it s main result is formulated in theorem 2 below although the answer to the conjecture in the general case is still unknown it has been shown to hold in many special cases a fairly detailed history of this subject will be contained in 14 one class of positive answers deals with cases in which one knows that f21 and 2 posses fairly large classes of holomorphic automorphisms that extend holomorphically across the boundaries like for instance reinhardt domains see e g h cartan 8 w kaup 16 and st bell 6 in these situations one even often does not need a smoothness assumption on the boundaries of 1 and 22 on the other hand the first result without a priori knowledge on automorphisms which gave c extendability was proved by ch fefferman in 1974 in 15 and says

type i intermittency in nonstationary systems models and human heart rate variability abstract type i intermittency occurs whenever a discrete dynamical system in a chaotic state is about to undergo a tangential bifurcation then long stretches of almost periodic behavior laminar phases occur these laminar phases are interspersed with chaotic behavior in type i intermittency simple models show that a characteristic u shaped probability distribution is obtained for the laminar phase length we have shown elsewhere that in some cases of pathology the laminar phase length distribution characteristic for type i intermittency may be obtained in human heart rate variability data the heart and its regulatory systems are usually presumed to be noisy although the effect of additive noise on the laminar phase distribution in type i intermittency is well known the effect of multiplicative noise has not been studied in this presentation we discuss the properties of classes of models of type i intermittency in which the control parameter is changed randomly within the specified parameter range we show that the properties of these models are importantly different from those obtained for type i intermittency in the presence of additive noise we also show how the two models help explain some of the features seen in the intermittency in human heart rate variability some technical problems with measuring laminar phase lengths in time series will also be discussed

experimental observation of the characteristic relations of type i intermittency in the presence of noise recently it has been reported that the characteristic relation of type i intermittency in the presence of noise is deformed nontrivially as the channel width epsilon changes from the positive region to the negative in order to verify it experimentally as a real phenomenon we study the characteristic relations both for epsilon 0 in a simple inductor resistor diode circuit that is under noisy circumstances the experimental results agree well with the theoretical expectation that the characteristic relations are proportional to epsilon 1 4 for epsilon 0 and proportional to exp alpha epsilon 3 2 for epsilon 0

the complex monge ampere equation on compact kahler manifolds we consider the complex monge ampere equation on a compact kahler manifold m g when the right hand side f has rather weak regularity in particular we prove that estimate of triangle phi and the gradient estimate hold when f is in w 1 p 0 for any p 0 2n as an application we show that there exists a classical solution in w 3 p 0 for the complex monge ampere equation when f is in w 1 p 0

the dirichlet problem for nonlinear second order elliptic equations ii complex monge ampere and uniformaly elliptic equations on considere le probleme de dirichlet pour des equations elliptiques non lineaires d ordre 2 pour une fonction reelle dans un domaine borne de r n a frontiere lisse

fine structure in the scaling of type i intermittency bifurcations abstract tangent bifurcations in maps with type i intermittency have been studied numerically and oscillations in the statistical properties were found as functions of the control parameter the average length of laminar events the lyapunov exponent and the averages of the dynamical variable were calculated for quadratic and quartic normal form tangencies in addition to the exponents of the monotonic critical behavior the frequency of the oscillations is also simply related to the nonlinear term of the maps