morse inequalities for fourier components of kohn rossi cohomology of cr manifolds with s 1 action let x be a compact connected cr manifold of dimension 2n 1 n ge 2 with a transversal cr s 1 action on x we study the fourier components of the kohn rossi cohomology with respect to the s 1 action by studying the szego kernel of the fourier components we establish the morse inequalities on x using the morse inequalities we have established on x we prove that there are abundant cr functions on x when x is weakly pseudoconvex and strongly pseudoconvex at a point

steffensen type inequalities for fuzzy integrals we provide several steffensen type inequalities for the sugeno integral the inequalities are of the form a b f g d a a a f d b or a b f g d a b b f d b where a b are constants is a fuzzy measure on r g a b 0 1 f a b r is nonincreasing or nondecreasing and b a a b g d we show that some of our sufficient conditions are also necessary

on steffensen pairs in the article we establish some general steffensen pairs by the use of mathematical induction and properties of the generalized weighted mean values

non realizable cr structures the examples given by nirenberg in 5 and 6 show that not all three dimensional cr manifolds may be realized as real hypersurfaces in 1172 these examples are perturbations of the cr structure on the sphere in this paper we extend his construction and show that any strictly pseudo convex realizable cr structure of dimension 3 may be perturbed so as to obtain a non realizable one this is done in sect 1 in sect 2 we show non realizability for higher dimensional structures provided the levi form is non degenerate and has only one positive eigenvalue in our last section we summarize cartan s approach to cr geometry from this viewpoint it was important to decide if nirenberg s example was really limited to perturbations of the sphere we have made this paper self contained and have provided proofs of several known results for instance lemma 1 l and the first part of lemma 1 5 were proved by lewy 2 and lemma 1 2 is the well known observation that a strictly pseudo convex hypersurface is convex in some coordinate system

on the stability of equivariant embedding of compact cr manifolds with circle action we prove the stability of the equivariant embedding of compact strictly pseudoconvex cr manifolds with transversal cr circle action under circle invariant deformations of the cr structures

testing the order of discrete markov chains using surrogate data tests of the null hypothesis that a symbolic sequence is of nth markov order are described these tests are based on resampling methods similar to the method of surrogate data used to test continuous time series for nonlinear structure the size and power of these symbolic surrogate tests are estimated using binary and heptary test sequences of known order conditioning on a sufficient statistic under the null hypothesis yields an estimated true size which closely approximates the nominal size and increases the power of the test with respect to the unconditional test we apply one method to a symbolic sequence derived from the rossler system and to a binary and heptary representation of an interspike interval sequence of an electroreceptor of the weakly electric eel apteronotus leptorhynchus

every real ellipsoid in mathbb c 2 admits cr umbilical points we prove that every real ellipsoid m c c 2 admits at least four umbilical points which can be compared to the result of webster that a generic real ellipsoid in c with n 3 does not admit any umbilical point

cr circle bundles and mizohata structures in this paper we study the relationship between cr structures on 3 dimensional manifolds and mizohata structures on 2 dimensional manifolds let x h 0 1 be a 3 dimensional strictly pseudoconvex cr manifold assume there is a free smooth s 1 action on x then x can be regarded as a principal s 1 bundle x m over a smooth 2 dimensional manifold m and assume the cr structure h 0 1 is invariant under the circle action first we show that the projection h 0 1 of h 0 1 into c tm induces a mizohata structure on m if x h 0 1 can be embedded into some c n by cr functions then the induced mizohata structure on m is locally integrable moreover we consider when a neighborhood of 1 p can be embedded into c 2 by cr functions second for every mizohata structure v on m we can construct a cr circle bundle x h 0 1 via a singular curvature form by using the theory of singular forms and singular connections over circle bundles described here this construction can be viewed as a generalization of geometric prequantization to a degenerate situation

cr structures with group action and extendability of cr functions part i presents results on local embedding of cr structures we consider an abstract cr manifold whose structure is invariant under a transversal lie group action we show that such a manifold can always be locally embedded in complex space as a generic submanifold the proof is based on selection of canonical coordinates and repeated use of the newlander nirenberg theorem 13 when the lie group is abelian the embedding can be given a particularly simple form let l 1 be the codimension of our submanifold called m throughout the paper it is then convenient to denote by n i the dimension of the ambient complex space and by zl z w 1 w t the complex coordinates we shall systematically write

umbilical points on three dimensional strictly pseudoconvex cr manifolds i manifolds with u 1 action the question of existence of umbilical points in the cr sense on compact three dimensional strictly pseudoconvex cr manifolds was raised in the seminal paper by chern and moser acta math 133 219 271 1974 in the present paper we consider compact three dimensional strictly pseudoconvex cr manifolds that possess a free transverse action by the circle group u 1 we show that every such cr manifold m has at least one orbit of umbilical points provided that the riemann surface x m u 1 is not a torus in particular every compact circular and strictly pseudoconvex hypersurface in mathbb c 2 has at least one circle of umbilical points the existence of umbilical points in the case where x is a torus is left open in general but it is shown that if such an m has additional symmetries in a certain sense then it must possess umbilical points as well

addendum to subelliptic spin c dirac operators iii we prove the relative index conjecture which in turn implies that the set of embeddable deformations of a strictly pseudoconvex cr structure on a compact 3 manifold is closed in the c 1 topology

entropy and local uncertainty of data from sensory neurons we present an empirical comparison between neural interspike interval sequences obtained from two different kinds of sensory receptors both differ in their internal structure as well as in the strength of correlations and the degree of predictability found in the respective spike trains as a further tool in this context we suggest the local uncertainty assigning a well defined predictability to individual spikes the local uncertainty is demonstrated to reveal significant patterns within the interspike interval sequences even when its overall structure is almost random our approach is based on the concept of symbolic dynamics and information theory

guessing probability distributions from small samples we propose a new method for the calculation of the statistical properties e g the entropy of unknown generators of symbolic sequences the probability distributionp k of the elementsk of a population can be approximated by the frequenciesf k of a sample provided the sample is long enough so that each elementk occurs many times our method yields an approximation if this precondition does not hold for a givenf k we recalculate the zipf ordered probability distribution by optimization of the parameters of a guessed distribution we demonstrate that our method yields reliable results

prediction and entropy of nonlinear dynamical systems and symbolic sequences with lro abstract following shannon we introduce higher order entropies and derive dynamic entropies the nth order dynamic entropy conditional entropy is a measure of the uncertainty of the next state which follows after the observation of n foregoing states the asymptotic behaviour of the dynamic entropies at large n is studied for several nonlinear model systems and for symbolic sequences with long range order lro for example we investigate 1d maps texts dna strings and time series it is shown that the existence of long correlations improves the possibility of predictions characteristic scaling laws for the higher order shannon entropies and the conditional entropies are derived and a new interpolation formula is tested finally instead of the dynamic entropies which yield mean values of the uncertainty predictability we investigate the local values of the uncertainty predictability

sur des varietes de cauchy riemann dont la forme de levi a une valeur propre positive for certain real hypersurfaces in the projective space of signature 1 n we study the filling problem for small deformations of the cr structure the other signatures being well understood we characterize the deformations which are fillable and prove that they have infinite codimension in the set of all cr structures this result generalizes the cases of the 3 sphere and of signature 1 1 to higher dimension

real ellipsoids and double valued reflection in complex space we develop the theory of double valued reflection for a real ellipsoidal hypersurface m in complex euclidean n space this leads to a pair of meromorphic involutions on the complexification of m the main problem is to understand the dynamics of their composition which is a reversible contact trnasformation by means of the segre polar correspondence this map is transformed into a kind of generalized complex null cone billiard map relative to two nonsingular n dimensional complex quadrics in projective n 1 space next we develop a generalization of classical confocal ellipsoidal coordinates which we use together with the maurer cartan equations of the orthogonal group to demonstrate the integrable nature of the dynamics of the reversible map we show that certain complex integral curves of the characteristic systems of suitably invariant contact forms of the structure are invariant by the map the integration of these systems is in turn reduced to solving systems of abelian differential equations by means of generalized jacobi inversion finally we indicate how to choose the contact form so that the real loci of these characteristic curves will be closed curves the complex curves will then yield generalized stationary curves in the sense of lempert for the ellipsoidal domain

extremal discs and the regularity of cr mappings in higher codimension we develop a local theory of extremal analytic discs attached to a strictly pseudoconvex manifold of higher codimension in complex space we then apply the results to prove the regularity of cr homeomorphisms of such manifolds introduction in his celebrated paper lempert li introduced extremal analytic discs as geodesies of the kobayashi metric on a convex domain d c cn let a be the unit disc in the complex plane an analytic disc a d is called extremal if 0 attains its maximum among all analytic discs in d with the same values and directions at 0 g a extremal discs are important holomorphic invariants of d their boundaries lie in the boundary bd of the domain and form an invariant family of curves for the cr structure of bd lempert li l2 applies the extremal discs to the regularity of biholomorphic mappings monge amp re equations and normal forms of convex domains other applications include deformations and embeddibility of cr structures bl bd and polynomial hulls s

mercer and wu srivastava generalisations of steffensen s inequality in this paper generalisations of steffensen s inequality obtained by pecaric mercer and wu srivastava are related some of this generalisations are showed to be equivalent and using wu srivastava refinement of steffensen s inequality refined versions of pecaric and mercer s results are obtained using wu srivastava sharpened and generalized version of mercer s result sharpened version of generalization given by pecaric is obtained furthermore weaker conditions for some generalisations and refinements are obtained moreover functionals defined as the difference between the left hand and the right hand side of generalizations and refinements of steffensen s inequality are studied new stolarsky type means using several families of exponentially convex functions related to defined functionals are generated

a relative index on the space of embeddable cr structures i we study the problem of embeddability for three dimensional cr manifolds let m m0b denote a compact embeddable strictly pseudoconvex cr manifold and os the orthogonal projection on ker 0 b if 16b denotes a deformation of this cr structure then 1 b is embeddable if and only if

a variant of jensen steffensen s inequality and quasi arithmetic means a variant of jensen steffensen s inequality is proved necessary and sufficient conditions for the equality in jensen steffensen s inequality are established several inequalities involving more than two monotonic functions and generalized quasi arithmetic means with not only positive weights are proved it is shown that such generalized quasi arithmetic means have the same comparability properties as those with positive weights

on the riemann mapping theorem we prove a generalization of the riemann mapping theorem if a bounded simply connected domain q with connected smooth boundary has the spherical boundary then it is biholomorphic to the unit ball

quasiconformal mappings on the heisenberg group on etudie les applications conformes et quasi conformes sur le groupe de heisenberg la metrique invariante a gauche naturelle sur cet espace est non isotrope sous des dilatations et ne peut se decrire par une metrique de riemann ceci engendre de nouveaux phenomenes

monge ampere equations and moduli spaces of manifolds of circular type abstract a bounded manifold of circular type is a complex manifold m of dimension n admitting a bounded exhaustive real function u defined on m minus a point x o so that a it is a smooth solution on m x o to the monge ampere equation d d c u n 0 b x o is a singular point for u of logarithmic type and e u extends smoothly on the blow up of m at x o c d d c e u 0 at any point of m x o this class of manifolds naturally includes all smoothly bounded strictly linearly convex domains and all smoothly bounded strongly pseudoconvex circular domains of c n a set of modular parameters for bounded manifolds of circular type is considered in particular for each biholomorphic equivalence class of them it is proved the existence of an essentially unique manifold in normal form it is also shown that the class of normalizing maps for an n dimensional manifold m is a new holomorphic invariant with the following property it is parameterized by the points of a finite dimensional real manifold of dimension n 2 when m is a non convex circular domain while it is of dimension n 2 2 n when m is a strictly linearly convex domain new characterizations of the circular domains and of the unit ball are also obtained

context tree modeling of observed symbolic dynamics modern techniques invented for data compression provide efficient automated algorithms for the modeling of the observed symbolic dynamics we demonstrate the relationship between coding and modeling motivating the well known minimum description length mdl principle and give concrete demonstrations of the context tree weighting and context tree maximizing algorithms the predictive modeling technique obviates many of the technical difficulties traditionally associated with the correct mdl analyses these symbolic models representing the symbol generating process as a finite state automaton with probabilistic emission probabilities provide excellent and reliable entropy estimations the resimulations of estimated tree models satisfying the mdl model selection criterion are faithful to the original in a number of measures the modeling suggests that the automated context tree model construction could replace fixed order word lengths in many traditional forms of empirical symbolic analysis of the data we provide an explicit pseudocode for implementation of the context tree weighting and maximizing algorithms as well as for the conversion to an equivalent markov chain

metriques autoduales sur la boule a conformal metric on a 4 ball induces on the boundary 3 sphere a conformal metric and a trace free second fundamental form conversely such a data on the 3 sphere is the boundary of a unique selfdual conformal metric defined in a neighborhood of the sphere in this paper we characterize the conformal metrics and trace free second fundamental forms on the 3 sphere close to the standard round metric which are boundaries of selfdual conformal metrics on the whole 4 ball when the data on the boundary is reduced to a conformal metric the trace free part of the second fundamental form vanishes one may hope to find in the conformal class of the filling metric an einstein metric with a pole of order 2 on the boundary we determine which conformal metrics on the 3 sphere are boundaries of such selfdual einstein metrics on the 4 ball in particular this implies the positive frequency conjecture of lebrun the proof uses twistor theory which enables to translate the problem in terms of complex analysis this leads us to prove a criterion for certain integrable cr structures of signature 1 1 to be fillable by a complex domain finally we solve an analogous higher dimensional problem selfdual einstein metrics are replaced by quaternionic kahler metrics and conformal structures on the boundary by quaternionic contact structures previously introduced by the author in contrast with the 4 dimensional case we prove that any small deformation of the standard quaternionic contact structure on the 4m 1 sphere is the boundary of a quaternionic kahler metric on the 4m ball

symmetries and other transformations of the complex monge ampere equation on definit des transformations tangentes en utilisant la structure symplectique holomorphe du fibre cotangent holomorphe

modular data and regularity of monge ampere exhaustions and of kobayashi distance regularity properties of intrinsic objects for a large class of stein manifolds namely of monge ampere exhaustions and kobayashi distance is interpreted in terms of modular data the results lead to a construction of an infinite dimensional family of convex domains with squared kobayashi distance of prescribed regularity properties a new sharp refinement of stoll s characterization of mathbb c n is also given

lie group structures on automorphism groups of real analytic cr manifolds given any real analytic cr manifold m we provide general conditions on m guaranteeing that the group of all its global real analytic cr automorphisms aut cr m is a lie group in an appropriate topology in particular we obtain a lie group structure for aut cr m when m is an arbitrary compact real analytic hypersurface embedded in some stein manifold

on the bellman generalization of steffensen s inequality iii soit f x une fonction non negative non croissante sur a b et soit u une fonction convexe croissante pour u 0 avec 0 0 si g x est une fonction non decroissante non negative sur a b telle qu il existe la fonction non negative g 1 x definie par l equation g 1 x g x g 1 x 1 et que a b g 1 t dt 1 alors l inegalite suivante est valable a b f t g t dt a b g t dt 1 a a f t dt ou a b g t d

parametrization of local cr automorphisms by finite jets and applications the goal of this work is to study the structure of the group of all local realanalytic cr automorphisms of a germ of a real analytic hypersurface m p in c or equivalently of all local biholomorphisms h c p c p mapping the germ m p into itself we also deal with the more general situation of a realanalytic cr manifold of arbitrary codimension see 2 for every fixed point p m we refer to this group of local biholomorphisms as the stability group of m p and denote it by aut m p endowed with the topology of uniform convergence on compact neighbourhoods of p it becomes a topological group this natural inductive limit topology can be described in the following way a sequence of germs of biholomorphisms hj aut m p converges to another such germ h aut m p if there exists a neighbourhood u of p in c to which all mappings hj extend holomorphically and such that hj h uniformly on u this group arises in a variety of different circumstances one of the most important ones where it naturally appears is the biholomorphic equivalence problem which consists in deciding whether two given germs of real analytic hypersurfaces are biholomorphically equivalent the biholomorphic equivalence problem has been studied since the beginning of the last century when poincare 37 observed probably for the first time that two real hypersurfaces in c for n 2 are not necessarily locally equivalent via a holomorphic transformation though they are via a c transformation today the biholomorphic equivalence problem for strongly pseudoconvex or levi nondegenerate hypersurfaces is pretty well understood based on the works of e cartan 14 15 in dimension two and the results of tanaka 42 and chern moser 17 in arbitrary dimension thanks to the works initiated in the 1970s a number of remarkable properties of the stability group of strongly pseudoconvex real analytic hypersurfaces have been discovered see e g the surveys by vitushkin 45 huang 30 and baouendi ebenfelt and rothschild 5 for complete discussions on this matter these properties depend on whether the germ m p can be biholomorphically mapped onto a piece of the unit sphere if there exists such a biholomorphism one says that m p is spherical for instance if m p is not spherical the group aut m p is

strictly pseudoconvex domains in c n the theory which we have developed so far will be expanded in two ways 1 in this chapter we shall use the cauchy fantappie calculus to transform the bochner martinelli kernel into a kernel adapted to strictly pseudoconvex domains in n and prove a number of finiteness and vanishing theorems for these domains 2 in the next chapter we shall pass from n to arbitrary hermitian manifolds and establish the analogue of the bmk formula in that context obtaining along the way the hodge decomposition theorem on compact manifolds both lines of thought will merge in the subsequent chapters and lead to a solution of the levi problem and finiteness theorems for strictly pseudoconvex manifolds as well as to a solution of the overline partial neumann problem to be formulated later

on the singularities of the szegő projections on lower energy forms let x be an abstract not necessarily compact orientable cr manifold of dimension 2n 1 n geqslant 2 let box q b be the gaffney extension of kohn laplacian on 0 q forms we show that the spectral function of box q b admits a full asymptotic expansion on the non degenerate part of the levi form as a corollary we deduce that if x is compact and the levi form is non degenerate of constant signature on x then the spectrum of box q b in 0 infty consists of point eigenvalues of finite multiplicity moreover we show that a certain microlocal conjugation of the associated szeg kernel admits an asymptotic expansion under a local closed range condition as applications we establish the szeg kernel asymptotic expansions on some weakly pseudoconvex cr manifolds and on cr manifolds with transversal cr s 1 actions by using these asymptotics we establish some local embedding theorems on cr manifolds and we give an analytic proof of a theorem of lempert asserting that a compact strictly pseudoconvex cr manifold of dimension three with a transversal cr s 1 action can be cr embedded into mathbb c n for some n in mathbb n

extensions of steffensen s inequality we offer a new proof of the well known steffensen inequality whose context is sufficiently general that it engenders a number of extensions

symbolic dynamics of noisy chaos abstract one model of randomness observed in physical systems is that low dimensional deterministic chaotic attractors underly the observations a phenomenological theory of chaotic dynamics requires an accounting of the information flow from the observed system to the observer the amount of information available in observations and just how this information affects predictions of the system s future behavior in an effort to develop such a description we discuss the information theory of highly discretized observations of random behavior metric entropy and topological entropy are well defined invariant measures of such an attractor s level of chaos and are computable using symbolic dynamics real physical systems that display low dimensional dynamics are however inevitably coupled to high dimensional randomless e g thermal noise we investigate the effects of such fluctuations coupled to deterministic chaotic systems in particular the metric entropy s response to the fluctuations we find that the entropy increases with a power law in the noise level and that the convergence of the entropy and the effect of fluctuations can be cast as a scaling theory we also argue that in addition to the metric entropy there is a second scaling invariant quantity that characterizes a deterministic system with added fluctuations i0 the maximum average information obtainable about the initial condition that produces a particular sequence of measurements or symbols

embeddability for three dimensional cr manifolds a question of principal interest in the theory of compact three dimensional cr manifolds is to understand when a given strictly pseudoconvex cr structure can be realized by an embedding in c we call a structure for which this is possible embeddable and nonembeddable otherwise this question is only of interest in three dimensions as a theorem of boutet de monvel states that any strictly pseudoconvex cr structure on a compact 2d 1 manifold is realizable as an embedding in some c provided d 1 on the other hand there is a classical example of rossi which shows that an arbitrarily small real analytic perturbation of the standard structure on the three sphere may fail to be embeddable ros later nirenberg constructed w perturbations for which the associated ob equation fails to have even local solutions n in fact this situation is generic in the f topology it is now understood principally from the work of eliashberg that cr structures with

on three dimensional cauchy riemann manifolds of the almost complex tensor of c2 multiplication by i the collection of these subspaces hp m forms a bundle hm c tm called the horizontal or contact bundle indeed strict pseudoconvexity of m implies that the plane field hpm is nondegenerate i e defines a contact structure the almost complex tensor of c2 then restricts to a bundle endomorphism j hm hm such that j2 id the bundle hm c tm together with this endomorphism j defines the cr structure of m slightly more generally let d d u m be a four dimensional compact manifold with boundary m interior d endowed with a smooth almost complex tensor which is integrable on d if d is strictly pseudoconvex i e in a neighborhood of any p e m there is a smooth strictly plurisubharmonic function u negative on d 0 on m but du 0 on m then m inherits a strictly pseudoconvex cr structure from d in this case we shall say that m bounds a strictly pseudoconvex surface this leads us to the abstract definition a strictly pseudoconvex threedimensional cr manifold is a compact manifold m without boundary dim m 3 endowed with a contact structure hm hpm p e m c tm and an endomorphism j of hm such that j2 id throughout this paper unless otherwise stated we shall be working with infinitely differentiable objects thus m hm j are assumed to be smooth in this sense furthermore a differentiable function f m c is a cr function if for any p e m the restriction df ihm hpm tf p c

a further refinement of a jordan type inequality and its application abstract by introducing taylor polynomials a new sharpened and generalized version of jordan s inequality is established the result is then used to obtain a substantially more refined inequality of jordan type moreover an application of the results presented here toward the improvement of the yang le inequality is also considered in this paper

moduli for pointed convex domains a moduli space for the class of pointed strictly linearly convex domains in n is obtained it is shown that the space of pointed smoothly bounded strictly linearly convex domains with a fixed indicatrix is parameterized by a class of deformations of the cr structure of the boundary of the indicatrix these deformations are constructed by using the circular representation of a domain to pull back its complex structure tensor to the indicatrix a careful study of the pull back structure shows that the allowable deformations are parameterized by a class of complex hamiltonian vector fields the proof of this fact is based on the folland stein estimates for the bar partial b complex of the boundary of the indicatrix

some improvements and generalizations of steffensen s integral inequality abstract in this paper the authors present several interesting results on improvements of steffensen s integral inequality moreover an error in a generalization of steffensen s inequality due to mercer p r mercer extensions of steffensen s inequality j math anal appl 246 2000 325 329 is corrected