bayesian estimation of dirichlet mixture model with variational inference in statistical modeling parameter estimation is an essential and challengeable task estimation of the parameters in the dirichlet mixture model dmm is analytically intractable due to the integral expressions of the gamma function and its corresponding derivatives we introduce a bayesian estimation strategy to estimate the posterior distribution of the parameters in dmm by assuming the gamma distribution as the prior to each parameter we approximate both the prior and the posterior distribution of the parameters with a product of several mutually independent gamma distributions the extended factorized approximation method is applied to introduce a single lower bound to the variational objective function and an analytically tractable estimation solution is derived moreover there is only one function that is maximized during iterations and therefore the convergence of the proposed algorithm is theoretically guaranteed with synthesized data the proposed method shows the advantages over the em based method and the previously proposed bayesian estimation method with two important multimedia signal processing applications the good performance of the proposed bayesian estimation method is demonstrated

bayesian approaches to gaussian mixture modeling a bayesian based methodology is presented which automatically penalizes overcomplex models being fitted to unknown data we show that with a gaussian mixture model the approach is able to select an optimal number of components in the model and so partition data sets the performance of the bayesian method is compared to other methods of optimal model selection and found to give good results the methods are tested on synthetic and real data sets

stein compacts in levi flat hypersurfaces we explore connections between geometric properties of the levi foliation of a levi flat hypersurface m and holomorphic convexity of compact sets in m or bounded in part by m applications include extendability of cauchy riemann functions solvability of the b equation approximation of cauchy riemann and holomorphic functions and global regularity of the neumann operator

weighted boundary limits of the generalized kobayashi royden metrics on weakly pseudoconvex domains the purpose of this paper is to study the existence of weighted boundary limits of the generalized kobayashi royden metrics on weakly pseudoconvex domains in cn and to explore the connections between the limits and the levi invariants the main result extends graham s result on strongly pseudoconvex domains to a large class of weakly pseudoconvex domains

on the analysis of the 1 1 evolutionary algorithm many experimental results are reported on all types of evolutionary algorithms but only few results have been proved a step towards a theory on evolutionary algorithms in particular the so called 1 1 evolutionary algorithm is performed linear functions are proved to be optimized in expected time o nlnn but only mutation rates of size 1 n can ensure this behavior for some polynomial of degree 2 the optimization needs exponential time the same is proved for a unimodal function both results were not expected by several other authors finally a hierarchy result is proved moreover methods are presented to analyze the behavior of the 1 1 evolutionary algorithm

holder estimates on cr manifolds with a diagonalizable levi form a for all e 0 the operator cl maps lip s po into lip s 2 m e po here lip s p denotes the space of square integrable forms of degree p q with 1 g q otheoperators o l o l q and0 map lip s p into lip s l m e po c for all e o the operators 8b8zo 8z8ho 1 q a aa cl 8 8 8 q 8 and 8 0 ab map lip s p into lip s p hebre aiag web asm acts on forms of type p q with 1 q 0 e zf f is a square integrable function on a with f e lip p then s f e lip s e p for all e 0 where s is the orthogonal projection on square integrable cr functions

convergence and divergence of formal cr mappings let m subset mathbb c n be a generic real analytic submanifold of finite type m subset mathbb c n be a real analytic set and p in m where we assume that n n geqslant 2 let h mathbb c n p to mathbb c n be a formal holomorphic mapping sending m into m and let mathcal e m denote the set of points in m through which there passes a complex analytic subvariety of positive dimension contained in m we show that if h does not send m into mathcal e m then h must be convergent as a consequence we derive the convergence of all formal holomorphic mappings when m does not contain any complex analytic subvariety of positive dimension answering by this a long standing open question in the field more generally we establish necessary conditions for the existence of divergent formal maps even when the target real analytic set is foliated by complex analytic subvarieties allowing us to settle additional convergence problems such as e g for transversal formal maps between levi non degenerate hypersurfaces and for formal maps with range in the tube over the light cone

markov chain analysis of evolutionary algorithms on onemax function from coupon collector s problem to 1 1 ea abstract the theoretical investigation of evolutionary algorithms eas has increased our understanding of the computational mechanism of algorithms onemax is a test function most frequently and deeply studied in the field of eas in this work a method is presented for describing the runtime properties of 1 1 ea on onemax this method is motivated by the work of erdos and renyi treating the coupon collector s problem they showed that the success probability of the coupon collector s problem is given by a function of double exponential form and that the number of uncollected coupons follows the poisson distribution today the double exponential function is called gumbel function which is one of three fundamental functions in extreme value statistics we introduce an algorithm that is a variant of the 1 1 ea first order evolutionary algorithm fo ea fo ea takes into account only the effect of single bit mutations in the 1 1 ea which in general includes multiple bit mutations we modified the method of erdos and renyi to apply fo ea we apply the gumbel distribution for calculating the success probability of the 1 1 ea on onemax this method turns out to give a sufficiently reliable estimation for success probabilities even in the tail region

a parametrix for the neumann problem on pseudoconvex domains of finite type abstract we construct a parametrix for the neumann problem on any pseudoconvex domain of finite type which reduces completely the study of isotropic lp sobolev and holder estimates to those for b and b we also establish the maximal nonisotropic estimates in the case of d q domains and provide a precise comparison of the bergman and szego projections and of the canonical solutions to with respect to these two operators the argument is substantially simpler than the known ones for the cases of strongly pseudoconvex domains and pseudoconvex domains of finite type in c 2

hearing the type of a domain in c 2 with the neumann laplacian a smooth bounded pseudoconvex domain in c2 is of finite type if and only if the number of eigenvalues of its a neumann laplacian that are less than or equal to has at most polynomial growth as goes to infinity

a novel bayesian logistic discriminant model an application to face recognition the linear discriminant analysis lda is a linear classifier which has proven to be powerful and competitive compared to the main state of the art classifiers however the lda algorithm assumes the sample vectors of each class are generated from underlying multivariate normal distributions of common covariance matrix with different means i e homoscedastic data this assumption has restricted the use of lda considerably over the years authors have defined several extensions to the basic formulation of lda one such method is the heteroscedastic lda hlda which is proposed to address the heteroscedasticity problem another method is the nonparametric da nda where the normality assumption is relaxed in this paper we propose a novel bayesian logistic discriminant bld model which can address both normality and heteroscedasticity problems the normality assumption is relaxed by approximating the underlying distribution of each class with a mixture of gaussians hence the proposed bld provides more flexibility and better classification performances than the lda hlda and nda a subclass and multinomial versions of the bld are proposed the posterior distribution of the bld model is elegantly approximated by a tractable gaussian form using variational transformation and jensen s inequality allowing a straightforward computation of the weights an extensive comparison of the bld to the lda support vector machine svm hlda nda and subclass discriminant analysis sda performed on artificial and real data sets has shown the advantages and superiority of our proposed method in particular the experiments on face recognition have clearly shown a significant improvement of the proposed bld over the lda

the szegő projection sobolev estimates in regular domains the szego projection preserves global smoothness in weakly pseudoconvex domains that are regular in the sense of diederich fornxess and catlin it preserves local smoothness near boundary points of finite type the bergman projection being linked to the 8 problem plays an important role in several complex variables as demonstrated by recent applications to the boundary behavior of holomorphic maps 7 16 the szego projection is interesting both by analogy and by its connection with the ab problem and hence with unsolvable equations of lewy type while in strictly pseudoconvex domains the bergman and szego projections are both well understood in general much more is known about the bergman projection this is no surprise for the theory of the 8 problem is further advanced than the theory of the 8b problem nonetheless i believe that the szego projection ought not be any more mysterious than the bergman projection this paper is intended to support this contention the most important domains for which the bergman projection is known to be regular are domains with a good a neumann operator here it is shown that in such domains the szego projection also is regular more precisely in smooth bounded pseudoconvex domains satisfying catlin s property p the most general condition presently known that guarantees global regularity of the neumann operator the szego projection is shown to be globally regular theorem 4 1 moreover at boundary points of finite type the szego projection is locally regular theorem 5 1 in both cases the szego projection exactly preserves the degree of differentiability measured in sobolev norms the method of proof is not the obvious one of directly generalizing from the strictly peudoconvex case for there are serious obstacles to such a generalization for instance the approach of analysis on the heisenberg group used by phong and stein 29 to derive estimates for both the bergman and szego projections cannot yet be extended because the theory of singular integrals on weakly pseudoconvex surfaces is still in its early stages 28 15 similarly one does not know how to carry over the fourier integral operator approach of boutet de monvel and sjostrand 10 to the weakly pseudoconvex case explicit construction of cauchy fantappie forms received by the editors march 11 1986 paper presented april 11 1986 to the 826th meeting of the american mathematical society indianapolis special session on several complex variables 1980 mathematics subject classification 1985 revision primary 32a25 32h10 partially supported by national science foundation grants mcs 8201063 and dms 8501758 1987 american mathematical society 0002 9947 87 1 00 25 per page

optimal hoelder and l p estimates for b on the boundaries of real ellipsoids in c n let d be a real ellipsoid in cn n 3 with defining function p z en x k yk2k 1 zk xk iyk where nk mk e n in this paper we study the sharp holder and lp estimates for the solutions of the tangential cauchy riemann equations 0b on the boundary od of d using the integral kernel method in particular we proved that if a e l o j od such that oba 0 on ad in the distribution sense then there exists a u e a112m od satisfying obu a and iiuiia 2 9d 0 independent of a where al 2m od is the lipschitz space with exponent 21 and 2m maxl k f min 2nk 2mk is the type of the m k domain d

discriminant adaptive nearest neighbor classification nearest neighbour classification expects the class conditional probabilities to be locally constant and suffers from bias in high dimensions we propose a locally adaptive form of nearest neighbour classification to try to ameliorate this curse of dimensionality we use a local linear discriminant analysis to estimate an effective metric for computing neighbourhoods we determine the local decision boundaries from centroid information and then shrink neighbourhoods in directions orthogonal to these local decision boundaries and elongate them parallel to the boundaries thereafter any neighbourhood based classifier can be employed using the modified neighbourhoods the posterior probabilities tend to be more homogeneous in the modified neighbourhoods we also propose a method for global dimension reduction that combines local dimension information in a number of examples the methods demonstrate the potential for substantial improvements over nearest neighbour classification

necessary geometric and analytic conditions for general estimates in the bar partial neumann problem we show the geometric and analytic consequences of a general estimate in the bar partial neumann problem a gain in the estimate yields a bound in the type of the boundary that is in its order of contact with an analytic curve as well as in the rate of the bergman metric we also discuss the potential theoretical consequence a gain implies a lower bound for the levi form of a bounded weight

construction of p s h functions on weakly pseudoconvex domains let q c c be a smoothly bounded pseudoconvex domain and let a q denote the functions holomorphic on q and continuous on sl a point p e aq is a peak point if there is a function f e a q such that f p 1 and if z i 2 were obtained by range 19 20 and hakim and sibony 12 these results do not extend automatically to arbitrary weakly pseudoconvex boundary points of q even if the boundary is real analytic an example of kohn and nirenberg 17 shows that a p e aq does not always have a separating function which is real analytic at p similarly this example does not have a peak function which is e at p see fornaess 6 here we show how peak functions and separating functions may be constructed for a large class of weakly pseudoconvex points on domains in c the results are most complete for domains in c2 so we state them here for that case

hearing pseudoconvexity with the kohn laplacian a bounded domain in open image in new window with connected lipschitz boundary is pseudoconvex if the bottom of the essential spectrum of the kohn laplacian on the space of 0 q forms 1 q n 1 with l2 coefficients is positive

defining equations for real analytic real hypersurfaces in ⁿ a defining function for a real analytic real hypersurface can be utliquely written as 2 re h e where h is holomorphic and e contains no pure terms we study how h and e change when we perform a local biholomorphic change of coordinates or multiply by a unit one of the main results is necesary and sufficient conditions on the first nonvanishing homogeneous part of e expanded in terms of h beyond eoo that serve as obstructions to writing a defining equation as 2 re h e where e is independent of h we also find necessary pluriharmonic obstructions to doing this which arise from the easier case of attempting to straighten the hypersurface

drift analysis and average time complexity of evolutionary algorithms the computational time complexity is an important topic in the theory of evolutionary algorithms eas this paper reports some new results on the average time complexity of eas based on drift analysis some useful drift conditions for deriving the time complexity of eas are studied including conditions under which an ea will take no more than polynomial time in problem size to solve a problem and conditions under which an ea will take at least exponential time in problem size to solve a problem the paper first presents the general results and then uses several problems as examples to illustrate how these general results can be applied to concrete problems in analyzing the average time complexity of eas while previous work only considered 1 1 eas without any crossover the eas considered in this paper are fairly general which use a finite population crossover mutation and selection 2001 elsevier science b v all rights reserved

compactness in the neumann problem magnetic schrodinger operators and the aharonov bohm effect abstract compactness of the neumann operator in the neumann problem is studied for weakly pseudoconvex bounded hartogs domains in two dimensions a nonsmooth domain is constructed for which condition p fails to hold yet the kohn laplacian still has compact resolvent the main result in contrast is that for smoothly bounded hartogs domains the well known sufficient condition p is equivalent to compactness the analyses of compactness and condition p boil down to the asymptotic behavior of the lowest eigenvalues of two related sequences of schrodinger operators one with a magnetic field and one without parametrized by a fourier variable resulting from the hartogs symmetry the nonsmooth example is based on the aharonov bohm phenomenon of quantum physics for smooth domains not satisfying p we prove that there always exists an exceptional sequence of fourier variables for which the aharonov bohm effect is weak and thence that compactness fails to hold this sequence can be very sparse so that the lack of compactness is due to a rather subtle effect

on the existence of maximum likelihood estimates in logistic regression models summary the problems of existence uniqueness and location of maximum likelihood estimates in log linear models have received special attention in the literature haberman 1974 chapter 2 wedderburn 1976 silvapulle 1981 for multinomial logistic regression models we prove existence theorems by considering the possible patterns of data points which fall into three mutually exclusive and exhaustive categories complete separation quasicomplete separation and overlap our results suggest general rules for identifying infinite parameter estimates in log linear models for frequency tables

multiple imputation of industry and occupation codes in census public use samples using bayesian logistic regression abstract we describe methods used to create a new census data base that can be used to study comparability of industry and occupation classification systems this project represents the most extensive application of multiple imputation to date and the modeling effort was considerable as well hundreds of logistic regressions were estimated one goal of this article is to summarize the strategies used in the project so that researchers can better understand how the new data bases were created another goal is to show how modifications of maximum likelihood methods were made for the modeling and imputation phases of the project to multiply impute 1980 census comparable codes for industries and occupations in two 1970 census public use samples logistic regression models were estimated with flattening constants for many of the regression models considered the data were too sparse to support conventional maximum likelihood analysis so some alternative had to be employed these methods solve existence and

newton polyhedra and the bergman kernel the purpose of this paper is to study singularities of the bergman kernel at the boundary for pseudoconvex domains of finite type from the viewpoint of the theory of singularities under some assumptions on a domain in n 1 the bergman kernel b z of takes the form near a boundary point p b z frac phi w rho rho 2 2 d f hbox log 1 rho m f 1 where w is some polar coordinates on a nontangential cone with apex at p and means the distance from the boundary here admits some asymptotic expansion with respect to the variables 1 m and log 1 as 0 on the values of df 0 mf and m are determined by geometrical properties of the newton polyhedron of defining functions of domains and the limit of as 0 on is a positive constant depending only on the newton principal part of the defining function analogous results are obtained in the case of the szego kernel

generalized models and local invariants of kohn nirenberg domains the main obstruction for constructing holomorphic reproducing kernels of cauchy type on weakly pseudoconvex domains is the kohn nirenberg phenomenon i e nonexistence of supporting functions and local nonconvexifiability in this paper we give an explicit verifiable characterization of weakly pseudoconvex but locally nonconvexifiable hypersurfaces of finite type in dimension two it is expressed in terms of a generalized model which captures local geometry of the hypersurface both in the complex tangential and nontangential directions as an application we obtain a new class of nonconvexifiable pseudoconvex hypersurfaces with convex models

pseudoconvex regions of finite d angelo type in four dimensional almost complex manifolds let d be a j pseudoconvex region in a smooth almost complex manifold m j of real dimension four we construct a local peak j plurisubharmonic function at every point p bd of finite d angelo type as applications we give local estimates of the kobayashi pseudometric implying the local kobayashi hyperbolicity of d at p in case the point p is of d angelo type less than or equal to four or the approach is nontangential we provide sharp estimates of the kobayashi pseudometric

boundary behavior of the bergman kernel function on pseudoconvex domains with comparable levi form abstract let be a smoothly bounded pseudoconvex domain in c n and let z 0 b be a point of finite type we also assume that the levi form of b is comparable in a neighborhood of z0 then we get a quantity which bounds from above and below the bergman kernel function in a small constant and large constant sense

the bergman kernel on certain weakly pseudoconvex domains let c c be a bounded pseudoconvex domain with c p lur i superharmonic defining function 0 let az q be the space of square integrable ho lomorph ic functions on if 0 is any complete o r thonorma l basis for az g2 then the bergman kernel for o is given by k z w y o z o w except for some special cases 53 it is not possible at present to obta in a nice closed form for k however for domains of the form

compactness of the complex green operator on cr manifolds of hypersurface type the purpose of this article is to study compactness of the complex green operator on cr manifolds of hypersurface type we introduce cr pq a potential theoretic condition on 0 q forms that generalizes catlin s property pq to cr manifolds of arbitrary codimension we prove that if an embedded cr manifold of hypersurface type of real dimension at least five satisfies cr pq and cr pn 1 q then the complex green operator is a compact operator on the sobolev spaces h s 0 q m and h s 0 n 1 q m if 1 q n 2 and s 0 we use cr plurisubharmonic functions to build a microlocal norm that controls the totally real direction of the tangent bundle 1 introduction and results in this article we introduce property cr pq a potential theoretic condition on 0 q forms we show that if an embedded cr manifold of hypersurface type satisfies cr pq and cr pn 1 q then the complex green operator is a compact operator on the sobolev spaces h s 0 q m and h s 0 n 1 q m if 1 q n 2 we use cr plurisubharmonic functions to build a microlocal norm that controls the bad direction of the tangent bundle we first prove the closed range and compactness results on l 2 q m and use an elliptic regularization

on the efficient evaluation of probabilistic similarity functions for image retrieval probabilistic approaches are a promising solution to the image retrieval problem that when compared to standard retrieval methods can lead to a significant gain in retrieval accuracy however this occurs at the cost of a significant increase in computational complexity in fact closed form solutions for probabilistic retrieval are currently available only for simple probabilistic models such as the gaussian or the histogram we analyze the case of mixture densities and exploit the asymptotic equivalence between likelihood and kullback leibler kl divergence to derive solutions for these models in particular 1 we show that the divergence can be computed exactly for vector quantizers vqs and 2 has an approximate solution for gauss mixtures gms that in high dimensional feature spaces introduces no significant degradation of the resulting similarity judgments in both cases the new solutions have closed form and computational complexity equivalent to that of standard retrieval approaches

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

weighted pseudometric discriminatory power improvement using a bayesian logistic regression model based on a variational method in this paper we investigate the effectiveness of a bayesian logistic regression model to compute the weights of a pseudometric in order to improve its discriminatory capacity and thereby increase image retrieval accuracy in the proposed bayesian model the prior knowledge of the observations is incorporated and the posterior distribution is approximated by a tractable gaussian form using variational transformation and jensen s inequality which allow a fast and straightforward computation of the weights the pseudometric makes use of the compressed and quantized versions of wavelet decomposed feature vectors and in our previous work the weights were adjusted by the classical logistic regression model a comparative evaluation of the bayesian and classical logistic regression models is performed for content based image retrieval as well as for other classification tasks in a decontextualized evaluation framework in this same framework we compare the bayesian logistic regression model to some relevant state of the art classification algorithms experimental results show that the bayesian logistic regression model outperforms these linear classification algorithms and is a significantly better tool than the classical logistic regression model to compute the pseudometric weights and improve retrieval and classification performance finally we perform a comparison with results obtained by other retrieval methods

on the automorphism group of a certain infinite type domain in c2 abstract in this study we consider an infinite type domain p in c 2 the aim of this study is to investigate the holomorphic vector fields tangent to an infinite type model in c 2 vanishing at an infinite type point and to give an explicit description of the automorphism group of p

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

convex domains of finite type we construct bounded plurisubharmonic functions with maximally large hessians near the boundary of a smoothly bounded convex domain in cn as a corollary the equality of the order of contact of the boundary with complex analytic varieties the d angelo type and the order of contact with complex lines is demonstrated

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

on the automorphism group of the kohn nirenberg domain abstract in this paper we prove that there are no automorphism orbits of the kohn nirenberg domain accumulating at the origin

effective vanishing order of the levi determinant on a smooth domain in mathbb c n of finite d angelo q type at a point an effective upper bound for the vanishing order of the levi determinant rm coeff partial r wedge bar partial r wedge partial bar partial r n q at that point is given in terms of the d angelo q type the dimension of the space n and q itself the argument uses catlin s notion of a boundary system as well as techniques pioneered by john d angelo

stably interior points and the semicontinuity of the automorphism group we present the new semicontinuity theorem for automorphism groups if a sequence omega j of bounded pseudoconvex domains in mathbb c 2 converges to omega 0 in mathcal c infty topology where omega 0 is a bounded pseudoconvex domain in mathbb c 2 with its boundary mathcal c infty and of the d angelo finite type and with text aut omega 0 compact then there is an integer n 0 such that for every j n there exists an injective lie group homomorphism psi j text aut omega j rightarrow text aut omega 0 the method of our proof of this theorem is new that it simplifies the proof of the earlier semicontinuity theorems for bounded strongly pseudoconvex domains

towards an analytic framework for analysing the computation time of evolutionary algorithms in spite of many applications of evolutionary algorithms in optimisation theoretical results on the computation time and time complexity of evolutionary algorithms on different optimisation problems are relatively few it is still unclear when an evolutionary algorithm is expected to solve an optimisation problem efficiently or otherwise this paper gives a general analytic framework for analysing first hitting times of evolutionary algorithms the framework is built on the absorbing markov chain model of evolutionary algorithms the first step towards a systematic comparative study among different eas and their first hitting times has been made in the paper

subellipticity of the neumann problem on a weakly q pseudoconvex concave domain abstract for a domain d of c n which is weakly q pseudoconvex or q pseudoconcave we give a sufficient condition for subelliptic estimates for the neumann problem this extends to domains which are not necessarily pseudoconvex the results and the techniques of catlin 1987 3