rigidity of cr maps between shilov boundaries of bounded symmetric domains our goal is to establish what seems to be the first rigidity result for cr embeddings between shilov boundaries of bounded symmetric domains of higher rank the result states that any such cr embedding is the standard linear embedding up to cr automorphisms our basic assumption extends precisely the well known optimal bound for the rank one case there are no other restrictions on the ranks in particular the difficult case when the target rank is larger than the source rank is also allowed

hermitian analogues of hilbert s 17 th problem abstract we pose and discuss several hermitian analogues of hilbert s 17 th problem we survey what is known offer many explicit examples and some proofs and give applications to cr geometry we prove one new algebraic theorem a non negative hermitian symmetric polynomial divides a non zero squared norm if and only if it is a quotient of squared norms we also discuss a new example of putinar scheiderer

d angelo conjecture in the third gap interval we show that the d angelo conjecture holds in the third gap interval more precisely we prove that the degree of any rational proper holomorphic map from mathbb b n to mathbb b 4n 6 with n ge 7 is not more than 3

on the third gap for proper holomorphic maps between balls let f be a proper rational map from the complex ball mathbb b n into mathbb b n with n 7 and 3n 1 le n le 4n 7 then f is equivalent to a map g 0 dots 0 where g is a proper holomorphic map from mathbb b n into mathbb b 3n

symmetries in cr complexity theory abstract we introduce the hermitian invariant group f of a proper rational map f between the unit ball in complex euclidean space and a generalized ball in a space of typically higher dimension we use properties of the groups to define the crucial new concepts of essential map and the source rank of a map we prove that every finite subgroup of the source automorphism group is the hermitian invariant group of some rational proper map between balls we prove that f is non compact if and only if f is a totally geodesic embedding we show that f contains an n torus if and only if f is equivalent to a monomial map we show that f contains a maximal compact subgroup if and only if f is equivalent to the juxtaposition of tensor powers we also establish a monotonicity result the group after intersecting with the unitary group does not decrease when a tensor product operation is applied to a polynomial proper map we give a necessary condition for f when the target is a generalized ball to contain automorphisms that move the origin

factor analysis models via i divergence optimization given a positive definite covariance matrix formula see text of dimension n we approximate it with a covariance of the form formula see text where h has a prescribed number formula see text of columns and formula see text is diagonal the quality of the approximation is gauged by the i divergence between the zero mean normal laws with covariances formula see text and formula see text respectively to determine a pair h d that minimizes the i divergence we construct by lifting the minimization into a larger space an iterative alternating minimization algorithm aml a la csiszar tusnady as it turns out the proper choice of the enlarged space is crucial for optimization the convergence of the algorithm is studied with special attention given to the case where d is singular the theoretical properties of the aml are compared to those of the popular em algorithm for exploratory factor analysis inspired by the ecme a newton raphson variation on em we develop a similar variant of aml called acml and in a few numerical experiments we compare the performances of the four algorithms

rational proper holomorphic maps from b n into b 2 n in this paper we will investigate rational proper holomorphic maps from b n into b2 n when n 4

rigidity of proper holomorphic maps between bounded symmetric domains our first main result gives assumptions guaranteeing that proper holomorphic maps between cartan type i bounded symmetric domains have simple block matrix shape answering positively a question of mok the proof is based on the second main result establishing similar phenomenon for local cr maps between arbitrary boundary components of two bounded symmetric domains of the above type since boundary components other than shilov boundaries are levi degenerate our analysis is based on their 2 nondegeneracy combining levi forms with higher order tensors

extending proper holomorphic mappings of positive codimension in this paper we obtain results on holomorphic continuation of proper holomorphic mappings between pseudoconvex domains with real analytic boundaries in complex spaces of different dimensions equivalently we obtain results concerning the analyticity of cauchy riemann mappings between real analytic pseudoconvex hypersurfaces in complex spaces of different dimensions to begin with we recall the corresponding results for mappings of equidimensional domains let d and d be bounded pseudoconvex domains with smooth boundaries in c if the boundaries of d and d are strictly pseudoconvex or more generally of finite type in the sense of d angelo 15 then every proper holomorphic map of d onto d extends smoothly t o according to the results of bell and catlin 7 8 and diederich and forn ess 19 for mappings between strictly pseudoconvex domains this was proved by fefferman 26 and nirenberg webster and yang 37 if the boundaries of the pseudoconvex domains d d c c are real analytic then every proper holomorphic mapping of d onto d extends holomorphically to a neighborhood of 3 according to baouendi and rothschild 2 and diederich and forn ess 21 this reflection principle was first discovered by lewy 34 and pin6uk 38 for mappings between strictly pseudoconvex domains in the case of biholomorphic mappings between weakly pseudoconvex domains with realanalytic boundaries the result follows from the work of baouendi jacobowitz and treves 5 results in this direction were obtained in recent years by several authors see the papers 4 6 18 22 33 47 49 in this paper we are treating the case when the domains d and d have different dimensions to be specific we assume that d e c and d c c n are bounded pseudoconvex domains with real analytic boundaries and n n l in this situation a proper holomorphic map f d d need not be regular at the boundary for instance there exist proper holomorphic maps of balls of different dimensions

a reflection principle for proper holomorphic mappings of strongly pseudoconvex domains and applications this result is proved in 5 the techniques of 12 and 5 involve detailed computat ion of differential invariants on the boundary another paper philosophically related to the above but involving more elementary techniques is 1 in which proper holomorphic maps of b n to b are shown to be automorphisms more recently this result has been proved on strongly pseudoconvex domains in 2 and in 4 in the paper 3 which is the starting point for the present work completely elementary means are used to obtain a reflection principle for suitable maps of b to b k this in turn leads to a proof of the result of 12 assuming only c 2 smoothness to the boundary further detailed information is obtained about the maps even when k n 1 the elementary techniques of 3 may be adapted to use on strictly

the connection between bayesian estimation of a gaussian random field and rkhs reconstruction of a function from noisy data is key in machine learning and is often formulated as a regularized optimization problem over an infinite dimensional reproducing kernel hilbert space rkhs the solution suitably balances adherence to the observed data and the corresponding rkhs norm when the data fit is measured using a quadratic loss this estimator has a known statistical interpretation given the noisy measurements the rkhs estimate represents the posterior mean minimum variance estimate of a gaussian random field with covariance proportional to the kernel associated with the rkhs in this brief we provide a statistical interpretation when more general losses are used such as absolute value vapnik or huber specifically for any finite set of sampling locations that includes where the data were collected the maximum a posteriori estimate for the signal samples is given by the rkhs estimate evaluated at the sampling locations this connection establishes a firm statistical foundation for several stochastic approaches used to estimate unknown regularization parameters to illustrate this we develop a numerical scheme that implements a bayesian estimator with an absolute value loss this estimator is used to learn a function from measurements contaminated by outliers

a reflection principle with applications to proper holomorphic mappings j l holomorphic mappings from b into b k we then apply this principle to the problem of characterizing the proper holomorphic mappings from b into b k two such mappings f and g are said to be equivalent if there exist holomorphic automorphisms b and p of b and bk respectively such that f pogo49 there are at present three significant papers on the subject the first by alexander 1 proves that i f f is a proper holomorphic mapping of b into b then f is a holomorphic automorphism of b webster i 5 proved that for n 3 i f f is a proper holomorphic mapping of b into b 1 which is c3 then f maps b into a hyperplane in r hence using alexander s result and properties of the holomorphic automorphisms f is equivalent to the mapping zl z2 z zi z2 zn o alexander s example zl z2 z2 zlz2 z 2 shows that webster s result does not hold for n 2 in a recent paper faran 2 has characterized all proper holomorphic mappings f from b 2 to b 3 that satisfy the additional smoothness condition fec3 2 he proves that such mappings are equivalent to one of the four mappings z w z w 0 z w z zw w2 z w r z 2 l zw w2

homotopy equivalence for proper holomorphic mappings abstract we introduce several homotopy equivalence relations for proper holomorphic mappings between balls we provide examples showing that the degree of a rational proper mapping between balls in positive codimension is not a homotopy invariant in domain dimension at least 2 we prove that the set of homotopy classes of rational proper mappings from a ball to a higher dimensional ball is finite by contrast when the target dimension is at least twice the domain dimension it is well known that there are uncountably many spherical equivalence classes we generalize this result by proving that an arbitrary homotopy of rational maps whose endpoints are spherically inequivalent must contain uncountably many spherically inequivalent maps we introduce whitney sequences a precise analogue in higher dimensions of the notion of finite blaschke product in one dimension we show that terms in a whitney sequence are homotopic to monomial mappings and we establish an additional result about the target dimensions of such homotopies

super rigidity for cr embeddings of real hypersurfaces into hyperquadrics we consider holomorphic mappings sending a given levi nondegenerate pseudoconcave hypersurface m in into a nondegenerate hyperquadric of the same signature in and show that if m is sufficiently close to a hyperquadric in a certain sense then any two such mappings differ only by an automorphism of the hyperquadric

maps from the two ball to the three ball in 4 webster proved the following theorem let b z e c 7 2 3 is a proper holomorphic map which is c a up to the boundary of b then f is linear fractional the purpose of this paper is to extend this result to the case n 2 however the theorem is false when n 2 f need not be linear fractional so instead of trying to prove this false theorem we shall instead try to make a list of maps f bz b a which are c 3 up to the boundary of course given such an f we can get another such by composing f with an automorphism of b 2 and an automorphism of b a so we make the

regularity of mappings into classical domains we study the regularity and algebraicity for mappings into classical domains among other things we establish everywhere regularity and algebraicity results for c 2 cr maps into the set of smooth boundary points of a classical domain where the codimension can be arbitrarily large

non embeddable real algebraic hypersurfaces we study various classes of real hypersurfaces that are not embeddable into more special hypersurfaces in higher dimension such as spheres real algebraic compact strongly pseudoconvex hypersurfaces or compact pseudoconvex hypersurfaces of finite type we conclude by stating some open problems

embedding strictly pseudoconvex domains into balls every relatively compact strictly pseudocc nvex domain d with c2 boundary in a stein manifold can be embedded as a closed complex submanifold of a finite dimensional ball however for each n 2 there exist bounded strictly pseudoconvex domains d in cn with real analytic boundary such that no proper holomorphic map from d into any finite dimensional ball extends smoothly to d 0 introduction in this paper we study the representations of bounded strictly pseudoconvex domains d c cn if the boundary of d is of class ck k e 2 3 oo then by a theorem of fornaess 8 and khenkin 13 d can be mapped biholomorphically onto the intersection xn q of a bounded strictly convex domain q c cn with ck boundary and a closed complex submanifold x defirxed in a neighborhood of q in cn x intersecting the boundary of q transversally moreover the map f d x n q extends to a holomorphic map on a neighborhood of d the convex domain q depends on d hence a natural question is whether a similar result holds with q replaced by the unit ball b z z1l zn e cn z 2 e zjl2 n that intersects bbn transversally such that d is biholomorphically equivalent to xn e3ff this question has been mentioned by lempert 14 pinduk 20 bedford 2 and others our main result is that the answer to this question is negative in general if d is as above then every biholomorphism of d onto xne3n extends smoothly to d according to 3 however we will show that not all such domains d admit a proper holomorphic map into a finite dimensional ball that is smooth on d theorem 1 1 a similar local result was obtained independently by faran 7 we shall show that the answer to the question q is positive if we allow the intersections of complex submanifolds with strictly convex domains q c cn with real analytic boundaries theorem 1 2 thetheoremsoffornaess 8 andkhenkin 13 onlygiveanqwith smooth boundary received by the editors april 4t 1985 1980 mathematics subject classification primary 32h05 1research supported by a fellowship from the alfred p sloan foundation t 1986 american mathematical society 0002 9947 86 1 00 25 per page