a burns krantz type theorem for domains with corners in their pioneering paper bk94 burns andkrantzhave established a boundary version of the classical cartan s uniqueness theorem if f is holomorphic self map of a smoothly bounded strongly pseudoconvex domain d c such that f z z o z p 3 as z p for some p d then f z z furthermore they give an example showing that the exponent 3 in the above statement cannot be decreased the goal of this paper is twofold first to give purely local boundary uniqueness results for maps defined only on one side as germs at a boundary point and hence not necessarily sending any domain to itself and also under the weaker assumption that f z z o z p 3 holds only for z in a proper cone in d with vertexp such results have no analogues in one complex variable in contrast with the situation when a domain is preserved our second goal is to extend the above results from boundaries of domains to real submanifolds m c of higher codimension recall that m is called generic if its tangent space tpm at every point p m spans tpc over c i e tpc n tpm itpm here the usual replacement for a one sided neighborhood of a boundary point is a wedge with edgem at a pointp m in the direction of an open cone npm tpc tpm i e a domain w c such that for any pair of open cones npm with 0 and 0 there exists a neighborhood u of p in c such that m u u w and m u u contains a neighborhood of p in w where we identify npm with a fixed complement of tpm in tpc c furthermore we extend the notion of strong pseudoconvexity to wedges as follows we call a wedge w with edge m at p in the direction of strongly

the schwarz lemma at the boundary of the symmetrized bidisc abstract the symmetrized bidisc g 2 is defined by g 2 z 1 z 2 z 1 z 2 c 2 z 1 1 z 2 1 z 1 z 2 c it is a bounded inhomogeneous pseudoconvex domain without c 1 boundary and especially the symmetrized bidisc hasn t any strongly pseudoconvex boundary point and the boundary behavior of both caratheodory and kobayashi metrics over the symmetrized bidisc is hard to describe precisely in this paper we study the boundary schwarz lemma for holomorphic self mappings of the symmetrized bidisc g 2 and our boundary schwarz lemma in the paper differs greatly from the earlier related results

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

rigidity of holomorphic mappings and a new schwarz lemma at the boundary a rigidity theorem for holomorphic mappings in the nature of the uniqueness statement of the classical one variable schwarz lemma is proved at the boundary of a strongly pseudoconvex domain the result reduces to an interesting and apparently new result even in one complex dimension the theorem has a variety of geometric and analytic interpretations department of mathematics university of michigan ann arbor michigan 48109 department of mathematics box 1146 washington university in st louis st louis missouri 63130 this content downloaded from 40 77 167 14 on wed 15 jun 2016 05 36 04 utc all use subject to http about jstor org terms

bounded composition operators on weighted bergman spaces abstract we answer two questions asked by t l kriete 1998 in contemp math vol 213 pp 73 91 amer math soc providence concerning bounded composition operators on weighted bergman spaces of the unit disk the main result is the following if gi e hi for i 1 2 are weight functions in a certain range for which h 1 r h 2 r as r 1 then there is a self map of the unit disk such that the induced composition operator c maps a2g2 boundedly into itself but does not map a2g1 into itself

identity principles for commuting holomorphic self maps of the unit disc abstract let f g be two commuting holomorphic self maps of the unit disc if f and g agree at the common wolff point up to a certain order of derivatives no more than 3 if the wolff point is on the unit circle then f g

schwarz lemma at the boundary of strongly pseudoconvex domain in mathbb c n in this paper we establish a new type of the classical boundary schwarz lemma for holomorphic self mappings of strongly pseudoconvex domain in mathbb c n

comparison of invariant metrics and distances on strongly pseudoconvex domains and worm domains we prove that for a strongly pseudoconvex domain d subset mathbb c n the infinitesimal caratheodory metric g c z v and the infinitesimal kobayashi metric g k z v coincide if z is sufficiently close to bd and if v is sufficiently close to being tangential to bd also we show that every two close points of d sufficiently close to the boundary and whose difference is almost tangential to bd can be joined by a unique up to reparameterization complex geodesic of d which is also a holomorphic retract of d the same continues to hold if d is a worm domain as long as the points are sufficiently close to a strongly pseudoconvex boundary point we also show that a strongly pseudoconvex boundary point of a worm domain can be globally exposed this has consequences for the behavior of the squeezing function

julia theory for slice regular functions the theory of slice regular functions is nowadays widely studied and has found elegant applications to functional calculus for quaternionic linear operators and schur analysis however much less is known about their boundary behaviors in this paper we initiate the study of julia theory for slice regular functions more precisely we establish the quaternionic versions of the julia lemma the julia caratheodory theorem the boundary schwarz lemma the hopf lemma and the burns krantz rigidity theorem for slice regular self mappings of the open unit ball b h and of the right half space h we provide an explicit example to show that in quaternionic hopf lemma the slice derivative of a slice regular function f at the boundary fixed point may not be real in contrast to the complex version our result implies that the commonly believed version of the hopf lemma turns out to be totally wrong this new quaternionic version of the hopf lemma also improves osserman estimate even in the complex setting and is essentially the first significant theorem belonging to the theory of quaternions itself other than the theory of complex analysis since the result involves the lie brackets which reflect the non commutative feature of quaternions

boundary behavior of infinitesimal generators in the unit ball we prove a julia wolff caratheodory type theorem for infinitesimal generators on the unit ball in n moreover we study jets expansions at the boundary and give necessary and sufficient conditions on such jets for an infinitesimal generator to generate a group of automorphisms of the ball

stationary holomorphic discs and finite jet determination problems we construct a family of small analytic discs attached to levi non degenerate hypersurfaces in mathbb c n 1 which is globally biholomorphically invariant we then apply this technique to study unique determination problems along levi non degenerate hypersurfaces that are merely of class mathcal c 4 this method gives 2 jet determination results for germs of biholomorphisms cr diffeomorphisms as well as in the almost complex setting

extension operators via semigroups abstract the roper suffridge extension operator and its modifications are powerful tools to construct biholomorphic mappings with special geometric properties the first purpose of this paper is to analyze common properties of different extension operators and to define an extension operator for biholomorphic mappings on the open unit ball of an arbitrary complex banach space the second purpose is to study extension operators for starlike spirallike and convex in one direction mappings in particular we show that the extension of each spirallike mapping is a spirallike for a variety of linear operators a our approach is based on a connection of special classes of biholomorphic mappings defined on the open unit ball of a complex banach space with semigroups acting on this ball

boundary jets of holomorphic maps between strongly pseudoconvex domains abstract in this paper we consider jets taken at a fixed boundary point of germs of holomorphic diffeomorphisms which send one strongly pseudoconvex domain into another we completely describe possible first and second jets and conditions of extremality in terms of the chern moser normal forms of the domains

second angular derivatives and parabolic iteration in the unit disk in this paper we deal with second angular derivatives at denjoy wolff points for parabolic functions in the unit disc namely we study and analyze the existence and the dynamical meaning of this second angular derivative for instance we provide several characterizations of that existence in terms of the so called koenigs function it is worth pointing out that there are two quite different classes of parabolic iteration those with positive hyperbolic step and those with zero hyperbolic step in the first case the koenigs function is in the caratheodory class but in the second case it is even unknown if it is normal therefore the ideas and techniques to approach these two cases are really different in the end we also present several rigidity results related to the second angular derivatives at denjoy wolff points

surface consistent deconvolution using first principles i formulate surface consistent deconvolution as a problem in optimization to which i apply methods for nonlinear least squares to minimize processing artifacts i avoid data transformations such as fourier transforms or slant stacks my deconvolution filters are computed as approximate solutions to a large sparse least squares system useful solutions are obtained in a few iterations with each iteration equivalent in cost to performing a single trace deconvolution of the data the use of optimization removes a great deal of uncertainty about the results of surface consistent deconvolution if with a given choice of filter length etc a good deconvolution can be obtained it is obtained i illustrate the method and its limitations with three instructive field examples the first underlines the issues of uncertainty and reliability the other two show how surface consistent deconvolution can degrade as seismic data depart from the surface consistent model by small and large a

finite jet determination of cr mappings abstract we prove the following finite jet determination result for cr mappings given a smooth generic submanifold m c n n 2 that is essentially finite and of finite type at each of its points for every point p m there exists an integer l p depending upper semicontinuously on p such that for every smooth generic submanifold m c n of the same dimension as m if h 1 h 2 m p m are two germs of smooth finite cr mappings with the same l p jet at p then necessarily j p k h 1 j p k h 2 for all positive integers k in the hypersurface case this result provides several new unique jet determination properties for holomorphic mappings at the boundary in the real analytic case in particular it provides the finite jet determination of arbitrary real analytic cr mappings between real analytic hypersurfaces in c n of d angelo finite type it also yields a new boundary version of h cartan s uniqueness theorem if c n are two bounded domains with smooth real analytic boundary then there exists an integer k depending only on the boundary such that if h 1 h 2 are two proper holomorphic mappings extending smoothly up to near some point p and agreeing up to order k at p then necessarily h 1 h 2

boundary schwarz lemma for solutions to poisson s equation abstract let d be the closure of the unit disk d in c and g a continuous function of d in this paper we establish the schwarz lemma at the boundary for solutions to the poisson s equation f g

drawn k in a row games wu and huang 2005 12 and wu et al 2006 13 presented a generalized family of k in a row games called connect m n k p q two players black and white alternately place p stones on an mxn board in each turn black plays first and places q stones initially the player who first gets k consecutive stones of his her own horizontally vertically or diagonally wins both tie the game when the board is filled up with neither player winning a connect m n k p q game is drawn if neither has any winning strategy given p this paper derives the value k d r a w p such that connect m n k p q games are drawn for all k k d r a w p m 1 n 1 email protected email protected p as follows 1 k d r a w p 11 2 for all p 3 k d r a w p 3p 3d 1 where d is a logarithmic function of p so the ratio k d r a w p p is approximately 3 for sufficiently large p the first result was derived with the help of a program to our knowledge our k d r a w p values are currently the smallest for all email protected p 1000