on uniformly bounded bases in spaces of holomorphic functions the main result of the paper is the construction of explicit uniformly bounded bases in the spaces of complex homogenous polynomials on the unit ball of c 3 extending an earlier result of the author in the c 2 case

about boundary values in a omega assume that c n is a balanced bounded domain with a holomorphic support function e g strictly pseudoconvex domain with c 2 boundary we denote f 2 z 1 0 f e 2 it z 2 dt let e 0 and be a circular invariant borel probability measure on if g a and h is a continuous function on with g h on then we construct nonconstant functions f 1 f 2 a with g f 1 z h z g f 2 z max 1 h z for z and additionally if is a circular bounded strictly convex domain with c 2 boundary then we give the construction of f 3 o the holomorphic function with h g f 3 z 0 for all z where f denotes the radial limit of f we also construct f 4 a with g f 4 z h z for z in all cases we can make f i arbitrarily small on a given compact subset f c and make it vanish to a given order at the point 0

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

the coexistence of periodic almost periodic and chaotic attractors in the van der pol duffing oscillator in this paper are presented the computer simulation and the approximate theoretical analysis of the behaviour of the van der pol duffing forced oscillator at the passage through principal resonance at increasing and decreasing driving frequency almost periodic oscillations frequency locking transition to chaotic motion and the jump from the non resonant to the resonant state are observed and interpreted in the light of approximate analytical theory and by the use of topological concepts

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

strongly pseudoconvex domains as subvarieties of complex manifolds in this paper we obtain existence and approximation results for closed complex subvarieties that are normalized by strongly pseudoconvex stein domains our sufficient condition for the existence of such subvarieties in a complex manifold x is expressed in terms of the morse indices and the number of positive levi eigenvalues of an exhaustion function on x examples show that our conditions cannot be weakened in general we obtain optimal results for subvarieties of this type in complements of compact complex submanifolds with griffiths positive normal bundle in the projective case these generalize classical theorems of remmert bishop and narasimhan concerning proper holomorphic maps and embeddings to bbb c n bbb p n backslash bbb p n 1

proper holomorphic mappings of balanced domains in mathbb c n we extend a well known result about the unit ball by h alexander to a class of balanced domains in mathbb c n n 1 specifically we prove that any proper holomorphic self map of a certain type of balanced finite type domain in mathbb c n n 1 is an automorphism the main novelty of our proof is the use of a recent result of opshtein on the behaviour of the iterates of holomorphic self maps of a certain class of domains we use opshtein s theorem together with the tools made available by finiteness of type to deduce that the aforementioned map is unbranched the monodromy theorem then delivers the result

uniqueness and existence for bounded boundary value problems the existence and uniqueness of solutions for the boundary value problems with general linear point evaluation boundary conditions is established we assume that f is bounded and that there is uniqueness on a homogeneous problem and on the linear variational problems

on the c version of the reflection principle for mappings between cr manifolds we prove a smooth version of the classical schwarz reflection principle for cr mappings between an abstract cr manifold m and a generic cr manifold embedded in euclidean complex space as a consequence of our results we settle a conjecture of x huang formulated in 1994

interpolation and inner maps that preserve measure it is shown for n m 1 that there exist inner maps bn bm with boundary values bn bm such that m a n 1 a where n and m are the haar measures on bn and bm respectively and a bn is an arbitrary borel set

on the proper holomorphic equivalence for a class of pseudoconvex domains a complete and explicit description of the holomorphic proper mappings between weakly pseudoconvex domains of the class ap see below is given the study of proper holomorphic mappings i e holomorphic mappings which antitransform compact sets into compact sets 1 f d d2 d1 d2 c cn n 1 between bounded pseudoconvex domains with smooth boundary has had recently a considerable impulse and has been applied mainly to investigate a under which hypothesis there is no obstruction for the existence of such an f b the regularity up to the boundary of 1 c for which domains proper means in effect biholomorphic the first class of problems has been taken into consideration for instance in 8 5 4 and a typical result is for example that a strictly pseudoconvex domain cannot be mapped properly onto a weakly pseudoconvex domain about b see for instance 4 2 3 and the bibliography there included the property enunciated in c has been proved in the case d d2 unit ball see 1 and 9 when d and d2 are strictly pseudoconvex domains with in addition d2 simply connected see 7 and 4 and in the case d d2 and strictly pseudoconvex see 7 the aim of this note is to look at questions of type a and c above posed for the following class of weakly pseudoconvex domains ap z 1j z2p lzn 2pn p pn e z n precisely we prove theorem in order for a proper holomorphic mapping from ap onto a q to exist it is necessary and sufficient that plq p llql pnlqn e z furthermore assuming the only proper holomorphic mapping between a p and a q is up to biholomorphisms of a q f z zfl l zpnqn received by the editors october 25 1982 and in revised form march 29 1983 1980 mathematics subject classification primary 32h99 secondary 32f15 work performed with finanical aid from m p i di1984 american mathematical society 0002 9947 84 1 00 25 per page

disconjugacy disfocality and differentiation with respect to boundary conditions in particular under certain assumptions concerning the disconjugacy or disfocality of 1 we will consider the differentiability of solutions of 1 with respect to boundary conditions both boundary values and boundary points in so doing we will prove an analogue of a theorem of peano s which appears in hartman s text 1 theorem 3 1 p 951 with the consideration therein being with the differentiation of solutions of 1 with respect to initial conditions in this setting given a solution y x of 1 we are also interested in solutions of the linear nth order differential equation

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

on reducing submodules of hilbert modules with g n invariant kernels fix a bounded domain omega subset of c n and a positive definite kernel k on omega both invariant under g n the permutation group on n symbols let h subset of hol omega be the hilbert module determined by k we show that h splits into orthogonal direct sum of subspaces pph indexed by the partitions p of n we prove that each submodule pph is a locally free hilbert module of rank equal to square of the dimension chi p 1 of the irreducible representation corresponding to p given two partitions p and q we show that if chi p 1 not equal chi q 1 then the sub modules pph and pqh are not unitarily equivalent we prove that if h is a contractive analytic hilbert module on omega then the taylor joint spectrum of the n tuple of multiplication operators by elementary symmetric polynomials on pph is clos s omega where s c n c n is the symmetrization map it is then shown that this commuting tuple of operators defines a contractive homomorphism of the ring of symmetric polynomials cz gn in n variables equipped with the sup norm on clos s omega c 2018 elsevier inc all rights reserved

proper self maps of weakly pseudoconvex domains this theorem was proved by burns and shnider i 5 in the case that d is strongly pseudoconvex in the more general case pin6uk 1 9 corollary 2 has reduced the problem of showing that a proper mapping f d d is biholomorphic to proving that f is unbranched i e that det f 0 our interest in the theorem above stems from the fact that if f d t d 2 is a proper holomorphic mapping of a weakly pseudoconvex domain d 1 onto a second domain dz it is possible for f to branch if d1 d 2 this is contrary to what may occur if d 1 is assumed to be strongly pseudoconvex see alexander 1 pin6uk 9 10 fornaess 1 8 bell 1 3 diederich and fornaess 1 7 it has recently been proved in 1 4 and in i 6 that the mapping f in the theorem extends smoothly to this fact is basic to our proof the theorem may be shown to remain valid if 112 is replaced by a stein manifold see 2 but si ace the arguments of 4 and 6 do not carry over automatically to this case the methods used in 1 2 are more technical

a construction of inner functions on the unit ball in c p in their recent paper 1 hakim and sibony came very close to constructing inner functions on the unit ball in c e in fact only some minor modifications of their arguments are necessary to produce bona fide inner functions the present paper carries out these modifications since we need to point out some information not explicitly stated but clear from the context in their paper and to make the exposition self contained we repeat most of the material in their paper our lemma 1 for instance is identical to their lemma 3 and our application of lemma 1 in the proof of lemma 2 is the same as theirs in the proof of theorem 1 the following notations are used b is the unit ball in c p br z cp izl r we use the distance

on the regularity of cr mappings between cr manifolds of hypersurface type we prove smooth and analytic versions of the classical schwarz reflection principle for transversal cr mappings between two levi nondegenerate cr manifolds of hypersurface type

local embeddability of cr manifolds into spheres in this paper we solve local cr embeddability problem of smooth cr manifolds into spheres under a certain nondegeneracy condition on the chern moser s curvature tensor we state necessary and sufficient conditions for the existence of cr embeddings as finite number of equations and rank conditions on the chern moser s curvature tensors and their derivatives we also discuss the rigidity of those embeddings

a peak set of hausdorff dimension 2n 1 for the algebra a in the boundary of a domain with c boundary in c ⁿ

inner functions in the unit ball of cn abstract let b be the open unit ball of c n n 1 let i for inner be the set of all u h b that have u 1 a e on the boundary s of b aleksandrov proved recently that there exist nonconstant u i this paper strengthens his basic theorem and provides further information about i and the algebra q generated by i let xy be the finite linear span of products xy x x y y and let x be the norm closure in l l s of x some results set i is dense in the unit ball of h b in the compact open topology on s q q is weak dense in l q q does not contain h c s q h h h l when n 1 q h and q q l every unimodular l is a pointwise limit a e of products u v u i i the zeros of every 0 in the ball algebra but not of every h function can be matched by those of some u i as can any finite number of derivatives at 0 if however u cannot be bounded in b if u i is non constant

optimal multi dimensional ogy controller abstract a technique which generalizes the standard ogy control scheme and may be applied to higher dimensional systems to stabilize both saddle nodes and fully unstable fixed points is presented the proposed method is designed for nonlinear flows rather than maps and also provides a powerful technique to assess the controllability of nonlinear systems with ogy type controllers an optimal ogy control scheme designed for larger dimensional systems having improved convergence properties compared to the standard ogy controller is also presented numerical examples for the well known van der pol duffing oscillator are presented to illustrate the proposed control schemes

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

boundary data smoothness for solutions of nonlocal boundary value problems for second order differential equations abstract under certain conditions solutions of the boundary value problem y f x y y y x 1 y 1 and y x 2 i 1 m r i y i y 2 are differentiated with respect to boundary conditions where a x 1 1 m x 2 b r 1 r m r and y 1 y 2 r

complete proper holomorphic embeddings of strictly pseudoconvex domains into balls abstract we construct a complete proper holomorphic embedding from any strictly pseudoconvex domain with c 2 boundary in c n into the unit ball of c n for n large enough thereby answering a question of alarcon and forstneric 2013

sobolev estimates for the bar partial neumann operator on domains inc n admitting a defining function that is plurisubharmonic on the boundary neumann operator on domains inc n admitting a defining function that is plurisubharmonic on the boundary

some regularity theorems for cr mappings the purpose of this article is to study lipschitz cr mappings from an h extendible or semi regular hypersurface in mathbb c n under various assumptions on the target hypersurface it is shown that such mappings must be smooth a rigidity result for proper holomorphic mappings from strongly pseudoconvex domains is also proved

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

localization of the kobayashi metric and the boundary continuity of proper holomorphic mappings a classical theorem of carath6odory 8 states that every biholomorphic map f d d2 between domains in the complex plane c bounded by simple closed jordan curves extends to a homeomorphism of 31 onto 2 there are some well known generalizations of this result to domains in ie a if d t and d2 are bounded pseudoconvex domains in 112 with if2 boundary and if the infinitezimal kobayashi metric on d2 grows sufficiently fast near the boundary of d2 ko2 g x xl d z bdz for some e e 0 1 then every proper holomorphic map f dt d2 extends to a h61der continuous map of 51 onto 32 2 3 10 25 29 32 this holds in particular if 92 is strictly pseudoconvex or if it is pseudoconvex with real analytic boundary the same result holds if d2 is only piecewise smooth strongly pseudoconvex 31 further results treat exceptional cases such as the balls 1 reinhardt domains 5 24 domains with many symmetries 4 and analytically bounded hartogs domains in c 2 i1 biholomorphic maps between certain types of nonpseudoconvex domains with real analytic boundaries were treated in 13 and 27 besides the papers mentioned above there is a vast literature concerning smooth extension of proper holomorphic maps of smoothly bounded pseudoconvex domains 6 7 12 14 to mention just a few in the present paper we obtain some results on the continuous extension of proper holomorphic maps f d1 2 under local assumptions on the boundaries bd1 and bd2 one of the main results is

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

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

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

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

nonresonant hopf bifurcations of a controlled van der pol duffing oscillator abstract the trivial equilibrium of a van der pol duffing oscillator with a nonlinear feedback control may lose its stability via hopf bifurcations when the time delay involved in the feedback control reaches certain values nonresonant hopf hopf interactions may occur in the controlled van der pol duffing oscillator when the corresponding characteristic equation has two pairs of purely imaginary roots with the aid of normal form theory and centre manifold theorem as well as a perturbation method the dynamic behaviour of the nonresonant co dimension two bifurcation is investigated by studying the possible solutions and their stability of the four dimensional ordinary differential equations on the centre manifold in the vicinity of the nonresonant hopf bifurcation the oscillator may exhibit the initial equilibrium solution two periodic solutions as well as a quasi periodic solution on a two dimensional torus depending on the dummy unfolding parameters and nonlinear terms the analytical predictions are found to be in good agreement with the results of numerical integration of the original delay differential equation

right focal point boundary value problems for ordinary differential equations and variational equations where x g x such problems are called right focal point boundary value problems in the last few years a number of papers have appeared dealing with such boundary value problems most notable in the case when 1 is linear are papers by elias 11 muldowney 7 8 and peterson 19 io and for nonlinear considerations the papers by henderson 3 4 jackson 5 and peterson 11 121 in those papers several analogues of known results for conjugate type boundary value problems are considered for eq l we will have need of the hypotheses