Erratum et Addendum zu der Arbeit: Über exzeptionelle Mengen

Let X be a complex space and A?X a compact subspace. Let \(\tilde X\) be the blowing up of A in X and \(\tilde A\) ? \(\tilde X\) the resulting hyper-surface. Then the normal bundle of \(\tilde A\) in { \(\tilde X\) is weakly negative iff the normal bundle of the k-th infinitesimal neighborhood of A in X is weakly negative for all k?0. This corrects a theorem in [5].     

Cosymplectic quasi-Sasakian pseudo-Riemannian manifolds and coisotropic foliations

Si considera una varietà neutra \(\tilde M\) di dimensione 2m munita di una struttura conforme simplettica \(CS_p \left( {2m; R} \right) = \left( {\tilde \Omega , \tilde \upsilon } \right)\) . Vengono studiati i differenti problemi concernenti gli automorfismi infinitesimali della 2-forma quasi simplettica \(\tilde \Omega \) . Inoltre vengono formulate alcune proprietà di un fogliettamento con isotropoF _c su \(\tilde M\) .     

Hölderabschätzungen für Ableitungen von Lösungen der Gleichung\bar \partial u = f bei streng pseudokonvexem Rand

Using the local Kerzman kernel we prove regularity of solutions of \(\bar \partial \) u=f, where f is a \(\bar \partial \) -closed (0,1)-form in a strongly pseudoconvex domain G in ?^N. If f is in H^m,∞, then the solution is in \(\tilde C^{m,\mu } \) forμ<1, that is, the m-th derivatives are in C^o,μ/2 and in addition areμ-Hölder continuous on curves “parallel” to the holomorphic tangent bundle \(\tilde T\) ?G. If f is in C^m,α with o<α<1, then the solution is in \(\tilde C^{m,1 + \mu } \) forμ<α, that is, the m-th derivatives are in C^o,(1+μ/2, and they have first derivatives “parallel” to \(\tilde T\) ?G lying in \(\tilde C^{o,\mu } \) . We derive the same results for the global solution constructed by Grauert and Lieb, and similar estimates on complex manifolds.     

The CF table

Letf be a continuous function on the circle ¦z¦=1. We present a theory of the (untruncated) “Carathéodory-Fejér (CF) table” of best supremumnorm approximants tof in the classes \(\tilde R_{mn} \) of functions $${{\tilde r(z) = \sum\limits_{k = - \infty }^m {a_k z^k } } \mathord{\left/ {\vphantom {{\tilde r(z) = \sum\limits_{k = - \infty }^m {a_k z^k } } {\sum\limits_{k = 0}^n {b_k } z^k ,}}} \right. \kern-\nulldelimiterspace} {\sum\limits_{k = 0}^n {b_k } z^k ,}}$$ , where the series converges in 1< ¦z¦ <∞. (The casem=n is also associated with the names Adamjan, Arov, and Krein.) Our central result is an equioscillation-type characterization: \(\tilde r \in \tilde R_{mn} \) is the unique CF approximant \(\tilde r^* \) tof if and only if \(f - \tilde r\) has constant modulus and winding numberω≥ m+ n+1?δ on ¦z¦=1, whereδ is the “defect” of \(\tilde r\) . If the Fourier series off converges absolutely, then \(\tilde r^* \) is continuous on ¦z¦=1, andω can be defined in the usual way. For general continuousf, \(\tilde r^* \) may be discontinuous, andω is defined by a radial limit. The characterization theorem implies that the CF table breaks into square blocks of repeated entries, just as in Chebyshev, Padé, and formal Chebyshev-Padé approximation. We state a generalization of these results for weighted CF approximation on a Jordan region, and also show that the CF operator \(K:f \mapsto \tilde r^* \) is continuous atf if and only if (m, n) lies in the upper-right or lower-left corner of its square block.     

Resplendency and recursive definability inω-stable theories

We prove Theorem A.Every resplendent model of an ω-stable theory is homogeneous. As an application we obtain Theorem B.Suppose T is ω-stable, M ? T is recursively saturated and P ∈ S (M) is such that for all finite \(\bar m\) ∈ M, p ↑ \(\bar m\) is realized in M. Then there is a \(\bar c\) ∈ M and a definition d of p over \(\bar c\) such that d is recursive in t ( \(\bar c\) /Ø).     

\mathfrak{E} of topological spaces II

In this paper, we obtain analogues, in the situation of \(\mathfrak{E}\) -extensions, of Magill's theorem on lattices of compactifications. We define an epireflective subcategory of the categoryT 2 of all Hausdorff spaces to be admissive (respectively finitely admissive) if for any \(\mathfrak{E}\) -regular spaceX, every Hausdorff quotient of \(\beta _\mathfrak{E} X\) which is Urysohn on \(\beta _\mathfrak{E} X - X\) (respectively which is finitary on \(\beta _\mathfrak{E} X - X\) ) and which is identity onX, has \(\mathfrak{E}\) . We notice that there are many proper epireflective subcategories ofT 2 containing all compact spaces and which are admissive; there are many such which are not admissive but finitely admissive. We prove that when \(\mathfrak{E}\) is a finitely admissive epireflective subcategory ofT 2, then the lattices of finitary \(\mathfrak{E}\) -extensions of two spacesX andY are isomorphic if and only if \(\beta _\mathfrak{E} X - X\) and \(\beta _\mathfrak{E} Y - Y\) are homeomorphic. Further if \(\mathfrak{E}\) is admissive, then the lattices of Urysohn \(\mathfrak{E}\) -extensions ofX andY are isomorphic if and only if \(\beta _\mathfrak{E} X - X\) and \(\beta _\mathfrak{E} Y - Y\) are homeomorphic.     

Anisotropic function spaces. I: Hardy's inequality,decompositions

В статье рассматрива ются анизотропные пр остранства Бесова \(B_p^{\bar s} \) и Соболева \(W_p^{\bar s} \) н а плоскости и на единич ном круге, где 1<р<∞ и \(1< p< \infty \) И \(\bar s = (s_1 ,s_2 )\) . Основная цель состои т в доказательстве анизотропных нераве нств Харди и в изучени и соответствующих про странств \(\dot B_p^{\bar s} \) и \(\dot W_p^{\bar s} \) типа Бесова—Соболе ва. Эти результаты буд ут использованы во втор ой работе для точного описания следов упом янутых пространств н а плоских кривых.     

Invariant submanifolds and properCR foliations on a para-coKählerian manifold with concircular structure vector field

After recalling the basic properties of para-coKählerian manifolds \(\tilde M\) with concircular structure vector field ξ, the infinitesimal auto morphismsX of the structure 1-form \(\tilde \eta \) are considered. One of the results is that the Lie derivative of all powers of the structure 2-form \(\tilde \Omega ,\) i.e. \(\mathcal{L}x\tilde \Omega ^p ;p = 1,...,m,\) is exterior recurrent. Further two types of horizontal distributionsD _n which are normal to ξ. IfD _t (resp.D _n ) is involutive, the corresponding leafM _t (resp.M _n ) is a minimal submanifold of \(\tilde M\) . FurtherM _n is a symplectic submanifold and ξ is an umbilical normal section ofM _n . Finally proper immersion \(M \to \tilde M\) are discussed, whereM is aCR-sub-manifold whose horizontal distribution isD _t . It is shown that the vertical distribution is involutive, and the restriction of ξ toM is an symptotic direction. Some interesting special cases are treated.     

L 2 existence theorems for the\bar \partial _b - Neumann problem on strongly pseudoconvex CR manifoldsproblem on strongly pseudoconvex CR manifolds

LetM be the boundary of a strongly pseudoconvex domain in \(\mathbb{C}^n \) ,n≥4 and ω be an open subset inM such that ?ω is the intersection ofM with a flat hypersurface. We establish theL ² existence theorems of the \(\bar \partial _b - Neumann\) problem on ω. In particular, we prove that the \(\bar \partial _b - Laplacian\) \(\square _b = \bar \partial _b \bar \partial _b^* + \bar \partial _b^* \bar \partial _b \) equipped with a pair of natural boundary conditions, the so-called \(\bar \partial _b - Neumann\) boundary conditions, has closed range when it acts on (0,q) forms, 1≤q≤n?3. Thus there exists a bounded inverse operator for \(\square _b \) , the \(\bar \partial _b - Neumann\) operatorN _b, and we have the following Hodge decomposition theorem on ω for \(\bar \partial _b \bar \partial _b^* N_b \alpha + \bar \partial _b^* \bar \partial _b N_b \alpha \) , for any (0,q) form α withL ²(ω) coefficients. The proof depends on theL ^p regularity of the tangential Cauchy-Riemann operators \(\bar \partial _b u = \alpha \) on ω?M under the compatibility condition \(\bar \partial _b \alpha = 0\) , where α is a (p, q) form on ω, where 1≤q≤n?2. The interior regularity ofN _b follows from the fact that \(\square _b \) is subelliptic in the interior of ω. The operatorN _b induces natural questions on the regularity up to the boundary ?ω. Near the characteristic point of the boundary, certain compatibility conditions will be present. In fact, one can show thatN _b is not a compact operator onL ²(ω).     

Recovery of a function from the coefficients of its Dirichlet series

Let L(λ) be an entire function of exponential type, letγ(t) be the function associated with L(λ) in the sense of Borel, let \(\bar D\) be the smallest closed convex set containing all the singular points ofγ(t), let λ₀, λ₁, ..., λ_n, ... be the simple zeros of L(λ), and let A \(\bar D\) be the space of functions analytic on \(\bar D\) with the topology of the inductive limit. With an arbitraryf (z) ∈ A( \(\bar D\) ) we can associate the series whereC is a closed contour containing \(\bar D\) , on and inside of whichf (z) is analytic. We give a method of recoveringf (z) from the Dirichlet coefficientsa _n.     

Elliptische Differentialgleichungen mit variablen Koeffizienten in Gebieten mit unbeschränktem Rand

We discuss the spectrum of a symmetric elliptic differential operator A with domain \(\mathop {H^m }\limits^o (\Omega ) \cap H^{2m} (\Omega )\) in regions Ω with unbounded boundary \(\dot \Omega \) , where are \(\bar \Omega \) uniformely of class C^2m and on \(\dot \Omega \) the normal condition x·ν(x)≦μ for sufficient small positiveμ. We prove the A-priori-estimate \(\parallel u\parallel _{m,\Omega } \leqq c\parallel (l + r) (A - k)u\parallel _{o,\Omega } \) and show for all k>k, k≧0 suitable, there are no eigenvalues of A and by characterizing weighted Sobolev spaces with negative norm the existence of solutions \((l + r)_2 ^{ - 1} u \in \mathop H\limits^0{^m} (\Omega ) \cap H^{2m} (\Omega )\) of the equation (A?k)u=f, (1+r)f∈L²(Ω).     

A method of multipliers for mathematical programming problems with equality and inequality constraints

This paper deals with the numerical solution of the general mathematical programming problem of minimizing a scalar functionf(x) subject to the vector constraints φ(x)=0 and ψ(x)≥0. The approach used is an extension of the Hestenes method of multipliers, which deals with the equality constraints only. The above problem is replaced by a sequence of problems of minimizing the augmented penalty function Ω(x, λ, μ,k)=f(x)+λ ^T φ(x)+kφ ^T (x)φ(x) ?μ ^T \(\tilde \psi \) (x)+k \(\tilde \psi \) ^T (x) \(\tilde \psi \) (x). The vectors λ and μ, μ ≥ 0, are respectively the Lagrange multipliers for φ(x) and \(\tilde \psi \) (x), and the elements of \(\tilde \psi \) (x) are defined by \(\tilde \psi \) _(j)(x)=min[ψ_(j)(x), (1/2k) μ_(j)]. The scalark>0 is the penalty constant, held fixed throughout the algorithm. Rules are given for updating the multipliers for each minimization cycle. Justification is given for trusting that the sequence of minimizing points will converge to the solution point of the original problem.     

Positive\partial \bar \partial - closed currents and non-Kähler geometrycurrents and non-Kähler geometry

In this paper some new results on positive \(\partial \bar \partial - closed\) currents are applied to modifications \(f:\bar M \to M\) . The main result in this topic is that every smooth proper modification of a compact Kähler manifoldM is balanced. Moreover, under suitable hypotheses on the map, the Kähler degrees of \(\bar M\) corresponds to homological properties of the exceptional set of the modification. More examples ofp-Kähler manifolds are discussed in the last section of the paper.     

Finite separable field extensions with prescribed extensions of valuations. II

Let A¹,...,A^k be pairwise independent valuation rings of K. Prescribing extensions Δ _i ^j . of the value group Γ^j and extensions \(\mathfrak{L}_i^j\) of the residue field \(H^j\) of A^j (i=1,...,r^j) such that \(\sum\limits_{i = 1}^{r^j } {(\Delta _i^j :\Gamma ^j )} \cdot [\mathfrak{L}_i^j :H^j ] = n\) , we provide sufficient conditions for the existence of a separable field extension L of K of degree n with exactly r^j pairwise independent valuation rings B _i ^j lying over A^j, which have Δ _i ^j as value groups and \(\mathfrak{L}_i^j\) as residue fields.     

Strong approximation of empirical process with independent but non-identically distributed random variables

Let{Y_t,t=1,2,…} be independent random variables with continuous distribution functionsF_i(y).For any y,dencte s=F_t(y)=1/t sum from i=1 to t F_i(y).The empirical process is defind by t~(-1/2)R(s,t) whereR(s,t)=t(1/t sum from i=1 to t I_((?)_t(Y_i)≤s)-s)=sum from i=1 to t I_(?)-ts=sum from i=1 to t I_(?)-(?)_t(y)=sum from i=1 to t I_(Y_(?)≤y)-sum from i=1 to t F_i(y).The purpose of this paper is to investigate the asymptotic properties of the empirical processR(s,t).We shall prove that for some integer sequence {t_k},there is a (?)-process (?)(s,t) such that(?)|R(s,t_k)-(?)(s,t_k)|=O(t_k~(1/2)(log t_k)~(-1/4)(log log t_k)~(1/2))a.s.where (?)(s,t) is a two-parameter Gaussian process defined in §1.     

Die metrisierbaren linearen Teilräume des Raumes\mathfrak{O}_M von L. Schwartz

In Schwartz' terminology, a real or complex valued functionf, defined and infinitely differentiable on ? _n , belongs to \(\mathfrak{O}_M \) iff, as well as any of its derivatives, is at most of polynomial growth. The topology of \(\mathfrak{O}_M \) is defined by the seminorms sup{∣?(x)D ^p f(x)∣;x∈? ⁿ }, where ? belongs to \(\mathfrak{S}\) andD ^p is any derivative. It is well-known that \(\mathfrak{O}_M \) is non-metrisable. For any μ: ? ⁿ →?, let \(\mathfrak{B}_\mu \) be the space of all infinitely differentiable functionsf satisfying, for eachp, sup{∣(1+∣x∣²)^?μ(p) D ^p f(x)∣;x∈? ⁿ }<∞, with the obvious topology. These spaces, which are of very little use elsewhere in the theory of distributions, can be conveniently applied to characterise the metrisable linear subspaces of \(\mathfrak{O}_M \) : A linear subspace of \(\mathfrak{O}_M \) is metrisable if and only if it is, algebraically and topologically, a subspace of some \(\mathfrak{B}_\mu \) .     

Analytische Fortsetzbarkeit von zufälligen Potenzreihen bezüglich abhängiger Zufallsvariabler

Рассматриваются слу чайная величина \(\mathfrak{X} = (X_n (\omega ))\) , удовлетворяющая усл овиюE(X _n ⁴ )≦M, и соответствующ ий случайный степенн ой ряд \(f_x (z;\omega ) = \mathop \sum \limits_{n = 0}^\infty a_n X_n (\omega )z^n\) . Устанавливаются тео ремы непродолжимост и почти наверное:

1. дляf _x при условиях с лабой мультипликати вности на \(\mathfrak{X}\) ,
2. для \(f_{\tilde x}\) , где \(\mathop \mathfrak{X}\limits^ \sim = (\mathop X\limits^ \sim _n )\) есть подп оследовательность в \(\mathfrak{X}\) ,
3. для по крайней мере од ного из рядовf _x′ илиf _x″ , где \(\mathfrak{X}'\) и \(\mathfrak{X}''\) — некоторые п ерестановки \(\mathfrak{X}\) , выбираемые универс ально, т. е. независимо от коэффициентовa _n .

     

Exact and asymptotic estimates forn-widths of some classes of periodic functions

Let \(\tilde W_p^r : = \left\{ {f\left| {f \in C^{r - 1} } \right.} \right.\left[ {0,2\pi } \right],f^{(i)} (0) = f^{(i)} (2\pi ),i = 0, \ldots ,r - 1,f^{(r - 1)}\) , abs. cont. on [0, 2π] andf ^(r)∈L ^p[0, 2π]}, and set \(\tilde B_p^r : = \left\{ {f\left| {f \in \tilde W_p^r ,} \right.\left\| {f^{(r)} } \right\|_p \leqslant 1} \right\}\) . We find the exact Kolmogrov, Gel'fand, and linearn-widths of \(\tilde B_p^r\) inL ^p forn even and allp∈(1, ∞). The strong asymptotic estimates forn-widths of \(\tilde B_p^r\) inL ^p are also obtained.     

Einige aussagen über die reduktion kohärenter analytischer modulgarben

Let (X, ) be a complex space and \(\mathfrak{F}\) a coherent -module. In analogy to the reduction red one can define a reduction \(\mathfrak{F}\) _red= \(\mathfrak{F}\) / \(\mathfrak{F}\) ′, where \(\mathfrak{F}\) ′ ? \(\mathfrak{F}\) is the subsheaf of “nilvalent” elements of \(\mathfrak{F}\) . (Even if X is reduced, we may have \(\mathfrak{F}\) ′ ≠ 0.) We prove that \(\mathfrak{F}\) ′ is coherent. Therefore we can construct the sheaf \(\mathfrak{F}\) (²)=( \(\mathfrak{F}\) ′)′ of nilvalent elements with respect to \(\mathfrak{F}\) ′. Iterating this process, we get a sequence ( \(\mathfrak{F}\) ⁽ⁿ⁾)n∈N of subsheaves of \(\mathfrak{F}\) . We show that on every compact subset of X the sheaves \(\mathfrak{F}\) (ⁿ) vanish for n sufficiently large (Satz 2).