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 \(\tilde M\) .     

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.     

On normal control processes

Control processes of the form \(\dot x - A(t) x = B(t) u(t)\) , which are normal with respect to the unit ballB _{p′, r′} of the control spaceL ^p′([τ, T]),l _m ^r ′ are characterized in terms ofH(t)=X(T)X ^?1(t),B(t),X(t) any fundamental matrix solution of \(\dot x - A(t)x = 0\) , and directly in terms ofA, B, when bothA andB are independent oft.     

The extreme bodies in the set of plane convex bodies with a given width function

Let \(\bar K\) (w) denote the class of plane convex bodies having a width functionw. Examining the length measure of the boundary of a convex body in \(\bar K\) (w), a characterization is given for the extreme (indecomposable) bodies in \(\bar K\) (w). This is a generalization of the solution previously given by the author in Israel J. Math. (1974) for the case wherew′ is absolutely continuous.     

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.     

Faserungen,die aus Reguli mit gemeinsamer Berührprojektivität längs einer gemeinsamen Erzeugenden zusammengesetzt sind

Let Π_PP be a pappian projective 3-space and ? be a set of lines of Π_PP; we define:

u

a line g of Π_PP has the property R with respect to ?, if all lines of ? meeting g form a regulus

? has the property E ₃, if there exists a pencil \(\mathfrak{L}_0 \) of lines such that one line z of \(\mathfrak{L}_0 \) belongs to ? and all lines of { \(\mathfrak{L}_0 \backslash \left\{ z \right\}\) have the property R with respect to ?.

A spread with the property E ₃ (abbreviated E ₃-spread) is built up of reguli which have one line in common and the same tangent projectivity along their common line. We point out a method of constructing an E ₃-spread of Π_PP. This construction is applied to the real 3-space ?³ to generalize a result of D. Betten [2, S.327] and to prove that another result of D. Betten [3, S.140, Bsp. 2] yields E ₃-spreads. For each natural number n (∈?) we specify two E ₃-spreads \(\mathfrak{F}_n \) and \(\mathfrak{S}_n \) of ?³ such that two different elements of \(F: = \left\{ {\mathfrak{F}_n |n \in \mathbb{N}} \right\} \cup \left\{ {\mathfrak{S}_n |n \in \mathbb{N}} \right\}\) are not equivalent with respect to the collineation group of ?³ apart from \(\mathfrak{F}_1 \) each spread of F represents a 4-dimensional translation plane with a 6-dimensional collineation group. Finally, the properties R and E ₃ are used to characterize the elliptic linear line congruences of a pappian 3-space.     

Synchronization theory for forced oscillations in second-order systems

We consider differential equations of the form $$\ddot x + \in f(x,\dot x) + x = \in u$$ , where ε >0 is supposed to be small. For piecewise continuous controlsu(t), satisfying |u(t)| ≤ 1, we present sufficient conditions for the existence of 2π-periodic solutions with a given amplitude. We present a method for determining the limiting behavior of controlsū _ε for which the equation has a 2π-periodic solution with a maximum amplitude and for determining the limit of this maximum amplitude as ε tends to zero. The results are applied to the linear system \(\ddot x + \in \dot x + x = \in u\) , the Duffing equation \(\ddot x + \in (x - 1)\dot x + x = \in u\) , and the Van der Pol equation \(\ddot x + \in (x^2 - 1)\dot x + x = \in u\) .     

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 \(\mathfrak{B}_\mu \) .     

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.     

A characterisation of multiplicity

LetR be a ring with non-zero identity and unitary leftR-modules, while \(\mathcal{N}_R \) is the subcategory of NoetherianR-modules. Given a length functionL on \(\mathcal{N}_R \) and central elements α₁,...,α _n ofR we can define the multiplicity length functione _R (L|α₁,...α _n |) on \(\mathcal{N}_R \) with the same properties as the classical multiplicity. Here, we characterise multiplicity as the greatest length function which can be defined inductively in terms of a certain type of function on \(\mathcal{N}_R \) .     

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).     

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.     

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).     

A characterization of the Wishart exponential families by an invariance property

E is the space of real symmetric (d, d) matrices, andS and \(\bar S\) are the subsets ofE of positive definite and semipositive-definite matrices. Let there be ap in $$\Lambda = \left\{ {\frac{1}{2},1,\frac{3}{2}, \ldots \frac{{d - 1}}{2}} \right\} \cup \left] {\frac{{d - 1}}{2}, + \infty } \right[$$ The Wishart natural exponential family with parameterp is a set of probability distributions on \(\bar S\) defined by $$F_p = \{ \exp [ - \tfrac{1}{2}Tr(\Gamma x)](det\Gamma )^p \mu _p (dx);\Gamma \in S\} $$ where μ_p is a suitable measure on \(\bar S\) . LetGL(?^d) be the subset ofE of invertible matrices. Fora inGL(?^d), define the automorphismg _a ofE byg _a(x)=^t axa, where ^t a is the transpose ofa. The aim of this paper is to show that a natural exponential familyF onE is invariant byg _a for alla inGL(?^d) if and only if there existsp in Λ such that eitherF=F _p, orF is the image ofF _p byx??x. (Theorem).     

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].     

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.     

\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.     

Embeddings into power series rings

Let T be an ordered ring without divisors of zero, and letA be the set of archimedean subgroups of T generated by a Banaschewski functionτ. Let_XΠ_Δ R be the power series ring of the real numbers ? over the totally ordered semigroup Δ of archimedean classes of T, and letχ be the usual Banaschewski function on_XΠ_Δ R. The following are equivalent:

1. τ satisfies the additional condition; for convex subgroups P,Q of T, where
2. There exists a one-to-one homomorphism Γ:T→_XΠ_Δ R of ordered rings such that for every convex subgroup Q of_XΠ_Δ R, there exists a convex subgroup P of T such that \(\Gamma (P) \subseteq Q\) and \(\Gamma (\tau (P)) \subseteq \chi (Q)\) .