Generalized Einstein conditions on holomorphic Riemannian manifolds

Summary At first, a necessary and sufficient condition for a K?hler-Norden manifold to be holomorphic Einstein is found. Next, it is shown that the so-called (real) generalized Einstein conditions for K?hler-Norden manifolds are not essential since the scalarcurvature of such manifolds is constant. In this context, we study generalized holomorphic Einstein conditions. Using the one-to-one correspondence between K?hler-Norden structures and holomorphic Riemannian metrics, we establish necessary and sufficient conditions for K?hler-Norden manifolds to satisfy the generalized holomorphic Einstein conditions. And a class of new examples of such manifolds is presented. Finally, in virtue of the obtained results, we mention that Theorems 1 and 2 of H. Kim and J. Kim [10] are not true in general.     

Cardinality of the set of extreme extensions¶of a quasi-measure

Cardinals that arise as the number of extreme quasi-measure extensions of a quasi-measure [resp., measure] μ defined on an algebra [resp., σ-algebra] of sets to a larger algebra [resp., σ-algebra] of sets are characterized in the general case as well as under some natural assumptions on μ. Received: 19 September 2000     

Product numerical range in a space with tensor product structure

We study operators acting on a tensor product Hilbert space and investigate their product numerical range, product numerical radius and separable numerical range. Concrete bounds for the product numerical range for Hermitian operators are derived. Product numerical range of a non-Hermitian operator forms a subset of the standard numerical range containing the barycenter of the spectrum. While the latter set is convex, the product range needs not to be convex nor simply connected. The product numerical range of a tensor product is equal to the Minkowski product of numerical ranges of individual factors.     

Estimating a largest eigenvector by Lanczos and polynomial algorithms with a random start

We study Lanczos and polynomial algorithms with random start for estimating an eigenvector corresponding to the largest eigenvalue of an n × n large symmetric positive definite matrix. We analyze the two error criteria: the randomized error and the randomized residual error. For the randomized error, we prove that it is not possible to get distribution-free bounds, i.e., the bounds must depend on the distribution of eigenvalues of the matrix. We supply such bounds and show that they depend on the ratio of the two largest eigenvalues. For the randomized residual error, distribution-free bounds exist and are provided in the paper. We also provide asymptotic bounds, as well as bounds depending on the ratio of the two largest eigenvalues. The bounds for the Lanczos algorithm may be helpful in a practical implementation and termination of this algorithm. © 1998 John Wiley & Sons, Ltd.     

Chromatographic and spectroscopic investigation of the associative changes with the selected higher fatty alcohols

An attempt was made to compare usefulness of the ir spectroscopic technique with paper chromatography in the investigations concerning self-association of higher fatty alcohols. A conclusion was drawn that only the complementary application of both analytical methods better enabled to trace the establishing of association equilibria in the given experimental conditions.     

Reactions of the N,N-dialkylpyridylcarboxylic amides with lithium amides. Regioselective lithiation of N,N-diisopropylpyridylcarboxylic amides,a useful method for synthesis of 2,3- and 3,4-disubstituted pyridines

The behaviour of the N,N-dimethyl-, N,N-diethyl- and N,N-diisopropylpyridylcarboxylic amides in the reactions with Et₂NLi and iPr₂NLi, and a convenient way of synthesis of 2,3- and 3,4-disubstituted pyridines, by the direct lithiation of N,N-diisopropylamides leading subsequently to the ortho new CC bonds formation, are described.     

Immersions of spheres and algebraically constructible functions

Let Λ be an algebraic set and let (n is even) be a polynomial mapping such that for each there is r(λ) > 0 such that the mapping g _λ  =  g(· , λ) restricted to the sphere S ⁿ (r) is an immersion for every 0  <  r  <  r (λ), so that the intersection number I(g _λ|S ⁿ (r)) is defined. Then is an algebraically constructible function. I. Karolkiewicz and A. Nowel supported by the grant BW/5100-5-0286-7.     

On the projectivity of threefolds

Let be a smooth complete three-dimensional algebraic variety (defined over an algebraically closed field ). We show that is projective if it contains a divisor which is positive on the cone of effective curves.

     

On the definition of the Monge-Ampère operator in ℂ<Superscript>2</Superscript>

We show that if u is a plurisubharmonic function defined on an open subset of ² then the Monge-Ampère measure (dd^cu)² can be well defined if and only if u belongs to the Sobolev space W^1,2_loc().Partially supported by KBN Grant #2 P03A 028 19     

Linear Automorphisms that are Symplectomorphisms

Let K be the field of real or complex numbers. Let (X K²ⁿ,) be a symplectic vector space and take 0 < k < n,N =. Let L₁,...,L_N X be 2k-dimensionallinear subspaces which are in a sufficiently general position.It is shown that if F : X X is a linear automorphism whichpreserves the form ^k on all subspaces L₁,...,L_N, then F is an_k-symplectomorphism (that is, F^* = _k, where ). In particular, if K = R and k is odd then F mustbe a symplectomorphism. The unitary version of this theoremis proved as well. It is also observed that the set A_l,2r ofall l-dimensional linear subspaces on which the form has rank 2r is linear in the Grassmannian G(l,2n), that is, there isa linear subspace L such that A_l,2r = L G(l, 2n). In particular,the set A_l,2r can be computed effectively. Finally, the notionof symplectic volume is introduced and it is proved that itis another strong invariant.