Remarks on tree-level topological string theories

A few observations concerning topological string theories at the string-tree level are presented: (1) The tree-level, large phase space solution of an arbitrary model is expressed in terms of a variational problem, with an ``action'' equal, at the solution, to the one-point function of the puncture operator, and found by solving equations of Gauss-Manin type; (2) For $A_k$ Landau-Ginzburg models, an extension to large phase space of the usual residue formula for three-point functions is given.Comment: LaTeX2e, 8 page

Dense definiteness and boundedness of composition operators in L2L^2-spaces via inductive limits

The questions of dense definiteness and boundedness of composition operators in $L^2$-spaces are studied by means of inductive limits of operators. Methods based on projective systems of measure spaces and inductive limits of $L^2$-spaces are developed. Illustrative examples are presented

Quantum Spaces: Notes and Comments on a Lecture by S. L. Woronowicz

These notes are based on a lecture given by S. L. Woronowicz at the Institute of Mathematics, Polish Academy of Sciences.Comment: 11 pages, LaTe

Quantum Principal Fiber Bundles: Topological Aspects

We introduce the notion of locally trivial quantum principal bundles. The base space and total space are compact quantum spaces (unital $C^{\star}$-algebras), the structure group is a compact matrix quantum group. We prove that a quantum bundle admits sections if and only if it is trivial. Using a quantum version of \v{C}ech cocycles, we obtain a reconstruction theorem for quantum principal bundles. The classification of bundles over a given quantum space as a base space is reduced to the corresponding problem, but with an ordinary classical group playing the role of structure group. Some explicit examples are considered.Comment: 29 pages, LaTeX, IM PAN preprint # 51

New properties of Cauchy and event horizons

We present several recent results concerning Cauchy and event horizons. In the first part of the paper we review the differentiablity properties of the Cauchy and the event horizons. In the second part we discuss compact Cauchy horizons and summarize their main properties.Comment: 11 pages, Talk at 2nd World Congress on Non-linear Analysis (July 2000, Catania, Italy

Unbounded composition operators via inductive limits: cosubnormal operators with matrix symbols. II

The paper deals with unbounded composition operators with infinite matrix symbols acting in $L^2$-spaces with respect to the gaussian measure on $\mathbb{R}^\infty$. We introduce weak cohyponormality classes \EuScript{S}_{n,r}^* of unbounded operators and provide criteria for the aforementioned composition operators to belong to \EuScript{S}_{n,r}^*. Our approach is based on inductive limits of operators

Unbounded composition operators via inductive limits: cosubnormals with matrical symbols

We prove, by use of inductive techniques, that assorted unbounded composition operators in $L^2$-spaces with matrical symbols are cosubnormal.Comment: 8 page

Quasinormal extensions of subnormal operator-weighted composition operators in ℓ2\ell^2-spaces

We prove the subnormality of an operator-weighted composition operator whose symbol is a transformation of a discrete measure space and weights are multiplication operators in $L^2$-spaces under the assumption of existence of a family of probability measures whose Radon-Nikodym derivatives behave regular along the trajectories of the symbol. We build the quasinormal extension which is a weighted composition operator induced by the same symbol. We give auxiliary results concerning commutativity of operator-weighted composition operators with multiplication operators

Generalized strong curvature singularities and weak cosmic censorship in cosmological space-times

This paper is a further development of the approach to weak cosmic censorship proposed by the authors in Ref. 5. We state and prove a modified version of that work's main result under significantly relaxed assumptions on the asymptotic structure of space--time. The result, which imposes strong constraints on the occurrence of naked singularities of the strong curvature type, is in particular applicable to physically realistic cosmological models.Comment: Latex, 7 pages, no figures, to be published in Mod. Phys. Lett.

A note on kk-hyperreflexivity of Toeplitz-harmonic subspaces

The 2-hyperreflexivity of Toeplitz-harmonic type subspace generated by an isometry or a quasinormal operator is shown. The $k$-hyperreflexivity of the tensor product $\mathcal{S}\otimes \mathcal{V}$ of a $k$-hyperreflexive decomposable subspace $\mathcal{S}$ and an abelian von Neumann algebra $\mathcal{V}$ is established