A new coordinate of teichmüller space

By using the theory of quadratic differentials, we give a new coordinate to the Teichmüller space as well as the trajectory structures of a special class of Jenkins-Strebel quadratic differentials.     

Kreĭn space representation and Lorentz groups of analytic Hilbert modules

This paper aims to introduce some new ideas into the study of submodules in Hilbert spaces of analytic functions. The effort is laid out in the Hardy space over the bidisk H²(D²). A closed subspace M in H²(D²) is called a submodule if z _i M ? M (i = 1, 2). An associated integral operator (defect operator) C _M captures much information about M. Using a Kre?n space indefinite metric on the range of C _M , this paper gives a representation of M. Then it studies the group (called Lorentz group) of isometric self-maps of M with respect to the indefinite metric, and in finite rank case shows that the Lorentz group is a complete invariant for congruence relation. Furthermore, the Lorentz group contains an interesting abelian subgroup (called little Lorentz group) which turns out to be a finer invariant for M.     

An analog of the Dougall formula and of the de Branges–Wilson integral

The Ramanujan Journal - We derive a beta-integral over $${mathbb {Z}}times {mathbb {R}}$$ , which is a counterpart of the Dougall $$_5H_5$$ -formula and of the de Branges–Wilson integral,...     

A class of normal weighted composition operators on the Fock space of ℂ n

This paper characterizes a class of normal weighted composition operators and their spectrum on the Fock space of Cⁿ.     

A proof of uniqueness of the Gurariĭ space

We present a short and elementary proof of isometric uniqueness of the Gurari? space.     

Representations of Unitary Relations Between Kreĭn Spaces

The structure of unitary relations between Kreĭn spaces is investigated in geometrical terms. Two approaches are presented: The first approach relies on the so-called Weyl identity and the second approach is based on a graph decomposition of unitary relations. As a consequence of these investigations a quasi-block and a proper block representation of unitary operators are established. Both approaches yield also several new necessary and sufficient conditions for isometric relations to be unitary.     

A determinantal version of the frobenius-könig theorem

Let F be a field and let {d ₁,…,d_k } be a set of independent indeterminates over F. Let A(d ₁,…,d_k ) be an n × n matrix each of whose entries is an element of F or a sum of an element of F and one of the indeterminates in {d ₁,…,d_k }. We assume that no d ¹ appears twice in A(d ₁,…,d_k ). We show that if det A(d ₁,…,d_k ) = 0 then A(d ₁,…,d_k ) must contain an r × s submatrix B, with entries in F, so that r + s = n + p and rank B ? p ? 1: for some positive integer p.     

A weak version of the Lipman–Zariski conjecture

Markushevich and Tikhomirov provided a construction of an irreducible symplectic V-manifold of dimension 4, the relative compactified Prym variety of a family of curves with involution, which is a Lagrangian fibration with polarization of type (1,2). We give a characterization of the dual Lagrangian fibration. We also identify the moduli space of Lagrangian fibrations of this type and show that the duality defines a rational involution on it.     

A constructive version of the Sylvester–Gallai theorem

The Sylvester–Gallai Theorem, stated as a problem by James Joseph Sylvester in 1893, asserts that for any finite, noncollinear set of points on a plane, there exists a line passing through exactly two points of the set. First, it is shown that for the real plane \({{\mathbb{R}^{2}}}\) the theorem is constructively invalid. Then, a well-known classical proof is examined from a constructive standpoint, locating the nonconstructivities. Finally, a constructive version of the theorem is established for the plane \({{\mathbb{R}^{2}}}\); this reveals the hidden constructive content of the classical theorem. The constructive methods used are those proposed by Errett Bishop.     

Index formulae for subspaces of Kreîn spaces

For a subspaceS of a Kreîn spaceK and an arbitrary fundamental decompositionK=K ^?[+]K ⁺ ofK, we prove the index formula $$\kappa ^ - \left( \mathcal{S} \right) + \dim \left( {\mathcal{S}^ \bot \cap \mathcal{K}^ + } \right) = \kappa ^ + \left( {\mathcal{S}^ \bot } \right) + \dim \left( {\mathcal{S} \cap \mathcal{K}^ - } \right)$$ where κ^±(S) stands for the positive/negative signature ofS. The difference dim(S∩K ^?)?dim(S ^⊥∩K ⁺), provided it is well defined, is called the index ofS. The formula turns out to unify other known index formulac for operators or subspaces in a Kreîn space.     

A version of the Jensen–Johnsbråten coding at arbitrary level n≥ 3

We generalize, on higher projective levels, a construction of “incompatible” generic Δ¹ ₃ real singletons given by Jensen and Johnsbr?ten. Received: 3 November 1998 / Revised version: 23 April 2000 / Published online: 3 October 2001     

A central-limit-theorem version of the periodic Little’s law

Queueing Systems - We establish a central-limit-theorem (CLT) version of the periodic Little’s law (PLL) in discrete time, which complements the sample-path and stationary versions of the PLL...     

Quotients of the unit ball of ℂn for a free action of ℤ

LetB _n be the unit ball of ℂⁿ and ℤ ≅ Γ ⊂ AutB _n be generated by a parabolic element of AutB _n. We show that the quotientB _n/Γ is biholomorphic to a holomorphically convex domain of ℂⁿ, whose automorphism group is explicity described. It follows thatB _n/ℤ is Stein for any free action of ℤ. Investigation partially supported by University of Bologna. Funds for selected research topics. The second author was supported by an Instituto Nazionale di Alta Matematica grant.     

On the measurable chromatic number of a space of dimension n ≤ 24

This paper is devoted to the classical problem of finding the measurable chromatic number of n-dimensional Euclidean space, i.e., the value χ _m (? ⁿ ) equal to the least possible number of Lebesgue measurable sets that do not contain pairs of points at a distance of 1 and cover the whole space. Assuming that a certain hypothesis is true, we significantly improve the lower bounds for χ _m (? ⁿ ).     

Parameterization of the Extrapolations in the Kreĭn–Schwartz Theorem and the Entropy Maximizer: the Scalar Case

A Kre?n-de Branges-Kotani space $\mathbb{H }$ is associated to a given positive-definite distribution $Q$ on the finite interval $(-a,a)$ of the real line. The Kre?n-de Branges Theorem is applied to get a spectral measure of $\mathbb{H }$ . In this way a simple proof of the solubility of the Kre?n’s extension problem related to $Q$ (Kre?n–Schwartz Theorem) is obtained. A condition that guarantees the uniqueness of the extrapolation as well as a parameterization of the extrapolations by means of Schur functions are given. The choice of the Schur function $\eta \equiv 0$ in the parameterization is shown to produce an absolutely continuous measure which maximizes Burg’s entropy. The results are based on the coupling of the Arov–Grossman functional model with a Hilbert space operator built up from the multiplication operator on $\mathbb{H }$ .