A consequence of the celebrated Dvoretzky–Milman theorem is a strong law large numbers for d-dimensional random projections n-dimensional cube. It shows that, with respect to Hausdorff distance, uniform projection cube [−1/n,+1/n]n onto Rd converges almost surely centered Euclidean ball radius 2/π, as n→∞. We start by providing an alternative proof this via Artstein–Vitale compact sets. Then, e...

The "Krein" regularization method of quantum field theory is studied, inspired by the Krein space quantization and metric fluctuations. It was previously considered in one-loop approximation, this paper generalized to all orders perturbation theory. We directly recover physical results obtained starting from standard QFT imposing renormalization conditions. By applying our approach curved space...

0. Introduction We work over an algebraically closed eld k of characteristic 0. 0.1. Statement. In this paper, we use techniques of toric geometry to reprove the following theorem: Theorem 0.1. Let X be a projective variety of nite type over k, and let Z X be a proper closed subset. Let G Aut k (Z X) be a nite group. Then there is a G-equivariant modii-cation r : X 1 ! X such that X 1 is nonsin...

We solve the inverse problem from the spectral measure and the inverse three-spectra problem for the class of singular Krein strings on a finite interval with trace class resolvents. In particular, this includes a complete description of all possible spectral measures and three (Dirichlet) spectra associated with this class of Krein strings. The solutions of these inverse problems are obtained ...

In this note we give a Krein space interpretation of the celebrated Kalman-Yakubovich-Popov (KYP) Lemma by introducing state-space models driven by inputs that lie in an indeenite-metric space. Such state-space models can be considered as generalizations of standard stochastic state-space models driven by stationary stochastic processes (that lie in a deenite, or so-called Hilbert, space). In t...

Sign type spectra are an important tool in the investigation of spectral properties of selfadjoint operators in Krein spaces. It is our aim to show that also sign type spectra for normal operators in Krein spaces provide insight in the spectral nature of the operator: If the real part and the imaginary part of a normal operator in a Krein space have real spectra only and if the growth of the re...

A criterion for the numerical range of a linear operator acting in a Krein space to be a two-component hyperbolical disc is given, using the concept of support function. A characterization of the Krein space numerical range as a union of hyperbolical discs is obtained by a reduction to the two-dimensional case. We revisit a famous result of Ando concerning the inclusion relation W (A) ⊆ W (B) o...

We present a simple proof of the Markov–Krein identity for distributions of means of linear functionals of the Dirichlet process and its various generalizations. The key idea is to use the representation of the Dirichlet process as the normalized gamma process and fundamental properties of gamma processes. Bibliography: 19 titles. 1. Introduction: Dirichlet processes, gamma processes and the Ma...