Nahum Zobin

Noncommutative gauge theory without Lorentz violation

The most popular noncommutative field theories are characterized by a matrix parameter

Extension of smooth functions from finitely connected planar domains

{\ensuremath{\theta}}^{\ensuremath{\mu}\ensuremath{\nu}}

Coxeter groups and interpolation of operators

that violates Lorentz invariance. We consider the simplest algebra in which the

Birkhoff's theorem and convex hulls of Coxeter groups

\ensuremath{\theta}

Failure of the Weierstrass Preparation Theorem in quasi-analytic Denjoy–Carleman rings☆

parameter is promoted to an operator and Lorentz invariance is preserved. This algebra arises through the contraction of a larger one for which explicit representations are already known. We formulate a star product and construct the gauge-invariant Lagrangian for Lorentz-conserving noncommutative QED. Three-photon vertices are absent in the theory, while a four-photon coupling exists and leads to a distinctive phenomenology.

On eigenvalues and boundary curvature of the numerical range

Consider the Sobolev space W∞k(Ω) of functions with bounded kth derivatives defined in a planar domain. We study the problem of extendability of functions from W∞k(Ω) to the whole ℝ2 with preservation of class, i.e., surjectivity of the restriction operator W∞k(ℝ2) → W∞k(Ω).

A general theory of sufficient collections of norms with a prescribed semigroup of contractions

Let V be a finite dimensional real Euclidean space and let G be a finite irreducible group generated by orthogonal reflections across hyperplanes in V. We study interpolation of operators in G-invariant norms on V. A collection of G-invariant norms is called G-sufficient if any G-invariant norm is a strict interpolation norm for this collection. Using the general theory of sufficient collections we calculate explicitly two remarkable minimal sufficient collections and study their extremal properties.

Some remarks on quasi-equivalence of bases in Fréchet spaces

We formulate and partially prove a general conjecture regarding the facial structure of convex hulls of finite irreducible Coxeter groups.

Geometric structure of b 2,2-orbihedra and interpolation of operators

Abstract It is shown that Denjoy–Carleman quasi-analytic rings of germs of functions in two or more variables fail to satisfy the Weierstrass Preparation Theorem. The result is proven via a non-extension theorem.

Linear preservers of isomorphic types of lattices of invariant operator ranges

Abstract Let A be an n×n matrix. By Donoghues theorem, all corner points of its numerical range W(A) belong to the spectrum σ(A) . It is therefore natural to expect that, more generally, the distance from a point p on the boundary ∂ W(A) of W(A) to σ(A) should be in some sense bounded by the radius of curvature of ∂ W(A) at p . We establish some quantitative results in this direction.