Continuous automorphisms of Cremona groups

We show that if a group automorphism of a Cremona group of arbitrary rank is also a homeomorphism with respect to either the Zariski or the Euclidean topology, then it is inner up to a field automorphism of the base-field. Moreover, we show that a similar result holds if we consider groups of polynomial automorphisms of affine spaces instead of Cremona groups.

The order on the light cone and its induced topology

In this paper, we first correct a recent misconception about a topology that was suggested by Zeeman as a possible alternative to his fine topology. This misconception appeared while trying to establish the causality in the ambient boundary-ambient space cosmological model. We then show that this topology is actually the intersection topology (in the sense of Reed [The intersection topology w.r.t. the real line and the countable ordinals, Trans. Am. Math. Soc. 297(2) (1986) 509–520]) between the Euclidean topology on [Formula: see text] and the order topology whose order, namely horismos, is defined on the light cone and we show that the order topology from horismos belongs to the class of Zeeman topologies. These results accelerate the need for a deeper and more systematic study of the global topological properties of spacetime manifolds.

Finite dimensional locally convex cones

We study the locally convex cones which have finite dimension. We introduce the Euclidean convex quasiuniform structure on a finite dimensional cone. In special case of finite dimensional locally convex topological vector spaces, the symmetric topology induced by the Euclidean convex quasiuniform structure reduces to the known concept of Euclidean topology. We prove that the dual of a finite dimensional cone endowed with the Euclidean convex quasiuniform structure is identical with it?s algebraic dual.

An Approach to Classical Quantum Field Theory Based on the Geometry of Locally Conformally Flat Space-Time

This paper gives an introduction to certain classical physical theories described in the context of locally Minkowskian causal structures (LMCSs). For simplicity of exposition we consider LMCSs which have locally Euclidean topology (i.e., are manifolds) and hence are Möbius structures. We describe natural principal bundle structures associated with Möbius structures. Fermion fields are associated with sections of vector bundles associated with the principal bundles while interaction fields (bosons) are associated with endomorphisms of the space of fermion fields. Classical quantum field theory (the Dirac equation and Maxwell’s equations) is obtained by considering representations of the structure group K⊂SU(2,2) of a principal bundle associated with a given Möbius structure where K, while being a subset of SU(2,2), is also isomorphic to SL2,C×U(1). The analysis requires the use of an intertwining operator between the action of K on R4 and the adjoint action of K on su⁡(2,2) and it is shown that the Feynman slash operator, in the chiral representation for the Dirac gamma matrices, has this intertwining property.

CASIMIR EFFECT UNDER QUASI-PERIODIC BOUNDARY CONDITION INSPIRED BY NANOTUBES

When one studies the Casimir effect, the periodic (anti-periodic) boundary condition is usually taken to mimic a periodic (anti-periodic) structure for a scalar field living in a flat space with a non-Euclidean topology. However, there could be an arbitrary phase difference between the value of the scalar field on one endpoint of the unit structure and that on the other endpoint, such as the structure of nanotubes. Then, in this paper, a periodic condition on the ends of the system with an additional phase factor, which is called the "quasi-periodic" condition, is imposed to investigate the corresponding Casimir effect. And an attractive or repulsive Casimir force is found, whose properties depend on the phase angle value. Especially, the Casimir effect disappears when the phase angle takes a particular value. High dimensional spacetime case is also investigated.

Approximately Rationally or Elliptically Connected Varieties

We discuss a possible approach to the study of the vanishing of the Kobayashi pseudo-metric of a projective variety X, using chains of rational or elliptic curves contained in an arbitrarily small neighbourhood of X in projective space for the Euclidean topology.

A Study of Non-Euclidean s-Topology

The present paper focuses on the characterization of compact sets of Minkowski space with a non-Euclidean -topology which is defined in terms of Lorentz metric. As an application of this study, it is proved that the 2-dimensional Minkowski space with -topology is not simply connected. Also, it is obtained that the -dimensional Minkowski space with -topology is separable, first countable, path-connected, nonregular, nonmetrizable, nonsecond countable, noncompact, and non-Lindelöf.

Ideas from Zariski Topology in the Study of Cubical Homology

Cubical sets and their homology have been used in dynamical systems as well as in digital imaging. We take a fresh look at this topic, following Zariski ideas from algebraic geometry. The cubical topology is defined to be a topology in ℝd in which a set is closed if and only if it is cubical. This concept is a convenient frame for describing a variety of important features of cubical sets. Separation axioms which, in general, are not satisfied here, characterize exactly those pairs of points which we want to distinguish. The noetherian property guarantees the correctness of the algorithms. Moreover, maps between cubical sets which are continuous and closed with respect to the cubical topology are precisely those for whom the homology map can be defined and computed without grid subdivisions. A combinatorial version of the Vietoris–Begle theorem is derived. This theorem plays the central role in an algorithm computing homology of maps which are continuous with respect to the Euclidean topology.