65,807 research outputs found
Discrete space-time geometry and skeleton conception of particle dynamics
It is shown that properties of a discrete space-time geometry distinguish
from properties of the Riemannian space-time geometry. The discrete geometry is
a physical geometry, which is described completely by the world function. The
discrete geometry is nonaxiomatizable and multivariant. The equivalence
relation is intransitive in the discrete geometry. The particles are described
by world chains (broken lines with finite length of links), because in the
discrete space-time geometry there are no infinitesimal lengths. Motion of
particles is stochastic, and statistical description of them leads to the
Schr\"{o}dinger equation, if the elementary length of the discrete geometry
depends on the quantum constant in a proper way.Comment: 22 pages, 0 figure
Integrable discrete nets in Grassmannians
We consider discrete nets in Grassmannians which generalize
Q-nets (maps with planar elementary
quadrilaterals) and Darboux nets (-valued maps defined on the
edges of such that quadruples of points corresponding to
elementary squares are all collinear). We give a geometric proof of
integrability (multidimensional consistency) of these novel nets, and show that
they are analytically described by the noncommutative discrete Darboux system.Comment: 10 p
On Jordan's measurements
The Jordan measure, the Jordan curve theorem, as well as the other generic
references to Camille Jordan's (1838-1922) achievements highlight that the
latter can hardly be reduced to the "great algebraist" whose masterpiece, the
Trait\'e des substitutions et des equations alg\'ebriques, unfolded the
group-theoretical content of \'Evariste Galois's work. The present paper
appeals to the database of the reviews of the Jahrbuch \"uber die Fortschritte
der Mathematik (1868-1942) for providing an overview of Jordan's works. On the
one hand, we shall especially investigate the collective dimensions in which
Jordan himself inscribed his works (1860-1922). On the other hand, we shall
address the issue of the collectives in which Jordan's works have circulated
(1860-1940). Moreover, the time-period during which Jordan has been publishing
his works, i.e., 1860-1922, provides an opportunity to investigate some
collective organizations of knowledge that pre-existed the development of
object-oriented disciplines such as group theory (Jordan-H\"older theorem),
linear algebra (Jordan's canonical form), topology (Jordan's curve), integral
theory (Jordan's measure), etc. At the time when Jordan was defending his
thesis in 1860, it was common to appeal to transversal organizations of
knowledge, such as what the latter designated as the "theory of order." When
Jordan died in 1922, it was however more and more common to point to
object-oriented disciplines as identifying both a corpus of specialized
knowledge and the institutionalized practices of transmissions of a group of
professional specialists
On a class of integrable systems of Monge-Amp\`ere type
We investigate a class of multi-dimensional two-component systems of
Monge-Amp\`ere type that can be viewed as generalisations of heavenly-type
equations appearing in self-dual Ricci-flat geometry. Based on the
Jordan-Kronecker theory of skew-symmetric matrix pencils, a classification of
normal forms of such systems is obtained. All two-component systems of
Monge-Amp\`ere type turn out to be integrable, and can be represented as the
commutativity conditions of parameter-dependent vector fields. Geometrically,
systems of Monge-Amp\`ere type are associated with linear sections of the
Grassmannians. This leads to an invariant differential-geometric
characterisation of the Monge-Amp\`ere property.Comment: arXiv admin note: text overlap with arXiv:1503.0227
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