1,258 research outputs found
Pregeometric Concepts on Graphs and Cellular Networks as Possible Models of Space-Time at the Planck-Scale
Starting from the working hypothesis that both physics and the corresponding
mathematics have to be described by means of discrete concepts on the
Planck-scale, one of the many problems one has to face is to find the discrete
protoforms of the building blocks of continuum physics and mathematics. In the
following we embark on developing such concepts for irregular structures like
(large) graphs or networks which are intended to emulate (some of) the generic
properties of the presumed combinatorial substratum from which continuum
physics is assumed to emerge as a coarse grained and secondary model theory. We
briefly indicate how various concepts of discrete (functional) analysis and
geometry can be naturally constructed within this framework, leaving a larger
portion of the paper to the systematic developement of dimensional concepts and
their properties, which may have a possible bearing on various branches of
modern physics beyond quantum gravity.Comment: 16 pages, Invited paper to appear in the special issue of the Journal
of Chaos, Solitons and Fractals on: "Superstrings, M, F, S ... Theory" (M.S.
El Naschie, C. Castro, Editors
On the lower bound of the inner radius of nodal domains
We discuss the asymptotic lower bound on the inner radius of nodal domains that arise from Laplacian eigenfunctions \varphi _{\lambda} on a closed Riemannian manifold (M, g) . In the real-analytic case, we present an improvement of the currently best-known bounds, due to Mangoubi (Commun Partial Differ Equ 33:1611–1621, 2008; Can Math Bull 51(2):249–260, 2008). Furthermore, using recent results of Hezari (P Am Math Soc, 2016, https://doi.org/10.1090/proc/13766; Anal PDE 11(4):855–871, 2018), we obtain log-type improvements in the case of negative curvature and improved bounds for (M, g) possessing an ergodic geodesic flow
Simplex and Polygon Equations
It is shown that higher Bruhat orders admit a decomposition into a higher
Tamari order, the corresponding dual Tamari order, and a "mixed order." We
describe simplex equations (including the Yang-Baxter equation) as realizations
of higher Bruhat orders. Correspondingly, a family of "polygon equations"
realizes higher Tamari orders. They generalize the well-known pentagon
equation. The structure of simplex and polygon equations is visualized in terms
of deformations of maximal chains in posets forming 1-skeletons of polyhedra.
The decomposition of higher Bruhat orders induces a reduction of the
-simplex equation to the -gon equation, its dual, and a compatibility
equation
Aspects of Algebraic Quantum Theory: a Tribute to Hans Primas
This paper outlines the common ground between the motivations lying behind
Hans Primas' algebraic approach to quantum phenomena and those lying behind
David Bohm's approach which led to his notion of implicate/explicate order.
This connection has been made possible by the recent application of orthogonal
Clifford algebraic techniques to the de Broglie-Bohm approach for relativistic
systems with spin.Comment: 18 pages. No figure
Proof of Swiss Cheese Version of Deligne's Conjecture
For an associative algebra A we consider the pair "the Hochschild cochain
complex C*(A,A) and the algebra A". There is a natural 2-colored operad which
acts on this pair. We show that this operad is quasi-isomorphic to the singular
chain operad of Voronov's Swiss Cheese operad. This statement is the Swiss
Cheese version of the Deligne conjecture formulated by M. Kontsevich in
arXiv:math/9904055.Comment: The article was rewritten. Multiple suggestions of referees were
addressed. The paper is accepted for publication to IMR
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