Related papers: Infinitesimal Gunk
This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a…
In what has become a standard eternal inflation picture of the string landscape there are many problematic consequences and a difficulty defining probabilities for the occurrence of each type of universe. One feature in particular that…
In the present paper, we propose a new axiomatic approach to nonstandard analysis and its application to the general theory of spatial structures in terms of category theory. Our framework is based on the idea of internal set theory, while…
We survay some nice result concerning the irrationals with a metric space point of view.Here is ofcourse nothing new may be or an expert in this field.
We present a new kind of nontermination argument, called geometric nontermination argument. The geometric nontermination argument is a finite representation of an infinite execution that has the form of a sum of several geometric series.…
In the present paper we deal with (local) infinitesimal isometries of special sub-Riemannian manifolds (a contact and oriented sub-Riemannian manifold is called special if the Reeb vector field is an isometry). The objective of the paper is…
We prove that there is a residual subset of the Gromov-Hausdorff space (i.e. the space of all compact metric spaces up to isometry endowed with the Gromov-Hausdorff distance) whose points enjoy several unexpected properties. In particular,…
Can there be a structure space-type theory for an arbitrary class of ideals of a ring? The ideal spaces introduced in this paper allows such a study and our theory includes (but not restricted to) prime, maximal, minimal prime, strongly…
Geodesic incompleteness is a problem in both general relativity and string theory. The Weyl invariant Standard Model coupled to General Relativity (SM+GR), and a similar treatment of string theory, are improved theories that are…
We show that the field of complex numbers $\mathbb C$ contains non-zero infinitesimals by observing that $\mathbb C$ contains non-Archimedean subfields. Our observation is based on an old theorem in algebra due to E. Steinitz, discussed in…
We show how to decompose all separable ultrametric spaces into a "Lego" combinations of scaled versions of full simplices. To do this we introduce metric resolutions of large scale metric spaces, which describe how a space can be broken up…
This commentary considers non-standard analysis and a recently introduced computational methodology based on the notion of \G1 (this symbol is called \emph{grossone}). The latter approach was developed with the intention to allow one to…
The mathematical notion of incompleteness (eg of rational numbers, Turing-computable functions, and arithmetic proof) does not play a key role in conventional physics. Here, a reformulation of the kinematics of quantum theory is attempted,…
In Itamar Pitowsky's reading of the Gleason and the Kochen-Specker theorems, in particular, his Logical Indeterminacy Principle, the emphasis is on the value indefiniteness of observables which are not within the preparation context. This…
This paper provides a quantitative version of the recent result of Kn\"upfer and Muratov ({\it Commun. Pure Appl. Math.} {\bf 66} (2013), 1129--1162) concerning the solutions of an extension of the classical isoperimetric problem in which a…
In this note several computations of equivariant cohomology groups are performed. For the compactly supported equivariant cohomology, the notion of infinitesimal index developed in arXiv:1003.3525, allows to describe these groups in terms…
This paper deals with the subject of infinitesimal variations of Euclidean submanifolds with arbitrary dimension and codimension. The main goal is to establish a Fundamental theorem for these geometric objects. Similar to the theory of…
Quantum field theory has successfully generated a number of general conclusions. It seems meaningful to disclose the logical forms of these conclusions. The present paper reports two results. The first result shows the logic of local gauge…
Infinite Time Register Machines ($ITRM$'s) are a well-established machine model for infinitary computations. Their computational strength relative to oracles is understood, see e.g. Koepke (2009), Koepke and Welch (2011) and Koepke and…
The notion of an orthogonality space was recently rediscovered as an effective means to characterise the essential properties of quantum logic. The approach can be considered as minimalistic; solely the aspect of mutual exclusiveness is…