Related papers: Where Infinitesimals Come From ..
Common meadows are commutative and associative algebraic structures with two operations (addition and multiplication) with additive and multiplicative identities and for which inverses are total. The inverse of zero is an error term…
According to the inflationary scenario, all structure in the Universe can be traced back to quantum fluctuations of the metric and scalar field(s) during inflation. The seeds of this structure can be observed as classical anisotropies in…
The tensor powers of the vector representation associated to an infinite rank quantum group decompose into irreducible components with multiplicities independant of the infinite root system considered. Although the irreducible modules…
Singularities appear in numerous important mathematical models used in Physics. And in most of such cases singularities are involved in essentially nonlinear contexts. For more than four decades, general enough nonlinear theories of…
Generalized Functions play a central role in the understanding of differential equations containing singularities and nonlinearities. Introducing infinitesimals and infinities to deal with these obstructions leads to controversies…
An infinite magmatic bialgebra is a vector space endowed with an n-ary operation, and an n-ary cooperation, for each n, verifying some compatibility relations. We prove a rigidity theorem, analogue to the Hopf-Borel theorem for commutative…
The roots of a complex polynomial depend continuously on the coefficients; that is, an infinitesimal perturbation of the coefficients results in an infinitesimal perturbation of the roots. A short, straightforward proof of this is possible…
We describe arbitrary multiplicative differential forms on Lie groupoids infinitesimally, i.e., in terms of Lie algebroid data. This description is based on the study of linear differential forms on Lie algebroids and encompasses many known…
This expository paper treats the model theory of probability spaces using the framework of continuous $[0,1]$-valued first order logic. The metric structures discussed, which we call probability algebras, are obtained from probability…
Motivated by problems arising in the relative trace formula and arithmetic invariant theory we prove the existence of rational points on orbits arising from certain infinitesimal symmetric spaces. As an application, we prove analogous…
In this paper, we use some basic quasi-topos theory to study two functors: one adding infinitesimals of Fermat reals to diffeological spaces (which generalize smooth manifolds including singular spaces and infinite dimensional spaces), and…
We generalize the differential space concept as a tool for developing differential geometry, and enrich this geometry with infinitesimals that allow us to penetrate into the superfine structure of space. This is achieved by Yoneda embedding…
We associate reduced and full C*-algebras to arbitrary rings and study the inner structure of these ring C*-algebras. As a result, we obtain conditions for them to be purely infinite and simple. We also discuss several examples.…
Szemeredi's regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemeredi's regularity lemma in the context of abelian groups and use it to derive some…
Algebraic hyperstructures represent a natural extension of classical algebraic structures. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two…
An $integral$ of a group $G$ is a group $H$ whose derived group (commutator subgroup) is isomorphic to $G$. This paper discusses integrals of groups, and in particular questions about which groups have integrals and how big or small those…
We study the infinitesimal variation of Hodge structure associated with families of reduced algebraic curves with singularities. The analysis applies to curves beyond the nodal case and is not restricted to plane curves, encompassing curves…
First order formulas in a relational signature can be considered as operations on the relations of an underlying set, giving rise to multisorted algebras we call first order algebras. We present universal axioms so that an algebra satisfies…
Magnetic fields appear everywhere in the universe. From stars and galaxies, all the way to galaxy clusters and remote protogalactic clouds magnetic fields of considerable strength and size have been repeatedly observed. Despite their…
We find an infinite dimensional free algebra which lives at large N in any SU(N)-invariant action or Hamiltonian theory of bosonic matrices. The natural basis of this algebra is a free-algebraic generalization of Chebyshev polynomials and…