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Marius Buliga

Publications and source records attributed to Marius Buliga.

At least 37 records · Page 2Linked to original sources
arXiv2013

On graphic lambda calculus and the dual of the graphic beta move

This is a short description of graphic lambda calculus, with special emphasis on a duality suggested by the two different appearances of knot diagrams, in lambda calculus and emergent algebra sectors of the graphic lambda calculus respectively. This duality leads to the introduction of the dual of the graphic beta move. While the graphic beta move corresponds to beta reduction in untyped lambda calculus, the dual graphic beta move appears in relation to emergent algebras.

math.GT↗
arXiv2012

Graphic lambda calculus and knot diagrams

In arXiv:1207.0332 [cs.LO] was proposed a graphic lambda calculus formalism, which has sectors corresponding to untyped lambda calculus and emergent algebras. Here we explore the sector covering knot diagrams, which are constructed as macros over the graphic lambda calculus.

math.GT↗
arXiv2012

Sub-riemannian geometry from intrinsic viewpoint

Gromov proposed to extract the (differential) geometric content of a sub-riemannian space exclusively from its Carnot-Carathéodory distance. One of the most striking features of a regular sub-riemannian space is that it has at any point a metric tangent space with the algebraic structure of a Carnot group, hence a homogeneous Lie group. Siebert characterizes homogeneous Lie groups as locally compact groups admitting a contracting and continuous one-parameter group of automorphisms. Siebert result has not a metric character. In these notes I show that sub-riemannian geometry may be described by about 12 axioms, without using any a priori given differential structure, but using dilation structures instead. Dilation structures bring forth the other intrinsic ingredient, namely the dilations, thus blending Gromov metric point of view with Siebert algebraic one.

math.MG↗
arXiv2011

Emergent algebras

Inspired from research subjects in sub-riemannian geometry and metric geometry, we propose uniform idempotent right quasigroups and emergent algebras as an alternative to differentiable algebras. Idempotent right quasigroups (irqs) are related with racks and quandles, which appear in knot theory (the axioms of a irq correspond to the first two Reidemeister moves). To any uniform idempotent right quasigroup can be associated an approximate differential calculus, with Pansu differential calculus in sub-riemannian geometry as an example. An emergent algebra A over a uniform idempotent right quasigroup X is a collection of operations such that each operation emerges from X, meaning that it can be realized as a combination of the operations of the uniform irq X, possibly by taking limits which are uniform with respect to a set of parameters. Two applications are considered: we prove a bijection between contractible groups and distributive uniform irqs (uniform quandles) and that some symmetric spaces in the sense of Loos may be seen as uniform quasigroups with a distributivity property.

math.RA↗
arXiv2011

Maps of metric spaces

This is a pedagogical introduction covering maps of metric spaces, Gromov-Hausdorff distance and its "physical" meaning, and dilation structures as a convenient simplification of an exhaustive database of maps of a metric space into another. See arXiv:1103.6007 for the context.

math.MG↗
arXiv2011

Computing with space: a tangle formalism for chora and difference

What is space computing, simulation, or understanding? Converging from several sources, this seems to be something more primitive than what is usually meant by computation, something that was along with us since antiquity (the word "choros", "chora", denotes "space" or "place" and is seemingly the most mysterious notion from Plato, described in Timaeus 48e - 53c) which has to do with cybernetics and with the understanding of the front end visual system. It may have some unexpected applications, also. Here, inspired by Bateson (see Supplementary Material), I explore from the mathematical side the point of view that there is no difference between the map and the territory, but instead the transformation of one into another can be understood by using a formalism of tangle diagrams. This paper continues arXiv:1009.5028 "What is a space? Computations in emergent algebras and the front end visual system" and the arXiv:1007.2362 "Introduction to metric spaces with dilations".

math.MG↗
arXiv2011

Braided spaces with dilations and sub-riemannian symmetric spaces

Braided sets which are also spaces with dilations are presented and explored in this paper, in the general frame of emergent algebras arxiv:0907.1520. Examples of such spaces are the sub-riemannian symmetric spaces. Keywords: braided sets, quandles; emergent algebras; dilatation structures (spaces with dilations); sub-riemannian symmetric spaces.

math.GR↗
arXiv2010

More than discrete or continuous: a bird's view

I try to give mathematical evidence to the following equivalence, which is based on ideas from Plato (Timaeus): reality emerges from a more primitive, non-geometrical, reality in the same way as the brain construct (understands, simulates, transforms, encodes or decodes) the image of reality, starting from intensive properties (like a bunch of spiking signals sent by receptors in the retina), without any use of extensive (i.e. spatial or geometric) properties.

math.MG↗
arXiv2010

Blurred maximal cyclically monotone sets and bipotentials

Let X be a reflexive Banach space and Y its dual. In this paper we find necessary and sufficient conditions for the existence of a bipotential for a blurred maximal cyclically monotone graph. Equivalently, we find a necessary and sufficient condition on $ϕ\in Γ_{0}(X)$ for that the differential inclusion $y \in \bar{B}(ε) + \partial ϕ(x)$ can be put in the form $y \in \partial b(\cdot, y)(x)$, with $b$ a bipotential.

math.FA↗
arXiv2009

Uniform refinements, topological derivative and a differentiation theorem in metric spaces

For the importance of differentiation theorems in metric spaces (starting with Pansu Rademacher type theorem in Carnot groups) and relations with rigidity of embeddings see the section 1.2 in Cheeger and Kleiner paper arXiv:math/0611954 and its bibliographic references. Here we propose another type of differentiation theorem, which does not involve measures. It is therefore different from Rademacher type theorems. Instead, this differentiation theorem (and the concept of uniformly topological derivable function) is formulated in terms of filters in topological spaces.

math.MG↗
arXiv2009

Deformations of normed groupoids and differential calculus. First part

Differential calculus on metric spaces is contained in the algebraic study of normed groupoids with $δ$-structures. Algebraic study of normed groups endowed with dilatation structures is contained in the differential calculus on metric spaces. Thus all algebraic properties of the small world of normed groups with dilatation structures have equivalent formulations (of comparable complexity) in the big world of metric spaces admitting a differential calculus. Moreover these results non trivially extend beyond metric spaces, by using the language of groupoids.

math.MG↗