Multilayer networks as embodied consciousness interactions. A formal model approach.

An algebraic interpretation of multilayer networks is introduced in relation to conscious experience, brain and body. The discussion is based on a network model for undirected multigraphs with coloured edges whose elements are time-evolving multilayers, representing complex experiential brain-body networks. These layers have the ability to merge by an associative binary operator, accounting for biological composition. As an extension, they can rotate in a formal analogy to how the activity inside layers would dynamically evolve. Under consciousness interpretation, we also studied a mathematical formulation of splitting layers, resulting in a formal analysis for the transition from conscious to non-conscious activity. From this construction, we recover core structures for conscious experience, dynamical content and causal efficacy of conscious interactions, predicting topological network changes after conscious layer interactions. Our approach provides a mathematical account of coupling and splitting layers co-arising with more complex experiences. These concrete results may inspire the use of formal studies of conscious experience not only to describe it, but also to obtain new predictions and future applications of formal mathematical tools.

An algebraic treatment of the Askey biorthogonal polynomials on the unit circle

A joint algebraic interpretation of the biorthogonal Askey polynomials on the unit circle and of the orthogonal Jacobi polynomials is offered. It ties their bispectral properties to an algebra called the meta-Jacobi algebra $m\mathfrak {J}$ .

Nondeterministic Syntactic Complexity

We introduce a new measure on regular languages: their nondeterministic syntactic complexity. It is the least degree of any extension of the ‘canonical boolean representation’ of the syntactic monoid. Equivalently, it is the least number of states of any subatomic nondeterministic acceptor. It turns out that essentially all previous structural work on nondeterministic state-minimality computes this measure. Our approach rests on an algebraic interpretation of nondeterministic finite automata as deterministic finite automata endowed with semilattice structure. Crucially, the latter form a self-dual category.

Algebraic approach to Rump’s results on relations between braces and pre-lie algebras

In 2014, Wolfgang Rump showed that there exists a correspondence between left nilpotent right [Formula: see text]-braces and pre-Lie algebras. This correspondence, established using a geometric approach related to flat affine manifolds and affine torsors, works locally. In this paper, we explain Rump’s correspondence using only algebraic formulae. An algebraic interpretation of the correspondence works for fields of sufficiently large prime characteristic as well as for fields of characteristic zero.

Autonomy of Geometry

In this paper we present three aspects of the autonomy of geometry. (1) An argument for the geometric as opposed to the ‘geometric algebraic’ interpretation of Euclid’s Books I and II; (2) Hilbert’s successful project to axiomatize Euclid’s geometry in a first order geometric language, notably eliminating the dependence on the Archimedean axiom; (3) the independent conception of multiplication from a geometric as opposed to an arithmetic viewpoint.

Effect of Size Polydispersity on the Pitch of Nanorod Cholesterics

Many nanoparticle-based chiral liquid crystals are composed of polydisperse rod-shaped particles with considerable spread in size or shape, affecting the mesoscale chiral properties in, as yet, unknown ways. Using an algebraic interpretation of Onsager-Straley theory for twisted nematics, we investigate the role of length polydispersity on the pitch of nanorod-based cholesterics with a continuous length polydispersity, and find that polydispersity enhances the twist elastic modulus, K 2 , of the cholesteric material without affecting the effective helical amplitude, K t . In addition, for the infinitely large average aspect ratios considered here, the dependence of the pitch on the overall rod concentration is completely unaffected by polydispersity. For a given concentration, the increase in twist elastic modulus (and reduction of the helical twist) may be up to 50% for strong size polydispersity, irrespective of the shape of the unimodal length distribution. We also demonstrate that the twist reduction is reinforced in bimodal distributions, obtained by doping a polydisperse cholesteric with very long rods. Finally, we identify a subtle, non-monotonic change of the pitch across the isotropic-cholesteric biphasic region.