The Largest Subsemilattices of the Endomorphism Monoid of an Independence Algebra

An algebra \A is said to be an independence algebra if it is a matroid algebra and every map \al:X\to A, defined on a basis $X$ of \A, can be extended to an endomorphism of \A. These algebras are particularly well behaved generalizations of vector spaces, and hence they naturally appear in several branches of mathematics such as model theory, group theory, and semigroup theory. It is well known that matroid algebras have a well defined notion of dimension. Let \A be any independence algebra of finite dimension $n$, with at least two elements. Denote by \End(\A) the monoid of endomorphisms of \A. We prove that a largest subsemilattice of \End(\A) has either $2^{n-1}$ elements (if the clone of \A does not contain any constant operations) or $2^n$ elements (if the clone of \A contains constant operations). As corollaries, we obtain formulas for the size of the largest subsemilattices of: some variants of the monoid of linear operators of a finite-dimensional vector space, the monoid of full transformations on a finite set $X$, the monoid of partial transformations on $X$, the monoid of endomorphisms of a free $G$-set with a finite set of free generators, among others. The paper ends with a relatively large number of problems that might attract attention of experts in linear algebra, ring theory, extremal combinatorics, group theory, semigroup theory, universal algebraic geometry, and universal algebra.Comment: To appear in Linear Algebra and its Application

Isotropic quantum walks on lattices and the Weyl equation

We present a thorough classification of the isotropic quantum walks on lattices of dimension $d=1,2,3$ for cell dimension $s=2$. For $d=3$ there exist two isotropic walks, namely the Weyl quantum walks presented in Ref. [G. M. D'Ariano and P. Perinotti, Phys. Rev. A 90, 062106 (2014)], resulting in the derivation of the Weyl equation from informational principles. The present analysis, via a crucial use of isotropy, is significantly shorter and avoids a superfluous technical assumption, making the result completely general.Comment: 16 pages, 1 figur

Weak Liouville-Arnold Theorems & Their Implications

This paper studies the existence of invariant smooth Lagrangian graphs for Tonelli Hamiltonian systems with symmetries. In particular, we consider Tonelli Hamiltonians with n independent but not necessarily involutive constants of motion and obtain two theorems reminiscent of the Liouville-Arnold theorem. Moreover, we also obtain results on the structure of the configuration spaces of such systems that are reminiscent of results on the configuration space of completely integrable Tonelli Hamiltonians.Comment: 24 pages, 1 figure; v2 corrects typo in online abstract; v3 includes new title (was: A Weak Liouville-Arnold Theorem), re-arrangement of introduction, re-numbering of main theorems; v4 updates the authors' email and physical addresses, clarifies notation in section 4. Final versio

A new near octagon and the Suzuki tower

We construct and study a new near octagon of order $(2,10)$ which has its full automorphism group isomorphic to the group $\mathrm{G}_2(4){:}2$ and which contains $416$ copies of the Hall-Janko near octagon as full subgeometries. Using this near octagon and its substructures we give geometric constructions of the $\mathrm{G}_2(4)$-graph and the Suzuki graph, both of which are strongly regular graphs contained in the Suzuki tower. As a subgeometry of this octagon we have discovered another new near octagon, whose order is $(2,4)$.Comment: 24 pages, revised version with added remarks and reference

Noncommutativity and Discrete Physics

The purpose of this paper is to present an introduction to a point of view for discrete foundations of physics. In taking a discrete stance, we find that the initial expression of physical theory must occur in a context of noncommutative algebra and noncommutative vector analysis. In this way the formalism of quantum mechanics occurs first, but not necessarily with the usual interpretations. The basis for this work is a non-commutative discrete calculus and the observation that it takes one tick of the discrete clock to measure momentum.Comment: LaTeX, 23 pages, no figure

Uncovering the spatial structure of mobility networks

The extraction of a clear and simple footprint of the structure of large, weighted and directed networks is a general problem that has many applications. An important example is given by origin-destination matrices which contain the complete information on commuting flows, but are difficult to analyze and compare. We propose here a versatile method which extracts a coarse-grained signature of mobility networks, under the form of a $2\times 2$ matrix that separates the flows into four categories. We apply this method to origin-destination matrices extracted from mobile phone data recorded in thirty-one Spanish cities. We show that these cities essentially differ by their proportion of two types of flows: integrated (between residential and employment hotspots) and random flows, whose importance increases with city size. Finally the method allows to determine categories of networks, and in the mobility case to classify cities according to their commuting structure.Comment: 10 pages, 5 figures +Supplementary informatio

Stability and Invariant Random Subgroups

Consider $\operatorname{Sym}(n)$, endowed with the normalized Hamming metric $d_n$. A finitely-generated group $\Gamma$ is \emph{P-stable} if every almost homomorphism $\rho_{n_k}\colon \Gamma\rightarrow\operatorname{Sym}(n_k)$ (i.e., for every $g,h\in\Gamma$, $\lim_{k\rightarrow\infty}d_{n_k}( \rho_{n_k}(gh),\rho_{n_k}(g)\rho_{n_k}(h))=0$) is close to an actual homomorphism $\varphi_{n_k} \colon\Gamma\rightarrow\operatorname{Sym}(n_k)$. Glebsky and Rivera observed that finite groups are P-stable, while Arzhantseva and P\u{a}unescu showed the same for abelian groups and raised many questions, especially about P-stability of amenable groups. We develop P-stability in general, and in particular for amenable groups. Our main tool is the theory of invariant random subgroups (IRS), which enables us to give a characterization of P-stability among amenable groups, and to deduce stability and instability of various families of amenable groups.Comment: 24 pages; v2 includes minor updates and new reference