Code algebras, axial algebras and VOAs

Inspired by code vertex operator algebras (VOAs) and their representation theory, we define code algebras, a new class of commutative non-associative algebras constructed from binary linear codes. Let $C$ be a binary linear code of length $n$. A basis for the code algebra $A_C$ consists of $n$ idempotents and a vector for each non-constant codeword of $C$. We show that code algebras are almost always simple and, under mild conditions on their structure constants, admit an associating bilinear form. We determine the Peirce decomposition and the fusion law for the idempotents in the basis, and we give a construction to find additional idempotents, called the $s$-map, which comes from the code structure. For a general code algebra, we classify the eigenvalues and eigenvectors of the smallest examples of the $s$-map construction, and hence show that certain code algebras are axial algebras. We give some examples, including that for a Hamming code $H_8$ where the code algebra $A_{H_8}$ is an axial algebra and embeds in the code VOA $V_{H_8}$.Comment: 32 pages, including an appendi

Generating infinite monoids of cellular automata

For a group $G$ and a set $A$, let $\text{End}(A^G)$ be the monoid of all cellular automata over $A^G$, and let $\text{Aut}(A^G)$ be its group of units. By establishing a characterisation of surjunctuve groups in terms of the monoid $\text{End}(A^G)$, we prove that the rank of $\text{End}(A^G)$ (i.e. the smallest cardinality of a generating set) is equal to the rank of $\text{Aut}(A^G)$ plus the relative rank of $\text{Aut}(A^G)$ in $\text{End}(A^G)$, and that the latter is infinite when $G$ has an infinite decreasing chain of normal subgroups of finite index, condition which is satisfied, for example, for any infinite residually finite group. Moreover, when $A=V$ is a vector space over a field $\mathbb{F}$, we study the monoid $\text{End}_{\mathbb{F}}(V^G)$ of all linear cellular automata over $V^G$ and its group of units $\text{Aut}_{\mathbb{F}}(V^G)$. We show that if $G$ is an indicable group and $V$ is finite-dimensional, then $\text{End}_{\mathbb{F}}(V^G)$ is not finitely generated; however, for any finitely generated indicable group $G$, the group $\text{Aut}_{\mathbb{F}}(\mathbb{F}^G)$ is finitely generated if and only if $\mathbb{F}$ is finite.Comment: 11 page

Conflictos socio-ambientales y recursos hídricos en Guanacaste; una descripción desde el cambio en el estilo de desarrollo (1997-2006)

Resumen El presente estudio se avoca a estudiar las causas y las dinámicas de conflicto por el agua en Guanacaste y su relación con las características del estilo de desarrollo allí implantado. El estudio concluye advirtiendo una problemática que, si no es controlada pronto por entidades estatales en conjunción de otros actores políticos, se arriesga a un impacto ambiental irreversible sobre los recursos hídricos en la provincia, al tiempo que a una elevación notable de la conflictividad regional. Abstract This study advocates to analyze the causes and dynamics of conflicts over water in Guanacaste and its relationship with the characteristics of the style of development implanted there. This document concludes that the problem of water in this area, if not checked or controlled by the responsible entities and other interested political actors, constitutes a risk of irreversible environmental impact on these resources, whilst a probable and dramatic elevation of regional conflictivity

Code algebras which are axial algebras and their Z2\mathbb{Z}_2-gradings

A code algebra $A_C$ is a non-associative commutative algebra defined via a binary linear code $C$. We study certain idempotents in code algebras, which we call small idempotents, that are determined by a single non-zero codeword. For a general code $C$, we show that small idempotents are primitive and semisimple and we calculate their fusion law. If $C$ is a projective code generated by a conjugacy class of codewords, we show that $A_C$ is generated by small idempotents and so is, in fact, an axial algebra. Furthermore, we classify when the fusion law is $\mathbb{Z}_2$-graded. In doing so, we exhibit an infinite family of $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded axial algebras - these are the first known examples of axial algebras with a non-trivial grading other than a $\mathbb{Z}_2$-grading.Comment: 29 page

The number of configurations in the full shift with a given least period

For any group $G$ and any set $A$, consider the shift action of $G$ on the full shift $A^G$. A configuration $x \in A^G$ has \emph{least period} $H \leq G$ if the stabiliser of $x$ is precisely $H$. Among other things, the number of such configurations is interesting as it provides an upper bound for the size of the corresponding $\text{Aut}(A^G)$-orbit. In this paper we show that if $G$ is finitely generated and $H$ is of finite index, then the number of configurations in $A^G$ with least period $H$ may be computed using the M\"obius function of the lattice of subgroups of finite index in $G$. Moreover, when $H$ is a normal subgroup, we classify all situations such that the number of $G$-orbits with least period $H$ is at most $10$.Comment: 8 page

Life History and Phenology of \u3ci\u3ePhylloicus pulchrus\u3c/i\u3e (Trichoptera: Calamoceratidae) In a Tropical Rainforest Stream of Puerto Rico

Caddisflies are abundant, diverse, and important insects in freshwater ecosystems. However our knowledge on their life history is incomplete, in particular for the Neotropics. The objectives of this study were to describe the life history and phenology of Phylloicus pulchrus in the Luquillo Experimental Forest, Puerto Rico. Eggs and larvae were reared to determine the species lifespan and time in each instar. Larval instars were determined based on a head width vs. pronotal suture length correlation (N= 120). Larvae and benthic leaf litter were sampled monthly at a headwater stream for a year; all specimens were classified into instars based on their case size. Adult P. pulchrus were sampled monthly for a year with a light trap and at various times with a Malaise trap. Monthly environmental variables were related to species and sex abundance. There was a gradient of egg development where eggs (within compound masses) closest to the water were more developed. There were five larval instars and reared larvae showed longer development times and more variable body measurements in later instars. The best correlation for larval instar determination was case length-head width (Pearson= 0.90, P= 2.2e-16, N= 120). Phylloicus pulchrus has a multivoltine life cycle, with asynchronous larval development. Adult abundance was low. First to third instar larvae were influenced significantly by rainfall and rainfall seasonality had a negative significant effect on second instar larval abundance (ANOVA= 7.45, P= 0.02).Compound egg masses were probably oviposited by different females that gathered for oviposition. Phylloicus pulchrus follows the predominant developmental characteristic of Trichoptera of having five larval stages. Development times were longer than expected (longest times for a Phylloicus species) and may be an effect of laboratory rearing. The influence of rainfall (and seasonality) on different larval instars highlights the importance of this variable on early larval development. The cause of low adult abundance remains unclear, but may be related to low emergence rates and trap efficiency. Rev. Biol. Trop. 66(2): 814-825. Epub 2018 June 01