Modular Categories Associated to Unipotent Groups

Let G be a unipotent algebraic group over an algebraically closed field k of characteristic p > 0 and let l be a prime different from p. Let e be a minimal idempotent in D_G(G), the braided monoidal category of G-equivariant (under conjugation action) \bar{Q_l}-complexes on G. We can associate to G and e a modular category M_{G,e}. In this article, we prove that the modular categories that arise in this way from unipotent groups are precisely those in the class C_p^{\pm}.Comment: 26 page

Organic farming systems benefit biodiversity and natural pest regulation in white cabbage

Natural regulation of cabbage root flies works well in experimental organic cropping systems of white cabbage. Low input and complex organic systems benefit functional biodiversity by providing good living conditions to several groups of natural enemies. Intercropped green manure benefits large predators while small predatory beetles favour low input organic systems with bare soil between crop rows

Cleft Extensions and Quotients of Twisted Quantum Doubles

Given a pair of finite groups $F, G$ and a normalized 3-cocycle $\omega$ of $G$, where $F$ acts on $G$ as automorphisms, we consider quasi-Hopf algebras defined as a cleft extension $\Bbbk^G_\omega\#_c\,\Bbbk F$ where $c$ denotes some suitable cohomological data. When $F\rightarrow \overline{F}:=F/A$ is a quotient of $F$ by a central subgroup $A$ acting trivially on $G$, we give necessary and sufficient conditions for the existence of a surjection of quasi-Hopf algebras and cleft extensions of the type $\Bbbk^G_\omega\#_c\, \Bbbk F\rightarrow \Bbbk^G_\omega\#_{\overline{c}} \, \Bbbk \overline{F}$. Our construction is particularly natural when $F=G$ acts on $G$ by conjugation, and $\Bbbk^G_\omega\#_c \Bbbk G$ is a twisted quantum double $D^{\omega}(G)$. In this case, we give necessary and sufficient conditions that Rep($\Bbbk^G_\omega\#_{\overline{c}} \, \Bbbk \overline{G}$) is a modular tensor category.Comment: LaTex; 14 page

Non-commutative connections of the second kind

A connection-like objects, termed {\em hom-connections} are defined in the realm of non-commutative geometry. The definition is based on the use of homomorphisms rather than tensor products. It is shown that hom-connections arise naturally from (strong) connections in non-commutative principal bundles. The induction procedure of hom-connections via a map of differential graded algebras or a differentiable bimodule is described. The curvature for a hom-connection is defined, and it is shown that flat hom-connections give rise to a chain complex.Comment: 13 pages, LaTe

Experiencing local news online: audience practices and perceptions

This article explores how audiences experience local news online. It discusses the findings of an empirical study that examined why audiences consumed local news online, what sources they were most likely to access, how important distributing platforms were in local news use, and what users understood by local news. The research had a qualitative design applying diaries as its main method collecting data in the South-East of England in 2016 and 2017. The findings suggest that there is no shared understanding among audience members about what local news is in the digital environment. The study identified three predominant ways in which participants understood local news: as personally relevant or interesting information, as content produced by legacy local media brands, and as community engagement. The study also found that each of the different understandings of local news was linked to particular online news consumption and engagement patterns. The paper argues that audience perceptions of news should be studied alongside motivations for and practices of news engagement and consumption in order to gain a comprehensive understanding of audiences and news in the digital age

Some operators that preserve the locality of a pseudovariety of semigroups

It is shown that if V is a local monoidal pseudovariety of semigroups, then K(m)V, D(m)V and LI(m)V are local. Other operators of the form Z(m)(_) are considered. In the process, results about the interplay between operators Z(m)(_) and (_)*D_k are obtained.Comment: To appear in International Journal of Algebra and Computatio

Økologisk dyrkning af hvidkål fremmer biodiversitet og naturlig regulering af skadedyr

Naturlig regulering af kålfluer er effektiv i økologisk dyrkede hvidkålsparceller. Økologiske dyrkningssystemer med lavt input og høj strukturel kompleksitet skaber gode livsbetingelser for en række nyttedyr. Mellemafgrøder af foregående sæsons grøngødning gavner de store arter, mens små løbe- og rovbiller bliver tilgodeset i et økologisk system med bar jord mellem afgrøderækkerne

Green's Relations in Finite Transformation Semigroups

We consider the complexity of Green's relations when the semigroup is given by transformations on a finite set. Green's relations can be defined by reachability in the (right/left/two-sided) Cayley graph. The equivalence classes then correspond to the strongly connected components. It is not difficult to show that, in the worst case, the number of equivalence classes is in the same order of magnitude as the number of elements. Another important parameter is the maximal length of a chain of components. Our main contribution is an exponential lower bound for this parameter. There is a simple construction for an arbitrary set of generators. However, the proof for constant alphabet is rather involved. Our results also apply to automata and their syntactic semigroups.Comment: Full version of a paper submitted to CSR 2017 on 2016-12-1

Categorical Foundation of Quantum Mechanics and String Theory

The unification of Quantum Mechanics and General Relativity remains the primary goal of Theoretical Physics, with string theory appearing as the only plausible unifying scheme. In the present work, in a search of the conceptual foundations of string theory, we analyze the relational logic developed by C. S. Peirce in the late nineteenth century. The Peircean logic has the mathematical structure of a category with the relation $R_{ij}$ among two individual terms $S_i$ and $S_j$, serving as an arrow (or morphism). We introduce a realization of the corresponding categorical algebra of compositions, which naturally gives rise to the fundamental quantum laws, thus indicating category theory as the foundation of Quantum Mechanics. The same relational algebra generates a number of group structures, among them $W_{\infty}$. The group $W_{\infty}$ is embodied and realized by the matrix models, themselves closely linked with string theory. It is suggested that relational logic and in general category theory may provide a new paradigm, within which to develop modern physical theories.Comment: To appear in International Journal of Modern Physics