On Automorphisms of the Double Cover of a Circulant Graph

A graph $X$ is said to be unstable if the direct product $X \times K_2$ (also called the canonical double cover of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is nontrivially unstable if it is unstable, connected, and nonbipartite, and no two distinct vertices of $X$ have exactly the same neighbors. We find three new conditions that each imply a circulant graph is unstable. (These yield infinite families of nontrivially unstable circulant graphs that were not previously known.) We also find all of the nontrivially unstable circulant graphs of order $2p$, where $p$ is any prime number. Our results imply that there does not exist a nontrivially unstable circulant graph of order $n$ if and only if either $n$ is odd, or $n < 8$, or $n = 2p$, for some prime number $p$ that is congruent to $3$ modulo $4$.

Computational Modeling of the Thermal Behavior of a Greenhouse

The need for production of all kinds of crops in high quantities and over the entire year makes the agricultural sector one of the major energy consumers. The optimization of this consumption is essential to guarantee its sustainability. The implementation of greenhouses is a strategy that allows assurance of production needs and possesses large optimization potential for the process. This article studies different greenhouse structures by computational simulation using EnergyPlus and DesignBuilder. First, a comparison was performed between the computational results and the measured values from a greenhouse prototype at different operating conditions. Overall, the comparison shows that the computational tool can provide a reasonable prediction of the greenhouse thermal behavior, depending on the differences between the weather data modeled and observed. An outdoor air temperature difference of 16 °C can cause a difference of about 10 °C between the air temperature predicted and measured inside the greenhouse. Subsequently, a selected set of case studies was developed in order to quantify their influence on the thermal performance of the greenhouse, namely: the greenhouse configuration and orientation; the variation of indoor air renewal; changes to the characteristics of the roof; the effect of the thermal mass of the walls; and location of the greenhouse. The results show that a correct greenhouse orientation, together with a polyethylene double cover with a 13 mm air layer, a granite wall of 40 cm thickness on the north wall, and variable airflow rate, may lead to a reduction of the greenhouse energy consumption by 57%, if the greenhouse is located in Lisbon, or by 43%, if it is located in Ostersund, during the hottest months of the heating season.

2 × 2 Matrices: Manifolds, Realizations, Applications

Both geometric and wave optical models, as well as classical and quantum mechanics, realize linear transformations with matrices; for plane optics, these are 2×2 and of unit determinant. Students and some researchers could assume that the structure of this matrix group is fairly evident and hardly interesting. However, the properties and applications even of this lowest 2×2 case are already unexpectedly rich. While in mechanics they cover classical angular momentum, quantum spin, and represent ‘2+1’ relativity, in optical models they lead from the geometrical description of light propagation in the paraxial regime to wave optics via linear canonical transforms requiring a more penetrating view of their manifold structure and multiple covers. The purpose of this review article is to highlight the topological space of 2×2 matrices as it applies to classical versus quantum and wave models, to underline how the latter requires the double cover of the former, thus using 2×2 matrices as an alternative viewpoint of the quantization process, beside the traditional characterization by commutation and non-commutations of position and momentum.

Abelianisation of logarithmic $$\mathfrak {sl}_2$$-connections

We prove a functorial correspondence between a category of logarithmic $$\mathfrak {sl}_2$$ sl 2 -connections on a curve $${\mathsf {X}}$$ X with fixed generic residues and a category of abelian logarithmic connections on an appropriate spectral double cover "Equation missing". The proof is by constructing a pair of inverse functors $$\pi ^\text {ab}, \pi _\text {ab}$$ π ab , π ab , and the key is the construction of a certain canonical cocycle valued in the automorphisms of the direct image functor $$\pi _*$$ π ∗ .

Maximal families of nodal varieties with defect

In this paper we prove that a nodal hypersurface in $$\mathbf {P}^4$$ P 4 with positive defect has at least $$(d-1)^2$$ ( d - 1 ) 2 nodes, and if it has at most $$2(d-2)(d-1)$$ 2 ( d - 2 ) ( d - 1 ) nodes and $$d\ge 7$$ d ≥ 7 then it contains either a plane or a quadric surface. Furthermore, we prove that a nodal double cover of $$\mathbf {P}^3$$ P 3 ramified along a surface of degree 2d with positive defect has at least $$d(2d-1)$$ d ( 2 d - 1 ) nodes. We construct the largest dimensional family of nodal degree d hypersurfaces in $$\mathbf {P}^{2n+2}$$ P 2 n + 2 with positive defect for d sufficiently large.

Rank one sheaves over quaternion algebras on Enriques surfaces

Let X be an Enriques surface over the field of complex numbers. We prove that there exists a nontrivial quaternion algebra 𝓐 on X. Then we study the moduli scheme of torsion free 𝓐-modules of rank one. Finally we prove that this moduli scheme is an étale double cover of a Lagrangian subscheme in the corresponding moduli scheme on the associated covering K3 surface.

On the dimension of totally geodesic submanifolds in the Prym loci

In this paper we give a bound on the dimension of a totally geodesic submanifold of the moduli space of polarised abelian varieties of a given dimension, which is contained in the Prym locus of a (possibly) ramified double cover. This improves the already known bounds. The idea is to adapt the techniques introduced by the authors in collaboration with A. Ghigi and G. P. Pirola for the Torelli map to the case of the Prym maps of (ramified) double covers.