Manifold Neighbourhoods and a Conjecture of Adjamagbo

We verify a conjecture of P. Adjamagbo that if the frontier of a relatively compact subset $V_0$ of a manifold is a submanifold then there is an increasing family $\{V_r\}$ of relatively compact open sets indexed by the positive reals so that the frontier of each is a submanifold, their union is the whole manifold and for each $r\ge 0$ the subfamily indexed by $(r,\infty)$ is a neighbourhood basis of the closure of the $r^{\rm th}$ set. We use smooth collars in the differential category, regular neighbourhoods in the piecewise linear category and handlebodies in the topological category.

Cartesian Differential Categories as Skew Enriched Categories

We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind of enriched category. The base for the enrichment is the category of commutative monoids—or in a straightforward generalisation, the category of modules over a commutative rig k. However, the tensor product on this category is not the usual one, but rather a warping of it by a certain monoidal comonad Q. Thus the enrichment base is not a monoidal category in the usual sense, but rather a skew monoidal category in the sense of Szlachányi. Our first main result is that cartesian differential categories are the same as categories with finite products enriched over this skew monoidal base. The comonad Q involved is, in fact, an example of a differential modality. Differential modalities are a kind of comonad on a symmetric monoidal k-linear category with the characteristic feature that their co-Kleisli categories are cartesian differential categories. Using our first main result, we are able to prove our second one: that every small cartesian differential category admits a full, structure-preserving embedding into the cartesian differential category induced by a differential modality (in fact, a monoidal differential modality on a monoidal closed category—thus, a model of intuitionistic differential linear logic). This resolves an important open question in this area.

Quantum Mathematical Integrated Information Theory: From the Classical Version

This paper is an attempt to give a Quantum theory of Mathematical integrated information theory which is mathematical version of integrated information theory by Masafumi Oizumi, Larissa Albantakis, Giulio Tononi. Using the definitions of Classical Mathematical Integrated Information Theory. And considering that the Quantum theory is given by the functor which maps from a category whose objects is topology to a linear category whose objects are Hilbert spaces indexed with the objects from previous category. Also we will be using the definition of conditional density matrix to define repertoire.

Convenient antiderivatives for differential linear categories

Differential categories axiomatize the basics of differentiation and provide categorical models of differential linear logic. A differential category is said to have antiderivatives if a natural transformation , which all differential categories have, is a natural isomorphism. Differential categories with antiderivatives come equipped with a canonical integration operator such that generalizations of the Fundamental Theorems of Calculus hold. In this paper, we show that Blute, Ehrhard, and Tasson's differential category of convenient vector spaces has antiderivatives. To help prove this result, we show that a differential linear category – which is a differential category with a monoidal coalgebra modality – has antiderivatives if and only if one can integrate over the monoidal unit and such that the Fundamental Theorems of Calculus hold. We also show that generalizations of the relational model (which are biproduct completions of complete semirings) are also differential linear categories with antiderivatives.

Descent and Galois theory for Hopf categories

Descent theory for linear categories is developed. Given a linear category as an extension of a diagonal category, we introduce descent data, and the category of descent data is isomorphic to the category of representations of the diagonal category, if some flatness assumptions are satisfied. Then Hopf–Galois descent theory for linear Hopf categories, the Hopf algebra version of a linear category, is developed. This leads to the notion of Hopf–Galois category extension. We have a dual theory, where actions by dual linear Hopf categories on linear categories are considered. Hopf–Galois category extensions over groupoid algebras correspond to strongly graded linear categories.

Carbon Stocks of Linear Category of Trees Out Side Forest in Anantapuramu District, Andhra Pradesh, India

In the present study, carbon stocks of linear structures of trees outside forest in Anantapuramu district was estimated through sampling of 344 (0.1 ha) plots. A total of 4229 tree individuals belonging to 66 angiosperm species were enumerated in the sampled plots. The mean tree density is 122.8per ha; mean diameter at breast height 4.04 m; mean basal area 15.43 m2 ha-1.Mean volume of trees with >10 cm diameter is 15.50 m3 ha-1; mean total tree biomass is 120.81 tons ha-1.The mean carbon stock is 57.385 tons ha-1 and extrapolated biomass and carbon content for linear structures are 0.176 Mt and 0.083 Mt respectively. The carbon sequestration potential of trees outside forests of Anantapuramu district is estimated at 0.304 Mt.