Pushouts of extensions of groupoids by bundles of abelian groups

We analyse extensions $\Sigma$ of groupoids G by bundles A of abelian groups. We describe a pushout construction for such extensions, and use it to describe the extension group of a given groupoid G by a given bundle A. There is a natural action of Sigma on the dual of A, yielding a corresponding transformation groupoid. The pushout of this transformation groupoid by the natural map from the fibre product of A with its dual to the Cartesian product of the dual with the circle is a twist over the transformation groupoid resulting from the action of G on the dual of A. We prove that the full C*-algebra of this twist is isomorphic to the full C*-algebra of $\Sigma$, and that this isomorphism descends to an isomorphism of reduced algebras. We give a number of examples and applications.

A Knowledge Representation of the Miller-Urey Experiment

We present an ontology log-OLOG- representation of the classical Miller-Urey experiment, usually considered as a paradigm for spontaneous generation of biomolecules on the prebiotic Earth and also, as a key in understanding the chemical evolution phenomena linked to the origins of life. Ologging The Miler-Urey experiment enables us, through the categorical notion of fibre product or pullback, to define the concept of Biogenic Space, as a space containing low complexity biogenic units subjected to appropriate physical and chemical conditions, facilitating the synthesis of highly complex organic molecules. Also, we characterize the Biogenic Space as a concrete universal object that could be associated with the preconditions for life inside various structures in the universe such as exoplanets and exomoons located in habitable zones, but also in interstellar and intergalactic organic clouds.

A Simple Homemade Electrospinning for Nanoscale Fibres Production

A high technology often is required by electrospinning device to produce a nanoscale fibre. This requirement results in difficulties to get the device simple design and operate with reasonable price. This paper presents the work to implement design of the electrospinning system. This electrospinning system consists of a main construction body, an electrical power supply to provide a voltage source direct current form, a syringe pump and a flat collector. The main body are made of acrylic with knockdown construction since acrylic is an easy material to shape. The power supply has a simple main single switch with high ferite transformer to produce high dc voltage through a diode – capacitor filtering. A used medical syringe has been modified as an injection pump to spray the materials into fibre form. A simple flat metal has been reconstructed to be electrode – ground collector to receive fibre product. The test was carried out using polyaniline materials. The general parameters in the production process are resolution of the spraying rate µℓ / minute and the power supply provide electricity in kiloVolt. The result of fibre production observed by using SEM shows that the electrospinning device successfully produces fibres on a nanometer scale.

On the Hodge, Tate and Mumford-Tate Conjectures for Fibre Products of Families of Regular Surfaces with Geometric Genus 1

The Hodge, Tate and Mumford-Tate conjectures are proved for the fibre product of two non-isotrivial 1-parameter families of regular surfaces with geometric genus 1 under some conditions on degenerated fibres, the ranks of the N\'eron - Severi groups of generic geometric fibres and representations of Hodge groups in transcendental parts of rational cohomology.Let \(\pi_i:X_i\to C\quad (i = 1, 2)\) be a projective non-isotrivial family (possibly with degeneracies) over a smooth projective curve \(C\). Assume that the discriminant loci \(\Delta_i=\{\delta\in C\,\,\vert\,\, Sing(X_{i\delta})\neq\varnothing\} \quad (i = 1, 2)\) are disjoint, \(h^{2,0}(X_{ks})=1,\quad h^{1,0}(X_{ks}) = 0\) for any smooth fibre \(X_{ks}\), and the following conditions hold:\((i)\) for any point \(\delta \in \Delta_i\) and the Picard-Lefschetz transformation \( \gamma \in GL(H^2 (X_{is}, Q)) \), associated with a smooth part \(\pi'_i: X'_i\to C\setminus\Delta_i\) of the morphism \(\pi_i\) and with a loop around the point \(\delta \in C\), we have \((\log(\gamma))^2\neq0\);\((ii)\) the variety \(X_i \, (i = 1, 2)\), the curve \(C\) and the structure morphisms \(\pi_i:X_i\to C\) are defined over a finitely generated subfield \(k \hookrightarrow C\).If for generic geometric fibres \(X_{1s}\) \, and \, \(X_{2s}\) at least one of the following conditions holds: \((a)\) \(b_2(X_{1s})- rank NS(X_{1s})\) is an odd prime number, \(\quad\,\,\) \(b_2(X_{1s})- rank NS(X_{1s})\neq b_2(X_{2s})- rank NS(X_{2s})\); \((b)\) the ring \(End_{ Hg(X_{1s})} NS_ Q(X_{1s})^\perp\) is an imaginary quadratic field, \(\quad\,\, b_2(X_{1s})- rank NS(X_{1s})\neq 4,\) \(\quad\,\, End_{ Hg(X_{2s})} NS_ Q(X_{2s})^\perp\) is a totally real field or \(\,\, b_2(X_{1s})- rank NS(X_{1s})\,>\, b_2(X_{2s})- rank NS(X_{2s})\) ; \((c)\) \([b_2(X_{1s})- rank NS(X_{1s})\neq 4, \, End_{ Hg(X_{1s})} NS_ Q(X_{1s})^\perp= Q\); \(\quad\,\,\) \(b_2(X_{1s})- rank NS(X_{1s})\neq b_2(X_{2s})- rank NS(X_{2s})\),then for the fibre product \(X_1 \times_C X_2\) the Hodge conjecture is true, for any smooth projective \(k\)-variety \(X_0\) with the condition \(X_1 \times_C X_2\) \(\widetilde{\rightarrow}\) \(X_0 \otimes_k C\) the Tate conjecture on algebraic cycles and the Mumford-Tate conjecture for cohomology of even degree are true.