Graham Higman's PORC theorem

Graham Higman published two important papers in 1960. In the first of these papers he proved that for any positive integer $n$ the number of groups of order $p^{n}$ is bounded by a polynomial in $p$, and he formulated his famous PORC conjecture about the form of the function $f(p^{n})$ giving the number of groups of order $p^{n}$. In the second of these two papers he proved that the function giving the number of $p$-class two groups of order $p^{n}$ is PORC. He established this result as a corollary to a very general result about vector spaces acted on by the general linear group. This theorem takes over a page to state, and is so general that it is hard to see what is going on. Higman's proof of this general theorem contains several new ideas and is quite hard to follow. However in the last few years several authors have developed and implemented algorithms for computing Higman's PORC formulae in special cases of his general theorem. These algorithms give perspective on what are the key points in Higman's proof, and also simplify parts of the proof. In this note I give a proof of Higman's general theorem written in the light of these recent developments

Higman's PORC conjecture for a family of groups

We prove that the number of groups of order $p^n$ whose Frattini subgroup is central is for fixed $n$ a PORC (`polynomial on residue classes') function of $p$. This extends a result of G. Higman.Comment: 17 page

On the regularity conjecture for the cohomology of finite groups

Non peer reviewedPublisher PD

Mathematical modelling of operation modes and performance evaluation of an innovative small-scale concentrated solar organic Rankine cycle plant

In this paper an innovative small-scale concentrated solar 2 kWe organic Rankine cycle plant coupled with a phase change material storage tank equipped with reversible heat pipes is investigated using a simulation analysis. The plant, intended for residential applications, is going to be built and tested under the European funded H2020 Innova MicroSolar project executed by the consortium of several Universities and industrial organizations, led by Northumbria University. The authors of this work used the design of the integrated system, developed by the consortium, to preliminary estimate the overall performance of the system in order to provide useful information for its forthcoming real operation. In particular, according to the varying ambient conditions, the influence of different operation modes of the prototype plant are evaluated. The dynamic simulation analysis has shown an interesting performance of the system in terms of annual operating hours, power production and conversion efficiencies. More precisely, the organic Rankine cycle unit is able to operate for more than 3100 h/year, achieving the design performance when solar power is sufficiently high, producing about 5100 kWhe/year. For the considered operating set-point temperatures of the thermal energy storage, the plant is able to reach high conversion efficiency also when the organic Rankine cycle unit is supplied by discharging the energy stored in the storage tank, for about 800 h/year. Hence, the work has provided some useful insights into the best working conditions of such micro combined heat and power system to be integrated in residential buildings. Moreover, the analysis could serve as a general guide for the design and optimization of the mutual interactions of the different subsystems in small-scale concentrated solar organic Rankine cycle plants

Designing bio-mimetic variational porosity for tissue scaffolds

Reconstructing or repairing the damaged or diseased tissues with porous scaffolds to restore the mechanical, biological and chemical functions is one of the major tissue engineering strategies. Development of Solid Free Form (SFF) techniques and improvement in biomaterial properties by synergy have provided the leverage to fabricate controlled and interconnected porous scaffold structures. But homogeneous scaffolds with regular porosity do not provide all the biological and mechanical requirements of an ideal tissue scaffold. Thus achieving controllable, continuous, interconnected gradient porosity with reproducible and fabricatable design is critical for successful regeneration of the replaced tissue. In this research, a novel scaffold modeling approach has been proposed to achieve bio-mimetic tissue scaffolds. Firstly, the optimum filament deposition angle has been determined based on the internal heterogeneous regions and their locations. Then an area-weight based approach has been applied to generate the spatial porosity function to determine the filament deposition location for the desired bio-mimetic porosity. The proposed methodology has been implemented using computer simulation. A micro-nozzle biomaterial deposition system driven by NC motion control has been used to fabricate a sample designed structure

The HY5-PIF regulatory module coordinates light and temperature control of photosynthetic gene transcription

The ability to interpret daily and seasonal alterations in light and temperature signals is essential for plant survival. This is particularly important during seedling establishment when the phytochrome photoreceptors activate photosynthetic pigment production for photoautotrophic growth. Phytochromes accomplish this partly through the suppression of phytochrome interacting factors (PIFs), negative regulators of chlorophyll and carotenoid biosynthesis. While the bZIP transcription factor long hypocotyl 5 (HY5), a potent PIF antagonist, promotes photosynthetic pigment accumulation in response to light. Here we demonstrate that by directly targeting a common promoter cis-element (G-box), HY5 and PIFs form a dynamic activation-suppression transcriptional module responsive to light and temperature cues. This antagonistic regulatory module provides a simple, direct mechanism through which environmental change can redirect transcriptional control of genes required for photosynthesis and photoprotection. In the regulation of photopigment biosynthesis genes, HY5 and PIFs do not operate alone, but with the circadian clock. However, sudden changes in light or temperature conditions can trigger changes in HY5 and PIFs abundance that adjust the expression of common target genes to optimise photosynthetic performance and growth