Acknowledgement of priority - A fractional Helly theorem for boxes

In our recent paper [1] we prove a fractional Helly type theorem for boxes in Rd. This short note is to acknowledge priority: in 1980 Meir Katchalski [4] proved exactly the same result and in 1988 Jürgen Eckhoff [2] proved the same result in much more generality. In fact, Eckhoff established an upper bound theorem for the f -vectors of finite families of boxes in Rd from which his result is derived. Besides apologies for our ignorance we would like to mention that Eckhoff extended his results further in a more recent paper [3]

COLOURFUL AND FRACTIONAL (p, q)-THEOREMS

Abstract. Let p ≥ q ≥ d+1 be positive integers and let F be a finite family of convex sets in Rd. Assume that the elements of F are coloured with p colours. A p-element subset of F is heterochromatic if it contains exactly one element of each colour. The family F has the heterochromatic (p, q)-property if in every heterochromatic p-element subset there are at least q elements that have a point in common. We show that, under the heterochromatic (p, q)-condition, some colour class can be pierced by a finite set whose size we estimate from above in terms of d, p, and q. This is a colourful version of the famous (p, q)theorem. (We prove a colourful variant of the fractional Helly theorem along the way.) A fractional version of the same problem is when the (p, q)-condition holds for all but an α fraction of the p-tuples in F. We show that, in the case that d = 1, all but a β fraction of the elements of F can be pierced by p − q + 1 points. Here β depends on α and p, q, and β → 0 as α goes to zero. 1

Work-based learning at higher education level: value, practice and critique

Since the 1980s there has been significant growth in the engagement of higher education with workforce development, with among other things the emergence of a distinct if varied area of provision commonly referred to as work-based learning. Recent examination of practice and literature indicates a growing sophistication in the way that work-based learning is being theorised and facilitated in higher education, with its gradual emergence as a distinct field of practice and study supported by relevant pedagogies and concepts of curriculum. Tensions continue to exist between the demands and opportunities provided by the workplace and the need to develop capable practice, support personal development and maintain academic validity; however, universities are beginning to engage with these issues at a deeper level than that suggested by simple notions of employer engagement and skills development, and the evidence indicates that well-designed work-based programmes are both effective and robust

Every large point set contains many collinear points or an empty pentagon

We prove the following generalised empty pentagon theorem: for every integer l 2, every sufficiently large set of points in the plane contains ̀ collinear points or an empty pentagon. As an application, we settle the next open case of the "big line or big clique" conjecture of Kára, Pór, and Wood [Discrete Comput. Geom. 34(3):497-506, 2005]

Barley starch

This thesis examined barley amylopectin structure and looked for correlations between the structure and physical properties of starch. The structure of amylopectin and gelatinisation and retrogradation of starch were studied in 10 different barley cultivars/breeding lines with differing genetic background. Amylopectin is built up of thousands of chains of glucose monomers, organised into clusters. The detailed fine structure of amylopectin was studied by isolating clusters of amylopectin and their building blocks, which are the tightly branched units building up the clusters. Barley cultivars/breeding lines possessing the amo1 mutation had fewer long chains of DP≥38 in amylopectin and more large building blocks. The structure of building blocks was rather conserved between the different barley cultivars/breeding lines studied and was categorized into different size groups. These different building blocks were shown to be randomly distributed in the amylopectin molecule. The C-chains in amylopectin can be of any length and are a category of chains different from the B-chains. The backbone in amylopectin consists of a special type of B-chains which, when cleaved by α-amylase, become chains of a similar type to C-chains. Gelatinisation and retrogradation (recrystallisation of gelatinised starch) of barley starch was studied by differential scanning calorimetry. The amo1 mutation resulted in a broader gelatinisation temperature range and a higher enthalpy of retrogradation. Other structural features were also found to influence the physical properties of starch. Small clusters and denser structure of the building blocks resulted in higher gelatinisation temperature. Fast retrogradation was observed in barley which had amylopectin with shorter chains and many large building blocks consisting of many chains. Amylopectin structure was also studied in developing barley kernels. Three barley cultivars/breeding lines were grown in a phytotron and kernels were harvested at 9, 12 and 24 days after flowering. The results showed that amylopectin synthesized at later stages of development had a more tightly branched structure. Expression of the enzymes involved in starch biosynthesis is also known to change during endosperm development

Effects of a High-Frequency Augmented Acoustic Environment on Parvalbumin Immunolabeling in the Anteroventral Cochlear Nucleus of BDA/2J and C57BL/6J Mice

Neurons in the anteroventral cochlear nucleus (AVCN) of DBA/2J (D2) and C57BL/6J (B6) mice were immunohistochemically labeled for the calcium binding protein parvalbumin (PV). Prior to this, mice were treated for 12 h nightly with a “high-frequency” augmented acoustic environment (HAAE: repetitive bursts of a 70 dB sound pressure level, half-octave noise band centered at 20 kHz). This was done during the period that hearing loss occurs: pre-weaning to 55 days in D2 mice and weaning to 9 months in B6 mice. After HAAE treatment in D2 mice, high-frequency hearing loss was ameliorated and fewer PV-labeled neurons were found in the AVCN compared to untreated controls. HAAE treatment in B6 mice exacerbated high-frequency hearing loss, yet the number of PV-labeled AVCN neurons in treated mice did not differ significantly from that of control mice. The findings suggest that HAAE treatment provides relief from physiological stress caused by deprivation of auditory input from the impaired cochlea