BSRLM geometry working group: establishing a professional development network to support teachers using dynamic mathematics software GeoGebra

The embedding of technology into mathematics teaching is known to be a complex process. GeoGebra, an open-source dynamic mathematics software that incorporates geometry and algebra into a single package, is proving popular with teachers - yet solely having access to such technology can be insufficient for the successful integration of technology into teaching. This paper reports on aspects of an NCETM-funded project that involved nine experienced teachers collaborating in developing ways of providing professional development and support for other teachers across England in the use of GeoGebra in teaching mathematics. The participating teachers tried various approaches to better integrate the use of GeoGebra into the mathematics curriculum (especially in geometry) and they designed and led professional development workshops for other teachers. As a result, the project initiated a core group which has started to be a source of support and professional development for other teachers of mathematics in the use of GeoGebra

Rational Generalised Moonshine from Abelian Orbifoldings of the Moonshine Module

We consider orbifoldings of the Moonshine Module with respect to the abelian group generated by a pair of commuting Monster group elements with one of prime order $p=2,3,5,7$ and the other of order $pk$ for $k=1$ or $k$ prime. We show that constraints arising from meromorphic orbifold conformal field theory allow us to demonstrate that each orbifold partition function with rational coefficients is either constant or is a hauptmodul for an explicitly found modular fixing group of genus zero. We thus confirm in the cases considered the Generalised Moonshine conjectures for all rational modular functions for the Monster centralisers related to the Baby Monster, Fischer, Harada-Norton and Held sporadic simple groups. We also derive non-trivial constraints on the possible Monster conjugacy classes to which the elements of the orbifolding abelian group may belong.Comment: 40 pages, Improved versio

Conserved Helix-Flanking Prolines Modulate Intrinsically Disordered Protein:Target Affinity by Altering the Lifetime of the Bound Complex.

Appropriate integration of cellular signals requires a delicate balance of ligand-target binding affinities. Increasing the level of residual structure in intrinsically disordered proteins (IDPs), which are overrepresented in these cellular processes, has been shown previously to enhance binding affinities and alter cellular function. Conserved proline residues are commonly found flanking regions of IDPs that become helical upon interacting with a partner protein. Here, we mutate these helix-flanking prolines in p53 and MLL and find opposite effects on binding affinity upon an increase in free IDP helicity. In both cases, changes in affinity were due to alterations in dissociation, not association, rate constants, which is inconsistent with conformational selection mechanisms. We conclude that, contrary to previous suggestions, helix-flanking prolines do not regulate affinity by modulating the rate of complex formation. Instead, they influence binding affinities by controlling the lifetime of the bound complex

Genus Two Partition and Correlation Functions for Fermionic Vertex Operator Superalgebras I

We define the partition and $n$-point correlation functions for a vertex operator superalgebra on a genus two Riemann surface formed by sewing two tori together. For the free fermion vertex operator superalgebra we obtain a closed formula for the genus two continuous orbifold partition function in terms of an infinite dimensional determinant with entries arising from torus Szeg\"o kernels. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties. Using the bosonized formalism, a new genus two Jacobi product identity is described for the Riemann theta series. We compute and discuss the modular properties of the generating function for all $n$-point functions in terms of a genus two Szeg\"o kernel determinant. We also show that the Virasoro vector one point function satisfies a genus two Ward identity.Comment: A number of typos have been corrected, 39 pages. To appear in Commun. Math. Phy

Torus n-Point Functions for R\mathbb{R}-graded Vertex Operator Superalgebras and Continuous Fermion Orbifolds

We consider genus one n-point functions for a vertex operator superalgebra with a real grading. We compute all n-point functions for rank one and rank two fermion vertex operator superalgebras. In the rank two fermion case, we obtain all orbifold n-point functions for a twisted module associated with a continuous automorphism generated by a Heisenberg bosonic state. The modular properties of these orbifold n-point functions are given and we describe a generalization of Fay's trisecant identity for elliptic functions.Comment: 50 page

Structure, Dynamics, and Evolution of the Intrinsically Disordered p53 Transactivation Domain

in numerous disease states, including cancers and neurodegenerative diseases. All proteins are dynamic in nature, occupying a range of conformational flexibilities. This inherent flexibility is required for their function, with ordered proteins and IDPs representing the least flexible, and most flexible, respectively. As such IDPs possess little to no stable tertiary or secondary structure, they instead form broad ensembles of heterogeneous structures, which fluctuate over multiple time scales. Although IDPs often lack stable secondary structure they can assume a more stable structure in the presence of their binding partners in a coupled folding binding reaction. The phenomenon of the dynamic behavior of IDPs is believed to confer several functional advantages but remains poorly understood. To that end the dynamic and structural properties of a family of IDPs - p53 transactivation domains (TAD) was measured and compared with the sequence divergence. Interestingly we were able to find stronger correlations between the dynamic properties and the sequence divergence than between the structure and sequence, suggesting that the dynamic properties are the primary trait being xiii conserved by evolution. These correlations were strongest within clusters of the IDPs that correlated with known protein binding sites. Additionally, we show strong correlations between the several available disorder predictors and the backbone dynamics of this family of IDPs. This indicates the potential of predicting the dynamic behavior of proteins, which may be beneficial in future drug design. The limited number of atomic models currently determined for IDPs hampers understanding of how their amino acid sequences dictate the structural ensembles they adopt. The current dearth of atomic models for IDPs makes it difficult to test the following hypotheses: 1. The structural ensembles of IDPs are dictated by local interactions. 2. The structural ensembles of IDPs will be similar above a certain sequence identity threshold. Based on the premise that sequence determines structure, structural ensembles were determined and compared for a set of homologous IDPs. Utilizing orthologues allows for the identification of important structural features and behaviors by virtue of their conservation. A new methodology of creating ensembles was implemented that broadly samples conformational space. This allowed us to find recurring local structural features within the structural ensembles even between the more distantly related homologues that were processed. This method of ensemble creation is also the first method to show convergence of secondary structural characteristics between discrete ensembles