Calculating conjugacy classes in Sylow p-subgroups of finite Chevalley groups of rank six and seven

Let G(q) be a finite Chevalley group, where q is a power of a good prime p, and let U(q) be a Sylow p-subgroup of G(q). Then a generalized version of a conjecture of Higman asserts that the number k(U(q)) of conjugacy classes in U(q) is given by a polynomial in q with integer coefficients. In an earlier paper, the first and the third authors developed an algorithm to calculate the values of k(U(q)). By implementing it into a computer program using GAP, they were able to calculate k(U(q)) for G of rank at most 5, thereby proving that for these cases k(U(q)) is given by a polynomial in q. In this paper we present some refinements and improvements of the algorithm that allow us to calculate the values of k(U(q)) for finite Chevalley groups of rank six and seven, except E_7. We observe that k(U(q)) is a polynomial, so that the generalized Higman conjecture holds for these groups. Moreover, if we write k(U(q)) as a polynomial in q-1, then the coefficients are non-negative. Under the assumption that k(U(q)) is a polynomial in q-1, we also give an explicit formula for the coefficients of k(U(q)) of degrees zero, one and two.Comment: 16 page

On the coadjoint orbits of maximal unipotent subgroups of reductive groups

Let G be a simple algebraic group defined over an algebraically closed field of characteristic 0 or a good prime for G. Let U be a maximal unipotent subgroup of G and \u its Lie algebra. We prove the separability of orbit maps and the connectedness of centralizers for the coadjoint action of U on (certain quotients of) the dual \u* of \u. This leads to a method to give a parametrization of the coadjoint orbits in terms of so-called minimal representatives which form a disjoint union of quasi-affine varieties. Moreover, we obtain an algorithm to explicitly calculate this parametrization which has been used for G of rank at most 8, except E8. When G is defined and split over the field of q elements, for q the power of a good prime for G, this algorithmic parametrization is used to calculate the number k(U(q), \u*(q)) of coadjoint orbits of U(q) on \u*(q). Since k(U(q), \u*(q)) coincides with the number k(U(q)) of conjugacy classes in U(q), these calculations can be viewed as an extension of the results obtained in our earlier paper. In each case considered here there is a polynomial h(t) with integer coefficients such that for every such q we have k(U(q)) = h(q).Comment: 14 pages; v2 23 pages; to appear in Transformation Group

Hybrid Local-Order Mechanism for Inversion Symmetry Breaking

Using classical Monte Carlo simulations, we study a simple statistical mechanical model of relevance to the emergence of polarisation from local displacements on the square and cubic lattices. Our model contains two key ingredients: a Kitaev-like orientation-dependent interaction between nearest neighbours, and a steric term that acts between next-nearest neighbours. Taken by themselves, each of these two ingredients is incapable of driving long-range symmetry breaking, despite the presence of a broad feature in the corresponding heat capacity functions. Instead each component results in a "hidden" transition on cooling to a manifold of degenerate states, the two manifolds are different in the sense that they reflect distinct types of local order. Remarkably, their intersection---\emph{i.e.} the ground state when both interaction terms are included in the Hamiltonian---supports a spontaneous polarisation. In this way, our study demonstrates how local ordering mechanisms might be combined to break global inversion symmetry in a manner conceptually similar to that operating in the "hybrid" improper ferroelectrics. We discuss the relevance of our analysis to the emergence of spontaneous polarisation in well-studied ferroelectrics such as BaTiO$_3$ and KNbO$_3$.Comment: 8 pages, 8 figure

Help-seeking behaviors and mental well-being of first year undergraduate university students

University students demonstrate poor help-seeking behaviors for their mental health, despite often reporting low levels of mental well-being. The aims of this study were to examine the help-seeking intentions and experiences of first year university students in terms of their mental well-being, and to explore these students’ views on formal (e.g. psychiatrists) and informal (e.g. friends) help-seeking. Students from a university in the Republic of Ireland (n=220) completed an online questionnaire which focused on mental well-being and help-seeking behaviors. Almost a third of students had sought help from a mental health professional. Very few students reported availing of university/online supports. Informal sources of help were more popular than formal sources, and those who would avail and had availed of informal sources demonstrated higher well-being scores. Counselors were the source of professional help most widely used. General practitioners, chaplains, social workers, and family therapists were rated the most helpful. Those with low/average well-being scores were less likely to seek help than those with higher scores. Findings indicate the importance of enhancing public knowledge of mental health issues, and for further examination of students’ knowledge of help-seeking resources in order to improve the help-seeking behaviors and mental well-being of this population group

Orbital Dimer Model for Spin-Glass State in Y2_2Mo2_2O7_7

The formation of a spin glass usually requires both structural disorder and frustrated magnetic interactions. Consequently, the origin of spin-glass behaviour in Y$_2$Mo$_2$O$_7$ $-$ in which magnetic Mo$^{4+}$ ions occupy a frustrated pyrochlore lattice with minimal compositional disorder $-$ has been a longstanding question. Here, we use neutron and X-ray pair-distribution function (PDF) analysis to develop a disorder model that resolves apparent incompatibilities between previously-reported PDF, EXAFS and NMR studies and provides a new and physical mechanism for spin-glass formation. We show that Mo$^{4+}$ ions displace according to a local "2-in/2-out" rule on each Mo$_4$ tetrahedron, driven by orbital dimerisation of Jahn-Teller active Mo$^{4+}$ ions. Long-range orbital order is prevented by the macroscopic degeneracy of dimer coverings permitted by the pyrochlore lattice. Cooperative O$^{2-}$ displacements yield a distribution of Mo$-$O$-$Mo angles, which in turn introduces disorder into magnetic interactions. Our study demonstrates experimentally how frustration of atomic displacements can assume the role of compositional disorder in driving a spin-glass transition.Comment: 6 pages, 3 figure

Public attitudes to transport: Knowledge review of existing evidence

Report for Department for Transpor

Hubble Space Telescope Observations of the Oldest Star Clusters in the LMC

We present V, V-I color-magnitude diagrams (CMDs) for three old star clusters in the Large Magellanic Cloud (LMC): NGC 1466, NGC 2257 and Hodge 11. Our data extend about 3 magnitudes below the main-sequence turnoff, allowing us to determine accurate relative ages and the blue straggler frequencies. Based on a differential comparison of the CMDs, any age difference between the three LMC clusters is less than 1.5 Gyr. Comparing their CMDs to those of M 92 and M 3, the LMC clusters, unless their published metallicities are significantly in error, are the same age as the old Galactic globulars. The similar ages to Galactic globulars are shown to be consistent with hierarchial clustering models of galaxy formation. The blue straggler frequencies are also similar to those of Galactic globular clusters. We derive a true distance modulus to the LMC of (m-M)=18.46 +/- 0.09 (assuming (m-M)=14.61 for M 92) using these three LMC clusters.Comment: 22 pages; to be published in Ap

Behaviour of infants with Down syndrome and their mothers in the still-face procedure

There has been limited study of how the constitutional characteristics of infants with Down syndrome (DS) influence the patterning of their relations with caregivers. To assess natural and perturbed interactions between infants with DS and their mothers, we tested ten 6-month-old infants with DS and 20 typically developing (TD) 4-month-old of similar mental age. Participants were videotaped with their mothers in a natural face-to-face interaction, a brief period when the mothers adopted a still-face, and a subsequent reengagement phase. There was little to distinguish the infants in the initial phase of natural interaction, but the mothers of infants with DS were more likely to show assertive warmth, and unlike in the case of mothers of TD infants, high maternal directiveness tended to be associated with lower levels of infant looking and lack of fussing. During the still-face episode, infants of both groups showed reduced looking and smiling, although infants with DS tended to show lower levels of fussing and fewer in this group showed fussing in the reengagement phase. Therefore DS infants were somewhat similar to TD infants of comparable mental age in being responsive to the still-face procedure, but showed indications of group differences in intense emotional reactivity

Exploration of Antiferromagnetic CoO and NiO using Reverse Monte Carlo Total Neutron Scattering Refinement

The atomic and magnetic structures of CoO and NiO have been probed using reverse Monte Carlo refinements of neutron total scattering data. The results obtained show that the known magnetic structure for NiO can be recovered by the reverse Monte Carlo process starting from random spin configurations, but it is insensitive to the spin direction in the {111} ferromagnetic planes. Refinements of the magnetic structure of CoO starting from random spin configurations result in collinear or non-collinear magnetic structure, consistent with those reported by other techniques. Starting from an ordered collinear spin structure for CoO and NiO leads to different results than when starting from a random arrangement of spins, evidence for configurational bias that highlights the need to take care when selecting a starting model for reverse Monte Carlo refinements of magnetic structures

A near-field optical microscopy nanoarray

Multiplexing near-field scanning optical microscopy (NSOM) by the use of a nanoarray with parallel imaging is studied. The fabrication, characterization, and utilization of nanoarrays with {approximately} 100 nm diameter apertures spaced 500 nm center-to- center is presented. Extremely uniform nanoarrays with {approximately} 10{sup 8} apertures were fabricated by electron beam lithography and reactive ion etching. The nanoarrays were characterized by atomic force microscopy (AFM) and scanning electron microscopy (SEM). In this paper we utilize these nanoarrays in a laser-illuminated microscope with parallel detection on a charge- coupled device (CCD). Detection of B-phycoerythrin (B-PE) molecules using near-field illumination is presented. In principle, our system can be used to obtain high lateral resolution NSOM images over a wide-field of view (e.g. 50-100 {mu}m) within seconds