Abacus models for parabolic quotients of affine Weyl groups

We introduce abacus diagrams that describe minimal length coset representatives in affine Weyl groups of types B, C, and D. These abacus diagrams use a realization of the affine Weyl group of type C due to Eriksson to generalize a construction of James for the symmetric group. We also describe several combinatorial models for these parabolic quotients that generalize classical results in affine type A related to core partitions.Comment: 28 pages, To appear, Journal of Algebra. Version 2: Updated with referee's comment

The enumeration of fully commutative affine permutations

We give a generating function for the fully commutative affine permutations enumerated by rank and Coxeter length, extending formulas due to Stembridge and Barcucci--Del Lungo--Pergola--Pinzani. For fixed rank, the length generating functions have coefficients that are periodic with period dividing the rank. In the course of proving these formulas, we obtain results that elucidate the structure of the fully commutative affine permutations.Comment: 18 pages; final versio

Results and conjectures on simultaneous core partitions

An n-core partition is an integer partition whose Young diagram contains no hook lengths equal to n. We consider partitions that are simultaneously a-core and b-core for two relatively prime integers a and b. These are related to abacus diagrams and the combinatorics of the affine symmetric group (type A). We observe that self-conjugate simultaneous core partitions correspond to the combinatorics of type C, and use abacus diagrams to unite the discussion of these two sets of objects. In particular, we prove that (2n)- and (2mn+1)-core partitions correspond naturally to dominant alcoves in the m-Shi arrangement of type C_n, generalizing a result of Fishel--Vazirani for type A. We also introduce a major statistic on simultaneous n- and (n+1)-core partitions and on self-conjugate simultaneous (2n)- and (2n+1)-core partitions that yield q-analogues of the Coxeter-Catalan numbers of type A and type C. We present related conjectures and open questions on the average size of a simultaneous core partition, q-analogs of generalized Catalan numbers, and generalizations to other Coxeter groups. We also discuss connections with the cyclic sieving phenomenon and q,t-Catalan numbers.Comment: 17 pages; to appear in the European Journal of Combinatoric

Nanodiamond arrays on glass for quantification and fluorescence characterisation

Quantifying the variation in emission properties of fluorescent nanodiamonds is important for developing their wide-ranging applicability. Directed self-assembly techniques show promise for positioning nanodiamonds precisely enabling such quantification. Here we show an approach for depositing nanodiamonds in pre-determined arrays which are used to gather statistical information about fluorescent lifetimes. The arrays were created via a layer of photoresist patterned with grids of apertures using electron beam lithography and then drop-cast with nanodiamonds. Electron microscopy revealed a 90% average deposition yield across 3,376 populated array sites, with an average of 20 nanodiamonds per site. Confocal microscopy, optimised for nitrogen vacancy fluorescence collection, revealed a broad distribution of fluorescent lifetimes in agreement with literature. This method for statistically quantifying fluorescent nanoparticles provides a step towards fabrication of hybrid photonic devices for applications from quantum cryptography to sensing

Twelve Steps in the Way of Christ: A Postmodern Path for Spiritual Formation and Leadership Development

The goal of this paper is to help North American Church leaders stake a claim to postmodern faith by focusing on the celebrity and centrality of Christ and then calling, growing, maturing, and guiding believers into their own God-inspired journeys in the way of Christ. The postmodern Church’s decline, leadership crisis, and celebrity culture call forth a renewed emphasis on the Christ story as an ancient faith pilgrimage in order to help people mature and become leaders through the use of a new twelve-step path in the way of Christ. Many churched and unchurched pilgrims find themselves stuck in earlier stages of arrested spiritual development, unable to sustain growth, attain maturity as leaders, and finish their life well—thereby releasing their God-given gifts, passion, and destiny. A more dynamic and visual path of faith pilgrimage is required to capture interest and garner support from postmodern seekers of spirituality. The Twelve Steps in the Way of Christ draw on ancient pilgrimage and allegoric models, as well as Scripture and the life of Christ, to provide a clear understanding and direction for Christian followers and leaders. Both linear and cyclical, the steps offer distinctive ways to measure spirituality and a useful template for spiritual and leadership development yet allow enough room for one’s unique imprint and individual personality, life experience, and journey of faith. Today’s church leaders who seek to form faith in tangible, relational, and visual ways that appeal to postmoderns of the next generation will find the Twelve Steps a valuable tool and marker for spiritual assessment. The answer to the North American Church’s dilemma lies not in higher forms of worship or more sophisticated programs but in returning to the well-trod paths of becoming a people of pilgrimage and following the steps first made by the Son of God. Content Reader: Craig Miller, DMi

Structure of the doublet bands in doubly odd nuclei: The case of 128Cs^{128}Cs

The structure of the $\Delta J = 1$ doublet bands in $^{128}Cs$ is investigated within the framework of the Interacting Vector Boson Fermion Model (IVBFM). A new, purely collective interpretation of these bands is given on the basis of the used boson-fermion dynamical symmetry of the model. The energy levels of the doublet bands as well as the absolute $B(E2)$ and $B(M1)$ transition probabilities between the states of both yrast and yrare bands are described quite well. The observed odd-even staggering of both $B(M1)$ and $B(E2)$ values is reproduced by the introduction of an appropriate interaction term of quadrupole type, which produces such a staggering effect in the transition strengths. The calculations show that the appearance of doublet bands in certain odd-odd nuclei could be a consequence of the realization of a larger dynamical symmetry based on the non-compact supersymmetry group $OSp(2\Omega /12, R)$.Comment: 12 pages, 8 figure

Fast pick up technique for high quality heterostructures of bilayer graphene and hexagonal boron nitride

We present a fast method to fabricate high quality heterostructure devices by picking up crystals of arbitrary sizes. Bilayer graphene is encapsulated with hexagonal boron nitride to demonstrate this approach, showing good electronic quality with mobilities ranging from 17 000 cm^2/V/s at room temperature to 49 000 cm^2/V/s at 4.2 K, and entering the quantum Hall regime below 0.5 T. This method provides a strong and useful tool for the fabrication of future high quality layered crystal devices.Comment: 5 pages, 3 figure

Complement deficiencies limit CD20 monoclonal antibody treatment efficacy in CLL

Monoclonal antibodies (MAbs) form a central part of chronic lymphocytic leukaemia (CLL) treatment. We therefore evaluated whether complement defects in CLL patients reduced the induction of complement-dependent cytotoxicity (CDC) by using anti-CD20 MAbs rituximab (RTX) and ofatumumab (OFA). Ofatumumab elicited higher CDC levels than RTX in all CLL samples examined, particularly in poor prognosis cohorts (11q− and 17p−). Serum sample analyses revealed that 38.1% of patients were deficient in one or more complement components, correlating with reduced CDC responses. Although a proportion of patients with deficient complement levels initially induced high levels of CDC, on secondary challenge CDC activity in sera was significantly reduced, compared with that in normal human serum (NHS; P<0.01; n=52). In addition, a high CLL cell number contributed to rapid complement exhaustion. Supplementing CLL serum with NHS or individual complement components, particularly C2, restored CDC on secondary challenge to NHS levels (P<0.0001; n=9). In vivo studies revealed that complement components were exhausted in CLL patient sera post RTX treatment, correlating with an inability to elicit CDC. Supplementing MAb treatment with fresh-frozen plasma may therefore maintain CDC levels in CLL patients with a complement deficiency or high white blood cell count. This study has important implications for CLL patients receiving anti-CD20 MAb therapy

Controlling spin relaxation in hexagonal BN-encapsulated graphene with a transverse electric field

We experimentally study the electronic spin transport in hBN encapsulated single layer graphene nonlocal spin valves. The use of top and bottom gates allows us to control the carrier density and the electric field independently. The spin relaxation times in our devices range up to 2 ns with spin relaxation lengths exceeding 12 $\mu$m even at room temperature. We obtain that the ratio of the spin relaxation time for spins pointing out-of-plane to spins in-plane is $\tau_{\bot} / \tau_{||} \approx$ 0.75 for zero applied perpendicular electric field. By tuning the electric field this anisotropy changes to $\approx$0.65 at 0.7 V/nm, in agreement with an electric field tunable in-plane Rashba spin-orbit coupling

The Luminosity Dependence of Quasar Clustering

We investigate the luminosity dependence of quasar clustering, inspired by numerical simulations of galaxy mergers that incorporate black hole growth. These simulations have motivated a new interpretation of the quasar luminosity function. In this picture, the bright end of the quasar luminosity function consists of quasars radiating nearly at their peak luminosities, while the faint end consists mainly of very similar sources, but at dimmer phases in their evolution. We combine this model with the statistics of dark matter halos that host quasar activity. We find that, since bright and faint quasars are mostly similar sources seen in different evolutionary stages, a broad range in quasar luminosities corresponds to only a narrow range in the masses of quasar host halos. On average, bright and faint quasars reside in similar host halos. Consequently, we argue that quasar clustering should depend only weakly on luminosity. This prediction is in qualitative agreement with recent measurements of the luminosity dependence of the quasar correlation function (Croom et al. 2005) and the galaxy-quasar cross-correlation function (Adelberger & Steidel 2005). Future precision clustering measurements from SDSS and 2dF, spanning a large range in luminosity, should provide a strong test of our model.Comment: 9 pages, 4 figures, submitted to Ap