A divisibility property of binomial coefficients viewed as an elementary sieve

The triangular array of binomial coefficients 012301111212131331… is said to have undergone a j-shift if the r-th row of the triangle is shifted rj units to the right (r=0,1,2,…). Mann and Shanks have proved that in a 2-shifted array a column number c>1 is prime if and only if every entry in the c-th column is divisible by its row number. Extensions of this result to j-shifted arrays where j>2 are considered in this paper. Moreover, an analog of the criterion of Mann and Shanks [2] is given which is valid for arbitrary arithmetic progressions

Extension of a theorem of Cauchy and Jacobi

AbstractLet q and p be prime with q = a2 + b2 ≡ 1 (mod 4), a ≡ 1 (mod 4), and p = qf + 1. In the nineteenth century Cauchy (Mém. Inst. France 17 (1840), 249–768) and Jacobi (J. für Math. 30 (1846), 166–182) generalized the work of earlier authors, who had determined certain binomial coefficients (mod p) (see H. J. S. Smith, “Report on the Theory of Numbers,” Chelsea, 1964), by determining two products of factorials given by Πk kf! (mod p = qf + 1) where k runs through the quadratic residues and the quadratic non-residues (mod q), respectively. These determinations are given in terms of parameters in representations of ph or of 4ph by binary quadratic forms. A remarkable feature of these results is the fact that the exponent h coincides with the class number of the related quadratic field. In this paper C. R. Mathews' (Invent. Math. 54 (1979), 23–52) recent explicit evaluation of the quartic Gauss sum is used to determine four products of factorials (mod p = qf + 1, q ≡ 5 (mod 8) > 5), given by Πk kf! where k runs through the quartic residues (mod q) and the three cosets which may be formed with respect to this subgroup. These determinations appear to be considerably more difficult. They are given in terms of parameters in representations of 16ph by quaternary quadratic forms. Stickelberger's theorem is required to determine the exponent h which is shown to be closely related to the class number of the imaginary quartic field Q(i√2q + 2a√q), q = a2 + b2 ≡ 5 (mod 8), a odd

Seasonal effects to metallothionein responses to metal exposure in a naturalised population of Ruditapes philippinarum in a semi-enclosed estuarine environment

The Manila clam (Ruditapes philippinarum), an invasive species in Northern Europe, can be used as a bioindicator of metal pollution. Seasonal effects on metallothionein (MT) production have not been considered in this species at the northernmost extent of its European distribution. This study assesses the annual seasonal effects on MT and metal concentrations in R. philippinarum from Poole Harbour, UK. R. philippinarum were collected in winter, spring, summer, and autumn throughout 2015, and MT and metal concentrations, as well as biotic and abiotic variables, were quantified. During winter, linear regression analysis showed significant positive relationships between tissue metal and MT concentrations. However, during spring and summer, these relationships were mostly insignificant. MT concentrations during spring had significant positive relationships with tissue and whole weight. Significant positive relationships were also observed between MT and condition index, during summer. During spring and summer, biotic factors seem to override the role of MT as a detoxification mechanism for metal exposure in this species. This is probably due to an increase in MT concentration in spring caused by gametogenesis, associated with increased tissue weight as the gonads expand. A depletion of energy resources, or physical stressors such as heat, may be attributed to the reduced MT production in clams of poor body condition in summer. The evidence from this study suggests that MT may only be a useful biomarker of metal pollution during winter in R. philippinarum in the UK. This verifies the natural variability of MT in this species at high latitudes, and highlights the potential and limits to a widely available bioindicator of metal pollution

A laser based accelerator for ultracold atoms

We present first results on our implementation of a laser based accelerator for ultracold atoms. Atoms cooled to a temperature of 420 nK are confined and accelerated by means of laser tweezer beams and the atomic scattering is directly observed in laser absorption imaging. The optical collider has been characterized using Rb87 atoms in the |F=2,mF=2> state, but the scheme is not restricted to atoms in any particular magnetic substates and can readily be extended to other atomic species as well.Comment: (c) 2012 The Optical Society, 3 pages, 4 figures, 1 movie lin

Development of a novel bi-specific monoclonal antibody approach for tumour targeting

To overcome the disadvantages of bi-specific antibody methodologies in vivo, a novel antibody approach has been designed to improve tumour targeting and effector to target ratio. The technique involves biotinylated anti-CD3 Fab fragments and streptavidinylated anti-tumour monoclonal antibodies (mAbs) that can spontaneously form cross-links. We describe here a method for the direct cross-linking of sulphydryl-conjugated HMFG1 (anti-MUC1 mucin mAb) to streptavidin by sulphosuccinimidyl-4-(N-maleimidomethyl) cyclohexane- 1-carboxylate. Fab fragments generated by papain digestion of the 1452C11 antibody (anti-CD3 mAb without Fc to avoid peripheral activation of T-cells) were biotinylated with NHS-Iminobiotin. MUC1-transfected BALB/c breast cancer cell lines 413BCR and 425CCR and the parental cell line (410.4) were labelled with streptavidinylated mouse anti-MUC1 mucin mAb. BALB/c effector T-cells were separately labelled with biotinylated anti-CD3 Fab fragments (1452C11) and mixed with tumour cells in different effector to target ratios. Percentage of killing was assessed using the 51Cr cytotoxicity assay. Seventy per cent lysis was measured in the case of 413BCR (high MUC1 mucin expressor) and 40% in the case of 425CCR (low expressor) cell line. No lysis was apparent in the MUC1 negative cell line. These results demonstrate that the novel T-cell redirecting approach we have developed can produce effective immune lysis of target cells in vitro. © 1999 Cancer Research Campaig

An investigation of vocational progression pathways for young people and adults in building crafts and hospitality CoVEs: a London case study

This report was commissioned by the Learning and Skills Development Agency’s (LSDA’s) strategic programme of research and development, funded by the Learning and Skills Council (LSC). Project aims and objectives The principal aim of this project is to explore how vocational pathways have been developed and enhanced in two occupational areas – building crafts and hospitality – among 14–19 year olds and employed adults in London

Public perceptions of internet, familial and localised sexual grooming: predicting perceived prevalence and safety

This paper examines public perceptions of three sexual grooming types: computer-mediated sexual grooming (CMSG), familial sexual grooming (FSG) and localised sexual grooming (LSG). Using data from a national survey of 557 respondents from the United Kingdom, we tested models that predicted perceptions of the prevalence of CMSG, FSG and LSG and the perceived safety of internet, familial and localised grooming spaces. Media-related factors were the most significant in predicting higher levels of perceived prevalence of CMSG and disagreement in relation to safety of internet and public spaces. Knowledge of a grooming victim was most significant in predicting higher levels of perceived prevalence of FSG and LSG and higher levels disagreement in relation to the safety of the home. The findings suggest that the public express too little concern over familial sexual grooming and that initiatives should be introduced to make citizens more aware of the distinctions between types of sexual grooming behaviours, settings and offenders

Economies of space and the school geography curriculum

This paper is about the images of economic space that are found in school curricula. It suggests the importance for educators of evaluating these representations in terms of the messages they contain about how social processes operate. The paper uses school geography texts in Britain since the 1970s to illustrate the different ways in which economic space has been represented to students, before exploring some alternative resources that could be used to provide a wider range of representations of economic space. The paper highlights the continued importance of understanding the politics of school knowledge