Fast computation of Bernoulli, Tangent and Secant numbers

We consider the computation of Bernoulli, Tangent (zag), and Secant (zig or Euler) numbers. In particular, we give asymptotically fast algorithms for computing the first n such numbers in O(n^2.(log n)^(2+o(1))) bit-operations. We also give very short in-place algorithms for computing the first n Tangent or Secant numbers in O(n^2) integer operations. These algorithms are extremely simple, and fast for moderate values of n. They are faster and use less space than the algorithms of Atkinson (for Tangent and Secant numbers) and Akiyama and Tanigawa (for Bernoulli numbers).Comment: 16 pages. To appear in Computational and Analytical Mathematics (associated with the May 2011 workshop in honour of Jonathan Borwein's 60th birthday). For further information, see http://maths.anu.edu.au/~brent/pub/pub242.htm

Power-law distributions and Levy-stable intermittent fluctuations in stochastic systems of many autocatalytic elements

A generic model of stochastic autocatalytic dynamics with many degrees of freedom $w_i$ $i=1,...,N$ is studied using computer simulations. The time evolution of the $w_i$'s combines a random multiplicative dynamics $w_i(t+1) = \lambda w_i(t)$ at the individual level with a global coupling through a constraint which does not allow the $w_i$'s to fall below a lower cutoff given by $c \cdot \bar w$, where $\bar w$ is their momentary average and $0<c<1$ is a constant. The dynamic variables $w_i$ are found to exhibit a power-law distribution of the form $p(w) \sim w^{-1-\alpha}$. The exponent $\alpha (c,N)$ is quite insensitive to the distribution $\Pi(\lambda)$ of the random factor $\lambda$, but it is non-universal, and increases monotonically as a function of $c$. The "thermodynamic" limit, N goes to infty and the limit of decoupled free multiplicative random walks c goes to 0, do not commute: $\alpha(0,N) = 0$ for any finite $N$ while $\alpha(c,\infty) \ge 1$ (which is the common range in empirical systems) for any positive $c$. The time evolution of ${\bar w (t)}$ exhibits intermittent fluctuations parametrized by a (truncated) L\'evy-stable distribution $L_{\alpha}(r)$ with the same index $\alpha$. This non-trivial relation between the distribution of the $w_i$'s at a given time and the temporal fluctuations of their average is examined and its relevance to empirical systems is discussed.Comment: 7 pages, 4 figure

Conceptualising sustainability in UK urban Regeneration: a discursive Formation

Despite the wide usage and popular appeal of the concept of sustainability in UK policy, it does not appear to have challenged the status quo in urban regeneration because policy is not leading in its conceptualisation and therefore implementation. This paper investigates how sustainability has been conceptualised in a case-based research study of the regeneration of Eastside in Birmingham, UK, through policy and other documents, and finds that conceptualisations of sustainability are fundamentally limited. The conceptualisation of sustainability operating within urban regeneration schemes should powerfully shape how they make manifest (or do not) the principles of sustainable development. Documents guide, but people implement regeneration—and the disparate conceptualisations of stakeholders demonstrate even less coherence than policy. The actions towards achieving sustainability have become a policy ‘fix’ in Eastside: a necessary feature of urban policy discourse that is limited to solutions within market-based constraints

Couplings of light I=0 scalar mesons to simple operators in the complex plane

The flavour and glue structure of the light scalar mesons in QCD are probed by studying the couplings of the I=0 mesons $\sigma(600)$ and $f_0(980)$ to the operators $\bar{q}q$, $\alpha_s G^2$ and to two photons. The Roy dispersive representation for the $\pi\pi$ amplitude $t_0^0(s)$ is used to determine the pole positions as well as the residues in the complex plane. On the real axis, $t_0^0$ is constrained to solve the Roy equation together with elastic unitarity up to the K\Kbar threshold leading to an improved description of the $f_0(980)$. The problem of using a two-particle threshold as a matching point is discussed. A simple relation is established between the coupling of a scalar meson to an operator $j_S$ and the value of the related pion form-factor computed at the resonance pole. Pion scalar form-factors as well as two-photon partial-wave amplitudes are expressed as coupled-channel Omn\`es dispersive representations. Subtraction constants are constrained by chiral symmetry and experimental data. Comparison of our results for the $\bar{q}q$ couplings with earlier determinations of the analogous couplings of the lightest I=1 and $I=1/2$ scalar mesons are compatible with an assignment of the $\sigma$, $\kappa$, $a_0(980)$, $f_0(980)$ into a nonet. Concerning the gluonic operator $\alpha_s G^2$ we find a significant coupling to both the $\sigma$ and the $f_0(980)$.Comment: 31 pages, 5 figure

Stellar structure and compact objects before 1940: Towards relativistic astrophysics

Since the mid-1920s, different strands of research used stars as "physics laboratories" for investigating the nature of matter under extreme densities and pressures, impossible to realize on Earth. To trace this process this paper is following the evolution of the concept of a dense core in stars, which was important both for an understanding of stellar evolution and as a testing ground for the fast-evolving field of nuclear physics. In spite of the divide between physicists and astrophysicists, some key actors working in the cross-fertilized soil of overlapping but different scientific cultures formulated models and tentative theories that gradually evolved into more realistic and structured astrophysical objects. These investigations culminated in the first contact with general relativity in 1939, when J. Robert Oppenheimer and his students George Volkoff and Hartland Snyder systematically applied the theory to the dense core of a collapsing neutron star. This pioneering application of Einstein's theory to an astrophysical compact object can be regarded as a milestone in the path eventually leading to the emergence of relativistic astrophysics in the early 1960s.Comment: 83 pages, 4 figures, submitted to the European Physical Journal

Search for direct production of charginos and neutralinos in events with three leptons and missing transverse momentum in √s = 7 TeV pp collisions with the ATLAS detector

A search for the direct production of charginos and neutralinos in final states with three electrons or muons and missing transverse momentum is presented. The analysis is based on 4.7 fb−1 of proton–proton collision data delivered by the Large Hadron Collider and recorded with the ATLAS detector. Observations are consistent with Standard Model expectations in three signal regions that are either depleted or enriched in Z-boson decays. Upper limits at 95% confidence level are set in R-parity conserving phenomenological minimal supersymmetric models and in simplified models, significantly extending previous results

Measurement of the correlation between flow harmonics of different order in lead-lead collisions at √sNN = 2.76 TeV with the ATLAS detector

Correlations between the elliptic or triangular flow coefficients vm (m=2 or 3) and other flow harmonics vn (n=2 to 5) are measured using √sNN=2.76 TeV Pb+Pb collision data collected in 2010 by the ATLAS experiment at the LHC, corresponding to an integrated luminosity of 7 μb−1. The vm−vn correlations are measured in midrapidity as a function of centrality, and, for events within the same centrality interval, as a function of event ellipticity or triangularity defined in a forward rapidity region. For events within the same centrality interval, v3 is found to be anticorrelated with v2 and this anticorrelation is consistent with similar anticorrelations between the corresponding eccentricities, ε2 and ε3. However, it is observed that v4 increases strongly with v2, and v5 increases strongly with both v2 and v3. The trend and strength of the vm−vn correlations for n=4 and 5 are found to disagree with εm−εn correlations predicted by initial-geometry models. Instead, these correlations are found to be consistent with the combined effects of a linear contribution to vn and a nonlinear term that is a function of v22 or of v2v3, as predicted by hydrodynamic models. A simple two-component fit is used to separate these two contributions. The extracted linear and nonlinear contributions to v4 and v5 are found to be consistent with previously measured event-plane correlations

Search for invisible decays of the Higgs boson produced in association with a hadronically decaying vector boson in pp collisions at √s=8 TeV with the ATLAS detector

A search for Higgs boson decays to invisible particles is performed using 20.3 fb −1 of pp collision data at a centre-of-mass energy of 8 TeV recorded by the ATLAS detector at the Large Hadron Collider. The process considered is Higgs boson production in association with a vector boson (V=W or Z) that decays hadronically, resulting in events with two or more jets and large missing transverse momentum. No excess of candidates is observed in the data over the background expectation. The results are used to constrain VH production followed by H decaying to invisible particles for the Higgs boson mass range 115<mH<300 GeV. The 95 % confidence-level observed upper limit on σVH×BR(H→inv.) varies from 1.6 pb at 115 GeV to 0.13 pb at 300 GeV. Assuming Standard Model production and including the gg→H contribution as signal, the results also lead to an observed upper limit of 78 % at 95 % confidence level on the branching ratio of Higgs bosons decays to invisible particles at a mass of 125 GeV

Search for W′→tb→qqbb decays in pp collisions at √s=8 TeV with the ATLAS detector

A search for a massive W′ gauge boson decaying to a top quark and a bottom quark is performed with the ATLAS detector in pp collisions at the LHC. The dataset was taken at a centre-of-mass energy of √s=8 TeV and corresponds to 20.3 fb−1 of integrated luminosity. This analysis is done in the hadronic decay mode of the top quark, where novel jet substructure techniques are used to identify jets from high-momentum top quarks. This allows for a search for high-mass W′ bosons in the range 1.5–3.0 TeV. b-tagging is used to identify jets originating from b-quarks. The data are consistent with Standard Model background-only expectations, and upper limits at 95 % confidence level are set on the W′→tb cross section times branching ratio ranging from 0.16pb to 0.33pb for left-handed W′ bosons, and ranging from 0.10pb to 0.21pb for W′ bosons with purely right-handed couplings. Upper limits at 95 % confidence level are set on the W′-boson coupling to tb as a function of the W′ mass using an effective field theory approach, which is independent of details of particular models predicting a W′boson

Search for vectorlike B quarks in events with one isolated lepton, missing transverse momentum, and jets at √s = 8 TeV with the ATLAS detector

A search has been performed for pair production of heavy vectorlike down-type (B) quarks. The analysis explores the lepton-plus-jets final state, characterized by events with one isolated charged lepton (electron or muon), significant missing transverse momentum, and multiple jets. One or more jets are required to be tagged as arising from b quarks, and at least one pair of jets must be tagged as arising from the hadronic decay of an electroweak boson. The analysis uses the full data sample of pp collisions recorded in 2012 by the ATLAS detector at the LHC, operating at a center-of-mass energy of 8 TeV, corresponding to an integrated luminosity of 20.3 fb −1 . No significant excess of events is observed above the expected background. Limits are set on vectorlike B production, as a function of the B branching ratios, assuming the allowable decay modes are B → Wt/Zb/Hb. In the chiral limit with a branching ratio of 100% for the decay B → Wt, the observed (expected) 95% C.L. lower limit on the vectorlike B mass is 810 GeV (760 GeV). In the case where the vectorlike B quark has branching ratio values corresponding to those of an SU(2) singlet state, the observed (expected) 95% C.L. lower limit on the vectorlike B mass is 640 GeV (505 GeV). The same analysis, when used to investigate pair production of a colored, charge 5/3 exotic fermion T 5/3 , with subsequent decay T 5/3 → Wt, sets an observed (expected) 95% C.L. lower limit on the T 5/3 mass of 840 GeV (780 GeV)