The Supremum Norm of the Discrepancy Function: Recent Results and Connections

A great challenge in the analysis of the discrepancy function D_N is to obtain universal lower bounds on the L-infty norm of D_N in dimensions d \geq 3. It follows from the average case bound of Klaus Roth that the L-infty norm of D_N is at least (log N) ^{(d-1)/2}. It is conjectured that the L-infty bound is significantly larger, but the only definitive result is that of Wolfgang Schmidt in dimension d=2. Partial improvements of the Roth exponent (d-1)/2 in higher dimensions have been established by the authors and Armen Vagharshakyan. We survey these results, the underlying methods, and some of their connections to other subjects in probability, approximation theory, and analysis.Comment: 15 pages, 3 Figures. Reports on talks presented by the authors at the 10th international conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Sydney Australia, February 2011. v2: Comments of the referee are incorporate

Near-optimal mean value estimates for multidimensional Weyl sums

We obtain sharp estimates for multidimensional generalisations of Vinogradov's mean value theorem for arbitrary translation-dilation invariant systems, achieving constraints on the number of variables approaching those conjectured to be the best possible. Several applications of our bounds are discussed

Electron spin as a spectrometer of nuclear spin noise and other fluctuations

This chapter describes the relationship between low frequency noise and coherence decay of localized spins in semiconductors. Section 2 establishes a direct relationship between an arbitrary noise spectral function and spin coherence as measured by a number of pulse spin resonance sequences. Section 3 describes the electron-nuclear spin Hamiltonian, including isotropic and anisotropic hyperfine interactions, inter-nuclear dipolar interactions, and the effective Hamiltonian for nuclear-nuclear coupling mediated by the electron spin hyperfine interaction. Section 4 describes a microscopic calculation of the nuclear spin noise spectrum arising due to nuclear spin dipolar flip-flops with quasiparticle broadening included. Section 5 compares our explicit numerical results to electron spin echo decay experiments for phosphorus doped silicon in natural and nuclear spin enriched samples.Comment: Book chapter in "Electron spin resonance and related phenomena in low dimensional structures", edited by Marco Fanciulli. To be published by Springer-Verlag in the TAP series. 35 pages, 9 figure

Propylene Carbonate Reexamined: Mode-Coupling β\beta Scaling without Factorisation ?

The dynamic susceptibility of propylene carbonate in the moderately viscous regime above $T_{\rm c}$ is reinvestigated by incoherent neutron and depolarised light scattering, and compared to dielectric loss and solvation response. Depending on the strength of $\alpha$ relaxation, a more or less extended $\beta$ scaling regime is found. Mode-coupling fits yield consistently $\lambda=0.72$ and $T_{\rm c}=182$K, although different positions of the susceptibility minimum indicate that not all observables have reached the universal asymptotics

Quasi-Monte Carlo rules for numerical integration over the unit sphere S2\mathbb{S}^2

We study numerical integration on the unit sphere $\mathbb{S}^2 \subset \mathbb{R}^3$ using equal weight quadrature rules, where the weights are such that constant functions are integrated exactly. The quadrature points are constructed by lifting a $(0,m,2)$-net given in the unit square $[0,1]^2$ to the sphere $\mathbb{S}^2$ by means of an area preserving map. A similar approach has previously been suggested by Cui and Freeden [SIAM J. Sci. Comput. 18 (1997), no. 2]. We prove three results. The first one is that the construction is (almost) optimal with respect to discrepancies based on spherical rectangles. Further we prove that the point set is asymptotically uniformly distributed on $\mathbb{S}^2$. And finally, we prove an upper bound on the spherical cap $L_2$-discrepancy of order $N^{-1/2} (\log N)^{1/2}$ (where $N$ denotes the number of points). This slightly improves upon the bound on the spherical cap $L_2$-discrepancy of the construction by Lubotzky, Phillips and Sarnak [Comm. Pure Appl. Math. 39 (1986), 149--186]. Numerical results suggest that the $(0,m,2)$-nets lifted to the sphere $\mathbb{S}^2$ have spherical cap $L_2$-discrepancy converging with the optimal order of $N^{-3/4}$

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

Glutaminolysis and fumarate accumulation integrate immunometabolic and epigenetic programs in trained immunity

Induction of trained immunity (innate immune memory) is mediated by activation of immune and metabolic pathways that result in epigenetic rewiring of cellular functional programs. Through network-level integration of transcriptomics and metabolomics data, we identify glycolysis, glutaminolysis, and the cholesterol synthesis pathway as indispensable for the induction of trained immunity by ß-glucan in monocytes. Accumulation of fumarate, due to glutamine replenishment of the TCA cycle, integrates immune and metabolic circuits to induce monocyte epigenetic reprogramming by inhibiting KDM5 histone demethylases. Furthermore, fumarate itself induced an epigenetic program similar to ß-glucan-induced trained immunity. In line with this, inhibition of glutaminolysis and cholesterol synthesis in mice reduced the induction of trained immunity by ß-glucan. Identification of the metabolic pathways leading to induction of trained immunity contributes to our understanding of innate immune memory and opens new therapeutic avenues.Netherlands Organization for Scientific Research (NWO). B.N. is supported by an NHMRC (Australia) CJ Martin Early Career Fellowship. N.P.R. Netherlands Heart Foundation (2012T051). N.P.R. and M.G.N. received a H2020 grant (H2020-PHC-2015-667873-2) from the European Union (grant agreement 667837). Fundação para a Ciência e Tecnologia, FCT (IF/00735/2014 to A.C., IF/00021/2014 to R.S., RECI/BBB-BQB/0230/2012 to L.G.G., and SFRH/BPD/96176/2013 to C. Cunha). The NMR spectrometers are part of the National NMR Facility supported by FCT (RECI/BBB-BQB/0230/2012). The research leading to these results received funding from the Fundação para a Ciência e Tecnologia (FCT), cofunded by Programa Operacional Regional do Norte (ON.2—O Novo Norte); from the Quadro de Referência Estratégico Nacional (QREN) through the Fundo Europeu de Desenvolvimento Regional (FEDER) and from the Projeto Estratégico – LA 26 – 2013–2014 (PEst-C/SAU/LA0026/2013). NIH (DK43351 and DK097485) and Helmsley Trust. D.L.W. is supported, in part, by the NIH (GM53522, GM083016, GM119197, and C06RR0306551