On spherical averages of radial basis functions

A radial basis function (RBF) has the general form $$s(x)=\sum_{k=1}^{n}a_{k}\phi(x-b_{k}),\quad x\in\mathbb{R}^{d},$$ where the coefficients a 1,…,a n are real numbers, the points, or centres, b 1,…,b n lie in ℝ d , and φ:ℝ d →ℝ is a radially symmetric function. Such approximants are highly useful and enjoy rich theoretical properties; see, for instance (Buhmann, Radial Basis Functions: Theory and Implementations, [2003]; Fasshauer, Meshfree Approximation Methods with Matlab, [2007]; Light and Cheney, A Course in Approximation Theory, [2000]; or Wendland, Scattered Data Approximation, [2004]). The important special case of polyharmonic splines results when φ is the fundamental solution of the iterated Laplacian operator, and this class includes the Euclidean norm φ(x)=‖x‖ when d is an odd positive integer, the thin plate spline φ(x)=‖x‖2log  ‖x‖ when d is an even positive integer, and univariate splines. Now B-splines generate a compactly supported basis for univariate spline spaces, but an analyticity argument implies that a nontrivial polyharmonic spline generated by (1.1) cannot be compactly supported when d>1. However, a pioneering paper of Jackson (Constr. Approx. 4:243–264, [1988]) established that the spherical average of a radial basis function generated by the Euclidean norm can be compactly supported when the centres and coefficients satisfy certain moment conditions; Jackson then used this compactly supported spherical average to construct approximate identities, with which he was then able to derive some of the earliest uniform convergence results for a class of radial basis functions. Our work extends this earlier analysis, but our technique is entirely novel, and applies to all polyharmonic splines. Furthermore, we observe that the technique provides yet another way to generate compactly supported, radially symmetric, positive definite functions. Specifically, we find that the spherical averaging operator commutes with the Fourier transform operator, and we are then able to identify Fourier transforms of compactly supported functions using the Paley–Wiener theorem. Furthermore, the use of Haar measure on compact Lie groups would not have occurred without frequent exposure to Iserles’s study of geometric integration

Pseudoscalar Glueball, eta'-meson and its Excitation in the Chiral Effective Lagrangian

A generalization of the chiral effective lagrangian of order $p^2$ is proposed which involves the $\eta'$-meson, its excitation, and the pseudoscalar (PS) glueball. Model-independent constraints are found for the contributions to the lagrangian of the above singlet states. Those allow one to independently identify the nature of these singlet states in the framework of the approach. The mixing among the iso-singlet states (including $\eta^8$-state) is analysed, and the hierarchy of the mixing angles is described which is defined by the chiral and large-$N_c$ expansions. The recent PCAC results are reproduced, which are related to the problem of the renormalization-group invariant description of the $\eta'$ and the PS glueball, and a further analysis of this problem is performed.Comment: 19 pages LaTeX, no figures. Revised version accepted in Phis.Rev.

Stochastic Gravity: A Primer with Applications

Stochastic semiclassical gravity of the 90's is a theory naturally evolved from semiclassical gravity of the 70's and 80's. It improves on the semiclassical Einstein equation with source given by the expectation value of the stress-energy tensor of quantum matter fields in curved spacetimes by incorporating an additional source due to their fluctuations. In stochastic semiclassical gravity the main object of interest is the noise kernel, the vacuum expectation value of the (operator-valued) stress-energy bi-tensor, and the centerpiece is the (stochastic) Einstein-Langevin equation. We describe this new theory via two approaches: the axiomatic and the functional. The axiomatic approach is useful to see the structure of the theory from the framework of semiclassical gravity. The functional approach uses the Feynman-Vernon influence functional and the Schwinger-Keldysh close-time-path effective action methods which are convenient for computations. It also brings out the open systems concepts and the statistical and stochastic contents of the theory such as dissipation, fluctuations, noise and decoherence. We then describe the application of stochastic gravity to the backreaction problems in cosmology and black hole physics. Intended as a first introduction to this subject, this article places more emphasis on pedagogy than completeness.Comment: 46 pages Latex. Intended as a review in {\it Classical and Quantum Gravity

Interplay of LFV and slepton mass splittings at the LHC as a probe of the SUSY seesaw

We study the impact of a type-I SUSY seesaw concerning lepton flavour violation (LFV) both at low-energies and at the LHC. The study of the di-lepton invariant mass distribution at the LHC allows to reconstruct some of the masses of the different sparticles involved in a decay chain. In particular, the combination with other observables renders feasible the reconstruction of the masses of the intermediate sleptons involved in $\chi_2^0\to \tilde \ell \,\ell \to \ell \,\ell\,\chi_1^0$ decays. Slepton mass splittings can be either interpreted as a signal of non-universality in the SUSY soft breaking-terms (signalling a deviation from constrained scenarios as the cMSSM) or as being due to the violation of lepton flavour. In the latter case, in addition to these high-energy processes, one expects further low-energy manifestations of LFV such as radiative and three-body lepton decays. Under the assumption of a type-I seesaw as the source of neutrino masses and mixings, all these LFV observables are related. Working in the framework of the cMSSM extended by three right-handed neutrino superfields, we conduct a systematic analysis addressing the simultaneous implications of the SUSY seesaw for both high- and low-energy lepton flavour violation. We discuss how the confrontation of slepton mass splittings as observed at the LHC and low-energy LFV observables may provide important information about the underlying mechanism of LFV.Comment: 50 pages, 42 eps Figures, typos correcte

A Scaling Hypothesis for Modulated Systems

We propose a scaling hypothesis for pattern-forming systems in which modulation of the order parameter results from the competition between a short-ranged interaction and a long-ranged interaction decaying with some power $\alpha$ of the inverse distance. With L being a spatial length characterizing the modulated phase, all thermodynamic quantities are predicted to scale like some power of L. The scaling dimensions with respect to L only depend on the dimensionality of the system d and the exponent \alpha. Scaling predictions are in agreement with experiments on ultra-thin ferromagnetic films and computational results. Finally, our scaling hypothesis implies that, for some range of values \alpha>d, Inverse-Symmetry-Breaking transitions may appear systematically in the considered class of frustrated systems.Comment: 13 pages, 6 figures, expanded versio

Linear Response, Validity of Semi-Classical Gravity, and the Stability of Flat Space

A quantitative test for the validity of the semi-classical approximation in gravity is given. The criterion proposed is that solutions to the semi-classical Einstein equations should be stable to linearized perturbations, in the sense that no gauge invariant perturbation should become unbounded in time. A self-consistent linear response analysis of these perturbations, based upon an invariant effective action principle, necessarily involves metric fluctuations about the mean semi-classical geometry, and brings in the two-point correlation function of the quantum energy-momentum tensor in a natural way. This linear response equation contains no state dependent divergences and requires no new renormalization counterterms beyond those required in the leading order semi-classical approximation. The general linear response criterion is applied to the specific example of a scalar field with arbitrary mass and curvature coupling in the vacuum state of Minkowski spacetime. The spectral representation of the vacuum polarization function is computed in n dimensional Minkowski spacetime, and used to show that the flat space solution to the semi-classical Einstein equations for n=4 is stable to all perturbations on distance scales much larger than the Planck length.Comment: 22 pages: This is a significantly expanded version of gr-qc/0204083, with two additional sections and two new appendices giving a complete, explicit example of the semi-classical stability criterion proposed in the previous pape

Weak Decays Beyond Leading Logarithms

We review the present status of QCD corrections to weak decays beyond the leading logarithmic approximation including particle-antiparticle mixing and rare and CP violating decays. After presenting the basic formalism for these calculations we discuss in detail the effective hamiltonians for all decays for which the next-to-leading corrections are known. Subsequently, we present the phenomenological implications of these calculations. In particular we update the values of various parameters and we incorporate new information on m_t in view of the recent top quark discovery. One of the central issues in our review are the theoretical uncertainties related to renormalization scale ambiguities which are substantially reduced by including next-to-leading order corrections. The impact of this theoretical improvement on the determination of the Cabibbo-Kobayashi-Maskawa matrix is then illustrated in various cases.Comment: 229 pages, 32 PostScript figures (included); uses RevTeX, epsf.sty, rotate.sty, rmpbib.sty (included), times.sty (included; requires LaTeX 2e); complete PostScript version available at ftp://feynman.t30.physik.tu-muenchen.de/pub/preprints/tum-100-95.ps.gz or ftp://feynman.t30.physik.tu-muenchen.de/pub/preprints/tum-100-95.ps2.gz (scaled down and rotated version to print two pages on one sheet of paper

Stochastic Gravity: Theory and Applications

Whereas semiclassical gravity is based on the semiclassical Einstein equation with sources given by the expectation value of the stress-energy tensor of quantum fields, stochastic semiclassical gravity is based on the Einstein-Langevin equation, which has in addition sources due to the noise kernel. In the first part, we describe the fundamentals of this new theory via two approaches: the axiomatic and the functional. In the second part, we describe three applications of stochastic gravity theory. First, we consider metric perturbations in a Minkowski spacetime, compute the two-point correlation functions of these perturbations and prove that Minkowski spacetime is a stable solution of semiclassical gravity. Second, we discuss structure formation from the stochastic gravity viewpoint. Third, we discuss the backreaction of Hawking radiation in the gravitational background of a black hole and describe the metric fluctuations near the event horizon of an evaporating black holeComment: 100 pages, no figures; an update of the 2003 review in Living Reviews in Relativity gr-qc/0307032 ; it includes new sections on the Validity of Semiclassical Gravity, the Stability of Minkowski Spacetime, and the Metric Fluctuations of an Evaporating Black Hol

Stochastic Gravity: Theory and Applications

Whereas semiclassical gravity is based on the semiclassical Einstein equation with sources given by the expectation value of the stress-energy tensor of quantum fields, stochastic semiclassical gravity is based on the Einstein-Langevin equation, which has in addition sources due to the noise kernel.In the first part, we describe the fundamentals of this new theory via two approaches: the axiomatic and the functional. In the second part, we describe three applications of stochastic gravity theory. First, we consider metric perturbations in a Minkowski spacetime: we compute the two-point correlation functions for the linearized Einstein tensor and for the metric perturbations. Second, we discuss structure formation from the stochastic gravity viewpoint. Third, we discuss the backreaction of Hawking radiation in the gravitational background of a quasi-static black hole.Comment: 75 pages, no figures, submitted to Living Reviews in Relativit

Progesterone after previous preterm birth for prevention of neonatal respiratory distress syndrome (PROGRESS): a randomised controlled trial

Background: Neonatal respiratory distress syndrome, as a consequence of preterm birth, is a major cause of early mortality and morbidity during infancy and childhood. Survivors of preterm birth continue to remain at considerable risk of both chronic lung disease and long-term neurological handicap. Progesterone is involved in the maintenance of uterine quiescence through modulation of the calcium-calmodulin-myosin-light-chain-kinase system in smooth muscle cells. The withdrawal of progesterone, either actual or functional is thought to be an antecedent to the onset of labour. While there have been recent reports of progesterone supplementation for women at risk of preterm birth which show promise in this intervention, there is currently insufficient data on clinically important outcomes for both women and infants to enable informed clinical decision-making. The aims of this randomised, double blind, placebo controlled trial are to assess whether the use of vaginal progesterone pessaries in women with a history of previous spontaneous preterm birth will reduce the risk and severity of respiratory distress syndrome, so improving their infant's health, without increasing maternal risks. Methods Design: Multicentred randomised, double blind, placebo-controlled trial. Inclusion Criteria: pregnant women with a live fetus, and a history of prior preterm birth at less than 37 weeks gestation and greater than 20 weeks gestation in the immediately preceding pregnancy, where onset of labour occurred spontaneously, or in association with cervical incompetence, or following preterm prelabour ruptured membranes. Trial Entry & Randomisation: After obtaining written informed consent, eligible women will be randomised between 18 and 23+6 weeks gestation using a central telephone randomisation service. The randomisation schedule prepared by non clinical research staff will use balanced variable blocks, with stratification according to plurality of the pregnancy and centre where planned to give birth. Eligible women will be randomised to either vaginal progesterone or vaginal placebo. Study Medication & Treatment Schedules: Treatment packs will appear identical. Woman, caregivers and research staff will be blinded to treatment allocation. Primary Study Outcome: Neonatal Respiratory Distress Syndrome (defined by incidence and severity). Sample Size: of 984 women to show a 40% reduction in respiratory distress syndrome from 15% to 9% (p = 0.05, 80% power). Discussion: This is a protocol for a randomised trial.Jodie M. Dodd, Caroline A. Crowther, Andrew J. McPhee, Vicki Flenady, and Jeffrey S. Robinso