Level crossings and other level functionals of stationary Gaussian processes

This paper presents a synthesis on the mathematical work done on level crossings of stationary Gaussian processes, with some extensions. The main results [(factorial) moments, representation into the Wiener Chaos, asymptotic results, rate of convergence, local time and number of crossings] are described, as well as the different approaches [normal comparison method, Rice method, Stein-Chen method, a general $m$-dependent method] used to obtain them; these methods are also very useful in the general context of Gaussian fields. Finally some extensions [time occupation functionals, number of maxima in an interval, process indexed by a bidimensional set] are proposed, illustrating the generality of the methods. A large inventory of papers and books on the subject ends the survey.Comment: Published at http://dx.doi.org/10.1214/154957806000000087 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the second moment of the number of crossings by a stationary Gaussian process

Cram\'{e}r and Leadbetter introduced in 1967 the sufficient condition $$\frac{r''(s)-r''(0)}{s}\in L^1([0,\delta],dx),\qquad \delta>0,$$ to have a finite variance of the number of zeros of a centered stationary Gaussian process with twice differentiable covariance function $r$. This condition is known as the Geman condition, since Geman proved in 1972 that it was also a necessary condition. Up to now no such criterion was known for counts of crossings of a level other than the mean. This paper shows that the Geman condition is still sufficient and necessary to have a finite variance of the number of any fixed level crossings. For the generalization to the number of a curve crossings, a condition on the curve has to be added to the Geman condition.Comment: Published at http://dx.doi.org/10.1214/009117906000000142 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Nuclear Structure Studies at ISOLDE and their Impact on the Astrophysical r-Process

The focus of the present review is the production of the heaviest elements in nature via the r-process. A correct understanding and modeling requires the knowledge of nuclear properties far from stability and a detailed prescription of the astrophysical environment. Experiments at CERN/ISOLDE have played a pioneering role in exploring the characteristics of nuclear structure in terms of masses and beta-decay properties. Initial examinations paid attention to far unstable nuclei with magic neutron numbers related to r-process peaks, while present activities are centered on the evolution of shell effects with the distance from the valley of stability. We first show in site-independent applications the effect of both types of nuclear properties on r-process abundances. Then, we explore the results of calculations related to two different `realistic' astrophysical sites, (i) the supernova neutrino wind and (ii) neutron star mergers. We close with a list of remaining theoretical and experimental challenges needed to overcome for a full understanding of the nature of the r-process, and the role CERN/ISOLDE can play in this process.Comment: LATEX, 38 pages, 16 figures, submitted to Hyperfine Interaction

Hermite polynomial expansion for non-smooth functionals of stationary Gaussian processes: Crossings and extremes

AbstractWe propose a new method to get the Hermite polynomial expansion of crossings of any level by a stationary Gaussian process, as well as the one of the number of maxima in an interval, under some assumptions on the spectral moments of the process

Charged-Particle and Neutron-Capture Processes in the High-Entropy Wind of Core-Collapse Supernovae

The astrophysical site of the r-process is still uncertain, and a full exploration of the systematics of this process in terms of its dependence on nuclear properties from stability to the neutron drip-line within realistic stellar environments has still to be undertaken. Sufficiently high neutron to seed ratios can only be obtained either in very neutron-rich low-entropy environments or moderately neutron-rich high-entropy environments, related to neutron star mergers (or jets of neutron star matter) and the high-entropy wind of core-collapse supernova explosions. As chemical evolution models seem to disfavor neutron star mergers, we focus here on high-entropy environments characterized by entropy $S$, electron abundance $Y_e$ and expansion velocity $V_{exp}$. We investigate the termination point of charged-particle reactions, and we define a maximum entropy $S_{final}$ for a given $V_{exp}$ and $Y_e$, beyond which the seed production of heavy elements fails due to the very small matter density. We then investigate whether an r-process subsequent to the charged-particle freeze-out can in principle be understood on the basis of the classical approach, which assumes a chemical equilibrium between neutron captures and photodisintegrations, possibly followed by a $\beta$-flow equilibrium. In particular, we illustrate how long such a chemical equilibrium approximation holds, how the freeze-out from such conditions affects the abundance pattern, and which role the late capture of neutrons originating from $\beta$-delayed neutron emission can play.Comment: 52 pages, 31 figure

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Nucleosynthesis Basics and Applications to Supernovae

This review concentrates on nucleosynthesis processes in general and their applications to massive stars and supernovae. A brief initial introduction is given to the physics in astrophysical plasmas which governs composition changes. We present the basic equations for thermonuclear reaction rates and nuclear reaction networks. The required nuclear physics input for reaction rates is discussed, i.e. cross sections for nuclear reactions, photodisintegrations, electron and positron captures, neutrino captures, inelastic neutrino scattering, and beta-decay half-lives. We examine especially the present state of uncertainties in predicting thermonuclear reaction rates, while the status of experiments is discussed by others in this volume (see M. Wiescher). It follows a brief review of hydrostatic burning stages in stellar evolution before discussing the fate of massive stars, i.e. the nucleosynthesis in type II supernova explosions (SNe II). Except for SNe Ia, which are explained by exploding white dwarfs in binary stellar systems (which will not be discussed here), all other supernova types seem to be linked to the gravitational collapse of massive stars (M$>$8M$_\odot$) at the end of their hydrostatic evolution. SN1987A, the first type II supernova for which the progenitor star was known, is used as an example for nucleosynthesis calculations. Finally, we discuss the production of heavy elements in the r-process up to Th and U and its possible connection to supernovae.Comment: 52 pages, 20 figures, uses cupconf.sty (included); to appear in "Nuclear and Particle Astrophysics", eds. J. Hirsch., D. Page, Cambridge University Pres

Development of Acid-Sensitive Platinum(II) Complexes With Protein-Binding Properties

Four new protein-binding platinum(II) complexes, 10, 11, 21, 22, in which the dichloroplatinum moiety is coordinated either to a carbon-substituted or a nitrogen-substituted ethylene diamino ligand, were prepared in ten-step syntheses. According to pH-dependent stability studies with strictly related compounds, 11 and 22 exhibit acid-sensitive properties