Simple proof of gauge invariance for the S-matrix element of strong-field photoionization

The relationship between the length gauge (LG) and the velocity gauge (VG) exact forms of the photoionization probability amplitude is considered. Our motivation for this paper comes from applications of the Keldysh-Faisal-Reiss (KFR) theory, which describes atoms (or ions) in a strong laser field (in the nonrelativistic approach, in the dipole approximation). On the faith of a certain widely-accepted assumption, we present a simple proof that the well-known LG form of the exact photoionization (or photodetachment) probability amplitude is indeed the gauge-invariant result. In contrast, to obtain the VG form of this probability amplitude, one has to either (i) neglect the well-known Goeppert-Mayer exponential factor (which assures gauge invariance) during all the time evolution of the ionized electron or (ii) put some conditions on the vector potential of the laser field.Comment: The paper was initially submitted (in a previous version) on 16 October 2006 to J. Phys. A and rejected. This is the extended version (with 2 figures), which is identical to the paper published online on 12 December 2007 in Physica Script

Nonparametric estimation of scalar diffusions based on low frequency data

We study the problem of estimating the coefficients of a diffusion (X_t,t\geq 0); the estimation is based on discrete data X_{n\Delta},n=0,1,...,N. The sampling frequency \Delta^{-1} is constant, and asymptotics are taken as the number N of observations tends to infinity. We prove that the problem of estimating both the diffusion coefficient (the volatility) and the drift in a nonparametric setting is ill-posed: the minimax rates of convergence for Sobolev constraints and squared-error loss coincide with that of a, respectively, first- and second-order linear inverse problem. To ensure ergodicity and limit technical difficulties we restrict ourselves to scalar diffusions living on a compact interval with reflecting boundary conditions. Our approach is based on the spectral analysis of the associated Markov semigroup. A rate-optimal estimation of the coefficients is obtained via the nonparametric estimation of an eigenvalue-eigenfunction pair of the transition operator of the discrete time Markov chain (X_{n\Delta},n=0,1,...,N) in a suitable Sobolev norm, together with an estimation of its invariant density.Comment: Published at http://dx.doi.org/10.1214/009053604000000797 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Validity and worth in the science curriculum: learning school science outside the laboratory

It is widely acknowledged that there are problems with school science in many developed countries of the world. Such problems manifest themselves in a progressive decline in pupil enthusiasm for school science across the secondary age range and the fact that fewer students are choosing to study the physical sciences at higher levels and as careers. Responses to these developments have included proposals to reform the curriculum, pedagogy and the nature of pupil discussion in science lessons. We support such changes but argue from a consideration of the aims of science education that secondary school science is too rooted in the science laboratory; substantially greater use needs to be made of out-of-school sites for the teaching of science. Such usage should result in a school science education that is more valid and more motivating and is better at fulfilling defensible aims of school science education. Our contention is that laboratory-based school science teaching needs to be complemented by out-of-school science learning that draws on the actual world (e.g. through fieldtrips), the presented world (e.g. in science centres, botanic gardens, zoos and science museums) and the virtual worlds that are increasingly available through information and communications technologies (ICT)

Imagining the World: The Significance of Religious Worldviews for Science Education

This paper begins by examining whether ‘science’ and ‘religion’ can better be seen as distinct or related worldviews, focusing particularly on scientific and religious understandings of biodiversity. I then explore how people can see the natural world, depending on their worldview, by looking at two contrasting treatments of penguin behaviour, namely that provided in the film March of the Penguins and in the children’s book And Tango Makes Three. I end by drawing some initial conclusions as to what might and what might not be included about religion in school science lessons. Science educators and teachers need to take account of religious worldviews if some students are better to understand the compass of scientific thinking and some of science’s key conclusions. It is perfectly possible for a science teacher to be respectful of the worldviews that students occupy, even if these are scientifically limited, while clearly and non-apologetically helping them to understand the scientific worldview on a particular issue

The use of ethical frameworks by students following a new science course for 16-18 year-olds

There has been a move in recent years towards the greater inclusion of social and ethical issues within science courses. This paper examines a new context-based course for 16-18 year-olds (Salters-Nuffield Advanced Biology) who are studying biology in England and Wales. The course is taught through contexts and has an emphasis on social issues and the development of ethical reasoning. Examination of a sample of reports written by students in 2005 as part of the course's summative assessment shows that utilitarian ethical reasoning is used widely and that the other ethical frameworks to which students are introduced in the course – rights and duties, autonomy and virtue ethics – are used substantially less often. In addition, students mostly argue anthropocentrically though many of them argue ecocentrically and/or biocentrically too

The relationship between evolutionary biology and religion

Belief in creationism and intelligent design is widespread and gaining in significance in a number of countries. This article examines the characteristics of science and of religions and the possible relationship between science and religion. I argue that creationism is sometimes best seen not as a misconception but as a worldview. In such instances, the most to which a science educator (whether in school, college or university) can normally aspire is to ensure that students with creationist beliefs understand the scientific position. In the short term, the scientific worldview is unlikely to supplant a creationist one for students who are firm creationists. We can help students to find their evolutionary biology courses interesting and intellectually challenging without their being threatening. Effective teaching in this area can not only help students learn about the theory of evolution but better to appreciate the way science is done, the procedures by which scientific knowledge accumulates, the limitations of science and the ways in which scientific knowledge differs from other forms of knowledge

Dust Devil Tracks

Dust devils that leave dark- or light-toned tracks are common on Mars and they can also be found on the Earth’s surface. Dust devil tracks (hereinafter DDTs) are ephemeral surface features with mostly sub-annual lifetimes. Regarding their size, DDT widths can range between ∼1 m and ∼1 km, depending on the diameter of dust devil that created the track, and DDT lengths range from a few tens of meters to several kilometers, limited by the duration and horizontal ground speed of dust devils. DDTs can be classified into three main types based on their morphology and albedo in contrast to their surroundings; all are found on both planets: (a) dark continuous DDTs, (b) dark cycloidal DDTs, and (c) bright DDTs. Dark continuous DDTs are the most common type on Mars. They are characterized by their relatively homogenous and continuous low albedo surface tracks. Based on terrestrial and martian in situ studies, these DDTs most likely form when surficial dust layers are removed to expose larger-grained substrate material (coarse sands of ≥500 μm in diameter). The exposure of larger-grained materials changes the photometric properties of the surface; hence leading to lower albedo tracks because grain size is photometrically inversely proportional to the surface reflectance. However, although not observed so far, compositional differences (i.e., color differences) might also lead to albedo contrasts when dust is removed to expose substrate materials with mineralogical differences. For dark continuous DDTs, albedo drop measurements are around 2.5 % in the wavelength range of 550–850 nm on Mars and around 0.5 % in the wavelength range from 300–1100 nm on Earth. The removal of an equivalent layer thickness around 1 μm is sufficient for the formation of visible dark continuous DDTs on Mars and Earth. The next type of DDTs, dark cycloidal DDTs, are characterized by their low albedo pattern of overlapping scallops. Terrestrial in situ studies imply that they are formed when sand-sized material that is eroded from the outer vortex area of a dust devil is redeposited in annular patterns in the central vortex region. This type of DDT can also be found in on Mars in orbital image data, and although in situ studies are lacking, terrestrial analog studies, laboratory work, and numerical modeling suggest they have the same formation mechanism as those on Earth. Finally, bright DDTs are characterized by their continuous track pattern and high albedo compared to their undisturbed surroundings. They are found on both planets, but to date they have only been analyzed in situ on Earth. Here, the destruction of aggregates of dust, silt and sand by dust devils leads to smooth surfaces in contrast to the undisturbed rough surfaces surrounding the track. The resulting change in photometric properties occurs because the smoother surfaces have a higher reflectance compared to the surrounding rough surface, leading to bright DDTs. On Mars, the destruction of surficial dust-aggregates may also lead to bright DDTs. However, higher reflective surfaces may be produced by other formation mechanisms, such as dust compaction by passing dust devils, as this may also cause changes in photometric properties. On Mars, DDTs in general are found at all elevations and on a global scale, except on the permanent polar caps. DDT maximum areal densities occur during spring and summer in both hemispheres produced by an increase in dust devil activity caused by maximum insolation. Regionally, dust devil densities vary spatially likely controlled by changes in dust cover thicknesses and substrate materials. This variability makes it difficult to infer dust devil activity from DDT frequencies. Furthermore, only a fraction of dust devils leave tracks. However, DDTs can be used as proxies for dust devil lifetimes and wind directions and speeds, and they can also be used to predict lander or rover solar panel clearing events. Overall, the high DDT frequency in many areas on Mars leads to drastic albedo changes that affect large-scale weather patterns