Scaling of the Critical Function for the Standard Map: Some Numerical Results

The behavior of the critical function for the breakdown of the homotopically non-trivial invariant (KAM) curves for the standard map, as the rotation number tends to a rational number, is investigated using a version of Greene's residue criterion. The results are compared to the analogous ones for the radius of convergence of the Lindstedt series, in which case rigorous theorems have been proved. The conjectured interpolation of the critical function in terms of the Bryuno function is discussed.Comment: 26 pages, 3 figures, 13 table

The Near-Infrared Spectrograph (NIRSpec) on the James Webb Space Telescope: I. Overview of the instrument and its capabilities

We provide an overview of the design and capabilities of the near-infrared spectrograph (NIRSpec) onboard the James Webb Space Telescope. NIRSpec is designed to be capable of carrying out low-resolution ($R\!=30\!-330$) prism spectroscopy over the wavelength range $0.6-5.3\!~\mu$m and higher resolution ($R\!=500\!-1340$ or $R\!=1320\!-3600$) grating spectroscopy over $0.7-5.2\!~\mu$m, both in single-object mode employing any one of five fixed slits, or a 3.1$\times$3.2 arcsec$^2$ integral field unit, or in multiobject mode employing a novel programmable micro-shutter device covering a 3.6$\times$3.4~arcmin$^2$ field of view. The all-reflective optical chain of NIRSpec and the performance of its different components are described, and some of the trade-offs made in designing the instrument are touched upon. The faint-end spectrophotometric sensitivity expected of NIRSpec, as well as its dependency on the energetic particle environment that its two detector arrays are likely to be subjected to in orbit are also discussed

The Near-Infrared Spectrograph (NIRSpec) on the James Webb Space Telescope: I. Overview of the instrument and its capabilities

We provide an overview of the design and capabilities of the near-infrared spectrograph (NIRSpec) onboard the James Webb Space Telescope. NIRSpec is designed to be capable of carrying out low-resolution ($R\!=30\!-330$) prism spectroscopy over the wavelength range $0.6-5.3\!~\mu$m and higher resolution ($R\!=500\!-1340$ or $R\!=1320\!-3600$) grating spectroscopy over $0.7-5.2\!~\mu$m, both in single-object mode employing any one of five fixed slits, or a 3.1$\times$3.2 arcsec$^2$ integral field unit, or in multiobject mode employing a novel programmable micro-shutter device covering a 3.6$\times$3.4~arcmin$^2$ field of view. The all-reflective optical chain of NIRSpec and the performance of its different components are described, and some of the trade-offs made in designing the instrument are touched upon. The faint-end spectrophotometric sensitivity expected of NIRSpec, as well as its dependency on the energetic particle environment that its two detector arrays are likely to be subjected to in orbit are also discussed

The James Webb Space Telescope Mission

Twenty-six years ago a small committee report, building on earlier studies, expounded a compelling and poetic vision for the future of astronomy, calling for an infrared-optimized space telescope with an aperture of at least $4m$. With the support of their governments in the US, Europe, and Canada, 20,000 people realized that vision as the $6.5m$ James Webb Space Telescope. A generation of astronomers will celebrate their accomplishments for the life of the mission, potentially as long as 20 years, and beyond. This report and the scientific discoveries that follow are extended thank-you notes to the 20,000 team members. The telescope is working perfectly, with much better image quality than expected. In this and accompanying papers, we give a brief history, describe the observatory, outline its objectives and current observing program, and discuss the inventions and people who made it possible. We cite detailed reports on the design and the measured performance on orbit.Comment: Accepted by PASP for the special issue on The James Webb Space Telescope Overview, 29 pages, 4 figure

How gold is the golden ratio?

We discuss the well-known importance of the golden ratio in Science and Art with few examples: its theoretical value is often taken as a good model in many applications. How good is this model in practice? We agree with many authors that more precision is needed giving, when possible, a measure of the best possible approximation. We deal with several definitions and representations, also comparing this irrational number and its rational approximations to other similar constants

Numeri in un foglio di carta

Dato un normale foglio di carta per fotocopie, di formato A4, con semplici piegature è possibile verificare alcune proprietà aritmetiche sul rapporto (radice di 2) tra le misure dei suoi lati. Si dimostra quindi che dal foglio A4, con solo tre pieghe, si può costruire un rettangolo "aureo", il cui rapporto tra i lati sia f = (Sqrt{5}+1)/2. Si verifica anche come lo sviluppo in frazioni continue di radice di 2 e f siano legati alla proprietà comune ai due rettangoli: dopo aver tagliato un quadrato costruito sul lato più corto per due volte, si ottiene un rettangolo simile a quello di partenza.From a regular A4 sheet of paper, with simple folds, it is possible to verify some arithmetical properties on the ratio of its sides (square root of 2). A proof is given that, starting with an A4 sheet and with just 3 folds, it is possible to construct a "golden rectangle" in wich the ratio of the sides is the golden section f = (Sqrt{5}+1)/2. Also, it is shown that the continued fraction representation of square root of 2 and f share a common property: cutting a maximal square twice on the smaller side gives a similar rectangle