From Euclidean Geometry to Knots and Nets

This document is the Accepted Manuscript of an article accepted for publication in Synthese. Under embargo until 19 September 2018. The final publication is available at Springer via https://doi.org/10.1007/s11229-017-1558-x.This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if (a) it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, (b) the information thus displayed is not metrical and (c) it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions.Peer reviewe

A multi-color optical survey of the orion nebula cluster. II. The H-R diagram

We present a new analysis of the stellar population of the Orion Nebula Cluster (ONC) based on multi-band optical photometry and spectroscopy.We study the color–color diagrams in BVI, plus a narrowband filter centered at 6200 Å, finding evidence that intrinsic color scales valid for main-sequence dwarfs are incompatible with the ONC in the M spectral-type range, while a better agreement is found employing intrinsic colors derived from synthetic photometry, constraining the surface gravity value as predicted by a pre-main-sequence isochrone.We refine these model colors even further, empirically, by comparison with a selected sample of ONC stars with no accretion and no extinction. We consider the stars with known spectral types from the literature, and extend this sample with the addition of 65 newly classified stars from slit spectroscopy and 182 M-type from narrowband photometry; in this way, we isolate a sample of about 1000 stars with known spectral type. We introduce a new method to self-consistently derive the stellar reddening and the optical excess due to accretion from the location of each star in the BVI color–color diagram. This enables us to accurately determine the extinction of the ONC members, together with an estimate of their accretion luminosities. We adopt a lower distance for the Orion Nebula than previously assumed, based on recent parallax measurements. With a careful choice of also the spectral-type–temperature transformation, we produce the new Hertzsprung–Russell diagram of the ONC population, more populated than previous works. With respect to previous works, we find higher luminosity for late-type stars and a slightly lower luminosity for early types. We determine the age distribution of the population, peaking from ~2 to ~3 Myr depending on the model. We study the distribution of the members in the mass–age plane and find that taking into account selection effects due to incompleteness, removes an apparent correlation between mass and age.We derive the initial mass function for low- and intermediate mass members of the ONC, which turns out to be model dependent and shows a turnover at M ≲ 0.2 M_⊙

Reconstructing the Star Formation History of the Galaxy

The evolution of the star formation rate in the Galaxy is one of the key ingredients quantifying the formation and determining the chemical and luminosity evolution of galaxies. Many complementary methods exist to infer the star formation history of the components of the Galaxy, from indirect methods for analysis of low-precision data, to new exact analytic methods for analysis of sufficiently high quality data. We summarise available general constraints on star formation histories, showing that derived star formation rates are in general comparable to those seen today. We then show how colour-magnitude diagrams of volume- and absolute magnitude-limited samples of the solar neighbourhood observed by Hipparcos may be analysed, using variational calculus techniques, to reconstruct the local star formation history. The remarkable accuracy of the data coupled to our maximum-likelihood variational method allows objective quantification of the local star formation history with a time resolution of ~ 50 Myr. Over the past 3Gyr, the solar neighbourhood star formation rate has varied by a factor of ~ 4, with characteristic timescale about 0.5Gyr, possibly triggered by interactions with spiral arms.Comment: 12 pages, Proc. of the Sept. 20-24, 1999 Vulcano Workshop ``The chemical evolution of the Milky Way: stars vs. clusters'', eds. F. Matteucci & F. Giovanell

Star formation histories of dwarf galaxies from the Colour-Magnitude diagrams of their resolved stellar populations

In this tutorial paper we summarize how the star formation (SF) history of a galactic region can be derived from the colour-magnitude diagram (CMD) of its resolved stars. The procedures to build synthetic CMDs and to exploit them to derive the SF histories (SFHs) are described, as well as the corresponding uncertainties. The SFHs of resolved dwarf galaxies of all morphological types, obtained from the application of the synthetic CMD method, are reviewed and discussed. In short: 1) Only early-type galaxies show evidence of long interruptions in the SF activity; late-type dwarfs present rather continuous, or gasping, SF regimes; 2) A few early-type dwarfs have experienced only one episode of SF activity concentrated at the earliest epochs, whilst many others show extended or recurrent SF activity; 3) No galaxy experiencing now its first SF episode has been found yet; 4) No frequent evidence of strong SF bursts is found; 5) There is no significant difference in the SFH of dwarf irregulars and blue compact dwarfs, except for the current SF rates. Implications of these results on the galaxy formation scenarios are briefly discussed.Comment: 29 pages. Tutorial Review to appear in the special issue "Dwarf-Galaxy Cosmology" in Advances in Astronom

On the structure of problem variability: From feature diagrams to problem frames

Requirements for product families are expressed in terms of commonality and variability. This distinction allows early identification of an appropriate software architecture and opportunities for software reuse. Feature diagrams provide intuitive notations and techniques for representing requirements in product line development. In this paper, we observe that feature diagrams tend to obfuscate three important descriptions: requirements, domain properties and specifications. As a result, feature diagrams do not adequately capture the problem structures that underlie variability, and inform the solution structures of their complexity. With its emphasis on separation of the three descriptions, the problem frames approach provides a conceptual framework for a more detailed analysis of variability and its structure. With illustrations from an example, we demonstrate how problem frames analysis of variability can augment feature diagrams

Developing BCS ideas in the former Soviet Union

The essay is an attempt to re-create the wonderful scientific atmosphere that emerged after the basic BCS ideas first arrived in Russia in 1957. It summarizes the most significant contributions to the microscopic theory of superconductivity by Russian physicists during the next few years that have given the theory its modern form.Comment: 20 pages, published in: BCS: 50 YEARS, edited by Leon N Cooper and Dmitri Feldman, World Scientific, 201

Introduction to Collider Physics

This is a set of four lectures presented at the Theoretical Advanced Study Institute (TASI-09) in June 2009. The goal of the lectures is to introduce students to some of the basic ideas and tools required for theoretical analysis of collider data. Several examples of Standard Model processes at electron-positron and hadron colliders are considered to illustrate these ideas. In addition, a general strategy for formulating searches for physics beyond the Standard Model is outlined. The lectures conclude with a brief survey of recent, ongoing and future searches for the Higgs boson and supersymmetric particles.Comment: 47 pages, 34 figures, contributed to TASI-09 proceedings. One reference added in v