Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow

Fermat, Leibniz, Euler, and Cauchy all used one or another form of approximate equality, or the idea of discarding "negligible" terms, so as to obtain a correct analytic answer. Their inferential moves find suitable proxies in the context of modern theories of infinitesimals, and specifically the concept of shadow. We give an application to decreasing rearrangements of real functions.Comment: 35 pages, 2 figures, to appear in Notices of the American Mathematical Society 61 (2014), no.

Demonstrating coherent control in 85Rb2 using ultrafast laser pulses: a theoretical outline of two experiments

Calculations relating to two experiments that demonstrate coherent control of preformed rubidium-85 molecules in a magneto-optical trap using ultrafast laser pulses are presented. In the first experiment, it is shown that pre-associated molecules in an incoherent mixture of states can be made to oscillate coherently using a single ultrafast pulse. A mechanism that can transfer molecular population to more deeply bound vibrational levels is used in the second. Optimal parameters of the control pulse are presented for the application of the mechanism to molecules in a magneto-optical trap. The calculations make use of an experimental determination of the initial state of molecules photoassociated by the trapping lasers in the magneto-optical trap and use shaped pulses consistent with a standard ultrafast laser system.Comment: 8 pages, 9 figures, PRA, 80, 033403 (2009

Detection of C3_{3} in Diffuse Interstellar Clouds

The smallest polyatomic carbon chain, C$_{3}$, has been identified in interstellar clouds (A$_{v}\sim$1 mag) towards $\zeta$ Ophiuchi, 20 Aquilae, and $\zeta$ Persei by detection of the origin band in its $A^{1}\Pi_{u}-X^{1}\Sigma^{+}_{g}$ electronic transition, near 4052\AA. Individual rotational lines were resolved up to $J$=30 enabling the rotational level column densities and temperature distributions to be determined. The inferred limits for the total column densities ($\sim$1 to 2$\times10^{12}$ cm$^{-2}$) offer a strong incentive to laboratory and astrophysical searches for the longer carbon chains. Concurrent searches for C$_2^{+}$, C$_2^{-}$ and C$_3^{-}$ were negative but provide sensitive estimates for their maximum column densities.Comment: 14 pages, 5 figures, 3 tables. To appear in the Astrophysical Journa

Assessing the Impact of Field-of-Use Restrictions in Patent Licensing Agreements: The Ethical Pharmaceutical Industry in the United States, 1950–1962

[EN]There are a number of strategies employed by companies to limit price competition, including patenting. This article investigates patent licensing restrictions as a strategy to erode price competition, using mainly information gleaned from the 1960–1962 Kefauver Committee hearings. The article deals with the pharmaceutical industry, which is one of the few sectors in which patents are essential to the development and introduction of innovations. The current study adds to a body of literature that has yielded mixed results with respect to the role of patents in this industry. The main contribution of this research is that restrictive licensing clauses, specifically field-of-use restrictions, are found to be relevant in eroding price competition in the institutional market. However, in the retail ethical market, price competition was absent even when no field-of-use restrictions were included in licensing contracts, although product competition was relevant between patented drugs

Observation of the onset of torsion-induced, mode-specific dissipative intramolecular vibrational redistribution (IVR)

Evidence is found showing that coupling with vibration-torsion (“vibtor”) levels of both in-plane and out-of-plane vibrations is instrumental in causing dissipative intramolecular vibrational redistribution (IVR). Both zero-electron-kinetic-energy (ZEKE) spectroscopy and two-dimensional laser-induced fluorescence (2D-LIF) spectroscopy are employed to investigate a series of bands located ∼1200 cm−1 above the S1 ← S0 origin in p-fluorotoluene. Transitions in this wavenumber region have been the focus of a number of studies probing IVR. By recording both ZEKE and 2D-LIF spectra, a prepared S1 population is projected onto both the ground state cation and ground state neutral energy states, respectively, giving added confidence to the assignments. The spectral region under discussion is dominated by a pair of fundamental bands, but for the first time, we present explicit evidence that this is complicated by contributions from a number of overtones and combinations, including vibtor levels. We deduce that very different extents of coupling are present across a 60 cm−1 window of the spectrum, even though the density of states is similar; in particular, one of the fundamentals couples efficiently to the increasing bath of levels, while one does not. We explain this by the influence of serendipitous near-coincidences of same-symmetry levels

Vibration-modified torsional potentials and vibration-torsion (“vibtor”) levels in the m-fluorotoluene cation

Zero-kinetic-energy (ZEKE) spectra are presented for m-fluorotoluene, employing different low-lying (≤350 cm−1) intermediate torsional and vibration-torsional (“vibtor”) levels of the S1 state. The adiabatic ionization energy (AIE) is found to be 71 997 ± 5 cm−1 (8.9265 ± 0.0006 eV). It is found that the activity in the ZEKE spectra varies greatly for different levels and is consistent with the assignments of the S1 levels deduced in the recent fluorescence study of Stewart et al. [J. Chem. Phys. 150, 174303 (2019)]. For cation torsional levels, the most intense band corresponds to changes in the torsional quantum number, in line with the known change in the phase of the torsional potential upon ionization. This leads to the observation of an unprecedented number of torsions and vibtor levels, with the pronounced vibtor activity involving out-of-plane vibrations. Interactions between levels involving torsions are discussed, with evidence presented, for the first time it is believed, for modification of a torsional potential induced by a vibration. Also, we discuss the possibility of distortion of the methyl group leading to a change from G6 molecular symmetry to Cs point group symmetry

Identification of separate isoenergetic routes for vibrational energy flow in p-fluorotoluene

A deceptively simple feature in the S1 ← S0 spectrum of p-fluorotoluene (pFT), 1013 cm−1 above the origin, is studied using both zero-electron-kinetic-energy (ZEKE) and two-dimensional laser-induced fluorescence (2D-LIF) spectroscopy. It is found to consist of a cornucopia of overlapped transitions to eigenstates that arise from numerous interacting levels. A significant variation in the activity is seen employing both the ZEKE and 2D-LIF techniques. Detailed insight into the complicated spectra can be achieved, owing to the large number of vibrational wavenumbers that have been previously determined for the S0, S1, and D0+ states, summarized herein. It is found that the activity is dominated by two overtones, which are individually interacting with other levels, so providing largely independent routes for vibrational energy flow at the same internal energy. Additionally, other weak features located 900–1050 cm−1 above the origin are examined