Incidental Results Research Articles

On the Mean Number of Curve Crossings by Non-Stationary Normal Processes

In this paper we consider the mean number of crossings of an arbitrary curve, in a given time $T$, by a non-stationary normal process. A formula (2) is obtained for this and sufficient conditions for its validity given. These conditions concern the behaviour of the covariance function of the normal process. Incidental results include sufficient conditions for continuity of a normal (non-stationary) process $x(t)$ and also for $x(t)$ to satisfy a Holder condition.

Lamb Shift in the Helium Atom

The calculation, first attempted by Kabir and Salpeter, of the mean excitation energy entering in the Lamb shift of the helium ground state is redone by a quite different approach. Our answer, $\mathrm{ln}[\frac{{k}_{0}}{{r}_{y}}]=4.370\ifmmode\pm\else\textpm\fi{}0.004$, leaves theory and experiment on the ionization energy of helium in agreement within the experimental uncertainty of \ifmmode\pm\else\textpm\fi{}0.15 ${\mathrm{cm}}^{\ensuremath{-}1}$. Incidental results are given for the electrostatic polarizability of He and ${\mathrm{H}}^{\ensuremath{-}}$ ground states and there is appended a new discussion of the construction of higher angular momentum eigenfunctions for the three-body problem.

Parallax Theory in Islamic Astronomy

Previous articleNext article No AccessParallax Theory in Islamic AstronomyE. S. KennedyE. S. Kennedy Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by Isis Volume 47, Number 1Mar., 1956 Publication of the History of Science Society Article DOIhttps://doi.org/10.1086/348454 Views: 9Total views on this site Citations: 13Citations are reported from Crossref Copyright 1956 History of Science Society, Inc.PDF download Crossref reports the following articles citing this article:José Chabás, Bernard R. Goldstein Ibn al-Kammād’s Muqtabis zij and the astronomical tradition of Indian origin in the Iberian Peninsula, Archive for History of Exact Sciences 69, no.66 (Aug 2015): 577–650.https://doi.org/10.1007/s00407-015-0158-8S. Mohammad Mozaffari Annular Eclipses and Considerations About Solar and Lunar Angular Diameters in Medieval Astronomy, (Oct 2014): 119–142.https://doi.org/10.1007/978-3-319-07614-0_9Julio Samsó Ibn al-Raqqām’s al-Zij al-Mustawfi in MS Rabat National Library 2461, (Sep 2013): 297–328.https://doi.org/10.1007/978-3-642-36736-6_16S. Mohammad Mozaffari Wābkanawīʼs prediction and calculations of the annular solar eclipse of 30 January 1283, Historia Mathematica 40, no.33 (Aug 2013): 235–261.https://doi.org/10.1016/j.hm.2013.04.001Hamid-Reza Giahi Yazdi Al-Khwārizmī and annular solar eclipse, Archive for History of Exact Sciences 65, no.55 (Apr 2011): 499–517.https://doi.org/10.1007/s00407-011-0080-7Petra Heijden, Ian T. Durham, İhsan Fazlıoğlu, Narahari Achar, Jürgen Hamel, Adriaan Blaauw, Petra G. Schmidl, Ian T. Durham, Derek Jensen, Glenn A. Walsh, Mary Woods Scott, Jordan D. Marché, Adam Jared Apt, Woodruff T. Sullivan, Gillian Knapp, Setsuro Ikeyama, Glen M. Cooper, Alexander A. Gurshtein, Meltem Akbas, David A. King, Y. Tzvi Langermann, Behnaz Hashemipour, Emilia Calvo, Mohammed Abattouy, Glen Van Brummelen, Sonja Brentjes, Nicholas Campion, Marvin Bolt, Naoshi Fukushima, Glen M. Cooper, Richard A. Jarrell, Roland Wielen, Roland Wielen, Roland Wielen, Roland Wielen, Joseph F. MacDonnell, Paul Charbonneau, Frank K. Edmondson, Thomas A. Dobbins, Virginia Trimble, Christof A. Plicht, Richard A. Jarrell, Katherine Bracher, Christof A. Plicht, Virginia Trimble, Thomas Hockey, Jordan D. Marché, Victor K. Abalakin, László Szabados, Jiȓí Grygar, Horst Kant, Thomas A. Dobbins, Virginia Trimble, Mihkel Joeveer, Thomas A. Dobbins, William Sheehan, Daniel Kolak, Fathi Habashi, Hartmut Frommert, Thomas A. Dobbins, Steven J. Dick, Thomas R. Williams, Len Berggren, Daniel W. E. Green, Ursula B. Marvin, Brian Luzum Kāshī: Ghiyāth (al‐Milla wa‐) al‐Dīn Jamshīd ibn Masҁūd ibn Maḥmūd al‐Kāshī [al‐Kāshānī], (Jan 2007): 613–615.https://doi.org/10.1007/978-0-387-30400-7_743Jos� Chab�s, Bernard R. Goldstein Andalusian astronomy: al-Z�j al-Muqtabis of Ibn al-Kamm�d, Archive for History of Exact Sciences 48, no.11 (Jan 1994): 1–41.https://doi.org/10.1007/BF00374421Benno Dalen A Statistical Method for Recovering Unknown Parameters from Medieval Astronomical Tables, Centaurus 32, no.22 (Jul 1989): 85–145.https://doi.org/10.1111/j.1600-0498.1989.tb00842.xMARIE-THÉRÈSE DEBARNOT The Z?j of Habash al-H?sib: A Survey of MS Istanbul Yeni Cami 784/2, Annals of the New York Academy of Sciences 500, no.1 From Deferent1 From Deferent (Jun 1987): 35–69.https://doi.org/10.1111/j.1749-6632.1987.tb37195.xO. NEUGEBAUER Three Copernican Tables, Centaurus 12, no.22 (Jun 1968): 97–106.https://doi.org/10.1111/j.1600-0498.1968.tb00081.x G. J. Toomer A Survey of the Toledan Tables, Osiris 15 (Oct 2015): 5–174.https://doi.org/10.1086/368631H. Dingle 41. Commission de L’histoire de L’astronomie, Transactions of the International Astronomical Union 10 (Apr 2016): 610–618.https://doi.org/10.1017/S0251107X00021192 I. Bernard Cohen , and Katharine Strelsky Eighty-Second Critical Bibliography of The History of Science and Its Cultural Influences (To 1 January 1957) (Endorsed by the History of Science Division of the International Union of the History and Philosophy of Science), Isis 48, no.22 (Oct 2015): 189–280.https://doi.org/10.1086/348564

Entropy and Choice Time: The Effect of Frequency Unbalance on Choice-Response

A human subject making a sequence of choice-responses is considered as a channel transmitting information. Earlier work suggests that the rate of transmission is limited, and so that response time is proportional to the “entropy” of the source of signals. Entropy is reduced by unbalance in the relative frequency of the possible signals according to the formula, sum (p log p). Unbalance should therefore reduce average response time This prediction is tested in a card-sorting task. The subjects sorted playing-cards into classes in various ways; times taken were proportional to calculated entropy-per-card. Departures from the expected results occurred and were found to be due to differences in perceptual difficulty of discriminations. Some incidental results are mentioned.

The Theory of Spin Waves in Ferromagnetic Media

The theory of spin waves, leading to the Bloch ${T}^{\frac{3}{2}}$ law for the temperature variation of saturation magnetization, is discussed for ferromagnetic insulators and metals, with emphasis on its relation to the theory of the energy of the Bloch interdomain wall. The analysis indicates that spin-wave theory is of more general validity than the Heitler-London-Heisenberg model from which it was originally derived. Many properties of spin waves of long wavelength can be derived without specialized assumptions, by a field-theoretical treatment of the ferromagnetic material as a continuous medium in which the densities of the three components of spin are regarded as amplitudes of a quantized vector field. As applications, the effects of anisotropy energy and magnetic forces are calculated; and it is shown that the Holstein-Primakoff result for the field dependence of the saturation magnetization can be derived in an elementary manner. An examination of the conditions for validity of the field theory indicates that it should be valid for insulators, and probably also for metals, independently of any simplifying assumptions. The connection with the itinerant electron model of a metal is discussed; it appears that this model is incomplete in that it omits certain spin wave states which can be proved to exist, and that when these are included, it will yield both a magnetization reversal proportional to ${T}^{\frac{3}{2}}$ and a specific heat proportional to $T$. Incidental results include some insight into the relation between the exchange and Ising models for a two-dimensional lattice, an upper limit to the effective exchange integral, and a treatment of spin waves in rhombic lattices.

On the Theory of Spin Waves in Ferromagnetic Media

The theory of spin waves, leading to the Bloch ${T}^{\frac{3}{2}}$ law for the temperature variation of saturation magnetization, is discussed for ferromagnetic insulators and metals, with emphasis on its relation to the theory of the energy of the Bloch interdomain wall. The analysis indicates that spin-wave theory is of more general validity than the Heitler-London-Heisenberg model from which it was originally derived. Many properties of spin waves of long wavelength can be derived without specialized assumptions, by a field-theoretical treatment of the ferromagnetic material as a continuous medium in which the densities of the three components of spin are regarded as amplitudes of a quantized vector field. As applications, the effects of anisotropy energy and magnetic forces are calculated; and it is shown that the Holstein-Primakoff result for the field dependence of the saturation magnetization can be derived in an elementary manner. An examination of the conditions for validity of the field theory indicates that it should be valid for insulators, and probably also for metals, independently of any simplifying assumptions. The connection with the itinerant electron model of a metal is discussed; it appears that this model is incomplete in that it omits certain spin wave states which can be proved to exist, and that when these are included, it will yield both a magnetization reversal proportional to ${T}^{\frac{3}{2}}$ and a specific heat proportional to $T$. Incidental results include some insight into the relation between the exchange and Ising models for a two-dimensional lattice, an upper limit to the effective exchange integral, and a treatment of spin waves in rhombic lattices.

The Theoretical Constitution of Metallic Beryllium

Calculations of total energy as a function of lattice constant, and of some other properties, have been made for metallic beryllium, by a self-consistent field method. Since such calculations have not been made before for any but monovalent metals, the principal object was to find out how far the various assumptions which are usually made in them remain valid for higher valencies, and to test the practicability of certain methods of making the calculations. The theoretical values of binding energy, lattice constant, and compressibility agree reasonably well with experiment. The calculated work function, however, could be made to coincide with the experimental value only by assuming a surface double layer of over 5 volts, which seems impossibly large. This suggests a very large deviation of the exchange energy from the value for completely free electrons. An investigation of the behavior of the exchange energy yields an expression for the deviation from the free electron value which is valid for low electron densities, but not for those which occur in beryllium. The distribution of the electronic states in energy is found to be of the sort needed to account for the diamagnetism of beryllium. Concerning methods of calculation, it is shown that a rather complicated procedure is necessary to obtain quantitative results when a Hartree ion core field is used (as was done in the present case), and that construction of an empirical field is preferable. The assumption ${E}_{k}={E}_{0}+\frac{\ensuremath{\alpha}{\ensuremath{\hbar}}^{2}{k}^{2}}{2m}$, for the energy of an electron with vector $k$, cannot be used for calculations of lattice constant or compressibility for a divalent metal; it is therefore necessary to calculate directly the energies of states near the Fermi surface. This was done by the orthogonalized plane wave method, which is shown by tests to be capable of fairly high accuracy, though laborious. This method suggests a simple qualitative way of understanding a number of features of the electronic energy spectrum of a metal and its manner of variation with lattice constant. Incidental results include a proof that the interaction of the $1s$ shells is entirely negligible, and a calculation of the electrostatic interaction energy of the ions as a function of the $\frac{c}{a}$ ratio.

Artificial Radioactivity Produced by the Deuteron Bombardment of Nitrogen

This paper describes the work leading to the discovery and investigation of radio-oxygen, and also includes some incidental results on other nuclear disintegrations. It is shown that nitrogen bombarded with deuterons gives rise to a radioactive substance which emits positive electrons of maximum energy 1.2 mv, has a half-life of 126\ifmmode\pm\else\textpm\fi{}5 sec., and is found by chemical analysis to be an isotope of oxygen. The nuclear cross section for the activation of nitrogen at 2 mv deuteron energy is 6\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}27}$ ${\mathrm{cm}}^{2}$; that for the activation of carbon is three times as great. The neutrons expected to accompany the formation of radio-oxygen were found to be present. These results were obtained using gaseous nitrogen as a target; in some of the experiments use was made of the fact that the active product can be deposited on a metal surface by recoil.