W.Lang: Some links and scans

Formatting OEIS tabl and tabf arrays

Explanations:

'alist' is the string of integers, enclosed by brackets [ ] and separated by commas (insensitive to blanks, no comma at the end), representing a tabl or tabf sequence as defined in OEIS under 'Keywords'.
'flist' is used only for 'tabf' arrays, and the entries of this string give the sequence of step widths (the difference of the number of entries in consecutive rows, starting with the number of entries of the first row).
'offset' is a string with two integers: the row offset and the column offset of the array when formatted as matrix entries a(n,m) with n for rows, m for columns.
'A-number' will appear in the caption of the formatted output. Some number should be inserted, e.g. 0. If a 'tabl' or 'tabf' array from OEIS is to be formatted its A-number can be inserted (without the 'A', and leading zeros are irrelevant).

The output will be a complete matrix type array (the 'alist' string is cut off, depeding on the 'flist' string which in the tabl case is the sequence of ten 1's). If numbers become too big an error message will be given. Truncation of the 'alist' or the 'flist' should help to circumvent this.

Acknowledgements:

This application would not exist without advice and help from Bernd Feucht, Gerrit Jahn, Christoph Mayer and, especially, from Dr. Thomas Hahn.

2005

Alexander Braun (May 10, 1805, Regensburg - March 29, 1877, Berlin):
From his phyllotaxis work on fir cones ('Tannenzapfen') from 1831. (.jpg files)

" Vergleichende Untersuchung über die Ordnung der Schuppen an Tannenzapfen als Einleitung
zur Untersuchung der Blattstellung überhaupt "

Nova Acta Physico-Medica Academiae Caesareae Leopoldino-Carolinae.
Naturae curiosorum. 15 (1831) 195-402.

Journal page	Title page, p.195	Table, p.325
fir cone, fig. XIX	fir cone (detail)	Ph. ex divergentia 5/12 (parastichies), fig. XIL

Albert Einstein in Karlsruhe (pdf file) (ps.gz file)

2006

Short biography of
Karl Heun (April 3, 1859, Wiesbaden - January 10, 1929, Karlsruhe)
based on Michael von Renteln: Die Mathematiker an der TH Karlsruhe (1825-1945), 2. Auflage, Druckerei Ernst Grässer, Karlsruhe, 2002

For the Heun class of equations and applications to physics see the book of Sergei Yu Slavyanov and Wolfgang Lay: Special Functions, A Unified Theory Based on Singularities, Oxford University Press, 2000.

Albert Girard and the Waring formula

In Major Percy A. MacMahon's " Combinatorial Analysis ", Chelsea Publ. New York, 1960, one finds in Vol. I, p.6, the following footnote:
" Girard, Invention Nouvelle en l'Algèbre, Amsterdam 1629. The formula is often erroneously ascribed to Waring who gave it without proof in 1782. "
See also the on-line version Major Percy A. Macmahon's " Combinatory Analysis ", p.6.
This footnote is corrected in Vol. II, p. vii, " Note on Waring's Formula for the Sum of the Powers of the Roots of an Equation " .
Girard gave only the formulae for the first, second, third and fourth power (see scan nr. 8 below). Edward Waring (1744-1793) gave the general formula with inductive proof in his
" Meditationes Algebraicæ " . from 1770. See the edited and translated version by Dennis Weeks , American Mathematical Society, Providence , Rhode Island, ch. 1. (In the statement of Problem I the letters p,q,r,s,t,v,w and z appear in two different styles, and the + sign in the top line of the first bracket should be a - sign). See also " Translator's Notes " on pp. 34-36.

Some pages from Albert Girard's book " Invention Nouvelle en l'Algèbre ", Amsterdam 1629, reissued 1884, Leyden, by B. de Hoan

.pnm and .jpg files, produced from microfiches of Universitätsbibliothek Tübingen.

.pnm files		Title page
Scan 1	Scan 2	Scan 3	Scan 4	Scan 5
Scan 6	Scan 7	Scan 8	Scan 9	Scan 10
		Scan 11

.jpg files		Title page
Scan 1	Scan 2	Scan 3	Scan 4	Scan 5
Scan 6	Scan 7	Scan 8	Scan 9	Scan 10
		Scan 11

Some remarks, also on Girard's notation (pdf file). (ps.gz file)

2007

A geometrical construction to approximate Pi to 9 digits by
Srinivasa Ramanujan 22 XII 1887 - 26 IV 1920

Ramanujan construction (pdf file) (ps.gz file)

Zum Tode von Julius Wess (pdf file) (ps.gz file)

2011

Decomposition of Wythoff A- and B-numbers

Complementary Wythoff sequences

2020

Exercise on Hohmann transfer compared to another transfer

WS 1986/87, Übungen Theorie A, Blatt 10, Nr. 4 (W. Lang)
Exercise (in German) Solution (in German)
See also: NASA Science, Basics of Spaceflight, Hohmann Transfer Orbits