Subject: Re: [HM] arrow notation
From: Emili Bifet (bifet@inch.com)
Date: Wed May 24 2000 - 19:57:08 EDT
Julio Gonzalez Cabillon wrote:
> It may be worth recalling that Bourbaki as early as 1939 (!) used the
> arrow in element-to-element notation [ "la application x --> f(x)" ].
> Earlier instances of this notation can also be found. For instance,
> Oystein Ore in "L'Agebre Abstraite" [Paris: Hermann, 1936] wrote:
>
> "Nous dirons que deux systemes algebriques S et S' sont homomorphes
> (par rapport a l'addition et a la multiplication) s'il existe une
> correspondance a --> a' entre les elements de S et S' donnant a
> chaque element $a$ de S une image unique $a'$ dans S' telle que
> chaque element de S' soit l'image d'au moins un element de S et en
> outre telle que de a --> a', b --> b' on puisse conclure
>
> a + b --> a' + b' , ab --> a'b' ."
Dear Julio,
I'm afraid that it was not clear from my posting, but Riemann seems to
have used "the arrow in element-to-element notation" as early as his
Lectures of 1856/57 (cf. pages 67 and 68 of the Supplement in the Dover
edition of the Gesammelte Werke.)
Here follows a transcription of some passages. Note that Riemann is
dealing with the monodromy transformation, in this case a linear
operator on a complex vector space of dimension 2.
"II. Die Integrale einer linearen Differentialgleichung zweiter Ordnung
in einem Verzweigungspunkt. (Aus einer Vorlesung Wintersemester
1856/57.)"
"Ist a ein Verzweigunspunkt der Loesung einer linearen
Differentialgleichung zweiter Ordnung und geht, waehrend x sich im
positiven Sinn um a bewegt, z_1 ueber in z_3 und z_2 in z_4, was kurz
durch z_1 --> z_3 und z_2 --> z_4 angedeutet werden soll, so ist
z_3 = t z_1 + u z_2
(1)
z_4 = r z_1 + s z_2."
"Ist e [=epsilon] irgend eine Kostante, so ist
z_1 + e z_2 --> z_3 + e z_4."
...
"Da nun
z_1 (x - a)^{-\alpha} --> z_3 (x - a)^{-\alpha} e^{-2 \alpha \pi i},
so muss
z_1 (x - a)^{-\alpha} --> z_1 (x - a)^{-\alpha}
+ (z_1 + e z_2)k(x - a)^{-\alpha}.
"Da ferner
k
------- (x - a)^{-\alpha}(z_1 + e z_2)l(x - a) -->
2 \pi i
k
------- (x - a)^{-\alpha}(z_1 + e z_2)l(x - a)
2 \pi i
+ k (x - a)^{-\alpha}(z_1 + e
z_2),
..."
Best wishes,
Emili Bifet
PS Concerning the arrow notation in analysis, Whittaker & Watson write
in a footnote, at the beginning of Chapter II: "The arrow notation is
due to Leathem, _Camb. Math. Tracts,_ No. 1."
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