# Tag Info

## Hot answers tagged mathematics

16

The letter Ш (sha) of the Cyrillic alphabet is widely accepted in theoretical computer science as the symbol for the shuffle product, which gives the shuffle algebra. The same letter is also used to denote the Tate-Shafarevich group, but I'm not sure if it's really a standard (the letter was introduced by Cassels only in 1990 in 1962 instead of TS, see below ...

10

It is sometimes asserted that $\varnothing$ for the empty set was introduced by Bourbaki using a Danish and Norwegian letter. EDIT: The source is the Weil autobiography, cited in Jeff Miller's collection of the origins of mathematical expressions: André Weil (1906-1998) says in his autobiography that he was responsible for the symbol: Wisely, ...

9

There are several non-alphabetic symbols, the best known is the integral sign $\int$ and the Weierstrass $P$-function $\wp$. To be sure their origins are letters of Latin alphabet, but they are special stylized symbols, and as far as I know there is no computer code for them in the standard sets of computer characters. Strictly speaking they do not belong to ...

5

“Mathemata mathematicis scribuntur.” is the original Latin of Copernic which is easily translated as “Mathematics is written for mathematicians.” but Edward Rosen chose to translate this famous passage as “Astronomy is written for astronomers.” Obviously "astronomy" is not the author's word and also it is generally agreed that there was no ...

5

Good sources on the history of fixed point theorems are Park, Ninety Years of the Brouwer Fixed Point Theorem and Kumar, A Short Survey of the Development of Fixed Point Theory. According to both, early versions of fixed point theorem concerned self-maps, from a ball or some other set to itself. The first version for non-self maps is in Rothe's Zur Theorie ...

4

Yes, orthogonal matrices with complex entries appeared at least as early as 1900, in E. Cartan's classification of simple Lie algebras (and Lie groups). In many ways, the complex numbers could be replaced by any algebraically closed field of characteristic $0$. Thinking of complex orthogonal groups as real Lie groups ("forgetting" the complex structure) ...

4

They did. A natural way to treat such matrices is to introduce an indefinite inner product on $\mathbb{C}^n$, a non-degenerate bilinear form $(z,w):=z_1w_1+z_2w_2+\dots z_nw_n$, instead of the usual sesquilinear one. Then $A^T=A^{-1}$ is equivalent to $(Az,Aw)=(z,w)$, i.e. complex orthogonal matrices are isometries of this space. The "orthogonal"/"unitary" ...

3

According to Wikipedia, it is the Austrian mathematician Gerhard Wanner.

3

Lucky you! (or me :-) ). This question was answered a while back on Math.SE I found his original thoughts in the translated version of "Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum, volume 1", chapter 7. The translation is called "Foundations of Differential Calculus" and a link is found here https://...

3

According to Fowler's Ratio in Early Greek Mathematics, palindromic patterns in continued fractions of $\sqrt{p}:\sqrt{q}$ with primes $p>q$ were likely known already to Pythagoreans. In the modern times the interest in continued fractions was revived largely due to Euler's work (from 1731), although Wallis and Huygens worked on them earlier, see ...

3

The letter $\upsilon$ or $\Upsilon$ is the second letter of δύναμις, κύβος and δυναμοδύναμις (square, cube and bisquare) of the words the notation abbreviates. Being the same letter, it plays an additional role of identifying the power symbols as being of a kind, marking Diophantus's "numerical species" (squares, cubes, etc., multiplied by coefficients). ...

3

The cyrillic letter Ш (sha)is -- for obvious reasons when looking at the graph) also used to denote the "function" (well, it is a distribution if you want to be picky) given by the sum of integral displacements of the Dirac-delta function, see https://en.wikipedia.org/wiki/Dirac_comb

2

My vote goes to three out of {Michał Dziewicki, Stanisław Zaremba, Jan Śleszyński, Witold Wilkosz, Otto Nikodym, Leon Chwistek, Stanisław Bilski}; most probably the three whose names I put in bold. On May 10, 1915, Russell wrote to Ludwig Wittgenstein who was doing military service in Krakow: "If you have time, you should visit in Krakow a lonely old ...

2

I'd be willing to bet that quite a number of theorems were lost simply due to the Library of Alexandria getting destroyed. Take a look at this History Stack Exchange question and some of the answers. Particularly where it mentions that all mathematics before Euclid was lost, and that the mathematics of that age wasn't really surpassed until the 19th ...

1

Yes, he was, at least in a particular example. Near the end of his life, in 1808-09, Lagrange studied perturbative dynamics of a planet on an elliptic orbit, and derived what came to be called Hamiltonian equations for it in Second mémoire sur la théorie de la variation des constantes arbitraires dans les problèmes de mécanique, dans lequel on simplifie l'...

1

The earliest reference I can find to "banana brackets" is in: G. Malcolm. Data structures and program transformation. Science of Computer Programming, 14(2-3):255-280, October 1990. Where they are clearly crescent-moon/banana shaped symbols: ⦅...⦆. The later style using $($ and $|$ seems to be a typographic practicality, and is used by ...

1

The writings of Descartes and Viète do indeed suggest a certain "inferiority complex", believing that their own work was simply rediscovering Greek analytic methods which had been lost by the "barbarians". From Descartes' Rules for the Direction of the Mind: But when I afterwards bethought myself how it could be that the earliest pioneers of Philosophy ...

1

For a discussion of Cauchy's contribution, see pages 92-131 of T. Muir, The Theory of Determinants in the Historical Order of Development, vol. 1, second edition, MacMillan, London, 1906.

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