[Corpora-List] QM analogy and grammatical incompleteness

John Williams johnwhoever at wanadoo.fr
Tue Dec 20 09:26:00 CET 2005



> In broad strokes, the history of vectors and functional

analysis became

> very closely linked in the 1840s and 1850s, partly through

Hamilton's

> work on quaternions and the theory of analytic functions on

4-space.

> Functions over the real numbers form a vector space - you can

add two

> functions together, and multiply any function by a scalar. As

a result,

> mathematicians came to realize that Fourier analysis could be

described

> in vectors - each of the functions sin(nx) and cos(nx) (for x

a real

> number, n an integer) is a basis vector, and any piecewise

smooth

> function can be expanded (uniquely) as a vector, using these

functions

> as a basis. The Fourier series coefficients are thus

interpreted as the

> coordinates of a vector in this basis. This vector space is

clearly

> infinite-dimensional, because a Fourier series expansion can

be

> infinitely long. (Note again that this means you will never

work with

> complete information once you've quantized your functions.)


And I took the languages option at school so I wouldn't have to do maths
and physics. Sigh.

Merry Christmas everybody.

John Williams

--

John Williams
Sometime Corpus Lexicographer and English Teacher.

17 rue Thionville
31000 TOULOUSE
France
Tel: (+33) (0)5 61 99 03 86
Mob: (+33) (0)6 76 12 42 24








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