A First Course in Topology

"Let K be a field, and R = K[x_1, ..., x_n] a finitely generated K-algebra. Then R is a field implies R is algebraic over K."

"Let K be a field, and R = K[x_1, ..., x_n] a finitely generated K-algebra. Then R is a field implies R is algebraic over K."

Divisible Modules

"Let M be an R-module where R is a PID. We say that M is divisible if for each nonzero a in R, aM = M."

"Let M be a compact and orientable Riemannian manifold. Then, the number of linearly independent real harmonic forms of degree p is equal to the pth Betti number of M."

Mishikichikimikata Morukushirukikumei Molu Fumodoshi Chikukamirikitoji

    1.  Mishikichikimikata kitomikitekutochi katokatafuarikiari
  

    2.  Mekufu kiariaridokuari/mimotomikushitoari kito midoshishikutochi shimotaku

   
    3.   Zumodoshitota/tekikashifu/tutaji lushimorin chikukamirikitoji

    4.  Kashikukaari  molu nokashichimidotakashi kito chikukamirikitoji

    5.  Tekukunokushi kunanotamoshikachikito molu  kiariaridokuari shikakiarikute kito nomoshichilumotakimo

Maths Banter

Over the years I'm sure you've seen many types of proof, in your next lecture; see if you can spot these lesser-known types:

Proof by cumbersome notation:
Best done with access to at least four alphabets and special symbols.

Proof by reference to inaccessible literature:
The author cites a simple corollary of a theorem to be found in a privately circulated memoir of the Slovenian Philological Society, 1883.

Proof by importance:
A large body of useful consequences all follow from the proposition in question.

Proof by intimidation:
"Trivial."

Proof by metaproof:
A method is given to construct the desired proof. The correctness of the method is proved by any of these techniques.

Proof by vehement assertion:
It is useful to have some kind of authority relation to the audience.

Proof by semantic shift:
Some of the standard but inconvenient definitions are changed for the statement of the result.

Proof by accumulated evidence:
Long and diligent search has not revealed a counterexample.