Simfish/InquilineKea's Thoughts

a very interesting word
February 27, 2008, 11:35 pm
Filed under: math

[edit] Beauty in method

Mathematicians describe an especially pleasing method of proof as elegant. Depending on context, this may mean:

  • A proof that uses a minimum of additional assumptions or previous results.
  • A proof that is unusually succinct.
  • A proof that derives a result in a surprising way (e.g., from an apparently unrelated theorem or collection of theorems.)
  • A proof that is based on new and original insights.
  • A method of proof that can be easily generalized to solve a family of similar problems.

In the search for an elegant proof, mathematicians often look for different independent ways to prove a result—the first proof that is found may not be the best. The theorem for which the greatest number of different proofs have been discovered is possibly the Pythagorean theorem, with hundreds of proofs having been published.1 Another theorem that has been proved in many different ways is the theorem of quadratic reciprocityCarl Friedrich Gauss alone published eight different proofs of this theorem.

Conversely, results that are logically correct but involve laborious calculations, over-elaborate methods, very conventional approaches, or that rely on a large number of particularly powerful axioms or previous results are not usually considered to be elegant, and may be called ugly or clumsy. This is perhaps related to the notion of Occam’s Razor.

math notes
February 9, 2008, 12:38 pm
Filed under: main, math

Ratio test: if converges to a number < 1, then function decreases as an exponential. if converges to 0, then function decreases faster than exponential.if converges to 1, decreases slower than exponential.

lagrange’s theorem: GIVEN group and subgroups, figure out order of subgroups partial converse: sylow’s theorem, GIVEN groups, there EXIST subgroups

abstract algebra
January 13, 2008, 5:11 pm
Filed under: math

dummit foote prove that a+bSqrt[-5] is not euclidean domain merely by proving it is not PID. (sufficient but not necessary)

all euclidean domains must be PIDs.

info theory
January 10, 2008, 3:17 pm
Filed under: math

The fundamental assumption in the paper is that the source information is ergodic. With this
assumption, the paper proved the AEP property and capacity theorems. Therefore, one curiosity
is arisen that “what happens if the source is not ergodic?”. If the information is not ergodic, it
is reducible or periodic. If AEP property holds with this source(not ergodic), shannon’s capacity
theorem also satis¯es in this case because capacity theorem is not based on ergodic source but on
AEP property. Therefore, to ¯nd a source that is not ergodic and holds AEP property is one of
meaningful works. Following example is one of these sources.

Definition A stochastic process is said to be stationary if the joint
distribution of any subset of the sequence of random variables is invariant
with respect to shifts in the time index; that is,
Pr{X1 = x1,X2 = x2, . . . , Xn = xn}
= Pr{X1+l = x1,X2+l = x2, . . . , Xn+l = xn} (4.1)
for every n and every shift l and for all x1, x2, . . . , xn ∈ X.