Simfish/InquilineKea's Thoughts

things you’ll never see from an ad hoc (read: nearly all) curriculum
January 26, 2008, 10:19 am
Filed under: education

Pluralism for the win!




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.