Greetings from San Luis Obispo. I am on vacation this week having a great time!
So far I visited Morro Bay (what a sight - despite the presence of a power plant),hung out in downtown SLO, and visited Cal Poly (and it's library - I am going to hit the library at UC Santa Barbara tomorrow).
Some math tips I picked up from my visit:
Fuzzy sets are sets that allow degrees of membership. Instead of having a Yes/No decision of whether an object belongs in a set, degrees of acceptance are allowed.
Let A(x) → [0,1] where A(x) is the degree of acceptance.
The basic properties for normal sets work for fuzzy sets.
From the NIST Handbook of Mathematical Functions (that book is huge!):
Ways to calculate some functions:
Error Function:
erf(z) = 2/√π * ∫(e^(-t^2) dt, 0, z) = 2/√π * Σ((-1)^n * z^(2*n+1) /(n! * (2*n+1)), n=0 to infinity)
Γ(z) = ∫(e^-t * t^(z-1) dt,0,infinity) = (z-1)! = π /(sin (π*z) * Γ(1-z))
Γ(x) ≈ e^-x * x^x * √(2 π/x) * Σ(g_k/x^k, k=0 to infinity) where g_k is from a series. The first few terms are
g_1 = 1
g_2 = 1/12
g_3 = 1/288
g_4 = -139/51840
g_5 = -571/2488320
g_6 = 163879/2090188880
zeta(s) = Σ(n^-s, n=0 to infinity) = 2^(s-1)/Γ(s+1) * ∫(x^s/(sinh x)^2 dx, 0, infinity)
= 1/Γ(s) * ∫(x^(s-1)/(e^x - 1) dx, 0, infinity)
Wishing the best for everyone,
Eddie
