I'm testing an algorithm to find the global mimina of a function. Can someone give me a few examples of optimization test functions in 2 or 3 dimensions, like the Rastrigin function.
I'm hoping to find functions with several local minima.
I've been working on a decentralized algorithm for finding local minima. Can anyone give me a few examples of mappings of the form F:R→R that have multiple local minima. I'm having problems defining neighbourhood on mappings from R2→R, so I thought I'll test it out on single variable functions...
Yeah sorry, that's [a,a].
We have [0,r)\cup(r,1] in A, if r is any rational in [0,1]. Therefore the element {r} ε A. So I think we could generalize this to all r ε Q\cap[0,1].
Yeah singletons are in A. Since singletons of the form (a,a) exist in J, the single union of such elements should exist in A. That I assume implies, all the rationals in [0,1] are also in A.
This topic came up while studying measures on sub-intervals of [0,1]. The collection of all intervals in [0,1] is a semi-algebra, say J. Now from finite disjoint union of members of J let's say we form a set A.
I was able to prove that A is an algebra, since for any C,D ε A, C\capD and C^{c}...
I've made an error in the symbols used and didn't re-check carefully enough. I meant to write,
a \cap A = ∅
and
1 \cap A = ∅
not ≠ ∅.
Since the axiom of regularity requires the existence of atleast one element disjoint from the set itself, in either case we'd have an element...
Wouldn't that contradiction disprove the existence of sets that have only themselves as elements? It'd have to be shown that any set that contains itself, contains only itself as an element. I'm assuming this is where the axiom of pairing is required to complete the proof.
Hey all,
I was reading Terence Tao's text on analysis. After stating the axioms of pairing and regularity, he asks for proof of the statement that no set can be an element of itself, using the above two axioms. He has not defined any concepts like hierarchy or ranks.
I can see how,
A...