@SamuelYusim Why would one think of constructing such a polyhedron? I suppose it must be useful in some way - I'd like to understand the construction at a non-technical level.
@Balarka so I left out any mention of optimization in order to hype you up with geometry first, but you need to delve into the depths of linear programming to see why we would think of constructing a polyhedron.
So, a linear program (or LP for short) is just a problem where you need to optimize a linear function $c^Tx$ (the objective function) subject to a set of linear inequalities, which usually gets expressed as $Ax \leq b$. The set of linear inequalities defines an intersection of half spaces, which is exactly what a polyhedron is. So if you just find a way to rephrase your problem as an LP you're in a really good position.
A key fact about LP's is duality, which says that for any maximization problem, there's a natural minimization problem for which the objective function will give the same value when optimized, and vice versa. Duality is also constructive. Now from duality you have another free set of inequalities which define your object, and you can use this information to prove validity of algorithms, bounds, etc.
Let the Möbius transformation $T(z) = \frac{z-i}{z+i}$ be given. Find the image under the mapping of $T$ over the area $G = \{z: |z|<1 \} \cap \{z: |z-1|<1 \}$
however, in combinatorial optimization you usually (e.g., min-cost spanning tree) have to add the constraint that a solution vector has only integer values. this is called integer programming, and it's NP-hard :(
I am trying to solve the inverse of 16x16 matrix, but Mathemtica cannot handle it since it is not numerical variable, and it is very large.
I know that my matrix is symmetric with lots of zero, does it have any good method or approximation to solve this faster? I don't really need exact solutio...
Ping Li also wrote a paper on something related to that, but using Ricci curvature instead of sectional curvature, here
Then finally Kotschick gave me a really nice & coolsounding thingy which is an unfinished manuscript so I probably shouldn't talk too much about it. It's again something about characteristic classes and how they relate to certain geometric/topological conditions
@Axoren I have a question... Have you had computability theory? Because I will have a presentation at this subject and wanted to ask you if you have some topics in consideration.
@Evinda I've done a bit of computability theory, but I'm rather busy today. I was in a hiking accident last Friday and I've been trying to recover and catch up with work that hasn't gotten done as a result of me being in forced-bed rest.
If you've got minor questions, you can ask me a couple
@Evinda You could do a program-description presentation, where lots of wacky things happen when you use a program's description as input in odd ways. The Fixed-point theorem, the recursion theorem, quines, and multi-quines.
You'd only have to talk about each for 15 minutes and build up to each of them in order
@MikeMiller I am trying to prove if we have a banach space which is seperable then it has a subset A which is of 2nd category and has non-empty interior.
I was wondering do you have any hint for this problem ?
Let $Exp_t^{[y]} (x) $ denote the $y$ th iteration of the exponential function with base $t$ : $t^x.$
For example $Exp_t^{[1]} (x) = t^x. $
Let ~~ denote best fit.
Now as $x$ Goes to positive infinity and a pair $(a,b)$ with $e<a<b$ Is given , I wonder how to find the best fit base value $C$ ...
@MikeMiller So in every degree the harmonic forms are in the kernel at least, I guess that much is right. Then for other things to be in it I need something like $d^*d\alpha=-d d^*\beta$
For $\alpha,\beta \in A^n$
Then $d\alpha+d^*\beta$ would be in the kernel
I don't know how to investigate if this can happen or not
Let $Exp_t^{[y]} (x) $ denote the $y$ th iteration of the exponential function with base $t$ : $t^x.$
For example $Exp_t^{[1]} (x) = t^x. $
Let ~~ denote best fit.
Now as $x$ Goes to positive infinity and a pair $(a,b)$ with $e<a<b$ Is given , I wonder how to find the best fit base value $C$ ...
