Articles of simplex

Geometrical interpretation of simplices

I’ve just started doing simplices, where the $n$-simplex has been defined to be $$\Delta^n = \{x \in \mathbb{R}^{n+1}\mid x_i \geq 0, \sum x_i=1\}.$$ It’s easy to see that the $0$-simplex is the point $1$ in $\mathbb{R}^1$, the $1$-simplex is the line from $(1,0)$ to $(0,1)$ in $\mathbb{R}^2$, and the $2$-simplex is the triangle, including the […]

How to test if a feasible solution is optimal – Complementary Slackness Theorem – Linear Programming

I have this linear program $$\begin{cases} \text{max }z=&5x_1+7x_2-3x_3\\ &2x_1+4x_2-2x_3&\le8\\ &-x_1+x_2+2x_3&\le10\\ &x_1+2x_2-x_3&\le6\\ &x_1,\,x_2,\,x_3\ge0 \end{cases}$$ and a feasible solution of his dual $(D)$ is $y = {7/2,2,0}$. I need to find an optimal basis of $(P)$ and an optimal basis of $(D)$ using the complementary slackness theorem. I thought about assuming that $y$ is an optimal solution […]

Intuition for volume of a simplex being 1/n!

Consider the simplex determined by the origin, and $n$ unit basis vectors. The volume of this simplex is $\frac{1}{n!}$, but I am intuitively struggling to see why. I have seen proofs for this and am convinced, but I can’t help but think there must be a slicker or more intuitive argument for why this is […]

Why is this not a triangulation of the torus?

I refer to example 4, fig.3.6, p.17 of Munkres’ Algebraic Topology. He says the given triangulation scheme “does more than paste opposite edges together”. Not clear to me. For those who don’t have the book to hand, a rectangle is divided into 6 equal squares by a horizontal midline and two verticals; each square has […]

How to solve this operation research problem using dual simplex method?

Maximize $$ z = 2x_1 -x_2 +x_3$$ Subject to constraints $$2x_1 + 3x_2 -5x_3 \ge 4$$ $$-x_1 +9x_2 -x_3 \ge 3$$ $$4x_1 +6x_2 +3x_3 \le 8$$ And $x_1, x_2, x_3 \ge 0$ I managed to solve this through simplex method(by 2 stage method) but I was asked solve it using dual simplex method, I found […]

Network simplex method, leaving and entering variables

Could someone give me a hint on this question, which is a past exam question: Under what circumstances will an entering variable in the network simplex method be the same as the leaving variable? Thank you for your help.

How to find out whether linear programming problem is infeasible using simplex algorithm

So in, there is a mention about finding out whether linear programming (LP) problem is infeasible by simplex algorithm, but it does not actually go over it. How does one find whether LP is infeasible using simplex algorithm?

Uniform sampling of points on a simplex

I have this problem: I’m trying to sample the relation $$ \sum_{i=1}^N x_i = 1 $$ in the domain where $x_i>0\ \forall i$. Right now I’m just extracting $N$ random numbers $u_i$ from a uniform distribution $[0,1]$ and then I transform them into $x_i$ by using $$ x_i = \frac{u_i}{\sum_{i=1}^N{u_i}}. $$ This is correct but […]

Does every positive ray intersect a deformed simplex? A topological conjecture.

This question is motivated by my partial answer to a different question. I will use $\mathbb R_+$ to denote the set of nonnegative reals. Consider the standard simplex $\Delta^n=\{(x_0,\dots,x_n)\in\mathbb R_+^{n+1}:x_0+\dots+x_n=1\}.$ We are given a continuous function $f:\Delta^n\to\mathbb R_+^{n+1}$ with the property that it preserves zero coordinates, that is, if $x_i=0$ then $f(x)_i=0.$ Thus vertices of […]

Boundary/interior of $0$-simplex

In a $1$-simplex, it’s clear that the boundary is the set of the two $0$-simplices and that the interior is all the points in between them. But what about in a $0$-simplex? I’m asking because I know that the topological realisation of a simplex is the union of the interiors of simplices. This would suggest […]