If $f$ is a (not necessarily closed) path in $X$, prove that the $1$-chain $f$ is homologous to $-f^{-1}$ The book is suggesting to use the fact that the Hurewicz map $\phi : \pi_1(X, x_0) \rightarrow H_1(X)$ is a homomorphism and suggests this graph as a hint. I can see that $((f*f^{-1}) * f) * […]

I believe to be missing something important in the Simplex algorithm, because I can’t get it to work. From what I gather, there are three steps per iteration, given a matrix for a linear program in standard form: Look for negative terms in the objective function’s row. If you find one, look for the pivot […]

All I got is that $$12y_1 + 7y_2 + 10y_3 = 2(0) + 4(10.4) + 3(0) + 1(0.4)$$ and $y_2 = 0$ because $x_6$ is in basis. How do I find $y_1$ and $y_3$ without going through the simplex method? I took the dual and that doesn’t seem to help. I know that $y_1 = […]

I want to use the simplex algorithm. At the first step we want to determine which variable will enter the basis. To do that we pick the smallest negative number of the last row of the simplex table. Then we want to determine which variable will go out of the basis. For that we have […]

According to my understanding, Degeneracy in a linear optimization problem, occurs when the same extreme point of a bounded feasible region $X$ can be represented by more than one basis, that is not every unique basic feasible solution of the polyhedron is represented by a unique basis. This is detected in Simplex Algorithm when one […]

This question already has an answer here: Volume of $T_n=\{x_i\ge0:x_1+\cdots+x_n\le1\}$ 4 answers

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 […]

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 […]

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 […]

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 […]

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