# Questions tagged [simplex]

For questions on the $n$-simplex, an $n$-dimensional polytope with $n+1$ points.

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### maximum of a concave function over a convex constraint

Let us assume we have a continuous function ($f$) which is concave, and we want to find its maximum over a convex set i.e. \begin{equation} \int_{0}^{a}q(x)p(x)f(x)dx\leq f(\int_{0}^{a}p(x)q(x)dx) \...
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### Why should we minimize the sum of artificial variables in $2$ phase method? [closed]

In Phase $I$, if the LP is of the maximization type, why we do not maximize the sum of the artificial variables in Phase $I$?
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### How to divide a unit space into many simplices?

I'm sorry, it may be simple and stupid but I didn't find any relative solutions on the Internet. Given the unit hypercube $C$ in the Euclidean space $R^n$, how to divide (or we can say "partition", I ...
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### Intuition behind singular $n$-simplex

I came across this definition that a singular $n$-simplex in a topological space $X$ is a continuous map $\sigma\colon \Delta^n \to X$. Using this definition a few examples were put forward: a ...
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### Finding the normal of a simplex facet in n-dimensions

I am attempting to find a generalised formula for the normal of a simplex facet in n-dimensions. For example if I had the 2 dimensional simplex formed by the vertices ABC below. Then I want to find ...
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### How do you mathematically characterize an “enlarged probability simplex”?

We all know that the probability simplex can be described as the set $$\Delta = \left\{\theta \in \mathbb{R}^n| \sum\limits_{i = 1}^N \theta_i = 1, \theta_i \geq 0\right\}$$ and in $\mathbb{R}^3$ it ...
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### Looking for a bijective function mapping an n-simplex to itself

As part of a research question I am exploring, I need to find a bijective function on an n-simplex that maps the midpoint of each sub-simplex to itself. This includes all vertices, midpoints of edges, ...
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### Four dimensional integral by linear change of variables

I have the following problem: I have an integral of the following form (the integrand is not import) $$\int_0^{\ell_q}dx\int_0^{\ell_q}dy\int_0^{\ell_p}d\xi_1\int_0^{\ell_p}d\xi_2.$$ My aim is to find ...
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### How to implement ratio test if all entries in the pivot column are negative?

I am trying to find the adjacent bfs for the following tableau. So far I understand that it has an alternative solution because there is $x_4$ a nonbasic decision variable that has a 0 coefficient in ...
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### Are constants allowed within an objective function for a linear programming problem?

I've been taking a class on linear programming and have been working with a lot of different problems and methods of solving them. All this time however, I have rarely come across an objective ...
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### solid angles of an n-simplex

Do there exist formulae relating the n-th dimensional solid angles of an n-simplex to either the n-th order dihedral angles, the volume of the n-1 dimensional facets, or the side lengths of the ...
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### Need help with simplex noise skew transformation

I am reading a paper about simplex noise. http://knielsen-hq.org/simplex_noise_skew_factor.pdf For whatever reason I can't figure out the result they got here. My brain is just goin kapoot. To ...