# Questions tagged [simplex]

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

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### Does k dimension simplexes under certain conditions homeomorphism with k+1-sphere

Given n vertexes in $R^{2n+100}$ , and finite many k-simplexes with vertexes on the given points, and any face(k-1 simplex) of a k-simplex is joint face of even number of those k-simplex. If the ...
1 vote
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### Unbounded slack variables in linear programming problem

I have a linear programming problem: Maximise, $z = x_1 + x_2$ Subject to: $$x_1 + x_2 \ge 10$$ $$2x_1 + x_2 \le 40$$ $$x_1, x_2 \ge 0$$ When I construct the simplex tableau adding 2 slack ...
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### What's the hypersolid angle of a 5-cell (4d tetrahedron)?

It's known that the solid angle of the vertex of a regular tetrahedron is $\arccos(\frac{23}{27})$, or equivalently, $\frac\pi2-3\arcsin(\frac13)$ or $3\arccos(\frac13)-\pi$. (Trig identities are ...
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### Are the weights of an irreducible representation of a complex semisimple algebra induced by the weights of a representation of $SL(N)$ for some $N$?

If we consider the weights of a complex irreducible representation $V$ of a complex semisimple Lie algebra $\mathfrak{g}$, with respect to a chosen Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$, ...
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### Problem with the simplex method in a cost accounting problem

I have the following problem in my business class. I have done an error in my solutions but I don't know where it is. Can Someone help me? It's October, and the new management of Elektronik AG is ...
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### Faces of the cap product

Let $X$ be a topological space and $A\subseteq X$ an open subspace. Let $R$ be an associative unital ring. Define the cap product $$\cap\colon S^q(X,A;R)\otimes S_{p+q}(X,A;R)\rightarrow S_p(X;R)$$ on ...
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### Finding a mapping from the hypercube to a convex hull that conserves the uniform distribution

I am drawing points uniformly in a hypercube $x \in [-1,1]^n$ and I would like to find a map f(x) = y such that $||y||_1 \leq 1$ and that the uniform distribution is conserved. My own attempt at this ...
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### The triangle as a manifold with corners: How to choose a proper chart?

I want to show that a the standard 2-Simplex in $\mathbb{R^2}$, i.e. the set of all convex combinations of the three vectors $0, e_1, e_2$ can be viewed as a manifold with corners. However, I am ...
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### Integral of exponential function over an $n-1$ - simplex

I am trying to solve the following integral over the simplex (I'm not sure if there even is a closed form to be honest) $$\int_{\Delta^{n-1}}\prod\limits_{i = 1}^n x_{i}^{a_i}e^{-b_ix_i}dx_i$$ Where ...
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### Rounding of semidefinite programms using simplex

Suppose we have two unit vectors, $\mathbf{a}_1$ and $\mathbf{a}_2$, where $\mathbf{a}_1,\mathbf{a}_2\in\mathbb{R}^N$ and $\|\mathbf{a}_1\|_2 = \|\mathbf{a}_2\|_2 =1$. Also, we have their inner ...
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### Area of intersection between hypercube and hyperplane

To be short, I have an hypercube $C_\delta = [0, \delta]^{n+1}$ and an hyperplane $H_t = \{x \in \mathbb{R}^{n+1} : \|x\|_1 = t\}$. The quantity I'm looking for is the hyper-area of $C_\delta \cap H_1$...
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### Distribution of the largest gap between uniform random variables

I've tried to formulate my question in a previous topic but I terribly messed up my formulation, so I will create a new question to avoid any confusion. What I'm looking for is the distribution of the ...
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### Alternate optimal can be unbounded

Can it happen that at the optimal simplex table for a maximization problme, we have reduced cost of one of the nonbasic variable as 0, but no entry in its corresponding column at the optimum is ...
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### Ant on a Simplex problem, expected number

I recently encountered a quant interview question, which I think is not super hard, but still I am not very sure about my results. Could you guys give me some hints on it? The question goes as follows:...
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### Map onto reference Simplex properties

I have the affine Transformation from the unit simplex $T_0 = \{e_0,...,e_d\}$ , onto another non-degenerative simplex $T = \{a_0,...,a_d\}$. $$F(e_j)=a_j, F(T_0)=T$$ I know that $F(x)=Ax+a_0$ where A ...
1 vote
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### Computing the integral of a linear function over a polyhedron

I am currently looking for a way to compute the integral of an n-dimensional linear function over a convex region in the shape of an n-vertex polyhedron. So far I have tried the R package ...
1 vote
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### Solving a LFP-Problem with as Simplex subprocedure

I'm a computer science student, who is trying to solve a Linear-fractional programming (LFP) problem for an exercise in a course about Decision Support Systems. In the said exercise I need to provide ...
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### How do you solve minimization LP problem with dual method?

So i just started with the linear programming topic in my university. And while I was practicing, I found the next question: $$\min Z=3x_1+4x_2-x_3$$ $$\text{Subject to: }x_1+3x_2-x_3\ge1$$ 2x_1+x_2+...
1 vote
Given a minimilization problem of the form: $min \ c^t x$ ($c, x \ \in \mathbb{R}^n$) where the following constraints have to be met $Ax = b$ (A has dimensions mxn) $x \ge 0$ (in every entry) \$rank(...