# Questions tagged [simplex]

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

487 questions
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### How to solve simplex problem with x1 + x2 + x3 + x4 =1 as restriction?

So, there's this problem: maximize $$6x_1 + 8x_2 + 5x_3 + 9x_4$$ subject to $$x_1+x_2+x_3+x_4 = 1 \;\text{ knowing that }\; x_1, x_2, x_3, x_4\geq0$$ The solution is obvious: since the sum of ...
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### Can the simplex method be used for general monotonically increasing objective functions?

The simplex method relies on the fact that if you have a linear objective function and linear constraints, the optima must lie on one of the vertices of the simplex formed by the constraints. Let's ...
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### Is this a non-example of a $\Delta$-complex?

I know this is not a simplicial complex, but is it a $\Delta$-complex? Here is the definition of $\Delta$-complex from Hatcher: I don't believe any of the rules are broken.
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### Computing hyper area of a contrained simplex

Let $D_n [r , (a_1, b_1 ) , \ldots , (a_n, b_n) ] = \{ (x_1 , \ldots , x_n ) \in \mathbb R^n \mid \sum_i x_i = r \mbox{ and } b_i \geq x_i \geq a_i \, \forall i \}$, where $r \geq b_i \geq a_i \geq 0$...
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### How can I denote the set of probability distributions over a finite set?

I am trying to refer to the set of probability mass functions over a finite set $A$. If the elements in the set $A$ are numbered, referring to the simplex $\Delta^{|A|-1}$ would describe exactly what ...
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### Rank of $\ker \partial_i$ for $n$-simplex

Let $K=[v_0,v_1,\dots,v_n]$ be a $n$-simplex, and let $\partial_i$ denote the $i$th boundary map. I am trying to find out the rank of $\ker\partial_i$, for $0\leq i\leq n$. What I have tried so far ...
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### Invertible matrix and division (Simplex algorithm)

My book presents the following algebraic construction: LP: $z_{min}=cx$ $s.t.$ $Ax=b$ $x \ge 0$ Where A is a m*n matrix and x is a vector of variables. Beyond the meaning of the simplex ...
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### linearization of minimum function

I need to be advised in linear programming design. I have spent on this issue some time, I have reached my goal partially, but I am blocked by my lack of knowledge in some areas to go further. Namely, ...
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### Singular Homology: every $0$-chain is a $0$-circle

I am having trouble understanding this fact, that is deemed as trivial and thus not proved in most books. First of all, I understand that the boundary operator $\operatorname{Bdy}$ goes from $S_n$ to ...
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### Singular $n$-simplex, unknown notation, homology

I would like to know what is denoted in the singular simplices paragraph by $e_0,...,e_n$ here: $$[p_0,p_1,...,p_n]=[\sigma(e_0),...,\sigma(e_n)]$$ ? How this matches with simplicial identities $d_i$...
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### Simplex is closed

I define the simplex by $C=C(x_1,\dots,x_n)= \{\sum_{i=1}^{n} \lambda_i x_i : \lambda_i \ge 0 \wedge \sum_{i}^{n} \lambda_i = 1\}$. Now assume that $x_1,\dots,x_n$ are linearly independent in some ...