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 ...
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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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Geometric realization of simplicial sets via nondegenerate simplices
I have just started studying simplicial sets, and I was given the definition of geometric realization of a simplicial set X as $$|X| = (\coprod_{n \in \mathbb{N}} X_n \times \Delta_n) / \sim$$ with $(...
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What is an example of why considering the category of finite sequences is more useful than the set of all finite sequences?
This is a basic question about why, on a general level, category theory is so useful.
I am exploring some mathematical questions about finite sequences. I want to explore mathematical generalizations ...
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Halving planes of a tetrahedron
This nice article
Megan Martin, Cornelia A. Van Cott & Qiyu Zhang (2024) The Beauty of Halving it All, Math Horizons, 31:2, 14-17, DOI: 10.1080/10724117.2023.2249357.
shows that the envelope of ...
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Is there any 3-simplex in this figure?
I have a simple question. Is it possible to construct a 3-simplex (a, b, c, d) in the following figure?
My guess was that as we can generate an edge (a, c) by a linear combination of (a, d) and (a, b),...
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Linear Programming Simplex method issue with basis
I'm working on what I think is a fairly basic LP problem, the objective function and constraints are seen in the attached image. The task also specifies that the starting basis should be x1, x2, x3, ...
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Action of symmetry group on homology?
For a singular n-simplex $\alpha$ and a permutation $t\in S_{n}$, define $t\alpha$ to be the simplex with vertices permuted by $t$. Do we have $t\alpha$ homologous to $\text{sgn}(t)\alpha$ ? And does ...
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How to manipulate these constraints for 2-phase simplex method?
I am trying to solve an LP problem with 3 constraints: $$x_1 + 2x_2 ≥6\\2x_1+x_2≥6\\ x_1 + x_2≤6\\$$ I understand that I need to change these constraints into standard from and add slack variables as ...
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Volume of the probability simplex?
I think I might be misunderstanding the concept of a simplex and its volume.
Take the 2-dimensional simplex (a triangle) embedded in 3-dimensional space with vertices $(1,0,0)$, $(0,1,0)$, and $(0,0,1)...
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1
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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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3
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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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Dual simplex computational feasibility when applied to primal problems with many bound variables
Assuming primal simplex problem in form of
$$
\displaylines{
\begin{align}
(P) & \\
\text{min} \ & c^Tx \\
\text{s.t.} \ & Ax = b \\
& -\infty \lt l \le x \le u \lt \infty \\
A \ \...
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1
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How many probability vectors do I need to convexly generate any probability vector?
Fix a positive integer $n$, and let $B = \{p_1, \dots, p_{k}\}$ be a set of $k$ mutually linearly independent vectors in $\mathbb{R}^{2^n}$, where each $p_i \in B$ is also a probability vector in the ...
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Center of a simplex given its vertices
We have been given $n+1$ points $v_0$,$v_1$,$\dots$,$v_n$ in general position in $n$ dimensions where $n\geq2$
what is the formula for the center of the simplex?
If say some $k+1$ of the points (...
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Sensitivity analysis adding new constraint to the simplex method final tableau
I don't know how to do this exercise:
Minimize-2x1 + x2 -x3
s.t x1+2x2+x3<=8
-x1+x2-2x3<=4
x1,x2,x3>=0
Final table:
Base z | x1 | x2 | x3 | x4 | x5 | RHS
z | 1 | 0 | 3 | 3 | 2 | 0 | 16
x1 | ...
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An example of an Alexander dual of a simplex
I'm trying to learn to take a dual of a simplex by understanding the first page of this paper. Here it defines an Alexander dual of a simplicial complex as the set of subsets of vertex set such that ...
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Another multi-dimensional integral with applications in Random Matrix Theory.
This question generalizes my other question here.
Let $\vec{a}:= \left( a_j\right)_{j=1}^N \in {\mathbb N}^N$ and let
$\vec{A} := \left(A_j\right)_{j=1}^N$ . We define a following multidimensional ...
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How to add slack variables to a system with only equalities to zero?
I have a programming problem, which I've expressed as the following system of equations. I'm trying to solve these equations using the Simplex method.
...
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A multidimensional integral involving an exponential and power functions over a simplex.
Let $N \ge 2$ be an integer and let $\vec{A}:= \left( A_j \right)_{j=1}^N \in {\mathbb R}^N_+$. Consider a following integral:
\begin{equation}
{\mathfrak I}_N(\vec{A}) := \int\limits_{{\mathbb R}_{\...
- 11k
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Reducing artificial variable needed for LPP
Given that I have a question of an objective function to minimize or maximize and I have a constraint for the same such that when converting to equation form for using simplex method would require an ...
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How to integrate a Dirichlet PDF on the simplex
I have a random 3-part composition that follows a Dirichlet distribution with concentration parameter $\alpha$. The PDF could look like this:
If I want to answer the question "what is the ...
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1
answer
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What is the Geometric Interpretation of the Addition of n-Singular Simplices?
Product of paths is a very geometric operation - it is the concatenation of the paths.
I'm trying to phrase the analogue for the addition operation of chains of n-simplices (and of the induced ...
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Dirichlet distribution restricted to affine subspace
This question is similar to Conditional distribution of subvector of a Dirichlet random variable, but I can't tell if it applies to my situation.
Suppose I have:
$X\in \Delta^{n-1}$ with $X\sim Dir(\...
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Let K be a simplicial complex and σ a simplex of K. Show that the link lk(σ) is a subcomplex of K, if lk(σ) is not empty.
I wanna know if I choose the right way to do my proof? it's correct?
To show that the link $\textrm{Lk}(\sigma)$ is a subcomplex of $\mathcal{K}$, if $\textrm{Lk}(\sigma)$ is not empty, we need to ...
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Does this decomposition hold in every dimension for the $n$-cube?
I have the need to cut up a cube ($n$-dimensional cube in fact, but let's stick to the dimension 3 for the moment) in a "wise" way. I come from a totally different field so I know basically ...
- 3,252
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Why do we introduce artificial variable for equality constraints in two phase simplex method?
Say we have a linear program where we wish to maximise some function Z, and we have the following equality constraints:
$3x_1 + 2x_2 + x_3 + 2x_4 = 225$
$x_1 + x_2 + x_3 + x_4 = 117$
$4x_1 + 3x_2 + ...
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Prove that there is unique simplician map $f\colon \sigma \to \tau$, whose restiction if $f_{0}$.
In Lee's book ''Introduction to Topological manifolds'' we have the following definition :
Def: A map $f\colon \sigma\to \tau$ between two simplices is call simplicial map if it is the restriction of ...
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How is the minimum embedding dimension of an abstract simplicial complex related to its simplices?
I was working on an example of an abstract simplicial complex and its geometric realization and I had a question. I think the question makes more sense when explained through the example. Some ...
- 177
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rational bijections on n-simplices
I want to study the following bijections on the n-simplex $\{(t_1, t_2, ..., t_n) \;|\; 0 < t_1 < ... < t_n < 1\}$
$$\sigma_k(\mathbf{t}) = (t_1, t_2, ..., t_k, \frac{t_{k}}{t_{n}}, \frac{...
- 381
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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 ...
- 2,627
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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 ...
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upper bound on the number of vertexes of the intersection of two simplices
Let $C_1, ..., C_{n+1} \in R^{n}$ be the vertices of a simplex and $D_1, ..., D_{n+1} \in \mathbb{R}^n$ form another simplex. Then what is the upper bound on the number of the vertices of $$\text{conv}...
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Understanding the Beck-Chevalley condition (II)
The older question here on the site asks about the intuitive meaning of the Beck-Chevalley condition. Accidentally one of the answers has caught my eye.
It made me wonder if it is possible to describe ...
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Criterion on a region to be a triangle in the three dimensional space.
Given the real numbers $a_1,a_2,b_1,b_2,c_1,c_2\in [0,1]$. Suppose we know that the set $$\begin{cases}x+y+z=1 \\a_1\le x \le a_2\\ b_1\le y \le b_2\\ c_1\le z \le c_2\\ \end{cases}$$ is a surface (...
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Sampling points from a space defined by bounded inequality
I have the problem: I'd like to generate a uniform sampling of points from a space defined by the bounded constraints and linear inequality described below:
$$lb_1 \leq x_1 \leq ub_1$$
$$lb_2 \leq x_2 ...
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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 ...
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1
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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+...
- 21
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0
answers
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How is the term for the change of the cost vector in the simplex method derived?
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(...
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1
answer
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Is the Simplex Method an exact, approximate or a heuristic method?
I'd like to understand whether the simplex method is an exact method (like Branch&Bound, Branch&Cut...), an approximate method or a heuristic method.
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