Questions tagged [simplex]

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

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Simplicial complex [closed]

I started to learn about "simplicial complex" and read about applications but it was very difficult for me to understand these applications, my question is as below what is the importance ...
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The norm in the n-simplex

İ have this question and my attempt to solve it as the following : Consider an n-simplex $[p_0, p_1,...,p_n]$. If $a,b \in [p_0, > p_1,...,p_n]$, then show that $||a − b|| \leq \sup_i ||a − p_i||$....
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Simplex : minimum of objective function is zero when ..

I run simplex (hopefully right) with break ties rules and everything for a minimisation problem. If I end up with the same base made of variables that are not in the cost function and there's no ...
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Singular complex as a $\Delta$-complex model

I'm trying to understand singular complexes via Hatcher page 108. Here is my understanding so far and where I'm not getting it. Given a space $X$, a singular n-simplex is just a $\sigma \in C^1(\Delta^...
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Understanding the definition of $\Delta$-complexes.

I'm reading Hatcher's Algebraic Topology and on page 103 he gives a definition of a $\Delta$-complex which requires, among other things: A $\Delta$-complex structure on a space $X$ is a collection of ...
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How to calculate the centroid of a Polytope?

Given a polytope is divided into simplexes, is it correct to calculate the centroid of the polytope as the average sum of its simplex centroid coordinates
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Another puzzling identity that arose from integrating over eigenvalues of Wishart matrices.

Let $n \ge 2$ and let $T > n $ be integers. We consider a sample covariance matrix, i.e. $c := {\bar C} \cdot Y \cdot Y^T \cdot {\bar C}^T \quad (1)$ where $Y $ is a $n \times T$ random matrix with ...
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Division of $n$-dimensional simplex [closed]

How many congruent $n$-dimensional simplexes can an $n$-dimensional simplex be divided into? Obviously, a triangle can be divided into $k^2\left(k\in N^*\right)$ congruent triangles (by dividing the ...
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Derivative of the flow for ODEs on for simplexes?

Does the "well-known result in standard ODE's theory" cited below hold for a derivative on a simplex? Also, is there a name for this result (or somewhere I can learn more about it)? Source: ...
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Maximize $\sum_k \frac{p_k}{\sum_{j \geq k} p_j}$ over the probability simplex?

Suppose that $p_1, \dots, p_n$ are nonnegative real numbers such that $p_1 + \cdots + p_n = 1$; denote the corresponding set of vectors by $\Delta_n$. I am interested in the following function, $f \...
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Understanding the statement "the boundary of the boundary of a simplex is zero" [duplicate]

Suppose that we have a triangle simplex $S=(p_{1}p_{2}p_{3})$, then the boundary of $S$ $\partial S$ is defined by $\partial S=p_{1}p_{2}+p_{2}p_{3}+p_{3}p_{1}$, then $\partial \partial S=p_{1}-p_{2}+...
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How to calculate minimax objective with simplex method?

Min Z enter image description here xi>=0 for i=1,2...,12 Can anyone tell me how to calculate this objective function in the simplex method? If there is no x in the objective function, then how do I ...
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Dimension of the union of a 4-dimensional and 3-dimensional simplex

Say I'm given two simplices: one is a 4-simplex and the other is a 3-simplex, and I've proven their intersection is non-empty. How do I now explain what the dimension of their union is? I understand ...
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Why do I get different inequalities from the same linear programming system?

This is from section 7.1 of Dasgupta's Algorithm book: I attempt to arrive at the optimal solution by using the existing inequalities. $$x_2 \leq 300$$ $$x_1 + x_2 + x_3 \leq 400$$ $$4x_2 + 12x_3 \leq ...
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Maximun of a Linear Functional on a Tetrahedron

I am looking for some help framing the solution to this question formally. Let $v_0, v_1, v_2, v_3$ be vertices of a 3-simplex T in $\mathbb{R}^3$ Let $f:\mathbb{R}^3\rightarrow\mathbb{R}$ be a ...
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Two questions about Cayley-Menger determinant

Let $\mathbf p_1, \ldots,\mathbf p_n$ be $n$ points in $\Re^d$ and let $k$ be the maximum number of affinely independent points among them. Consider the following matrix $D$ whose determinant is known ...
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Help in finding a linear functional from a 3-simplex

I am trying to come up with a linear functional $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ from a 3-simplex with vertices at $(0,0,0), (1,0,0), (0,1,0)$ and $(0,0,1)$ such that the larger value of $f$ ...
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Is this function upper-semi continuous?

Let $(A,\mathcal A)$ be a measure space, let $\mathcal P(A)=\{ q : (A, \mathcal A, q)\text{ is a probability space}\}$, i.e. the simplex associated to $(A,\mathcal A)$. We equip $\mathcal P(A)$ with ...
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Is this (not necessarily compact) convex set the closed convex hull of it's extreme points?

Let $(\mathcal X,\Sigma_X)$ be a measure space, let $\mathcal S_X=\{ q : (\mathcal X, \Sigma_X, q)\text{ is a probability space}\}$, i.e. the simplex associated to $(\mathcal X,\Sigma_X)$. We equip $\...
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Simplex method Tableau Row operations why can't I multiply reduced cost by -1?

I am a little confused... Lets say I have this tableau where I have to make sure my reduced cost (Row 3) is non-negative, why can't I simply multiply Row 3 by -1/R3? Afterwards, since all my reduced ...
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1 answer
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On the Aitchison Geometry

This question is for people who know the Aitchison Geometry - I'm working on a (more mathematical and not statistical) paper on the Aitchison Geometry and I try to understand how ellipses (or any ...
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Uniform distribution over order simplex

Consider the set of all vectors $x \in [0,1]^K$ that are monotone, i.e. $0\leq x_1\leq x_2\leq ... \leq x_K\leq 1$. This set is known as orthoscheme or order simplex. Is there a formula for the ...
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Can this set be not separable ? Can it be not compact?

Let $(\mathcal X,\Sigma_X)$ be a measure space, let $\mathcal S_X=\{ q : (\mathcal X, \Sigma_X, q)\text{ is a probability space}\}$, i.e. the simplex associated to $(\mathcal X,\Sigma_X)$. We equip $\...
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Optimal Degenerate Basic Feasible Solution with Positive Reduced Cost

For a standard form linear program that minimizes an objective function as follows: $$ \begin{aligned} \text{minimize } &c^T x \\ \text{subject to } & Ax =0, x \ge 0 \end{aligned} $$ a ...
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Degenerate basic solutions that correspond to only one basis

On page 60 of the book Introduction to Linear Programming[1], the author mentioned: In the case of a degenerate basic solution [in the standard form polyhedron], more than $n-m$ of the constraints $...
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What is the "perturbation operator" referred to here?

Here's an excerpt from this paper: I can't get access to the book the citation references. Does anyone know what this operator might be? $\mathbf{p}$ is the joint pmf of two Bernoulli rvs: \begin{...
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4 votes
1 answer
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On simplicial complexes and their geometric realization

Simplicial complexes can be defined in two different way, i.e. either abstractly as purely combinatorial objects, or embedded in Euclidean space. Let me briefly mention which definitions I use exactly:...
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Heronian triangles and tetrahedra exist—how about Heronian 4-simplices?

Inspired by the paper Heronian Tetrahedra Are Lattice Tetrahedra by Susan H. Marshall and Alexander R. Perlis, I started thinking about higher dimensional Heronian simplices. Heronian simplices are ...
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General program to generate computation to calculate circumsphere for a given dimension $d$?

I am aware of: circumcenter of the $n$-simplex Circumsphere of a tetrahedron But am wondering if there is an optimal (computationally) approach to generating the programs to compute the cirumcentres ...
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1 answer
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Derivation of exponential (in a general algebra)

There is a curious formula relating the derivation of an exponential in some, possibly noncommutative (associative) algebra, namely $$ D(e^K) = \int_0^1 e^{(1-s)K}D(K)e^{sK} ds $$ where $D$ is a ...
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Baby rudin example 10.32

This is the definition which we need for this example : . There is the example: I don't understand why is $\partial\Sigma$ equal to $\Sigma(\partial D)$ and then why is $\Sigma$($\partial D$) equal ...
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Every compact subset can intersect only finitely many simplices

Well my question is almost in the title: how can I prove that every compact subset C of |K| can intersect only finitely many simplices, where K is a infinite simplicial complex? In particular given C ...
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1 vote
1 answer
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Is $S_{n}(X) \oplus S_{n}(Y) \cong S_{n}(X \amalg Y)$?

Question: Is $S_{n}(X) \oplus S_{n}(Y) \cong S_{n}(X \amalg Y)$? Proof attempt: Here's my argument: Suppose that $X$ and $Y$ are disjoint topological spaces. Note that if $\sigma : \Delta^{n} \...
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Geometric Realization of Standard Combinatorial $n$-Simplex is Standard Topological $n$-Simplex

Let $\Delta$ be the simplex category with objects $[n]=\{0,...,n\}$, $n\geq 0$, and morphisms ordering preserving functions $[n]\rightarrow [m]$. Then let the standard categorical $n$-simplex be $Hom_{...
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Infeasible solution to a canonical LPP

I've been given the question Solve the canonical linear programming problems by using the simplex algorithm. ...
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3 votes
1 answer
135 views

Why is a Simplex "Triangular"?

I was reading the Wikipedia definition of a Geometric Simplex (https://en.wikipedia.org/wiki/Simplex): We are told that a Simplex in more than 2 dimensions always has a triangular shape: However, ...
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1 vote
1 answer
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Volume of a $n$-simplex in $(n+k)$-dimensions

From here the formula for calculating the volume of a $n$-simplex in an $n$-dimensional space is given. Please how does one find the volume of the same simplex existing in a $(n+k)$-dimensional space ...
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Embed k-skelton of simplex into Euclidean space

Let $\Delta^n=\{(t_0,t_1,\dots,t_n)\in \mathbb{R}^{n+1} \mid \sum t_i=1,t_i\geq 0\}$ be the $n$-dimensional simplex. Let $X$ be the $k$-skelton of this simplex. What is the minimal m such that there ...
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Weighted volume of simplices making up a cube not equal to volume of cube?

I am trying to implement an algorithm to calculate the volume of a polyhedron by dividing it into simplices with their apex as some arbitrary vertex of the polyhedron and then summing up the volume of ...
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2 votes
1 answer
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Geometric centre of a simplex

Let be $S_n(1)=$$\{(x_1, x_2, ... ,x_n)\in \mathbb R^n : x_j\geq 0, x_1 + x_2 + ... + x_n \leq 1\}$ the standard simplex. My task is to calculate the geometric centre C with Lebesgue integration. I ...
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What is meant by the geodesic distance between two points on a rips complex?

I am trying to reimplement this paper and in the surface segmentation section (Section 8 Paragraph 4) of the paper, the author speak of "the geodesic distance in the Rips graph from p to the ...
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2 votes
0 answers
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Closed form for simplex homeomorphism

I have a question regarding homeomorphisms from compact convex spaces to the standard simplex. I know that every compact convex subset of $\mathbb{R}^{n}$ with a non-empty interior is homeomorphic to ...
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2 votes
1 answer
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Prove a feasible point is optimal for an LP using complementary slackness

Prove that $(2,0,0)$ is the optimal solution to this problem. P) Minimize $2x_1+5x_2+7x_3$ subject to constraints: $7x_1+6x_2+3x_3-s_1=14$ $2x_1+4x_2+5x_3+s_2=4$ Where: $x_1,x_2,x_3 \ge 0$ This ...
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0 answers
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Interchanged Domain of multiple integral under n-simplex

Suppose that we have a multiple integral under $n$-dimensional simplex as follows: \begin{align*} \underbrace{\int_0^1 dx_n \int_0^{x_n} dx_{n-1} \cdots \int_0^{x_2} dx_1}_{n\text{ times}}\, f(x_i) ...
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1 vote
0 answers
61 views

How to calculate the volume of the image of a simplex by a general linear transformation??

It is well known that if a linear map $H$ is bijective then we have $$Vol(S)=\text{det}H \, Vol(\mathcal{X})$$ Now I want to know the case when $H$ is a surjective map. How to calculate the volume of ...
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1 answer
68 views

Prove that constant 1-simplex is a 1-boundary

I have been given the following question: Given a topological space $X$, for any $x_0 \in X$ we have defined $\varphi_{x_0}:\sigma_1 \rightarrow X: (t_0,t_1) \rightarrow x_0$. Prove that $\varphi_{x_0}...
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2 votes
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show that the reverse of the optimization test is not always true using an example

We have the following theorem: For a BFS $x^{0}$, if $z_{j}-c_{j} \leq 0, \forall j \in J_{N}$ (N is the set of all non-basic variables), Then $x^{0}$ is an optimized solution for this problem. Now ...
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1 vote
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Intuition for orientation of a simplex (in 3 dimensions)

In trying to begin to learn basic homological algebra, i am confronted with orientation of simplices. The definition seems unmotivated and unintuitive: for $n$-simplices with $n \in \{-1,0,1,2\}$, it ...
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Covering $n$-simplex with $k$-subsets to produce a lower $m$-simplex, $m<n$?

Let vertices of an $n$-simplex be labeled $\{x_1,x_2,...,x_n\}$ and let the $k$-subsets or $k$-intersections ($k \leq n$)be identified as $x_{i_1} \cap x_{i_2} \cap ... x_{i_k}=x_{i_1}x_{i_2}...x_{i_k}...
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2 votes
3 answers
452 views

Do we distinguish two singular simplices if they have different vertex orders?

We define a $\textbf{singular $n$-simplex}$ in $X$ to be a continuous map $\sigma:\Delta^n\to X$ where $\Delta^n$ is the standard $n$-simplex. Now, as an example, Let $X$ be a singleton $\{p\}$. Then ...
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