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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35 views

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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1answer
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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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1answer
67 views

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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Linear programming simplex method pivot points illogical?

so I'm struggling to find how the pivot points were found in the following optimization problem: $$Minimize\ z = a + b + c$$ $$Subject\ to:$$ $$a - b - c ≤ 0$$ $$a + b + c ≥ 4$$ $$a + b - c = 2$$ $$...
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Comparing interior angles and dihedral angles in tetrahedra

Let $S\subset\Bbb R^3$ be a tetrahedron (not necessarily regular, just the convex hull of any four points in general position). Let $v,e,\sigma\subset S$ be a vertex, an edge and a face of $S$, so ...
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Transfer homomorphism in Algebraic topology

I am studying Hatcher's Algebraic topology. I am reading 3G about Transfer homomorphism. But most of the results are deemed obvious and I don't understnad why. "Let $\pi:\tilde X\to X$ be an $n$-...
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Barycentric subdivision of an affine $n$-simplex $\Sigma^n$ definition?

From Rotman's Algebraic Topology: The barycentric subdivision of an affince $n$-simplex $\Sigma^n$, denoted by $\text{Sd} \space \Sigma^n$, is a family of affine $n$-simplexes defined inductively ...
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How to find vertices of a $5$-dimensional simplex, where the vertices are formed by zeroes and ones?

I'm trying to find the vertices of a simplex in the $5$-dimensional space, where the vertices are formed by only zeros and ones, similarly to these coordinates that represent a simplex in the $7$-...
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Simplex of states in a $C^*$- algebra

Let $A$ be a unital $C^*$- algebra. What does it mean by a simplex in the space of states on a $C^*$- algebra. I know that the space $S(A)$ of all states is a compact convex set in weak * topology ...
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When can a regular simplex be inscribed in a regular hypercube?

A cube's vertices can be trivially divided into two sets, each forming the vertices of a regular tetrahedron. I was wondering if such a construction could be generalized to higher dimensions. I've ...
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Maximum of concave function over probability simplex

Let $F:\mathcal{C}\subset\mathbb{R}^n\to\mathbb{R}$ be a continuous function defined over $$ \mathcal{C}=\left\{x\in\mathbb{R}^n:x_i\geq 0\,\forall i,\,\sum_{i=1}^nx_i=1\right\}. $$ In addition, $F$ ...
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Polynomial integral inequality over a simplex

I have the following integral defined over the standard D-dimensional simplex: \begin{equation} \int_{{\triangle}_{D}} \prod_{i=1}^D \sqrt{(1+ x_i - x_i^2)}d\mathbf{x} \qquad \text{s.t.} \sum_{i=1}^...
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A reference to a change of variables

As an answer to Definite integral over a simplex, user StubbornAtom gave a very nice solution that uses a certain change of variables. I have seen it somewhere in a book but unfortunately cannot find ...
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Solving Linear Problems by SIMPLEX with unknown values

During this quarantine, a teacher of mine opted to start using Winston's Operations Research as support material for the now virtual classes This includes such problems as this one. I get what the ...
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Finding Optimal Dual Solution without solving de Dual Problem (knowing the Optimal Primal Solution)

I've been asked to solve this problem using the Dual Simplex Method. $$ \begin{array}{ll} \min: & x_1+6x_2+3x_3 \\ \text{s.t}: & -4x_1-4x_2-2x_3+x_4=-18 \\ & 2x_1+2x_2-4x_3+x_5 = -16 \\ ...
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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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1answer
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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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Max min problem for matrix product $Ax$ over simplex

Let $A = (a_{ij})_{1 \le i, j \le n}$ be a square matrix with all $a_{ij} \ge 0$. Moreover, assume $x \in \mathbb{R}^n$ is constrained to $$\Delta = \{x\in \mathbb{R}^n \mid x_1 + \cdots + x_n = 1, ...
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If $f$ and $g$ are homotopic, then $H_n(f) = H_n(g)$ proof question

From Rotman's Algebraic Topology: Theorem : If $f$ and $g$ are homotopic, then $H_n(f) = H_n(g)$. It suffices to construct homomorphisms $P_n^X: S_n(X) \rightarrow S_{n+1}(X \times I)$ with $\...
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Trying to do Fourier-Motzkin elimination in linear programming

I'm trying to implement Fourier Motzkin elimination in this problem, but I don't know how to proceed. Thanks for your help.
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23 views

Can you describe a $3$-simplex in $\mathbb{R}^3$?

I suppose that the standard $3$-simplex must be defined in $\mathbb{R}^4$ because you need four independientes unit vectors. But, since a $n$-simplex is generate by $n+1$ affinely independient points ...
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Definition and examples of orientations of $\Delta^n$ that are “the same” and “opposite”?

From Rotman's "Algebraic Topology": Two orientations of $\Delta^n$ are the same if, as permutations of $\{e_0, \dots, e_n\}$, they have the same parity (i.e. both are even or both are odd); ...
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Distance from point to closest intersection of 2 hyperplanes

I'm working in an unit simplex of size $S$, where hyperplanes are defined as $S$-dimensional vectors $H = \left \langle h_1, ..., h_S \right \rangle$ where each $h_i$ defines the height of $H$ at that ...
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28 views

Minimize $\sum_{i=1}^p (y_i-x_i)^2 $ such that $\sum_{i=1}^{p'} y_i^2 - R^{2} \le 0$

I'm solving the following optimization problem. Could you please verify if my proof is correct or contains logical mistake? Thank you so much! Let $x = (x_1,\ldots,x_p) \in \mathbb R^p$, $p' \le p$,...
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Minimize $\frac{1}{2}\sum_{i=1}^p (y_i-x_i)^2$ such that $\sum_{i=1}^p y_i - 1=0$ and $\forall i \in [\![ p ]\!]: -y_i \le 0$ by KKT method [duplicate]

I asked how to solve this optimization here. I found this approach by combining @Royi's idea in his answer with KKT's conditions. Personally, I feel my formulation is clearer and easier to understand. ...
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Proximal mapping and projection

Show that for $x = [x_1; x_2; \ldots; x_n]$, $$ \max_{1 \le i \le n} \{x_i\} = \max_{y \in C} \{x*y\}, $$ where $C$ is the unit simplex defined by $$C = \{y \in \mathbb{R}^n| y > 0, 1^Ty = 1\}.$$ ...
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Simplex Method Optimization

For part D of question 1: How do we know which column to pivot next? In my understanding I need to make all the numbers in top row positive, so I would go from the most negative number (-6), find the ...
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33 views

Inertia tensor/second moments/covariance matrix of an $n$-dimensional simplex

Given a measurable region $S\subset\mathbb R^n$, we define a symmetric linear transformation $M$ by a volume integral $$M(v)=\frac{\int_{x\in S}\,x(x\cdot v)\,dV}{\int_{x\in S}\,dV}$$ which has ...
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Is a degenerate simplex an incomplete simplex?

A $n$-simplex is degenerate if its Cayley-Menger determinant is $0$, i.e. its $n$-dimensional volume is $0$. Is such a simplex not a full simplex, a somehow "handicapped" object? Does a degenerate $...
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Expected tetrahedron volume from normal distribution

Two equivalent formulas for the volume of a random tetrahedron are given. Further on you can find an interesting conjecture for the expected volume that shall be proved. Tetrahedron volume Given ...
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Questions relating to explanation of polyhedra and simplexes

I am told that the definition of a polyhedron is $$P = \{ x \vert a_j^T x \le b_j, j = 1, \dots, m, c_j^T x = d_j, j = 1, \dots, p \}.$$ I am then told that the compact notation is $$P = \{ x \vert ...
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Consider a linear program with unrestricted variables is it possible to show there is no basic feasible solution

Consider the following Linear Program min z = $\sum_{j=1}^{n}c_jx_j$ $s.t.$ $\sum_{j=1}^{n}a_{ij}x_j=b_i$ i = 1, ..,m where the variables $x_j$ are unrestricted we then replace $x_j=x_{1j} ...
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Show that there is no feasible solution when unrestricted variables when converted to standard form

Consider the following Linear Program min z = $\sum_{j=1}^{n}c_jx_j$ $s.t.$ $\sum_{j=1}^{n}a_{ij}x_j=b_i$ i = 1, ..,m where the variables $x_j$ are unrestricted we then replace $x_j=x_{1j} ...
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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 ...
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n-dimensional simplex $\Delta^n$: every cycle without boundary is a boundary

This question is concerned with the evaluation of the homology groups $H_k(\Delta^n)=H_k(\Delta^n,\mathbb Z)$, where $\Delta ^n$ is a $n$-dimensional simplex. Since $\Delta^n$ is retractable, $H_k(\...
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Question about faces of a simplex and relation to complexes

I'm reading Hatcher's book on algebraic topology, p103: Let $[v_0, \dots, v_n]$ be an $n$-simplex. A face of $[v_0, \dots, v_n]$ is the $(n-1)$-simplex obtained by deleting one vertex $v_i$ from the ...
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1answer
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What are we allowed to do when proving deformation retraction of simplex?

I'm trying to show that the $3$-simplex with the edge identifications $[v_0, v_1] \sim [v_2, v_3]$ and $[v_0, v_2] \sim [v_1, v_3]$ deformation retracts onto the torus. I have a couple of potential ...
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Generalization of the sum of angles in simplices

It is well known that the three angles in a triangle sum to $\pi$. A similar statement is true for the tetrahedron. If we use the notation that $\theta_{ij}$ denotes the dihedral angle for an ...
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Why does $\partial^2 = 0$ in simplicial homology?

Let $\sigma:\Delta^n\rightarrow{X}$ be a map from the standard n-simplex to a topological space $X$. Now let us define the boundary operator as: $$\partial(\sigma)=\sum_{j=0}^{n}{(-1)^{j}\sigma}i_j,$$...
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confusion with the jth face of a n-simplex

Let $[x_0,...x_n]$ is an n-simplex of $\mathbb{R}^{n+1}$. Then I have a definition for the j$^{th}$ face as the following: $\{{\sum_{i=0}^{n}{t_ix_i}}\in{[x_0,...,x_n]}|t_j=0\}$ This makes sense in ...
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102 views

what is meant by a non-degenerate n - simplex

Let $[x_1,...,x_n]$ be an $n$ - simplex. It is non-degenerate if $\{x_1,...,x_n\}$ is not contained in an affine subspace of $\mathbb{R}^n$. In the example $[(0,1),(1,0)]$ the standard 2 simplex. Is ...
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41 views

Number of 1d- and 2d-cells in barycentric subdivision

I'm supposed to compute the number of 1- and 2-dimensional faces of a $k$-simplex after one step of barycentric subdivision. I already figured out that a triangle splits into $6=3!$ triangles as can ...
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1answer
53 views

Find the smallest $N$ admitting an embedding $\Delta^p\times \Delta^q\to \Delta^N$

Denote by $\Delta^n$ the standard $n$-simplex. What is the smallest integer $N$ so that there is a simplicial embedding $\Delta^p\times \Delta^q\to \Delta^N$. In particular, $\Delta^1\cong [0,1]$ and ...
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41 views

Does simplex assumption make Exponentiated Gradient Method applicable only to a specific optimization problem?

According to page 1 of this lecture here, and page 12 of here, I see that there is always an assumption of a simplex feasible region when using Exponentiated Gradient (EG) Method to solve convex ...

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