Questions tagged [simplex]
For questions on the $n$-simplex, an $n$-dimensional polytope with $n+1$ points.
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Beginner Linear optimization problem - Simplex method
I'm asked to solve the following optimization problem. So far I've only learned the simplex algorithm and I'm not sure what I'm doing wrong but the Z value only gets worse and never gets better.
The ...
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Is there a concise method for shifting the integration bounds of polytopes in multidimensional integrals?
I'm looking to better understand a change of variables/u-substitution for multidimensional integrals over regions determined by polytopes. For example, I know that
$$ \int^{1}_{0} dz \int^{1-z}_{0} ...
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Generalized variance for probability vectors
I am trying to compute something like the "generalized variance" for a bunch of probability vectors (vectors with non-negative entries which add up to 1). Let the vectors have length $n$. ...
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Maximum number of vertices when dividing the $d$-dimensional simplex into $n$ convex polytopes.
The question is essentially in the title, but to be more precise: I would like to find the maximum number of vertices (i.e. distinct points of intersection) produced when splitting up the $d$-...
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The First Singular Homology Group $H_1(X)$ and the Fundamental Group
THEOREM. Let $X$ be a topological space and let $x_0\in X$ a point. The map $\varphi\colon\pi_1(X,x_0)\to H_1(X)$ defined by $[\sigma]_{\simeq}\mapsto[\sigma]_{\sim}$ is well defined and it is a ...
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Set dual with half-spaces [closed]
Let $X\subseteq \mathbb{R}^d,$ we define the set dual to
$X$, denoted by $X^*$, as follows:
$$X^*=\{y\in \mathbb{R}^d \mid \langle y,x\rangle\leq 1, \forall x\in X\}.$$
Geometrically, $X^*$ is the ...
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Maximum and concavity of function
I have a function
\begin{align}
G(x_1,x_2,x_3)=\int_0^{2\pi}\int_0^{2\pi}\int_0^{2\pi} \frac{d\theta_1d\theta_2d\theta_3}{(2\pi)^3} p(\vec{x},\vec{\theta}) \log\left( p(\vec{x}, \vec{\theta})\right)
\...
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Simplicial Generalization of Pythagoras
I recently heard about a claim that
For a triangle in 3-space, its area squared equals the sum of squares of areas of its projections onto three pairwise orthogonal planes.
I currently don't have ...
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Concave maximization over $d$-dimensional simplex.
Can either an analytic solution or the dual be characterized for the following concave maximization:
$v:= \underset{w \in \Delta_d}{\max} \sum^{d}_{i=1}\frac{1}{\sqrt{1+b_i/w_i}}$
where $\Delta_d$ ...
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Confusion about lemma 18.9 in Lee’s introduction to smooth manifolds (a map that is smooth on each face on a simplex is smooth)
In the proof of the lemma, during the induction step, a new function is chosen. However, this new function ignores the k-th coordinate in the input of two terms. My confusion is that even if a point ...
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What is the centre of mass of subset of a simplex with simple linear bounds
Let $\mathcal{R} \subset \mathcal{S}$ be the sub-region of the probability simplex with dimension $k-1$ defined by:
$$
\mathcal{R} = \{\mathbf{x} \in \mathbb{R}^k | \sum_{i = 1}^k x_i = 1, 0 \leq x_1 ...
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Left or right for 2 points on a line, clockwise or counterclockwise for 3 points in the plane. What is the analogue for 4 points in space?
For $a, b \in \mathbb{R},$ there is a notion of left or right. As a society, we agreed $a$ is left of $b$ if $a<b$ and otherwise $a$ is right of $b.$ It could've been the other way and nothing ...
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Name of $d$-simplex with "orthogonal" complementary subsimplices
Three-dimensional space allows for the following sequence of tetrahedra:
The regular tetrahedron with $d+1$ vertices
The pyramid whose base is a triangle with $d$ vertices centered at $0$ in $\{x_3=0\...
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Reading $B^{-1}$ from simplex table
In uni I'm following a course on optimalisation and I have come across a problem. I am given the following minimalisation problem:
and the corresponding final Simplex table:
I now need to determine ...
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Verify the formula for the the 3-simplex
One book I'm reading defines the regular 3-dimensional simplex as a subset of $\mathbb{R}^4$ as follows:
$$
\{\mathbf{x}=(x_1, x_2, x_3, x_4)\in\mathbb{R}^4: x_1+x_2+x_3+x_4=1, x_i\ge 0, i=\{ 1, 2, 3 ,...
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Triangulating Product of Simplicial Complexes
I am currently working on a problem for which I believe the following result is crucial.
The result of this problem was discussed in this post.
Product of simplicial complexes?
However it is not ...
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PDF of a fixed variable in the probability simplex
Let $\mathcal{P}_n$ be the probability simplex in $\mathbb{R}^n$, that is, the simplex whose vertices are the standard basis vectors. Then $\mathcal{P}_n$ is the set of all $n$-tuples of nonnegative ...
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The physical motivation of simplex
I read that homology, cohomology, and simplex emerged due to physical motivation on our country's blog. However, I cannot attach a link because my country is not an English-speaking country. For ...
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Every subset of vertex set defines a face of a polytope then it is a simplex
As the title suggests, I am trying to prove:
Prove that if any subset of the vertex set of a polytope defines a face, then the
polytope is a simplex.
For polytope $P$ with $n$ vertices $\{v_1,...,...
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Open mapping theorem for affine maps between Choquet simplices
I was curious if an analogue of the open mapping theorem existed for affine maps between compact convex spaces. I'm interested in a question like the following:
Suppose that $\mathcal{K}, \mathcal{L}$...
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What arrangements of face angles and side lengths uniquely determines an $n$-simplex up to isometry?
I am looking for a generalization of the result in plane geometry that triangles are determined up to isometry by 3 parameters in the following arrangements: side-side-side, side-angle-side, and angle-...
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Triangulation Around a Point in Any Dimension
Given a set of randomly distributed points in n-dimensional space, I am looking for a way to algorithmically find the optimal simplex (no sharp angles if there are multiple options) surrounding a ...
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How to show that a projecting an n-dimensional simplex decrease its volume?
Let $[v_{0},v_{1},\cdots,v_{n}]$ be an $n-$simplex in $n$ dimensional space. Let $H$ be the projection of $v_{0}$ onto the hyperplane spanned by $\{v_{1},\cdots ,v_{n}\}$. Prove that $|[v_{0},v_{1},\...
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The higher dimension of the law of sines. Which version is correct?
Recently I have interested in the law of sines in higher dimension, so I found the result from Wikipedia with the link below:
https://en.wikipedia.org/wiki/Law_of_sines#Higher_dimensions
The statement ...
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Convexity of (Probability) Simplex - Understanding the question
Take the $n-1$-simplex, $\Delta$, whose vertices are the $n$ standard unit vectors and let $\mu \in$ int $\Delta$. Define a probability distribution with finite support and barycenter $\mu, P_k$, to ...
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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$$ with $(...
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Halving planes of a tetrahedron
This nice article
Megan Martin, Cornelia A. Van Cott & Qiyu Zhang (2024) The Beauty of Halving it All, Math Horizons, 31:2, 14-17, DOI: 10.1080/10724117.2023.2249357.
shows that the envelope of ...
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Is there any 3-simplex in this figure?
I have a simple question. Is it possible to construct a 3-simplex (a, b, c, d) in the following figure?
My guess was that as we can generate an edge (a, c) by a linear combination of (a, d) and (a, b),...
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Linear Programming Simplex method issue with basis
I'm working on what I think is a fairly basic LP problem, the objective function and constraints are below. The task also specifies that the starting basis should be $x_1, x_2, x_3, x_5, x_6$ where ...
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Action of symmetry group on homology?
For a singular n-simplex $\alpha$ and a permutation $t\in S_{n}$, define $t\alpha$ to be the simplex with vertices permuted by $t$. Do we have $t\alpha$ homologous to $\text{sgn}(t)\alpha$ ? And does ...
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How to manipulate these constraints for 2-phase simplex method?
I am trying to solve an LP problem with 3 constraints: $$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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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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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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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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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}_{\...
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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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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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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 ...
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