Questions tagged [configuration-space]
Configuration spaces refer to topological spaces that consist of ordered or unordered subsets of a topological space, of a given (finite) cardinality.
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Natural metric(s) on the unordered configuration space of a metric space
Let $(X, d)$ be a metric space, where $X$ is a non-empty set and $d$ is a metric on $X$. Consider the configuration space $C_n(X)$ of $n$ distinct points in $X$. Is there a natural metric on $C_n(X) / ...
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Similarity between Configuration Spaces and Monad of little cube Operad.
I am trying to understand May's Recognition Principle, specifically its proof.
I will now recall some definition, which can be found in [Geometry of Iterated Loop Space] (or, a concise survey of Maru ...
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Why do unit quaternions span $S^3$ space, while rigid body configuration represented by them span $S^2 \times S^1$?
I'm reading "Modern Robotics" by Lynch and Park and came to realize that the rigid body configuration space is not $\mathbb{R}^3 \times S^3$, but rather $\mathbb{R}^3 \times S^2 \times S^1$ (...
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Configuration space of points on the REAL line
Can anyone kindly provide me with some intro or reference to the configuration space of $n$ points on the real line? The configuration space of $n$ points in $R^d$ is embedded in
$(R^d)^n \times (S^{d-...
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Compute the homology of this configuration space.
Let $U\subset V$ be finite labeling sets, and $K:\mathbb S^1\to\mathbb R^3$ be a knot. Consider the configuration space with points labeled $U$ lying on the knot, to make this space connected we fix ...
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The configuration space of 3 unordered points in $\mathbb{R}^2$ with distinct distances
Let $(X, d)$ be a metric space and let $n \in \mathbb{N}$. Define $A_n(X) = \{ (x_i)_{i=1}^n \in X^n \mid \forall i \neq j: x_i \neq x_j \}$ to be the space of $n$ ordered distinct points in $X$. ...
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Is this quotient space a manifold?
Consider the configuration space $\text{Conf}_n(\mathbb R^k)$, and consider the subgroup $G=\mathbb R^k\rtimes \mathbb R^{\times}\leq \mathbb R^k\rtimes \text{GL}(k,\mathbb R)=\text{Aff}(\mathbb R^k)$ ...
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Cohomology classes induced by embeddings
I'm currently studying this article by Farber about Configuration Spaces and Motion Planning Algorithms for a seminar. I'm having some trouble with some arguments because I'm also learning singular (...
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Which parameters can we choose in order to solve this triangle construction issue
This is follow-on of a question asked yesterday, with real work shown under the form of sketches but not understandable. Visibly, the asker isn't used to formulate mathematics with sentences (his/her ...
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What is the geometric realization of the configuration simplicial space?
Take a path-connected space $X$ and consider the family of non-empty, ordered configuration spaces $(\text{Conf}_{\bullet+1}(X))$. This collection of spaces forms a semi-simplicial space with face ...
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What is meant by Einstein's point configuration argument, and is it correct?
My question is: what does the quoted passage mean, and is it correct?
This is from The Meaning of Relativity by Albert Einstein. Many years ago I took it to a math professor at UT Austin. He went ...
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Why is the space of all frames in $\mathbb{R}^3$ the space $\mathbb{R}^3 \times \operatorname{SO}(3)$?
The following regards the textbook "Mathematical Methods of Classical Mechanics" by V.I. Arnold.
Definition. The configuration space of a system of $n$ points is the direct product of $n$ ...
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Energy increasing over time?
I'm reading some lecture notes on differential geometry with focus on Newtonian mechanics and applications to fluid mechanics. One theorem claims that the total energy of a system is decreasing. ...
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Configuration spaces to moduli spaces
In Segal's paper on Mapping Configuration spaces to moduli spaces, I'm not understanding what the map $\Phi$ is, explicitly.
Also in section 2, he goes on to say $M_{g,2} \simeq BHomeo^{+}(F_{g,2}; \...
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How to draw the following configuration space (manifold)?
I am studying configuration spaces for robots. For example, the configuration of a two-linked robot can be described as $\mathbf{q}=(\theta_1,\theta_2)$ and the configuration space is a torus ($\...
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