Questions tagged [iterated-integral]
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Are these two concepts of a differential form on the loop space equivalent?
Notation:
Let $X$ denote a smooth manifold (without boundary) and define $LX = C^{\infty}(S^1, X)$ to be the loop space on $X$.
In the context of loop space homology and the supersymmetric path ...
2
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Infinite sum of iterated integrals of matrix products
Originally asked over at Stackexchange (https://math.stackexchange.com/questions/4169812/infinite-sum-of-iterated-integrals-of-matrix-products), but this forum was deemed more appropriate.
The problem:...
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Is it possible to express $\int_{0+\epsilon}^{1-\epsilon}\left(\sqrt{1-x^2}^{\sqrt{1-x^2}^{\cdots}}\right) dx$ in elementary functions?
let $\epsilon >0$, I tried to evaluate $\int_{0+\epsilon}^{1-\epsilon}\left(\sqrt{1-x^2}^{\sqrt{1-x^2}^{\cdots}}\right) dx$ , using the fact $x= \cos t$ yield to have integrand using $\sin $ ...
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Time ordered integral involving beta function:
Any help on unpacking integrals of the following type, would be
helpful: $$ \int_0^1 \int_0^r r^a (1-r)^b t^n (1-t)^m dr dt $$ where $a, b, n, m \in \mathbb{N}$ and $0 \le t \le 1$.
Edit/...
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Bar construction and space of homotopy invariant functionals
I am introducing myself to the topic of iterated integrals using these notes
http://www.ihes.fr/%7Ebrown/ColombiaNotes7.pdf
On pag 22 is defined the following operator
$$ D : (X^1)^n \to (\mathcal{A}...
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Iterated integral with a irregular path
For the proof of Fundamental Lemma 3.1 on the page 400 of K.T. Chen's 1957 paper Integration of paths--A faithful representation of paths by noncommutative
formal power series, it requires the path $\...
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How to analytically evaluate this n-dimensional iterated integral?
I would very much appreciate any suggestions and/or pointers to references relevant for the analytic evaluation of the following n-dimensional iterated integral
$$\int_{-\infty}^{+\infty}dx_1\int_{-\...
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Understand rough path iterated integral and how to compute it numerically?
The "signature" of rough path theory is defined by iterated integral as
$s(k)=\int_{0 \le u_1 \le \cdots \le u_k \le t} \mathrm{d}X_{u_1} \otimes \cdots \otimes \mathrm{d}X_{u_k}$
in witch $X(t)$ is ...
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Chen's iterated integrals and free loop space
I recently found out about Chen's iterated integrals for paths in a differentiable manifold, and I was wondering if an analogous construction exists for free loops, i.e. a set of variables one ...
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Restriction of iterated integrals
For a smooth manifold $X$, let $B_s(X)$ denote the space of iterated integrals of length at most $s$. Here we consider iterated integrals as functions on the path space $PX$. Fix a base point $x_0$ ...
user1688
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Understanding iterated integrals
I have encountered iterated integrals on papers dealing with multizeta values, polylogarithms etc.. Since then I am trying to figure out the motivations and purpose of the theory.
It seems the ...
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