shuffle algebra
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2022 ◽  
Vol 99 ◽  
pp. 103406
Author(s):  
Laura Colmenarejo ◽  
Joscha Diehl ◽  
Miruna-Ştefana Sorea

Author(s):  
Ido Efrat

Abstract For a prime number p and a free profinite group S on the basis X, let $S_{\left (n,p\right )}$ , $n=1,2,\dotsc ,$ be the p-Zassenhaus filtration of S. For $p>n$ , we give a word-combinatorial description of the cohomology group $H^2\left (S/S_{\left (n,p\right )},\mathbb {Z}/p\right )$ in terms of the shuffle algebra on X. We give a natural linear basis for this cohomology group, which is constructed by means of unitriangular representations arising from Lyndon words.


Author(s):  
Yu Zhao

Abstract In this paper, we define the $K$-theoretic Hall algebra for dimension $0$ coherent sheaves on a smooth projective surface, prove that the algebra is associative, and construct a homomorphism to a shuffle algebra introduced by Negut [ 10].


Author(s):  
Charles Curry ◽  
Kurusch Ebrahimi–Fard ◽  
Simon J. A. Malham ◽  
Anke Wiese

We construct an efficient integrator for stochastic differential systems driven by Lévy processes. An efficient integrator is a strong approximation that is more accurate than the corresponding stochastic Taylor approximation, to all orders and independent of the governing vector fields. This holds provided the driving processes possess moments of all orders and the vector fields are sufficiently smooth. Moreover, the efficient integrator in question is optimal within a broad class of perturbations for half-integer global root mean-square orders of convergence. We obtain these results using the quasi-shuffle algebra of multiple iterated integrals of independent Lévy processes.


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