Zeroing the Output of a Nonlinear System Without Relative Degree

The goal of this paper is to establish some facts concerning the problem of zeroing the output of an input-output system that does not have relative degree. The approach taken is to work with systems that have a Chen-Fliess series representation. The main result is that a class of generating series called primely nullable series provides the building blocks for solving this problem using the shuffle algebra. It is shown that the shuffle algebra on the set of generating polynomials is a unique factorization domain so that any polynomial can be uniquely factored modulo a permutation into its irreducible elements for the purpose of identifying nullable factors. This is achieved using the fact that this shuffle algebra is isomorphic to the symmetric algebra over the vector space spanned by Lyndon words. A specific algorithm for factoring generating polynomials into its irreducible factors is presented based on the Chen-Fox-Lyndon factorization of words

Algebraic Birkhoff decomposition and its applications

Central in the Hopf algebra approach to the renormalization of perturbative quantum field theory of Connes and Kreimer is their Algebraic Birkhoff Decomposition. In this tutorial article, we introduce their decomposition and prove it by the Atkinson Factorization in Rota-Baxter algebra. We then give some applications of this decomposition in the study of divergent integrals and multiple zeta values.Comment: 39 pages. To appear in "Automorphic Forms and Langlands Program

Dual bases for non commutative symmetric and quasi-symmetric functions via monoidal factorization

In this work, an effective construction, via Sch\"utzenberger's monoidal factorization, of dual bases for the non commutative symmetric and quasi-symmetric functions is proposed

Renormalization: a quasi-shuffle approach

In recent years, the usual BPHZ algorithm for renormalization in perturbative quantum field theory has been interpreted, after dimensional regularization, as a Birkhoff decomposition of characters on the Hopf algebra of Feynman graphs, with values in a Rota-Baxter algebra of amplitudes. We associate in this paper to any such algebra a universal semi-group (different in nature from the Connes-Marcolli "cosmical Galois group"). Its action on the physical amplitudes associated to Feynman graphs produces the expected operations: Bogoliubov's preparation map, extraction of divergences, renormalization. In this process a key role is played by commutative and noncommutative quasi-shuffle bialgebras whose universal properties are instrumental in encoding the renormalization process

On a conjecture by Pierre Cartier about a group of associators

In \cite{cartier2}, Pierre Cartier conjectured that for any non commutative formal power series $\Phi$ on $X=\{x_0,x_1\}$ with coefficients in a \Q-extension, $A$, subjected to some suitable conditions, there exists an unique algebra homomorphism $\varphi$ from the \Q-algebra generated by the convergent polyz\^etas to $A$ such that $\Phi$ is computed from $\Phi_{KZ}$ Drinfel'd associator by applying $\varphi$ to each coefficient. We prove $\varphi$ exists and it is a free Lie exponential over $X$. Moreover, we give a complete description of the kernel of polyz\^eta and draw some consequences about a structure of the algebra of convergent polyz\^etas and about the arithmetical nature of the Euler constant

Direct QR factorizations for tall-and-skinny matrices in MapReduce architectures

The QR factorization and the SVD are two fundamental matrix decompositions with applications throughout scientific computing and data analysis. For matrices with many more rows than columns, so-called "tall-and-skinny matrices," there is a numerically stable, efficient, communication-avoiding algorithm for computing the QR factorization. It has been used in traditional high performance computing and grid computing environments. For MapReduce environments, existing methods to compute the QR decomposition use a numerically unstable approach that relies on indirectly computing the Q factor. In the best case, these methods require only two passes over the data. In this paper, we describe how to compute a stable tall-and-skinny QR factorization on a MapReduce architecture in only slightly more than 2 passes over the data. We can compute the SVD with only a small change and no difference in performance. We present a performance comparison between our new direct TSQR method, a standard unstable implementation for MapReduce (Cholesky QR), and the classic stable algorithm implemented for MapReduce (Householder QR). We find that our new stable method has a large performance advantage over the Householder QR method. This holds both in a theoretical performance model as well as in an actual implementation