An evaluation of DNA-damage response and cell-cycle pathways for breast cancer classification

Accurate subtyping or classification of breast cancer is important for ensuring proper treatment of patients and also for understanding the molecular mechanisms driving this disease. While there have been several gene signatures proposed in the literature to classify breast tumours, these signatures show very low overlaps, different classification performance, and not much relevance to the underlying biology of these tumours. Here we evaluate DNA-damage response (DDR) and cell cycle pathways, which are critical pathways implicated in a considerable proportion of breast tumours, for their usefulness and ability in breast tumour subtyping. We think that subtyping breast tumours based on these two pathways could lead to vital insights into molecular mechanisms driving these tumours. Here, we performed a systematic evaluation of DDR and cell-cycle pathways for subtyping of breast tumours into the five known intrinsic subtypes. Homologous Recombination (HR) pathway showed the best performance in subtyping breast tumours, indicating that HR genes are strongly involved in all breast tumours. Comparisons of pathway based signatures and two standard gene signatures supported the use of known pathways for breast tumour subtyping. Further, the evaluation of these standard gene signatures showed that breast tumour subtyping, prognosis and survival estimation are all closely related. Finally, we constructed an all-inclusive super-signature by combining (union of) all genes and performing a stringent feature selection, and found it to be reasonably accurate and robust in classification as well as prognostic value. Adopting DDR and cell cycle pathways for breast tumour subtyping achieved robust and accurate breast tumour subtyping, and constructing a super-signature which contains feature selected mix of genes from these molecular pathways as well as clinical aspects is valuable in clinical practice.Comment: 28 pages, 7 figures, 6 table

Rota-Baxter algebras and new combinatorial identities

The word problem for an arbitrary associative Rota-Baxter algebra is solved. This leads to a noncommutative generalization of the classical Spitzer identities. Links to other combinatorial aspects, particularly of interest in physics, are indicated.Comment: 8 pages, improved versio

Time-ordering and a generalized Magnus expansion

Both the classical time-ordering and the Magnus expansion are well-known in the context of linear initial value problems. Motivated by the noncommutativity between time-ordering and time derivation, and related problems raised recently in statistical physics, we introduce a generalization of the Magnus expansion. Whereas the classical expansion computes the logarithm of the evolution operator of a linear differential equation, our generalization addresses the same problem, including however directly a non-trivial initial condition. As a by-product we recover a variant of the time ordering operation, known as T*-ordering. Eventually, placing our results in the general context of Rota-Baxter algebras permits us to present them in a more natural algebraic setting. It encompasses, for example, the case where one considers linear difference equations instead of linear differential equations

Generalized shuffles related to Nijenhuis and TD-algebras

Shuffle and quasi-shuffle products are well-known in the mathematics literature. They are intimately related to Loday's dendriform algebras, and were extensively used to give explicit constructions of free commutative Rota-Baxter algebras. In the literature there exist at least two other Rota-Baxter type algebras, namely, the Nijenhuis algebra and the so-called TD-algebra. The explicit construction of the free unital commutative Nijenhuis algebra uses a modified quasi-shuffle product, called the right-shift shuffle. We show that another modification of the quasi-shuffle product, the so-called left-shift shuffle, can be used to give an explicit construction of the free unital commutative TD-algebra. We explore some basic properties of TD-operators and show that the free unital commutative Nijenhuis algebra is a TD-algebra. We relate our construction to Loday's unital commutative dendriform trialgebras, including the involutive case. The concept of Rota-Baxter, Nijenhuis and TD-bialgebras is introduced at the end and we show that any commutative bialgebra provides such objects.Comment: 20 pages, typos corrected, accepted for publication in Communications in Algebr

Exponential renormalization

Moving beyond the classical additive and multiplicative approaches, we present an "exponential" method for perturbative renormalization. Using Dyson's identity for Green's functions as well as the link between the Faa di Bruno Hopf algebra and the Hopf algebras of Feynman graphs, its relation to the composition of formal power series is analyzed. Eventually, we argue that the new method has several attractive features and encompasses the BPHZ method. The latter can be seen as a special case of the new procedure for renormalization scheme maps with the Rota-Baxter property. To our best knowledge, although very natural from group-theoretical and physical points of view, several ideas introduced in the present paper seem to be new (besides the exponential method, let us mention the notions of counterfactors and of order n bare coupling constants).Comment: revised version; accepted for publication in Annales Henri Poincar

Dendriform-Tree Setting for Fully Non-commutative Fliess Operators

This paper provides a dendriform-tree setting for Fliess operators with matrix-valued inputs. This class of analytic nonlinear input-output systems is convenient, for example, in quantum control. In particular, a description of such Fliess operators is provided using planar binary trees. Sufficient conditions for convergence of the defining series are also given