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This manuscript stands at the interface between combinatorial Hopf algebra theory and renormalization theory. Its plan is as follows: Section 1 is the introduction, and contains as well an elementary invitation to the subject. The rest of part I, comprising Sections 2-6, is devoted to the basics of Hopf algebra theory and examples, in ascending level of complexity. Part II turns around the all-important Faà di Bruno Hopf algebra. Section 7 contains a first, direct approach to it. Section 8 gives applications of the Faà di Bruno algebra to quantum field theory and Lagrange reversion. Section 9 rederives the related Connes-Moscovici algebras. In Part III we turn to the Connes-Kreimer Hopf algebras of Feynman graphs and, more generally, to incidence bialgebras. In Section 10 we describe the first. Then in Section 11 we give a simple derivation of (the properly combinatorial part of) Zimmermann's cancellation-free method, in its original diagrammatic form. In Section 12 general incidence algebras are introduced, and the Faà di Bruno bialgebras are described as incidence bialgebras. In Section 13, deeper lore on Rota's incidence algebras allows us to reinterpret Connes-Kreimer algebras in terms of distributive lattices. Next, the general algebraic-combinatorial proof of the cancellation-free formula for antipodes is ascertained; this is the heart of the paper. The structure results for commutative Hopf algebras are found in Sections 14 and 15. An outlook section very briefly reviews the coalgebraic aspects of quantization and the Rota-Baxter map in renormalization. Part I: Basic Combinatorial Hopf Algebra Theory 1 Why Hopf algebras?Quantum field theory (QFT) aims to describe the fundamental phenomena of physics at the shortest scales, that is, higher energies. In spite of many practical successes, QFT is mathematically a problematic construction. Many of its difficulties are related to the need for renormalization. This complicated process is at present required to make sense of quantities very naturally defined, that we are however unable to calculate without incurring infinities. The complications are of both analytical and combinatorial nature.Since the work by Joni and Rota [1] on incidence coalgebras, the framework of Hopf algebras (a dual concept to groups in the spirit of noncommutative geometry) has been recognized as a very sophisticated one at our disposal, for formalizing the art of combinatorics. Now, recent developments (from 1998 on) have placed Hopf algebras at the heart of a noncommutative geometry approach to physics. Rather unexpectedly, but quite naturally, akin Hopf algebras appeared in two previously unrelated contexts: perturbative renormalization in quantum field theories [2-4] and index formulae in noncommutative geometry [5].Even more recently, we have become aware of the neglected coalgebraic side of the quantization procedure [6,7]. Thus, even leaving aside the role of "quantum symmetry groups" in conformal field theory, Hopf algebra is invading QFT from both ends, both at the foundational a...

The receptor for advanced glycation end products (RAGE), firstly described in 1992, is a single-transmembrane and multiligand member of the immunoglobulin protein family. RAGE engagement produces activation of multiple intracellular signaling mechanisms involved in several inflammation-associated clinical entities, such as diabetes, cancer, renal and heart failures, as well as neurodegenerative diseases. Although RAGE expression has been extensively reported in many cancer types, it is now emerging as a relevant element that can continuously fuel an inflammatory milieu at the tumor microenvironment, thus changing our perception of its contribution to cancer biology. In this review, we will discuss the role of multiligand/RAGE axis, particularly at the multicellular cross talk established in the inflammatory tumor microenvironment. A better understanding of its contribution may provide new targets for tumor management and risk assessment.

The phase-space approach to quantization of systems whose symmetry group is compact and semisimple is developed from two basic principles: covariance and traciality. This generalizes results and methods already implemented for spin systems [J. C. Várilly and J. M. Gracia-Bondía, Ann. Phys. 190, 107 (1989)]. The twisted product of phase-space functions is shown to be the image of group convolution in the context of a novel Fourier theory on the coadjoint orbits.

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