Network calculus is a deterministic queuing theory that has gained increasing attention in recent time. Founded on min-plus algebra it resorts to intuitive convolution formulae for efficient concatenation of servers and derivation of related performance bounds. Yet, the pessimistic worst-case analysis of deterministic network calculus gave rise to probabilistic counterparts that aim at utilizin...

A linear system theory is developped for a class of continuous and discrete Systems subject to Synchronization and Saturations that we call S3. This class provides a generalization of a particular class of timed discrete event systems which can be modeled as event graphs. The development is based on a linear modeling of S3 in the min-plus algebra. This allows us to extend elementary notions and...

These notes are an extended version of a series of lectures given at Bogota from 2nd to 6th december 2002. They aim to present a self-contained introduction to the Hopf-algebraic techniques which appear in the work of A. Connes and D. Kreimer on renormalization in Quantum Field Theory [CK1], [CK2], [BF]... Our point of view consists in revisiting a substantial part of their work in the abstract...

In this paper, we generalize to the context of algebras some recent results on existence common hypercyclic vectors for families products backward shift operators. We also give, in a multi-dimensional setting, positive answer question raised by F. Bayart, D. Papathanasiou and author about algebra ℓ1(N) with convolution product family shifts (Bw(λ))λ>0 induced weights wn(λ)=1+λ/n.

We study the ring Γ of all functions N → K, endowed with the usual convolution product. Γ, which we call the ring of number-theoretic functions, is an inverse limit of the “truncations” Γn = { f ∈ Γ ∀m > n : f(m) = 0 } . Each Γn is a zero-dimensional, finitely generated K-algebra, which may be expressed as the quotient of a finitely generated polynomial ring with a stable (after reversing the o...

Suppose that G is a locally compact abelian group, and write M(G) for the algebra of bounded, regular, complex-valued measures under convolution. A measure μ ∈ M(G) is said to be idempotent if μ ∗ μ = μ, or alternatively if μ̂ takes only the values 0 and 1. The Cohen-Helson-Rudin idempotent theorem states that a measure μ is idempotent if and only if the set {γ ∈ Ĝ : μ̂(γ) = 1} belongs to the cos...

Introduction 145 1. Quantum affine algebra 150 2. Quiver variety 155 3. Stratification of M0 163 4. Fixed point subvariety 167 5. Hecke correspondence and induction of quiver varieties 169 6. Equivariant K-theory 174 7. Freeness 178 8. Convolution 185 9. A homomorphism Uq(Lg)→ KGw×C ∗ (Z(w))⊗Z[q,q−1] Q(q) 192 10. Relations (I) 194 11. Relations (II) 202 12. Integral structure 214 13. Standard m...

In this paper, we generalize Haagerup’s inequality [H] (on convolution norm in the free group) to a very general context of R-diagonal elements in a tracial von Neumann algebra; moreover, we show that in this “holomorphic” setting, the inequality is greatly improved from its originial form. We give an elementary combinatorial proof of a very special case of our main result, and then generalize ...

A nonzero 2-cocycle Γ ∈ Z(g,R) on the Lie algebra g of a compact Lie group G defines a twisted version of the Lie-Poisson structure on the dual Lie algebra g∗, leading to a Poisson algebra C∞(g∗(Γ)). Similarly, a multiplier c ∈ Z (G, U(1)) on G which is smooth near the identity defines a twist in the convolution product on G, encoded by the twisted group C∗algebra C∗(G, c). Further to some supe...

For a class of large-N multi-matrix models, we identify a group G that plays the same role as the group of loops on space-time does for Yang-Mills theory. G is the spectrum of a commutative shuffle-deconcatenation Hopf algebra that we associate to correlations. G is the exponential of the free Lie algebra. The generating series of correlations is a function on G and satisfies quadratic equation...