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Commutative shuffle products are known to be intimately related to universal formulas for products, exponentials and logarithms in group theory as well as in the theory of free Lie algebras, such as, for instance, the Baker-Campbell-Hausdorff formula or the analytic expression of a Lie group law in exponential coordinates in the neighbourhood of the identity. Non-commutative shuffle products happen to have similar properties with respect to pre-Lie algebras. However, the situation is more complex since in the non-commutative framework three exponential-type maps and corresponding logarithms are naturally defined. This results in several new formal group laws together with new operations - for example a new notion of adjoint action particularly well fitted to the new theory. These developments are largely motivated by various constructions in non-commutative probability theory. The second part of the article is devoted to exploring and deepening this perspective. We illustrate our approach by revisiting universal products from a group-theoretical viewpoint, including additive convolution in monotone, free and boolean probability, as well as the Bercovici-Pata bijection and subordination products., Comment: revised manuscript

Misspecifications (i.e. errors on the parameters) of state space models lead to incorrect inference of the hidden states. This paper studies weakly nonlin-ear state space models with additive Gaussian noises and proposes a method for detecting and correcting misspecifications. The latter induce a biased estimator of the hidden state but also happen to induce correlation on innovations and other residues. This property is used to find a well-defined objective function for which an optimisation routine is applied to recover the true parameters of the model. It is argued that this method can consistently estimate the bias on the parameter. We demonstrate the algorithm on various models of increasing complexity.

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.

We present AMIGOS-- A dataset for Multimodal research of affect, personality traits and mood on Individuals and GrOupS. Different to other databases, we elicited affect using both short and long videos in two social contexts, one with individual viewers and one with groups of viewers. The database allows the multimodal study of the affective responses, by means of neuro-physiological signals of individuals in relation to their personality and mood, and with respect to the social context and videos' duration. The data is collected in two experimental settings. In the first one, 40 participants watched 16 short emotional videos. In the second one, the participants watched 4 long videos, some of them alone and the rest in groups. The participants' signals, namely, Electroencephalogram (EEG), Electrocardiogram (ECG) and Galvanic Skin Response (GSR), were recorded using wearable sensors. Participants' frontal HD video and both RGB and depth full body videos were also recorded. Participants emotions have been annotated with both self-assessment of affective levels (valence, arousal, control, familiarity, liking and basic emotions) felt during the videos as well as external-assessment of levels of valence and arousal. We present a detailed correlation analysis of the different dimensions as well as baseline methods and results for single-trial classification of valence and arousal, personality traits, mood and social context. The database is made publicly available., Comment: 14 pages, Transaction on Affective Computing

The theory of cumulants is revisited in the "Rota way", that is, by following a combinatorial Hopf algebra approach. Monotone, free, and boolean cumulants are considered as infinitesimal characters over a particular combinatorial Hopf algebra. The latter is neither commutative nor cocommutative, and has an underlying unshuffle bialgebra structure which gives rise to a shuffle product on its graded dual. The moment-cumulant relations are encoded in terms of shuffle and half-shuffle exponentials. It is then shown how to express concisely monotone, free, and boolean cumulants in terms of each other using the pre-Lie Magnus expansion together with shuffle and half-shuffle logarithms., Comment: final version

Motivated by recent work on mixtures of classical and free probabilities, we introduce and study the notion of $\epsilon$-noncrossing partitions. It is shown that the set of such partitions forms a lattice, which interpolates as a poset between the poset of partitions and the one of noncrossing partitions. Moreover, $\epsilon$-cumulants are introduced and shown to characterize the notion of $\epsilon$-independence.

The theory of pictures between posets is known to encode much of the combinatorics of symmetric group representations and related topics such as Young diagrams and tableaux. Many reasons, com-binatorial (e.g. since semi-standard tableaux can be viewed as double quasi-posets) and topological (quasi-posets identify with finite topolo-gies) lead to extend the theory to quasi-posets. This is the object of the present article.

Many features of classical Lie theory generalize to the broader context of algebras over Hopf operads. However, this idea remains largely to be developed systematically. Quasi-shuffle algebras provide for example an interesting illustration of these phenomena, but have not been investigated from this point of view.The notion of quasi-shuffle algebras can be traced back to the beginings of the theory of Rota--Baxter algebras, but was developed systematically only recently, starting essentially with Hoffman's work, that was motivated by multizeta values (MZVs) and featured their bialgebra structure. Many partial results on the fine structure of quasi-shuffle bialgebras have been obtained since then but, besides the fact that each of these articles features a particular point of view, they fail to develop systematically a complete theory.This article builds on these various results and develops the analog theory, for quasi-shuffle algebras, of the theory of descent algebras and their relations to free Lie algebras for classical enveloping algebras.

The present article deals with various generating series and group schemes (not necessarily affine ones) associated with MZVs. Our developments are motivated by Ecalle's mould calculus approach to the latter. We propose in particular a Hopf algebra--type encoding of symmetril moulds and introduce a new resummation process for MZVs.

Traditional methods for motion estimation estimate the motion field F between a pair of images as the one that minimizes a predesigned cost function. In this paper, we propose a direct method and train a Convolutional Neural Network (CNN) that when, at test time, is given a pair of images as input it produces a dense motion field F at its output layer. In the absence of large datasets with ground truth motion that would allow classical supervised training, we propose to train the network in an unsupervised manner. The proposed cost function that is optimized during training, is based on the classical optical flow constraint. The latter is differentiable with respect to the motion field and, therefore, allows backpropagation of the error to previous layers of the network. Our method is tested on both synthetic and real image sequences and performs similarly to the state-of-the-art methods., Comment: Submitted to ICIP 2016

In this work we deal with the problem of high-level event detection in video. Specifically, we study the challenging problems of i) learning to detect video events from solely a textual description of the event, without using any positive video examples, and ii) additionally exploiting very few positive training samples together with a small number of ``related'' videos. For learning only from an event's textual description, we first identify a general learning framework and then study the impact of different design choices for various stages of this framework. For additionally learning from example videos, when true positive training samples are scarce, we employ an extension of the Support Vector Machine that allows us to exploit ``related'' event videos by automatically introducing different weights for subsets of the videos in the overall training set. Experimental evaluations performed on the large-scale TRECVID MED 2014 video dataset provide insight on the effectiveness of the proposed methods., Comment: Image and Vision Computing Journal, Elsevier, 2015, accepted for publication

Three equivalent methods allow to compute the antipode of the Hopf algebras of Feynman diagrams in perturbative quantum field theory (QFT): the Dyson-Salam formula, the Bogoliubov formula, and the Zimmermann forest formula. Whereas the first two hold generally for arbitrary connected graded Hopf algebras, the third one requires extra structure properties of the underlying Hopf algebra but has the nice property to reduce drastically the number of terms in the expression of the antipode (it is optimal in that sense).The present article is concerned with the forest formula: we show that it generalizes to arbitrary right-handed polynomial Hopf algebras. These Hopf algebras are dual to the enveloping algebras of preLie algebras -a structure common to many combinatorial Hopf algebras which is carried in particular by the Hopf algebras of Feynman diagrams.

The construction of integrity bases for invariant and covariant polynomials built from aset of three dimensional vectors under the SO(3) and O(3) symmetries is presented. Thispaper is a follow--up to our previous work that dealt with a set of two dimensional vectorsunder the action of the SO(2) and O(2) groups [G. Dhont and B. I. Zhilinski{\i}, J. Phys. A:Math. Theor., 46, 455202 (2013)]. The expressions of the Molien generating functions asone rational function are a useful guide to build integrity bases for the rings of invariantsand the free modules of covariants. The structure of the non--free modules of covariants ismore complex. In this case, we write the Molien generating function as a sum of rationalfunctions and show that its symbolic interpretation leads to the concept of generalizedintegrity basis. The integrity bases and generalized integrity bases for O(3) are deducedfrom the SO(3) ones. The results are useful in quantum chemistry to describe the potentialenergy or multipole moment hypersurfaces of molecules. In particular, the generalizedintegrity bases that are required for the description of the electric and magnetic quadrupolemoment hypersurfaces of tetratomic molecules are given for the first time.

The aim of this work is to outline in some detail the use of combinatorial algebra in planar quantum field theory. Particular emphasis is given to the relations between the different types of planar Green's functions. The key object is a Hopf algebra which is naturally defined on non-commuting sources, and the fact that its genuine unshuffle coproduct splits into left- and right unshuffle half-coproduts. The latter give rise to the notion of unshuffle bialgebra. This setting allows to describe the relation between planar full and connected Green's functions by solving a simple linear fixed point equation. A modification of this linear fixed point equation gives rise to the relation between planar connected and one-particle irreducible Green's functions. The graphical calculus that arises from this approach also leads to a new understanding of functional calculus in planar QFT, whose rules for differentiation with respect to non-commuting sources can be translated into the language of growth operations on planar rooted trees. We also include a brief outline of our approach in the framework of non-planar theories.

In this paper we propose a supervised initialization scheme for cascaded face alignment based on explicit head pose estimation. We first investigate the failure cases of most state of the art face alignment approaches and observe that these failures often share one common global property, i.e. the head pose variation is usually large. Inspired by this, we propose a deep convolutional network model for reliable and accurate head pose estimation. Instead of using a mean face shape, or randomly selected shapes for cascaded face alignment initialisation, we propose two schemes for generating initialisation: the first one relies on projecting a mean 3D face shape (represented by 3D facial landmarks) onto 2D image under the estimated head pose; the second one searches nearest neighbour shapes from the training set according to head pose distance. By doing so, the initialisation gets closer to the actual shape, which enhances the possibility of convergence and in turn improves the face alignment performance. We demonstrate the proposed method on the benchmark 300W dataset and show very competitive performance in both head pose estimation and face alignment., Comment: Accepted by BMVC2015

We consider stochastic differential systems driven by continuous semimartingales and governed by non-commuting vector fields. We prove that the logarithm of the flowmap is an exponential Lie series. This relies on a natural change of basis to vector fields for the associated quadratic covariation processes, analogous to Stratonovich corrections. The flowmap can then be expanded as a series in compositional powers of vector fields and the logaritm of the flowmap can thus be expanded in the Lie algebra of vector fields. Further, we give a direct self-contained proof of the corresponding Chen-Strichartz formula which provides an explicit formula for the Lie series coefficients. Such exponential Lie series are important in the development of strong Lie group integration schemes that ensure approximate solutions themselves lie in any homogeneous manifold on which the solution evolves., Comment: 25 pages

For stochastic systems driven by continuous semimartingales an explicit formula for the logarithm of the Ito flow map is given. A similar formula is also obtained for solutions of linear matrix-valued SDEs driven by arbitrary semimartingales. The computation relies on the lift to quasi-shuffle algebras of formulas involving products of Ito integrals of semimartingales. Whereas the Chen-Strichartz formula computing the logarithm of the Stratonovich flow map is classically expanded as a formal sum indexed by permutations, the analogous formula in Ito calculus is naturally indexed by surjections. This reflects the change of algebraic background involved in the transition between the two integration theories.

In this paper, we propose a maximum margin classifier that deals with uncertainty in data input. More specifically, we reformulate the SVM framework such that each training example can be modeled by a multi-dimensional Gaussian distribution described by its mean vector and its covariance matrix -- the latter modeling the uncertainty. We address the classification problem and define a cost function that is the expected value of the classical SVM cost when data samples are drawn from the multi-dimensional Gaussian distributions that form the set of the training examples. Our formulation approximates the classical SVM formulation when the training examples are isotropic Gaussians with variance tending to zero. We arrive at a convex optimization problem, which we solve efficiently in the primal form using a stochastic gradient descent approach. The resulting classifier, which we name SVM with Gaussian Sample Uncertainty (SVM-GSU), is tested on synthetic data and five publicly available and popular datasets; namely, the MNIST, WDBC, DEAP, TV News Channel Commercial Detection, and TRECVID MED datasets. Experimental results verify the effectiveness of the proposed method., Comment: IEEE Transactions on Pattern Analysis and Machine Intelligence. (c) 2017 IEEE. DOI: 10.1109/TPAMI.2017.2772235 Author's accepted version. The final publication is available at http://ieeexplore.ieee.org/document/8103808/

Free cumulants were introduced by Speicher as a proper analog of classical cumulants in Voiculescu's theory of free probability. The relation between free moments and free cumulants is usually described in terms of Moebius calculus over the lattice of non-crossing partitions. In this work we explore another approach to free cumulants and to their combinatorial study using a combinatorial Hopf algebra structure on the linear span of non-crossing partitions. The generating series of free moments is seen as a character on this Hopf algebra. It is characterized by solving a linear fixed point equation that relates it to the generating series of free cumulants. These phenomena are explained through a process similar to (though different from) the arborification process familiar in the theory of dynamical systems, and originating in Cayley's work.

Do object part localization methods produce bilaterally symmetric results on mirror images? Surprisingly not, even though state of the art methods augment the training set with mirrored images. In this paper we take a closer look into this issue. We first introduce the concept of mirrorability as the ability of a model to produce symmetric results in mirrored images and introduce a corresponding measure, namely the \textit{mirror error} that is defined as the difference between the detection result on an image and the mirror of the detection result on its mirror image. We evaluate the mirrorability of several state of the art algorithms in two of the most intensively studied problems, namely human pose estimation and face alignment. Our experiments lead to several interesting findings: 1) Surprisingly, most of state of the art methods struggle to preserve the mirror symmetry, despite the fact that they do have very similar overall performance on the original and mirror images; 2) the low mirrorability is not caused by training or testing sample bias - all algorithms are trained on both the original images and their mirrored versions; 3) the mirror error is strongly correlated to the localization/alignment error (with correlation coefficients around 0.7). Since the mirror error is calculated without knowledge of the ground truth, we show two interesting applications - in the first it is used to guide the selection of difficult samples and in the second to give feedback in a popular Cascaded Pose Regression method for face alignment., Comment: 8 pages, 9 figures

Recent experimental studies confirm the prevalence of the widely known performance anomaly problem in current Wi-Fi networks, and report on the severe network utility degradation caused by this phenomenon. Although a large body of work addressed this issue, we attribute the refusal of prior solutions to their poor implementation feasibility with off-the-shelf hardware and their imprecise modelling of the 802.11 protocol. Their applicability is further challenged today by very high throughput enhancements (802.11n/ac) whereby link speeds can vary by two orders of magnitude. Unlike earlier approaches, in this paper we introduce the first rigorous analytical model of 802.11 stations' throughput and airtime in multi-rate settings, without sacrificing accuracy for tractability. We use the proportional-fair allocation criterion to formulate network utility maximisation as a convex optimisation problem for which we give a closed-form solution. We present a fully functional light-weight implementation of our scheme on commodity access points and evaluate this extensively via experiments in a real deployment, over a broad range of network conditions. Results demonstrate that our proposal achieves up to 100\% utility gains, can double video streaming goodput and reduces TCP download times by 8x., Comment: 21 pages, 11 figures

Free cumulants were introduced as the proper analog of classical cumulants in the theory of free probability. There is a mix of similarities and differences, when one considers the two families of cumulants. Whereas the combinatorics of classical cumulants is well expressed in terms of set partitions, the one of free cumulants is described, and often introduced in terms of non-crossing set partitions. The formal series approach to classical and free cumulants also largely differ. It is the purpose of the present article to put forward a different approach to these phenomena. Namely, we show that cumulants, whether classical or free, can be understood in terms of the algebra and combinatorics underlying commutative as well as non-commutative (half-)shuffles and (half-)unshuffles. As a corollary, cumulants and free cumulants can be characterized through linear fixed point equations. We study the exponential solutions of these linear fixed point equations, which display well the commutative, respectively non-commutative, character of classical, respectively free, cumulants., Comment: updated and revised version; accepted for publication in PRSA

This article analyses a new class of advanced particle Markov chain Monte Carlo algorithms recently introduced by Andrieu, Doucet, and Holenstein (2010). We present a natural interpretation of these methods in terms of well known unbiasedness properties of Feynman-Kac particle measures, and a new duality with many-body Feynman-Kac models. This perspective sheds a new light on the foundations and the mathematical analysis of this class of methods. A key consequence is the equivalence between the backward and ancestral particle Markov chain Monte Carlo methods, and Gibbs sampling of a many-body Feynman-Kac target distribution. Our approach also presents a new stochastic differential calculus based on geometric combinatorial techniques to derive explicit non-asymptotic Taylor type series of the semigroup of a class of particle Markov chain Monte Carlo models around their invariant measures with respect to the population size of the auxiliary particle sampler. These results provide sharp quan- titative estimates of the convergence properties of conditional particle Markov chain models with respect to the time horizon and the size of the systems. We illustrate the implication of these results with sharp estimates of the contraction coefficient and the Lyapunov exponent of conditional particle samplers, and explicit and non-asymptotic Lp-mean error decompositions of the law of the random states around the limiting invariant measure. The abstract framework developed in the article also allows the design of natural extensions to island (also called SMC2) type particle methodologies.

Finite topological spaces are in bijective correspondence with preorders on finite sets. We undertake their study using combinatorial tools that have been developed to investigate general discrete structures. A particular emphasis will be put on recent topological and combinatorial Hopf algebra techniques. We will show that the linear span of finite spaces carries generalized Hopf algebraic structures that are closely connected with familiar constructions and structures in topology (such as the one of cogroups in the category of associative algebras that has appeared e.g. in the study of loop spaces of suspensions). The most striking results that we obtain are certainly that the linear span of finite spaces carries the structure of the enveloping algebra of a $B\_\infty$--algebra, and that there are natural (Hopf algebraic) morphisms between finite spaces and quasi-symmetric functions. In the process, we introduce the notion of Schur-Weyl categories in order to describe rigidity theorems for cogroups in the category of associative algebras and related structures, as well as to account for the existence of natural operations (graded permutations) on them.

With the increasing availability of flexible wireless 802.11 devices, the potential exists for users to selfishly manipulate their channel access parameters and gain a performance advantage. Such practices can have a severe negative impact on compliant stations. To enable access points to counteract these selfish behaviours and preserve fairness in wireless networks, in this paper we propose a policing mechanism that drives misbehaving users into compliant operation without requiring any cooperation from clients. This approach is demonstrably effective against a broad class of misbehaviours, soundly-based, i.e. provably hard to circumvent and amenable to practical implementation on existing commodity hardware., Comment: 14 pages, 18 figures

We investigate deformations of the shuffl e Hopf algebra structure Sh(A) which can be de fined on the tensor algebra over a commutative algebra A. Such deformations, leading for example to the quasi-shuffl e algebra QSh(A), can be interpreted as natural transformations of the functor Sh, regarded as a functor from commutative nonunital algebras to coalgebras. We prove that the monoid of natural endomophisms of the functor Sh is isomorphic to the monoid of formal power series in one variable without constant term under composition, so that in particular, its natural automorphisms are in bijection with formal diffeomorphisms of the line. These transformations can be interpreted as elements of the Hopf algebra of word quasi-symmetric functions WQSym, and in turn de fine deformations of its structure. This leads to a new embedding of free quasi-symmetric functions into WQSym, whose relevance is illustrated by a simple and transparent proof of Goldberg's formula for the coe cients of the Hausdorff series.

Invariant theory provides more efficient tools, such as Molien generating functions and integrity bases, than basic group theory, that relies on projector techniques for the construction of symmetry--adapted polynomials in the symmetry coordinates of a molecular system, because it is based on a finer description of the mathematical structure of the latter. The present article extends its use to the construction of polynomial bases which span possibly, non--totally symmetric irreducible representations of a molecular symmetry group. Electric or magnetic observables can carry such irreducible representations, a common example is given by the electric dipole moment surface. The elementary generating functions and their corresponding integrity bases, where both the initial and the final representations are irreducible, are the building blocks of the algorithm presented in this article, which is faster than algorithms based on projection operators only. The generating functions for the full initial representation of interest are built recursively from the elementary generating functions. Integrity bases which can be used to generate in the most economical way symmetry--adapted polynomial bases are constructed alongside in the same fashion. The method is illustrated in detail on XY4 type of molecules. Explicit integrity bases for all five possible final irreducible representations of the tetrahedral group have been calculated and are given in the supplemental material associated with this paper.

The usual time-ordering operation and the corresponding time-ordered exponential play a fundamental role in physics and applied mathematics. In this work we study a new approach to the understanding of time-ordering relying on recent progress made in the context of enveloping algebras of pre-Lie algebras. Various general formulas for pre-Lie and Rota-Baxter algebras are obtained in the process. Among others, we recover the noncommutative analog of the classical Bohnenblust-Spitzer formula, and get explicit formulae for operator products of time-ordered exponentials.

Gian-Carlo Rota suggested in one of his last articles the problem of developing a theory around the notion of integration algebras, complementary to the already existing theory of differential algebras. This idea was mainly motivated by Rota's deep appreciation for Kuo-Tsai Chen's seminal work on iterated integrals. As a starting point for such a theory of integration algebras Rota proposed to consider a particular operator identity first introduced by the mathematician Glen Baxter. Later it was coined Rota-Baxter identity. In this article we briefly recall basic properties of Rota--Baxter algebras, and present a concise review of recent work with a particular emphasis of noncommutative aspects.

Motivated by a recent surge of interest for Dynkin operators in mathematical physics and by problems in the combinatorial theory of dynamical systems, we propose here a systematic study of logarithmic derivatives in various contexts. In particular, we introduce and investigate generalizations of the Dynkin operator for which we obtain Magnus-type formulas.

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.

We focus in this text on the adaptation to the study of shuffles of the main combinatorial tool in the theory of free Lie algebras, namely the existence of a universal algebra of endomorphisms for tensor and other cocommutative Hopf algebras: the family of Solomon's descent algebras of type A. We show that there exists similarly a natural endomorphism algebra for commutative shuffle algebras, which is a natural extension of the Malvenuto-Reutenauer Hopf algebra of permutations, or algebra of free quasi-symmetric functions. We study this new algebra for its own, establish freeness properties, study its generators, bases, and also feature its relations to the internal structure of shuffle algebras., Comment: 19 pages

In 802.11 WLANs, adapting the contention parameters to network conditions results in substantial performance improvements. Even though the ability to change these parameters has been available in standard devices for years, so far no adaptive mechanism using this functionality has been validated in a realistic deployment. In this paper we report our experiences with implementing and evaluating two adaptive algorithms based on control theory, one centralized and one distributed, in a large-scale testbed consisting of 18 commercial off-the-shelf devices. We conduct extensive measurements, considering different network conditions in terms of number of active nodes, link qualities and traffic generated. We show that both algorithms significantly outperform the standard configuration in terms of total throughput. We also identify the limitations inherent in distributed schemes, and demonstrate that the centralized approach substantially improves performance under a large variety of scenarios, which confirms its suitability for real deployments., Comment: 23 pages, 16 figures

We study the internal structure of enveloping algebras of preLie algebras. We show in particular that the canonical projections arising from the Poincar\'e-Birkhoff-Witt theorem can be computed explicitely. They happen to be closely related to the Magnus formula for matrix differential equations. Indeed, we show that the Magnus formula provides a way to compute the canonical projection on the preLie algebra. Conversely, our results provide new insights on classical problems in the theory of differential equations and on recent advances in their combinatorial understanding.

This article aims at advancing the recently introduced exponential method for renormalisation in perturbative quantum field theory. It is shown that this new procedure provides a meaningful recursive scheme in the context of the algebraic and group theoretical approach to renormalisation. In particular, we describe in detail a Hopf algebraic formulation of Bogoliubov's classical R-operation and counterterm recursion in the context of momentum subtraction schemes. This approach allows us to propose an algebraic classification of different subtraction schemes. Our results shed light on the peculiar algebraic role played by the degrees of Taylor jet expansions, especially the notion of minimal subtraction and oversubtractions., Comment: revised version

Recent developments in quantum chemistry, perturbative quantum field theory, statistical physics or stochastic differential equations require the introduction of new families of Feynman-type diagrams. These new families arise in various ways. In some generalizations of the classical diagrams, the notion of Feynman propagator is extended to generalized propagators connecting more than two vertices of the graphs. In some others (introduced in the present article), the diagrams, associated to noncommuting product of operators inherit from the noncommutativity of the products extra graphical properties. The purpose of the present article is to introduce a general way of dealing with such diagrams. We prove in particular a "universal" linked cluster theorem and introduce, in the process, a Feynman-type "diagrammatics" that allows to handle simultaneously nonlocal (Coulomb-type) interactions, the generalized diagrams arising from the study of interacting systems (such as the ones where the ground state is not the vacuum but e.g. a vacuum perturbed by a magnetic or electric field, by impurities...) or Wightman fields (that is, expectation values of products of interacting fields). Our diagrammatics seems to be the first attempt to encode in a unified algebraic framework such a wide variety of situations. In the process, we promote two ideas. First, Feynman-type diagrammatics belong mathematically to the theory of linear forms on combinatorial Hopf algebras. Second, linked cluster-type theorems rely ultimately on M\"obius inversion on the partition lattice. The two theories should therefore be introduced and presented accordingly

The Hopf algebra of word-quasi-symmetric functions ($\WQSym$), a noncommutative generalization of the Hopf algebra of quasi-symmetric functions, can be endowed with an internal product that has several compatibility properties with the other operations on $\WQSym$. This extends constructions familiar and central in the theory of free Lie algebras, noncommutative symmetric functions and their various applications fields, and allows to interpret $\WQSym$ as a convolution algebra of linear endomorphisms of quasi-shuffle algebras. We then use this interpretation to study the fine structure of quasi-shuffle algebras (MZVs, free Rota-Baxter algebras...). In particular, we compute their Adams operations and prove the existence of generalized Eulerian idempotents, that is, of a canonical left-inverse to the natural surjection map to their indecomposables, allowing for the combinatorial construction of free polynomial generators for these algebras., Comment: 18 pages

We present the first representation of the general term of the Rayleigh-Schr\"odinger series for quasidegenerate systems. Each term of the series is represented by a tree and there is a straightforward relation between the tree and the analytical expression of the corresponding term. The combinatorial and graphical techniques used in the proof of the series expansion allow us to derive various resummation formulas of the series. The relation with several combinatorial objects used for special cases (degenerate or non-degenerate systems) is established., Comment: 16 pages, 3 figures

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 Poincare

The convergence of U-statistics has been intensively studied for estimators based on families of i.i.d. random variables and variants of them. In most cases, the independence assumption is crucial [Lee90, de99]. When dealing with Feynman-Kac and other interacting particle systems of Monte Carlo type, one faces a new type of problem. Namely, in a sample of N particles obtained through the corresponding algorithms, the distributions of the particles are correlated -although any finite number of them is asymptotically independent with respect to the total number N of particles. In the present article, exploiting the fine asymptotics of particle systems, we prove convergence theorems for U-statistics in this framework.

The computational complexity of time-dependent perturbation theory is well-known to be largely combinatorial whatever the chosen expansion method and family of parameters (combinatorial sequences, Goldstone and other Feynman-type diagrams...). We show that a very efficient perturbative expansion, both for theoretical and numerical purposes, can be obtained through an original parametrization by trees and generalized iterated integrals. We emphasize above all the simplicity and naturality of the new approach that links perturbation theory with classical and recent results in enumerative and algebraic combinatorics. These tools are applied to the adiabatic approximation and the effective Hamiltonian. We prove perturbatively and non-perturbatively the convergence of Morita's generalization of the Gell-Mann and Low wavefunction. We show that summing all the terms associated to the same tree leads to an utter simplification where the sum is simpler than any of its terms. Finally, we recover the time-independent equation for the wave operator and we give an explicit non-recursive expression for the term corresponding to an arbitrary tree., Comment: 22 pages, 2 figures

The CMS experiment will collect data from the proton-proton collisions delivered by the Large Hadron Collider (LHC) at a centre-of-mass energy up to 14 TeV. The CMS trigger system is designed to cope with unprecedented luminosities and LHC bunch-crossing rates up to 40 MHz. The unique CMS trigger architecture only employs two trigger levels. The Level-1 trigger is implemented using custom electronics, while the High Level Trigger (HLT) is based on software algorithms running on a large cluster of commercial processors, the Event Filter Farm. We present the major functionalities of the CMS High Level Trigger system as of the starting of LHC beams operations in September 2008. The validation of the HLT system in the online environment with Monte Carlo simulated data and its commissioning during cosmic rays data taking campaigns are discussed in detail. We conclude with the description of the HLT operations with the first circulating LHC beams before the incident occurred the 19th September 2008.

We design exact polynomial expansions of a class of Feynman--Kac particle distributions. These expansions are finite and are parametrized by coalescent trees and other related combinatorial quantities. The accuracy of the expansions at any order is related naturally to the number of coalescences of the trees. Our results include an extension of the Wick product formula to interacting particle systems. They also provide refined nonasymptotic propagation of chaos-type properties, as well as sharp $\mathbb{L}_p$-mean error bounds, and laws of large numbers for $U$-statistics., Comment: Published in at http://dx.doi.org/10.1214/08-AAP565 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The algebraic structure of iterated integrals has been encoded by Chen. Formally, it identifies with the shuffle and Lie calculus of Lyndon, Ree and Sch\"utzenberger. It is mostly incorporated in the modern theory of free Lie algebras. Here, we tackle the problem of unraveling the algebraic structure of computations of effective Hamiltonians. This is an important subject in view of applications to chemistry, solid state physics, quantum field theory or engineering. We show, among others, that the correct framework for these computations is provided by the hyperoctahedral group algebras. We define several structures on these algebras and give various applications. For example, we show that the adiabatic evolution operator (in the time-dependent interaction representation of an effective Hamiltonian) can be written naturally as a Picard-type series and has a natural exponential expansion., Comment: Minor corrections. Some misleading notations and typos in the first version have been fixed

Renormalization techniques in perturbative quantum field theory were known, from their inception, to have a strong combinatorial content emphasized, among others, by Zimmermann's celebrated forest formula. The present article reports on recent advances on the subject, featuring the role played by the Dynkin operators (actually their extension to the Hopf algebraic setting) at two crucial levels of renormalization, namely the Bogolioubov recursion and the renormalization group (RG) equations. For that purpose, an iterated integrals toy model is introduced to emphasize how the operators appear naturally in the setting of renormalization group analysis. The toy model, in spite of its simplicity, captures many key features of recent approaches to RG equations in pQFT, including the construction of a universal Galois group for quantum field theories.

The valuation of counterparty risk for single name credit derivatives requires the computa- tion of joint distributions of default times of two default-prone entities. For a Merton-type model, we derive some formulas for these joint distribu- tions. As an application, closed formulas for counterparty risk on a CDS or for a first-to-default swap on two underlyings are obtained.

In quantum field theory (QFT), the vacuum expectation value of a normal product of creation and annihilation operators is always zero. This simple property paves the way to the classical treatment of perturbative QFT. This is no longer the case in the presence of initial correlations, that is if the vacuum is replaced by a general state. As a consequence, the combinatorics of correlated systems such as the ones occurring in many-body physics is more complex than that of quantum field theory and the general theory has made very slow progress. Similar observations hold in statistical physics or quantum probability for the perturbation series arising from the study of non Gaussian measures. In this work, an analysis of the Hopf algebraic aspects of quantum field theory is used to derive the structure of Green functions in terms of connected and one-particle irreducible Greeen functions for perturbative QFT in the presence of initial correlations., Comment: 22 pages, 2 figures, slightly edited version of the published paper

This paper introduces a new Lie-theoretic approach to the computation of counterterms in perturbative renormalization. Contrary to the usual approach, the devised version of the Bogoliubov recursion does not follow a linear induction on the number of loops. It is well-behaved with respect to the Connes-Kreimer approach: that is, the recursion takes place inside the group of Hopf algebra characters with values in regularized Feynman amplitudes. (Paradigmatically, we use dimensional regularization in the minimal subtraction scheme, although our procedure is generalizable to other schemes.) The new method is related to Zassenhaus' approach to the Baker-Campbell-Hausdorff formula for computing products of exponentials. The decomposition of counterterms is parametrized by a family of Lie idempotents known as the Zassenhaus idempotents. It is shown, inter alia, that the corresponding Feynman rules generate the same algebra as the graded components of the Connes-Kreimer beta-function. This further extends previous work of ours (together with Jose M. Gracia-Bondia) on the connection between Lie idempotents and renormalization procedures, where we constructed the Connes-Kreimer beta-function by means of the classical Dynkin idempotent., Comment: 11 pages, improved version, to appear in Lie Theory and Its Applicatons in Physics VII ed. V.K. Dobrev et al., Heron Press, Sofia, 2008

Dendriform structures arise naturally in algebraic combinatorics (where they allow, for example, the splitting of the shuffle product into two pieces) and through Rota-Baxter algebra structures (the latter appear, among others, in differential systems and in the renormalization process of pQFT). We prove new combinatorial identities in dendriform dialgebras that appear to be strongly related to classical phenomena, such as the combinatorics of Lyndon words, rewriting rules in Lie algebras, or the fine structure of the Malvenuto-Reutenauer algebra. One of these identities is an abstract noncommutative, dendriform, generalization of the Bohnenblust-Spitzer identity and of an identity involving iterated Chen integrals due to C.S. Lam., Comment: 16 pages, LaTeX. Concrete examples and applications added

The Bogoliubov recursion is a particular procedure appearing in the process of renormalization in perturbative quantum field theory. It provides convergent expressions for otherwise divergent integrals. We develop here a theory of functional identities for noncommutative Rota-Baxter algebras which is shown to encode, among others, this process in the context of Connes-Kreimer's Hopf algebra of renormalization. Our results generalize the seminal Cartier-Rota theory of classical Spitzer-type identities for commutative Rota-Baxter algebras. In the classical, commutative, case, these identities can be understood as deriving from the theory of symmetric functions. Here, we show that an analogous property holds for noncommutative Rota-Baxter algebras. That is, we show that functional identities in the noncommutative setting can be derived from the theory of noncommutative symmetric functions. Lie idempotents, and particularly the Dynkin idempotent play a crucial role in the process. Their action on the pro-unipotent groups such as those of perturbative renormalization is described in detail along the way., Comment: improved version, accepted for publication in the Journal of Noncommutative Geometry