a geometric approach to variance analysis in system identification this paper addresses the problem of quantifying the model error variance error in estimates of dynamic systems it is shown that under very general conditions the asymptotic in data length covariance of an estimated system property represented by a smooth function of estimated system parameters can be interpreted in terms of an orthogonal projection of a certain function associated with the property of interest onto a subspace determined by the model structure and experimental conditions the presented geometric approach simplifies structural analysis of the model variance and this is illustrated by analyzing the influence of inputs and sensors on the model accuracy

l 1 stability estimates for n x n conservation laws let u t f u x 0 be a strictly hyperbolic n times n system of conservation laws each characteristic field being linearly degenerate or genuinely nonlinear in this paper we explicitly define a functional phi phi u v equivalent to the l 1 distance which is almost decreasing i e phi big u t v t big phi big u s v s big leq o ve cdot t s quad hbox for all t s geq 0 for every pair of e approximate solutions u v with small total variation generated by a wave front tracking algorithm the small parameter e here controls the errors in the wave speeds the maximum size of rarefaction fronts and the total strength of all non physical waves in u and in v from the above estimate it follows that front tracking approximations converge to a unique limit solution depending lipschitz continuously on the initial data in the vec l 1 norm this provides a new proof of the existence of the standard riemann semigroup generated by a n n system of conservation laws

vanishing viscosity solutions of nonlinear hyperbolic systems we consider the cauchy problem for a strictly hyperbolic n n system in one space dimension ut a u ux 0 assuming that the initial data have small total variation we show that the solutions of the viscous approximations ut a u ux euxx are defined globally in time and satisfy uniform bv estimates independent of e moreover they depend continuously on the initial data in the l 1 distance with a lipschitz constant independent of t e letting e 0 these viscous solutions converge to a unique limit depending lipschitz continuously on the initial data in the conservative case where a df is the jacobian

weak solutions of general systems of hyperbolic conservation laws in this paper we establish the existence theory for general system of hyperbolic conservation laws and obtain the uniform l1 boundness for the solutions the existence theory generalizes the classical glimm theory for systems for which each characteristic field is either genuinely nonlinear or linearly degenerate in the sense of lax we construct the solutions by the glimm scheme through the wave tracing method one of the key elements is a new way of measuring the potential interaction of the waves of the same characteristic family involving the angle between waves a new analysis is introduced to verify the consistency of the wave tracing procedure the entropy functional is used to study the l1 boundedness

the unique limit of the glimm scheme we introduce the definitions of a standard riemann semigroup and of a viscosity solution for a nonlinear hyperbolic system of conservation laws for a class including general 2 2 systems it is proved that the solutions obtained by a wavefront tracking algorithm or by the glimm scheme are precisely the semigroup trajectories in particular these solutions are unique and depend lipschitz continuously on the initial data in the l1 norm

existence theory for hyperbolic systems of conservation laws with general flux functions to begin with we assume that the flux function f u is piecewise genuinely nonlinear in the sense that it exhibits finitely many at most p say points of lack of genuine nonlinearity along each wave curve importantly our analysis applies to arbitrary large p in the sense that the constant c restricting the total variation is independent of p second by an approximation argument we prove that the existence theory above extends to general flux functionsf u that can be approached by a sequence of piecewise genuinely nonlinear flux functions fe u

on the pre lie algebra of specified feynman graphs we study the pre lie algebra of specified feynman graphs v t and we define a pre lie structure on its doubling space f t we prove that f t is pre lie module on v t and we find some relations between the two pre lie structures also we study the enveloping algebras of two pre lie algebras denoted respectively by d t and h t and we prove that d t is a module bialgebra on h t we study the pre lie algebra of specified feynman graphs v t and we define a pre lie structure on its doubling space f t we prove that f t is pre lie module on v t and we find some relations between the two pre lie structures also we study the enveloping algebras of two pre lie algebras denoted respectively by d t and h t and we prove that d t is a module bialgebra on h t

rota baxter paired modules and their constructions from hopf algebras abstract in this paper we introduce the concept of a rota baxter paired module to study rota baxter modules without necessarily a rota baxter operator we obtain two characterizations of rota baxter paired modules and give some basic properties of rota baxter paired modules beginning with the connection between the notion of integrals in the representations of hopf algebra and of the notion of an integral algebra as a motivation and special case of rota baxter algebra of weight zero we obtain a large number of rota baxter paired modules from hopf related algebras and modules including semisimple hopf algebras weak hopf algebras long bialgebras quasitriangular hopf algebras weak hopf modules dimodules and doi hopf modules some of them give new examples of rota baxter algebras

optimal sampling schedule for parameter estimation of linear models with unknown but bounded measurement errors the problem of optimal sampling design for parameter estimation when data are generated by linear models is addressed the measurements are assumed to be corrupted by an unknown but bounded additive noise the sampling design assumes that the number of samples is unconstrained and no replication is allowed two main results are shown 1 for particular classes of linear models the optimal number of measurements is equal to the number of parameters as in the statistical context 2 the uncertainty intervals of the parameter estimates are bounded from above by quantities that can be computer a priori knowing only the model and the error structure

noncommutative poisson bialgebras abstract in this paper we introduce the notion of a noncommutative poisson bialgebra and establish the equivalence between matched pairs manin triples and noncommutative poisson bialgebras using quasi representations and the corresponding cohomology theory of noncommutative poisson algebras we study coboundary noncommutative poisson bialgebras which leads to the introduction of the poisson yang baxter equation a skew symmetric solution of the poisson yang baxter equation naturally gives a coboundary noncommutative poisson bialgebra rota baxter operators more generally o operators on noncommutative poisson algebras and noncommutative pre poisson algebras are introduced by which we construct skew symmetric solutions of the poisson yang baxter equation in some special noncommutative poisson algebras obtained from these structures

decay estimate for a new measure of rarefaction waves in general systems of conservation laws abstract the sharp decay estimate of rarefaction waves in terms of a partial ordering among positive measures is not only interesting in itself but also crucial in the study of convergence rate of vanishing viscosity approximations cf bressan and yang 2004 10 such an estimate is well established for genuinely nonlinear system of conservation laws cf bressan and yang 2004 9 but similar result is not available for non genuinely nonlinear system in this paper we give a new measure about the rarefaction waves in addition a sharp decay estimate of the new measure is given for the cubic nonlinear system of conservation laws

algebraic renormalisation of regularity structures we give a systematic description of a canonical renormalisation procedure of stochastic pdes containing nonlinearities involving generalised functions this theory is based on the construction of a new class of regularity structures which comes with an explicit and elegant description of a subgroup of their group of automorphisms this subgroup is sufficiently large to be able to implement a version of the bphz renormalisation prescription in this context this is in stark contrast to previous works where one considered regularity structures with a much smaller group of automorphisms which lead to a much more indirect and convoluted construction of a renormalisation group acting on the corresponding space of admissible models by continuous transformations our construction is based on bialgebras of decorated coloured forests in cointeraction more precisely we have two hopf algebras in cointeraction coacting jointly on a vector space which represents the generalised functions of the theory two twisted antipodes play a fundamental role in the construction and provide a variant of the algebraic birkhoff factorisation that arises naturally in perturbative quantum field theory

preordered forests packed words and contraction algebras abstract we introduce the notions of preordered and heap preordered forests generalizing the construction of ordered and heap ordered forests we prove that the algebras of preordered and heap preordered forests are hopf for the cut coproduct and we construct a hopf morphism to the hopf algebra of packed words moreover we define another coproduct on the preordered forests given by the contraction of edges finally we give a combinatorial description of morphism defined on hopf algebras of forests with values in the hopf algebras of shuffles or quasi shuffles

integrable renormalization i the ladder case in recent years a hopf algebraic structure underlying the process of renormalization in quantum field theory was found it led to a birkhoff factorization for regularized hopf algebra characters i e for feynman rules in this work we would like to show that this birkhoff factorization finds its natural formulation in terms of a classical r matrix coming from a rota baxter structure underlying the target space of the regularized hopf algebra characters working in the rooted tree hopf algebra the simple case of the hopf subalgebra of ladder trees is treated in detail the extension to the general case i e the full hopf algebra of rooted trees or feynman graphs is briefly outlined

well posedness theory for hyperbolic conservation laws the paper presents a well posedness theory for the initial value problem for a general system of hyperbolic conservation laws we will start with the refinement of glimm s existence theory and discuss the principle of nonlinear through wave tracing our main goal is to introduce a nonlinear functional for two solutions with the property that it is equivalent to the l1 x distance between the two solutions and is time decreasing moreover the functional is constructed explicitly in terms of the wave patterns of the solutions through the nonlinear superposition it consists of a linear term measuring the l1 x distance a quadratic term measuring the coupling of waves and distance and a generalized entropy functional 1999 john wiley sons inc

a lie theoretic approach to renormalization motivated by recent work of connes and marcolli based on the connes kreimer approach to renormalization we augment the latter by a combinatorial lie algebraic point of view our results rely both on the properties of the dynkin idempotent one of the fundamental lie idempotents in the theory of free lie algebras and on properties of hopf algebras encapsulated in the notion of associated descent algebras besides leading very directly to proofs of the main combinatorial aspects of the renormalization procedures the new techniques give rise to an algebraic approach to the galois theory of renormalization in particular they do not depend on the geometry underlying the case of dimensional regularization and the riemann hilbert correspondence this is illustrated with a discussion of the bphz renormalization scheme

on the riemann problem for non conservative hyperbolic systems we consider the construction and the properties of the riemann solver for the hyperbolic system

hopf algebras of rooted forests cocyles and free rota baxter algebras the hopf algebra and the rota baxter algebra are the two algebraic structures underlying the algebraic approach of connes and kreimer to renormalization of perturbative quantum field theory in particular the hopf algebra of rooted trees serves as the baby model of feynman graphs in their approach and can be characterized by certain universal properties involving a hochschild 1 cocycle decorated rooted trees have also been applied to study feynman graphs we will continue the study of universal properties of various spaces of decorated rooted trees with such a 1 cocycle leading to the concept of a cocycle hopf algebra we further apply the universal properties to equip a free rota baxter algebra with the structure of a cocycle hopf algebra

a nonparametric polynomial identification algorithm for the hammerstein system almost all existing hammerstein system nonparametric identification algorithms can recover the unknown system nonlinear element up to an additive constant and one functional value of the nonlinearity is usually assumed to be known to make the constant solvable to overcome this defect in this paper a new nonparametric polynomial identification algorithm for the hammerstein system is proposed which extends the idea in the author s previous work 1993 1994 on the hammerstein system identification to a more general and practical case where no functional value of the system nonlinearity is known a priori convergence and convergence rates in both uniform and global senses are established and simulation studies demonstrate the effectiveness and advantage of the new algorithm

rota baxter operators on 4 dimensional complex simple associative algebras rota baxter operators or relations were introduced to solve certain analytic and combinatorial problems and then applied to many fields in mathematics and mathematical physics in this paper we commence to study the rota baxter operators of weight zero on a 4 dimensional complex simple associative algebra which is isomorphic to the algebra of all 2x2 matrices over the field of complex numbers such operators satisfy the operator form of the classical yang baxter equation on the general linear lie algebra gl 2 c we provide two different solving methods one by hand to show some tricks for solving a large nonlinear system and another one by computer algebra to show the possibility of solving higher dimensional cases

decay estimation in nonlinear hyperbolic system of conservation laws abstract in this paper we study some decay estimates in nonlinear hyperbolic system of conservation laws this research is not only interesting in itself but also crucial in studying the large time behavior problem by introducing a proper glimm functional we obtain some useful decay estimates which are proved helpful in obtaining decay rates of the admissible solutions to nonlinear hyperbolic conservation laws as t

parametric representation of rank d tensorial group field theory abelian models with kinetic term sum_s p_s mu we consider the parametric representation of the amplitudes of abelian models in the so called framework of rank d tensorial group field theory these models are called abelian because their elds live on

renormalization in quantum field theory and the riemann hilbert problem i the hopf algebra structure of graphs and the main theorem this paper gives a complete selfcontained proof of our result announced in 6 showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the riemann hilbert problem we shall first show that for any quantum field theory the combinatorics of feynman graphs gives rise to a hopf algebra which is commutative as an algebra it is the dual hopf algebra of the enveloping algebra of a lie algebra whose basis is labelled by the one particle irreducible feynman graphs the lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa the corresponding lie group g is the group of characters of we shall then show that using dimensional regularization the bare unrenormalized theory gives rise to a loop where c is a small circle of complex dimensions around the integer dimension d of space time our main result is that the renormalized theory is just the evaluation at z d of the holomorphic part of the birkhoff decomposition of we begin to analyse the group g and show that it is a semi direct product of an easily understood abelian group by a highly non trivial group closely tied up with groups of diffeomorphisms the analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title

next to k leading log expansions by chord diagrams green functions in a quantum field theory can be expanded as bivariate series in the coupling and a scale parameter the leading logs are given by the main diagonal of this expansion i e the subseries where the coupling and the scale parameter appear to the same power then the next to leading logs are listed by the next diagonal of the expansion where the power of the coupling is incremented by one and so on we give a general method for deriving explicit formulas and asymptotic estimates for any hbox next to k leading log expansion for a large class of single scale green functions these green functions are solutions to dyson schwinger equations that are known by previous work to be expressible in terms of chord diagrams we look in detail at the green function for the fermion propagator in massless yukawa theory as one example and the green function of the photon propagator in quantum electrodynamics as a second example as well as giving general theorems our methods are combinatorial but the consequences are physical giving information on which terms dominate and on the dichotomy between gauge theories and other quantum field theories

renormalization of gauge fields a hopf algebra approach we study the connes kreimer hopf algebra of renormalization in the case of gauge theories we show that the ward identities and the slavnov taylor identities in the abelian and non abelian case respectively are compatible with the hopf algebra structure in that they generate a hopf ideal consequently the quotient hopf algebra is well defined and has those identities built in this provides a purely combinatorial and rigorous proof of compatibility of the slavnov taylor identities with renormalization

optimal input design for worst case system identification in l 1 l 2 l abstract in this paper we examine optimal sequences that generate worst case parameters estimation errors in the l 1 l 2 and l norm context for algorithms identifying linear time invariant discrete time finite impulse response systems excited by bounded sequences and with l norm measurement error

a combinatorial hopf algebra for nonlinear output feedback control systems abstract in this work a combinatorial description is provided of a faa di bruno type hopf algebra which naturally appears in the context of fliess operators in nonlinear feedback control theory it is a connected graded commutative and non cocommutative hopf algebra defined on rooted circle trees a cancellation free forest formula for its antipode is given

the structure of renormalization hopf algebras for gauge theories i representing feynman graphs on bv algebras we study the structure of renormalization hopf algebras of gauge theories we identify certain hopf subalgebras in them whose character groups are semidirect products of invertible formal power series with formal diffeomorphisms this can be understood physically as wave function renormalization and renormalization of the coupling constants respectively after taking into account the slavnov taylor identities for the couplings as generators of a hopf ideal we find hopf subalgebras in the corresponding quotient as well in the second part of the paper we explain the origin of these hopf ideals by considering a coaction of the renormalization hopf algebras on the batalin vilkovisky bv algebras generated by the fields and couplings constants the so called classical master equation satisfied by the action in the bv algebra implies the existence of the above hopf ideals in the renormalization hopf algebra finally we exemplify our construction by applying it to yang mills gauge theory

non commutative hopf algebra of formal diffeomorphisms abstract this paper deals with two hopf algebras which are the non commutative analogues of two different groups of formal power series the first group is the set of invertible series with the group law being multiplication of series while the second is the set of formal diffeomorphisms with the group law being a composition of series the motivation to introduce these hopf algebras comes from the study of formal series with non commutative coefficients invertible series with non commutative coefficients still form a group and we interpret the corresponding new non commutative hopf algebra as an alternative to the natural hopf algebra given by the co ordinate ring of the group which has the advantage of being functorial in the algebra of coefficients for the formal diffeomorphisms with non commutative coefficients this interpretation fails because in this case the composition is not associative anymore however we show that for the dual non commutative algebra there exists a natural co associative co product defining a non commutative hopf algebra moreover we give an explicit formula for the antipode which represents a non commutative version of the lagrange inversion formula and we show that its coefficients are related to planar binary trees then we extend these results to the semi direct co product of the previous hopf algebras and to series in several variables finally we show how the non commutative hopf algebras of formal series are related to some renormalization hopf algebras which are combinatorial hopf algebras motivated by the renormalization procedure in quantum field theory and to the renormalization functor given by the double tensor algebra on a bi algebra

nonabelian generalized lax pairs the classical yang baxter equation and postlie algebras we generalize the classical study of generalized lax pairs the related mathcal o operators and the modified classical yang baxter equation by introducing the concepts of nonabelian generalized lax pairs extended mathcal o operators and the extended classical yang baxter equation we study in this context the nonabelian generalized r matrix ansatz and the related double lie algebra structures relationship between extended mathcal o operators and the extended classical yang baxter equation is established especially for self dual lie algebras this relationship allows us to obtain an explicit description of the manin triples for a new class of lie bialgebras furthermore we show that a natural structure of postlie algebra is behind mathcal o operators and fits in a setup of the triple lie algebra that produces self dual nonabelian generalized lax pairs

a wavefront tracking algorithm for n n nongenuinely nonlinear conservation laws we introduce a wavefront tracking algorithm for n n hyperbolic systems of conservation lawsut f u x 0 that admits characteristic fields that are neither genuinely nonlinear nor linearly degenerate in the sense of lax instead we assume that for any nongenuinely nonlinear ith characteristic family the derivative of the ith eigenvalue i u of df u in the direction of the ith right eigenvector ri u vanishes on a single n 1 dimensional hypersurface in the u space transversal to the field ri u systems that fulfill this type of assumptions are of particular interest in studying elastodynamic or rigid heat conductors at low temperature the first proof of the existence of weak solutions for nongenuinely nonlinear systems was given by t p liu mem amer math soc 30 1981 no 240 based on a glimm scheme our construction here provides an alternative method for establishing the global existence of weak solutions for such systems moreover relying on the stability analysis developed in ancona and marson preprint s i s s a i s a s 27 99 11 1999 and preprint 2000 we show that these solutions are entropy admissible in the sense of lax

optimal experiment design for regression polynomial models identification in this paper the problem of optimal experimental design for parameter identification of static non linear blocks is addressed non linearities are assumed to be polynomial and represented according to the vandermonde base the optimality problem is formulated in a set membership context and the cost functions to be minimized are the worst case parameter uncertainties closed form optimal input sequences are derived when the input u is allowed to vary on a given interval u a u b since optimal input sequences are in general not invariant to base changes results and criteria for representing polymomials with different bases still preserving the optimal set of input levels derived from the vandermonde parameterization are introduced as well finally numerical results are reported showing the effectiveness of using optimal input sequences especially when identifying some block described dynamic models that include in their structure static non linearities such as hammerstein and lpv models in su

holder continuous rough paths by fourier normal ordering we construct in this article an explicit geometric rough path over arbitrary d dimensional paths with finite 1 variation for any alpha in 0 1 the method may be coined as fourier normal ordering since it consists in a regularization obtained after permuting the order of integration in iterated integrals so that innermost integrals have highest fourier frequencies in doing so there appear non trivial tree combinatorics which are best understood by using the structure of the hopf algebra of decorated rooted trees in connection with the chen or multiplicative property and of the hopf shuffle algebra in connection with the shuffle or geometric property holder continuity is proved by using besov norms the method is well suited in particular in view of applications to probability theory see the companion article 34 for the construction of a rough path over multidimensional fractional brownian motion with hurst index 1 4 or 35 for a short survey in that case

perturbative quantum field theory via vertex algebras in this paper we explain how perturbative quantum field theory can be formulated in terms of a version of vertex algebras our starting point is the wilson zimmermann operator product expansion ope following ideas of a previous paper s hollands e print arxiv 0802 2198 we consider a consistency essentially associativity condition satisfied by the coefficients in this expansion we observe that the information in the ope coefficients can be repackaged straightforwardly into vertex operators and that the consistency condition then has essentially the same form as the key condition in the theory of vertex algebras we develop a general theory of perturbations of the algebras that we encounter similar in nature to the hochschild cohomology describing the deformation theory of ordinary algebras the main part of the paper is devoted to the question how one can calculate the perturbations corresponding to a given interaction lagrangian such as 4 in practice using the consistency condition an

feynman integrals and motives of configuration spaces we formulate the problem of renormalization of feynman integrals and its relation to periods of motives in configuration space instead of momentum space the algebro geometric setting is provided by the wonderful compactifications overline conf gamma x of arrangements of subvarieties associated to the subgraphs of a feynman graph with x a quasi projective variety the motive and the class in the grothendieck ring are computed explicitly for these wonderful compactifications in terms of the motive of x and the combinatorics of the feynman graph using recent results of li li the pullback to the wonderful compactification of the form defined by the unrenormalized feynman amplitude has singularities along a hypersurface whose real locus is contained in the exceptional divisors of the iterated blowup that gives the wonderful compactification a regularization of the feynman integrals can be obtained by modifying the cycle of integration by replacing the divergent locus with a leray coboundary the ambiguities are then defined by poincare residues while these residues give periods associated to the cohomology of the exceptional divisors and their intersections the regularized integrals give rise to periods of the hypersurface complement in the wonderful compactification

normal coordinates and primitive elements in the hopf algebra of renormalization we introduce normal coordinates on the infinite dimensional group g introduced by connes and kreimer in their analysis of the hopf algebra of rooted trees we study the primitive elements of the algebra and show that they are generated by a simple application of the inverse poincare lemma given a closed left invariant 1 form on g for the special case of the ladder primitives we find a second description that relates them to the hopf algebra of functionals on power series with the usual product either approach shows that the ladder primitives are given by the schur polynomials the relevance of the lower central series of the dual lie algebra in the process of renormalization is also discussed leading to a natural concept of k primitiveness which is shown to be equivalent to the one already in the literature

groupoids and faa di bruno formulae for green functions in bialgebras of trees we prove a faa di bruno formula for the green function in the bialgebra of p trees for any polynomial endofunctor p the formula appears as relative homotopy cardinality of an equivalence of groupoids

renormalization in quantum field theory and the riemann hilbert problem ii the β function diffeomorphisms and the renormalization group we showed in part i that the hopf algebra of feynman graphs in a given qft is the algebra of coordinates on a complex infinite dimensional lie group g and that the renormalized theory is obtained from the unrenormalized one by evaluating at 0 the holomorphic part of the riemann hilbert decomposition 1 of the loop g provided by dimensional regularization we show in this paper that the group g acts naturally on the complex space x of dimensionless coupling constants of the theory more precisely the formula g 0 gz 1 z 3 3 2 for the effective coupling constant when viewed as a formal power series does define a hopf algebra homomorphism between the hopf algebra of coordinates on the group of formal diffeomorphisms to the hopf algebra this allows first of all to read off directly without using the group g the bare coupling constant and the renormalized one from the riemann hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms this shows that renormalization is intimately related with the theory of non linear complex bundles on the riemann sphere of the dimensional regularization parameter it also allows to lift both the renormalization group and the function as the asymptotic scaling in the group g this exploits the full power of the riemann hilbert decomposition together with the invariance of under a change of unit of mass this not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group g for the full higher pole structure of minimal subtracted counterterms in terms of the residue

from dynamical systems to renormalization in this paper we study logarithmic derivatives associated to derivations on completed graded lie algebra as well as the existence of inverses these logarithmic derivatives when invertible generalize the exp log correspondence between a lie algebra and its lie group such correspondences occur naturally in the study of dynamical systems when dealing with the linearization of vector fields and the non linearizability of a resonant vector fields corresponds to the non invertibility of a logarithmic derivative and to the existence of normal forms these concepts stemming from the theory of dynamical systems can be rephrased in the abstract setting of lie algebra and the same difficulties as in perturbative quantum field theory pqft arise here surprisingly one can adopt the same ideas as in pqft with fruitful results such as new constructions of normal forms with the help of the birkhoff decomposition the analogy goes even further locality of counter terms choice of a renormalization scheme and shall lead to more interactions between dynamical systems and quantum field theory

o operators on hom lie algebras o operators also known as relative rota baxter operators on lie algebras have several applications in integrable systems and the classical yang baxter equations in this article we study o operators on hom lie algebras we define a cochain complex for o operators on hom lie algebras with respect to a representation any o operator induces a hom pre lie algebra structure we express the cochain complex of an o operator in terms of the specific hom lie algebra cochain complex if the structure maps in a hom lie algebra and its representation are invertible then we can extend the above cochain complex to a deformation complex for o operators by adding the space of zero cochains subsequently we study formal deformations of o operators on regular hom lie algebras in terms of the deformation cohomology in the end we deduce deformations of s rota baxter operators of weight 0 and skew symmetric r matrices on hom lie algebras as particular cases of o operators on hom lie algebras

the semigroup generated by 2 2 conservation laws consider the cauchy problem for a strictly hyperbolic 2 2 system of conservation laws in one space dimension ie1 01 assuming that each characteristic field is either linearly degenerate or genuinely nonlinear this paper develops a new algorithm based on wave front tracking which yields a cauchy sequence of approximate solutions converging to a unique limit depending continuously on the initial data the solutions that we obtain constitute a semigroup s defined on a set ie1 02 of integrable functions with small total variation for some lipschitz constant l we have the estimate ie1 03

dimensional regularization in position space and a forest formula for epstein glaser renormalization we reformulate dimensional regularization as a regularization method in position space and show that it can be used to give a closed expression for the renormalized time ordered products as solutions to the induction scheme of epstein glaser this closed expression which we call the epstein glaser forest formula is analogous to zimmermann s forest formula for bph renormalization for scalar fields the resulting renormalization method is always applicable we compute several examples we also analyze the hopf algebraic aspects of the combinatorics our starting point is the main theorem of renormalization of stora and popineau and the arising renormalization group as originally defined by stuckelberg and petermann

on the invariance of residues of feynman graphs we use simple iterated one loop graphs in massless yukawa theory and qed to pose the following question what are the symmetries of the residues of a graph under a permutation of places to insert subdivergences the investigation confirms partial invariance of the residue under such permutations the highest weight transcendental is invariant under such a permutation for qed this result is gauge invariant i e the permutation invariance holds for any gauge computations are done making use of the hopf algebra structure of graphs and employing ginac to automate the calculations

rota baxter operators on sl 2 c and solutions of the classical yang baxter equation we explicitly determine all rota baxter operators of weight zero on sl 2 c under the cartan weyl basis for the skew symmetric operators we give the corresponding skew symmetric solutions of the classical yang baxter equation in sl 2 c confirming the related study by semenov tian shansky in general these rota baxter operators give a family of solutions of the classical yang baxter equation in the six dimensional lie algebra sl 2 c ad sl 2 c they also give rise to three dimensional pre lie algebras which in turn yield solutions of the classical yang baxter equation in other six dimensional lie algebras

the signed permutation group on feynman graphs the feynman rules assign to every graph an integral which can be written as a function of a scaling parameter l assuming l for the process under consideration is very small so that contributions to the renormalization group are small we can expand the integral and only consider the lowest orders in the scaling the aim of this article is to determine specific combinations of graphs in a scalar quantum field theory that lead to a remarkable simplification of the first non trivial term in the perturbation series it will be seen that the result is independent of the renormalization scheme and the scattering angles to achieve that goal we will utilize the parametric representation of scalar feynman integrals as well as the hopf algebraic structure of the feynman graphs under consideration moreover we will present a formula which reduces the effort of determining the first order term in the perturbation series for the specific combination of graphs to a minimum

some new properties of chebyshev polynomials this paper deals with the problem of the polynomial interpolation of data subject to bounded perturbations in particular we show that interpolation on the chebyshev polynomial extrema minimizes the diameter of the set of the vectors of the coefficients of all possible polynomials interpolating the perturbed data in doing so some new properties of the chebyshev polynomials are obtained as well some of the proposed results are of direct interest in system identification theory when considering the optimal input design for the identification of non linear block described dynamic systems such as hammerstein and wiener models

hopf algebras cyclic cohomology and the transverse index theorem in this paper we solve a longstanding internal problem of noncommutative geometry namely the computation of the index of transversally elliptic operators on foliations we show that the computation of the local index formula for transversally hypoelliptic operators can be settled thanks to a very specific hopf algebra associated to each integer codimension this hopf algebra reduces transverse geometry to a universal geometry of affine nature the structure of this hopf algebra its relation with the lie algebra of formal vector fields as well as the computation of its cyclic cohomology are done in the present paper in which we also show that under a suitable unimodularity condition the cosimplicial space underlying the hochschild cohomology of a hopf algebra carries a highly nontrivial cyclic structure

free tri dendriform family algebras abstract in this paper we first prove that a rota baxter family algebra indexed by a semigroup amounts to an ordinary rota baxter algebra structure on the tensor product with the semigroup algebra we show that the same phenomenon arises for dendriform and tridendriform family algebras then we construct free dendriform family algebras in terms of typed decorated planar binary trees finally we generalize typed decorated rooted trees to typed valently decorated schroder trees and use them to construct free tridendriform family algebras

optimal worst case estimation for lpv fir models with bounded errors abstract in this paper discrete time linear parameter varying lpv models with finite impulse response fir dynamic structure are considered measurement errors are assumed to be bounded in such condition optimal input sequences minimizing the worst case parameter uncertainties are derived the main result of this paper consists in finding optimal worst case input sequences for lpv fir models these are obtained suitably combining the optimal design results for standing alone invariant fir models and standing alone nonlinear memoryless blocks the quite relevant improvement obtained using optimal input sequences instead of random sequences is shown by simulation

decomposition spaces incidence algebras and mobius inversion i basic theory abstract this is the first in a series of papers devoted to the theory of decomposition spaces a general framework for incidence algebras and mobius inversion where algebraic identities are realised by taking homotopy cardinality of equivalences of groupoids a decomposition space is a simplicial groupoid satisfying an exactness condition weaker than the segal condition expressed in terms of active and inert maps in just as the segal condition expresses composition the new exactness condition expresses decomposition and there is an abundance of examples in combinatorics after establishing some basic properties of decomposition spaces the main result of this first paper shows that to any decomposition space there is an associated incidence coalgebra spanned by the space of 1 simplices and with coefficients in groupoids we take a functorial viewpoint throughout emphasising conservative ulf functors these induce coalgebra homomorphisms reduction procedures in the classical theory of incidence coalgebras are examples of this notion and many are examples of decalage of decomposition spaces an interesting class of examples of decomposition spaces beyond segal spaces is provided by hall algebras the waldhausen s construction of an abelian or stable infinity category is shown to be a decomposition space in the second paper in this series we impose further conditions on decomposition spaces to obtain a general mobius inversion principle and to ensure that the various constructions and results admit a homotopy cardinality in the third paper we show that the lawvere menni hopf algebra of mobius intervals is the homotopy cardinality of a certain universal decomposition space two further sequel papers deal with numerous examples from combinatorics note the notion of decomposition space was arrived at independently by dyckerhoff and kapranov 17 who call them unital 2 segal spaces our theory is quite orthogonal to theirs the definitions are different in spirit and appearance and the theories differ in terms of motivation examples and directions

qed hopf algebras on planar binary trees abstract in this paper we describe the hopf algebras on planar binary trees used to renormalize the feynman propagators of quantum electrodynamics and the coaction which describes the renormalization procedure both structures are related to some semi direct coproduct of hopf algebras

etude de l algebre de lie double des arbres enracines decores the double lie algebra ld of rooted trees decorated by a set d is introduced generalising the construction of connes and kreimer it is shown that it is a simple lie algebra its derivations and its automorphisms are described as well as some central extensions finally the category of lowest weight modules is introduced and studied

from rota baxter algebras to pre lie algebras rota baxter algebras were introduced to solve some analytic and combinatorial problems and have appeared in many fields in mathematics and mathematical physics rota baxter algebras provide a construction of pre lie algebras from associative algebras in this paper we give all rota baxter operators of weight 1 on complex associative algebras in dimension 3 and their corresponding pre lie algebras

renormalization of multiple zeta values abstract multiple zeta values mzvs in the usual sense are the special values of multiple variable zeta functions at positive integers their extensive studies are important in both mathematics and physics with broad connections and applications in contrast very little is known about the special values of multiple zeta functions at non positive integers since the values are usually undefined we define and study multiple zeta functions at integer values by adapting methods of renormalization from quantum field theory and following the hopf algebra approach of connes and kreimer this definition of renormalized mzvs agrees with the convergent mzvs and extends the work of ihara kaneko zagier on renormalization of mzvs with positive arguments we further show that the important quasi shuffle stuffle relation for usual mzvs remains true for the renormalized mzvs

on the convergence rate of vanishing viscosity approximations given a strictly hyperbolic genuinely nonlinear system of conservation laws we prove the a priori boundu t u e t l1 o 1 1 t e ln e on the distance between an exact bv solution u and a viscous approximation u e letting the viscosity coefficient e 0 in the proof starting from u we construct an approximation of the viscous solution u e by taking a mollification u e and inserting viscous shock profiles at the locations of finitely many large shocks for each fixed e error estimates are then obtained by introducing new lyapunov functionals that control interactions of shock waves in the same family and also interactions of waves in different families c 2004 wiley periodicals inc

deformations of associative rota baxter operators abstract rota baxter operators and more generally o operators on associative algebras are important in probability combinatorics associative yang baxter equation and splitting of algebras using a method of uchino we construct an explicit graded lie algebra whose maurer cartan elements are given by o operators this allows us to construct cohomology for an o operator this cohomology can also be seen as the hochschild cohomology of a certain algebra with coefficients in a suitable bimodule next we study linear and formal deformations of an o operator which are governed by the above defined cohomology we introduce nijenhuis elements associated with an o operator which give rise to trivial deformations as applications we conclude deformations of weight zero rota baxter operators associative r matrices and averaging operators

structure identification of nonlinear dynamic systems a survey on input output approaches abstract in the past several methods have been elaborated for the identification of nonlinear dynamic systems most of the methods assume that the structure of the system is given a priori therefore they are in reality parameter estimation algorithms and structure identification is thus usually performed by repeated parameter estimation however in nonlinear system theory several methods are known to determine the structure of a system in this paper structure identification of block oriented especially cascade models of semi linear dynamic models with signal dependent parameters and of nonlinear dynamic models being linear in the parameters will be considered different structure selection methods are summarized based on step and impulse tests frequency response measurements correlation analysis repeated reproducible tests and normal operating data

birkhoff type decompositions and the baker campbell hausdorff recursion we describe a unification of several apparently unrelated factorizations arising from quantum field theory vertex operator algebras combinatorics and numerical methods in differential equations the unification is given by a birkhoff type decomposition that was obtained from the baker campbell hausdorff formula in our study of the hopf algebra approach of connes and kreimer to renormalization in perturbative quantum field theory there we showed that the birkhoff decomposition of connes and kreimer can be obtained from a certain baker campbell hausdorff recursion formula in the presence of a rota baxter operator we will explain how the same decomposition generalizes the factorization of formal exponentials and uniformization for lie algebras that arose in vertex operator algebra and conformal field theory and the even odd decomposition of combinatorial hopf algebra characters as well as the lie algebra polar decomposition as used in the context of the approximation of matrix exponentials in ordinary differential equations

on the stability of some groups of formal diffeomorphisms by the birkhoff decomposition abstract let g be the group of one parameter identity tangent diffeomorphisms on the line whose coefficients are formal laurent series in the parameter e with a pole of finite order at 0 it is well known that the birkhoff decomposition can be defined in such a group we investigate the stability of the birkhoff decomposition in subgroups of g and give a formula for this decomposition these results are strongly related to renormalization in quantum field theory since it was proved by a connes and d kreimer that after dimensional regularization the unrenormalized effective coupling constants are the image by a formal identity tangent diffeomorphism of the coupling constants of the theory in the massless 6 3 theory this diffeomorphism is in g and its birkhoff decomposition gives directly the bare coupling constants and the renormalized coupling constants

bv solutions of the semidiscrete upwind scheme consider the strictly hyperbolic system of conservation laws u t f u x 0 1 where u n f n n smooth the system is said to be strictly hyperbolic provided that the jacobian matrix a u df u has n distinct eigenvalues 1 u n u

the structure of the ladder insertion elimination lie algebra we continue our investigation into the insertion elimination lie algebra open image in new window of feynman graphs in the ladder case emphasizing the structure of this lie algebra relevant for future applications in the study of dyson schwinger equations we work out the relation to the classical infinite dimensional lie algebra open image in new window and we determine the cohomology of open image in new window

deformations and homotopy theory of relative rota baxter lie algebras we determine the l infty algebra that controls deformations of a relative rota baxter lie algebra and show that it is an extension of the dg lie algebra controlling deformations of the underlying mathsf lie mathsf rep pair by the dg lie algebra controlling deformations of the relative rota baxter operator consequently we define the cohomology of relative rota baxter lie algebras and relate it to their infinitesimal deformations a large class of relative rota baxter lie algebras is obtained from triangular lie bialgebras and we construct a map between the corresponding deformation complexes next the notion of a homotopy relative rota baxter lie algebra is introduced we show that a class of homotopy relative rota baxter lie algebras is intimately related to pre lie infty algebras

an algebraic birkhoff decomposition for the continuous renormalization group this paper aims at presenting the first steps towards a formulation of the exact renormalization group equation in the hopf algebra setting of connes and kreimer it mostly deals with some algebraic preliminaries allowing us to formulate perturbative renormalization within the theory of differential equations the relation between renormalization formulated as a change of boundary condition for a differential equation and an algebraic birkhoff decomposition for rooted trees is explicited