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We propose an inertial forward–backward splitting algorithm to compute a zero of a sum of two monotone operators allowing for stochastic errors in the computation of the operators. More precisely, we establish almost sure convergence in real Hilbert spaces of the sequence of iterates to an optimal solution. Then, based on this analysis, we introduce two new classes of stochastic inertial primal–dual splitting methods for solving structured systems of composite monotone inclusions and prove their convergence. Our results extend to the stochastic and inertial setting various types of structured monotone inclusion problems and corresponding algorithmic solutions. Application to minimization problems is discussed.

We discuss structural aspects of the functional renormalisation group. Flows for a general class of correlation functions are derived, and it is shown how symmetry relations of the underlying theory are lifted to the regularised theory. A simple equation for the flow of these relations is provided. The setting includes general flows in the presence of composite operators and their relation to standard flows, an important example being NPI quantities. We discuss optimisation and derive a functional optimisation criterion. Applications deal with the interrelation between functional flows and the quantum equations of motion, general Dyson-Schwinger equations. We discuss the combined use of these functional equations as well as outlining the construction of practical renormalisation schemes, also valid in the presence of composite operators. Furthermore, the formalism is used to derive various representations of modified symmetry relations in gauge theories, as well as to discuss gauge-invariant flows. We close with the construction and analysis of truncation schemes in view of practical optimisation., Comment: 58 pages, no figures, minor changes

We consider the bosonic sl ( 2 ) sector of the maximally supersymmetric N = 4 SYM model and show that anomalous dimension of the twist-3 single-trace composite operators built of scalar fields, recently calculated up to the four-loop order, can be generated by a compact reciprocity respecting evolution kernel.

We construct a long-range Baxter equation encoding anomalous dimensions of composite operators in the SL ( 2 | 1 ) sector of N = 4 supersymmetric Yang–Mills theory. The formalism is based on the analytical Bethe ansatz. We compare predictions of the Baxter equations for short operators with available multiloop perturbative calculations.

We discuss how the ordinary renormalization group (RG) equations arise in the context of Wilson's exact renormalization group (ERG) as formulated by Polchinski. We consider the phi4 theory in four dimensional euclidean space as an example, and introduce a particular scheme of parameterizing the solutions of the ERG equations. By analyzing the scalar composite operators of dimension two and four, we show that the parameters obey mass independent RG equations. We conjecture the equivalence of our parameterization scheme with the MS scheme for dimensional regularization., Comment: 18 pages, 3 figures, LaTeX2e; small changes made to enhance clarity, especially in sect. 3

Matrix model describing the anomalous dimensions of composite operators in N = 4 super Yang–Mills theory up to one-loop level is considered at finite temperature. We compute the thermal effective action for this model, which we define as the log of the partition function restricted to the states of given fixed length and spin. The result is obtained in the limit of high as well as low temperature.

We employ a new strategy for a non perturbative determination of the renormalized energy momentum tensor. The strategy is based on the definition of suitable lattice Ward identities probed by observables computed along the gradient flow. The new set of identities exhibits many interesting qualities, arising from the UV finiteness of flowed composite operators. In this paper we show how this method can be used to non perturbatively renormalize the energy momentum tensor for a SU(3) Yang-Mills theory, and report our numerical results. We employ a new strategy for a non perturbative determination of the renormalized energy momentum tensor. The strategy is based on the definition of suitable lattice Ward identities probed by observables computed along the gradient flow. The new set of identities exhibits many interesting qualities, arising from the UV finiteness of flowed composite operators. In this paper we show how this method can be used to non perturbatively renormalize the energy momentum tensor for a SU(3) Yang-Mills theory, and report our numerical results.

In this paper, the local Poincare inequality and embedding inequality are proved first. Then the global embedding inequality of composite operators for differential forms on $L^{\varphi}$ -averaging domains with $L^{\varphi}$ -norm is established. Some examples are also given to illustrate applications.

In this paper, we obtain Poincare-type inequalities for the composite operator acting on differential forms and establish the $L^{p}$ , Lipschitz, and BMO norm estimates. We also give the weighted versions of the comparison theorems for the $L^{p}$ , Lipschitz, and BMO norms.

The NLO evolution equations for quadrupole and double dipole operators have been obtained within the high energy operator expansion method. The corresponding quasi-conformal evolution equations for the composite operators were constructed.

The effect of finite isospin chemical potential to the effective masses of the mesons at finite temperature is investigated in the framework of the O(4) linear sigma model with explicit chiral symmetry breaking. We present a mechanism to include the isospin chemical potential in the model. By using the Cornwall-Jackiw-Tomboulis method of composite operators, we obtain a set gap equations for the effective masses of the mesons and get the numerical results in the Hartree approximation. We find that the introduction of the chemical potential only affects the mass of the charged pions and sigma, while there is almost NO effects on the mass of neutral pions., Comment: 11 pages, 6 figures; Version to appear in J. Phys. G

We investigate various perturbative properties of the deformed N=4 SYM theory. We carry out a three-loops calculation of the chiral matter superfield propagator and derive the condition on the couplings for maintaining finiteness at this order. We compute the 2-, 3- and 4-point functions of composite operators of dimension 2 at two loops. We identify all the scalar operators (chiral and non-chiral) of bare dimension 4 with vanishing one-loop anomalous dimension. We compute some 2- and 3-point functions of these operators at two loops and argue that the observed finite corrections cannot be absorbed by a finite renormalization of the operators.

We compute the pole mass of the gluon in QCD from the local composite operator formalism at two loops in the MSbar renormalization scheme. For Yang-Mills theory an estimate of the mass at two loops is 2.13 Lambda_MSbar.

Instanton contributions to the anomalous dimensions of gauge-invariant composite operators in the N =4 supersymmetric SU(N) Yang–Mills theory are studied in the one-instanton sector. Independent sets of scalar operators of bare dimension �0 ≤ 5 are constructed in all the allowed representations of the SU(4) R-symmetry group and their two-point functions are computed in the semiclassical approximation. Analysing the moduli space integrals the sectors in which the scaling dimensions receive non-perturbative contributions are identified. The requirement that the integrations over the fermionic collective coordinates which arise in the instanton background are saturated leads to non-renormalisation properties for a large class of operators. Instanton-induced corrections to the scaling dimensions are found only for dimension 4 SU(4) singlets and for dimension 5 operators in the 6 of SU(4). In many cases the non-renormalisation results are argued to be specific to operators of small dimension, but for some special sectors it is shown that they are valid for arbitrary dimension. Comments are also made on the implications of the results on the form of the instanton contributions to the dilation operator of the theory and on the possibility of realising its action on the instanton moduli space.

The recently developed energy-scale-dependent Composite Operator Method is applied to the single-impurity Anderson model. A fully self-consistent solution is given and analyzed. At very low temperatures, the density of states presents, on the top of the high-energy background, a Kondo-like peak whose parameter dependence is discussed in detail. The proposed method reproduces the exact results known in the literature with very low numerical effort and it is applicable for arbitrary values of the external parameters.

The local composite operator $A_{\mu}^{2}$ is analysed within the algebraic renormalization in Yang-Mills theories in linear covariant gauges. We establish that it is multiplicatively renormalizable to all orders of perturbation theory. Its anomalous dimension is computed to two-loops in the MSbar scheme., Comment: 10 pages, LaTeX, final version to appear in Phys. Lett. B

The Green's function formalism in Condensed Matter Physics is reviewed within the equation of motion approach. Composite operators and their Green's functions naturally appear as building blocks of generalized perturbative approaches and require fully self-consistent treatments in order to be properly handled. It is shown how to unambiguously set the representation of the Hilbert space by fixing both the unknown parameters, which appear in the linearized equations of motion and in the spectral weights of non-canonical operators, and the zero-frequency components of Green's functions in a way that algebra and symmetries are preserved. To illustrate this procedure some examples are given: the complete solution of the two-site Hubbard model, the evaluation of spin and charge correlators for a narrow-band Bloch system, the complete solution of the three-site Heisenberg model, and a study of the spin dynamics in the Double-Exchange model., Comment: 20 RevTeX4 pages, 4 embedded figures

We discuss a general method of constructing the products of composite operators using the exact renormalization group formalism. Considering mainly the Wilson action at a generic fixed point of the renormalization group, we give an argument for the validity of short distance expansions of operator products. We show how to compute the expansion coefficients by solving differential equations, and test our method with some simple examples., Comment: 31 pages, 5 figures, LaTeX2e, references added and Appendix B revised

In this work we propose a systematic way to compute the logarithmic divergences of composite operators in the pure spinor description of the $AdS_5\times S^5$ superstring. The computations of these divergences can be summarized in terms of a dilatation operator acting on the local operators. We check our results with some important composite operators of the formalism., Comment: 35 pages

Gauge-invariant perturbation theory is an extension of ordinary perturbation theory which describes strictly gauge-invariant states in theories with a Brout-Englert-Higgs effect. Such gauge-invariant states are composite operators which have necessarily only global quantum numbers. As a consequence, flavor is exchanged for custodial quantum numbers in the standard model, recreating the fermion spectrum in the process. Here, we study the implications of such a description, possibly also for the generation structure of the standard model. In particular, this implies that scattering processes are essentially bound-state-bound-state interactions, and require a suitable description. We analyze the implications for the pair-production process $e^+e^-\to{\bar f}f$ at a linear collider to leading order. We show how ordinary perturbation theory is recovered as the leading contribution. Developing a suitable PDF-type language, we also assess the impact of sub-leading contributions. We find that only for very heavy fermions in the final state, especially top quarks, sizable corrections could emerge. This gives an interesting, possibly experimentally testable, scenario for the formal field theory underlying the electroweak sector of the standard model., 11 pages, 2 figures, revised version: We corrected a calculation in Sec.3 and reworked/improved the presentation

We construct the most general composite operators of ${\mathcal N} = 4$ SYM in Lorentz harmonic chiral (≈twistor) superspace. The operators are built from the SYM supercurvature which is nonpolynomial in the chiral gauge prepotentials. We reconstruct the full nonchiral dependence of the supercurvature. We compute all tree-level MHV form factors via the LSZ reduction procedure with on-shell states made of the same supercurvature. We construct the most general composite operators of N = 4 SYM in Lorentz harmonic chiral ($\approx$ twistor) superspace. The operators are built from the SYM supercurvature which is nonpolynomial in the chiral gauge prepotentials. We reconstruct the full nonchiral dependence of the supercurvature. We compute all tree-level MHV form factors via the LSZ redcution procedure with on-shell states made of the same supercurvature.

The anomalous dimension of single and multi-trace composite operators of scalar fields is shown to vanish at all orders of the perturbative series. The proof hold for theories with N=2 supersymmetry with any number of hypermultiplets in a generic representation of the gauge group. It then applies to the finite N=4 theory as well as to non conformal N=2 models.

We show that the effective potential for local composite operators is a useful object in studing dynamical symmetry breaking by calculating the effective potential for the local composite operators $\bar{\psi} \psi$ and $\phi^2$ in the Gross-Neveu (GN) and O(N) models, respectively. Since the effective potential for local composite operators can be calculated by using the Cornwall-Jackiw-Tomboulis (CJT) effective potential in theory with additional bare mass terms, we show that divergences in the effective potential for local composite operators are the same as in the CJT effective potential. We compare the results obtained with the results give by the auxiliary field method., Comment: 17 pages, LaTeX, the analysis of the O(N) model with finite cut-off have been reconsidered and the corresponding important reference have been added

We discuss semi-inclusive Deep Inelastic Scattering (DIS) in the z -> 1 limit, in particular the relationship between fracture functions, generalised cut vertices and Green functions of the composite operators arising in the OPE. The implications, in the spin-polarised case, for testing whether the "proton spin" effect is target-independent are explored. Explicit calculations in (phi^3)_6 theory are presented which are consistent with our observations., 22 pages, 25 figures, LaTeX 2e; uses graphics package

The one-dimensional Hubbard model at half-filling is studied in the framework of the Composite Operator Method using a static approximation. A solution characterized by strong antiferromagnetic correlations and a gap for any nonzero on-site interaction U is found. The corresponding ground-state energy, double occupancy and specific heat are in excellent agreement with those obtained within the Bethe Ansatz. These results show that the Composite Operator Method is an appropriate framework for the half-filled Hubbard chain and can be applied to evaluate properties, like the correlation functions, which cannot be obtained by means of the Bethe Ansatz, except for some limiting cases., 7 pages, 3 embedded Postscript figures, EuroTeX, submitted to EuroPhysics Letters

The composite operator effective potential is compared with the conventional Dyson-Schwinger method as a calculational tool for (2+1)-dimensional quantum electrodynamics. It is found that when the fermion propagator ansatz is put directly into the effective potential, it reproduces exactly the usual gap equations derived in the Dyson-Schwinger approach., 9 pages LATEX, axodraw.sty figures incorporated

We calculate the singular part of the sound attenuation and dispersion near the liquid-gas critical point on the critical isochore above T c . It is shown that the corresponding scaling function is related to the correlation function 2ψ2> of two composite operators of the Halperin-Hohenberg–Siggia model H.1 Using the renormalization group (RG) and ∊-expansion we obtain this scaling function in second order in ∊.

The finite temperature effective potential for a scalar field with Φ6 interaction is calculated by extending the CJT formalism for composite operators. It is found that unrenormalized terms appear in the effective potential due to the presence of an unrenormalized mass term. Nonzero turning points are obtained both for positive and negativeλ. High temperature expansion is performed and the results are analysed numerically. Graphical analysis indicates symmetry restoration whenT→0.

We analyse the one loop effective action of the Gribov-Zwanziger Lagrangian and use the local composite operator formalism to include the most general Becchi-Rouet-Stora-Tyutin (BRST) dimension two mass operator for the localizing ghost fields. We show that the energetically favourable colour channel corresponds to what is known as the R direction., Comment: 9 latex pages

AbdellatifMoudafiL.S.I.S, Aix Marseille Universit´e,U.F.R Sciences, Domaine Universitaire de Saint-J´erˆome.Avenue Escadrille Normandie-Niemen,13397 MARSEILLE CEDEX 20,abdellatif.moudafi@lsis.orgABSTRACTBased in a very recent paper by Micchelli et al. [8], we present an algorithmic approachfor computing the resolvent of composite operators: the composition of a monotoneoperator and a continuous linear mapping. The proposed algorithm can be used, forexample, for solving problems arising in image processing and traffic equilibrium. Fur-thermore, our algorithm gives an alternative to Dykstra-like method for evaluating theresolvent of the sum of two maximal monotone operators.RESUMENBasados en un art´iculo reciente de Micchelli et al. [8], presentamos una manera al-gor´itmica para calcular la resolvente de operadores compuestos: la composici´on de unoperador mono´tono y una aplicaci´on lineal continua. El algoritmo propuesto puede us-arse, por ejemplo, para resolver problemas que aparecen en procesamiento de ima´genesy equilibrio de transito. Adem´as, nuestro algoritmo entrega una alternativa a m´etodostipo Dykstra para evaluar al resolvente de la suma de dos operadores mono´tonos max-imales.Keywords and Phrases: 49J53, 65K10, 49M37, 90C25.2010 AMS Mathematics Subject Classification: maximalmonotoneoperators,KM-algorithm,Yosida regularization, Douglas-Rachford algorithm, Dykstra-like method.

International audience; We propose a variable metric forward-backward splitting algorithm and prove its convergence in real Hilbert spaces. We then use this framework to derive primal-dual splitting algorithms for solving various classes of monotone inclusions in duality. Some of these algorithms are new even when specialized to the fixed metric case. Various applications are discussed.

In consideration of the lowest dimensional condensate contributions to the free quark propagator, we obtain the nonperturbative quark propagator with the nonvanishing vacuum average value for quark composite operator [\(q) over bar q\]. Using the [\(q) over bar q\]-corrected quark propagator and basing on the simple quark-parton model, we discuss the nonperturbative effect in the nucleon structure function. it is shown that the nonperturbative effect modifies the conventional quark-parton model formula of the nucleon structure function at finite Q(2) and suggests a nontrivial Q(2)-dependence in the Gottfried sum.

We calculate a fully renormalizable effective action for local composite operators in the U(N) Gross-Neveu model, recently introduced by one of us, to two loop order. We obtain results for the massgap in one and two loops which are optimized using the Grunberg method of effective charges. At one loop the accuracy is of the order of 2% or less (except for N = 2). At two loops the accuracy is significantly improved for N ≥ 5.

Within the composite operator formalism we derive a formula for the pion decay constant $f_{\pi}$, as defined directly from the residue at the pion pole of the meson propagator, rather than from the matrix element of the axial current. The calculation is performed under some simplifying assumptions, and we verify the complete consistency with soft-pion results, in particular with the Adler-Dashen relation. The formula one obtains for (the pole-defined) $f_{\pi}^2$ differs from the previous Pagels-Stokar expression by an additive term, and it still provides $f_{\pi}^2$ in terms of the quark self-energy. We make some numerical estimates leading to $(30 \div 40)%$ deviation for $f_{\pi}^2$ with respect to the Pagels-Stokar formula., Comment: 10 pages, latex, no figures

It has been shown by several authors that a certain class of composite operators with many fields and gradients endangers the stability of nontrivial fixed points in 2+eps expansions for various models. This problem is so far unresolved. We investigate it in the N-vector model in an 1/N-expansion. By establishing an asymptotic naive addition law for anomalous dimensions we demonstrate that the first orders in the 2+eps expansion can lead to erroneous interpretations for high--gradient operators. While this makes us cautious against over--interpreting such expansions (either 2+eps or 1/N), the stability problem in the N-vector model persists also in first order in 1/N below three dimensions., Comment: 18 pages, 4 Postscript figures; revised version contains two additional references and "Note added in proof"

The method of the effective action for the composite operators $\Phi^2(x)$ and $\Phi^4(x)$ is applied to the termodynamics of the scalar quantum field with $\lambda\Phi^4$ interaction. An expansion of the finite temperature effective potential in powers of $\hbar$ provides successive approximations to the free energy with an effective mass and an effective coupling determined by the gap equations. The numerical results are studied in the space-time of one dimension, when the theory is equivalent to the quantum mechanics of an anharmonic oscillator. The approximations to the free energy show quick convergence to the exact result., Comment: 10 pages, plain Latex, 2 figures

We present new results on the problem of morphologically sampling binary images and image operators, thus enhancing the morphological sampling theory of Heijmans and Toet. These include sampling composite operators and approximating continuous translation invariant erosions, openings, and closings by discrete erosions, openings, and closings, respectively. The results presented in this paper have been already used in a theory of morphologically sampling random binary images modeled as random closed sets.

In order to expand the application of rough set operator, we analyzed two kinds of operator and the relationship between conversion structure based on the degree of rough set operator and variable precision rough set operator. By introducing a parameter, we give a Graded-Variable precision rough set model. And we get the Upper approximation operator and Lower approximation operator and some of its properties. This mod-el is feasible and effectiveness for dealing with fuzzy and uncertain knowledge in the information system. Experiments show that the model adds the concept of information understanding. It improved the accuracy of approximation classification and concept of accuracy in the wrong classification of the same degree.

We explore the gauge dependence of the effective average action within the functional renormalization group (FRG) approach. It is shown that in the framework of standard definitions of FRG for the Yang-Mills theory, the effective average action remains gauge-dependent on-shell, independent on the use of truncation scheme. Furthermore, we propose a new formulation of the FRG, based on the use of composite operators. In this case one can provide on-shell gauge-invariance for the effective average action and universality of $S$-matrix., Comment: 1+31 pages, no figures, the final version to appear in JHEP