Recent zbMATH articles in MSC 30Ehttps://www.zbmath.org/atom/cc/30E2022-05-16T20:40:13.078697ZWerkzeugThe discrete case of the mixed joint universality for a class of certain partial zeta-functionshttps://www.zbmath.org/1483.111912022-05-16T20:40:13.078697Z"Kačinskaitė, Roma"https://www.zbmath.org/authors/?q=ai:kacinskaite.roma"Matsumoto, Kohji"https://www.zbmath.org/authors/?q=ai:matsumoto.kohjiAuthors' abstract: We give a new type of mixed discrete joint universality properties, which is satisfied by a wide class of zeta-functions. We study the universality for a certain modification of Matsumoto zeta-functions \(\varphi_h(s)\) and a collection of periodic Hurwitz zeta-functions \(\zeta (s;\alpha;\mathfrak B)\) under the condition that the common difference of arithmetical progression \(h > 0\) is such that \(\exp \{ \frac{2\pi}h\}\) is a rational number and parameter \(\alpha\) is a transcendental number.
Reviewer: Anatoly N. Kochubei (Kyïv)Nonlinear conditions for ultradifferentiabilityhttps://www.zbmath.org/1483.260232022-05-16T20:40:13.078697Z"Nenning, David Nicolas"https://www.zbmath.org/authors/?q=ai:nenning.david-nicolas"Rainer, Armin"https://www.zbmath.org/authors/?q=ai:rainer.armin"Schindl, Gerhard"https://www.zbmath.org/authors/?q=ai:schindl.gerhardSummary: A remarkable theorem of Joris states that a function \(f\) is \(C^{\infty}\) if two relatively prime powers of \(f\) are \(C^{\infty}\). Recently, Thilliez showed that an analogous theorem holds in Denjoy-Carleman classes of Roumieu type. We prove that a division property, equivalent to Joris's result, is valid in a wide variety of ultradifferentiable classes. Generally speaking, it holds in all dimensions for non-quasianalytic classes. In the quasianalytic case we have general validity in dimension one, but we also get validity in all dimensions for certain quasianalytic classes.Analog of the Hadamard theorem and related extremal problems on the class of analytic functionshttps://www.zbmath.org/1483.300012022-05-16T20:40:13.078697Z"Akopyan, R. R."https://www.zbmath.org/authors/?q=ai:akopyan.roman-razmikovichSummary: We study several related extremal problems for analytic functions in a finitely connected domain \(G\) with rectifiable Jordan boundary \(\Gamma \). A sharp inequality is established between values of a function analytic in \(G\) and weighted means of its boundary values on two measurable subsets \(\gamma_1\) and \(\gamma_0=\Gamma\setminus\gamma_1\) of the boundary:
\[ |f(z_0)|\leq\mathcal{C}\,\|f\|^{\alpha}_{L^q_{\varphi_1}(\gamma_1)}\, \|f\|^{\beta}_{L^p_{\varphi_0}(\gamma_0)},\quad z_0\in G,\quad 0<q,p\leq\infty.\]
The inequality is an analog of Hadamard's three-circle theorem and the Nevanlinna brothers' two-constant theorem. In the case of a doubly connected domain \(G\) and \(1\leq q,p\leq\infty \), we study the cases where the inequality provides the value of the modulus of continuity for a functional of analytic extension of a function from the part \(\gamma_1\) of the boundary to a given point of the domain. In these cases, the corresponding problem of optimal recovery of a function from its approximate boundary values on \(\gamma_1\) and the problem of the best approximation of a functional by bounded linear functionals are solved. The case of a simply connected domain \(G\) has been completely investigated previously.Weighted Chebyshev polynomials on compact subsets of the complex planehttps://www.zbmath.org/1483.300232022-05-16T20:40:13.078697Z"Novello, Galen"https://www.zbmath.org/authors/?q=ai:novello.galen"Schiefermayr, Klaus"https://www.zbmath.org/authors/?q=ai:schiefermayr.klaus"Zinchenko, Maxim"https://www.zbmath.org/authors/?q=ai:zinchenko.maximSummary: We study weighted Chebyshev polynomials on compact subsets of the complex plane with respect to a bounded weight function. We establish existence and uniqueness of weighted Chebyshev polynomials and derive weighted analogs of Kolmogorov's criterion, the alternation theorem, and a characterization due to Rivlin and Shapiro. We derive invariance of the Widom factors of weighted Chebyshev polynomials under polynomial pre-images and a comparison result for the norms of Chebyshev polynomials corresponding to different weights. Finally, we obtain a lower bound for the Widom factors in terms of the Szegő integral of the weight function and discuss its sharpness.
For the entire collection see [Zbl 1479.47003].Approximate properties of the \(p\)-Bieberbach polynomials in regions with simultaneously exterior and interior zero angleshttps://www.zbmath.org/1483.300312022-05-16T20:40:13.078697Z"Abdullayev, F. G."https://www.zbmath.org/authors/?q=ai:abdullayev.fahreddin-g"Imashkyzy, M."https://www.zbmath.org/authors/?q=ai:imashkyzy.meerim"Özkartepe, P."https://www.zbmath.org/authors/?q=ai:ozkartepe.naciye-pelinSummary: In this paper, we study the uniform convergence of \(p\)-Bieberbach polynomials in regions with a finite number of both interior and exterior zero angles at the boundary.Approximation of functions and all derivatives on compact setshttps://www.zbmath.org/1483.300682022-05-16T20:40:13.078697Z"Armeniakos, Sotiris"https://www.zbmath.org/authors/?q=ai:armeniakos.sotiris"Kotsovolis, Giorgos"https://www.zbmath.org/authors/?q=ai:kotsovolis.giorgos"Nestoridis, Vassili"https://www.zbmath.org/authors/?q=ai:nestoridis.vassiliSummary: In Mergelyan type approximation we uniformly approximate functions on compact sets \(K\) by polynomials or rational functions or holomorphic functions on varying open sets containing \(K\). In the present paper we consider analogous approximation, where uniform convergence on \(K\) is replaced by uniform approximation on \(K\) of all order derivatives.Induced fields in isolated elliptical inhomogeneities due to imposed polynomial fields at infinityhttps://www.zbmath.org/1483.300692022-05-16T20:40:13.078697Z"Calvo-Jurado, Carmen"https://www.zbmath.org/authors/?q=ai:calvo-jurado.carmen"Parnell, William J."https://www.zbmath.org/authors/?q=ai:parnell.william-jSummary: The Eshelby inhomogeneity problem plays a crucial role in the micromechanical analysis of the effective mechanical behaviour of inhomogeneous media since it provides a mechanism to predict interior fields associated with ellipsoidal inhomogeneities. In the context of linear elasticity, Eshelby showed that given an isolated elliptical (two dimensions) or ellipsoidal (three dimensions) inhomogeneity embedded in a homogeneous material of infinite extent, then for any uniform strain or traction imposed in the far field, the induced strain inside the inhomogeneity is also uniform. In the case of non-uniform far-field conditions, Eshelby showed that if the loading is a polynomial of order \(n\), the associated interior field is characterized by a polynomial of the same order. This is often called `Eshelby's polynomial conservation theorem'. Since then, the problem has been studied by many, but in most cases for the uniform loading scenario, i.e. when strains or tractions in the far field are uniform. However, in many applications, e.g. permittivity, conductivity, elasticity, etc., the case of non-uniform conditions is also of interest and furthermore, methods to deal with non-elliptical and non-ellipsoidal inhomogeneities are required. In this work, for prescribed non-uniform polynomial far-field conditions, we introduce a method to approximate interior fields for isolated inhomogeneities of elliptical shape. This subproblem is relevant for approximating effective properties of numerous composites since constituent inhomogeneities are often of this form, or limiting forms, e.g. layered and fibre reinforced composites. We verify that the obtained results agree with the polynomial conservation property and with results determined using conformal mappings or the classical circle inclusion theorem. We close with a discussion of how the method can be straightforwardly extended to the case of non-elliptical inhomogeneities.Regularization of a class of summary equationshttps://www.zbmath.org/1483.300702022-05-16T20:40:13.078697Z"Garif'yanov, F. N."https://www.zbmath.org/authors/?q=ai:garifyanov.farkhat-nurgayazovich"Strezhneva, E. V."https://www.zbmath.org/authors/?q=ai:strezhneva.elena-vasilevnaSummary: Let \(D\) be an arbitrary quadrangle with boundary \(\Gamma \). We consider a four-element linear summary equation. The solution is sought in the class of functions which are holomorphic outside \(D\) and vanish at infinity. The boundary values satisfy the Hölder condition on any compact set which does not contain the vertices. At the vertices, singularities at most of logarithmic order are allowed. The coefficients of the equation are holomorphic in \(D\) and their boundary values satisfy the Hölder condition on \(\Gamma \). The free term satisfies the same conditions. The solution is sought in the form of the Cauchy type integral over \(\Gamma\) with unknown density. To regularize the obtained functional equation, we use the Carleman problem. Previously, a Carleman shift is introduced on \(\Gamma \); it transfers each side to itself and reverses orientation; the midpoints of the sides are fixed under the shift. We indicate some applications of this summary equation to the problem of moments for entire functions of exponential type.The Dirichlet-Neumann boundary value problem for the inhomogeneous Bitsadze equation in a ring domainhttps://www.zbmath.org/1483.300712022-05-16T20:40:13.078697Z"Gençtürk, İlker"https://www.zbmath.org/authors/?q=ai:gencturk.ilkerSummary: In this study, by using some integral representations formulas, we study solvability conditions and explicit solution of the Dirichlet-Neumann problem, an example for a combined boundary value problem, for the Bitsadze equation in a ring domain.Polyanalytic boundary value problems for planar domains with harmonic Green functionhttps://www.zbmath.org/1483.300722022-05-16T20:40:13.078697Z"Begehr, Heinrich"https://www.zbmath.org/authors/?q=ai:begehr.heinrich"Shupeyeva, Bibinur"https://www.zbmath.org/authors/?q=ai:shupeyeva.bibinurThe authors characterize the solvability of three boundary value problems for the inhomogeneous polyanalytic equation in planar domains (having a harmonic Green function), namely the well-posed iterated Schwarz problem, and two over-determined iterated problems of Dirichlet and Neumann type. Solutions formulas are also obtained and, in particular, it is concluded that the polyanalytic Cauchy-Pompeiu representation formula provides the solution to the Dirichlet problem (for any degree \(n\), and in the cases for which the solution exists).
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)Asymptotics of the Riemann-Hilbert problem for the Somov model of magnetic reconnection of long shock waveshttps://www.zbmath.org/1483.300732022-05-16T20:40:13.078697Z"Bezrodnykh, S. I."https://www.zbmath.org/authors/?q=ai:bezrodnykh.sergei-i"Vlasov, V. I."https://www.zbmath.org/authors/?q=ai:vlasov.vladimir-ivanovichSummary: We consider the Riemann-Hilbert problem in a domain of complicated shape (the exterior of a system of cuts), with the condition of growth of the solution at infinity. Such a problem arises in the Somov model of the effect of magnetic reconnection in the physics of plasma, and its solution has the physical meaning of a magnetic field. The asymptotics of the solution is obtained for the case of infinite extension of four cuts from the given system, which have the meaning of shock waves, so that the original domain splits into four disconnected components in the limit. It is shown that if the coefficient in the condition of growth of the magnetic field at infinity consistently decreases in this case, then this field basically coincides in the limit with the field arising in the Petschek model of the effect of magnetic reconnection.Noether property and approximate solution of the Riemann boundary value problem on closed curveshttps://www.zbmath.org/1483.300742022-05-16T20:40:13.078697Z"Bory-Reyes, Juan"https://www.zbmath.org/authors/?q=ai:bory-reyes.juan|moreno-garcia.tania"Katz, David"https://www.zbmath.org/authors/?q=ai:katz.david-f|katz.david-bBased on classic methods, the authors analyse the Noether property of a Riemann boundary value problem in the Banach algebra of continuous functions over closed curves, as well as consequent approximate solutions (by using quasi-Fredholm operators).
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)On the splitting type of holomorphic vector bundles induced from regular systems of differential equationhttps://www.zbmath.org/1483.300752022-05-16T20:40:13.078697Z"Giorgadze, Grigori"https://www.zbmath.org/authors/?q=ai:giorgadze.grigory"Gulagashvili, Gega"https://www.zbmath.org/authors/?q=ai:gulagashvili.gegaSummary: We calculate the splitting type of holomorphic vector bundles on the Riemann sphere induced by a Fuchsian system of differential equations. Using this technique, we indicate the relationship between Hölder continuous matrix functions and a moduli space of vector bundles on the Riemann sphere. For second order systems with three singular points we give a complete characterization of the corresponding vector bundles by the invariants of Fuchsian system.On the number of linearly independent solutions of the Riemann boundary value problem on the Riemann surface of an algebraic functionhttps://www.zbmath.org/1483.300762022-05-16T20:40:13.078697Z"Kruglov, V. E."https://www.zbmath.org/authors/?q=ai:kruglov.vladislav-e|kruglov.viktor-eSummary: We suggest a modified solution to the Riemann boundary value problem on a Riemann surface of an algebraic function of genus \(\rho \). This allows us to to reduce the problem of finding the number \(l\) of linearly independent algebraic functions (LIAF), that are multiples of a fractional divisor \(Q\), to finding the number of LIAF that are multiples of an effective divisor \(J \) (\(\operatorname{ord}J = \rho\)); this provides a solution to the Jacobi inversion problem given in this paper. We study the case, where the exponents of the normal basis elements coincide, and solve the problem of finding the number of LIAF, multiples of an effective divisor. The definitions of conjugate points of Riemann surface and hyperorder of an effective divisor are introduced. Depending on the structure of divisor \(J\), exact formulas are obtained for number \(l\); they are expressed in terms of the order of divisor \(Q\), the hyperorder of divisor \(J\), and numbers \(\rho\) and \(n\), where \(n\) is the number of sheets of the algebraic Riemann surface.Free boundary problems in the spirit of Sakai's theoremhttps://www.zbmath.org/1483.300772022-05-16T20:40:13.078697Z"Vardakis, Dimitris"https://www.zbmath.org/authors/?q=ai:vardakis.dimitris"Volberg, Alexander"https://www.zbmath.org/authors/?q=ai:volberg.alexander-lSummary: A Schwarz function on an open domain \(\Omega\) is a holomorphic function satisfying \(S(\zeta)=\bar{\zeta}\) on \(\Gamma\), which is part of the boundary of \(\Omega\). \textit{M. Sakai} [Acta Math. 166, No. 3-4, 263--297 (1991; Zbl 0728.30007)] gave a complete characterization of the boundary of a domain admitting a Schwarz function. In fact, if \(\Omega\) is simply connected and \(\Gamma =\partial\Omega\cap D(\zeta_0,r)\), then \(\Gamma\) has to be regular real analytic (with possible cusps). Sakai's result has natural applications to 1) quadrature domains, 2) free boundary problem for \(\Delta u=1\) equation. In our scenarios \(\Gamma\) can be, respectively, from real-analytic to just \(C^\infty\), regular except for a harmonic-measure-zero set, or regular except finitely many points.Bounded extremal problems in Bergman and Bergman-Vekua spaceshttps://www.zbmath.org/1483.300892022-05-16T20:40:13.078697Z"Delgado, Briceyda B."https://www.zbmath.org/authors/?q=ai:delgado.briceyda-b"Leblond, Juliette"https://www.zbmath.org/authors/?q=ai:leblond.julietteSummary: We analyze Bergman spaces \(A_f^p(\mathbb{D})\) of generalized analytic functions of solutions to the Vekua equation \(\bar{\partial}w = (\bar{\partial}f/f)\bar{w}\) in the unit disc of the complex plane, for Lipschitz-smooth non-vanishing real valued functions \(f\) and \(1<p<\infty\). We consider a family of bounded extremal problems (best constrained approximation) in the Bergman space \(A^p(\mathbb{D})\) and in its generalized version \(A^p_f(\mathbb{D})\), that consists in approximating a function in subsets of \(\mathbb{D}\) by the restriction of a function belonging to \(A^p(\mathbb{D})\) or \(A^p_f(\mathbb{D})\) subject to a norm constraint. Preliminary constructive results are provided for \(p = 2\).On the investigation of isotropic thick-walled shellshttps://www.zbmath.org/1483.300902022-05-16T20:40:13.078697Z"Khvoles, A."https://www.zbmath.org/authors/?q=ai:khvoles.a-r|khvoles.alexander|khvoles.a-a"Zgenti, V."https://www.zbmath.org/authors/?q=ai:zgenti.v"Vashakmadze, T."https://www.zbmath.org/authors/?q=ai:vashakmadze.tamaz-s|vashakmadze.tamara-sSummary: We consider the problems of creating 2-dim models for thin-walled structures and satisfaction of boundary conditions when the generalized stress vector is given on the surfaces for elastic plates and shells. This problem was open also both for refined theories in the wide sense and hierarchical type models.A note on the phase retrieval of holomorphic functionshttps://www.zbmath.org/1483.300972022-05-16T20:40:13.078697Z"Perez, Rolando III"https://www.zbmath.org/authors/?q=ai:perez.rolando-iiiSummary: We prove that if \(f\) and \(g\) are holomorphic functions on an open connected domain, with the same moduli on two intersecting segments, then \(f=g\) up to the multiplication of a unimodular constant, provided the segments make an angle that is an irrational multiple of \(\pi\). We also prove that if \(f\) and \(g\) are functions in the Nevanlinna class, and if \(|f|=|g|\) on the unit circle and on a circle inside the unit disc, then \(f=g\) up to the multiplication of a unimodular constant.Complete interpolating sequences for small Fock spaceshttps://www.zbmath.org/1483.300992022-05-16T20:40:13.078697Z"Omari, Youssef"https://www.zbmath.org/authors/?q=ai:omari.youssefSummary: We give a characterization of complete interpolating sequences for the Fock spaces \(\mathcal{F}_\varphi^p\), \(1\leq p< \infty\), where \(\varphi(z)=\alpha(\log^+|z|)^2\), \(\alpha > 0\). Our results are analogous to the classical Kadets-Ingham's 1/4-Theorem on perturbation of Riesz bases of complex exponentials, and they answer a question asked by \textit{A. Baranov} et al. [J. Math. Pures Appl. (9) 103, No. 6, 1358--1389 (2015; Zbl 1315.30009)].Completely monotone sequences and harmonic mappingshttps://www.zbmath.org/1483.310052022-05-16T20:40:13.078697Z"Long, Bo-Yong"https://www.zbmath.org/authors/?q=ai:long.boyong"Sugawa, Toshiyuki"https://www.zbmath.org/authors/?q=ai:sugawa.toshiyuki"Wang, Qi-Han"https://www.zbmath.org/authors/?q=ai:wang.qihanSummary: In the present paper, we will study geometric properties of harmonic mappings whose analytic and co-analytic parts are (shifted) generated functions of completely monotone sequences.Necessary density conditions for \(d\)-approximate interpolation sequences in the Bargmann-Fock spacehttps://www.zbmath.org/1483.320042022-05-16T20:40:13.078697Z"Li, Haodong"https://www.zbmath.org/authors/?q=ai:li.haodong"Mitkovski, Mishko"https://www.zbmath.org/authors/?q=ai:mitkovski.mishkoSummary: Inspired by \textit{A. Olevskiĭ} and \textit{A. Ulanovskiĭ} [St. Petersbg. Math. J. 21, No. 6, 1015--1025 (2010); translation from Algebra Anal. 21, No. 6, 227--240 (2009; Zbl 1207.30038)], we introduce the concept of \(d\)-approximate interpolation in weighted Bargmann-Fock spaces as a natural extension of the classical concept of interpolation. We then show that d-approximate interpolation sets satisfy a density condition, similar to the one that classical interpolation sets satisfy. More precisely, we show that the upper Beurling density of any \(d\)-approximate interpolation set must be bounded from above by \(1/(1-d^2)\).Escaping Fatou components of transcendental self-maps of the punctured planehttps://www.zbmath.org/1483.370542022-05-16T20:40:13.078697Z"Martí-Pete, David"https://www.zbmath.org/authors/?q=ai:marti-pete.davidHolomorphic self-maps of the Riemann sphere are the most well-studied families of systems in complex dynamics (rational dynamics), followed by self-maps of the once punctured plane (transcendental dynamics). This paper concerns the study of holomorphic self-maps of the twice punctured Riemann sphere, a subject still in its early days.
In transcendental dynamics, an important role is played by the escaping set, which consists of those points which iterate to the essential singularity of the function. In the present setting, however, there are two essential singularities. The main result is that any possible way of escaping is possible for a wandering Fatou component. The author's main technique is approximation theory.
Reviewer: Kirill Lazebnik (New Haven)Complex best \(r\)-term approximations almost always exist in finite dimensionshttps://www.zbmath.org/1483.410102022-05-16T20:40:13.078697Z"Qi, Yang"https://www.zbmath.org/authors/?q=ai:qi.yang"Michałek, Mateusz"https://www.zbmath.org/authors/?q=ai:michalek.mateusz"Lim, Lek-Heng"https://www.zbmath.org/authors/?q=ai:lim.lek-hengSummary: We show that in finite-dimensional nonlinear approximations, the best \(r\)-term approximant of a function \(f\) almost always exists over \(\mathbb{C}\) but that the same is not true over \(\mathbb{R}\), i.e., the infimum \(\inf_{f_1, \dots, f_r \in D} \| f - f_1 - \dots - f_r \|\) is almost always attainable by complex-valued functions \(f_1, \ldots, f_r\) in \(D\), a set (dictionary) of functions (atoms) with some desired structures. Our result extends to functions that possess properties like symmetry or skew-symmetry under permutations of arguments. When \(D\) is the set of separable functions, this is the best rank-\(r\) tensor approximation problem. We show that over \(\mathbb{C}\), any tensor almost always has a unique best rank-\(r\) approximation. This extends to other notions of ranks such as symmetric and alternating ranks, to best \(r\)-block-terms approximations, and to best approximations by tensor networks. Applied to sparse-plus-low-rank approximations, we obtain that for any given \(r\) and \(k\), a general tensor has a unique best approximation by a sum of a rank-\(r\) tensor and a \(k\)-sparse tensor with a fixed sparsity pattern; a problem arising in covariance estimation of Gaussian model with \(k\) observed variables conditionally independent given \(r\) hidden variables. The existential (but not uniqueness) part of our result also applies to best approximations by a sum of a rank-\(r\) tensor and a \(k\)-sparse tensor with no fixed sparsity pattern, and to tensor completion problems.A novel approach to the computation of one-loop three- and four-point functions. I: The real mass casehttps://www.zbmath.org/1483.811122022-05-16T20:40:13.078697Z"Guillet, J. Ph"https://www.zbmath.org/authors/?q=ai:guillet.j-ph"Pilon, E."https://www.zbmath.org/authors/?q=ai:pilon.eric"Shimizu, Y."https://www.zbmath.org/authors/?q=ai:shimizu.yuji|shimizu.yasuhiro|shimizu.yasushi|shimizu.yuuko|shimizu.yoshiaki|shimizu.yuya|shimizu.yusuke|shimizu.yoshifumi-r|shimizu.yuichi|shimizu.yukiko-s|shimizu.yuma|shimizu.yoshinori|shimizu.yuuki|shimizu.yasutaka|shimizu.yoshimasa|shimizu.yoshimitsu|shimizu.yosuke|shimizu.yasuyuki|shimizu.yoshiyuki|shimizu.youichiro|shimizu.yuki"Zidi, M. S."https://www.zbmath.org/authors/?q=ai:zidi.m-sThe article presents a novel method for the computation of certain correlation functions in perturbative quantum field theory, namely the three and four point functions evaluated at one loop. It is the first part of a planned series of three articles.
The computation of Feynman diagrams is the most important theoretical tool in the search for new phenomena in high energy physics. At the high precision frontier theoretical computations are matched with experimental results. The continuous increase of experimental accuracy demands an equal development of theoretical methods. While numerous computations can be done numerically, there are structural limitations in computing power. Hence the need of analytical results, even partial.
The series of articles intends to present a method to evaluate certain generalized one-loop type N-point Feynman-type integrals, which enter as building blocks of more complicated amplitudes. The present paper focuses on three and four point functions, where the integrals have the form of a certain rational function integrated over a certain simplex-like domain. In particular the authors present their methods using as a proof of concept the one loop amplitudes in a scalar theory. The method efficiently trivializes the integrals over Feynman parameters as boundary terms, using a Stokes type identity, and obtains all the necessary analytic continuations in a systematic fashion. The paper consider the case with real internal masses, postponing the cases of complex or zero masses to two subsequent articles.
This method is illustrated in detail during the course of the paper, with technical proofs and results explained in 6 appendices. The paper is very clearly written and easy to read, and contains several interesting manipulations of Feynman integrals. The results and the methods explained are likely to prove quite useful in the evaluation of higher loop amplitudes. The paper is mostly aimed to expert in the field and assumes that the reader is already familiar with the computations of loop amplitudes.
Reviewer: Michele Cirafici (Trieste)