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Until the late 15th century, knowledge of Ptolemy’s Almagest in the Latin West was constituted by Gerard of Cremona’s translation from Arabic into Latin. The text of Gerard’s translation has been examined carefully and its dependence on two different Arabic versions is well studied. However, the tables of Gerard’s Latin Almagest have not been scrutinized, and the relation to their Arabic or Greek counterparts has not been examined. In this article, I will analyze the historical mathematical structure of tables in Gerard’s Latin Almagest translated from the Arabic in comparison to their Arabic and Greek precursors. While Gerard’s text has proved to be a faithful translation from Arabic templates, some of the tables will turn out to be different. Fundamental tables for, for example, the chord interpolation values, declination, and rising times appear to have been recomputed in order to match Ptolemy’s proofs and paradigm computations, which, in contrast, generally diverge in both Greek and Arabic tradition with the tables. It remains unclear if Gerard himself or someone in his company recalculated these tables and thus deliberately aimed to correct the ancient classic of astronomy. By a systematic analysis of these tables, I intend to provide a novel perspective on the medieval transmission and translation of knowledge, its cross-cultural exchange, and especially the practice of Gerard of Cremona and his collaborator(s).

We introduce a method to obtain the analytic solution of the higher-order Baxter equation for twist-two and twist-three operators of planar N = 4 SYM. Our result proofs the conjectured formula for the threeloop anomalous dimension of twist-two operators. As such we derive the maximally transcendental part of the corresponding three-loop QCD result from the maximal supersymmetric gauge theory in four dimension purely by methods of integrability. © 2009 Elsevier B.V. All rights reserved.

We identify the gauge theory dual of a spinning string of minimal energy with spins S_1, S_2 on AdS_5 and charge J on S^5. For this purpose we focus on a certain set of local operators with two different types of covariant derivatives acting on complex scalar fields. We analyse the corresponding nested Bethe equations for the ground states in the limit of large spins. The auxiliary Bethe roots form certain string configurations in the complex plane, which enable us to derive integral equations for the leading and sub-leading contribution to the anomalous dimension. The results can be expressed through the observables of the sl(2) sub-sector, i.e. the cusp anomaly f(g) and the virtual scaling function B_L(g), rendering the strong-coupling analysis straightforward. Furthermore, we also study a particular sub-class of these operators specialising to a scaling limit with finite values of the second spin at weak and strong coupling., 23 pages, 3 figures, minor changes, references added

We write an integral equation that incorporates finite corrections to the large spin asymptotics of N=4 SYM twist operators from the non-linear integral equation. We show that these corrections are an all-loop result, not affected by wrapping effects, and agree, after determining the strong coupling expansion, with string theory predictions.

We present a formula for the five-loop anomalous dimension of N=4 SYM twist-three operators in the sl(2) sector. We obtain its asymptotic part from the Bethe Ansatz and finite volume corrections from the generalized Luescher formalism, considering scattering processes of spin chain magnons with virtual particles that travel along the cylinder. The complete result respects the expected large spin scaling properties and passes non-trivial tests including reciprocity constraints. We analyze the pole structure and find agreement with a conjectured resummation formula. In analogy with the twist-two anomalous dimension at four-loops, wrapping effects are of order log^2 M/M^2 for large values of the spin., 19 pages

The spectrum of anomalous dimensions of gauge-invariant operators in maximally supersymmetric Yang-Mills theory is believed to be described by a long-range integrable spin chain model. We focus in this study on its $sl(2)$ subsector spanned by the twist-two single-trace Wilson operators, which are shared by all gauge theories, supersymmetric or not. We develop a formalism for the solution of the perturbative multiloop Baxter equation encoding their anomalous dimensions, using Wilson polynomials as basis functions and Mellin transform technique. These considerations yield compact results which allow analytical calculations of multiloop anomalous dimensions bypassing the use of the principle of maximal transcendentality. As an application of our method we analytically confirm the known four-loop result. We also determine the dressing part of the five-loop anomalous dimensions., Comment: 44 pages, v2: references added and typos fixed

We compute the anomalous dimensions of field strength operators Tr F^L in N=4 SYM from an asymptotic nested Bethe ansatz to all-loop order. Starting from the exact solution of the one-loop problem at arbitrary L, we derive a single effective integral equation for the thermodynamic limit of these dimensions. We also include the recently proposed phase factor for the S-matrix of the planar AdS/CFT system. The terms in the effective equation corresponding to, respectively, the nesting and the dressing are structurally very similar. This hints at the physical origin of the dressing phase, which we conjecture to arise from the hidden presence of infinitely many auxiliary Bethe roots describing a non-trivial "filled" structure of the theory's BPS vacuum. We finally show that the mechanism for creating effective nesting/dressing kernels is quite generic by also deriving the integral equation for the all-loop dimension of a certain one-loop so(6) singlet state., Comment: 38 pages, 2 figures. v2: References and appendix discussing the emulation of the dressing phase added