Analytic tadpole coefficients of one-loop integrals

One remaining problem of unitarity cut method for one-loop integral reduction is that tadpole coefficients can not be straightforward obtained through this way. In this paper, we reconsider the problem by applying differential operators over an auxiliary vector R. Using differential operators, we establish the corresponding differential equations for tadpole coefficients at the first step. Then using the tensor structure of tadpole coefficients, we transform the differential equations to the recurrence relations for undetermined tensor coefficients. These recurrence relations can be solved easily by iteration and we can obtain analytic expressions of tadpole coefficients for arbitrary one-loop integrals.

The Wilson-loop d log representation for Feynman integrals

We introduce and study a so-called Wilson-loop d log representation of certain Feynman integrals for scattering amplitudes in $$ \mathcal{N} $$ N = 4 SYM and beyond, which makes their evaluation completely straightforward. Such a representation was motivated by the dual Wilson loop picture, and it can also be derived by partial Feynman parametrization of loop integrals. We first introduce it for the simplest one-loop examples, the chiral pentagon in four dimensions and the three-mass-easy hexagon in six dimensions, which are represented by two- and three-fold d log integrals that are nicely related to each other. For multi-loop examples, we write the L-loop generalized penta-ladders as 2(L − 1)-fold d log integrals of some one-loop integral, so that once the latter is known, the integration can be performed in a systematic way. In particular, we write the eight-point penta-ladder as a 2L-fold d log integral whose symbol can be computed without performing any integration; we also obtain the last entries and the symbol alphabet of these integrals. Similarly we study the symbol of the seven-point double-penta-ladder, which is represented by a 2(L − 1)-fold integral of a hexagon; the latter can be written as a two-fold d log integral plus a boundary term. We comment on the relation of our representation to differential equations and resumming the ladders by solving certain integral equations.

Robust tracking control of an underwater vehicle and manipulator system based on double closed-loop integral sliding mode

A nonlinear robust control method for the trajectory tracking of the underwater vehicle and manipulator system that operates in the presence of external current disturbances is proposed using double closed-loop integral sliding mode control. The designed controller uses a double closed-loop control structure to track the desired trajectory in the joint space of the underwater vehicle and manipulator system, and its inner and outer loop systems use integral sliding surface to enhance the robustness of the whole system. Then, the continuous switching mode based on hyperbolic tangent function is used instead of the traditional discontinuous switching mode to reduce the chattering of the control input of the underwater vehicle and manipulator system. In addition, the control method proposed in this article does not need to estimate the uncertainties of the underwater vehicle and manipulator system control system through online identification, but also can ensure the robustness of the underwater vehicle and manipulator system motion control in underwater environment. Therefore, it is easier to be implemented on the embedded platform of the underwater vehicle and manipulator system and applied to the actual marine operation tasks. At last, the stability of the control system is proved by the Lyapunov theory, and its effectiveness and feasibility are verified by the simulation experiments in MATLAB software.