Integrability in general tetrad formalisms is reviewed, following and clarifying work of Papapetrou and Edgar. The integrability conditions are (combinations of) the Bianchi equations and their consequences. The introduction of additional constraints is considered. Recent results on the conservation of constraints in the 1+3 covariant formulation of cosmology are shown to follow from the Bianch...

We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Liénard equations and equations related with special functions such as Hypergeometric and Heun ones. We also study the Poincar...

the objective of this article is to derive the necessary optimality conditions, known as pontryagin's minimum principle, for fuzzy optimal control problems based on the concepts of differentiability and integrability of a fuzzy mapping that may be parameterized by the left and right-hand functions of its $alpha$-level sets.

Weak integrability of the central foliation 56 6. Intermediate Foliations 58 6.1. Non-integrability of intermediate distributions 58 6.2. Invariant families of local manifolds 59 6.3. Insufficient smoothness of intermediate foliations 64 7. Absolute Continuity 69 7.1. The holonomy map 69 7.2. Absolute continuity of local manifolds 75

Worldsheet techniques can be used to argue for the integrability of string theory on AdS 5 × S 5 /Z S , which is dual to the strongly coupled Z S-orbifold of N = 4 SYM. We analyze the integrability of these field theories in the perturbative regime and construct the relevant Bethe equations.

Quantum integrability of classical integrable systems given by quadratic Killing tensors on curved configuration spaces is investigated. It is proven that, using a “minimal” quantization scheme, quantum integrability is insured for a large class of classic examples. Preprint: CPT-2004/P.120 and LPTHE-04-33

The text contains introduction and preliminary definitions and results to my talk on category theory description of supersymmetries and integrability in string theory. In the talk I plan to present homological and homotopical algebra framework for Calabi-Yau supermanifolds and stacks in open and closed string theory. In the framework we investigate supersymmetries and integrability.

This is the second part of integrability analysis of cosmological models with scalar fields. Here, we study systems with conformal coupling, and show that apart from four cases, where explicit first integrals are known, the generic system is not integrable. We also comment on some chaotic properties of the system, and the issues of integrability restricted to the real domain.

The geometric theory of Lie systems is used to establish integrability conditions for several systems of differential equations, in particular some Riccati equations and Ermakov systems. Many different integrability criteria in the literature will be analysed from this new perspective, and some applications in physics will be given.

The generalized symmetry method is applied to a class of completely discrete equations including the Adler-Bobenko-Suris list. Assuming the existence of a generalized symmetry, we derive a few integrability conditions suitable for testing and classifying equations of this class. Those conditions are used at the end to test for integrability discretizations of some well-known hyperbolic equations.