Innovations in horse equipment during the early Middle Ages provided advantages to societies from the steppes, reshaping the social landscape of Eurasia. Comparatively little is known about the precise origin of these crucial advances, although the available evidence points to early adoption in East Asia. The authors present new archaeological discoveries from western and northern Mongolia, dating to the fourth and fifth centuries AD, including a wooden frame saddle with horse hide components from Urd Ulaan Uneet and an iron stirrup from Khukh Nuur. Together, these finds suggest that Mongolian groups were early adopters of stirrups and saddles, facilitating the expansion of nomadic hegemony across Eurasia and shaping the conduct of medieval mounted warfare.

We settle the question of how to compute the entry and leaving arcs for turnpikes in autonomous variational problems, in the one-dimensional case using the phase space of the vector field associated with the Euler equation, and the initial/final and/or the transversality condition. The results hinge on the realization that extremals are the contours of a well-known function and that the transversality condition is (generically) a curve. An approximation algorithm is presented, and an example is included for completeness.

We give a new characterisation of integrability of a planar vector field at the origin. This allows us to prove that the analytic systems where h, K, Ψ and ξ are analytic functions defined in the neighbourhood of O with K(O) ≠ 0 or Ψ(O) ≠ 0 and n ≥ 1, have a local analytic first integral at the origin. We show new families of analytically integrable systems that are held in the above class. In particular, this class includes all the nilpotent and generalised nilpotent integrable centres that we know.