Uniswap-and other constant product markets-appear to work well in practice despite their simplicity. In this paper, we give a simple formal analysis of constant product markets and their generalizations, showing that, under some common conditions, these markets must closely track the reference market price. We also show that Uniswap satisfies many other desirable properties and numerically demonstrate, via a large-scale agent-based simulation, that Uniswap is stable under a wide range of market conditions.

We derive and demonstrate a new capability for snake robots in which two behaviors-one for locomotion and the other for manipulation-are executed simultaneously on the same robot. This is done in two steps: 1) inverse kinematics via numerical optimization and 2) gait-based locomotion via modal decomposition. The result is an analytical representation of a multiple mode behavior that reduces online execution to simple parameterized control. This representation makes it possible to derive a feedback control law that enables reliable visual servoing using a snake robot while climbing a pole.

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