In this issue of Neuron, Wang et al. and Machold and Fishell present contrasting molecular fate maps of Math1, which redefine the derivatives of the embryonic rhombic lip and offer a conceptual overhaul of cerebellar and precerebellar development. These fate maps identify a common developmental thread linking diverse, functionally associated neurons and reveal an exquisite temporal organization...

It is shown that Witten's star product in string field theory, defined as the overlap of half strings, is equivalent to the Moyal star product involving the relativistic phase space of even string modes. The string field ψ A (x µ [σ]) can be rewritten as a phase space field of the even modes A (x µ 2n , x 0 , p µ 2n) , where x µ 2n are the positions of the even string modes, and p µ 2n are rela...

‎Let $R$ be a ring‎ ‎with the Jacobson radical $J(R)$ and let $picolon Rto R/J(R)$ be‎ the canonical map‎. ‎Then $pi$ induces an order preserving group homomorphism‎ ‎$K_0picolon K_0(R)to K_0(R/J(R))$ and an‎ ‎affine continuous map $S(K_0pi)$ between the state space $St(R/J(R))$ and the‎ ‎state space $St(R).$‎ ‎In this paper‎, ‎we consider the natural affine map $S(K_0pi).$ We give a condition ...

Let f be a continuous self-map of a smooth compact connected and simply-connected manifold of dimension m ≥ 3, r a fixed natural number. A topological invariant Dm r [f ], introduced in [5], is equal to the minimal number of r-periodic points for all smooth maps homotopic to f . In this paper we calculate D3 r [f ] for all self-maps of S3.