A$generalized polynomial$is a real-valued function which is obtained from conventional polynomials by the use of the operations of addition, multiplication, and taking the integer part; a$generalized polynomial mapping$is a vector-valued mapping whose coordinates are generalized polynomials. We show that any bounded generalized polynomial mapping$u$:$Z$^{$d$}→$R$^{$l$}has a representation$u$($n$) =$f$($ϕ$($n$)$x$),$n$∈$Z$^{$d$}, where$f$is a piecewise polynomial function on a compact nilmanifold$X$,$x$∈$X$, and$ϕ$is an ergodic$Z$^{$d$}-action by translations on$X$. This fact is used to show that the sequence$u$($n$),$n$∈$Z$^{$d$}, is well distributed on a piecewise polynomial surface $\mathcal{S}\subset\mathbf{R}^{l}$ (with respect to the Borel measure on $\mathcal{S}$ that is the image of the Lebesgue measure under the piecewise polynomial function defining $\mathcal{S}$ ). As corollaries we also obtain a von Neumann-type ergodic theorem along generalized polynomials and a result on Diophantine approximations extending the work of van der Corput and of Furstenberg–Weiss.

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In this paper we study segmented strings in AdS3 coupled to a background two-form whose field strength is proportional to the volume form. By changing the coupling, the theory interpolates between the Nambu-Goto string and the SL(2) Wess-Zumino-Witten model. In terms of the kink momentum vectors, the action is independent of the coupling and the classical theory reduces to a single discrete-time Toda-type theory. The WZW model is a singular point in coupling space where the map into Toda variables degenerates.

Motivated by the study of collapsing Calabi–Yau 3-folds with a Lefschetz K3 fibration, we construct a complete Calabi–Yau metric on C3 with maximal volume growth, which in the appropriate scale is expected to model the collapsing metric near the nodal point. This new Calabi–Yau metric has singular tangent cone at infinity C2/Z2×C, and its Riemannian geometry has certain non-standard features near the singularity of the tangent cone, which are more typical of adiabatic limit problems. The proof uses an existence result in H-J. Hein’s Ph.D. thesis to perturb an asymptotic approximate solution into an actual solution, and the main difficulty lies in correcting the slowly decaying error terms.

In this paper, we are concerned with the one-species VlasovPoissonBoltzmann system with a nonconstant background density in full space. There exists a stationary solution when the background density is a small perturbation of a positive constant state. We prove the nonlinear stability of solutions to the Cauchy problem near the stationary state in some Sobolev space without any time derivatives. This result is nontrivial even when the background density is a constant state. In the proof, the macroscopic balance laws are essentially used to deal with the a priori estimates on both the microscopic and macroscopic parts of the solution. Moreover, some interactive energy functionals are introduced to overcome difficulty stemming from the absence of time derivatives in the energy functional.