The Picard group of the forms of the affine line and of the additive group

We obtain an explicit upper bound on the torsion of the Picard group of the forms of ${\mathbb{A}}_k^1$ and their regular completions. We also obtain a sufficient condition for the Picard group of the forms of ${\mathbb{A}}_k^1$ to be nontrivial and we give examples of nontrivial forms of ${\mathbb{A}}_k^1$ with trivial Picard groups.     

Uniqueness of Fq-quadratic perfect nonlinear maps from Fq3 to Fq2

Let q be a power of an odd prime. We prove that all ${\mathbb{F}}_q$-quadratic perfect nonlinear maps from ${\mathbb{F}}_{q^3}$ to ${\mathbb{F}}_q^2$ are equivalent. We also give a geometric method to find the corresponding equivalence explicitly.     

Bifurcation and chaotic behavior of a discrete-time predator–prey system

The dynamics of a discrete-time predator–prey system is investigated in the closed first quadrant ${\mathbb{R}}_{+}^2$. It is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation in the interior of ${\mathbb{R}}_{+}^2$ by using a center manifold theorem and bifurcation theory. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as orbits of period 7, 14, 21, 63, 70, cascades of period-doubling bifurcation in orbits of period 2, 4, 8, quasi-periodic orbits and chaotic sets. These results show far richer dynamics of the discrete model compared with the continuous model. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.     

Determinacy of refinements to the difference hierarchy of co-analytic sets

In this paper we develop a technique for proving determinacy of classes of the form ${\omega }^2\text{-}{\mathrm{\Pi }}_1^1+\mathrm{\Gamma }$ (a refinement of the difference hierarchy on ${\mathrm{\Pi }}_1^1$ lying between ${\omega }^2\text{-}{\mathrm{\Pi }}_1^1$ and $({\omega }^2+1)\text{-}{\mathrm{\Pi }}_1^1$) from weak principles, establishing upper bounds for the determinacy-strength of the classes ${\omega }^2\text{-}{\mathrm{\Pi }}_1^1+{\mathrm{\Sigma }}_{\alpha }^0$ for all computable α and of ${\omega }^2\text{-}{\mathrm{\Pi }}_1^1+{{\mathrm{\Delta }}^1}_1$. This bridges the gap between previously known hypotheses implying determinacy in this region.     

Families of curves over P1 with 3 singular fibers

Suppose $f:S\to {\mathbb{P}}^1$ is a fibration of genus g with 3 singular fibers and two of them are semistable. We show that the Mordell–Weil group of f is finite, the surface S is rational and $2g??K_S^2?4g?4$. We construct some examples to show that such fibrations exist for infinitely many g.     

Comparing motives of smooth algebraic varieties

Given a perfect field of exponential characteristic e, the Cor-, $K_0^{\oplus }$-, $K_0$- and ${\mathbb{K}}_0$-motives of smooth algebraic varieties with $\mathbb{Z}[1/e]$-coefficients are shown to be locally quasi-isomorphic to each other. Moreover, it is proved that their triangulated categories of motives with $\mathbb{Z}[1/e]$-coefficients are equivalent. An application is given for the bivariant motivic spectral sequence.