We show that in every solutions $(D_1,D_2,x,y,m,n)$ of the equation $D_1 x^2 + D_2^m = 4 y^n$ with $n$ prime and coprime with the class-number of the imaginary quadratic field $\mathbf{Q}(\sqrt{-D_1D_2})$ one has $n \leq 5351$.

We answer two questions on Pell's equations and periods of quadratic irrationals raised by the second author while computing some cohomology groups.

After R. T. Farouki introduced, there has been vast researches on Pythagorean Hodograph curves. By the nice property that its velocity function is polynomial/rational, it gives us simpler formulae in computing geometrically driven objects --- offset curve/surface, pipe/canal surface --- which are widely used in CAGD. We present a geometric method to construct PH curves in higher dimensional Euclidean spaces.

In this paper, we introduce and study a new class of variational inclusions and resolvent equations, respectively, and establish the equivalence between the variational inclusions and the resolvent equations. Using the resolvent equations technique, we construct some new iterative algorithms for solving the classes of varitional inclusions and resolvent equations. Under suitable conditions, the convergence analyses of the iterative algorithms are also studied. Our results include several previously known results as special cases.

Blow-analytic equisingularity is a relatively young branch of mathematics, it has been developed over the last three decades. This paper answers some fundamental questions raised by Kuo-Milman [10] and Koike (private communication). We believe that the results we present here are of interest for those working in blow-analytic equisingularity. In particular we show that the singular loci of analytic functions germs, correspond even under arc-analytic equivalences. We also give easier proofs of some results in [1].

In this paper, using a recent result of Bilu, Hanrot and Voutier on primitive divisors, we prove that if $a = |V_r|$, $b = |U_r|$, $c = m^2 + 1$, and $b \equiv 3 \pmod{4}$ is a prime power, then the Diophantine equation $x^2 + b^y = c^z$ has only the positive integer solution $(x,y,z) = (a,2,r)$, where $r > 1$ is an odd integer, $m \in \mathbf{N}$ with $2 \mid m$ and the integers $U_r$, $V_r$ satisfy $(m + \sqrt{-1} )^r = V_r + U_r \sqrt{-1}$.