Under a pull-back approach given by T. Aikou and L. Kozma and firstly presented by D. Bao, S. S. Chern and Z. Shen, we introduce, in this paper, the concepts of almost contact and normal almost contact Finsler structures on the pull-back bundles. Properties of structures partly Sasakians are studied. Using the $hh$-curvature tensor of Chern connection given by D. Bao, S. S. Chern and Z. Shen, we obtain some characterizations of horizontal Sasakian Finsler structures and $K$-contact structures via the horizontal Ricci tensor and the flag curvature.

We quantize the simply-laced isomonodromy systems using the theory of Manin matrices and Talalaev’s quantum spectral curve method.

In this paper we consider the asymptotic expansion of the energy $I_{\alpha }^{\beta }(\gamma ,\mu ,s)$ associated with a nonlinear Schrödinger system with three wave interaction as $\beta \to \infty$ with $\alpha ={\beta }^{\kappa }$ for a given $\kappa \in ℝ$. In particular, we classify the asymptotic expansion formula into five cases for the parameter $\kappa$.

Let $a$, $b$, $c$ be pairwise relatively prime positive integers such that $a^2+b^r=c^2$ with $r$ positive integer. Then we show that the equation $x^2+b^m=c^n$ has the positive integer solution $(x,m,n)=(a,r,2)$ only under some conditions. The proof is based on elementary methods and Zsigmondy’s theorem.

In this paper, we show an arithmeticity of the ratio of the Petersson norm of the Hilbert-Siegel cusp form determined by the Ikeda lift from a Hilbert cusp form to that of the Hilbert cusp form. This is a generalization of the algebraicity of the ratio of the Petersson norm of the Siegel cusp form under the Ikeda lift of an elliptic cusp form to that of the elliptic cusp form due to Y. Choie and W. Kohnen.