In this short paper we give the sub-local upper bound for the sup norm of an automorphic form on PGL_n, whose associated automorphic representation has finite conductor C(π)=p^c with c→∞, and its local component at the place of ramification is a minimal vector belonging to an irreducible representation with generic induction datum.

In this paper we extend and deepen a shortlist for the 37th International Mathematical Olympiad (IMO) and propose the Frog Leap Commute Theorem and the Queue Polynomial. We explore the problem from the following aspects: (1) Make use of semi-invariants and propose the Frog Leap Commute Theorem. (2) Make extensions regarding frogs leaping to opposite directions on a straight line. (3) Research frogs leaping to the same direction on a straight line and solve the minimum number of frogs satisfying an infinite leap. (4) Extend the problem to leaps on a plane or in space. (5) Research and extend problems regarding frogs leaping on a circle. (6) Estimate the function c(n) and calculate the order of the function.

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One of the theorems of Bengt Stolt’s article “On the Diophantine Equation$u$^{2}−$Dv$^{2}=4$N$” is not quite correct in its entirety. A counter-example will be given to show this. A modification of the theorem which he was trying to prove will be given for certain special cases.

A reformulation of the planetary geostrophic equations (PGEs) with the inviscid balance equation is proposed and the existence of global weak solutions is established, provided that the mechanical force satisfies an integral constraint. There is only one prognostic equation for the temperature field, and the velocity field is statically determined by the planetary geostrophic balance combined with the incompressibility condition. Furthermore, the velocity profile can be accurately represented as a function of the temperature gradient. In particular, the vertical velocity depends only on the first-order derivative of the temperature. As a result, the bound for the L ∞ (0, t 1 ; L 2 ) ∩ L 2 (0, t 1 ; H 1 ) norm of the temperature field is sufficient to show the existence of the weak solution.