We give a formula for the number of admissible sequences for indecomposable serial rings with $n$ indecomposable projective modules whose minimum composition length is less than or equal to $m$. In particular, if $n=m$ is prime, we show that the number of such admissible sequences is $${{2n-1}\choose n} +{1\over n} \bigg[ (n-1)^2 - {{2n-2}\choose {n}}\bigg] .$$

This note ties together optimization problems from differential calculus, the Cantor function, and the Baire Category Theorem.

In this paper, we study the general third Painlevé equation $$y^{\prime \prime }=\frac{y^{\prime 2}}{y}-\frac{y^{\prime }}{x}+\frac{1}{x} (\alpha y^{2}+\beta )+\gamma y^{3}+\frac{\delta }{y}$$ where $\alpha$, $\beta$, $\gamma$, and $\delta$ are real parameters, discuss the boundedness of some solutions when $\gamma <0$ and $\delta >0$, and find an asymptotic representation of a group of oscillating solutions.

It is shown that the set of integer solutions of a single Diophantine equation, or of several simultaneous linear Diophantine equations, in an arbitrary number of variables, say $n$, is a group with respect to a suitably defined binary operation. Further, the aforesaid group is the direct sum of $n$ cyclic groups.

We prove that the determinant of the adjacency matrix of the wheel graph of even order is equal to the determinant of the adjacency matrix of the complete graph of the same order. We present counterexamples to two conjectures by Yong about simple undirected graphs.