The equitable chromatic number of a graph is the smallest integer $n$ for which the graph's vertex set can be partitioned into $n$ independent sets, each pair of which differs in size by at most 1. We develop a formula and a linear-time algorithm which compute the equitable chromatic number of an arbitrary complete multipartite graph. These results yield tractable solutions of certain scheduling problems.

Let ${\mathcal K}_o$ be a class of strictly convex compact bodies in $\mathbb R^n$ which are centrally symmetric with respect to the origin $o$ in its interior. Let $\overline{U}_i , \overline{U}\in {\mathcal K}_o$ for $i = 1, 2, \ldots$. In this paper, we prove that if $\partial \overline{U}_i =U_i$ converges to $\partial \overline{U}=U$ in the Hausdorff sense as $i$ tends to infinity, then the tangent space $T_o M^n (U_i)$ of the Minkowski space $M^n (U_i )$ converges to the tangent space $T_o M^n (U)$ of $M^n (U)$.

In a previous note [1], we showed how to linearize trigonometric polynomials using complex numbers, Euler's formula, and DeMoivre's formula. Here we explain why this gives, in fact, the Fourier series of these trigonometric polynomials.

This note discusses the implementation of Rémy's algorithm for generating unbiased random binary trees. We point out an error in a published implementation of the algorithm. The error is found by using the $\chi^{2}$-test. Moreover, a correct implementation of the algorithm is presented.

Characterizations of the functions $$f(z) = \frac {z} {(1 - z)^2} \text{ and } f(z) = \frac {1 + z} {(1 - z)^2}$$ are given. We also give counterexamples to show that some generalized problems on interpolation do not hold.

All rings are commutative rings with identity, and $J(R)$ denotes the Jacobson radical of the ring $R$. For any fixed integer $n\geq 1$, it is shown that the class of all $n$-stable rings is properly contained in the class of all $n+1$-stable rings. Results are given showing the connection between several types of rings whose finite sequences satisfy different stability conditions, some involving $J(R)$. It is shown that in the strongly $n$-stable case, it suffices to check whether the $n+1$-tuples satisfy the stable condition. In addition to other results and an example of a ring which is not $n$-stable for any integer $n\geq 1$, examples are given to show the distinction between the different types of stability cases. Finally, in the last section, some surjective mapping properties of a generalized form of $GL_{n}(R)$ and $SL_{n}(R)$ in connection to some stable conditions in the ring $R$ are investigated.