The primordial problems of linear algebra are the solution of a system of linear equations

and the solution of the eigenvalue problem

for the eigenvalues λk, and corresponding eigenvectors of a given matrix A.

In this brief exposition we collect several results on rings of formal power series with coefficients from a field or a ring with some special properties. The results that are catalogued below are mostly algebraic in nature.

Paul Halmos expressed [3, p. 110] the general dissatisfaction with the usual proofs of this famous and important theorem. They all make it seem like an accidental product of a computation. A more conceptual proof was devised by N. P. Dekker. In spite of the elegance of his proof, the one offered below may have some claim to be regarded as "the reason the theorem is true".

If f(x) is real-valued and continuous, it has the property that it takes on all intermediate values when it passes from one value to another. This means that whenever f(x1) and f(x2) are different and u is any number between them, then f(x) = u for at least one x between x1 and x2. We shall call this the Darboux property.

A subset of a topological space is called discrete iff every point in the space has a neighborhood which meets the set in at most one point. Discrete sets are useful for decomposing the images of certain maps and for generalizing closed maps. All discrete sets are closed iff the space is T1. As a result of characterizing discrete and countably discrete maps, theorems due to Vaĭnšteĭn and Engelking are extended to these maps.

This paper presents variants of the Holder inequality for integrals of functions (as well as for sums of real numbers) and its inverses. In these contexts, all possible transliterations and some extensions to more than two functions are also mentioned.

We prove that there exist infinitely many coprime numbers a, b, c with $a+b=c$ and $c>\operatorname {\mathrm {rad}}(abc)\exp (6.563\sqrt {\log c}/\log \log c)$. These are the most extremal examples currently known in the $abc$ conjecture, thereby providing a new lower bound on the tightest possible form of the conjecture. Our work builds on that of van Frankenhuysen (J. Number Theory 82(2000), 91–95) who proved the existence of examples satisfying the above bound with the constant $6.068$ in place of $6.563$. We show that the constant $6.563$ may be replaced by $4\sqrt {2\delta /e}$ where $\delta $ is a constant such that all unimodular lattices of sufficiently large dimension n contain a nonzero vector with $\ell _1$-norm at most $n/\delta $.

Let us call an equational class (variety) K of algebras permutable if and only if every pair of congruences on each K-algebra is permutable. Similarly, we will call K modular (distributive) if the congruence lattice of each K-algebra is modular (distributive).

In this article, we give generalizations of the well-known Fermat’s Little Theorem, Wilson’s theorem, and the little-known Gegenbauer’s theorem.

Let R be a ring such that, for each element a of R, there exists a positive integer n(a) > 1, depending on a, such that an(a)=a. Jacobson proved that such a ring R is necessarily commutative and the purpose of this paper is to give a proof of Jacobson’s Theorem that does not involve the use of the axiom of choice.

We take this opportunity to point out that all rings are assumed to be associative and that nothing beyond elementary ring theory is assumed in the proof to follow. Such ring theory can be found, for example, in [1].