The Prüfer Code is a bijection between the nn−2 trees on the vertex set [1, n] and the nn−2 strings in the set [1, n]n−2 (known as Prüfer strings of order n). Efficient linear-time algorithms for decoding (i.e., converting string to tree) and encoding (i.e., converting tree to string) are well-known. In this paper, we examine an improved decoding algorithm (due to Cho et al.) that scans the ele...

The Prüfer code is a bijection between trees on the vertex set [n] and strings on the set [n] of length n − 2 (Prüfer strings of order n). In this paper we examine the ‘locality’ properties of the Prüfer code, i.e. the effect of changing an element of the Prüfer string on the structure of the corresponding tree. Our measure for the distance between two trees T, T ∗ is ∆(T, T ∗) = n − 1 − |E(T )...

Let R be an associative ring. A map σ : R → R is called a ring endomorphism if σ(x+y) = σ(x)+σ(y) and σ(xy) = σ(x)σ(y) for all elements a,b ∈ R. A ring R is said to be rigid if it has only the trivial ring endomorphisms, that is, identity idR and zero 0R . Rigid left Artinian rings were described by Maxson [9] and McLean [11]. Friger [4, 6] has constructed an example of a noncommutative rigid r...

The method proposed here has been devised for solution of the spectral problem for the Lamé wave equation (see [2]), but extended lately to more general problems. This method is based on the phase function concept or the Prüfer angle determined by the Prüfer transformation cot θ(x) = y′(x)/y(x), where y(x) is a solution of a second order self-adjoint o.d.e. The Prüfer angle θ(x) has some useful...

In his article: “Untersuchungen über die Teilbarkeitseigenschaften in Körpern” J. Reine Angew. Math. 168, 1 36, 1932 [21], Heinz Prüfer introduced a new class of integral domains, namely those domains R in which all finitely generated ideals are invertible. He also proved that to verify this condition, it suffices to check that it holds for all two-generated ideals of R. This was the modest beg...

The classical Prüfer transformation has proved to be a useful tool in the study of Sturm-Liouville theory. In this paper we introduce the Prüfer transformation for self-adjoint difference equations and use it to obtain oscillation criteria and other results. We then offer an extension of this approach to the case of general symplectic systems on time scales. Time scales have been introduced in ...

All rings in this paper are commutative with unity; we will deal mainly with integral domains. Let R be a ring with total quotient ring K. A fractional ideal I of R is invertible if II−1 = R; equivalently, I is a projective module of rank 1 (see, e.g., [Eis95, Section 11.3]). Here, I−1 = (R : I) = {x ∈ K |xI ⊆ R}. Moreover, a projective R-module of rank 1 is isomorphic to an invertible ideal. (...

Prüfer transformation is a useful tool for study of second-order ordinary differential equations. There are many possible extensions of the original Prüfer transformation. We focus on a transformation suitable for study of boundary value problems for the p-Laplacian in the resonant case. The purpose of this paper is to establish its basic properties in deep detail.