A graph G is a couple (X, R) where X is a set, R ⊂ X × X. If G is an undirected graph without loops (R a symmetric irreflexive relation), we can interpret G as a couple (X, R), where R is a set of two-element subsets of X, i.e. . This interpretation is generalized in the notion of society.

A society is a couple (X, R), where ; a k-society is a society (X, R) with |A| = k for each A ∊ R.

A subset K of a lattice is said to be directed if for any a, b∊K there is c∊K with c ≥ a, b. A complete lattice L is called upper continuous if for every directed subset (aα) and every element b.

The following is a slight improvement of [4; Anmerkung 1. 11, p. 11].

A set E of edges of a graph G is said to be a dominating set of edges if every edge of G either belongs to E or is adjacent to an edge of E. If the subgraph 〈E〉 induced by E is a trail T, then T is called a dominating trail of G. Dominating circuits are defined analogously. A sufficient condition is given for a graph to possess a spanning (and thus dominating) circuit and a sufficient condition is given for a graph to possess a spanning (and thus dominating) trail between each pair of distinct vertices. The line graph L(G) of a graph G is defined to be that graph whose vertex set can be put in one-to-one correspondence with the edge set of G in such a way that two vertices of L(G) are adjacent if and only if the corresponding edges of G are adjacent. The existence of dominating trails and circuits is employed to present results on line graphs and second iterated line graphs, respectively.

Throughout this article p denotes a fixed number such that 1 ≤ p < ∞. The definition of a real Lp space associated with a measure space is well known. These spaces are Banach Spaces and, with the usual partial ordering of (equivalence classes of) functions, also Banach Lattices.

Let $R$ be a ring. The following results are proved. $\left( 1 \right)$ Every element of $R$ is a sum of an idempotent and a tripotent that commute if and only if $R$ has the identity ${{x}^{6}}\,=\,{{x}^{4}}$ if and only if $R\,\cong \,{{R}_{1}}\,\times \,{{R}_{2}}$, where ${{{R}_{1}}}/{J\left( {{R}_{1}} \right)}\;$ is Boolean with $U\left( {{R}_{1}} \right)$ a group of exponent $2$ and ${{R}_{2}}$ is zero or a subdirect product of ${{\mathbb{Z}}_{3}}^{,}s$. $\left( 2 \right)$ Every element of $R$ is either a sum or a difference of two commuting idempotents if and only if $R\,\cong \,{{R}_{1}}\,\times \,{{R}_{2}}$, where ${{{R}_{1}}}/{J\left( {{R}_{1}} \right)}\;$ is Boolean with $J\left( R \right)\,=\,0$ or $J\left( R \right)\,=\,\left\{ 0,\,2 \right\}$ and ${{R}_{2}}$ is zero or a subdirect product of ${{\mathbb{Z}}_{3}}^{,}s$. $\left( 3 \right)$ Every element of $R$ is a sum of two commuting tripotents if and only if $R\,\cong \,{{R}_{1}}\,\times \,{{R}_{2}}\,\times \,{{R}_{3}}$, where ${{{R}_{1}}}/{J\left( {{R}_{1}} \right)}\;$ is Boolean with $U\left( {{R}_{1}} \right)$ a group of exponent $2$ , ${{R}_{2}}$ is zero or a subdirect product of ${{\mathbb{Z}}_{3}}^{,}s$, and ${{R}_{3}}$ is zero or a subdirect product of ${{\mathbb{Z}}_{5}}^{,}s$.

Let $RG$ denote the group ring of the group $G$ over the ring $R$. Using an isomorphism between $RG$ and a certain ring of $n\,\times \,n$ matrices in conjunction with other techniques, the structure of the unit group of the group algebra of the dihedral group of order 8 over any finite field of chracteristic 2 is determined in terms of split extensions of cyclic groups.

In [A], Arhangel'skii showed that for any T2 space X, |X|≤2L(x)χ(x), where L(X) is the Lindelöf degree of X and χ(X) is the character of X.

In [B], Bell, Ginsburg and Woods improved this result, assuming normality, by showing that for T4 spaces X, |X|≤2wL(x)χ(x), where wL(X) is the weak Lindelöf degree of X.

We introduce below a new cardinal function aL(X), the almost Lindelöf degree of X, which agrees with L(X) on T3 spaces, but which is often smaller than L(X) on T2 spaces, and show that for T2 spaces X,

The main result of this paper is to prove that a generalization of the Principle of Dependent Choices, introduced by A. Levy [2; see also 1, Chapter 8], is equivalent to a form of Zorn's Lemma.

These pages summarize recent progress on the oscillation problem for semilinear elliptic partial differential equations of the form

(1)

in unbounded domains Ω in n-dimensional Euclidean space Rn. Our attention is restricted to the second order symmetric equation (1), and completeness is not attempted; the emphasis is on results obtained in the last five years.

It is shown that the nth center of a radical ring coincides with that of its adjoint group, from which a result of Jennings is sharpened and a conjecture of his is confirmed.

We construct a class of fermions (or bosons) by using a Clifford (or Weyl) algebra to get two families of irreducible representations for the extended affine Lie algebra $\widetilde{\mathfrak{g}{{\mathfrak{l}}_{N}}\left( {{\mathbb{C}}_{q}} \right)}$ of level (1, 0) (or (−1, 0)).

When I agreed to give the 1976 Jeffery-Williams Lecture to the Canadian Mathematical Congress, I did not envisage that it would be my lot to give what might now be called the first memorial lecture in this series of lectures. During the past year, Ralph Jeffery and Lloyd Williams both died, and their deaths brought forward many deserved tributes to these two wonderful men. My lecture today will forge a link with my mathematical past, a past in which both of these men played important roles.

It is shown that a nonstandard model of Peano arithmetic which has a full satisfaction class is necessarily recursively saturated.

A partially ordered set P has the fixed point property if every orderpreserving mapping f of P to P has a fixed point, that is, f(a) = a for some aϵP; call P fixed point free if P does not have the fixed point property.

We construct several examples of higher-dimensional Calabi–Yau manifolds and prove their modularity.

A new look at Manià's classical example of the Lavrentiev Phenomenon leads to several pertinent observations.

Finite factor groups of are called Hurwitz groups. Here we prove that apart from 2G2(3), G2(2), G2(3) and G2(4), all the groups 2G2(32n+1) and G2(q), q = pn, p € P, are Hurwitz groups. For the proof, (2, 3, 7) structure constants are calculated from the character tables [2], [7], and then with the lists of maximal subgroups [8], [5], the existence of generating triples is deduced.

There are several notions of Ricci curvature tensor in Finsler geometry and spray geometry. One of them is defined by the Hessian of the well-known Ricci curvature. In this paper we will introduce a new notion of Ricci curvature tensor and discuss its relationship with the Ricci curvature and some non-Riemannian quantities. Using this Ricci curvature tensor, we shall have a better understanding of these non-Riemannian quantities.

The maximum density of packings of a given type into the whole of a Euclidean space is defined to be the limit of the maximum density of such packings into a cube as the edge of the cube goes to infinity.

For E2 in particular, a number of well known results such as those due to A. Thue [1], L. Fejes-Toth [2], and C. A. Rogers [3] yield precise information about packings into the whole space. They are however of limited applicability to problems of finite packing in so-far as each requires some restriction upon the boundary of the configuration.

We prove that every finite distributive lattice can be represented as the congruence lattice of a finite (planar) semimodular lattice.