The complete tunnel axiom

The complete tunnel axiom (abbreviated CTA) is that there is a continuous function from the Stone-Čech remainder ω¹ of the integers onto a linearly ordered space, such that every point- inverse has empty interior. The concept of a complete tunnel through a space is defined and its presence in ω¹ is shown equivalent to the CTA. Various modifications of this concept are defined and some are shown equivalent to statements about ω¹ and about compactifications of the Mrówka-Isbell space Ψ. The strongest of these statements is shown to follow from CH, but the weakest is negated by the Proper Forcing Axiom (PFA).     

The complete optimal stars-clustering-tree problem

We consider the following complete optimal stars-clustering-tree problem: Given a complete graph G=(V,E) with a weight on every edge and a collection of subsets of V, we want to find a minimum weight spanning tree T such that each subset of the vertices in the collection induces a complete star in T. One motivation for this problem is to construct a minimum cost (weight) communication tree network for a collection of (not necessarily disjoint) groups of customers such that each group induces a complete star. As a result the network will provide a “group broadcast” property, “group fault tolerance” and “group privacy”. We present another motivation from database systems with replications. For the case where the intersection graph of the subsets is connected we present a structure theorem that describes all feasible solutions. Based on it we provide a polynomial algorithm for finding an optimal solution. For the case where each subset induces a complete star minus at most k leaves we prove that the problem is NP-hard.     

The edge-coloring of complete hypergraphs I

This paper contains and generalizes the solution of the following classical problem:If h | n then the h-element subsets of an n-element set can be partitioned into (_h?1^n?1) classes so that every class contains $\text{n}\text{h}$ disjoint h-element sets and every h-element set appears in exactly one class. A short formulation of this statement is: If h | n then the hypergraph $\text{K}{}_{\mathrm{n}}{}^{\mathrm{h}}$ is 1-factorizable. In this paper we study the factorization and edge-coloring problems of the hypergraph K_rxm^h (which is the complete, regular, h-uniform, r-partite hypergraph with m vertices in each of the r classes of vertices).     

The complete closure of a graph

We define the complete closure number cc(G) of a graph G of order n as the greatest integer k ≤ 2n ? 3 such that the kth Bondy-Chvátal closure Cl_k(G) is complete, and give some necessary or sufficient conditions for a graph to have cc(G) = k. Similarly, the complete stability cs(P) of a property P defined on all the graphs of order n is the smallest integer k such that if Cl_k(G) is complete then G satisfies P. For some properties P, we compare cs(P) with the classical stability s(P) of P and show that cs(P) may be far smaller than s(P). © 1993 John Wiley & Sons, Inc.     

The inducibility of complete bipartite graphs

For any s ≥ 1 and t ≥ (^S₂), we prove that among all graphs with n vertices the graph that contains the maximal number of induced copies of K_{t, t+s} for any fixed s ≥ 1 and t ≥ (^s₂) is K_{(n/2)+α(n/2)-α} for some function α = o(n). We show that this is not valid for t < (^s₂). Analogous results for complete multipartite graphs are also obtained.     

The bornology of cofinally complete subsets

Cofinal completeness of a metric space, which is a property between completeness and compactness, can be characterized in terms of a measure of local compactness functional [7]. Using this functional, we introduce and then study the variational notion of cofinally complete subset of a metric space.     

The equations of almost complete intersections

In this paper we inject four Hilbert functions in the determination of the defining equations of the Rees algebra of almost complete intersections of finite co-length. Because three of the corresponding modules are Artinian, some of these relationships are very effective, with the novel approach opening up tracks to the determination of the equations and also to processes of going from homologically defined sets of equations to higher degrees ones. While not specifically directed towards the extraction of elimination equations, it will show how some of these arise naturally.     

The random-cluster model on the complete graph

Summary The random-cluster model of Fortuin and Kasteleyn contains as special cases the percolation, Ising, and Potts models of statistical physics. When the underlying graph is the complete graph onn vertices, then the associated processes are called mean-field. In this study of the mean-field random-cluster model with parametersp=/n andq, we show that its properties for any value ofq(0, ) may be derived from those of an Erds-Rényi random graph. In this way we calculate the critical point _c (q) of the model, and show that the associated phase transition is continuous if and only ifq2. Exact formulae are given for _C (q), the density of the largest component, the density of edges of the model, and the free energy. This work generalizes earlier results valid for the Potts model, whereq is an integer satisfyingq2. Equivalent results are obtained for a fixed edge-number random-cluster model. As a consequence of the results of this paper, one obtains large-deviation theorems for the number of components in the classical random-graph models (whereq=1).     

The Nash sheaf of a complete resolution

Every three-dimensional complex algebraic variety with isolated singular point has a resolution factoring through the Nash blowup and the blowup of the maximal ideal over which the second Fitting ideal sheaf is locally principal. In such resolutions one can construct Hsiang–Pati coordinates and thus obtain generators for the Nash sheaf that are the differentials of monomial functions. Received: 9 April 2001 / Revised version: 26 July 2001     

Programming under uncertainty: The complete problem

We define the complete problem of a two-stage linear programming under uncertainty, to be:

$$\begin{gathered} Minimize z(x) = E_\xi \{ cx + q^ + y^ + + q^ - y^ - \} \hfill \\ subject to Ax = b \hfill \\ Tx + Iy^ + + Iy^ - = \xi \hfill \\ x \geqq 0,y^ + \geqq 0,y^ - \geqq 0 \hfill \\ \end{gathered} $$