Integrally Closed Domains with Only Finitely Many Star Operations

We prove that an integrally closed domain R admits only finitely many star operations if and only if R satisfies each of the following conditions: (1) R is a Prüfer domain with finite character, (2) all but finitely many maximal ideals of R are divisorial, (3) only finitely many maximal ideals of R contain a nonzero prime ideal that is contained in some other maximal ideal of R, and (4) if P ≠ (0) is the largest prime ideal contained in a (necessarily finite) collection of maximal ideals of R, then the prime spectrum of R/P is finite.     

Star operations and theD+M construction

In this paper, we study star operations on integral domains of the formD+M. In particular, we give an example of a star operation which is not comparable to thet-operation. Supported in part by a National Security Agency Grant.     

On v-Domains and Star Operations

Let * be a star operation on an integral domain D. Let f (D) be the set of all nonzero finitely generated fractional ideals of D. Call D a *-Prüfer (respectively, (*, v)-Prüfer) domain if (FF ^?1)* = D (respectively, (F ^v F ^?1)* = D) for all F ∈  f (D). We establish that *-Prüfer domains (and (*, v)-Prüfer domains) for various star operations * span a major portion of the known generalizations of Prüfer domains inside the class of v-domains. We also use Theorem 6.6 of the Larsen and McCarthy book [30 Larsen , M. D. , McCarthy , P. J. ( 1971 ). Multiplicative Theory of Ideals . New York : Academic Press . [Google Scholar]], which gives several equivalent conditions for an integral domain to be a Prüfer domain, as a model, and we show which statements of that theorem on Prüfer domains can be generalized in a natural way and proved for *-Prüfer domains, and which cannot be. We also show that in a *-Prüfer domain, each pair of *-invertible *-ideals admits a GCD in the set of *-invertible *-ideals, obtaining a remarkable generalization of a property holding for the “classical” class of Prüfer v-multiplication domains. We also link D being *-Prüfer (or (*, v)-Prüfer) with the group Inv*(D) of *-invertible *-ideals (under *-multiplication) being lattice-ordered.     

Functorial Properties of Star Operations

We define a universal star operation to be an assignment *: A ? * _A of a star operation * _A on A to every integral domain A. Prime examples of universal star operations include the divisorial closure star operation v, the t-closure star operation t, and the star operation w = F _∞ of Hedstrom and Houston. For any universal star operation *, we say that an extension B ? A of integral domains is *-ideal class linked if there is a group homomorphism Cl_{* A} (A) → Cl_{* B} (B) of star class groups induced by the map I ? (IB)^{* _B} on the set of * _A -ideals I of A. We study several natural subclasses of the class of *-ideal class linked extensions.     

Star Operations on Numerical Semigroups

It is proved that the number of numerical semigroups with a fixed number n of star operations is finite if n > 1. The result is then extended to the class of analytically irreducible residually rational one-dimensional Noetherian rings with finite residue field and integral closure equal to a fixed discrete valuation domain.     

Star operations on modules

We present a theory of (semi)star operations for torsion-free modules. This extends the analogous theory of star operations on domains as in [R. Gilmer, Multiplicative Ideal Theory, M. Dekker, New York, 1972] and its generalization to semistar operations studied in [A. Okabe, R. Matsuda, Semistar operations on integral domains, Math. J. Toyama Univ. 17 (1994) 1–21], and recovers some closure operations defined on modules. We investigate some properties of (semi)star operations on a given module over a domain D and their relation with the properties of some classes of semistar operations induced on D.Among other things, this leads to a connection between semistar operations on the D-module and localizing systems on the domain D.     

Cancellation properties in ideal systems: A classification of e.a.b. semistar operations

We give a classification of e.a.b. semistar (and star) operations by defining four different (successively smaller) distinguished classes. Then, using a standard notion of equivalence of semistar (and star) operations to partition the collection of all e.a.b. semistar (or star) operations, we show that there is exactly one operation of finite type in each equivalence class and that this operation has a range of nice properties. We give examples to demonstrate that the four classes of e.a.b. semistar (or star) operations we defined can all be distinct. In particular, we solve the open problem of showing that a.b. is really a stronger condition than e.a.b.     

Note on Star Operations Over Polynomial Rings

This article studies the notions of star and semistar operations over a polynomial ring. It aims at characterizing when every upper to zero in R[X] is a *-maximal ideal and when a *-maximal ideal Q of R[X] is extended from R, that is, Q = (Q ∩ R)[X] with Q ∩ R ≠0, for a given star operation of finite character * on R[X]. We also answer negatively some questions raised by Anderson–Clarke by constructing a Prüfer domain R for which the v-operation is not stable.     

Stability and Clifford Regularity with Respect to Star Operations

In the last few years, the concepts of stability and Clifford regularity have been fruitfully extended by using star operations. In this article we deepen the study of star stable and star regular domains and relate these two classes of domains to each other.     

The stable set problem and the lift-and-project ranks of graphs

We study the lift-and-project procedures for solving combinatorial optimization problems, as described by Lovász and Schrijver, in the context of the stable set problem on graphs. We investigate how the procedures' performances change as we apply fundamental graph operations. We show that the odd subdivision of an edge and the subdivision of a star operations (as well as their common generalization, the stretching of a vertex operation) cannot decrease the N₀-, N-, or N₊-rank of the graph. We also provide graph classes (which contain the complete graphs) where these operations do not increase the N₀- or the N-rank. Hence we obtain the ranks for these graphs, and we also present some graph-minor like characterizations for them. Despite these properties we give examples showing that in general most of these operations can increase these ranks. Finally, we provide improved bounds for N₊-ranks of graphs in terms of the number of nodes in the graph and prove that the subdivision of an edge or cloning a vertex can increase the N₊-rank of a graph.Research of these authors was supported in part by a PREA from Ontario, Canada and research grants from NSERC.Mathematics Subject Classification (2000): 0C10, 90C22, 90C27, 47D20     

Existence of positive radial solutions for a nonlinear elliptic problem in annulus domains

In this paper, we study a nonlinear elliptic problem in an annulus domain. For which, the nonexistence of nontrivial solutions has been obtained by some authors in star‐shaped domain. Although we note that it may admit non‐trivial solutions if the domain is non‐star shaped. With the use of a variational method, we establish the existence of positive radial solutions in an annulus domain. On the basis of this, we also extend this result to the periodic problem of the corresponding degenerate parabolic problem. Copyright © 2012 John Wiley & Sons, Ltd.     

6‐Star‐Coloring of Subcubic Graphs

A star coloring of an undirected graph G is a proper vertex coloring of G (i.e., no two adjacent vertices are assigned the same color) such that no path on four vertices is 2‐colored. The star chromatic number of G is the smallest integer k for which G admits a star coloring with k colors. In this paper, we prove that every subcubic graph is 6‐star‐colorable. Moreover, the upper bound 6 is best possible, based on the example constructed by Fertin, Raspaud, and Reed (J Graph Theory 47(3) (2004), 140–153).     

On the Determination of Star and Convex Bodies by Section Functions

The i th section function of a star body in ⁿ gives the i -dimensional volumes of its sections by i -dimensional subspaces. It is shown that no star body is determined among all star bodies, up to reflection in the origin, by any of its i th section functions. Moreover, the set of star bodies that are determined among all star bodies, up to reflection in the origin, by their i th section functions for all i , is a nowhere dense set. The determination of convex bodies in this sense is also studied. The results complement and contrast with recent results on the determination of convex bodies by i th projection functions. The paper continues the development of the dual Brunn—Minkowski theory initiated by Lutwak. Received December 4, 1996, and in revised form April 14, 1997.     

The Star Chromatic Numbers of Some Planar Graphs Derived from Wheels

The notion of the star chromatic number of a graph is a generalization of the chromatic number. In this paper, we calculate the star chromatic numbers of three infinite families of planar graphs. The first two families are derived from a 3-or 5-wheel by subdivisions, their star chromatic numbers being 2+2/(2n + 1), 2+3/(3n + 1), and 2+3(3n−1), respectively. The third family of planar graphs are derived from n odd wheels by Hajos construction with star chromatic numbers 3 + 1/n, which is a generalization of one result of Gao et al. Received September 21, 1998, Accepted April 9, 2001.     

The star function for meromorphic functions of several complex variables

We define an analogue of the Baernstein star function for a meromorphic function f in several complex variables. This function is subharmonic on the upper half-plane and encodes some of the main functionals attached to f. We then characterize meromorphic functions admitting a harmonic star function.     

Directed star decompositions of the complete directed graph

An (s, t)-directed star is a directed graph with s + t + 1 vertices and s + t arcs; s vertices have indegree zero and outdegree one, t have indegree one and outdegree zero, and one has indegree s and outdegree t. An (s, t)-directed star decomposition is a partition of the arcs of a complete directed graph of order n into (s, t)-directed starsx. We establish necessary and sufficient conditions on s, t, and n for an (s, t)-directed star decomposition of order n to exist.