We consider the relationship of two fixed point theorems for direction-preserving discrete correspondences. We show that, for space of no more than three dimensions, the fixed point theorem [5] of Iimura, Murota and Tamura, on integrally convex sets can be derived from Chen and Deng’s fixed point theorem [1] on lattices by expanding every direction-preserving discrete correspondence over an int...

Let G be a torsion-free abelian group of type (0, 0, 0, . . . ) and R an integrally closed integral domain with quotient field K. We show that every divisorial ideal (respectively, t-ideal) J of the group ring R[X;G] is of the form J = hIR[X;G] for some h ∈ K[X;G] and a divisorial ideal (respectively, t-ideal) I of R. Consequently, there are natural monoid isomorphisms Cl(R) ∼= Cl(R[X;G]) and C...

By a finite open Riemann surface is meant a proper, open, connected subset of a compact Riemann surface W whose boundary T is also the boundary of W— X and consists of a finite number of closed analytic arcs. Since W—X has an interior we may employ the Riemann-Roch Theorem to show that B(X) has enough functions to separate points and provide each point in X with a local uniformizer. Such a surf...

In this paper we show that a finite symmetric game has a pure strategy equilibrium if the payoff functions of players are integrally concave (the negative of the integrally convex functions due to Favati and Tardella [Convexity in nonlinear integer programming, Ricerca Operativa, 1990, 53:3–44]). Since the payoff functions of any two-strategy game are integrally concave, this generalizes the re...

A canonically defined mod 2 linear dependency current is associated to each collection ν of sections, ν 1 , ..., ν m , of a real rank n vector bundle. This current is supported on the linear dependency set of ν. It is defined whenever the collection ν satisfies a weak measure theoretic condition called " atomicity ". Essentially any reasonable collection of sections satisfies this condition, va...

A well-known theorem in network flow theory states that for a strongly connected digraph D = (V, A) there exists a set of directed cycles the incidence vectors of which form a basis for the circulation space of D and integrally span the set of integral circulations; that is, every integral circulation can be written as an integral combination of these vectors. In this paper, we extend this resu...

We compute the first André-Quillen homology modules for the simple over-rings of integrally closed domains and study an ideal theoretic condition arising from the vanishing of H1. André-Quillen (co)homology is known to be a powerful tool in characterizing various classes of rings or morphisms between noetherian rings. Regular and complete intersection local rings, regular, (formally) smooth or ...

New and old results on closed polynomials, i.e., such polynomials f ∈ k[x1, . . . , xn]\k that the subalgebra k[f ] is integrally closed in k[x1, . . . , xn], are collected in the paper. Using some properties of closed polynomials we prove the following factorization theorem: Let f ∈ k[x1, . . . , xn] \ k, where k is algebraically closed. Then for all but finite number μ ∈ k the polynomial f + ...

In discrete convex analysis, the scaling and proximity properties for the class of L-convex functions were established more than a decade ago and have been used to design efficient minimization algorithms. For the larger class of integrally convex functions of n variables, we show here that the scaling property only holds when n ≤ 2, while a proximity theorem can be established for any n, but o...

An integer partition λ ` n corresponds, via its Ferrers diagram, to an artinian monomial ideal I ⊂ C[x, y] with dimC C[x, y]/I = n. If λ corresponds to an integrally closed ideal we call it concave . We study generating functions for the number of concave partitions, unrestricted or with at most r parts. 1. concave partitions By an integer partition λ = (λ1, λ2, λ3, . . . ) we mean a weakly dec...