Abstract Integrally convex functions constitute a fundamental function class in discrete analysis, including M-convex functions, L-convex and many others. This paper aims at rather comprehensive survey of recent results on integrally with some new technical results. Topics covered this include characterizations integral sets operations optimality criteria for minimization proximity-scaling algo...

Let I be a monomial ideal in the polynomial ring S = K[x1, . . . , xn]. We study the Stanley depth of the integral closure I of I. We prove that for every integer k ≥ 1, the inequalities sdepth(S/Ik) ≤ sdepth(S/I) and sdepth(Ik) ≤ sdepth(I) hold. We also prove that for every monomial ideal I ⊂ S there exist integers k1, k2 ≥ 1, such that for every s ≥ 1, the inequalities sdepth(S/I1) ≤ sdepth(S...

In this article, we delve into the properties possessed by algebras, which we have termed seeds, that map to big Cohen-Macaulay algebras. We will show that over a complete local domain of positive characteristic any two big Cohen-Macaulay algebras map to a common big Cohen-Macaulay algebra. We will also strengthen Hochster and Huneke’s “weakly functorial” existence result for big Cohen-Macaulay...

1. In the first theorem of [3], Wiegandt incorrectly states that a semimodule is complete if and only if it is divisible. (See §2 for definitions.) In this paper we show (in §3) that his theorem should read: a semimodule is complete if and only if it is a divisible group. Divisible semimodules which are not necessarily groups are discussed in §4, in connection with Wiegandt's second theorem. It...

Suppose M is a maximal ideal of a commutative integral domain R and that some power Mn of M is finitely generated. We show that M is finitely generated in each of the following cases: (i) M is of height one, (ii) R is integrally closed and htM = 2, (iii) R = K[X; S̃] is a monoid domain over a field K, where S̃ = S ∪ {0} is a cancellative torsion-free monoid such that ⋂∞ m=1 mS = ∅, and M is the m...

We investigate the structure of prime ideals of finite height in polynomial extension rings of a commutative unitary ring R. We consider the question of finite generation of such prime ideals. The valuative dimension of prime ideals of R plays an important role in our considerations. If X is an infinite set of indeterminates over R, we prove that every prime ideal of R[X] of finite height is fi...

Solid-state bonding methods, e.g., diffusion bonding and pressure welding, are becoming common manufacture and repair techniques for gas turbine engine components. Effective NDE inspection techniques are crucial to the utilization of this approach due to the high stresses on the bond plane associated with jet engine operation. Recently we have examined ultrasonic techniques for assessing bond q...

We show, under suitable hypothesis which are sharp in a certain sense, that the core of an m-primary ideal in a regular local ring of dimension d is equal to the adjoint (or multiplier) ideal of its d-th power. This generalizes the fundamental formula for the core of an integrally closed ideal in a two dimensional regular local ring due to Huneke and Swanson. We also find a generalization of th...

A monoid M is a Cale monoid with base Q if for every nonunit x ∈ M there exists a positive integer n such that xn factors uniquely up to order and associates as elements from Q ⊆ M\M. An integral domain D is a Cale domain with base Q, if its multiplicative monoid of nonzero elements is a Cale monoid with base Q. We explore the basic properties of Cale monoids and integral domains. In particular...

In this paper, we develop a perturbation analysis for stability spectra (Lyapunov exponents and Sacker-Sell spectrum) for products of operators on a Hilbert space (both real and complex) based upon the discrete QR technique. Error bounds are obtained in both the integrally separated and non-integrally separated cases that correspond to distinct and multiple eigenvalues, respectively, for a sing...