Since Seidenberg’s (1953-54) papers [35, 36] and Jaffard’s (1960) pamphlet [28] on the dimension theory of commutative rings, the literature abounds in works exploring the prime ideal structure of polynomial rings, including four pioneering articles by Arnold and Gilmer on dimension sequences [3, 4, 5, 6]. Of particular interest is Bastida-Gilmer’s (1973) precursory article [8] which establishe...

Abstract We introduce the concept of pre-Jaffard family , a generalization Jaffard families obtained by substituting locally finite hypothesis with much weaker compactness hypothesis. From any such family, we construct sequence overrings starting domain that allows to decompose stable semistar operations and singular length functions in more cases than what is allowed families. also apply one-d...

We prove a necessary and sufficient criterion for the ring of integer-valued polynomials to behave well under localization. Then, we study how Picard group Int(D) quotient P ( D ) : = Pic Int / $\mathcal {P}(D):=\mathrm{Pic}(\mathrm{Int}(D))/\mathrm{Pic}(D)$ in relation Jaffard, weak pre-Jaffard families; particular, show that ≃ ⨁ T {P}(D)\simeq \bigoplus \mathcal {P}(T)$ when ranges Jaffard fa...

let $d$ be an integral domain and $star$ a semistar operation stable and of finite type on it. we define the semistar dimension (inequality) formula and discover their relations with $star$-universally catenarian domains and $star$-stably strong s-domains. as an application, we give new characterizations of $star$-quasi-pr"{u}fer domains and um$t$ domains in terms of dimension ine...

Let $D$ be an integral domain and $star$ a semistar operation stable and of finite type on it. We define the semistar dimension (inequality) formula and discover their relations with $star$-universally catenarian domains and $star$-stably strong S-domains. As an application, we give new characterizations of $star$-quasi-Pr"{u}fer domains and UM$t$ domains in terms of dimension inequal...

All rings and algebras considered in this paper are commutative with identity elements and, unless otherwise specified, are to be assumed to be non-trivial. All ringhomomorphisms are unital. Let k be a field. We denote the class of commutative k−algebras with finite transcendence degree over k by C. Also, we shall use t.d.(A) to denote the transcendence degree of a k−algebra A over k, A[n] to d...

Recently orthonormal Wilson bases with good time–frequency localization have been constructed by Daubechies, Jaffard, and Journé. We extend this construction to Wilson sets and frames with arbitrary oversampling (or redundancy). We state conditions under which dual Weyl–Heisenberg (WH) sets induce dual Wilson sets, and we formulate duality conditions in the time domain and frequency domain. We ...

Let JR be a commutative ring with identity of finite (Krull) dimension n0, and for each positive integer /c, let nk be the dimension of the polynomial ring R = R[XU . . . , Xk] in k indeterminates over R. The sequence {wjiio * Ud the dimension sequence for R, and the sequence {di}fLl9 where dt = nt — ni_1 for each i, is called the difference sequence for R. We are concerned with a determination...

We study the efficient numerical solution of infinite matrix equations Au = f for a matrix A in the Jaffard algebra. These matrices appear naturally via frame discretizations in many applications such as Gabor analysis, sampling theory, and quasi-diagonalization of pseudo-differential operators in the weighted Sjöstrand class. The proposed algorithm has two main features: firstly, it converges ...