Relative polynomial closure and monadically Krull monoids of integer-valued polynomials

Let D be a Krull domain and Int(D) the ring of integer-valued polynomials on D. For any f in Int(D), we explicitly construct a divisor homomorphism from [f], the divisor-closed submonoid of Int(D) generated by f, to a finite sum of copies of (N_0,+). This implies that [f] is a Krull monoid. For V a discrete valuation domain, we give explicit divisor theories of various submonoids of Int(V). In the process, we modify the concept of polynomial closure in such a way that every subset of D has a finite polynomially dense subset. The results generalize to Int(S,V), the ring of integer-valued polynomials on a subset, provided S doesn't have isolated points in v-adic topology.Comment: 12 pages; v.2 contains corrections, in that some necessary conditions on those subsets S, for which we consider integer-valued polynomials on subsets, are impose

On the convergence of double integrals and a generalized version of Fubini's theorem on successive integration

Let the function f: \bar{\R}^2_+ \to \C be such that f\in L^1_{\loc} (\bar{\R}^2_+). We investigate the convergence behavior of the double integral \int^A_0 \int^B_0 f(u,v) du dv \quad {\rm as} \quad A,B \to \infty,\leqno(*) where $A$ and $B$ tend to infinity independently of one another; while using two notions of convergence: that in Pringsheim's sense and that in the regular sense. Our main result is the following Theorem 3: If the double integral (*) converges in the regular sense, or briefly: converges regularly, then the finite limits $$\lim_{y\to \infty} \int^A_0 \Big(\int^y_0 f(u,v) dv\Big) du =: I_1 (A)$$ and $$\lim_{x\to \infty} \int^B_0 \Big(\int^x_0 f(u,v) du) dv = : I_2 (B)$$ exist uniformly in $0<A, B <\infty$, respectively; and $$\lim_{A\to \infty} I_1(A) = \lim_{B\to \infty} I_2 (B) = \lim_{A, B \to \infty} \int^A_0 \int^B_0 f(u,v) du dv.$$ This can be considered as a generalized version of Fubini's theorem on successive integration when f\in L^1_{\loc} (\bar{\R}^2_+), but $f\not\in L^1 (\bar{\R}^2_+)$

Erratum: Solution of periodic Poisson's equation and the Hartree-Fock approach for solids with extended electron states: application to linear augmented plane wave method

Erratum to Int. J. Quantum Chem. v. 89, pp. 57-85 (2002)Comment: 1 pag

Pr\"ufer intersection of valuation domains of a field of rational functions

Let $V$ be a rank one valuation domain with quotient field $K$. We characterize the subsets $S$ of $V$ for which the ring of integer-valued polynomials ${\rm Int}(S,V)=\{f\in K[X] \mid f(S)\subseteq V\}$ is a Pr\"ufer domain. The characterization is obtained by means of the notion of pseudo-monotone sequence and pseudo-limit in the sense of Chabert, which generalize the classical notions of pseudo-convergent sequence and pseudo-limit by Ostrowski and Kaplansky, respectively. We show that ${\rm Int}(S,V)$ is Pr\"ufer if and only if no element of the algebraic closure $\overline{K}$ of $K$ is a pseudo-limit of a pseudo-monotone sequence contained in $S$, with respect to some extension of $V$ to $\overline{K}$. This result expands a recent result by Loper and Werner.Comment: to appear in J. Algebra. All comments are welcome. Keywords: Pr\"ufer domain, pseudo-convergent sequence, pseudo-limit, residually transcendental extension, integer-valued polynomia

Wnt signaling during tooth replacement in zebrafish (Danio rerio) : pitfalls and perspectives

The canonical (13-catenin dependent) Wnt signaling pathway has emerged as a likely candidate for regulating tooth replacement in continuously renewing dentitions. So far, the involvement of canonical Wnt signaling has been experimentally demonstrated predominantly in amniotes. These studies tend to show stimulation of tooth formation by activation of the Wnt pathway, and inhibition of tooth formation when blocking the pathway. Here, we report a strong and dynamic expression of the soluble V\int inhibitor dickkopfl (dkkl) in developing zebrafish (Danio rerio) tooth germs, suggesting an active repression of V\int signaling during morphogenesis and cytodifferentiation of a tooth, and derepression of Wnt signaling during start of replacement tooth formation. To further analyse the role of Wnt signaling, we used different gain-of-function approaches. These yielded disjunct results, yet none of them indicating enhanced tooth replacement. Thus, masterblind (mbl) mutants, defective in axinl, mimic overexpression of Mt, but display a normally patterned dentition in which teeth are replaced at the appropriate times and positions. Activating the pathway with LICI had variable outcomes, either resulting in the absence, or the delayed formation, of first-generation teeth, or yielding a regular dentition with normal replacement, but no supernumerary teeth or accelerated tooth replacement. The failure so far to influence tooth replacement in the zebrafish by perturbing Wnt signaling is discussed in the light of (i) potential technical pitfalls related to dose- or time-dependency, (ii) the complexity of the canonical V\int pathway, and (iii) species-specific differences in the nature and activity of pathway components. Finally, we emphasize the importance of in-depth knowledge of the wild-type pattern for reliable interpretations. It is hoped that our analysis can be inspiring to critically assess and elucidate the role of V\int signaling in tooth development in polyphyodonts