Heilbronn gave a sufficient condition for a number field with a totally ramified prime to fail to be norm-Euclidean. We say that Heilbronn’s criterion applies to a polynomial f if it applies to the number field $K=\mathbb {Q}[x]/(f)$ generated by f. Suppose $n\geq 3$ is odd and $p\geq 5$ is prime with $\gcd (p-1,n)=1$. Let $\mathcal {F}_{p,n}$ denote the collection of monic polynomials $f\in \mathbb {Z}[x]$ of degree n that are Eisenstein at the prime p. Order the polynomials by the natural height $\operatorname {\mathrm {Ht}}(f)$. Define $\delta _{p,n}(X)$ to be the proportion of polynomials $f\in \mathcal {F}_{p,n}$ with $\operatorname {\mathrm {Ht}}(f)\leq X$ for which Heilbronn’s criterion applies. Then,

$$ \begin{align*} \liminf_{X\to\infty} \delta_{p,n}(X)\geq \max\{\tfrac{2}{27},\;1-\varepsilon(p)\}, \end{align*} $$

where $\varepsilon (p)\to 0$ and is effectively computable. In particular, the lower density tends to $1$ as $p\to \infty $ uniformly in n. We also give a version of this result where we weaken the condition on $\gcd (p-1,n)$. As a corollary, we show that given an integer $n\geq 2$, a positive proportion of Eisenstein polynomials of degree n fail to generate norm-Euclidean fields.

Let $f:M\to M$ be a homeomorphism over a compact Riemannian manifold, ergodic with respect to a measure $\mu $ defined on the completion of the Borel $\sigma $-algebra, and $\mathcal F$ a f-invariant one-dimensional continuous foliation of M by $C^1$-leaves. Then, if f preserves a continuous $\mathcal {F}$-arc length system, then we only have three possibilities for the conditional measures of $\mu $ along $\mathcal F$, namely: (i) they are atomic for almost every leaf; or (ii) for almost every leaf, they are equivalent to the measure $\unicode{x3bb} _x$ induced by the invariant arc-length system over $\mathcal F$; or (iii) for almost every leaf, their support is a nowhere dense, perfect subset of the leaf. Furthermore, we show that restricted to ergodic partially hyperbolic diffeomorphism with one-dimensional topological neutral center direction, we are able to eliminate the third case obtaining a dichotomy.

This article estimates separate individual and partisan incumbencyeffects in Uruguay’s regional elections (1971–2020). It contributes tothe limited evidence on incumbency effects in developing countries andto the emerging Differences-in-Discontinuities (Diff-in-Disc)literature, which addresses endogeneity and disentangles candidatefrom party effects within the same institutional setting. Exploitingconstitutionally mandated term limits and strong electoralenforcement, we identify clean causal effects using close electionsunder open and closed races. Results show a large and statisticallysignificant individual incumbency effect of approximately 74%,alongside non-significant partisan effects. These findings suggestthat incumbency in Uruguay is primarily personal rather thanparty-based. The results contribute to debates on personalization ofpolitics, proportional representation, and institutional developmentin Latin America, highlighting how strong democratic institutions cancoexist with highly individualized electoral dynamics.

This paper explores the ontological relationship between descriptive and normative by drawing on the perspectives of Ludwig Wittgenstein and Martin Heidegger. Through Wittgenstein’s concept of “grammar” and Heidegger’s notion of das Man, we see that normativity shapes human perception and interpretation, making descriptive neutrality unattainable. Descriptions are always informed by norms and norms evolve by descriptions. This intertwined relationship has significant implications for business ethics, since ethical conflicts can now be reframed as lack of normative references. Ultimately, the paper proposes a perspective of moral perspectivism.

Limpets are abundant and ecologically important gastropods in intertidal and some riverine ecosystems, yet their role in trematode transmission remains comparatively understudied. We investigated trematode infection in intertidal limpets from Otago Harbour, New Zealand, using a molecular approach. Two species were identified: the avian schistosome Gigantobilharzia cf. patagonensis in Siphonaria australis and Acanthoparyphium sp. A metacercariae in Notoacmea sp. Gigantobilharzia cf. patagonensis was detected at all five sampling sites, with prevalence ranging from 2.6% to 100%. Haplotype network analysis using cox1 revealed high haplotype diversity and a star-like topology, suggesting a recent population expansion. This study expanded the known diversity of marine schistosomes in the region and raises potential public health concerns with cercarial dermatitis. We also conducted a literature synthesis further demonstrating that limpets contribute to trematode transmission across 12 superfamilies and 23 families, with distinct parasitism profiles between freshwater and marine environments. These findings highlight limpets as overlooked but significant hosts in trematode ecology and emphasize the need for broader surveys and definitive host screening to resolve incomplete life cycles and assess epidemiological risks in coastal ecosystems.

We propose a bio-psycho-socio-anthropological (BPSA) model that mainstreams cultural formulation and power-aware practice across training, interventions and evaluation. Contrasting it with transcultural and community models, we show UK examples where embedding cultural insight in statutory decisions reduced detentions and improved engagement, aligning with national equality priorities.

The thesis of this article is that the understanding of the vocation of an artist in the writings of Jacques Maritain emerges as to develop habitus (practical virtues of the intellect) in order to direct their inspirations in order to make beautiful things that convey the spiritual heritage of the nation and the civilization and inspire others to contemplate God. This vocation to be an associate of God in creating beautiful works is a powerful reminder of the close relationship between all personal vocations and the common good. First I explore Maritain’s conversion to Catholicism, including through the arts. I next clarify Maritain’s Thomistic understanding of art as one expression of the illuminating intellect. I then review Maritain’s writings about the arts in education. I conclude with theological and pastoral reflections on the vocation of an artist.

We characterize dynamics of every distortion element in the group of diffeomorphisms of the 2-sphere that has at least two fixed points and another recurrent point. The key result is that if f is such a diffeomorphism, then the homeomorphism $\check {f}_{\mathrm { ann}}$, which is a lift of the homeomorphism of the closed annulus $\overline {\mathcal {A}}$ obtained from $\mathbb {S}^2$ by blowing up two fixed points of f to the universal covering space of $\overline {\mathcal {A}}$, has a unique rotation number. Moreover, we find the differential of such a distortion element in the group of diffeomorphisms of the 2-sphere at each fixed point up to conjugacy.

We propose a deep reinforcement learning (RL) framework designed to optimize the hedging of specific, user-defined risk factors—referred to as targeted risks—in financial instruments affected by multiple sources of uncertainty. Our methodology uses Shapley value decompositions to establish source of risk grouping’s contribution to the projected contract cash flows, providing a clear attribution of the profit and loss to distinct risk categories. Leveraging this decomposition, we apply deep RL to hedge only the targeted risks, while leaving non-targeted risks mostly unaffected. In addition, we introduce a joint neural network architecture in which the agent network utilizes risk estimates from a risk measurement neural network to stabilize the hedging strategy, taking into account local risk dynamics. Numerical experiments show that our approach outperforms traditional methods, such as delta hedging and traditional deep hedging, significantly reducing targeted risks in variable annuities while maintaining flexibility for broader applications.