On the mod pp cohomology for GL⁡2\operatorname{GL}_2

Let $p$ be a prime number and $F$ a totally real number field unramified at places above $p$. Let $\bar{r}:\operatorname{Gal}(\bar F/F)\rightarrow\operatorname{GL}_2(\bar{\mathbb{F}_p})$ be a modular Galois representation which satisfies the Taylor-Wiles hypothesis and some technical genericity assumptions. For $v$ a fixed place of $F$ above $p$, we prove that many of the admissible smooth representations of $\operatorname{GL}_2(F_v)$ over $\bar{\mathbb{F}_p}$ associated to $\bar{r}$ in the corresponding Hecke-eigenspaces of the mod $p$ cohomology have Gelfand--Kirillov dimension $[F_v:\mathbb{Q}_p]$. This builds on and extends the work of Breuil-Herzig-Hu-Morra-Schraen and Hu-Wang, giving a unified proof in all cases ($\bar{r}$ either semisimple or not at $v$).Comment: 49 pages, 5 tables. arXiv admin note: substantial text overlap with arXiv:2009.03127 by other author

Approximate Capacities of Two-Dimensional Codes by Spatial Mixing

We apply several state-of-the-art techniques developed in recent advances of counting algorithms and statistical physics to study the spatial mixing property of the two-dimensional codes arising from local hard (independent set) constraints, including: hard-square, hard-hexagon, read/write isolated memory (RWIM), and non-attacking kings (NAK). For these constraints, the strong spatial mixing would imply the existence of polynomial-time approximation scheme (PTAS) for computing the capacity. It was previously known for the hard-square constraint the existence of strong spatial mixing and PTAS. We show the existence of strong spatial mixing for hard-hexagon and RWIM constraints by establishing the strong spatial mixing along self-avoiding walks, and consequently we give PTAS for computing the capacities of these codes. We also show that for the NAK constraint, the strong spatial mixing does not hold along self-avoiding walks

Product quality risk perceptions and decisions: contaminated pet food and lead-painted toys.

In the context of the recent recalls of contaminated pet food and lead-painted toys in the United States, we examine patterns of risk perceptions and decisions when facing consumer product-caused quality risks. Two approaches were used to explore risk perceptions of the product recalls. In the first approach, we elicited judged probabilities and found that people appear to have greatly overestimated the actual risks for both product scenarios. In the second approach, we applied the psychometric paradigm to examine risk perception dimensions concerning these two specific products through factor analysis. There was a similar risk perception pattern for both products: they are seen as unknown risks and are relatively not dread risks. This pattern was also similar to what prior research found for lead paint. Further, we studied people's potential actions to deal with the recalls of these two products. Several factors were found to be significant predictors of respondents' cautious actions for both product scenarios. Policy considerations regarding product quality risks are discussed. For example, risk communicators could reframe information messages to prompt people to consider total risks packed together from different causes, even when the risk message has been initiated due to a specific recall event

A Study of Spatial Writing in John Keats’s Lamia

The 19th English romantic poet John Keats constructed a large number of spaces to assist the narrative process in his narrative poem Lamia. If we re-analyze this poem from a spatial perspective, we will find that the poet is using spatial changes to promote the development of the storyline, and reflect the protagonist’s mental journey from the subtle changes of the scene. This essay aims to explore the special significance of its spatial construction and its role in promoting the development of narrative, and analyze the power discourse contained in the poem