372 research outputs found

    Approximate Capacities of Two-Dimensional Codes by Spatial Mixing

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    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

    On the mod pp cohomology for GL2\operatorname{GL}_2

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    Let pp be a prime number and FF a totally real number field unramified at places above pp. Let rˉ:Gal(Fˉ/F)GL2(Fpˉ)\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 vv a fixed place of FF above pp, we prove that many of the admissible smooth representations of GL2(Fv)\operatorname{GL}_2(F_v) over Fpˉ\bar{\mathbb{F}_p} associated to rˉ\bar{r} in the corresponding Hecke-eigenspaces of the mod pp cohomology have Gelfand--Kirillov dimension [Fv:Qp][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 (rˉ\bar{r} either semisimple or not at vv).Comment: 49 pages, 5 tables. arXiv admin note: substantial text overlap with arXiv:2009.03127 by other author

    A Study of Spatial Writing in John Keats’s Lamia

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    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