The Bernstein Projector Determined by a Weak Associate Class of Good Cosets

Let ${\text G}$ be a reductive group over a $p$-adic field $F$ of characteristic zero, with $p \gg 0$, and let $G={\text G}(F)$. In [ 15], J.-L. Kim studied an equivalence relation called weak associativity on the set of unrefined minimal $K$-types for ${\text G}$ in the sense of A. Moy and G. Prasad. Following [ 15], we attach to the set $\overline{\mathfrak{s}}$ of good $K$-types in a weak associate class of positive-depth unrefined minimal $K$-types a ${G}$-invariant open and closed subset $\mathfrak{g}_{\overline{\mathfrak{s}}}$ of the Lie algebra $\mathfrak{g} = {\operatorname{Lie}}({\text G})(F)$, and a subset $\tilde{{G}}_{\overline{\mathfrak{s}}}$ of the admissible dual $\tilde{{G}}$ of ${G}$ consisting of those representations containing an unrefined minimal $K$-type that belongs to $\overline{\mathfrak{s}}$. Then $\tilde{{G}}_{\overline{\mathfrak{s}}}$ is the union of finitely many Bernstein components of ${G}$, so that we can consider the Bernstein projector $E_{\overline{\mathfrak{s}}}$ that it determines. We show that $E_{\overline{\mathfrak{s}}}$ vanishes outside the Moy–Prasad ${G}$-domain ${G}_r \subset{G}$, and reformulate a result of Kim as saying that the restriction of $E_{\overline{\mathfrak{s}}}$ to ${G}_r\,$, pushed forward via the logarithm to the Moy–Prasad ${G}$-domain $\mathfrak{g}_r \subset \mathfrak{g}$, agrees on $\mathfrak{g}_r$ with the inverse Fourier transform of the characteristic function of $\mathfrak{g}_{\overline{\mathfrak{s}}}$. This is a variant of one of the descriptions given by R. Bezrukavnikov, D. Kazhdan, and Y. Varshavsky in [8] for the depth-$r$ Bernstein projector.

Bridging Inference Based Sentence Linking Model for Semantic Coherence

As the various social Medias emerge on the web, how to link the large scale of unordered short texts with semantic coherence is becoming a practical problem since these short texts have vast decentralized topics, weak associate relations, abundant noise and large redundancy. The challenging issues to solve the above problem includes what knowledge foundation supports sentence linking process and how to link these unordered short texts for pursuing well coherence. Herein, the authors develop bridging inference based sentence linking model by simulating human beings' discourse bridging process, which narrows semantic coherence gaps between short texts. Such model supports linking process by implicit and explicit knowledge and proposes different bridging inference schemas to guide the linking process. The bridging inference based linking process under different schemas generates different semantic coherence including central semantics, concise semantics and layered semantics etc. To validate the bridging inference based sentence linking model, the authors conduct some experiments. Experimental results confirm that the proposed bridging inference based sentence linking process increases semantic coherence. The model can be used in short-text origination, e-learning, e-science, web semantic search, and online question-answering system in the future works.