On certain Iwahori representations of unramified U(2,1)U(2, 1) in characteristic pp

Let $F$ be a non-archimedean local field of odd residue characteristic $p$. Let $G$ be the unramified unitary group $U(2, 1)(E/F)$, and $K$ be a maximal compact open subgroup of $G$. For an $\overline{\mathbf{F}}_p$-smooth representation $\pi$ of $G$ containing a weight $\sigma$ of $K$, we follow the work of Hu (\cite{Hu12}) to attach $\pi$ a certain $I_K$-subrepresentation, where $I_K$ is the Iwahori subgroup in $K$. In terms of such an $I_K$-subrepresentation, we prove a sufficient condition for $\pi$ to be non-finitely presented. We determine such an $I_K$-subrepresentation explicitly, when $\pi$ is either a spherical universal Hecke module or an irreducible principal series.Comment: New and revised version: 1) some irrelevant parts are removed. 2) some arguments are modified. 3) main results remain unchange

A remark on the simple cuspidal representations of GL(n, F)

Let $F$ be a non-archimedean local field of residue characteristic $p$, $G$ be the group $GL(n, F)$. In this note, under the assumption $(n, p)=1$, we show a simple cuspidal representation $\pi$ (that with normalized level $\frac{1}{n}$) of $G$ is determined uniquely up to isomorphism by the local constants of $\chi\circ \text{det}\otimes \pi$ for all characters $\chi$ of $F^\times$.Comment: 9 pages. This is version 2, and a detailed proof of Lemma 2.2 is included. Submitte

Freeness of spherical Hecke modules of unramified U(2,1)U(2,1) in characteristic pp

Let $F$ be a non-archimedean local field of odd residue characteristic $p$. Let $G$ be the unramified unitary group $U(2, 1)(E/F)$ in three variables, and $K$ be a maximal compact open subgroup of $G$. For an irreducible smooth representation $\sigma$ of $K$ over $\overline{\mathbf{F}}_p$, we prove that the compactly induced representation $\text{ind}^G _K \sigma$ is free of infinite rank over the spherical Hecke algebra $\mathcal{H}(K, \sigma)$.Comment: Final version, to appear in Journal of Number Theor

Hecke modules and supersingular representations of U(2,1)

Let F be a nonarchimedean local field of odd residual characteristic p. We classify finite-dimensional simple right modules for the pro-p-Iwahori-Hecke algebra $\mathcal{H}_C(G,I(1))$, where G is the unramified unitary group U(2,1)(E/F) in three variables. Using this description when C is the algebraic closure of $\mathbb{F}_p$, we define supersingular Hecke modules and show that the functor of I(1)-invariants induces a bijection between irreducible nonsupersingular mod-p representations of G and nonsupersingular simple right $\mathcal{H}_C(G,I(1))$-modules. We then use an argument of Paskunas to construct supersingular representations of G.Comment: 36 pages. Article shortened, results unchange

Better Long-Range Dependency By Bootstrapping A Mutual Information Regularizer

In this work, we develop a novel regularizer to improve the learning of long-range dependency of sequence data. Applied on language modelling, our regularizer expresses the inductive bias that sequence variables should have high mutual information even though the model might not see abundant observations for complex long-range dependency. We show how the `next sentence prediction (classification)' heuristic can be derived in a principled way from our mutual information estimation framework, and be further extended to maximize the mutual information of sequence variables. The proposed approach not only is effective at increasing the mutual information of segments under the learned model but more importantly, leads to a higher likelihood on holdout data, and improved generation quality. Code is released at https://github.com/BorealisAI/BMI.Comment: Camera-ready for AISTATS 202

Symmetrizable intersection matrices and their root systems

In this paper we study symmetrizable intersection matrices, namely generalized intersection matrices introduced by P. Slodowy such that they are symmetrizable. Every such matrix can be naturally associated with a root basis and a Weyl root system. Using $d$-fold affinization matrices we give a classification, up to braid-equivalence, for all positive semi-definite symmetrizable intersection matrices. We also give an explicit structure of the Weyl root system for each $d$-fold affinization matrix in terms of the root system of the corresponding Cartan matrix and some special null roots

Analysis of diffusion and trapping efficiency for random walks on non-fractal scale-free trees

We study discrete random walks on the NFSFT and provide new methods to calculate the analytic solutions of the MFPT for any pair of nodes, the MTT for any target node and MDT for any source node. Further more, using the MTT and the MDT as the measures of trapping efficiency and diffusion efficiency respectively, we compare the trapping efficiency and diffusion efficiency for any two nodes of NFSFT and find the best (or worst) trapping sites and the best (or worst) diffusion sites. Our results show that: the two hubs of NFSFT is the best trapping site, but it is also the worst diffusion site, the nodes which are the farthest nodes from the two hubs are the worst trapping sites, but they are also the best diffusion sites. Comparing the maximum and minimum of MTT and MDT, we found that the ratio between the maximum and minimum of MTT grows logarithmically with network order, but the ratio between the maximum and minimum of MTT is almost equal to $1$. These results implie that the trap's position has great effect on the trapping efficiency, but the position of source node almost has no effect on diffusion efficiency. We also conducted numerical simulation to test the results we have derived, the results we derived are consistent with those obtained by numerical simulation.Comment: 23 pages, 4 figure

Investigations on Knowledge Base Embedding for Relation Prediction and Extraction

We report an evaluation of the effectiveness of the existing knowledge base embedding models for relation prediction and for relation extraction on a wide range of benchmarks. We also describe a new benchmark, which is much larger and complex than previous ones, which we introduce to help validate the effectiveness of both tasks. The results demonstrate that knowledge base embedding models are generally effective for relation prediction but unable to give improvements for the state-of-art neural relation extraction model with the existing strategies, while pointing limitations of existing methods

Connecting Language and Knowledge with Heterogeneous Representations for Neural Relation Extraction

Knowledge Bases (KBs) require constant up-dating to reflect changes to the world they represent. For general purpose KBs, this is often done through Relation Extraction (RE), the task of predicting KB relations expressed in text mentioning entities known to the KB. One way to improve RE is to use KB Embeddings (KBE) for link prediction. However, despite clear connections between RE and KBE, little has been done toward properly unifying these models systematically. We help close the gap with a framework that unifies the learning of RE and KBE models leading to significant improvements over the state-of-the-art in RE. The code is available at https://github.com/billy-inn/HRERE.Comment: Camera-ready for NAACL HLT 201

Numerical Algorithms for 1-d Backward Stochastic Differential Equations: Convergence and Simulations

In this paper we study different algorithms for backward stochastic differential equations (BSDE in short) basing on random walk framework for 1-dimensional Brownian motion. Implicit and explicit schemes for both BSDE and reflected BSDE are introduced. Then we prove the convergence of different algorithms and present simulation results for different types of BSDEs.Comment: 29 pages, 8 figure