71,140 research outputs found

### On certain Iwahori representations of unramified $U(2, 1)$ in characteristic $p$

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 $U(2,1)$ in characteristic $p$

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

### 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

### 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

### 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

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