2,745 research outputs found
Approximate and pseudo-amenability of various classes of Banach algebras
We continue the investigation of notions of approximate amenability that were
introduced in work of the second and third authors. It is shown that every
boundedly approximately contractible Banach algebra has a bounded approximate
identity.
Among our other results, it is shown that the Fourier algebra of the free
group on two generators is not approximately amenable. Further examples are
obtained of -semigroup algebras which are approximately amenable but
not amenable; using these, we show that bounded approximate amenability need
not imply sequential approximate amenability. Results are also given for Segal
subalgebras of , where is a locally compact group, and the algebras
of -pseudofunctions on a discrete group (of which
the reduced -algebra is a special case).Comment: 35 pages, revision of Jan '08 preprint. Abstract and MSC added;
bibliograpy updated; slight tweaks to Section 4; and correction of a few
typos. The final version is to appear in J. Funct. Ana
Characterizing Jordan derivations of matrix rings through zero products
Let \Mn be the ring of all matrices over a unital ring
, let be a 2-torsion free unital \Mn-bimodule and
let D:\Mn\rightarrow \mathcal{M} be an additive map. We prove that if
D(\A)\B+ \A D(\B)+D(\B)\A+ \B D(\A)=0 whenever \A,\B\in \Mn are such that
\A\B=\B\A=0, then D(\A)=\delta(\A)+\A D(\textbf{1}), where
\delta:\Mn\rightarrow \mathcal{M} is a derivation and lies in
the centre of . It is also shown that is a generalized
derivation if and only if D(\A)\B+ \A D(\B)+D(\B)\A+ \B D(\A)-\A
D(\textbf{1})\B-\B D(\textbf{1})\A=0 whenever \A\B=\B\A=0. We apply this
results to provide that any (generalized) Jordan derivation from \Mn into a
2-torsion free \Mn-bimodule (not necessarily unital) is a (generalized)
derivation. Also, we show that if \varphi:\Mn\rightarrow \Mn is an additive
map satisfying \varphi(\A \B+\B \A)=\A\varphi(\B)+\varphi(\B)\A \quad (\A,\B
\in \Mn), then \varphi(\A)=\A\varphi(\textbf{1}) for all \A\in \Mn, where
lies in the centre of \Mn. By applying this result we
obtain that every Jordan derivation of the trivial extension of \Mn by \Mn
is a derivation.Comment: To appear in Mathematica Slovac
2n-Weak module amenability of semigroup algebras
Let be an inverse semigroup with the set of idempotents . We prove
that the semigroup algebra is always -weakly module amenable
as an -module, for any , where acts on
trivially from the left and by multiplication from the right.Comment: arXiv admin note: text overlap with arXiv:1207.4514 by other author
A Theoretically Grounded Application of Dropout in Recurrent Neural Networks
Recurrent neural networks (RNNs) stand at the forefront of many recent
developments in deep learning. Yet a major difficulty with these models is
their tendency to overfit, with dropout shown to fail when applied to recurrent
layers. Recent results at the intersection of Bayesian modelling and deep
learning offer a Bayesian interpretation of common deep learning techniques
such as dropout. This grounding of dropout in approximate Bayesian inference
suggests an extension of the theoretical results, offering insights into the
use of dropout with RNN models. We apply this new variational inference based
dropout technique in LSTM and GRU models, assessing it on language modelling
and sentiment analysis tasks. The new approach outperforms existing techniques,
and to the best of our knowledge improves on the single model state-of-the-art
in language modelling with the Penn Treebank (73.4 test perplexity). This
extends our arsenal of variational tools in deep learning.Comment: Added clarifications; Published in NIPS 201
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