93,857 research outputs found

### Gravitational fields with a non Abelian bidimensional Lie algebra of symmetries

Vacuum gravitational fields invariant for a bidimensional non Abelian Lie
algebra of Killing fields, are explicitly described. They are parameterized
either by solutions of a transcendental equation (the tortoise equation) or by
solutions of a linear second order differential equation on the plane.
Gravitational fields determined via the tortoise equation, are invariant for a
3-dimensional Lie algebra of Killing fields with bidimensional leaves. Global
gravitational fields out of local ones are also constructed.Comment: 8 pagese, latex, no figure

### Vacuum Einstein metrics with bidimensional Killing leaves. I-Local aspects

The solutions of vacuum Einstein's field equations, for the class of
Riemannian metrics admitting a non Abelian bidimensional Lie algebra of Killing
fields, are explicitly described. They are parametrized either by solutions of
a transcendental equation (the tortoise equation), or by solutions of a linear
second order differential equation in two independent variables. Metrics,
corresponding to solutions of the tortoise equation, are characterized as those
that admit a 3-dimensional Lie algebra of Killing fields with bidimensional
leaves.Comment: LateX file, 33 pages, 2 figure

### BattRAE: Bidimensional Attention-Based Recursive Autoencoders for Learning Bilingual Phrase Embeddings

In this paper, we propose a bidimensional attention based recursive
autoencoder (BattRAE) to integrate clues and sourcetarget interactions at
multiple levels of granularity into bilingual phrase representations. We employ
recursive autoencoders to generate tree structures of phrases with embeddings
at different levels of granularity (e.g., words, sub-phrases and phrases). Over
these embeddings on the source and target side, we introduce a bidimensional
attention network to learn their interactions encoded in a bidimensional
attention matrix, from which we extract two soft attention weight distributions
simultaneously. These weight distributions enable BattRAE to generate
compositive phrase representations via convolution. Based on the learned phrase
representations, we further use a bilinear neural model, trained via a
max-margin method, to measure bilingual semantic similarity. To evaluate the
effectiveness of BattRAE, we incorporate this semantic similarity as an
additional feature into a state-of-the-art SMT system. Extensive experiments on
NIST Chinese-English test sets show that our model achieves a substantial
improvement of up to 1.63 BLEU points on average over the baseline.Comment: 7 pages, accepted by AAAI 201

### Estimating hyperbolicity of chaotic bidimensional maps

We apply to bidimensional chaotic maps the numerical method proposed by
Ginelli et al. to approximate the associated Oseledets splitting, i.e. the set
of linear subspaces spanned by the so called covariant Lyapunov vectors (CLV)
and corresponding to the Lyapunov spectrum. These subspaces are the analog of
linearized invariant manifolds for non-periodic points, so the angles between
them can be used to quantify the degree of hyperbolicity of generic orbits;
however, being such splitting non invariant under smooth transformations of
phase space, it is interesting to investigate the properties of transversality
when coordinates change, e.g. to study it in distinct dynamical systems. To
illustrate this issue on the Chirikov-Taylor standard map we compare the
probability densities of transversality for two different coordinate systems;
these are connected by a linear transformation that deforms splitting angles
through phase space, changing also the probability density of almost-zero
angles although complete tangencies are in fact invariant. This is completely
due to the PDF transformation law and strongly suggests that any statistical
inference from such distributions must be generally taken with care.Comment: 14 pages, 23 figures (This paper is for the IJBC Special Issue edited
by Prof. Gregoire Nicolis, Prof. Marko Robnik, Dr. Vassilis Rothos and Dr.
Haris Skokos

### Bidimensional Inequalities with an Ordinal Variable

We investigate the normative foundations of two empirically implementable dominance criteria for comparing distributions of two attributes, where the first one is cardinal while the second is ordinal. The criteria we consider are Atkinson and Bourguignon\'s (1982) first quasi-ordering and a generalization of Bourguignon\'s (1989) ordered poverty gap criterion. In each case we specify the restrictions to be placed on the individual utility functions, which guarantee that all utility-inequality averse welfarist ethical observers will rank the distributions under comparison in the same way as the dominance criterion. We also identify the elementary inequality reducing transformations successive applications of which permit to derive the dominating distribution from the dominated one.Normative Analysis, Utilitarianism, Welfarism, Bidimensional Stochastic Dominance, Inequality Reducing Transformations

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