467 research outputs found
Generic Automorphisms and Green Fields
We show that the generic automorphism is axiomatisable in the green field of
Poizat (once Morleyised) as well as in the bad fields which are obtained by
collapsing this green field to finite Morley rank. As a corollary, we obtain
"bad pseudofinite fields" in characteristic 0. In both cases, we give geometric
axioms. In fact, a general framework is presented allowing this kind of
axiomatisation. We deduce from various constructibility results for algebraic
varieties in characteristic 0 that the green and bad fields fall into this
framework. Finally, we give similar results for other theories obtained by
Hrushovski amalgamation, e.g. the free fusion of two strongly minimal theories
having the definable multiplicity property. We also close a gap in the
construction of the bad field, showing that the codes may be chosen to be
families of strongly minimal sets.Comment: Some minor changes; new: a result of the paper (Cor 4.8) closes a gap
in the construction of the bad fiel
Endoscopic transfer of orbital integrals in large residual characteristic
This article constructs Shalika germs in the context of motivic integration,
both for ordinary orbital integrals and kappa-orbital integrals. Based on
transfer principles in motivic integration and on Waldspurger's endoscopic
transfer of smooth functions in characteristic zero, we deduce the endoscopic
transfer of smooth functions in sufficiently large residual characteristic.Comment: 33 page
Local integrability results in harmonic analysis on reductive groups in large positive characteristic
Let be a connected reductive algebraic group over a non-Archimedean local
field , and let be its Lie algebra. By a theorem of Harish-Chandra, if
has characteristic zero, the Fourier transforms of orbital integrals are
represented on the set of regular elements in by locally constant
functions, which, extended by zero to all of , are locally integrable. In
this paper, we prove that these functions are in fact specializations of
constructible motivic exponential functions. Combining this with the Transfer
Principle for integrability [R. Cluckers, J. Gordon, I. Halupczok, "Transfer
principles for integrability and boundedness conditions for motivic exponential
functions", preprint arXiv:1111.4405], we obtain that Harish-Chandra's theorem
holds also when is a non-Archimedean local field of sufficiently large
positive characteristic. Under the hypothesis on the existence of the mock
exponential map, this also implies local integrability of Harish-Chandra
characters of admissible representations of , where is an
equicharacteristic field of sufficiently large (depending on the root datum of
) characteristic.Comment: Compared to v2/v3: some proofs simplified, the main statement
generalized; slightly reorganized. Regarding the automatically generated text
overlap note: it overlaps with the Appendix B (which is part of
arXiv:1208.1945) written by us; the appendix and this article cross-reference
each other, and since the set-up is very similar, some overlap is unavoidabl
TorusE: Knowledge Graph Embedding on a Lie Group
Knowledge graphs are useful for many artificial intelligence (AI) tasks.
However, knowledge graphs often have missing facts. To populate the graphs,
knowledge graph embedding models have been developed. Knowledge graph embedding
models map entities and relations in a knowledge graph to a vector space and
predict unknown triples by scoring candidate triples. TransE is the first
translation-based method and it is well known because of its simplicity and
efficiency for knowledge graph completion. It employs the principle that the
differences between entity embeddings represent their relations. The principle
seems very simple, but it can effectively capture the rules of a knowledge
graph. However, TransE has a problem with its regularization. TransE forces
entity embeddings to be on a sphere in the embedding vector space. This
regularization warps the embeddings and makes it difficult for them to fulfill
the abovementioned principle. The regularization also affects adversely the
accuracies of the link predictions. On the other hand, regularization is
important because entity embeddings diverge by negative sampling without it.
This paper proposes a novel embedding model, TorusE, to solve the
regularization problem. The principle of TransE can be defined on any Lie
group. A torus, which is one of the compact Lie groups, can be chosen for the
embedding space to avoid regularization. To the best of our knowledge, TorusE
is the first model that embeds objects on other than a real or complex vector
space, and this paper is the first to formally discuss the problem of
regularization of TransE. Our approach outperforms other state-of-the-art
approaches such as TransE, DistMult and ComplEx on a standard link prediction
task. We show that TorusE is scalable to large-size knowledge graphs and is
faster than the original TransE.Comment: accepted for AAAI-1
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