661 research outputs found
The Igusa modular forms and ``the simplest'' Lorentzian Kac--Moody algebras
We find automorphic corrections for the Lorentzian Kac--Moody algebras with
the simplest generalized Cartan matrices of rank 3:
A_{1,0} =
2 0 -1
0 2 -2
-1 -2 2
and
A_{1,I} =
2 -2 -1
-2 2 -1
-1 -1 2
For A_{1,0} this correction is given by the Igusa Sp_4(Z)-modular form
\chi_{35} of weight 35, and for A_{1,I} by a Siege modular form of weight 30
with respect to a 2-congruence subgroup. We find infinite product or sum
expansions for these forms. Our method of construction of \chi_{35} leads to
the direct construction of Siegel modular forms by infinite product expansions,
whose divisors are the Humbert surfaces with fixed discriminants. Existence of
these forms was proved by van der Geer in 1982 using some geometrical
consideration. We announce a list of all hyperbolic symmetric generalized
Cartan matrices A of rank 3 such that A has elliptic or parabolic type, A has a
lattice Weyl vector, and A contains the affine submatrix \tilde{A}_1.Comment: 40 pages, no figures. AMS-Te
Reflection groups in hyperbolic spaces and the denominator formula for Lorentzian Kac--Moody Lie algebras
This is a continuation of our "Lecture on Kac--Moody Lie algebras of the
arithmetic type" \cite{25}.
We consider hyperbolic (i.e. signature ) integral symmetric bilinear
form (i.e. hyperbolic lattice), reflection group
, fundamental polyhedron \Cal M of and an acceptable
(corresponding to twisting coefficients) set P({\Cal M})\subset M of vectors
orthogonal to faces of \Cal M (simple roots). One can construct the
corresponding Lorentzian Kac--Moody Lie algebra {\goth g}={\goth
g}^{\prime\prime}(A(S,W,P({\Cal M}))) which is graded by .
We show that \goth g has good behavior of imaginary roots, its denominator
formula is defined in a natural domain and has good automorphic properties if
and only if \goth g has so called {\it restricted arithmetic type}. We show
that every finitely generated (i.e. P({\Cal M}) is finite) algebra {\goth
g}^{\prime\prime}(A(S,W_1,P({\Cal M}_1))) may be embedded to {\goth
g}^{\prime\prime}(A(S,W,P({\Cal M}))) of the restricted arithmetic type. Thus,
Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type is a
natural class to study.
Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type have the
best automorphic properties for the denominator function if they have {\it a
lattice Weyl vector }. Lorentzian Kac--Moody Lie algebras of the
restricted arithmetic type with generalized lattice Weyl vector are
called {\it elliptic}Comment: Some corrections in Sects. 2.1, 2.2 were done. They don't reflect on
results and ideas. 31 pages, no figures. AMSTe
Neural Distributed Autoassociative Memories: A Survey
Introduction. Neural network models of autoassociative, distributed memory
allow storage and retrieval of many items (vectors) where the number of stored
items can exceed the vector dimension (the number of neurons in the network).
This opens the possibility of a sublinear time search (in the number of stored
items) for approximate nearest neighbors among vectors of high dimension. The
purpose of this paper is to review models of autoassociative, distributed
memory that can be naturally implemented by neural networks (mainly with local
learning rules and iterative dynamics based on information locally available to
neurons). Scope. The survey is focused mainly on the networks of Hopfield,
Willshaw and Potts, that have connections between pairs of neurons and operate
on sparse binary vectors. We discuss not only autoassociative memory, but also
the generalization properties of these networks. We also consider neural
networks with higher-order connections and networks with a bipartite graph
structure for non-binary data with linear constraints. Conclusions. In
conclusion we discuss the relations to similarity search, advantages and
drawbacks of these techniques, and topics for further research. An interesting
and still not completely resolved question is whether neural autoassociative
memories can search for approximate nearest neighbors faster than other index
structures for similarity search, in particular for the case of very high
dimensional vectors.Comment: 31 page
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