69,124 research outputs found
Approximation properties of simple Lie groups made discrete
In this paper we consider the class of connected simple Lie groups equipped
with the discrete topology. We show that within this class of groups the
following approximation properties are equivalent: (1) the Haagerup property;
(2) weak amenability; (3) the weak Haagerup property. In order to obtain the
above result we prove that the discrete group GL(2,K) is weakly amenable with
constant 1 for any field K.Comment: 15 pages. Final version. To appear in J. Lie Theor
Infinitely many solutions to the Yamabe problem on noncompact manifolds
We establish the existence of infinitely many complete metrics with constant scalar curvature on prescribed conformal classes on certain noncompact product manifolds. These include products of closed manifolds with constant positive scalar curvature and simply-connected symmetric spaces of noncompact or Euclidean type; in particular, , , , and , . As a consequence, we obtain infinitely many periodic solutions to the singular Yamabe problem on , for all , the maximal range where nonuniqueness is possible. We also show that all Bieberbach groups in are periods of bifurcating branches of solutions to the Yamabe problem on , ,
Complex dimensions and their observability
We show that the dimension of spacetime becomes complex-valued when its
short-scale geometry is invariant under a discrete scaling symmetry. This
characteristic can generically arise in quantum gravities, for instance, in
those based on combinatorial or multifractal structures or as the partial
breaking of continuous dilation symmetry in any conformal-invariant theory.
With its infinite scale hierarchy, discrete scale invariance overlaps with the
traditional separation between ultraviolet and infrared physics and it can
leave an all-range observable imprint, such as a pattern of log oscillations
and sharp features in the cosmic microwave background primordial power
spectrum.Comment: 6 pages, 1 figure. v2: discussion slightly expande
Discrete integrable systems generated by Hermite-Pad\'e approximants
We consider Hermite-Pad\'e approximants in the framework of discrete
integrable systems defined on the lattice . We show that the
concept of multiple orthogonality is intimately related to the Lax
representations for the entries of the nearest neighbor recurrence relations
and it thus gives rise to a discrete integrable system. We show that the
converse statement is also true. More precisely, given the discrete integrable
system in question there exists a perfect system of two functions, i.e., a
system for which the entire table of Hermite-Pad\'e approximants exists. In
addition, we give a few algorithms to find solutions of the discrete system.Comment: 20 page
Automorphisms of algebras and Bochner`s property for discrete vector orthogonal polynomials
We construct new families of discrete vector orthogonal polynomials that have
the property to be eigenfunctions of some difference operator. They are
extensions of Charlier, Meixner and Kravchuk polynomial systems. The ideas
behind our approach lie in the studies of bispectral operators. We exploit
automorphisms of associative algebras which transform elementary (vector)
orthogonal polynomial systems which are eigenfunctions of a difference operator
into other systems of this type. While the extension of Charlier polynomilas is
well known it is obtained by different methods. The extension of Meixner
polynomial system is new.Comment: 14 pages. arXiv admin note: text overlap with arXiv:1512.0389
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