4,446 research outputs found
Redundancy of minimal weight expansions in Pisot bases
Motivated by multiplication algorithms based on redundant number
representations, we study representations of an integer as a sum , where the digits are taken from a finite alphabet
and is a linear recurrent sequence of Pisot type with
. The most prominent example of a base sequence is the
sequence of Fibonacci numbers. We prove that the representations of minimal
weight are recognised by a finite automaton and obtain an
asymptotic formula for the average number of representations of minimal weight.
Furthermore, we relate the maximal order of magnitude of the number of
representations of a given integer to the joint spectral radius of a certain
set of matrices
On depth zero L-packets for classical groups
By computing reducibility points of parabolically induced representations, we construct, to within at most two unramified quadratic characters, the Langlands parameter of an arbitrary depth zero irreducible cuspidal representation (Formula presented.) of a classical group (which may be not-quasi-split) over a non-archimedean local field of odd residual characteristic. From this, we can explicitly describe all the irreducible cuspidal representations in the union of one, two, or four (Formula presented.) -packets, containing (Formula presented.). These results generalize the work of DeBacker–Reeder (in the case of classical groups) from regular to arbitrary tame Langlands parameters
Towards MKM in the Large: Modular Representation and Scalable Software Architecture
MKM has been defined as the quest for technologies to manage mathematical
knowledge. MKM "in the small" is well-studied, so the real problem is to scale
up to large, highly interconnected corpora: "MKM in the large". We contend that
advances in two areas are needed to reach this goal. We need representation
languages that support incremental processing of all primitive MKM operations,
and we need software architectures and implementations that implement these
operations scalably on large knowledge bases.
We present instances of both in this paper: the MMT framework for modular
theory-graphs that integrates meta-logical foundations, which forms the base of
the next OMDoc version; and TNTBase, a versioned storage system for XML-based
document formats. TNTBase becomes an MMT database by instantiating it with
special MKM operations for MMT.Comment: To appear in The 9th International Conference on Mathematical
Knowledge Management: MKM 201
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