6,092 research outputs found
Higher Segal spaces I
This is the first paper in a series on new higher categorical structures
called higher Segal spaces. For every d > 0, we introduce the notion of a
d-Segal space which is a simplicial space satisfying locality conditions
related to triangulations of cyclic polytopes of dimension d. In the case d=1,
we recover Rezk's theory of Segal spaces. The present paper focuses on 2-Segal
spaces. The starting point of the theory is the observation that Hall algebras,
as previously studied, are only the shadow of a much richer structure governed
by a system of higher coherences captured in the datum of a 2-Segal space. This
2-Segal space is given by Waldhausen's S-construction, a simplicial space
familiar in algebraic K-theory. Other examples of 2-Segal spaces arise
naturally in classical topics such as Hecke algebras, cyclic bar constructions,
configuration spaces of flags, solutions of the pentagon equation, and mapping
class groups.Comment: 221 page
Euclidean distance geometry and applications
Euclidean distance geometry is the study of Euclidean geometry based on the
concept of distance. This is useful in several applications where the input
data consists of an incomplete set of distances, and the output is a set of
points in Euclidean space that realizes the given distances. We survey some of
the theory of Euclidean distance geometry and some of the most important
applications: molecular conformation, localization of sensor networks and
statics.Comment: 64 pages, 21 figure
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