24,010 research outputs found
The Euler characteristic of an enriched category
We define Euler characteristic of a category enriched by a monoidal model
category. If a monoidal model category V is equipped with Euler characteristic
that is compatible with weak equivalences and fibrations in V, then our Euler
characteristic is also compatible with weak equivalences and fibrations in the
model structure induced by that of V. In particular, we focus on the case of
topological categories; that is, categories enriched by the category of
topological spaces. As its application, we obtain the ordinary Euler
characteristic of a cellular stratified space X by computing the Euler
characteristic of the face category C(X) induced from X
Valuations in Nilpotent Minimum Logic
The Euler characteristic can be defined as a special kind of valuation on
finite distributive lattices. This work begins with some brief consideration on
the role of the Euler characteristic on NM algebras, the algebraic counterpart
of Nilpotent Minimum logic. Then, we introduce a new valuation, a modified
version of the Euler characteristic we call idempotent Euler characteristic. We
show that the new valuation encodes information about the formul{\ae} in NM
propositional logic
Finiteness obstructions and Euler characteristics of categories
We introduce notions of finiteness obstruction, Euler characteristic,
L^2-Euler characteristic, and M\"obius inversion for wide classes of
categories. The finiteness obstruction of a category Gamma of type (FP) is a
class in the projective class group K_0(RGamma); the functorial Euler
characteristic and functorial L^2-Euler characteristic are respectively its
RGamma-rank and L^2-rank. We also extend the second author's K-theoretic
M\"obius inversion from finite categories to quasi-finite categories. Our main
example is the proper orbit category, for which these invariants are
established notions in the geometry and topology of classifying spaces for
proper group actions. Baez-Dolan's groupoid cardinality and Leinster's Euler
characteristic are special cases of the L^2-Euler characteristic. Some of
Leinster's results on M\"obius-Rota inversion are special cases of the
K-theoretic M\"obius inversion.Comment: Final version, accepted for publication in the Advances in
Mathematics. Notational change: what was called chi(Gamma) in version 1 is
now called chi(BGamma), and chi(Gamma) now signifies the sum of the
components of the functorial Euler characteristic chi_f(Gamma). Theorem 5.25
summarizes when all Euler characteristics are equal. Minor typos have been
corrected. 88 page
Euler Characteristic in Odd Dimensions
It is well known that the Euler characteristic of an odd dimensional compact
manifold is zero. An Euler complex is a combinatorial analogue of a compact
manifold. We present here an elementary proof of the corresponding result for
Euler complexes
Manifolds with odd Euler characteristic and higher orientability
It is well-known that odd-dimensional manifolds have Euler characteristic
zero. Furthemore orientable manifolds have an even Euler characteristic unless
the dimension is a multiple of . We prove here a generalisation of these
statements: a -orientable manifold (or more generally Poincar\'e complex)
has even Euler characteristic unless the dimension is a multiple of ,
where we call a manifold -orientable if the Stiefel-Whitney class
vanishes for all (). More generally, we show that for a
-orientable manifold the Wu classes vanish for all that are not a
multiple of . For , -orientable manifolds with odd Euler
characteristic exist in all dimensions , but whether there exist a
4-orientable manifold with an odd Euler characteristic is an open question.Comment: 12 pages, main theorem extended in this versio
Manifolds with odd Euler characteristic and higher orientability
It is well-known that odd-dimensional manifolds have Euler characteristic
zero. Furthemore orientable manifolds have an even Euler characteristic unless
the dimension is a multiple of . We prove here a generalisation of these
statements: a -orientable manifold (or more generally Poincar\'e complex)
has even Euler characteristic unless the dimension is a multiple of ,
where we call a manifold -orientable if the Stiefel-Whitney class
vanishes for all (). More generally, we show that for a
-orientable manifold the Wu classes vanish for all that are not a
multiple of . For , -orientable manifolds with odd Euler
characteristic exist in all dimensions , but whether there exist a
4-orientable manifold with an odd Euler characteristic is an open question.Comment: 12 pages, main theorem extended in this versio
Streaming Algorithm for Euler Characteristic Curves of Multidimensional Images
We present an efficient algorithm to compute Euler characteristic curves of
gray scale images of arbitrary dimension. In various applications the Euler
characteristic curve is used as a descriptor of an image.
Our algorithm is the first streaming algorithm for Euler characteristic
curves. The usage of streaming removes the necessity to store the entire image
in RAM. Experiments show that our implementation handles terabyte scale images
on commodity hardware. Due to lock-free parallelism, it scales well with the
number of processor cores. Our software---CHUNKYEuler---is available as open
source on Bitbucket.
Additionally, we put the concept of the Euler characteristic curve in the
wider context of computational topology. In particular, we explain the
connection with persistence diagrams
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