39,933 research outputs found
The random graph
Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique
(and highly symmetric) countably infinite random graph. This graph, and its
automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul
Erd\H{o}s
Sums over Graphs and Integration over Discrete Groupoids
We show that sums over graphs such as appear in the theory of Feynman
diagrams can be seen as integrals over discrete groupoids. From this point of
view, basic combinatorial formulas of the theory of Feynman diagrams can be
interpreted as pull-back or push-forward formulas for integrals over suitable
groupoids.Comment: 27 pages, 4 eps figures; LaTeX2e; uses Xy-Pic. Some ambiguities
fixed, and several proofs simplifie
Direct sums and products in topological groups and vector spaces
We call a subset of an abelian topological group : (i)
provided that for every open neighbourhood of one
can find a finite set such that the subgroup generated by
is contained in ; (ii) if, for every
family of integer numbers, there exists such that the
net \left\{\sum_{a\in F} z_a a: F\subseteq A\mbox{ is finite}\right\}
converges to ; (iii) provided that and for every neighbourhood of there exists a neighbourhood of
such that, for every finite set and each set of integers, implies that for all
. We prove that: (1) an abelian topological group contains a direct
product (direct sum) of -many non-trivial topological groups if and
only if it contains a topologically independent, absolutely (Cauchy) summable
subset of cardinality ; (2) a topological vector space contains
as its subspace if and only if it has an infinite
absolutely Cauchy summable set; (3) a topological vector space contains
as its subspace if and only if it has an
multiplier convergent series of non-zero elements.
We answer a question of Hu\v{s}ek and generalize results by
Bessaga-Pelczynski-Rolewicz, Dominguez-Tarieladze and Lipecki
Algebraic K-theory of group rings and the cyclotomic trace map
We prove that the Farrell-Jones assembly map for connective algebraic
K-theory is rationally injective, under mild homological finiteness conditions
on the group and assuming that a weak version of the Leopoldt-Schneider
conjecture holds for cyclotomic fields. This generalizes a result of
B\"okstedt, Hsiang, and Madsen, and leads to a concrete description of a large
direct summand of in terms
of group homology. In many cases the number theoretic conjectures are true, so
we obtain rational injectivity results about assembly maps, in particular for
Whitehead groups, under homological finiteness assumptions on the group only.
The proof uses the cyclotomic trace map to topological cyclic homology,
B\"okstedt-Hsiang-Madsen's functor C, and new general isomorphism and
injectivity results about the assembly maps for topological Hochschild homology
and C.Comment: To appear in Advances in Mathematics. 77 page
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