40,131 research outputs found
Joins for (Augmented) Simplicial Sets
We introduce a notion of join for (augmented) simplicial sets generalising
the classical join of geometric simplicial complexes. The definition comes
naturally from the ordinal sum on the base simplicial category .Comment: 8 page
The interplay of classes of algorithmically random objects
We study algorithmically random closed subsets of , algorithmically
random continuous functions from to , and algorithmically
random Borel probability measures on , especially the interplay
between these three classes of objects. Our main tools are preservation of
randomness and its converse, the no randomness ex nihilo principle, which say
together that given an almost-everywhere defined computable map between an
effectively compact probability space and an effective Polish space, a real is
Martin-L\"of random for the pushforward measure if and only if its preimage is
random with respect to the measure on the domain. These tools allow us to prove
new facts, some of which answer previously open questions, and reprove some
known results more simply.
Our main results are the following. First we answer an open question of
Barmapalias, Brodhead, Cenzer, Remmel, and Weber by showing that
is a random closed set if and only if it is the
set of zeros of a random continuous function on . As a corollary we
obtain the result that the collection of random continuous functions on
is not closed under composition. Next, we construct a computable
measure on the space of measures on such that
is a random closed set if and only if
is the support of a -random measure. We also establish a
correspondence between random closed sets and the random measures studied by
Culver in previous work. Lastly, we study the ranges of random continuous
functions, showing that the Lebesgue measure of the range of a random
continuous function is always contained in
Tommy Prince: Warrior
Over the last decade, there has been a flurry of interest in the Aboriginal men and women who served in the world wars and Korea. No one is more famous than Sergeant Thomas George Prince, MM (1919–77), one of the most decorated non-commissioned officers in Canadian military history. Yet he remains, to most Canadians, an unknown figure
Dynamical Systems on Networks: A Tutorial
We give a tutorial for the study of dynamical systems on networks. We focus
especially on "simple" situations that are tractable analytically, because they
can be very insightful and provide useful springboards for the study of more
complicated scenarios. We briefly motivate why examining dynamical systems on
networks is interesting and important, and we then give several fascinating
examples and discuss some theoretical results. We also briefly discuss
dynamical systems on dynamical (i.e., time-dependent) networks, overview
software implementations, and give an outlook on the field.Comment: 39 pages, 1 figure, submitted, more examples and discussion than
original version, some reorganization and also more pointers to interesting
direction
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