451 research outputs found
A construction which relates c-freeness to infinitesimal freeness
We consider two extensions of free probability that have been studied in the
research literature, and are based on the notions of c-freeness and
respectively of infinitesimal freeness for noncommutative random variables. In
a 2012 paper, Belinschi and Shlyakhtenko pointed out a connection between these
two frameworks, at the level of their operations of 1-dimensional free additive
convolution. Motivated by that, we propose a construction which produces a
multi-variate version of the Belinschi-Shlyakhtenko result, together with a
result concerning free products of multi-variate noncommutative distributions.
Our arguments are based on the combinatorics of the specific types of cumulants
used in c-free and in infinitesimal free probability. They work in a rather
general setting, where the initial data consists of a vector space given
together with a linear map . In this setting, all
the needed brands of cumulants live in the guise of families of multilinear
functionals on , and our main result concerns a certain transformation
on such families of multilinear functionals.Comment: Version 2: Minor revision, added reference
Matroids, Feynman categories, and Koszul duality
We show that various combinatorial invariants of matroids such as Chow rings
and Orlik--Solomon algebras may be assembled into "operad-like" structures.
Specifically, one obtains several operads over a certain Feynman category which
we introduce and study in detail. In addition, we establish a Koszul-type
duality between Chow rings and Orlik--Solomon algebras, vastly generalizing a
celebrated result of Getzler. This provides a new interpretation of
combinatorial Leray models of Orlik--Solomon algebras.Comment: Should be an almost final versio
Natural Communication
In Natural Communication, the author criticizes the current paradigm of specific goal orientation in the complexity sciences. His model of "natural communication" encapsulates modern theoretical concepts from mathematics and physics, in particular category theory and quantum theory. The author is convinced that only by looking to the past is it possible to establish continuity and coherence in the complexity science
Symmetry in Finite Combinatorial Objects: Scalable Methods and Applications.
Symmetries of combinatorial objects are known to complicate search algorithms, but such obstacles can often be removed by detecting symmetries early and discarding symmetric subproblems. Canonical labeling of combinatorial objects facilitates easy equivalence checking through quick matching. All existing canonical-labeling software also finds symmetries, but the fastest symmetry-finding software does not perform canonical labeling. In this thesis, we describe highly scalable symmetry-detection algorithms for two widely-used combinatorial objects: graphs and Boolean functions. Our algorithms are based on a decision tree that combines elements of group-theoretic computation with branching and backtracking search. Moreover, we contrast the search for graph symmetries and a canonical labeling to dissect typical algorithms and identify their similarities and differences. We develop a novel approach to graph canonical labeling where symmetries are found first and then used to speed up the canonical-labeling routines. Empirical results are given for graphs with millions of vertices and Boolean functions with hundreds of I/Os, where our algorithms can often find all symmetry group generators or a canonical labeling in seconds.PHDComputer Science & EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/100003/1/hadik_1.pd
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