414,436 research outputs found
Counting the spanning trees of the 3-cube using edge slides
We give a direct combinatorial proof of the known fact that the 3-cube has
384 spanning trees, using an "edge slide" operation on spanning trees. This
gives an answer in the case n=3 to a question implicitly raised by Stanley. Our
argument also gives a bijective proof of the n=3 case of a weighted count of
the spanning trees of the n-cube due to Martin and Reiner.Comment: 17 pages, 9 figures. v2: Final version as published in the
Australasian Journal of Combinatorics. Section 5 shortened and restructured;
references added; one figure added; some typos corrected; additional minor
changes in response to the referees' comment
Note on islands in path-length sequences of binary trees
An earlier characterization of topologically ordered (lexicographic)
path-length sequences of binary trees is reformulated in terms of an
integrality condition on a scaled Kraft sum of certain subsequences (full
segments, or islands). The scaled Kraft sum is seen to count the set of
ancestors at a certain level of a set of topologically consecutive leaves is a
binary tree.Comment: 4 page
Boosting insights in insurance tariff plans with tree-based machine learning methods
Pricing actuaries typically operate within the framework of generalized
linear models (GLMs). With the upswing of data analytics, our study puts focus
on machine learning methods to develop full tariff plans built from both the
frequency and severity of claims. We adapt the loss functions used in the
algorithms such that the specific characteristics of insurance data are
carefully incorporated: highly unbalanced count data with excess zeros and
varying exposure on the frequency side combined with scarce, but potentially
long-tailed data on the severity side. A key requirement is the need for
transparent and interpretable pricing models which are easily explainable to
all stakeholders. We therefore focus on machine learning with decision trees:
starting from simple regression trees, we work towards more advanced ensembles
such as random forests and boosted trees. We show how to choose the optimal
tuning parameters for these models in an elaborate cross-validation scheme, we
present visualization tools to obtain insights from the resulting models and
the economic value of these new modeling approaches is evaluated. Boosted trees
outperform the classical GLMs, allowing the insurer to form profitable
portfolios and to guard against potential adverse risk selection
A lower bound for nodal count on discrete and metric graphs
According to a well-know theorem by Sturm, a vibrating string is divided into
exactly N nodal intervals by zeros of its N-th eigenfunction. Courant showed
that one half of Sturm's theorem for the strings applies to the theory of
membranes: N-th eigenfunction cannot have more than N domains. He also gave an
example of a eigenfunction high in the spectrum with a minimal number of nodal
domains, thus excluding the existence of a non-trivial lower bound. An analogue
of Sturm's result for discretizations of the interval was discussed by
Gantmacher and Krein. The discretization of an interval is a graph of a simple
form, a chain-graph. But what can be said about more complicated graphs? It has
been known since the early 90s that the nodal count for a generic eigenfunction
of the Schrodinger operator on quantum trees (where each edge is identified
with an interval of the real line and some matching conditions are enforced on
the vertices) is exact too: zeros of the N-th eigenfunction divide the tree
into exactly N subtrees. We discuss two extensions of this result in two
directions. One deals with the same continuous Schrodinger operator but on
general graphs (i.e. non-trees) and another deals with discrete Schrodinger
operator on combinatorial graphs (both trees and non-trees). The result that we
derive applies to both types of graphs: the number of nodal domains of the N-th
eigenfunction is bounded below by N-L, where L is the number of links that
distinguish the graph from a tree (defined as the dimension of the cycle space
or the rank of the fundamental group of the graph). We also show that if it the
genericity condition is dropped, the nodal count can fall arbitrarily far below
the number of the corresponding eigenfunction.Comment: 15 pages, 4 figures; Minor corrections: added 2 important reference
Combinatorics of Rooted Trees and Hopf Algebras
We begin by considering the graded vector space with a basis consisting of
rooted trees, graded by the count of non-root vertices. We define two linear
operators on this vector space, the growth and pruning operators, which
respectively raise and lower grading; their commutator is the operator that
multiplies a rooted tree by its number of vertices. We define an inner product
with respect to which the growth and pruning operators are adjoint, and obtain
several results about the multiplicities associated with each operator.
The symmetric algebra on the vector space of rooted trees (after a degree
shift) can be endowed with a coproduct to make a Hopf algebra; this was defined
by Kreimer in connection with renormalization. We extend the growth and pruning
operators, as well as the inner product mentioned above, to Kreimer's Hopf
algebra. On the other hand, the vector space of rooted trees itself can be
given a noncommutative multiplication: with an appropriate coproduct, this
gives the Hopf algebra of Grossman and Larson. We show the inner product on
rooted trees leads to an isomorphism of the Grossman-Larson Hopf algebra with
the graded dual of Kreimer's Hopf algebra, correcting an earlier result of
Panaite.Comment: 19 pages; final revision has minor corrections, slightly expanded
sect. 4 and additional reference
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