718 research outputs found
The space of ultrametric phylogenetic trees
The reliability of a phylogenetic inference method from genomic sequence data
is ensured by its statistical consistency. Bayesian inference methods produce a
sample of phylogenetic trees from the posterior distribution given sequence
data. Hence the question of statistical consistency of such methods is
equivalent to the consistency of the summary of the sample. More generally,
statistical consistency is ensured by the tree space used to analyse the
sample.
In this paper, we consider two standard parameterisations of phylogenetic
time-trees used in evolutionary models: inter-coalescent interval lengths and
absolute times of divergence events. For each of these parameterisations we
introduce a natural metric space on ultrametric phylogenetic trees. We compare
the introduced spaces with existing models of tree space and formulate several
formal requirements that a metric space on phylogenetic trees must possess in
order to be a satisfactory space for statistical analysis, and justify them. We
show that only a few known constructions of the space of phylogenetic trees
satisfy these requirements. However, our results suggest that these basic
requirements are not enough to distinguish between the two metric spaces we
introduce and that the choice between metric spaces requires additional
properties to be considered. Particularly, that the summary tree minimising the
square distance to the trees from the sample might be different for different
parameterisations. This suggests that further fundamental insight is needed
into the problem of statistical consistency of phylogenetic inference methods.Comment: Minor changes. This version has been published in JTB. 27 pages, 9
figure
The simplicial boundary of a CAT(0) cube complex
For a CAT(0) cube complex , we define a simplicial flag complex
, called the \emph{simplicial boundary}, which is a
natural setting for studying non-hyperbolic behavior of . We compare
to the Roller, visual, and Tits boundaries of
and give conditions under which the natural CAT(1) metric on
makes it (quasi)isometric to the Tits boundary.
allows us to interpolate between studying geodesic
rays in and the geometry of its \emph{contact graph} , which is known to be quasi-isometric to a tree, and we characterize
essential cube complexes for which the contact graph is bounded. Using related
techniques, we study divergence of combinatorial geodesics in using
. Finally, we rephrase the rank-rigidity theorem of
Caprace-Sageev in terms of group actions on and
and state characterizations of cubulated groups with
linear divergence in terms of and .Comment: Lemma 3.18 was not stated correctly. This is fixed, and a minor
adjustment to the beginning of the proof of Theorem 3.19 has been made as a
result. Statements other than 3.18 do not need to change. I thank Abdul
Zalloum for the correction. See also: arXiv:2004.01182 (this version differs
from previous only by addition of the preceding link, at administrators'
request
On several varieties of cacti and their relations
Motivated by string topology and the arc operad, we introduce the notion of
quasi-operads and consider four (quasi)-operads which are different varieties
of the operad of cacti. These are cacti without local zeros (or spines) and
cacti proper as well as both varieties with fixed constant size one of the
constituting loops. Using the recognition principle of Fiedorowicz, we prove
that spineless cacti are equivalent as operads to the little discs operad. It
turns out that in terms of spineless cacti Cohen's Gerstenhaber structure and
Fiedorowicz' braided operad structure are given by the same explicit chains. We
also prove that spineless cacti and cacti are homotopy equivalent to their
normalized versions as quasi-operads by showing that both types of cacti are
semi-direct products of the quasi-operad of their normalized versions with a
re-scaling operad based on R>0. Furthermore, we introduce the notion of
bi-crossed products of quasi-operads and show that the cacti proper are a
bi-crossed product of the operad of cacti without spines and the operad based
on the monoid given by the circle group S^1. We also prove that this particular
bi-crossed operad product is homotopy equivalent to the semi-direct product of
the spineless cacti with the group S^1. This implies that cacti are equivalent
to the framed little discs operad. These results lead to new CW models for the
little discs and the framed little discs operad.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-13.abs.htm
Interconnection networks for parallel and distributed computing
Parallel computers are generally either shared-memory machines or distributed- memory machines. There are currently technological limitations on shared-memory architectures and so parallel computers utilizing a large number of processors tend tube distributed-memory machines. We are concerned solely with distributed-memory multiprocessors. In such machines, the dominant factor inhibiting faster global computations is inter-processor communication. Communication is dependent upon the topology of the interconnection network, the routing mechanism, the flow control policy, and the method of switching. We are concerned with issues relating to the topology of the interconnection network. The choice of how we connect processors in a distributed-memory multiprocessor is a fundamental design decision. There are numerous, often conflicting, considerations to bear in mind. However, there does not exist an interconnection network that is optimal on all counts and trade-offs have to be made. A multitude of interconnection networks have been proposed with each of these networks having some good (topological) properties and some not so good. Existing noteworthy networks include trees, fat-trees, meshes, cube-connected cycles, butterflies, MΓΆbius cubes, hypercubes, augmented cubes, k-ary n-cubes, twisted cubes, n-star graphs, (n, k)-star graphs, alternating group graphs, de Bruijn networks, and bubble-sort graphs, to name but a few. We will mainly focus on k-ary n-cubes and (n, k)-star graphs in this thesis. Meanwhile, we propose a new interconnection network called augmented k-ary n- cubes. The following results are given in the thesis.1. Let k β₯ 4 be even and let n β₯ 2. Consider a faulty k-ary n-cube Q(^k_n) in which the number of node faults f(_n) and the number of link faults f(_e) are such that f(_n) + f(_e) β€ 2n - 2. We prove that given any two healthy nodes s and e of Q(^k_n), there is a path from s to e of length at least k(^n) - 2f(_n) - 1 (resp. k(^n) - 2f(_n) - 2) if the nodes s and e have different (resp. the same) parities (the parity of a node Q(^k_n) in is the sum modulo 2 of the elements in the n-tuple over 0, 1, βββ , k - 1 representing the node). Our result is optimal in the sense that there are pairs of nodes and fault configurations for which these bounds cannot be improved, and it answers questions recently posed by Yang, Tan and Hsu, and by Fu. Furthermore, we extend known results, obtained by Kim and Park, for the case when n = 2.2. We give precise solutions to problems posed by Wang, An, Pan, Wang and Qu and by Hsieh, Lin and Huang. In particular, we show that Q(^k_n) is bi-panconnected and edge-bipancyclic, when k β₯ 3 and n β₯ 2, and we also show that when k is odd, Q(^k_n) is m-panconnected, for m = (^n(k - 1) + 2k - 6β / β_2), and (k -1) pancyclic (these bounds are optimal). We introduce a path-shortening technique, called progressive shortening, and strengthen existing results, showing that when paths are formed using progressive shortening then these paths can be efficiently constructed and used to solve a problem relating to the distributed simulation of linear arrays and cycles in a parallel machine whose interconnection network is Q(^k_n) even in the presence of a faulty processor.3. We define an interconnection network AQ(^k_n) which we call the augmented k-ary n-cube by extending a k-ary n-cube in a manner analogous to the existing extension of an n-dimensional hypercube to an n-dimensional augmented cube. We prove that the augmented k-ary n-cube Q(^k_n) has a number of attractive properties (in the context of parallel computing). For example, we show that the augmented k-ary n-cube Q(^k_n) - is a Cayley graph (and so is vertex-symmetric); has connectivity 4n - 2, and is such that we can build a set of 4n - 2 mutually disjoint paths joining any two distinct vertices so that the path of maximal length has length at most max{{n- l)k- (n-2), k + 7}; has diameter [(^k) / (_3)] + [(^k - 1) /( _3)], when n = 2; and has diameter at most (^k) / (_4) (n+ 1), for n β₯ 3 and k even, and at most [(^k)/ (_4) (n + 1) + (^n) / (_4), for n ^, for n β₯ 3 and k odd.4. We present an algorithm which given a source node and a set of n - 1 target nodes in the (n, k)-star graph S(_n,k) where all nodes are distinct, builds a collection of n - 1 node-disjoint paths, one from each target node to the source. The collection of paths output from the algorithm is such that each path has length at most 6k - 7, and the algorithm has time complexity O(k(^3)n(^4))
Property A and CAT(0) cube complexes
Property A is a non-equivariant analogue of amenability defined for metric spaces. Euclidean spaces and trees are examples of spaces with Property A. Simultaneously generalising these facts, we show that finite-dimensional CAT(0) cube complexes have Property A. We do not assume that the complex is locally finite. We also prove that given a discrete group acting properly on a finite-dimensional CAT(0) cube complex the stabilisers of vertices at infinity are amenable
Contracting Boundaries of CAT(0) Spaces
As demonstrated by Croke and Kleiner, the visual boundary of a CAT(0) group
is not well-defined since quasi-isometric CAT(0) spaces can have
non-homeomorphic boundaries. We introduce a new type of boundary for a CAT(0)
space, called the contracting boundary, made up rays satisfying one of five
hyperbolic-like properties. We prove that these properties are all equivalent
and that the contracting boundary is a quasi-isometry invariant. We use this
invariant to distinguish the quasi-isometry classes of certain right-angled
Coxeter groups.Comment: 27 pages, 8 figure
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