Department of Mathematics
http://hdl.handle.net/1957/13737
2015-04-22T01:48:12ZInequalities for positive rank and crank moments of overpartitions
http://hdl.handle.net/1957/55598
Inequalities for positive rank and crank moments of overpartitions
Larsen, Acadia; Rust, Alexa; Swisher, Holly
In recent work, Andrews, Chan, and Kim extend a result of Garvan about even rank
and crank moments of partitions to positive moments. In a similar fashion we extend a
result of Mao about even rank moments of overpartitions. We investigate positive Dyson-rank,
M₂-rank, first residual crank, and second residual crank moments of overpartitions.
In particular, we prove a conjecture of Mao which states that the positive Dyson-rank
moments are larger than the positive M₂-rank moments. We also prove some additional
inequalities involving rank and crank moments of overpartitions, including an interlacing
property.
This is an author's peer-reviewed final manuscript, as accepted by the publisher. The published article is copyrighted by the World Scientific Publishing Company and can be found at: http://www.worldscientific.com/worldscinet/ijnt
2014-12-01T00:00:00ZSEK: sparsity exploiting k-mer-based estimation of bacterial community composition
http://hdl.handle.net/1957/55141
SEK: sparsity exploiting k-mer-based estimation of bacterial community composition
Chatterjee, Saikat; Koslicki, David; Dong, Siyuan; et al.
MOTIVATION: Estimation of bacterial community composition from
a high-throughput sequenced sample is an important task in
metagenomics applications. Since the sample sequence data
typically harbors reads of variable lengths and different levels of
biological and technical noise, accurate statistical analysis of such
data is challenging. Currently popular estimation methods are
typically very time consuming in a desktop computing environment.
RESULTS: Using sparsity enforcing methods from the general sparse
signal processing field (such as compressed sensing), we derive
a solution to the community composition estimation problem by a
simultaneous assignment of all sample reads to a pre-processed
reference database. A general statistical model based on kernel
density estimation techniques is introduced for the assignment task
and the model solution is obtained using convex optimization tools.
Further, we design a greedy algorithm solution for a fast solution. Our
approach offers a reasonably fast community composition estimation
method which is shown to be more robust to input data variation than
a recently introduced related method.
AVAILABILITY: A platform-independent Matlab implementation of the
method is freely available at http://www.ee.kth.se/ctsoftware; source
code that does not require access to Matlab is currently being tested
and will be made available later through the above website.
This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Bioinformatics following peer review. The definitive publisher-authenticated version, Chatterjee, S., Koslicki, D., Dong, S., Innocenti, N., Cheng, L., Lan, Y., ... & Corander, J. (2014). SEK: Sparsity exploiting k-mer-based estimation of bacterial community composition. Bioinformatics, 30(17), 2423-2431. doi:10.1093/bioinformatics/btu320, is available online at: http://bioinformatics.oxfordjournals.org/content/30/17/2423.
The published article is copyrighted by the Author(s) and published by Oxford University Press.
2014-09-01T00:00:00ZCoding sequence density estimation via topological pressure
http://hdl.handle.net/1957/55140
Coding sequence density estimation via topological pressure
Koslicki, David; Thompson, Daniel J.
We give a new approach to coding sequence (CDS) density
estimation in genomic analysis based on the topological pressure, which
we develop from a well known concept in ergodic theory. Topological
pressure measures the ‘weighted information content’ of a finite word,
and incorporates 64 parameters which can be interpreted as a choice
of weight for each nucleotide triplet. We train the parameters so that
the topological pressure fits the observed coding sequence density on
the human genome, and use this to give ab initio predictions of CDS
density over windows of size around 66,000bp on the genomes of Mus
Musculus, Rhesus Macaque and Drososphilia Melanogaster. While the
differences between these genomes are too great to expect that training
on the human genome could predict, for example, the exact locations of
genes, we demonstrate that our method gives reasonable estimates for
the ‘coarse scale’ problem of predicting CDS density.
Inspired again by ergodic theory, the weightings of the nucleotide
triplets obtained from our training procedure are used to define a probability
distribution on finite sequences, which can be used to distinguish
between intron and exon sequences from the human genome of lengths
between 750bp and 5,000bp. At the end of the paper, we explain the
theoretical underpinning for our approach, which is the theory of Thermodynamic
Formalism from the dynamical systems literature. Mathematica
and MATLAB implementations of our method are available at
http://sourceforge.net/projects/topologicalpres/.
This is an author's peer-reviewed final manuscript, as accepted by the publisher. The published article is copyrighted by Springer and can be found at: http://link.springer.com/journal/285
2015-01-01T00:00:00ZHomogeneity groups of ends of open 3-manifolds
http://hdl.handle.net/1957/55121
Homogeneity groups of ends of open 3-manifolds
Garity, Dennis J.; Repovš, Dušan
For every finitely generated abelian group G, we construct an irreducible
open 3-manifold M[subscript G] whose end set is homeomorphic to a Cantor set and
whose homogeneity group is isomorphic to G. The end homogeneity group
is the group of self-homeomorphisms of the end set that extend to homeomorphisms
of the 3-manifold. The techniques involve computing the embedding
homogeneity groups of carefully constructed Antoine-type Cantor sets
made up of rigid pieces. In addition, a generalization of an Antoine Cantor
set using infinite chains is needed to construct an example with integer homogeneity
group. Results about the local genus of points in Cantor sets and
about the geometric index are also used.
This is the publisher’s final pdf. The published article is copyrighted by the Mathematical Sciences Publishers and can be found at: http://msp.org/pjm/2014/269-1/index.xhtml.
2014-07-15T00:00:00Z