280 research outputs found
Developments in the Khintchine-Meinardus probabilistic method for asymptotic enumeration
A theorem of Meinardus provides asymptotics of the number of weighted
partitions under certain assumptions on associated ordinary and Dirichlet
generating functions. The ordinary generating functions are closely related to
Euler's generating function for partitions, where
. By applying a method due to Khintchine, we extend Meinardus'
theorem to find the asymptotics of the coefficients of generating functions of
the form for sequences , and
general . We also reformulate the hypotheses of the theorem in terms of
generating functions. This allows us to prove rigorously the asymptotics of
Gentile statistics and to study the asymptotics of combinatorial objects with
distinct components.Comment: 28 pages, This is the final version that incorporated referee's
remarks.The paper will be published in Electronic Journal of Combinatoric
Asymptotics for incidence matrix classes
We define {\em incidence matrices} to be zero-one matrices with no zero rows
or columns. A classification of incidence matrices is considered for which
conditions of symmetry by transposition, having no repeated rows/columns, or
identification by permutation of rows/columns are imposed. We find asymptotics
and relationships for the number of matrices with ones in these classes as
.Comment: updated and slightly expanded versio
Asymptotic enumeration of 2-covers and line graphs
In this paper we find asymptotic enumerations for the number of line graphs
on -labelled vertices and for different types of related combinatorial
objects called 2-covers.
We find that the number of 2-covers, , and proper 2-covers, , on
both have asymptotic growth where is the th Bell number, while the number of
restricted 2-covers, , restricted, proper 2-covers on , , and
line graphs , all have growth
In our proofs we use probabilistic arguments for the unrestricted types of
2-covers and and generating function methods for the restricted types of
2-covers and line graphs
First Occurrence in Pairs of Long Words: A Penney-ante Conjecture of Pevzner
Suppose X1, X2, is a sequence of independent and identically distributed random elements whose values are taken in a finite set S of size |S| ≥ 2 with probability distribution (X = s) = p(s) > 0 for s S. Pevzner has conjectured that for every probability distribution there exists an N > 0 such that for every word A with letters in S whose length is at least N, there exists a second word B of the same length as A, such that the event that B appears before A in the sequence X1, X2, has greater probability than that of A appearing before B. In this paper it is shown that a distribution satisfies Pevzner's conclusion if and only if the maximum value of , p, and the secondary maximum c satisfy the inequality . For |S| = 2 or |S| = 3, the inequality is true and the conjecture holds. If , then the conjecture is true when A is not allowed to consist of pure repetitions of that unique element for which the distribution takes on its mod
Meinardus' theorem on weighted partitions: extensions and a probabilistic proof
We give a probalistic proof of the famous Meinardus' asymptotic formula for
the number of weighted partitions with weakened one of the three Meinardus'
conditions, and extend the resulting version of the theorem to other two
classis types of decomposable combinatorial structures, which are called
assemblies and selections. The results obtained are based on combining
Meinardus' analytical approach with probabilistic method of Khitchine.Comment: The version contains a few minor corrections.It will be published in
Advances in Applied Mathematic
Single-cell mutational burden distributions in birth-death processes
Genetic mutations are footprints of cancer evolution and reveal critical
dynamic parameters of tumour growth, which otherwise are hard to measure in
vivo. The mutation accumulation in tumour cell populations has been described
by various statistics, such as site frequency spectra (SFS) from bulk or
single-cell data, as well as single-cell division distributions (DD) and
mutational burden distributions (MBD). An integrated understanding of these
distributions obtained from different sequencing information is important to
illuminate the ecological and evolutionary dynamics of tumours, and requires
novel mathematical and computational tools. We introduce dynamical matrices to
analyse and unite the SFS, DD and MBD based on a birth-death process. Using the
Markov nature of the model, we derive recurrence relations for the expectations
of these three distributions. While recovering classic exact results in
pure-birth cases for the SFS and the DD through our new framework, we also
derive a new expression for the MBD as well as approximations for all three
distributions when death is introduced, confirming our results with stochastic
simulations. Moreover, we demonstrate a natural link between the SFS and the
single-cell MBD, and show that the MBD can be regenerated through the DD.
Surprisingly, the single-cell MBD is mainly driven by the stochasticity arising
in the DD, rather than the extra stochasticity in the number of mutations at
each cell division.Comment: 27 pages, 6 figure
The Probability of Non-Existence of a Subgraph in a Moderately Sparse Random Graph
We develop a general procedure that finds recursions for statistics counting
isomorphic copies of a graph in the common random graph models and . Our results apply when the average degrees of the
random graphs are below the threshold at which each edge is included in a copy
of . This extends an argument given earlier by the second author for
with a more restricted range of average degree. For all strictly
balanced subgraphs , our results gives much information on the
distribution of the number of copies of that are not in large "clusters"
of copies. The probability that a random graph in has no copies
of is shown to be given asymptotically by the exponential of a power
series in and , over a fairly wide range of . A corresponding result
is also given for , which gives an asymptotic formula for the
number of graphs with vertices, edges and no copies of , for the
applicable range of . An example is given, computing the asymptotic
probability that a random graph has no triangles for in and for in , extending results of the
second author.Comment: 44 page
A Meinardus theorem with multiple singularities
Meinardus proved a general theorem about the asymptotics of the number of
weighted partitions, when the Dirichlet generating function for weights has a
single pole on the positive real axis. Continuing \cite{GSE}, we derive
asymptotics for the numbers of three basic types of decomposable combinatorial
structures (or, equivalently, ideal gas models in statistical mechanics) of
size , when their Dirichlet generating functions have multiple simple poles
on the positive real axis. Examples to which our theorem applies include ones
related to vector partitions and quantum field theory. Our asymptotic formula
for the number of weighted partitions disproves the belief accepted in the
physics literature that the main term in the asymptotics is determined by the
rightmost pole.Comment: 26 pages. This version incorporates the following two changes implied
by referee's remarks: (i) We made changes in the proof of Proposition 1; (ii)
We provided an explanation to the argument for the local limit theorem. The
paper is tentatively accepted by "Communications in Mathematical Physics"
journa
Nonlinear wavelength conversion in photonic crystal fibers with three zero dispersion points
In this theoretical study, we show that a simple endlessly single-mode
photonic crystal fiber can be designed to yield, not just two, but three
zero-dispersion wavelengths. The presence of a third dispersion zero creates a
rich phase-matching topology, enabling enhanced control over the spectral
locations of the four-wave-mixing and resonant-radiation bands emitted by
solitons and short pulses. The greatly enhanced flexibility in the positioning
of these bands has applications in wavelength conversion, supercontinuum
generation and pair-photon sources for quantum optics
Asymptotic enumeration of incidence matrices
We discuss the problem of counting {\em incidence matrices}, i.e. zero-one
matrices with no zero rows or columns. Using different approaches we give three
different proofs for the leading asymptotics for the number of matrices with
ones as . We also give refined results for the asymptotic
number of incidence matrices with ones.Comment: jpconf style files. Presented at the conference "Counting Complexity:
An international workshop on statistical mechanics and combinatorics." In
celebration of Prof. Tony Guttmann's 60th birthda
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