33,867 research outputs found
Combinatorial problems related to sequences with repeated entries
Student Number : 9708525G -
PhD thesis -
School of Mathematics -
Faculty of ScienceSequences of numbers have important applications in the field of Computer Science.
As a result they have become increasingly regarded in Mathematics, since analysis
can be instrumental in investigating algorithms.
Three concepts are discussed in this thesis, all of which are concerned with ‘words’
or ‘sequences’ of natural numbers where repeated letters are allowed:
• The number of distinct values in a sequence with geometric distri-
bution
In Part I, a sample which is geometrically distributed is considered, with the
objective of counting how many different letters occur at least once in the
sample. It is concluded that the number of distinct letters grows like log n as
n → ∞. This is then generalised to the question of how many letters occur
at least b times in a word.
• The position of the maximum (and/or minimum) in a sequence
with geometric distribution
Part II involves many variations on the central theme which addresses the
question: “What is the probability that the maximum in a geometrically distributed
sample occurs in the first d letters of a word of length n?” (assuming
d ≤ n). Initially, d is considered fixed, but in later chapters d is allowed to
grow with n. It is found that for 1 ≤ d = o(n), the results are the same as
when d is fixed.
• The average depth of a key in a binary search tree formed from a
sequence with repeated entries
Lastly, in Part III, random sequences are examined where repeated letters
are allowed. First, the average left-going depth of the first one is found,
and later the right-going path to the first r if the alphabet is {1, . . . , r} is
examined. The final chapter uses a merge (or ‘shuffle’) operator to obtain
the average depth of an arbitrary node, which can be expressed in terms of
the left-going and right-going depths
Structural origin of the midgap electronic states and the Urbach tail in pnictogen-chalcogenide glasses
We determine the electronic density of states for computationally-generated
bulk samples of amorphous chalcogenide alloys AsSe. The samples
were generated using a structure-building algorithm reported recently by us
({J. Chem. Phys.} , 114505). Several key features of the calculated
density of states are in good agreement with experiment: The trend of the
mobility gap with arsenic content is reproduced. The sample-to-sample variation
in the energies of states near the mobility gap is quantitatively consistent
with the width of the Urbach tail in the optical edge observed in experiment.
Most importantly, our samples consistently exhibit very deep-lying midgap
electronic states that are delocalized significantly more than what would be
expected for a deep impurity or defect state; the delocalization is highly
anisotropic. These properties are consistent with those of the topological
midgap electronic states that have been proposed by Zhugayevych and Lubchenko
as an explanation for several puzzling opto-electronic anomalies observed in
the chalcogenides, including light-induced midgap absorption and ESR signal,
and anomalous photoluminescence. In a complement to the traditional view of the
Urbach states as a generic consequence of disorder in atomic positions, the
present results suggest these states can be also thought of as intimate pairs
of topological midgap states that cannot recombine because of disorder.
Finally, samples with an odd number of electrons exhibit neutral, spin
midgap states as well as polaron-like configurations that consist of a charge
carrier bound to an intimate pair of midgap states; the polaron's
identity---electron or hole---depends on the preparation protocol of the
sample.Comment: submitted to J Phys Chem
A Monte Carlo exploration of threefold base geometries for 4d F-theory vacua
We use Monte Carlo methods to explore the set of toric threefold bases that
support elliptic Calabi-Yau fourfolds for F-theory compactifications to four
dimensions, and study the distribution of geometrically non-Higgsable gauge
groups, matter, and quiver structure. We estimate the number of distinct
threefold bases in the connected set studied to be . The
distribution of bases peaks around . All bases encountered
after "thermalization" have some geometric non-Higgsable structure. We find
that the number of non-Higgsable gauge group factors grows roughly linearly in
of the threefold base. Typical bases have isolated gauge
factors as well as several larger connected clusters of gauge factors with
jointly charged matter. Approximately 76% of the bases sampled contain
connected two-factor gauge group products of the form SU(3)SU(2), which
may act as the non-Abelian part of the standard model gauge group.
SU(3)SU(2) is the third most common connected two-factor product group,
following SU(2)SU(2) and SU(2), which arise more frequently.Comment: 38 pages, 22 figure
The Adaptive Sampling Revisited
The problem of estimating the number of distinct keys of a large
collection of data is well known in computer science. A classical algorithm
is the adaptive sampling (AS). can be estimated by , where is
the final bucket (cache) size and is the final depth at the end of the
process. Several new interesting questions can be asked about AS (some of them
were suggested by P.Flajolet and popularized by J.Lumbroso). The distribution
of is known, we rederive this distribution in a simpler way.
We provide new results on the moments of and . We also analyze the final
cache size distribution. We consider colored keys: assume that among the
distinct keys, do have color . We show how to estimate
. We also study colored keys with some multiplicity given by
some distribution function. We want to estimate mean an variance of this
distribution. Finally, we consider the case where neither colors nor
multiplicities are known. There we want to estimate the related parameters. An
appendix is devoted to the case where the hashing function provides bits with
probability different from
Quantum Fluctuations of a Coulomb Potential as a Source of Flicker Noise
The power spectrum of quantum fluctuations of the electromagnetic field
produced by an elementary particle is determined. It is found that in a wide
range of practically important frequencies the power spectrum of fluctuations
exhibits an inverse frequency dependence. The magnitude of fluctuations
produced by a conducting sample is shown to have a Gaussian distribution around
its mean value, and its dependence on the sample geometry is determined. In
particular, it is demonstrated that for geometrically similar samples the power
spectrum is inversely proportional to the sample volume. It is argued also that
the magnitude of fluctuations induced by external electric field is
proportional to the field strength squared. A comparison with experimental data
on flicker noise measurements in continuous metal films is made.Comment: 11 pages, substantially corrected and extende
Geometrically necessary dislocation densities in olivine obtained using high-angular resolution electron backscatter diffraction
© 2016 The AuthorsDislocations in geological minerals are fundamental to the creep processes that control large-scale geodynamic phenomena. However, techniques to quantify their densities, distributions, and types over critical subgrain to polycrystal length scales are limited. The recent advent of high-angular resolution electron backscatter diffraction (HR-EBSD), based on diffraction pattern cross-correlation, offers a powerful new approach that has been utilised to analyse dislocation densities in the materials sciences. In particular, HR-EBSD yields significantly better angular resolution (<0.01°) than conventional EBSD (~0.5°), allowing very low dislocation densities to be analysed. We develop the application of HR-EBSD to olivine, the dominant mineral in Earths upper mantle by testing (1) different inversion methods for estimating geometrically necessary dislocation (GND) densities, (2) the sensitivity of the method under a range of data acquisition settings, and (3) the ability of the technique to resolve a variety of olivine dislocation structures. The relatively low crystal symmetry (orthorhombic) and few slip systems in olivine result in well constrained GND density estimates. The GND density noise floor is inversely proportional to map step size, such that datasets can be optimised for analysing either short wavelength, high density structures (e.g. subgrain boundaries) or long wavelength, low amplitude orientation gradients. Comparison to conventional images of decorated dislocations demonstrates that HR-EBSD can characterise the dislocation distribution and reveal additional structure not captured by the decoration technique. HR-EBSD therefore provides a highly effective method for analysing dislocations in olivine and determining their role in accommodating macroscopic deformation
New activity pattern in human interactive dynamics
We investigate the response function of human agents as demonstrated by
written correspondence, uncovering a new universal pattern for how the reactive
dynamics of individuals is distributed across the set of each agent's contacts.
In long-term empirical data on email, we find that the set of response times
considered separately for the messages to each different correspondent of a
given writer, generate a family of heavy-tailed distributions, which have
largely the same features for all agents, and whose characteristic times grow
exponentially with the rank of each correspondent. We furthermore show that
this universal behavioral pattern emerges robustly by considering weighted
moving averages of the priority-conditioned response-time probabilities
generated by a basic prioritization model. Our findings clarify how the range
of priorities in the inputs from one's environment underpin and shape the
dynamics of agents embedded in a net of reactive relations. These newly
revealed activity patterns might be present in other general interactive
environments, and constrain future models of communication and interaction
networks, affecting their architecture and evolution.Comment: 15 pages, 7 figure
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