IST Austria: PubRep (Institute of Science and Technology)
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On the size of chromatic Delaunay mosaics
No full textGiven a locally finite set A⊆Rd and a coloring χ:A→{0,1,…,s}, we introduce the chromatic Delaunay mosaic of χ, which is a Delaunay mosaic in Rs+d that represents how points of different colors mingle. Our main results are bounds on the size of the chromatic Delaunay mosaic, in which we assume that d and s are constants. For example, if A is finite with n=#A, and the coloring is random, then the chromatic Delaunay mosaic has O(n⌈d/2⌉) cells in expectation. In contrast, for Delone sets and Poisson point processes in Rd, the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in R2 all colorings of a dense set of n points have chromatic Delaunay mosaics of size O(n). This encourages the use of chromatic Delaunay mosaics in applications
The Hamilton space of pseudorandom graphs
No full textWe show that if n is odd and p>=Clog n/n, then with high probability Hamilton cycles in G(n,p) span its cycle space. More generally, we show this holds for a class of graphs satisfying certain natural pseudorandom properties. The proof is based on a novel idea of parity-switchers, which can be thought of as analogues of absorbers in the context of cycle spaces. As another application of our method, we show that Hamilton cycles in a near-Dirac graph G, that is, a graph G with odd n vertices and minimum degree n/2+C for sufficiently large constant C, span its cycle space
ISTA Thesis
No full textMutation rates represent the net result of complex interactions among various
cellular processes and can dramatically influence the evolutionary fate of
microbial populations. However, many popular techniques used to study
mutations are subject to the confounding effects of heredity and the subtleties
of adaptation to selection, all of which make it difficult to observe any dynamic
responses of mutation rates to fitness challenges. Furthermore, in spite of the
ubiquity of quorum sensing systems across the bacterial domain and relevance
for many physiological behaviors, the effects of such mechanisms on mutation
rate and adaptation remain poorly understood. In the following work, I
present the development of a microfluidic droplet-based method to measure
single base-pair mutation rates in growing populations of the bacterium
Escherichia coli. I use this method to observe a stress-induced increase in
mutation rate that is mediated by luxS, a highly conserved bacterial quorum
sensing component. I also show that the aforementioned increase in mutation
rate, and its associated control by luxS, corresponds to a higher degree of
adaptability under competitive environments
