The Stretch - Length Tradeoff in Geometric Networks: Average Case and Worst Case Study

Consider a network linking the points of a rate-$1$ Poisson point process on the plane. Write \Psi^{\mbox{ave}}(s) for the minimum possible mean length per unit area of such a network, subject to the constraint that the route-length between every pair of points is at most $s$ times the Euclidean distance. We give upper and lower bounds on the function \Psi^{\mbox{ave}}(s), and on the analogous "worst-case" function \Psi^{\mbox{worst}}(s) where the point configuration is arbitrary subject to average density one per unit area. Our bounds are numerically crude, but raise the question of whether there is an exponent $\alpha$ such that each function has $\Psi(s) \asymp (s-1)^{-\alpha}$ as $s \downarrow 1$.Comment: 33 page

Meanders: Exact Asymptotics

We conjecture that meanders are governed by the gravitational version of a c=-4 two-dimensional conformal field theory, allowing for exact predictions for the meander configuration exponent \alpha=\sqrt{29}(\sqrt{29}+\sqrt{5})/12, and the semi-meander exponent {\bar\alpha}=1+\sqrt{11}(\sqrt{29}+\sqrt{5})/24. This result follows from an interpretation of meanders as pairs of fully packed loops on a random surface, described by two c=-2 free fields. The above values agree with recent numerical estimates. We generalize these results to a score of meandric numbers with various geometries and arbitrary loop fugacities.Comment: new version with note added in proo

Calculation of 3D genome structures for comparison of chromosome conformation capture experiments with microscopy: An evaluation of single-cell Hi-C protocols.

Single-cell chromosome conformation capture approaches are revealing the extent of cell-to-cell variability in the organization and packaging of genomes. These single-cell methods, unlike their multi-cell counterparts, allow straightforward computation of realistic chromosome conformations that may be compared and combined with other, independent, techniques to study 3D structure. Here we discuss how single-cell Hi-C and subsequent 3D genome structure determination allows comparison with data from microscopy. We then carry out a systematic evaluation of recently published single-cell Hi-C datasets to establish a computational approach for the evaluation of single-cell Hi-C protocols. We show that the calculation of genome structures provides a useful tool for assessing the quality of single-cell Hi-C data because it requires a self-consistent network of interactions, relating to the underlying 3D conformation, with few errors, as well as sufficient longer-range cis- and trans-chromosomal contacts

On the Pricing of Step-Up Bonds in the European Telecom Sector

This paper investigates the pricing of step-up bonds, i.e. corporate bonds with provisions stating that the coupon payments increase as the credit rating level of the issuer declines. To assess the risk-neutral rating transition probabilities necessary to price these bonds, we introduce a new calibration method within the reduced-form rating-based model of Jarrow, Lando, and Turnbull (1997). We also treat split ratings and adjust for rating outlook. Step-up bonds have been issued in large amounts in the European telecom sector, and we find that, through most of the sample, step-up bonds issued by the two largest issuers have traded at a discount relative to comparable fixed-coupon bonds from the same issuers. Our findings cannot be attributed to traditional liquidity factors, and they suggest that issuing step-up bonds increased the cost of capital for the issuers. Keywords: defaultable bonds, step-up coupons, rating-based models JEL classification: G12, G1

Quantitative single-molecule microscopy reveals that CENP-A(Cnp1) deposition occurs during G2 in fission yeast

The inheritance of the histone H3 variant CENP-A in nucleosomes at centromeres following DNA replication is mediated by an epigenetic mechanism. To understand the process of epigenetic inheritance, or propagation of histones and histone variants, as nucleosomes are disassembled and reassembled in living eukaryotic cells, we have explored the feasibility of exploiting photo-activated localization microscopy (PALM). PALM of single molecules in living cells has the potential to reveal new concepts in cell biology, providing insights into stochastic variation in cellular states. However, thus far, its use has been limited to studies in bacteria or to processes occurring near the surface of eukaryotic cells. With PALM, one literally observes and 'counts' individual molecules in cells one-by-one and this allows the recording of images with a resolution higher than that determined by the diffraction of light (the so-called super-resolution microscopy). Here, we investigate the use of different fluorophores and develop procedures to count the centromere-specific histone H3 variant CENP-A(Cnp1) with single-molecule sensitivity in fission yeast (Schizosaccharomyces pombe). The results obtained are validated by and compared with ChIP-seq analyses. Using this approach, CENP-A(Cnp1) levels at fission yeast (S. pombe) centromeres were followed as they change during the cell cycle. Our measurements show that CENP-A(Cnp1) is deposited solely during the G2 phase of the cell cycle

Confidence sets for continuous-time rating

This paper addresses the estimation of default probabilities and associated confidence sets with special focus on rare events. Research on rating transition data has documented a tendency for recently downgraded issuers to be at an increased risk of experiencing further downgrades compared to issuers that have held the same rating for a longer period of time. To capture this non-Markov effect we introduce a continuous-time hidden Markov chain model in which downgrades firms enter into a hidden, ’excited’ state. Using data from Moody’s we estimate the parameters of the model, and conclude that both default probabilities and confidence sets are strongly influenced by the introduction of hidden excited states

Quantification of DNA-associated proteins inside eukaryotic cells using single-molecule localization microscopy

Development of single-molecule localization microscopy techniques has allowed nanometre scale localization accuracy inside cells, permitting the resolution of ultra-fine cell structure and the elucidation of crucial molecular mechanisms. Application of these methodologies to understanding processes underlying DNA replication and repair has been limited to defined in vitro biochemical analysis and prokaryotic cells. In order to expand these techniques to eukaryotic systems, we have further developed a photo-activated localization microscopy-based method to directly visualize DNA-associated proteins in unfixed eukaryotic cells. We demonstrate that motion blurring of fluorescence due to protein diffusivity can be used to selectively image the DNA-bound population of proteins. We designed and tested a simple methodology and show that it can be used to detect changes in DNA binding of a replicative helicase subunit, Mcm4, and the replication sliding clamp, PCNA, between different stages of the cell cycle and between distinct genetic backgrounds

The Jones polynomial and graphs on surfaces

The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobas-Riordan-Tutte polynomial generalizes the Tutte polynomial of planar graphs to graphs that are embedded in closed oriented surfaces of higher genus. In this paper we show that the Jones polynomial of any link can be obtained from the Bollobas-Riordan-Tutte polynomial of a certain oriented ribbon graph associated to a link projection. We give some applications of this approach.Comment: 19 pages, 9 figures, minor change

Scope for Credit Risk Diversification

This paper considers a simple model of credit risk and derives the limit distribution of losses under different assumptions regarding the structure of systematic risk and the nature of exposure or firm heterogeneity. We derive fat-tailed correlated loss distributions arising from Gaussian risk factors and explore the potential for risk diversification. Where possible the results are generalised to non-Gaussian distributions. The theoretical results indicate that if the firm parameters are heterogeneous but come from a common distribution, for sufficiently large portfolios there is no scope for further risk reduction through active portfolio management. However, if the firm parameters come from different distributions, then further risk reduction is possible by changing the portfolio weights. In either case, neglecting parameter heterogeneity can lead to underestimation of expected losses. But, once expected losses are controlled for, neglecting parameter heterogeneity can lead to overestimation of risk, whether measured by unexpected loss or value-at-risk