Embedding laws in diffusions by functions of time

We present a constructive probabilistic proof of the fact that if $B=(B_t)_{t\ge0}$ is standard Brownian motion started at $0$, and $\mu$ is a given probability measure on $\mathbb{R}$ such that $\mu(\{0\})=0$, then there exists a unique left-continuous increasing function $b:(0,\infty)\rightarrow\mathbb{R}\cup\{+\infty\}$ and a unique left-continuous decreasing function $c:(0,\infty)\rightarrow\mathbb{R}\cup\{-\infty\}$ such that $B$ stopped at $\tau_{b,c}=\inf\{t>0\vert B_t\ge b(t)$ or $B_t\le c(t)\}$ has the law $\mu$. The method of proof relies upon weak convergence arguments arising from Helly's selection theorem and makes use of the L\'{e}vy metric which appears to be novel in the context of embedding theorems. We show that $\tau_{b,c}$ is minimal in the sense of Monroe so that the stopped process $B^{\tau_{b,c}}=(B_{t\wedge\tau_{b,c}})_{t\ge0}$ satisfies natural uniform integrability conditions expressed in terms of $\mu$. We also show that $\tau_{b,c}$ has the smallest truncated expectation among all stopping times that embed $\mu$ into $B$. The main results extend from standard Brownian motion to all recurrent diffusion processes on the real line.Comment: Published at http://dx.doi.org/10.1214/14-AOP941 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

An Optimal Skorokhod Embedding for Diffusions

Given a Brownian motion $B_t$ and a general target law $\mu$ (not necessarily centered or even integrable) we show how to construct an embedding of $\mu$ in $B$. This embedding is an extension of an embedding due to Perkins, and is optimal in the sense that it simultaneously minimises the distribution of the maximum and maximises the distribution of the minimum among all embeddings of $\mu$. The embedding is then applied to regular diffusions, and used to characterise the target laws for which a $H^p$-embedding may be found.Comment: 22 pages, 4 figure

Large-scale compression of genomic sequence databases with the Burrows-Wheeler transform

Motivation The Burrows-Wheeler transform (BWT) is the foundation of many algorithms for compression and indexing of text data, but the cost of computing the BWT of very large string collections has prevented these techniques from being widely applied to the large sets of sequences often encountered as the outcome of DNA sequencing experiments. In previous work, we presented a novel algorithm that allows the BWT of human genome scale data to be computed on very moderate hardware, thus enabling us to investigate the BWT as a tool for the compression of such datasets. Results We first used simulated reads to explore the relationship between the level of compression and the error rate, the length of the reads and the level of sampling of the underlying genome and compare choices of second-stage compression algorithm. We demonstrate that compression may be greatly improved by a particular reordering of the sequences in the collection and give a novel `implicit sorting' strategy that enables these benefits to be realised without the overhead of sorting the reads. With these techniques, a 45x coverage of real human genome sequence data compresses losslessly to under 0.5 bits per base, allowing the 135.3Gbp of sequence to fit into only 8.2Gbytes of space (trimming a small proportion of low-quality bases from the reads improves the compression still further). This is more than 4 times smaller than the size achieved by a standard BWT-based compressor (bzip2) on the untrimmed reads, but an important further advantage of our approach is that it facilitates the building of compressed full text indexes such as the FM-index on large-scale DNA sequence collections.Comment: Version here is as submitted to Bioinformatics and is same as the previously archived version. This submission registers the fact that the advanced access version is now available at http://bioinformatics.oxfordjournals.org/content/early/2012/05/02/bioinformatics.bts173.abstract . Bioinformatics should be considered as the original place of publication of this article, please cite accordingl

Data-Discriminants of Likelihood Equations

Maximum likelihood estimation (MLE) is a fundamental computational problem in statistics. The problem is to maximize the likelihood function with respect to given data on a statistical model. An algebraic approach to this problem is to solve a very structured parameterized polynomial system called likelihood equations. For general choices of data, the number of complex solutions to the likelihood equations is finite and called the ML-degree of the model. The only solutions to the likelihood equations that are statistically meaningful are the real/positive solutions. However, the number of real/positive solutions is not characterized by the ML-degree. We use discriminants to classify data according to the number of real/positive solutions of the likelihood equations. We call these discriminants data-discriminants (DD). We develop a probabilistic algorithm for computing DDs. Experimental results show that, for the benchmarks we have tried, the probabilistic algorithm is more efficient than the standard elimination algorithm. Based on the computational results, we discuss the real root classification problem for the 3 by 3 symmetric matrix~model.Comment: 2 table

The bosonic Kondo effect

The Kondo effect is associated with the formation of a many-body ground state that contains a quantum-mechanical entanglement between a (localized) fermion and the free fermions. We show that a bosonic version of the Kondo effect can occur in degenerate atomic Fermi gases near the Feshbach resonance. We also discuss how this bosonic Kondo effect can be observed experimentally.Comment: 4 pages, 2 figures, some references added, some removed. More comments adde

High-Temperature Transport Properties of the Zintl Phases Yb_(11)GaSb_9 and Yb_(11)InSb_9

Two rare-earth Zintl phases, Yb_(11)GaSb_9 and Yb_(11)InSb_9, were synthesized in high-temperature self-fluxes of molten Ga and In, respectively. Structures were characterized by both single-crystal X-ray diffraction and powder X-ray diffraction and are consistent with the published orthorhombic structure, with the space group Iba2. High-temperature differential scanning calorimetry (DSC) and thermal gravimetry (TG) measurements reveal thermal stability to 1300 K. Seebeck coefficient and resistivity measurements to 1000 K are consistent with the hypothesis that Yb_(11)GaSb_9 and Yb_(11)InSb_9 are small band gap semiconductors or semimetals. Low doping levels lead to bipolar conduction at high temperature, preventing a detailed analysis of the transport properties. Thermal diffusivity measurements yield particularly low lattice thermal conductivity values, less than 0.6 W/m K for both compounds. The low lattice thermal conductivity suggests that Yb_(11)MSb_9 (M = Ga, In) has the potential for high thermoelectric efficiency at high temperature if charge-carrier doping can be controlled

On alpha stable distribution of wind driven water surface wave slope

We propose a new formulation of the probability distribution function of wind driven water surface slope with an $\alpha$-stable distribution probability. The mathematical formulation of the probability distribution function is given under an integral formulation. Application to represent the probability of time slope data from laboratory experiments is carried out with satisfactory results. We compare also the $\alpha$-stable model of the water surface slopes with the Gram-Charlier development and the non-Gaussian model of Liu et al\cite{Liu}. Discussions and conclusions are conducted on the basis of the data fit results and the model analysis comparison.Comment: final version of the manuscript: 25 page

Euler characteristic of coherent sheaves on simplicial torics via the Stanley-Reisner ring

We combine work of Cox on the total coordinate ring of a toric variety and results of Eisenbud-Mustata-Stillman and Mustata on cohomology of toric and monomial ideals to obtain a formula for computing the Euler characteristic of a Weil divisor D on a complete simplicial toric variety in terms of graded pieces of the Cox ring and Stanley-Reisner ring. The main point is to use Alexander duality to pass from the toric irrelevant ideal, which appears in the computation of the Euler characteristic of D, to the Stanley-Reisner ideal of the fan, which is used in defining the Chow ring. The formula also follows from work of Maclagan-Smith.Comment: 9 pages 1 figur

Scanning tunneling microscopy and kinetic Monte Carlo investigation of Cesium superlattices on Ag(111)

Cesium adsorption structures on Ag(111) were characterized in a low-temperature scanning tunneling microscopy experiment. At low coverages, atomic resolution of individual Cs atoms is occasionally suppressed in regions of an otherwise hexagonally ordered adsorbate film on terraces. Close to step edges Cs atoms appear as elongated protrusions along the step edge direction. At higher coverages, Cs superstructures with atomically resolved hexagonal lattices are observed. Kinetic Monte Carlo simulations model the observed adsorbate structures on a qualitative level.Comment: 8 pages, 7 figure