Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces

Reproducing kernel Hilbert spaces (RKHSs) play an important role in many statistics and machine learning applications ranging from support vector machines to Gaussian processes and kernel embeddings of distributions. Operators acting on such spaces are, for instance, required to embed conditional probability distributions in order to implement the kernel Bayes rule and build sequential data models. It was recently shown that transfer operators such as the Perron-Frobenius or Koopman operator can also be approximated in a similar fashion using covariance and cross-covariance operators and that eigenfunctions of these operators can be obtained by solving associated matrix eigenvalue problems. The goal of this paper is to provide a solid functional analytic foundation for the eigenvalue decomposition of RKHS operators and to extend the approach to the singular value decomposition. The results are illustrated with simple guiding examples

Diffusion maps tailored to arbitrary non-degenerate Ito processes

We present two generalizations of the popular diffusion maps algorithm. The first generalization replaces the drift term in diffusion maps, which is the gradient of the sampling density, with the gradient of an arbitrary density of interest which is known up to a normalization constant. The second generalization allows for a diffusion map type approximation of the forward and backward generators of general Ito diffusions with given drift and diffusion coefficients. We use the local kernels introduced by Berry and Sauer, but allow for arbitrary sampling densities. We provide numerical illustrations to demonstrate that this opens up many new applications for diffusion maps as a tool to organize point cloud data, including biased or corrupted samples, dimension reduction for dynamical systems, detection of almost invariant regions in flow fields, and importance sampling

Family‐ and population‐level responses to atmospheric CO2 concentration: gas exchange and the allocation of C, N, and biomass in Plantago lanceolata (Plantaginaceae)

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/142197/1/ajb21080.pd

Swift J045106.8-694803: A Highly Magnetised Neutron Star in the Large Magellanic Cloud

We report the analysis of a highly magnetised neutron star in the Large Magellanic Cloud (LMC). The high mass X-ray binary pulsar Swift J045106.8-694803 has been observed with Swift X-ray telescope (XRT) in 2008, The Rossi X-ray Timing Explorer (RXTE) in 2011 and the X-ray Multi-Mirror Mission - Newton (XMM-Newton) in 2012. The change in spin period over these four years indicates a spin-up rate of 5.010.06 s/yr, amongst the highest observed for an accreting pulsar. This spin-up rate can be accounted for using Ghosh and Lambs (1979) accretion theory assuming it has a magnetic field of (1.2 +/= 0.20/0.7) x 10(exp 14) Gauss. This is over the quantum critical field value. There are very few accreting pulsars with such high surface magnetic fields and this is the first of which to be discovered in the LMC. The large spin-up rate is consistent with Swift Burst Alert Telescope (BAT) observations which show that Swift J045106.8-694803 has had a consistently high X-ray luminosity for at least five years. Optical spectra have been used to classify the optical counterpart of Swift J045106.8-694803 as a B0-1 III-V star and a possible orbital period of 21.631 +/- 0.005 days has been found from MACHO optical photometry

АНАЛИТИЧЕСКОЕ РЕШЕНИЕ НДС ДЛИННОГО ПОЛОГО ЦИЛИНДРА ПОД ДЕЙСТВИЕМ ТЕРМОРАДИАЦИОННОЙ НАГРУЗКИ

The paper considers specific features of a stress-strain state of structure elements which have cylindrical form and which are subjected to the high temperature field, neutron irradiation. An analytical solution concerning displacement and stress dependences on a cylinder radius has been obtained and curve dependences have been constructed in the paper..Рассмотрены особенности напряженно-деформированного состояния элементов конструкций цилиндрической формы, находящихся под действием поля высоких температур, нейтронного облучения. Получено аналитическое решение зависимостей перемещений и напряжений от радиуса цилиндра. Построены кривые зависимости

A weak characterization of slow variables in stochastic dynamical systems

We present a novel characterization of slow variables for continuous Markov processes that provably preserve the slow timescales. These slow variables are known as reaction coordinates in molecular dynamical applications, where they play a key role in system analysis and coarse graining. The defining characteristics of these slow variables is that they parametrize a so-called transition manifold, a low-dimensional manifold in a certain density function space that emerges with progressive equilibration of the system's fast variables. The existence of said manifold was previously predicted for certain classes of metastable and slow-fast systems. However, in the original work, the existence of the manifold hinges on the pointwise convergence of the system's transition density functions towards it. We show in this work that a convergence in average with respect to the system's stationary measure is sufficient to yield reaction coordinates with the same key qualities. This allows one to accurately predict the timescale preservation in systems where the old theory is not applicable or would give overly pessimistic results. Moreover, the new characterization is still constructive, in that it allows for the algorithmic identification of a good slow variable. The improved characterization, the error prediction and the variable construction are demonstrated by a small metastable system

Epigenetic silencing of DSC3 is a common event in human breast cancer

INTRODUCTION: Desmocollin 3 (DSC3) is a member of the cadherin superfamily of calcium-dependent cell adhesion molecules and a principle component of desmosomes. Desmosomal proteins such as DSC3 are integral to the maintenance of tissue architecture and the loss of these components leads to a lack of adhesion and a gain of cellular mobility. DSC3 expression is down-regulated in breast cancer cell lines and primary breast tumors; however, the loss of DSC3 is not due to gene deletion or gross rearrangement of the gene. In this study, we examined the prevalence of epigenetic silencing of DSC3 gene expression in primary breast tumor specimens. METHODS: We used bisulfite genomic sequencing to analyze the methylation state of the DSC3 promoter region from 32 primary breast tumor specimens. We also used a quantitative real-time RT-PCR approach, and analyzed all breast tumor specimens for DSC3 expression. Finally, in addition to bisulfite sequencing and RT-PCR, we used an in vivo nuclease accessibility assay to determine the chromatin architecture of the CpG island region from DSC3-negative breast cancer cells lines. RESULTS: DSC3 expression was downregulated in 23 of 32 (72%) breast cancer specimens comprising: 22 invasive ductal carcinomas, 7 invasive lobular breast carcinomas, 2 invasive ductal carcinomas that metastasized to the lymph node, and a mucoid ductal carcinoma. Of the 23 specimens showing a loss of DSC3 expression, 13 (56%) were associated with cytosine hypermethylation of the promoter region. Furthermore, DSC3 expression is limited to cells of epithelial origin and its expression of mRNA and protein is lost in a high proportion of breast tumor cell lines (79%). Lastly, DNA hypermethylation of the DSC3 promoter is highly correlated with a closed chromatin structure. CONCLUSION: These results indicate that the loss of DSC3 expression is a common event in primary breast tumor specimens, and that DSC3 gene silencing in breast tumors is frequently linked to aberrant cytosine methylation and concomitant changes in chromatin structure

Resonances in a chaotic attractor crisis of the Lorenz Flow

Local bifurcations of stationary points and limit cycles have successfully been characterized in terms of the critical exponents of these solutions. Lyapunov exponents and their associated covariant Lyapunov vectors have been proposed as tools for supporting the understanding of critical transitions in chaotic dynamical systems. However, it is in general not clear how the statistical properties of dynamical systems change across a boundary crisis during which a chaotic attractor collides with a saddle. This behavior is investigated here for a boundary crisis in the Lorenz flow, for which neither the Lyapunov exponents nor the covariant Lyapunov vectors provide a criterion for the crisis. Instead, the convergence of the time evolution of probability densities to the invariant measure, governed by the semigroup of transfer operators, is expected to slow down at the approach of the crisis. Such convergence is described by the eigenvalues of the generator of this semigroup, which can be divided into two families, referred to as the stable and unstable Ruelle--Pollicott resonances, respectively. The former describes the convergence of densities to the attractor (or escape from a repeller) and is estimated from many short time series sampling the state space. The latter is responsible for the decay of correlations, or mixing, and can be estimated from a long times series, invoking ergodicity. It is found numerically for the Lorenz flow that the stable resonances do approach the imaginary axis during the crisis, as is indicative of the loss of global stability of the attractor. On the other hand, the unstable resonances, and a fortiori the decay of correlations, do not flag the proximity of the crisis, thus questioning the usual design of early warning indicators of boundary crises of chaotic attractors and the applicability of response theory close to such crises