Discrete Circular Distributions with Applications to Shared Orthologs of Paired Circular Genomes

For structural comparisons of paired prokaryotic genomes,an important topic in synthetic and evolutionary biology,the locations of shared orthologous genes(henceforth orthologs)are observed as binned data.This and other data,e.g.,wind directions recorded at monitoring sites and intensive care unit arrival times on the 24-hour clock,are counted in binned circular arcs,thus modeling them by discrete circular distributions(DCDs)is required.We propose a novel method to construct a DCD from a base continuous circular distribution(CCD).The probability mass function is defined to take the normalized values of the probability density function at some pre-fixed equidistant points on the circle.Five families of constructed DCDs which have normalizing constants in closed form are presented.Simulation studies show that DCDs outperform the corresponding CCDs in modeling grouped(discrete)circular data,and minimum chi-square estimation outperforms maximum likelihood estimation for parameters.We apply the constructed DCDs,invariant wrapped Poisson and wrapped discrete skew Laplace to compare the structures of paired bacterial genomes.Specifically,discrete four-parameter wrapped Cauchy(nonnegative trigonometric sums)distribution models multi-modal shared orthologs in Clostridium(Sulfolobus)better than the others considered,in terms of AIC and Freedman’s goodness-of-fit test.The result that different DCDs fit the shared orthologs is consistent with the fact they belong to two kingdoms.Nevertheless,these prokaryotes have a common favored site around 70°on the unit circle;this finding is important for building synthetic prokaryotic genomes in synthetic biology.These DCDs can also be applied to other binned circular data.

EDGEWORTH EXPANSION FOR CIRCULAR DISTRIBUTION

The paper discusses the Edgeworth expansion for the mean direction =μ(Fn) of directional data and its 'Studentization'. As an application of the results,we give the corresponding result of von Mises population.

THEORETICAL INVESTIGATION ON STATE OF LOAD DISTRIBUTION AMONG CONTACT BEARINGS OF DOUBLE CIRCULAR ARC HELICAL GEARS

A dynamic analysis approcach to investigate the state of load distribution among contact bearings of double circular are belical gears is proposed and a computer program is developed for calculating the factor of iced distribution based on the theory of W-N gears. The changing situations of load distribution among contact bearings of the gears influenced by main issues are analyzed by a series of parametric studies.

The influence of nonlinear shear stress on partially averaged Navier-Stokes(PANS)method

In most of partially averaged Navier-Stokes（PANS）methods,the Reynolds stress is solved by a linear hypothesis isotropic model.They could not capture all kinds of vortexes in tubomachineries.In this paper,a PANS model is modified from the RNG k-？turbulence model and is used to investigate the influence of the nonlinear shear stress on the simulation of the high pressure gradient flows and the large curvature flows.Comparisons are made between the result obtained by using the PANS model modified from the RNG k-？model and that obtained by using the nonlinear PANS methods.The flow past a curved rectangular duct is calculated by using the PANS methods.The obtained nonlinear shear stress agrees well with the experimental results,especially in the high pressure gradient region.The calculation results show that the nonlinear PANS methods are more reliable than the linear PANS methods for the high pressure gradient flows,the large curvature flows,and they can be used to capture complex vortexes in a turbomachinary.