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Two supercongruences related to multiple harmonic sums
No full texthttp://dx.doi.org/10.1017/S000497271200033
High resolution simulations of a tornadic storm affecting Sydney
Get PDFOn 16 December 2015 a severe thunderstorm and associated tornado affected Sydney causing widespread damage and insured losses of 550m, 2016.
E. R. Mansell, C. L. Ziegler, and E. C. Bruning. Simulated electrification of a small thunderstorm with two-moment bulk microphysics. J. Atmos. Sci., 67:171–194, 2010. doi:10.1175/2009JAS2965.1.
R. C. Miller. Notes on analysis and severe-storm forecasting procedures of the Air Force Global Weather Central, volume 200. Air Weather Service, 1972. URL https://apps.dtic.mil/sti/citations/AD0744042.
H. Morrison, J. A. Curry, and V. I. Khvorostyanov. A new double-moment microphysics parameterization for application in cloud and climate models. Part I: Description. J. Atmos. Sci., 62:1665–1677, 2005. doi:10.1175/JAS3446.1.
H. Morrison, G. Thompson, and V. Tatarskii. Impact of cloud microphysics on the development of trailing stratiform precipitation in a simulated squall line: Comparison of one- and two-moment schemes. Month. Wea. Rev., 137:991–1007, 2009. doi:10.1175/2008MWR2556.1.
J. G. Powers, J. B. Klemp, W. C. Skamarock, C. A. Davis, J. Dudhia, D. O. Gill, J. L. Coen, D. J. Gochis, R. Ahmadov, S. E. Peckham, et al. The Weather Research and Forecasting Model: Overview, system efforts, and future directions. Bull. Am. Meteor. Soc., 98:1717–1737, 2017. doi:10.1175/BAMS-D-15-00308.1.
H. Richter, A. Protat, J. Taylor, and J. Soderholm. Doppler radar and storm environment observations of a maritime tornadic supercell in Sydney, Australia. In Preprints, 28th Conf. on Severe Local Storms, Portland OR, Amer. Meteor. Soc. P, 2016.
W. C. Skamarock, J. B. Klemp, J. Dudhia, D. O. Gill, Z. Liu, J. Berner, W. Wang, J. G. Powers, M. G. Duda, D. Barker, and X.-Y. Huang. A description of the advanced research WRF Model version 4. Technical report, 2019.
Storm Prediction Center. The Enhanced Fujita Scale (EF Scale), 2014. URL https://www.spc.noaa.gov/efscale/.
R. A. Warren, H. A. Ramsay, S. T. Siems, M. J. Manton, J. R. Peter, A. Protat, and A. Pillalamarri. Radar-based climatology of damaging hailstorms in Brisbane and Sydney, Australia. Quart. J. Roy. Meteor. Soc., 146:505–530, 2020. doi:10.1002/qj.3693
A variant of Cauchy's argument principle for analytic functions which applies to curves containing zeroes
No full texthttp://dx.doi.org/10.1017/S000497271200033
Uniqueness of extendable temperatures
No full textLet E and D be open subsets of Rn+1 such that D is a compact subset of E, and let v be a supertemperature on E. A temperature u on D is called extendable by v if there is a supertemperature w on E such that w = u on D and w = v on E\D. We know that either there is a unique temperature extendable by v, or there are infinitely many. We also know that a neces- sary condition for uniqueness is that the generalised solution of the Dirichlet problem on D corresponding to the restriction of v to ∂eD, is equal to the greatest thermic minorant of v on D. In this paper, we first give a condition for non-uniqueness and an example to show that this necessary condition is not sufficient. We then give a uniqueness theorem involving the thermal and cothermal fine topologies, and deduce a corollary involving only parabolic and coparabolic tusks
Confidence intervals constructed by model averaging and bootstrap smoothing
No full textWe assess the performance of confidence intervals constructed by model averaging and bootstrap smoothing, using a simple testbed situation of two nested linear regression models with either known error variance or unknown error variance. The assessment is carried out by deriving computationally convenient exact expressions for the coverage probability and scaled expected length where the scaling is with respect to the usual confidence interval, with the same minimum coverage, based on the full model.
We assessed the properties of the confidence interval centered on the model averaged estimator proposed by Buckland et al. (1997) with half-width proportional to the standard
error of the estimate put forward by Buckland et al. (1997) of their formula (9). We considered both the finite sample case and the asymptotic case. Our results show that this model averaged confidence interval cannot be generally recommended, although it performs well when the error degrees of freedom is 1.
We also assessed the properties of the confidence intervals centered on the bootstrap smoothed estimator with half-width proportional to the estimate of (a) the standard deviation of this estimator and (b) the delta method approximation, derived by Efron (2014), to the standard deviation of this estimator. We show that, except for the case of unknown error variance and small error degrees of freedom, the confidence interval centered on the bootstrap smoothed estimator does not have scaled expected length that is substantially less than 1, when the simpler model is correct. For this reason, we sought a formula for the data-based width that leads to improved performance of the confidence interval centered on the bootstrap smoothed estimator. Since this search failed, it raises the following question. Is there any formula for the data-based width of such a confidence interval so that it performs well? To answer this question, we compute the performance bound due to Kabaila & Kong (2016)
On statistical properties of non-linear functionals of random fields
No full textThis thesis studies the asymptotic behaviour of non-linear transformations of Gaussian vector random fields
via integral functionals and solutions of stochastic partial differential equations.
The components of vector fields considered in this thesis admit different types of dependence properties.
These random fields and their transformations play a central role in the asymptotic theory and various statistical applications. They can produce non-Gaussian limit theorems.
The thesis also investigates stochastic hyperbolic diffusion equations with random initial conditions given by random~fields.
First, the reduction principle for non-linear functionals of Gaussian vector random fields is derived.
The components of such fields can exhibit different behaviour under long-range dependence.
It is shown that the asymptotic distributions of integral functionals of Gaussian vector random fields are not necessarily determined by the leading components at the Hermite rank level.
The concept of dominant components which determine the asymptotic distribution is discussed.
The non-central limit theorem for the first Minkowski functionals of Fisher-Snedecor random fields is derived.
Next, the asymptotics of non-linear functionals of vector random fields with strongly and weakly dependent components are studied.
A version of the reduction theorem is provided which proves that the asymptotic distributions of such functionals are not determined by their Hermite ranks.
These functionals converge in distribution to random variables which are determined by sums of the multiple Wiener-It\^{o} stochastic integrals.
Applications to the first Minkowski functionals of Student random fields are provided.
Last, the solutions of hyperbolic diffusion equations with random initial conditions given by random fields are investigated.
The restriction of solutions to the unit sphere is considered.
The dependence structures of spherical hyperbolic diffusion random fields are studied.
The approximations of the exact solution random fields are investigated.
The smoothness properties of the exact solution and its approximation are obtained
A note on derived length and character degrees
No full textIsaacs and Seitz have conjectured that the derived length of a finite solvable group G is bounded by the cardinality of the set of all irreducible character degrees of G, namely that the inequality holds for , where is the derived length of and is the set of all irreducible character degrees of . In this paper, we first prove that the inequality holds for if the degrees of nonlinear monolithic characters of having same kernels are distinct. Also, we show that the conjecture is true in the case when , where is the set of all nonlinear monolithic characters of . Additionally, we give some sufficient conditions for the inequality related to monolithic characters or real-valued irreducible characters of when the commutator subgroup is supersolvable
Existence and box dimension of general recurrent fractal interpolation functions
No full textThe notion of recurrent fractal interpolation functions (RFIFs) was introduced by Barnsley, Elton and Hardin in 1989. Roughly speaking, the graph of an RFIF is the invariant set of a recurrent iterated function system on . In this paper, we generalize the definition of RFIFs in the following sense: iterated functions in the recurrent system need not to be contractive with respect to the first variable. We also obtain the box dimensions of all self-affine RFIFs in our general setting
On -congruent numbers over real number fields
No full textThe notion of -congruent numbers is a generalization of congruent numbers, where one considers triangles with an angle such that is a rational number. In this article, we discuss a criterion for a natural number to be a -congruent number over certain real number fields