Aggregate implements an efficient fast Fourier transform (FFT)-based algorithm to approximate compound probability distributions. Leveraging FFT-based methods offers advantages over recursion and simulation-based approaches, providing speed and accuracy to otherwise time-consuming calculations. Combining user-friendly features and an expressive domain-specific language called DecL, Aggregate enables practitioners and nonprogrammers to work with complex distributions effortlessly. The software verifies the accuracy of its FFT-based numerical approximations by comparing their first three moments to those calculated analytically from the specified frequency and severity. This moment-based validation, combined with carefully chosen default parameters, allows users without in-depth knowledge of the underlying algorithm to be confident in the results. Aggregate supports a wide range of frequency and severity distributions, policy limits and deductibles, and reinsurance structures and has applications in pricing, reserving, risk management, teaching, and research. It is written in Python.

Polynuclear hydroxy-Al cations were prepared by partially neutralizing dilute solutions of aluminum chloride. These cations were introduced in the interlayer space of montmorillonite by cation exchange, which formed heat-stable pillars between the silicate layers. Polynuclear hydroxy-Al was preferentially adsorbed on montmorillonite compared with monomer-Al; the maximum amount adsorbed was ∼400 meq/100 g of montmorillonite. Of this amount 320 meq was non-exchangeable. The 001 X-ray powder diffraction reflection of the polynuclear hydroxy-Al-montmorillonite complex was at 27 Å, with four additional higher-order basal reflections, giving an average d(001) value of 28.4 Å. This complex was thermally stable to 700°C. An analysis of the basal reflections by the Fourier transform method indicated that the 28-Å complex had a relatively regular interstratified structure of 9.6- and 18.9-Å component layers with a mixing ratio of 0.46:0.54. This ratio implies that the hydroxy-Al pillars occupied every second layer. Considering the relatively small amount of Al adsorbed and the thermally stable nature of the structure, the hydroxy-Al pillars must have been sparsely but homogeneously distributed in the interlayer space.

Synthetic aluminous hematites and goethites have been examined by Fourier-transform infrared spectroscopy. For aluminous hematites prepared at 950°C a linear relationship exists between Al content and the location of the band near 470 cm−1, up to 10 mole % Al substitution which is shown to be the solubility limit. The spectra of aluminous goethites prepared in two different ways are qualitatively similar to each other, but differ as to the relationship between the position of the band near 900 cm−1 and the Al content. The spectra of the two series of hematites produced by calcining the goethites at 590°C also show a strong dependence of band position and intensity on the goethite preparative method.

The notion of indicator of an analytic function, that describes the function’s growth along rays, was introduced by Phragmen and Lindelöf. Trigonometric convexity is a defining property of the indicator. For multivariate cases, an analogous property of trigonometric convexity was not known so far. We prove the property of trigonometric convexity for the indicator of multivariate analytic functions, introduced by Ivanov. The results that we obtain are sharp. Derivation of a multidimensional analogue of the inverse Fourier transform in a sector and obtaining estimates on its decay is an important step of our proof.

Halloysite is used for targeted delivery of drugs and other biomolecules. Renewed interest in examination by X-ray diffraction (XRD) to predict the size of particles that can be loaded onto the nanotubes has resulted. Anhydrous halloysite consists of spiraled tubules the length and diameter of which can be determined by measurement using an electron microscope. In spite of ample evidence regarding the spiral structure of halloysite, current programs to evaluate the structure of halloysite nanotubes consider it to be a hollow tube or a cylinder which prevents accurate prediction of its structure and leads to misinformation about the sizes of materials that can be loaded onto the nanotubes. The overall objective of the current study was to derive equations to estimate the structure of halloysite nanotubes which take into consideration its spiral structure. The study of Fourier transform either by electron diffraction or XRD led to the measurement of the spiral thickness and the nature of the spiral. Calculations of the nanotube dimensions may determine the ability of these carriers to allow the mechanical delivery of certain drugs. Here the structure of hydrated halloysite (hollow cylindrical tubes with a doughnut-like cross-section) and anhydrous halloysite (spiraled or helical structure) are described as previously reported in the literature. The Fourier transform of the spiraled structure was selected based on three different kinds of spirals: the Archimedean spiral, the Power spiral, and the Logarithmic spiral. Programs used to define the crystal structure of materials and to calculate the Fourier transform need to take the spiral structure into consideration.

In this article, we study the recent development of the qualitative uncertainty principle on certain Lie groups. In particular, we consider that if the Weyl transform on certain step-two nilpotent Lie groups is of finite rank, then the function has to be zero almost everywhere as long as the nonvanishing set for the function has finite measure. Further, we consider that if the Weyl transform of each Fourier–Wigner piece of a suitable function on the Heisenberg motion group is of finite rank, then the function has to be zero almost everywhere whenever the nonvanishing set for each Fourier–Wigner piece has finite measure.

This article is the second within a three-part series on Fourier ptychography, which is a computational microscopy technique for high-resolution, large field-of-view imaging. While the first article laid out the basics of Fourier ptychography, this second part sheds light on its algorithmic ingredients. We present a non-technical discussion of phase retrieval, which allows for the synthesis of high-resolution images from a sequence of low-resolution raw data. Fourier ptychographic phase retrieval can be carried out on standard, widefield microscopy platforms with the simple addition of a low-cost LED array, thus offering a convenient alternative to other phase-sensitive techniques that require more elaborate hardware such as differential interference contrast and digital holography.

Fourier ptychography is an emerging computational microscopy technique that can generate gigapixel-scale images of biological samples. With only the addition of a low-cost LED array to a standard digital microscope and a reconstruction algorithm, Fourier ptychography overcomes the fundamental trade-off between a microscope's resolution and field-of-view without any moving parts. This article is the first in a three-part series that aims to introduce the fundamentals of the technology to the broader microscopy community and beyond, using intuitive explanations.

We develop a new analytical solution of a three-dimensional atmospheric pollutant dispersion. The main idea is to subdivide vertically the planetary boundary layer into sub-layers, where the wind speed and eddy diffusivity assume average values for each sub-layer. Basically, the model is assessed and validated using data obtained from the Copenhagen diffusion and Prairie Grass experiments. Our findings show that there is a good agreement between the predicted and observed crosswind-integrated concentrations. Moreover, the calculated statistical indices are within the range of acceptable model performance.