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A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse factors. As a result, it manages to reduce the complexity of computing the DFT from , which arises if one simply applies the definition of DFT, to , where is the data size. The difference in speed can be enormous, especially for long data sets where N may be in the thousands or millions. In the presence of round-off error, many FFT algorithms are much more accurate than evaluating the DFT definition directly or indirectly. There are many different FFT algorithms based on a wide range of published theories, from simple complex-number arithmetic to group theory and number theory.
Spectroscopy is the general field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter waves and acoustic waves can also be considered forms of radiative energy, and recently gravitational waves have been associated with a spectral signature in the context of the Laser Interferometer Gravitational-Wave Observatory (LIGO)
X-ray crystallography is the experimental science determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to diffract into many specific directions. By measuring the angles and intensities of these diffracted beams, a crystallographer can produce a three-dimensional picture of the density of electrons within the crystal. From this electron density, the mean positions of the atoms in the crystal can be determined, as well as their chemical bonds, their crystallographic disorder, and various other information.
Surface science is the study of physical and chemical phenomena that occur at the interface of two phases, including solid–liquid interfaces, solid–gas interfaces, solid–vacuum interfaces, and liquid–gas interfaces. It includes the fields of surface chemistry and surface physics. Some related practical applications are classed as surface engineering. The science encompasses concepts such as heterogeneous catalysis, semiconductor device fabrication, fuel cells, self-assembled monolayers, and adhesives. Surface science is closely related to interface and colloid science. Interfacial chemistry and physics are common subjects for both. The methods are different. In addition, interface and colloid science studies macroscopic phenomena that occur in heterogeneous systems due to peculiarities of interfaces.
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Neutron diffraction or elastic neutron scattering is the application of neutron scattering to the determination of the atomic and/or magnetic structure of a material. A sample to be examined is placed in a beam of thermal or cold neutrons to obtain a diffraction pattern that provides information of the structure of the material. The technique is similar to X-ray diffraction but due to their different scattering properties, neutrons and X-rays provide complementary information: X-Rays are suited for superficial analysis, strong x-rays from synchrotron radiation are suited for shallow depths or thin specimens, while neutrons having high penetration depth are suited for bulk samples.
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The Cooley–Tukey algorithm, named after J. W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size in terms of N1 smaller DFTs of sizes N2, recursively, to reduce the computation time to O(N log N) for highly composite N. Because of the algorithm's importance, specific variants and implementation styles have become known by their own names, as described below.
The following timeline of algorithms outlines the development of algorithms since their inception.
Diffuse reflection is the reflection of light or other waves or particles from a surface such that a ray incident on the surface is scattered at many angles rather than at just one angle as in the case of specular reflection. An ideal diffuse reflecting surface is said to exhibit Lambertian reflection, meaning that there is equal luminance when viewed from all directions lying in the half-space adjacent to the surface.
Gopalasamudram Narayanan Ramachandran, or G.N. Ramachandran, FRS was an Indian physicist who was known for his work that led to his creation of the Ramachandran plot for understanding peptide structure. He was the first to propose a triple-helical model for the structure of collagen. He subsequently went on to make other major contributions in biology and physics.
Extended X-ray absorption fine structure (EXAFS), along with X-ray absorption near edge structure (XANES), is a subset of X-ray absorption spectroscopy (XAS). Like other absorption spectroscopies, XAS techniques follow Beer's law. The X-ray absorption coefficient of a material as a function of energy is obtained using X-rays of a narrow energy resolution are directed at a sample and the incident and transmitted x-ray intensity is recorded as the incident x-ray energy is incremented.
Cornelius (Cornel) Lanczos was a Hungarian-American and later Hungarian-Irish mathematician and physicist, who was born in Székesfehérvár, Fejér County, Kingdom of Hungary on February 2, 1893, and died on June 25, 1974. According to György Marx he was one of The Martians.
Phase retrieval is the process of algorithmically finding solutions to the phase problem. Given a complex signal , of amplitude , and phase :
Particle size analysis, particle size measurement, or simply particle sizing, is the collective name of the technical procedures, or laboratory techniques which determines the size range, and/or the average, or mean size of the particles in a powder or liquid sample.
Fourier-transform infrared spectroscopy (FTIR) is a technique used to obtain an infrared spectrum of absorption or emission of a solid, liquid or gas. An FTIR spectrometer simultaneously collects high-resolution spectral data over a wide spectral range. This confers a significant advantage over a dispersive spectrometer, which measures intensity over a narrow range of wavelengths at a time.
The multislice algorithm is a method for the simulation of the elastic interaction of an electron beam with matter, including all multiple scattering effects. The method is reviewed in the book by Cowley. The algorithm is used in the simulation of high resolution Transmission electron microscopy micrographs, and serves as a useful tool for analyzing experimental images. Here we describe relevant background information, the theoretical basis of the technique, approximations used, and several software packages that implement this technique. Moreover, we delineate some of the advantages and limitations of the technique and important considerations that need to be taken into account for real-world use.
Prof. RNDr. Jiří Horáček, DrSc. is a Czech theoretical physicist, professor at Charles University in Prague, where he works at the Institute of Theoretical Physics, which is a part of the Faculty of Mathematics and Physics. He was the director of the Institute in 2003–2011. He works in the field of theoretical atomic and molecular physics with a special focus on numerical solutions to integral and differential equations of scattering theory and on numerical analytic continuation methods localizing the poles of scattering quantities related to resonances.
Namrata Vaswani is an Indian-American electrical engineer known for her research in compressed sensing, robust principal component analysis, signal processing, statistical learning theory, and computer vision. She is a Joseph and Elizabeth Anderlik Professor in Electrical and Computer Engineering at Iowa State University, and a professor of mathematics at Iowa State.
References
- ↑ Jensen, Erling; MacKintosh, Allan; Swenson, Clayton (1984). "Gordon C. Danielson". Physics Today. 37 (1): 94–95. Bibcode:1984PhT....37a..94J. doi:10.1063/1.2916066 . Retrieved 19 August 2019.
- ↑ "Gordon C. Danielson papers". Iowa University. Retrieved 19 August 2019.
- ↑ Danielson, G. C., and C. Lanczos, "Some improvements in practical Fourier analysis and their application to X-ray scattering from liquids," J. Franklin Inst.233, 365–380 and 435–452 (1942).
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