Sample-optimal average-case sparse Fourier Transform in two dimensions

We propose a new technique, using femtosecond Fourier-transform spectral interferometry, to measure the second-order nonlinear response of a material in two dimensions of frequency. We show numerically the specific and unique information obtained from such a two-dimensional measurement. The technique is demonstrated by measuring the second-order phase-matching map of two non-resonant nonlinear crystals.

Download Full-text

Electron-density maps

Outline of Crystallography for Biologists ◽

10.1093/oso/9780198510512.003.0016 ◽

2002 ◽

Author(s):

David Blow

Keyword(s):

Fourier Transform ◽

Electron Density ◽

Three Dimensional ◽

Two Dimensions ◽

Unit Cell ◽

Density Maps ◽

Wave Cycle ◽

Density Map ◽

The Fourier Transform ◽

Electron Density Map

When everything has been done to make the phases as good as possible, the time has come to examine the image of the structure in the form of an electron-density map. The electron-density map is the Fourier transform of the structure factors (with their phases). If the resolution and phases are good enough, the electron-density map may be interpreted in terms of atomic positions. In practice, it may be necessary to alternate between study of the electron-density map and the procedures mentioned in Chapter 10, which may allow improvements to be made to it. Electron-density maps contain a great deal of information, which is not easy to grasp. Considerable technical effort has gone into methods of presenting the electron density to the observer in the clearest possible way. The Fourier transform is calculated as a set of electron-density values at every point of a three-dimensional grid labelled with fractional coordinates x, y, z. These coordinates each go from 0 to 1 in order to cover the whole unit cell. To present the electron density as a smoothly varying function, values have to be calculated at intervals that are much smaller than the nominal resolution of the map. Say, for example, there is a protein unit cell 50 Å on a side, at a routine resolution of 2Å. This means that some of the waves included in the calculation of the electron density go through a complete wave cycle in 2 Å. As a rule of thumb, to represent this properly, the spacing of the points on the grid for calculation must be less than one-third of the resolution. In our example, this spacing might be 0.6 Å. To cover the whole of the 50 Å unit cell, about 80 values of x are needed; and the same number of values of y and z. The electron density therefore needs to be calculated on an array of 80×80×80 points, which is over half a million values. Although our world is three-dimensional, our retinas are two-dimensional, and we are good at looking at pictures and diagrams in two dimensions.

Download Full-text

Diffraction by crystals

Outline of Crystallography for Biologists ◽

10.1093/oso/9780198510512.003.0009 ◽

2002 ◽

Author(s):

David Blow

Keyword(s):

Fourier Transform ◽

Three Dimensional ◽

Incident Beam ◽

Two Dimensions ◽

Three Dimensions ◽

Dimensional Structure ◽

Angle Of Incidence ◽

X Ray ◽

Max Von Laue ◽

The Fourier Transform

In Chapter 4 many two-dimensional examples were shown, in which a diffraction pattern represents the Fourier transform of the scattering object. When a diffracting object is three-dimensional, a new effect arises. In diffraction by a repetitive object, rays are scattered in many directions. Each unit of the lattice scatters, but a diffracted beam arises only if the scattered rays from each unit are all in phase. Otherwise the scattering from one unit is cancelled out by another. In two dimensions, there is always a direction where the scattered rays are in phase for any order of diffraction (just as shown for a one-dimensional scatterer in Fig. 4.1). In three dimensions, it is only possible for all the points of a lattice to scatter in phase if the crystal is correctly oriented in the incident beam. The amplitudes and phases of all the scattered beams from a three-dimensional crystal still provide the Fourier transform of the three-dimensional structure. But when a crystal is at a particular angular orientation to the X-ray beam, the scattering of a monochromatic beam provides only a tiny sample of the total Fourier transform of its structure. In the next section, we are going to find what is needed to allow a diffracted beam to be generated. We shall follow a treatment invented by Lawrence Bragg in 1913. Max von Laue, who discovered X-ray diffraction in 1912, used a different scheme of analysis; and Paul Ewald introduced a new way of looking at it in 1921. These three methods are referred to as the Laue equations, Bragg’s law and the Ewald construction, and they give identical results. All three are described in many crystallographic text books. Bragg’s method is straightforward, understandable, and suffices for present needs. I had heard J.J. Thomson lecture about…X-rays as very short pulses of radiation. I worked out that such pulses…should be reflected at any angle of incidence by the sheets of atoms in the crystal as if these sheets were mirrors.…It remained to explain why certain of the atomic mirrors in the zinc blende [ZnS] crystal reflected more powerfully than others.

Download Full-text

Numerical Simulation of Non-Gaussian Wave Elevation and Kinematics Based on Two-Dimensional Fourier Transform

Volume 1: Offshore Technology; Offshore Wind Energy; Ocean Research Technology; LNG Specialty Symposium ◽

10.1115/omae2006-92014 ◽

2006 ◽

Author(s):

Xiang Yuan Zheng ◽

Torgeir Moan ◽

Ser Tong Quek

Keyword(s):

Fourier Transform ◽

Wave Interaction ◽

Wave Theory ◽

Two Dimensions ◽

Second Order ◽

Interaction Matrix ◽

Random Wave ◽

Two Dimensional ◽

Near Surface ◽

Wave Elevation

The one-dimensional Fast Fourier Transform (FFT) has been extensively applied to efficiently simulate Gaussian wave elevation and water particle kinematics. The actual sea elevation/kinematics exhibit non-Gaussianities that mathematically can be represented by the second-order random wave theory. The elevation/kinematics formulation contains double-summation frequency sum and difference terms which in computation make the dynamic analysis of offshore structural response prohibitive. This study aims at a direct and efficient two-dimensional FFT algorithm for simulating the frequency sum terms. For the frequency difference terms, inverse FFT and FFT are respectively implemented across the two dimensions of the wave interaction matrix. Given specified wave conditions, not only the wave elevation but kinematics and associated Morison force are simulated. Favorable agreements are achieved when the statistics of elevation/kinematics are compared with not only the empirical fits but the analytical solutions developed based on modified eigenvalue/eigenvector approach, while the computation effort is very limited. In addition, the stochastic analyses in both time-and frequency domains show that the near-surface Morison force and induced linear oscillator response exhibits stronger non-Gaussianities by involving the second-order wave effects.

Download Full-text

TIME OPTIMAL ALGORITHMS FOR BLACK HOLE SEARCH IN RINGS

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830911001346 ◽

2011 ◽

Vol 03 (04) ◽

pp. 457-471 ◽

Cited By ~ 8

Author(s):

B. BALAMOHAN ◽

P. FLOCCHINI ◽

A. MIRI ◽

N. SANTORO

Keyword(s):

Black Hole ◽

Time Complexity ◽

Time Cost ◽

Ring Networks ◽

Worst Case ◽

Average Case ◽

Case Complexity ◽

Average Case Complexity ◽

Time Optimal ◽

Optimal Average

In a network environment supporting mobile entities (called robots or agents), a black hole is a harmful site that destroys any incoming entity without leaving any visible trace. The black-hole search problit is the task of a team of k > 1 mobile entities, starting from the same safe location and executing the same algorithm, to determine within finite time the location of the black hole. In this paper, we consider the black hole search problit in asynchronous ring networks of n nodes, and focus on time complexity. It is known that any algorithm for black-hole search in a ring requires at least 2(n - 2) time in the worst case. The best known algorithm achieves this bound with a team of n - 1 agents with an average time cost of 2(n - 2), equal to the worst case. In this paper, we first show how the same number of agents using 2 extra time units in the worst case, can solve the problit in only [Formula: see text] time on the average. We then prove that the optimal average case complexity of [Formula: see text] can be achieved without increasing the worst case using 2(n - 1) agents. Finally, we design an algorithm that achieves asymptotically optimal both worst and average case time complexities itploying an optimal team of k = 2 agents, thus improving on the earlier results that required O(n) agents.

Download Full-text

Non-Gaussian Random Wave Simulation by Two-Dimensional Fourier Transform and Linear Oscillator Response to Morison Force

Journal of Offshore Mechanics and Arctic Engineering ◽

10.1115/1.2783888 ◽

2007 ◽

Vol 129 (4) ◽

pp. 327-334 ◽

Cited By ~ 6

Author(s):

Xiang Yuan Zheng ◽

Torgeir Moan ◽

Ser Tong Quek

Keyword(s):

Fourier Transform ◽

Time Domain ◽

Two Dimensions ◽

Second Order ◽

Linear Oscillator ◽

Interaction Matrix ◽

Random Wave ◽

Two Dimensional ◽

Near Surface ◽

Non Gaussian

The one-dimensional fast Fourier transform (FFT) has been applied extensively to simulate Gaussian random wave elevations and water particle kinematics. The actual sea elevations/kinematics exhibit non-Gaussian characteristics that can be represented mathematically by a second-order random wave theory. The elevations/kinematics formulations contain frequency sum and difference terms that usually lead to expensive time-domain dynamic analyses of offshore structural responses. This study aims at a direct and efficient two-dimensional FFT algorithm for simulating the frequency sum terms. For the frequency-difference terms, inverse FFT and forward FFT are implemented, respectively, across the two dimensions of the wave interaction matrix. Given specified wave conditions, the statistics of simulated elevations/kinematics compare well with not only the empirical fits but also the analytical solutions based on a modified eigenvalue/eigenvector approach, while the computational effort of simulation is very economical. In addition, the stochastic analyses in both time domain and frequency domain show that, attributable to the second-order nonlinear wave effects, the near-surface Morison force and induced linear oscillator response are more non-Gaussian than those subjected to Gaussian random waves.

Download Full-text