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The origin‐free modulus sum function SM refines the set Φ of phases of the structure factors by maximizing the coincidence between the experimental origin‐free modulus function and the calculated one in terms of the ρ(Φ)2 density function [Rius, J. (1993). Acta Cryst. A49, 406–409]. Maximization is normally achieved through the recursive application of a Fourier‐based algorithm. The purpose of the present study is: (i) to show that ρ(Φ)2 can be replaced by |ρ(Φ)| in SM; (ii) to illustrate the viability of the corresponding phasing algorithm with experimental data. [ABSTRACT FROM AUTHOR]

The standard method of joint probability distribution functions, so crucial for the development of direct methods, has been revisited and updated. It consists of three steps: identification of the reflections which may contribute to the estimation of a given structure invariant or seminvariant, calculation of the corresponding joint probability distribution, and derivation of the conditional distribution of the invariant or seminvariant phase given the values of some diffracted amplitudes. In this article the conditional distributions are derived directly without passing through the second step. A good feature of direct methods is that they may work in the absence of any prior information: that is also their weakness. Different types of prior information have been taken into consideration: interatomic distances, interatomic vectors, Patterson peaks, structural model. The method of directly deriving the conditional distributions has been applied to those cases. Some new formulas have been obtained estimating two‐, three‐ and four‐phase invariants. Special attention has been dedicated to the practical aspects of the new formulas, in order to simplify their possible use in direct phasing procedures. Direct methods techniques are revisited and new mathematical approaches are described. [ABSTRACT FROM AUTHOR]

Demonstrates how Patterson function signal generated by dipolar field in inhomogenous liquids can provide structural information without the use of any model structure. Use of the field to probe the structure of heterogenous structures; Techniques in obtaining spin density.

Abstract The X-ray diffraction patterns of quill and hair, as well as other trichocyte keratin appendages, contain meridional reflections that can be indexed on an axial repeat of 470 Å. Unusually, however, many of the expected orders are not observed. A possible explanation, proposed by Fraser and MacRae (1983), was that the intermediate filaments (IF) that constitute the fibrillar component of the filament/matrix texture consist of 4-chain protofilaments arranged on a surface lattice subject to a helical dislocation. The radial projection of the resulting 8-protofilament ribbon was defined in terms of a two-dimensional unit cell characterized by vectors ( a, b ) with axial projections z a ∼ 74 Å and z b ∼ 198 Å. This situation resembles that found in microtubules, where helical dislocations in subunit packing are also encountered, leading to a so-called "seam" along their length (Metoz and Wade, 1997). In keratin, however, the protofilaments are helical so the seam is inclined to the axis of the IF. Here we report details of the Patterson function that provides independent evidence for both the helical dislocation and the dimensions of the surface lattice. In addition, the observed meridional X-ray amplitudes have been compared with those predicted by various models of the axial distribution of electron density. A new model, adapted from one previously proposed, fits the data significantly better than has heretofore proved possible. An interpretation of the model in terms of either specific keratin-associated-protein (KAP) binding or the retention of IF symmetry by a portion of the head and/or tail domains is suggested. [ABSTRACT FROM AUTHOR]

Gauss quadrature points are not nested so search for quadrature rules with nested points and similar efficiency are important. A well-studied source of candidates are the Kronrod-Patterson extensions. Under suitable conditions, it is possible to build towers of nested rules. We investigate this topic further and give a detailed description of the algorithms used for constructing such iterative extensions. Our new implementation combines several important ideas spread out in theoretical research papers. We apply the resulting algorithms to the classical orthogonal polynomials and build sparse high-dimensional quadrature rules for each class. [ABSTRACT FROM AUTHOR]

Coherent X-ray diffraction was used to measure the type, quantity and the relative distances between stacking faults along the growth direction of two individual wurtzite GaAs nanowires grown by metalorganic vapour epitaxy. The presented approach is based on the general property of the Patterson function, which is the autocorrelation of the electron density as well as the Fourier transformation of the diffracted intensity distribution of an object. Partial Patterson functions were extracted from the diffracted intensity measured along the direction in the vicinity of the wurtzite Bragg peak. The maxima of the Patterson function encode both the distances between the fault planes and the type of the fault planes with the sensitivity of a single atomic bilayer. The positions of the fault planes are deduced from the positions and shapes of the maxima of the Patterson function and they are in excellent agreement with the positions found with transmission electron microscopy of the same nanowire. [ABSTRACT FROM AUTHOR]

In this paper we introduced the RH-regularity condition of six dimensional matrix. Matrix summability is one of the important tool used to characterize sequence spaces. In 2004 Patterson presented such a characterization of bounded double sequence using four dimensional matrix. Our main aim is to extend Patterson result in triple sequence spaces using six dimensional matrix transformations. [ABSTRACT FROM AUTHOR]

The paper deals with | F|2, A and D Patterson functions. The information available in the various volumes of International Tables for Crystallography is detailed. A and D are defined and the reason for their use is explained. The symmetry of these functions is explained. Calculation by symmetry-adapted Patterson functions and the Fast-Fourier algorithm are described. There are two examples. There is a very useful table presenting the relevant information. [ABSTRACT FROM AUTHOR]

This paper reviews the number-theoretic concept of diaphony, a measure of uniform distribution for number sequences and point sets based on a Fourier theory approach, and its relation to crystallographic concepts like the largest interplanar spacing of a lattice, the structure-factor equation and the Patterson function. [ABSTRACT FROM AUTHOR]

This paper presents a completely new method for the calculation of expectations (and thus joint probability distributions) of structure factors or phase invariants. As an example, a first approximation of the expectation of the triplet invariant (up to a constant) is given and a complex number is obtained. Instead of considering the atomic vector positions or reciprocal vectors as the fundamental random variables, the method samples over all functions (distributions) with a given number of atoms and given Patterson function. The aim of this paper was to explore the feasibility of the method, so the easiest problem was chosen: the calculation of the expectation value of the triplet invariant in P1. Calculation of the joint probability distribution of the triplet is not performed here but will be done in the future. [ABSTRACT FROM AUTHOR]

In 1934, Arthur Lindo Patterson showed that a map of interatomic vectors is obtainable from measured X-ray diffraction data without phase information. Such maps were interpretable for simple crystal structures, but proliferation and overlapping of peaks caused confusion as the number of atoms increased. Since the peak height of a vector between two particular atoms is related to the product of their atomic numbers, a complicated structure could effectively be reduced to a simple one by including just a few ‘heavy’ atoms (of high atomic number) since their interatomic vectors would stand out from the general clutter. Once located, these atoms provide approximate phases for Fourier syntheses that reveal the locations of additional atoms. Surveys of small-molecule structures in the Cambridge Structural Database during the periods 1936–1969, when Patterson methods were commonly used, and 1980–2013, dominated by direct methods, demonstrate large differences in the abundance of certain elements. The ‘moderately heavy’ elements K, Rb, As and Br are the heaviest elements in the structure more than 3 times as often in the early period than in the recent period. Examples are given of three triumphs of the heavy atom method and two initial failures that had to be overcome. [ABSTRACT FROM PUBLISHER]

X-ray crystallography and NMR are complementary tools in structural biology. However, it is often difficult to use NMR structures as search models in molecular replacement (MR) to phase crystallographic data. In this study, a new approach is reported utilizing a molecular envelope of NMR structures for MR phasing with the program FSEARCH at low resolution (about 6 Å). Several targets with both crystallographic and NMR structures available have been tested. FSEARCH was able to find the correct translation and orientation of the search model in the crystallographic unit cell, while conventional MR procedures were unsuccessful. [ABSTRACT FROM AUTHOR]

The crystal structure of sarmientite, Fe3+2(AsO4)(SO4)(OH)·5H2O, from the type locality (Santa Elena mine, San Juan Province, Argentina), was solved and refined from in-house powder diffraction data (CuKα1,2radiation). It is monoclinic, space group P21/n, with unit-cell dimensions a= 6.5298(1), b= 18.5228(4), c= 9.6344(3) Å, β = 97.444(2)°, V= 1155.5(5) Å3, and Z= 4. The structure model was derived from cluster-based Patterson-function direct methods and refined by means of the Rietveld method to Rwp= 0.0733 (χ2= 2.20). The structure consists of pairs of octahedral-tetrahedral (Fe–As) chains at (y,z) = (0,0) and (½,½), running along a. There are two symmetry-independent octahedral Fe sites. The Fe1 octahedra share two corners with the neighbouring arsenate groups. Both individual chains are related by a symmetry centre and joined by two symmetry-related Fe2 octahedra. Each Fe2 octahedron shares three corners with double-chain polyhedra (O3, O4 with arsenate groups; the O8 hydroxyl group with the Fe1 octahedron) and one corner (O11) with the monodentate sulfate group. The coordination of the Fe2 octahedron is completed by two H2O molecules (O9 and O10). There is also a complex network of H bonds that connects polyhedra within and among chains. Raman and infrared spectra show that (SO4)2–tetrahedra are strongly distorted. [ABSTRACT FROM AUTHOR]

The Sardinas-Patterson's test for codes has contributed many effective testing algorithms to the development of theory of codes, formal languages, etc. However, we will show that a modification of this test proposed in this paper can deduce more effective testing algorithms for codes. As a consequence, we establish a quadratic algorithm that, given as input a regular language X defined by a tuple (φ, M, B), where φ : A* → M is a monoid morphism saturating X, M is a finite monoid, $B \subseteq M$, X = φ-1(B), decides in time complexity $\mathcal{O}(n^2)$ whether X is a code, where n = Card(M). Specially, n can be chosen as the finite index of X. A quadratic algorithm for testing of ⋄-codes is also established. [ABSTRACT FROM AUTHOR]

The Src homology 3 (SH3) domain is a small, noncatalytic domain with a conserved sequence of about 60 amino-acid residues that interacts with proline-rich peptides to form a protein complex. In this study, the C-terminal SH3 domain of human Tks4 (residues 853-911) was expressed, purified and crystallized. X-ray diffraction data were collected to 2.3 Å resolution. The crystal belonged to the trigonal space group P3121 (or P3221), with unit-cell parameters a = b = 83.87, c = 108.44 Å, α = β = 90, γ = 120°. Calculating the self-rotation and the native Patterson function did not lead to the detection of any noncrystallographic translational symmetry. Six, seven or eight protein molecules are likely to be present in the asymmetric unit, resulting in a Matthews coefficient and approximate solvent content of 2.71 Å3 Da−1 and 55%, 2.32 Å3 Da−1 and 47%, and 2.03 Å3 Da−1 and 39%, respectively. To solve the crystal structure of the C-terminal SH3 domain of human Tks4, the isomorphous replacement method is presently being utilized. [ABSTRACT FROM AUTHOR]

Monalysin was recently described as a novel pore-forming toxin (PFT) secreted by the Drosophila pathogen Pseudomonas entomophila. Recombinant monalysin is multimeric in solution, whereas PFTs are supposed to be monomeric until target membrane association. Monalysin crystals were obtained by the hanging-drop vapour-diffusion method using PEG 8000 as precipitant. Preliminary X-ray diffraction analysis revealed that monalysin crystals belonged to the monoclinic space group C2, with unit-cell parameters a = 162.4, b = 146.2, c = 144.4 Å, β = 122.8°, and diffracted to 2.85 Å resolution using synchrotron radiation. Patterson self-rotation analysis and Matthews coefficient calculation indicate that the asymmetric unit contains nine copies of monalysin. Heavy-atom derivative data were collected and a Ta6Br14 cluster derivative data set confirmed the presence of ninefold noncrystallographic symmetry. [ABSTRACT FROM AUTHOR]

The structure of the Si(111)-(5×2)-Au surface, one of the long-standing problems in surface science, has been solved by means of Weissenberg reflection high-energy electron diffraction. The arrangement of the Au atoms and their positions with respect to the substrate were determined from a three-dimensional Patterson function with a lateral resolution of 0.3 Å based on a large amount of diffraction data. The new structural model consists of six Au atoms in a 5×2 unit, which agrees with the recently confirmed Au coverage of 0.6 ML [ I. Barke et al. Phys. Rev. B 79 155301 (2009)]. The model has a distinct ×2 periodicity, and includes a Au dimer. The model is also compatible with previously obtained STM images. [ABSTRACT FROM AUTHOR]

This paper presents a shuffled frog leaping algorithm(SFLA) for the single-mode resource-constrained project scheduling problem where activities can be divided into equant units and interrupted during processing. Each activity consumes 0-3 types of resources which are renewable and temporarily not available due to resource vacations in each period. The presence of scarce resources and precedence relations between activities makes project scheduling a difficult and important task in project management. A recent popularmetaheuristic shuffled frog leaping algorithm, which is enlightened by the predatory habit of frog group in a small pond, is adopted to investigate the project makespan improvement on Patterson benchmark sets which is composed of different small and medium size projects. Computational results demonstrate the effectiveness and efficiency of SFLA in reducing project makespan and minimizing activity splitting number within an average CPU runtime, 0.521 second. This paper exposes all the scheduling sequences for each project and shows that of the 23 best known solutions have been improved. [ABSTRACT FROM AUTHOR]

The cross-correlation function between the target and a model electron density, denoted as the C map, has been crystallographically characterized. In particular, a study of its interatomic vectors and of their relation with the Patterson vectors has been undertaken. Since the C map is not available during the phasing process, the C′ map, its centric modification, is considered. It may be computed at any stage of the phasing process and shows properties that are very useful for the crystal structure determination process. It has been combined with the implication transformation method and with vector-superposition techniques for performing the Patterson deconvolution and obtaining an initial model for dual-space recycling. While Patterson methods are traditionally considered to be more efficient for structures containing heavy atoms, the C map extends their potential to light-atom structures ( i.e. containing atoms not heavier than O). [ABSTRACT FROM AUTHOR]

We consider some applications (old and new) of the Patterson-Selberg zeta function to 3D gravity with a negative cosmological constant.We also consider 2D black hole vacua with a parabolic generator of their holonomy. This article is part of a special issue of Journal of Physics A:Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'. [ABSTRACT FROM AUTHOR]

The direct methods origin-free modulus sum function [Rius (1993). Acta Cryst. A 49, 406-409] includes in its definition the structure factor G(Φ) of the squared crystal structure expressed in terms of Φ, the set of ϕ phases of the normalized structure factors E's of the crystal structure of unit-cell volume V. Here the simpler sum function variant = ∑ H E− H∫ VδP,Δ(Φ)exp( i2π Hr)d V extended over all H reflections is introduced which involves no G's and in which the δP,Δ function corresponds to δP = FT−1{()exp[ iϕ H(Φ)]} (where FT = Fourier transform) with all values smaller than Δ = 2.5σP equated to zero ( is the variance of δP calculable from the experimental intensities). The new phase estimates are obtained by Fourier transforming δP,Δ. This iterative phasing method (δ recycling) only requires calculation of Fourier transforms at two stages. Since δM≃δP/2, similar arguments are valid for δM = FT−1[( E H−〈 E〉)exp( iϕ H)] from which the corresponding phasing function follows. [ABSTRACT FROM AUTHOR]

A fast Fourier transform algorithm is introduced into the method recently defined for calculating powder diffraction patterns by means of the Debye scattering equation (DSE) [Thomas (2010). Acta Cryst. A 66, 64-77]. For this purpose, conventionally used histograms of interatomic distances are replaced by compound transmittance functions. These may be Fourier transformed to partial diffraction patterns, which sum to give the complete diffraction pattern. They also lead to an alternative analytical expression for the DSE sum, which reveals its convergence behaviour. A means of embedding the DSE approach within the reciprocal-lattice-structure-factor method is indicated, with interpolation methods for deriving the peak profiles of nanocrystalline materials outlined. Efficient calculation of transmittance functions for larger crystallites requires the Patterson group symmetry of the crystals to be taken into account, as shown for α- and β-quartz. The capability of the transmittance functions to accommodate stacking disorder is demonstrated by reference to kaolinite, with a fully analytical treatment of disorder described. Areas of future work brought about by these developments are discussed, specifically the handling of anisotropic atomic displacement parameters, inverse Fourier transformation and the incorporation of instrumental (diffractometer) parameters. [ABSTRACT FROM AUTHOR]

Abstract: The molar volume and compressibility of (glycylglycine+CuCl2) in aqueous ethanol mixtures have been obtained at four different temperatures T =(288.15 to 318.15)K from ultrasonic velocity and density measurements. Excess molar volumes were found to be negative throughout the composition range indicating notable changes in hydrogen bonding and electrostatic interactions. Using the Prigogine–Flory–Patterson theory, quantitative estimation of different contributions, i.e. interactional, free volume, and P ∗ effect to V E have been obtained. The molar isentropic compressibility has been computed using the ultrasonic velocity and excess volume data. The trends in are affected by the size of the molecule leading to negative contributions. In order to compare the theoretical values of ultrasonic velocity, the equations of Nomoto and Junjie were used and found to predict the experimental data very well. [Copyright &y& Elsevier]

This paper defines homometry in the rather general case of locally-compact topological groups, and proposes new cases of its musical use. For several decades, homometry has raised interest in computational musicology and especially set-theoretical methods, and in an independent way and with different vocabulary in crystallography and other scientific areas. The link between these two approaches was only made recently, suggesting new interesting musical applications and opening new theoretical problems. We present some old and new results on homometry, and give perspective on future research assisted by computational methods. We assume from the reader's basic knowledge of groups, topological groups, group algebras, group actions, Lebesgue integration, convolution products, and Fourier transform. [ABSTRACT FROM AUTHOR]

This paper describes phase-retrieval approaches in music by focusing on the particular case of the cyclic groups (beltway problem). After presenting some old and new results on phase retrieval, we introduce the extended phase retrieval for a generalized musical Z-relation. This concept is accompanied by mathematical definitions and motivations from computer-aided composition. We assume from the reader basic knowledge of groups, topological groups, group algebras, group actions, Lebesgue integration, convolution products, and Fourier transform. [ABSTRACT FROM AUTHOR]

The article discusses the study on the use of Patterson-function direct methods (DM) in determining the structure of organic compounds from powder diffraction data. It says that the method is based on the structure factors (Φ) maximization of the sum function of Sp. It cites the optimization of Sp sum with Patterson-function tangent formula (TF) by the Sliding-Fast Fourier Transform (S-FFT) algorithm. It shows that Patterson-function DM can deal with powder data.

The article presents a study on the use of Patterson vectors molecular-replacement (PVMR) method in tracking helix fragments. It states that PVMR is calculated by equating a search model in a real space with Patterson cross and self vectors. It mentions that the equal matches of Patterson vectors are added to get the signal-to-noise ratio. Moreover, findings of the study show that PVMR is suitable in setting small helix fragments as structural root for molecular substitution.

The article offers information on a study which aims to investigate the intensity difference in Friedel opposites on the analysis of small-molecule single-crystal structure using the Patterson functions. It says that Patterson functions give clear view with regards to resonant scattering of the crystal structures and on the difference on the model and observed intensities. The study used least square refinement on the space group determination in the crystal structure.