Use Of Symmetry Research Articles

Crystallography and phase equilibria a review: Part II—Space groups and structure

• The lengthy discussion of the structure determination of the Ni10Zr7 phase illustrates the utility of symmetry in reducing work. Had there been no use of symmetry, the determination of the structures would have required the determination of the positional coordinates of 68 atoms, or a total of 204 positional parameters. The inclusion of the symmetry constraints reduced this number to 24. This is a reduction in number in excess of eightfold.

Principle of an a priori use of symmetries in the theory of nonlinear waves

The principle of an a priori use of symmetries is proposed as a new approach to solving nonlinear problems on the basis of a reasonable complication of mathematical models. Such a complication often causes an additional symmetry and, hence, opens up possibilities for finding new analytical solutions. The application of group analysis to the problems of nonlinear acoustics is outlined. The potentialities of the proposed approach are illustrated by exact solutions, which are of interest for wave theory.

Theory of periodic states for feedback-controlled photorefractive nonlinear systems

Certain feedback loops can be used in photorefractive optical schemes to implement periodic states and create spatial gratings characterized by extremely high or low diffraction efficiencies. This highly nonlinear phenomenon is studied both experimentally and numerically. An analytical method is developed for analyzing periodic states with the use of symmetries of time-dependent diffraction equations and fast feedback response. The method is applied to describe the properties of periodic states, including their spatial structure, diffraction-efficiency oscillation period and amplitude, and characteristics of feedback-controlled strong phase modulation.

Method of Length Measurement Symmetrical about the Laboratory and Moving Reference Systems in the Special Theory of Relativity

With the use of symmetry it is demonstrated that for equal moduli of non-simultaneity of fixing the ends of the measurable length in both reference systems, the coordinate difference is equal to the length itself. The identity of scales of both reference systems is proved. A method of calculating the coordinate difference for non-simultaneities that differ from those in the symmetrical measurement scheme is described. The method is based on the use of “experimental” non-simultaneities and obviates the need for the application of the Lorentz transformation of coordinates. It is demonstrated quantitatively that there are no false discontinuities in a closed flow when an increase in the particle flow density is proved for a stationary observer and that the observable sizes of a rotating disk remain unchanged.

SYMMETRY AND GOAL-DIRECTED MOTION IN MUSIC BY BÉLA BARTÓK AND GEORGE CRUMB

Two of the most important goals for music composition during the past century were innovation and originality. In reaching for these goals, many composers abandoned tonal protocol, looking instead for new ways to generate coherence in their music. In the absence of traditional voice leading based on tonic/dominant harmony, some composers gravitated toward the use of symmetry. By symmetry I mean the projection of a series of events above and below, or forward and backward from a central axis, with a one-to-one correspondence between events on one side of the axis and those on the other. In this article, I will focus primarily on vertical symmetry in pitch space. Because horizontal symmetry often accompanies vertical symmetry, however, it cannot be set aside altogether. George Crumb's La luna está muerta, muerta… from Night of the Four Moons, for example, contains several symmetrical pitch collections, arranged in the manner of a balanced arch-form. As Ex. 1 illustrates, four pitch sets, comprising a single symmetrical interval series, occur over the course of the composition. The pitch sets in the first half of the composition, moreover, are presented in reverse order – and, hence, upside down – in the second half, transferring symmetry into the linear dimension; the events in the second half of the piece, that is, mirror those in the first half, just as the intervals above the pitch axis mirror those below it in each of the pitch collections.

Plane strain formability analysis of automotive body structures using DYNA2D

At the early design phase, formability evaluation of automotive panels is usually needed before the complete geometry is defined in order to differentiate between design alternatives. The designs of these panels at early phases are rapidly changing with few defined sections. This creates the need for fast formability evaluation techniques with minimum geometric definition, modeling, analysis time and effort to effectively guide the design for manufacturability process. This paper introduces the plain strain approach as a tool for formability prediction of automotive body structural panels at early design phases. The plane strain approach was introduced based on the assumption that for a certain section location of a stamped panel member, the minor strains are relatively small and negligible in comparison with the major in-section strains. For automotive panels this approach can save time, cost and effort in analyzing sheet metal formability, since only few sections of the entire panel need be analyzed. It is also the most logical approach, when only few sections are defined and their geometries are rapidly changing. During analysis, the state of plane strain can be easily induced on a narrow strip of the sectional analysis model, through the use of symmetry and boundary conditions. The accuracy and applicability of the plane strain approach was investigated by comparing the DYNA2D analytical results with the measured results of actual automotive body side panels. The results of the study were satisfactory and suggested that the plane strain approach can be used to evaluate the formability of the panel members with fairly constant cross-section along the length. When the members have cross-sectional variation along the length, several sections are needed to obtain meaningful results. For the plane strain approach to achieve accurate strain distribution over an entire simulated section, the geometric modeling of the draw bead is necessary.

The use of symmetry in the search for canted ferromagnetism, its application to the molecular magnets Mn[N(CN) 2] 2 and Fe[N(CN) 2] 2

Canted ferromagnets present a important class of molecular magnets. In these materials the uncompensated magnetisation, created by the DzyaloshinskyMoriya interaction, has the particular property of being compatible with the symmetry of the underlying antiferromagnetic Hamiltonian. If the ordering transition is continuous, Landau theory and simple symmetry arguments about the crystal structure can be used to determine whether such parasitic ferromagnetism is possible, and therefore to aid the systematic search for new molecular ferromagnets.

The maximum-entropy method in superspace.

One of the applications of the maximum-entropy method (MEM) in crystallography is the reconstruction of the electron density from phased structure factors. Here the application of the MEM to incommensurately modulated crystals and incommensurate composite crystals is considered. The MEM is computed directly in superspace, where the electron density in the (3+d)-dimensional unit cell (d > 0) is determined from the scattering data of aperiodic crystals. Periodic crystals (d = 0) are treated as a special case of the general formalism. The use of symmetry in the MEM is discussed and an efficient algorithm is proposed for handling crystal symmetry. The method has been implemented into a computer program BayMEM and applications are presented to the electron density of the periodic crystal NaV(2)O(5) and the electron density of the incommensurate composite crystal (LaS)(1.14)NbS(2). The MEM in superspace is shown to provide a model-independent estimate of the shapes of the modulation functions of incommensurate crystals. The discrete character of the electron density is found to be the major source of error, limiting the accuracy of the reconstructed modulation functions to approximately 10% of the sizes of the pixels. MaxEnt optimization using the Cambridge and Sakata-Sato algorithms are compared. The Cambridge algorithm is found to perform better than the Sakata-Sato algorithm, being faster, always reaching convergence, and leading to more reliable density maps. Nevertheless, the Sakata-Sato algorithm leads to similar density maps, even in cases where it does not reach complete convergence.

A Symmetric Boundary Element Method/Finite Element Method Coupling Procedure for Two-Dimensional Elastodynamic Problems

In this paper, a symmetric collocation boundary element method (SCBEM)/finite element method (FEM) coupling procedure is given and applied to a two-dimensional elastodynamic problem. The use of symmetry for the boundary element method not only saves memory storage but also enables the employment of efficient symmetric equation solvers. This is especially important for BEM/FEM coupling procedure. Compared with the symmetric Galerkin boundary element method (SGBEM) where double-space integration should be carried out, SCBEM is easier and faster.

New developments in the symmetry-adapted algorithm of the Polarizable Continuum Model.

We present recent developments in the symmetry implementation of the Polarizable Continuum Model (PCM). The structure of the matrix, which defines the PCM solvent response, is examined, and we demonstrate how this matrix can be transformed to a block diagonal form where each block belongs to different irreducible representations of the molecular point group. This development is especially important at the Multi-configurational Self-Consistent Field (MCSCF) level where symmetry is needed to avoid problems with symmetry breaking in the wave function and facilitate the optimization of electronic excited states. Moreover, although only the totally symmetric part of the solvent interaction is needed for energy calculations, in response or perturbation theory calculations of molecular properties, other irreps play an important role and the classification of solvent interaction terms by irrep is, therefore, desirable. In addition, the use of symmetry reduces the computational cost. The implementation presented here is illustrated with a series of calculations of absorption and emission processes in solution on the diazines pyrazine, pyrimidine, and pyridazine. These calculations allow us to illustrate both formal aspects of the implementation such as the choice of active spaces based on orbital symmetry as well as practical aspects such as the speed-up of the calculation.

Aesthetic judgments of novel graphic patterns: analyses of individual judgments.

Aesthetic judgments were investigated using a combined nomothetic and idiographic approach. Participants judged novel graphic patterns with respect to their own personal definitions of "beauty." Judgment analysis was employed to derive individual case models of judgment strategies as well as a group model. As predicted, symmetry had the highest correlations with aesthetic judgments of beauty. Stimulus complexity was the second-highest correlate of a positive evaluation. Thus, there was agreement at the group level. The judgment analyses, however, indicated substantial individual differences. These included use of symmetry or complexity cues that were contrary to the main group use, e.g., a few participants considered nonsymmetric patterns more beautiful. These findings suggest that exclusive consideration of the group model would have leveled the individual differences and been misleading. The group model is significant; however, the individual judgment analyses represent individual patterns of judgment in a notedly more accurate way.

Symmetric collocation BEM/FEM coupling procedure for 2-D dynamic structural-acoustic interaction problems

This paper presents a symmetric collocation BEM (SCBEM)/FEM coupling procedure applicable to 2-D time domain structural–acoustic interaction problems. The use of symmetry for BEM not only saves memory storage but also enables the employment of efficient symmetric equation solvers, especially for BEM/FEM coupling procedure. Compared with symmetric Galerkin BEM (SGBEM) where double boundary integration should be carried out, SCBEM can reduce significantly the computing cost. Two numerical examples are included to illustrate the effectiveness and accuracy of the proposed method.

Erosions in inflammatory polyarthritis are symmetrical regardless of rheumatoid factor status: results from a primary care-based inception cohort of patients.

Symmetry is considered an important criterion for the differentiation of rheumatoid arthritis (RA) from other forms of inflammatory polyarthritis (IP), particularly those that are seronegative. Because of the inclusion of symmetry in the diagnostic and classification process, however, its true occurrence in RA cannot be assessed. As a surrogate, peripheral inflammatory arthropathies associated with rheumatoid factor production may be more likely to be symmetrical. We examined the degree of symmetry of erosions in an unselected cohort of patients with IP and tested the hypothesis that the presence of rheumatoid factor (RF) is associated with greater symmetry. All patients registered with The Norfolk Arthritis Register (NOAR; a UK primary-care based cohort of patients with IP with annual follow-up) and who had radiographs performed at the fifth anniversary from notification were included in the analysis. Radiographs of the hands and feet were read using the Larsen method; a score of 2 or more in any particular joint indicated an erosion. Log-linear modelling was used to determine the symmetry of erosions between right and left for the following joint groups: wrists, metacarpophalangeal joints, proximal interphalangeal joints and metatarsophalangeal joints. Log-linear modelling was also used to determine the influence of RF on symmetry. Five hundred and thirty-seven patients contributed to the analysis. The median time to performing radiographs was 69 months (interquartile range 65.5-74.8) from the onset of symptoms. A total of 212 (39%) patients had erosive disease. Overall, IP was found to be a symmetrical disease. Despite there being more erosions in RF-positive patients, there was no greater excess of symmetry in RF-positive compared with RF-negative patients. Radiographically, IP is a symmetrical disease irrespective of RF status. The use of symmetry as an important feature in identifying subgroups of patients with IP, such as RA, is challenged.

Finite Element Analysis of Nanoscale Thermal Measurements of Superlattices

ABSTRACTA finite element analysis applicable to two- and three-dimensional heat flow in samples of arbitrary geometry and composition is presented for use in a thermal wave experiment. The finite element formulation is summarized, including the use of symmetry to simplify the problem, and the governing differential equations for the heat transport are found to be in the form of the Helmholtz equation for the specific case of a modulated heat source. Simulated data for a Nb/Si superlattice is calculated using the finite element code and is shown to agree with predictions from an analytical model, validating the approach taken.

Electron-phonon interaction and ultrasonic attenuation in the ruthenate and cuprate superconductors

This article derives an electron-phonon interaction suitable for interpreting ultrasonic attenuation measurements in the ruthenate and cuprate superconductors. The huge anisotropy found experimentally (Lupien et al., 2001) in Sr2RuO4 in the normal state is accounted for in terms of the layered square-lattice structure of Sr2RuO4, and the dominant contribution to the attenuation in Sr2RuO4 is found to be due to electrons in the gamma band. The experimental data in the superconducting state is found to be inconsistent with vertical lines nodes in the gap in either (100) or (110) planes. Also, a general method, based on the use of symmetry, is developed to allow for the analysis of ultrasonic attenuation experiments in superconductors in which the electronic band structure is complicated or not known. Our results, both for the normal-state anisotropy, and relating to the positions of the gap nodes in the superconducting state, are different from those obtained from analyses using a more traditional model for the electron-phonon interaction in terms of an isotropic electron stress tensor. Also, a brief discussion of the ultrasonic attenuation in UPt3 is given.

Normal mode analysis of large systems with icosahedral symmetry: Application to (Dialanine)60 in full and reduced basis set implementations

A group theoretical method is described for calculating the normal modes of icosahedral systems such as viruses. The use of symmetry reduces the size of the matrices that have to be diagonalized from 60N×60N to 5N×5N (where N is the number of degrees of freedom of every subunit) while preserving complete accuracy. The method includes algorithms to calculate the normal modes, the atomic fluctuations and cross-correlations, and the projections of normal modes on vectors of interest. The correctness and accuracy of the method are verified using a model system, consisting of 60 Dialanine peptides arranged in a nonbonded icosahedral complex. The effects of using reduced basis sets in the normal mode calculations are examined. Reduced basis sets, especially those consisting of dihedral and/or bond angles, are shown to have relatively small effects on the frequencies, spatial fluctuations, and directions of the normal mode displacements. The current implementation allows accurate reduced basis normal mode calculations on icosahedral virus molecules with moderately powerful computers.

Finding and using local symmetry in identifying lower domain movements in hexon subunits of the herpes simplex virus type 1 B capsid

A characteristic of virus assembly is the use of symmetry to construct a complex capsid from a limited number of different proteins. Many spherical viruses display not only icosahedral symmetry, but also local symmetries, which further increase the redundancy of their structural proteins. We have developed a computational procedure for evaluating the quality of these local symmetries that allows us to probe the extent of local structural variations among subunits. This type of analysis can also provide orientation parameters for carrying out non-icosahedral averaging of quasi-equivalent subunits during three-dimensional structural determination. We have used this procedure to analyze the three types of hexon (P, E and C) in the 8.5 Å resolution map of the herpes simplex virus type 1 (HSV-1) B capsid, determined by electron cryomicroscopy. The comparison of the three hexons showed that they have good overall 6-fold symmetry and are almost identical throughout most of their lengths. The largest difference among the three lies near the inner surface in a region of about 34 Å in thickness. In this region, the P hexon displays slightly lower 6-fold symmetry than the C and E hexons. More detailed analysis showed that parts of two of the P hexon subunits are displaced counterclockwise with respect to their expected 6-fold positions. The most highly displaced subunit interacts with a subunit from an adjacent P hexon (P′). Using the local 6-fold symmetry axis of the P hexon as a rotation axis, we examined the geometrical relationships among the local symmetry axes of the surrounding capsomeres. Deviations from exact symmetry are also found among these local symmetry axes. The relevance of these findings to the process of capsid assembly is considered.

The geometric integration of scale-invariant ordinary and partial differential equations

This review paper examines a synthesis of adaptive mesh methods with the use of symmetry to solve ordinary and partial differential equations. It looks at the effectiveness of numerical methods in preserving geometric structures of the underlying equations such as scaling invariance, conservation laws and solution orderings. Studies are made of a series of examples including the porous medium equation and the nonlinear Schrödinger equation.

Exploiting symmetry when verifying transistor-level circuits by symbolic trajectory evaluation: a summary

Summary form only given. We describe the use of symmetry for verification of transistor-level circuits by symbolic trajectory evaluation (STE). We present a new formulation of STE which allows a succinct description of symmetry properties in circuits. Symmetries in circuits are classified as structural symmetries, arising from similarities in circuit structure, data symmetries, arising from similarities in the handling of data values, and mixed structural-data symmetries. We use graph isomorphism testing and symbolic simulation to verify the symmetries in the original circuit. Using conservative approximations, we partition a circuit to expose the symmetries in its components, and construct reduced system models which can be verified efficiently. Introducing X-drivers into switch-level circuits simplifies the task of creating conservative approximations of switch-level circuits. Our empirical results show that exploiting symmetry with conservative approximations can allow one to verify systems several orders of magnitude larger than otherwise possible. We present results of verifying Static Random Access Memory (SRAM) circuits with up to 1.5 million transistors.

Applications of symmetry to problem solving

Symmetry is an important mathematical concept which plays an extremely important role as a problem-solving technique. Nevertheless, symmetry is rarely used in secondary school in solving mathematical problems. Several investigations demonstrate that secondary school mathematics teachers are not aware enough of the importance of this elegant problem-solving tool. In this paper we present examples of problems from several branches of mathematics that can be solved using different types of symmetry. Teachers' attitudes and beliefs regarding the use of symmetry in the solutions of these problems are discussed.