Use Of Symmetry Research Articles

The Gaussian formula and the elusive fourth principal ray

The Gaussian formula in geometrical optics can be derived without relying on specific mechanisms such as reflection and refraction. The extensive use of symmetries in this derivation clarifies the deep analogy between mirrors and lenses, and reveals the natural existence of four, not three, principal rays for both mirrors and lenses.

On the symmetry conditions for laminated fibre-reinforced composite structures

In this paper the use of symmetries in problems concerned with laminated fibre-reinforced composite structures is examined systematically. Incorrect and incomplete treatments of this subject have appeared in the literature and these are clarified herein. A model of a complete structure is not needed provided that the geometry and the loading of the structure show the required symmetry properties, even though the boundary conditions introduced can vary from one symmetry to another. The extent of the reduction in the size of the problem depends upon the laminate layout scheme. Particular comments are made on the proper use of symmetries in geometrically nonlinear problems. Further applications of symmetries in analytical approaches which lead to a reduction in the number of the dimensions of the problem are also discussed.

Symbolic exploitation of symmetry in numerical pathfollowing

Parameter-dependent systems of nonlinear equations with symmetry are treated by a combination of symbolic and numerical computations. In the symbolic part of the algorithm the complete analysis of the symmetry occurs, and it is here that symmetrical normal forms, symmetry reduced systems, and block diagonal Jacobians are computed. Given a particular problem, the symbolic algorithm can create and compute through the list of possible bifurcations thereby forming a so-called tree of decisions correlated to the different types of symmetry breaking bifurcation points. The remaining part of the algorithm deals with the numerical pathfollowing based on the implicit reparameterization as suggested and worked out by Deutlhard, Fiedler, and Kunkel. The symmetry breaking bifurcation points are computed using recently developed augmented systems incorporating the use of symmetry.

Quantum chemistry by random walk: Exact treatment of many-electron systems

We report an improved Monte Carlo method for quantum chemistry which permits the exact treatment of many-electron systems. The method combines many of the best features of earlier fixed-node, released-node, and positive/negative cancellation methods with new ideas for relocation after node crossing, self-cancellations, multiple cancellations, maximum use of symmetry in promoting cancellations, and rigorous evaluation of energies using importance sampling with trial wave functions. The method is illustrated with applications to the problems of the first excited state of a particle in a two-dimensional box, the two-electron system of excited H2 3Σ+u, and the three-electron system of linear symmetric HHH, the intermediate for the reaction H+H2→H2+H.

Rigorous Approach to Polarimetric Radar Modeling of Hydrometeor Orientation Distributions

Abstract A Muceller-matrix-based approach is described for calculating polarimetric radar parameters such as reflectivity, differential reflectivity, linear depolarization ratio, propagation differential phase, and backscatter differential phase. Particle scattering is calculated using the T matrix, which is ideally suited for spheroids in the Mie region. The T matrix, is calculated once for a given wavelength, particle size, shape, and dielectric constant, making full use of symmetry through use of a body coordinate system attached to the particle. For various orientations of the particle symmetry axis and for various radar elevation angles, a separate laboratory coordinate system is used. The scattering amplitudes and polarization basis vectors are transformed between the two coordinate systems using geometrical transformations. Two particle orientation distributions, which are two-dimensional in θ, ϕ, are considered. These are the Gaussian and simple harmonic in θ with random ϕ. Sample calculations are...

Development of B-spline techniques for dynamic ion-metal interaction theories

A three-dimensional time-dependent mean-field (TDMF) treatment for this problem is made computationally feasible by careful use of symmetries, incorporation of B-spline techniques and operator splitting concepts in the development of the method. The TDMF theory used is an adaptation of time-dependent Hartree-Fock theory used in atomic and nuclear physics for dealing with similar short-duration, strong, nonlinear collision interactions. Results of preliminary numerical studies are described.

Using harmonious houses for visual pairwise comparison of multiple criteria alternatives

In this paper, we consider the problem of evaluating decision alternatives, which are described by means of several criteria. Our aim is to develop an approach which enables a decision-maker to present value information on the basis ofa visual representation. The underlying ideas used in the approach are based on the use of symmetry and harmony. These principles are operationalized by using “harmonious houses”. The theoretical approach is implemented as a decision support system, which makes pairwise comparisons possible between alternatives. To perform sensitivity analysis is also possible. In our pilot tests, the idea has proven very promising.

Solution of fluid–structure interaction problems using a coupled finite element and variational boundary element technique

This work is concerned with the numerical modeling of elasto-acoustic problems applied to thin shells and specifically curved plates. A finite element formulation of the elastic problem is coupled to a variational boundary element solution of the acoustic problem. This solution to the acoustic problem, proposed by Mariem and Hamdi [J. B. Mariem and M. A. Hamdi, Int. J. Num. Methods. Eng. 24, 1251–1267 (1987)], is implemented using high-order isoparametric elements, and attention is given to techniques of accelerating the numerical implementation for solutions at multiple frequencies. These techniques include a reduction in the problem size through the use of symmetry and, more importantly, the interpolation of the fluid impedance matrix within the frequency regime. The advantages in using this variational formulation are, first, the manner in which a highly singular integral operator is made amenable to numerical approximation, second, its application to nonclosed thin shells, and, third, its numerical implementation leads to the formulation of a symmetrical fluid matrix.

Consistent force field modeling of matrix isolated molecules. V. Minimum energy path potential to the conformer conversion of 1,2-difluoroethane: Ar 364, ab initio calculation of electric multipole moments and electric polarization contribution to the conversion barrier

In this paper results of consistent force field modeling (CFF) of the potential function to conversion of the gauche (g) to the trans (t) conformer of 1,2-difluoroethane (DFE) isolated in an argon matrix will be reported. Starting point are locally stable configurations gDFE:Ar 364 (defect GH1) and tDFE:Ar 364 (TH1) obtained in previous work from CFF modeling of a cube shaped Ar 364 fragment containing one DFE molecule in its center. Using the dihedral angle of DFE as an independent parameter the minimum energy path of the conversion process gDFE:Ar 364→tDFE:Ar 364 will be determined by CFF energy minimization. Determination of the minimum energy path is found to require large numbers of energy minimization steps and to lead to a rather complicated motion of the molecule with respect to the crystal fragment. Surprisingly the molecule-matrix interactions lead to a reduction of the g-t barrier by ≈500 cal/mol and to a stabilization of the trans species by ≈500 cal/mol. This finding is a consequence of a delicate interplay of matrix-molecule and matrix-matrix interactions. Calculation of the electric polarization energy (induced dipole-first-order polarization approximation) is based on extended ab initio calculations of dipole and quadrupole moments and a bond polarizability estimate of the first-order polarizability of DFE as a function of the internal rotation angle, on Fourier expansion of multipole components and use of symmetry for reduction of the order of the linear system defining the (self-consistent) induced dipole moments of all Ar atoms. Electric polarization is found to alter the potential function of the conversion process in a profound way: the g-t barrier and the t-g energy difference are increased to ≈3000 cal/mol and to ≈1500 cal/mol respectively (≈2500 and ≈530 cal/mol respectively for free DFE). Further applications of the technique developed in this work to related problems of matrix isolated molecules, e.g., vibrational matrix shifts will be discussed.

N-dimensional symmetries and their applications in digital filters

N-dimensional ( N-D) digital signal processing always results in enormous data processing and numerous parameters to be determined. The use of symmetries existing in the desired frequency specifications largely reduces the independent parameters, and furthermore, reduces the computation in the implementation. N-D digital filter symmetries existing in the frequency domain are developed and are related to numerator and denominator polynomials C( z 1, …, z N ). Constraints for identify magnitude ∥ C( rme jω 1, …, e jω N )∥ symmetries, anti-identity phase θ( ω 1,…, ω N ) symmetries, and conjugate complex frequency response C(e jω 1,…e j ω N symmetries, are developed. The necessary and sufficient conditions for a polynomial C( z 1, …, z N ) to satisfy a certain symmetry constraint are proposed. The newly developed symmetry constraints are used to derive special forms of discrete polynomials C( z 1, …, z N ) possessing these symmetries. The stability constraint is then imposed on the newly derived polynomial forms. These polynomial forms can be used in the design (in the frequency domain) and implementation of FIR and IIR digital filters, and will provide great saving of computation work and simplified hardware architecture.

Application of a perspective cursor as a 3D locator device

The functionality of a 3D cursor is described. It can be used as a 3D locator device in such applications as interactive path planning (robotics), motion specification in 3D computer animation design, and wireframe modelling. With respect to the last of these applications, it is important that the cursor can be used either in constructive mode, as a set of three mutually orthogonal rulers, or in trace mode to copy a 2D perspective view of an object while reconstruction its 3D shape. As a special feature, the cursor may be equipped with a mirror of locations in scenes that are (partially) symmetric, but also allows 3D symmetric objects to be entered from 2D (perspective) views, while making use of symmetry as a clue for reconstructing z-coordinates.

Space ⇌ color ⇌ symmetry

The author refers to the direct or indirect use of symmetry in her art. She uses straight lines, all of which begin from the same point, in order to transmit in design and color a beam of light. She works colors in their positive and negative aspects. In other works she interrupts her planes and divests the cut points, thus, giving the impression of a refraction which, in the similar elements, is like two successive reflections. Her compositions with the linear perspective are either developed symmetrically in a vertical axis or in parallel removal of this axis.

Efficient computation of the discrete pseudo-Wigner distribution

A description is given of a novel algorithm, the fast Fourier transform in part (FFTP), for the computation of the discrete pseudo-Wigner distribution (DPWD). The FFTP computes the cosine and sine parts of the discrete Fourier transform (DFT) separately by employing real inverse sinusoidal twiddle factors. Unlike the conventional methods which directly utilize the complex DFT, the FFTP yields real output since the DPWD is always real. In addition, the new method reduces the computational cost by making full use of symmetries and removing redundancies in the FFTP computation. The authors also describe a simple algorithm for computing the discrete Hilbert transform (DHT) to produce the nonaliased DPWD. A pipeline structure for real-time and a bulk processing technique for offline implementations of the method are presented. >

Helical polymers within density functional formalism

A method for calculating the electronic ground state properties of a single, infinite, periodic, helical polymer with a straight polymer axis is presented. Here, any material which can be considered as formed by parallel macromolecules interacting only weakly is considered a polymer with these characteristics. The method is based on the density functional formalism in a local approximation and on the Bom-Oppenheimer approximation. The single-particle eigenfunctions are expanded in Linearized Muffin-Tin Orbitals (LMTO's), but the muffin-tin approximation is solely used in defining the basis functions. In the discussion special emphasis is put on the use of symmetry and it is demonstrated that with an atom-centered LMTO basis set of limited size the full symmetry can be used, and that detailed and accurate investigations of general helical polymers thereby can be carried through. Applications of the method to sulphur and selenium helices, carbon disclenide, polysilene, hydrogen fluoride, and sulphur nitride are reported.

Quasi-particle bands of a graphite monolayer

The quasi-particle band structure of a graphite monolayer has been calculated by a Green's function method in second order of Møller-Plesset perturbation theory. Rules for the use of symmetry in correlation calculations are given. The XPS spectrum has been calculated within the Gelius model and compared to the experimental spectrum of graphite.

The use of symmetry in reciprocal space integrations. Asymmetric units and weighting factors for numerical integration procedures in any crystal symmetry

AbstractA systematic collection of spatial domains for reciprocal space integrations is derived for all possible crystal symmetries. This set can be used as a simpler alternative to the conventional Brillouin zones. The analysis is restricted to integrations where the function in the integrand satisfies inversion symmetry in k space. In this case only 24 different spatial domains have to be defined in order to allow for k space integrations in the 230 different crystal symmetries. A graphic representation of the asymmetric unit for each of the 24 integration domains is given. Special positions and the associated weighting factors required for numerical integrations in theoretical solid‐state approaches are tabulated.

Bidding Models—Symmetry and State of Information

Several competitive bidding models since the late 1960s have been founded, criticized, and even rejected, based on incorrectly stated arguments concerning the apparent symmetry of the competitive positions of the prospective bidders. The proliferation of these arguments has led to confusion about the validity of the proposed models. This paper: (1) Illustrates the correct use of symmetry in competitive bidding as a function of available information and control; (2) explains the differences among nonsymmetric states of information that lead to apparent symmetry in the case of only two competitors; (3) presents and explains the assumptions that lead to Friedman's general bidding model; (4) proves the probabilistic validity of Friedman's model by using correctly stated arguments of symmetry; and (5) provides the foundations for understanding the limitations of other models.

NMR solvent peak suppression by nonlinear excitation

Most existing NMR solvent peak suppression sequences provide a satisfactory dependence of the intensity of excited signals on frequency but poor phase characteristics. In practice this leads to spectral distortions which generally become more severe as the frequency selectivity of the sequence is increased. However, it is shown that by working well outside the linear response regime, excitation schemes which combine high frequency selectivity with good phase properties may be devised. Sequences of six rectangular radio-frequency pulses were discovered using a combination of coherent averaging theory to treat the near-resonant behavior and numerical simulation further from resonance. Extensive use of symmetry greatly simplifies both the coherent averaging calculations and the numerical simulations. The new pulse sequences have been given the acronym NERO (nonlinear excitation rejecting on-resonance). Experimental spectra of an enzyme in dilute aqueous solution are shown.

Translation functions. Note on the use of symmetry in the minimization of structure-independent spurious maxima

In a previous paper [Langs (1985). Acta Cryst. A41, 578-582] formulae were derived to reduce the magnitudes of structure-independent spurious peaks which appear in translation syntheses. The present note describes how crystallographic symmetry may be exploited to simplify these calculations by a factor of a hundredfold. The solution maximum produced by this new formulation is often more than ten times larger than the largest spurious background peak.

The use of symmetry in bifurcation calculations and its application to the Bénard problem

Some aspects of the use of symmetry in bifurcation calculations are discussed. First, it is shown how substantial reductions in cost are obtained in computing symmetry-breaking bifurcation points by exploiting the underlying symmetry of a problem. This is demonstrated in a finite-element calculation of 2-dimensional Benard convection in a finite cavity. Second, the stability of paths of symmetry-breaking bifurcation points, which occur when a second parameter in the problem varies, is investigated. A criterion is established for deciding whether intersection of such paths is allowed by symmetry constraints.