Diagonal Symmetry Research Articles

Walks in the quarter plane: Kreweras’ algebraic model

We consider planar lattice walks that start from (0,0), remain in the first quadrant i,j≥0, and are made of three types of steps: North-East, West and South. These walks are known to have remarkable enumerative and probabilistic properties: • they are counted by nice numbers [Kreweras, Cahiers du B.U.R.O 6 (1965) 5–105], • the generating function of these numbers is algebraic [Gessel, J. Statist. Plann. Inference 14 (1986) 49–58], • the stationary distribution of the corresponding Markov chain in the quadrant has an algebraic probability generating function [Flatto and Hahn, SIAM J. Appl. Math. 44 (1984) 1041–1053]. These results are not well understood, and have been established via complicated proofs. Here we give a uniform derivation of all of them, which is more elementary that those previously published. We then go further by computing the full law of the Markov chain. This helps to delimit the border of algebraicity: the associated probability generating function is no longer algebraic, unless a diagonal symmetry holds. Our proofs are based on the solution of certain functional equations, which are very simple to establish. Finding purely combinatorial proofs remains an open problem.

Abelian and Non-Abelian Branes in WZW Models and Gerbes

We discuss how gerbes may be used to set up a consistent Lagrangian approach to the WZW models with boundary. The approach permits to study in detail possible boundary conditions that restrict the values of the fields on the worldsheet boundary to brane submanifolds in the target group. Such submanifolds are equipped with an additional geometric structure that is summarized in the notion of a gerbe module and includes a twisted Chan-Paton gauge field. Using the geometric approach, we present a complete classification of the branes that conserve the diagonal current-algebra symmetry in the WZW models with simple, compact but not necessarily simply connected target groups. Such symmetric branes are supported by a discrete series of conjugacy classes in the target group and may carry Abelian or non-Abelian twisted gauge fields. The latter situation occurs for the conjugacy classes with fundamental group ℤ2×ℤ2 in SO(4n)/ℤ2. The branes supported by such conjugacy classes have to be equipped with a projectively flat twisted U(2) gauge field in one of the two possible WZW models differing by discrete torsion. We show how the geometric description of branes leads to explicit formulae for the boundary partition functions and boundary operator product coefficients in the WZW models with non-simply connected target groups.

３次元凹凸曲面上における対称パターン検出の効率分析

Based on the ideal observer analysis, we investigated sampling properties of image information used by human visual system, for symmetrical pattern discrimination on 3D bumpy surface. There were three models of ideal observer (IO) to perform the task: 2D-IO using 2D projection image (i.e., retinal image), 2.5D-IO using image transformed to canonical view, and 3D-IO using recovered pattern image of 2D plane. We measured discrimination thresholds on the task for each IO model and subjects, and calculated human statistical efficiency relative to each ideal observer. The results indicated for the detection of a diagonal symmetry in the bumpy surface that human performance was similar to 3D-IO. This implies that human observers use the structure of the bumpy surface to detect the diagonal symmetry.

Phase Transitions in Li[sub 1−δ]Ni[sub 0.5]Mn[sub 1.5]O[sub 4] during Cycling at 5 V

Structural change of LiNi 0 . 5 Mn 1 . 5 O 4 with two different crystallographic structures of Fd3m and P4 3 32 was studied during Li extraction. Ex situ X-ray diffraction and transmission electron microscopy studies on partially delithiated Li 1 - δ Ni 0 . 5 Mn 1 . 5 O 4 revealed that LiNi 0 . 5 Mn 1 . 5 O 4 (Fd3m) structure lost its characteristic diagonal glide symmetry, suggesting a structural transition arising from the possible migration of cations during oxidation from Ni 2 + to Ni 4 + . For the Li 1 - δ Ni 0 . 5 Mn 1 . 5 O 4 (P4 3 32), ordering of cations was destroyed by random migration of cations during Li extraction, resulting in the formation of intermediate phase with the Fd3m structure. The phase transitions were reversible, meaning that migration of cations is highly reversible during lithiation/delithiation.

Design of 2-D Linear Phase Digital Filters Using Schur Decomposition and Symmetries

In this paper we present a new and numerically efficient technique for designing 2-D linear phase octagonally symmetric digital filters using Schur decomposition method (SDM) and the diagonal symmetry of the 2-D impulse response specifications. This technique is based on two steps. First, the 2-D impulse response matrix is decomposed into a parallel realization of k sections, each comprising two cascaded linear phase SISO 1-D FIR digital filters. It is shown that using the symmetry property of the 2-D impulse response matrix and the fact that the left and right eigenspaces obtained by SDM are transpose of each other, the design problem of two 1-D digital filters is reduced to the design problem of only one 1-D digital filter in each section. In the second step, the 2-D linear phase FIR filter is converted to 2-D IIR filter. This is done by converting the constituent N-order 1-D FIR filters to an n-order IIR filters, where n < N using the given model reduction algorithm. The reduced order IIR filters are obtained without computing the balancing transformation, but by finding the orthonormal eigenspaces associated with the largest eigenvalues of the cross-Gramian matrix W CO. Two design examples are given to illustrate the advantages of the proposed technique.

SPECSIM2: a program for spectral simulation of anisotropic two-dimensional random fields

The authors describe the structure and implementation of SPECSIM2, an ANSI Fortran-77 program for simulating anisotropic two-dimensional random fields via a spectral approach. This new program overcomes the limitation of the previously published version (Pardo-Igúzquiza and Chica-Olmo, Comput. Geosci. 20 (4) (1994) 597), which was confined to the direct generation of isotropic fields with any anisotropy imposed by a subsequent co-ordinate transformation. The diagonal symmetry of the two-dimensional covariance function with respect to the origin, i.e. C( x, y)= C(− x,− y) and C( x,− y)= C(− x, y), implies that its Fourier transform (i.e. the spectral density) is real (the imaginary part vanishes). This observation leads to a straightforward direct simulation of anisotropic fields by the spectral method. Examples are given to demonstrate the performance of the method using semi-variograms with or without nugget effects and with simple or nested structures, as well random fields with several structures that have different anisotropy angles.

Schur Decomposition and Model Reduction in the Design of Two-Dimensional Separable-Denominator Digital Filters

A new and numerically efficient technique for designing two-dimensional (2-D) separable-denominator digital filters using Schur decomposition (SD) and the diagonal symmetry of the 2-D impulse response specifications. This technique is based on two steps: first, the 2-D impulse response specifications are decomposed into two cascaded specifications, representing SIMO and MISO 1-D digital filters. By using the symmetry property of the 2-D impulse response matrix and that the left and right eigenspaces obtained by SD are a transpose of each other, the design problem of the two 1-D digital filters is reduced to the design problem of only one 1-D digital filter. Thus, the computational effort is reduced by approximately half. Further computational reduction is obtained by applying the SD to only part of the 2-D impulse response matrix, the leading principal minor submatrix, and from the symmetry property of the class of filters that are considered here, the decomposition of the impulse response matrix is obtained. In the second step, a model reduction algorithm is used to design a filter that approximates the 1-D specifications obtained from the first step. Two design examples illustrate the advantages of the proposed technique.

The Structure of the Log Odds-Ratios in Non-Independence and Symmetry Diagonal Models for Square Contingency Tables

This paper examines the structure of the interaction term when expressed as the log ofthe odds-ratios ij in the log-linear model formulation of non-independence and symmetry diagonalmodels applied to an I x I contingency table having ordinal classificatory variables. The class ofprincipal diagonal, diagonal band, full diagonal and diagonal-parameters symmetry models areexamined in the study. The general structure of Φij is examined for each class of diagonal models.The 5 x 5 British Social Mobility data (Glass, 1954) will be employed as an example in this paper.The structure of the log odds-ratios are formulated in terms of the parameter estimates obtained from the application of SAS PROC GENMOD. Goodman in several of his papers on the subject has also derived some of these log odds ratios but our focus here is the structure of these odds, when viewed from the non-standard log-linear perspective (Lawal, 2001, 2002).

Characterization of diagonal symmetry: location unknown, and a test based on allied U-processes

For a (possibly multivariate) distribution F, a characterization of (diagonal) symmetry is made with respect to a kernel of degree 2; this is incorporated in the formulation of appropriate U-processes that provide the access to a suitable test statistic for testing the hypothesis of diagonal symmetry when the location is treated as unknown. Asymptotic properties of the test are studied.

The Mori–Tanaka stiffness tensor: diagonal symmetry, complex fibre orientations and non-dilute volume fractions

This paper considers the diagonal symmetry of the stiffness tensors predicted using the Mori-Tanaka (MT) approach. Since the MT approach may yield asymmetric stiffness tensors this paper considers an alternative approach to ensure the diagonal symmetry. Furthermore, an extension of the MT approach to non-dilute volume fractions is considered. It is shown that the MT approach by Benveniste [Mech. Mater. 6 (1987) 147], may not yield diagonally symmetric stiffness tensors in situations when a statistical fibre orientation function is incorporated. Only when the inclusions are spherical and randomly distributed or fully aligned will the MT predicted stiffness tensor be diagonally symmetric. The extension to non-dilute volume fractions is seen to lie within the Hashin–Shtrikman–Walpole (HSW) bounds and comparison to other approaches and experiments found in the literature shows very good agreement.

A CHARACTERISTIC MEASURE OF ASYMMETRY AND ITS APPLICATION FOR TESTING DIAGONAL SYMMETRY

A new test for diagonal symmetry (around 0) is proposed based on a characteristic property of diagonally symmetric distributions. The sequence of statistics leads to a consistent test against general alternatives.

Retrieval of symmetrical image blur using zero sheets

The analytically continued Fourier transform of a two-dimensional image vanishes to zero on a two-dimensional surface embedded in a four-dimensional space. This surface uniquely characterises the image and is known as a ‘zero sheet’. Since the manipulation of a function in four-dimensional space is cumbersome, the projections of zero sheets, known as ‘zero tracks’ are calculated. This knowledge of zero sheets can be extended to a number of practical applications including image processing. Image restoration can be realised without prior knowledge of the point spread function, i.e. blind deconvolution is possible even when only a single blurred image is given. If the blurred image concerned contains a point spread function with diagonal symmetry, the point zeros calculated row-wise and column-wise contain some similarities, which supports retrieval of both the true image and a point spread function. This novel scheme performs the separation effectively in the absence of contamination.

Mode Truncation Effects in an Explicit Mode Elastic Characterization Method

The second author and colleagues developed an elastic identification method based on model data obtained experimentally from free transverse vibration of impacted freely supported plates. Application of the method to thin, square isotropic metal and composite plates is described in this paper. For these modes, some additively and subtractively coupled modes arising from diagonal symmetry of geometry and boundary conditions (thus termed diagonal modes) are important. The focus of this paper is on how the number of modes used to represent the vibration affects accuracy. One-, three-, and six-term optimized Rayleigh displacement expressions are used for plate vibration, including the diagonal modes. Sensitivity analysis of the frequencies to each elastic constant is included. Results show that the optimal three-term approximation gives results comparable to the normal 36-term Rayleigh expansion, while the optimal six-term compares favorably with the 64-term Rayleigh expansion.

Permutation Tests for Reflected Symmetry

The paper presents a permutation procedure for testing reflected (or diagonal) symmetry of the distribution of a multivariate variable. The test statistics are based in empirical characteristic functions. The resulting permutation tests are strictly distribution free under the null hypothesis that the underlying variables are symmetrically distributed about a center. Furthermore, the permutation tests are strictly valid if the symmetric center is known and are asymptotic valid if the center is an unknown point. The equivalence, in the large sample sense, between the tests and their permutation counterparts are established. The power behavior of the tests and their permutation counterparts under local alternative are investigated. Some simulations with small sample sizes (⩽20) are conducted to demonstrate how the permutation tests works.

Schindler, Symmetry and the Free Public Library, 1920

Schindler's Free Public Library competition project of 1920 is analysed. Archival documents are interpreted and an analytic model is constructed. Schindler's standing as a Modernist is considered and his early career is rehearsed, especially his involvement with Frank Lloyd Wright. The formal analysis focuses on the use of symmetry. An interpretation of Schindler's employment of the diagonal axis in the Library is proffered. Historical precedents for diagonal butterfly symmetry in the English free school and the Arts and Crafts movement are examined. A method of analysis using the partial ordering of subsymmetries of the square is applied to the floor plans In conclusion, the influence of the Library project on Schindler's later work is surveyed.

Temporal stride patterns in horses with experimentally induced fore‐ or hindlimb lameness

SummaryIn a kinematic study of equine lameness, temporal stride variables were investigated for their significance as diagnostic indicators and as a means of compensation for lameness. Kinematics of 11 clinically sound Dutch Warmblood horses were recorded while walking (1.6 m/s) and trotting (3.5 m/s) on a treadmill. A transient lameness model, provoking pain on the hoof sole, was used to induce 3degreesof fore‐ and hindlimb lameness. Intra‐ and interlimb stride variables as well as left‐to‐right symmetry were analysed using a 2 way analysis of variance (significance level: P&lt;0.05). At the walk, stride duration was not affected by fore‐ or hindlimb lameness, while intra‐ and interlimb timing showed minor changes. At the trot, the stride duration in forelimb lameness decreased by 3.1% and relative stance duration of all limbs became longer (5.3% in forelimbs, 1.5% and 3.2% in hindlimbs). The diagonal advanced placement of the forelimbs increased and the duration of the suspension phase following the lame diagonal decreased by 61%. Left‐to‐right symmetry decreased for the suspension phase and contralateral advanced placement, while the stance durations remained symmetrical. In hindlimb lameness the stride duration at the trot decreased by 3.7% and the stance duration of the sound hind limb increased by 2.7%. Diagonal advanced placement, suspension phase and symmetry variables remained unaffected. It was concluded that compensation for lameness by changing temporal stride patterns is more pronounced in fore than in hindlimb lameness. The maintenance of symmetry in most stride variables in mild lamenesses limits their value as lameness indicators.

Molecular structure of monomeric uranium tetrachloride determined by gas electron diffraction at 900 K, gasphase infrared spectroscopy and quantum-chemical density-functional calculations

The gas electron diffraction pattern of monomeric UCl4 has been recorded with a nozzle temperature of 900 K. The data are in good agreement with a model of Td symmetry and a U–Cl bond distance of ra= 250.3(3) pm. The root-mean-square vibrational amplitudes are l(U–Cl)= 8.9(3) pm and l(Cl ⋯ Cl)= 34.3(10) pm. The gas-phase infrared absorption spectra have been recorded from 25 to 3400 cm–1 at temperatures ranging from 700 to 900 K, and the t2 stretching and deformation modes have been assigned at ν3 337.4 and ν4 71.7 cm–1 respectively. The molecular structure of UCl4 has been optimised by density-functional calculations under D2d symmetry. It is tetrahedral with bond distance re= 251 pm. The calculations yield values for the IR-active modes which are in good agreement with the experimental counterparts. The ‘best values’ for the two IR-inactive frequencies were obtained by refining a diagonal symmetry force field to the four calculated (density functional) frequencies, the two observed frequencies and the vibrational amplitudes obtained by gas electron diffraction. The entropies of gaseous UCl4 calculated from these normal modes and a symmetry number of 12 corresponding to Td symmetry are in good agreement with the experimental counterparts.

Revision of the Family Spironemidae Doflein (Protista, Hemimastigophora), With Description of Two New Species, Spironema Terricola N. Sp. and Stereonema Geiseri N, G., N. Sp.

ABSTRACT. Two new hemimastigophoran flagellates are described using light and electron microscopy, and the family Spironemidae is revised. Spironema terricola n. sp. occurs in soil from the Grand Canyon (southwest USA). It moves in a conspicuously euglenoid manner and differs from S. multiciliatum Klebs by its vermiform shape and shorter kinetics. Spironema terricola is similar to Goodey's Spironema multiciliatum from soil in England. However, Goodey's vermiform species has a very elongate nucleus and is thus neither identical with S. terricola, which has a roundish nucleus, nor with Klebs' lanceolate S. multiciliatum; we consider it a new species, Spironema goodeyi n. sp, Stereonema geiseri n. g., n. sp. was discovered in the Aufwuchs (periphyton) of a river in Bavaria, Germany. the new genus differs from Spironema by its a contractility, and from Hemimastix by its shorter kinetics and less plicate cortex. the fine structure of Spironema and Stereonema is very similar to that of Hemimastix Foissner et al., viz., the cortex is composed of two plates having diagonal symmetry and the flagellated basal bodies have associated a short and a long microtubular ribbon. All species have unique extrusomes of the same type. the main differences between the three genera and five species recognized are contractility, length of kinetics, body size, shape of cell and nucleus, and particulars of the cortex and extrusomes. the phylogenetic relationships of the Hemimastigophora are still uncertain. However, the diagonal symmetry of the cortical plates and the pronounced euglenoid movement of Spironema spp. suggest a common ancestor with euglenids.

Symmetry study on 2-D complex analog and digital filter functions

The symmetry properties of 2-D complex polynomials are analyzed in this paper. The characteristics of a polynomial possessing different standard symmetries in their magnitude and phase responses are studied. The nature of constraints that are imposed by the defined symmetries on analog and digital polynomials is discussed. The various classes of complex polynomials possessing the different (quadrantal, diagonal, rotational, and octagonal) symmetries and antisymmetries in their magnitude responses and/or phase responses are tabulated.

On diagonal and elastic symmetry of the approximate effective stiffness tensor of heterogeneous media

T he existence of diagonal symmetry in estimates of overall stiffness tensors of heterogeneous media is examined for several micromechanical models. The dilute approximation gives symmetric estimates for all matrix-based multiphase media. The Mori-Tanaka and the self-consistent methods do so for all two-phase systems, but only for those multiphase systems where the dispersed inclusions have a similar shape and alignment. However, the differential schemes associated with the self-consistent method can predict diagonally symmetric overall stiffness and compliance for multiphase systems of arbitrary phase geometry. A related question is raised about the equivalence of two possible approaches to evaluation of the overall thermal stress and strain tensors. A direct estimate follows from each of the above models, whereas L evin's results [ Mechanics of Solids 2, 58 (1967)] permit an indirect evaluation in terms of the estimated overall mechanical properties or concentration factors and phase thermoelastic moduli. These two results are shown to coincide for those systems and models which return diagonally symmetric estimates of the overall stiffness. Finally, model predictions of the overall elastic symmetry of composite media are discussed with regard to the spatial distribution of the phases.