Academic literature on the topic 'Trigonometric orthogonal principle'

We analyze the double scaling limit of unitary matrix models in terms of trigonometric orthogonal polynomials on the circle. In particular we find a compact formulation of the string equation at the kth multicritical point in terms of pseudodifferential operators and a corresponding action principle. We also relate this approach to the mKdV hierarchy which appears in the analysis in terms of conventional orthogonal polynomials on the circle.

Consideration is given to dynamic behavior of cylindrical pressure pipe with elastic boundary conditions. Based on Sanders’ shell theory and Hamilton principle, the system equations are established for integrating the uniform distributed pressure into the elastic boundary condition. In the analytical formulation, the Rayleigh–Ritz method with a set of displacement shape functions is used to deduce mass, damping, and stiffness matrices of the pipe system. The displacements in three directions are represented by the characteristic orthogonal polynomial series and trigonometric functions which are satisfied with the elastic boundary conditions, which are represented as four sets of independent springs placed at the ends including three sets of linear springs and one set of rotational spring. The pressure pipe always suffers a uniform distributed pressure in radial direction. To verify the accuracy and reliability of the present method, several numerical examples with classical boundary condition, including free and simply supported supports are listed and comparisons are made with open literature. Then the influences of boundary restraint stiffness and the distributed pressure on natural frequency and the forced vibration response are studied: The natural frequencies increase significantly as the restraint stiffness or the distributed pressure increases. Compared to the rotational spring stiffness, the stiffnesses of axial, radial, and circumferential springs have more significant effect on natural frequency. And the lower modes are more sensitive on restraint stiffness than higher modes. But the variation of natural frequency with respect to the spring stiffness decreases monotonically with the increasing distributed pressure. The forced vibration response is also affected by the restraint stiffness.

The fortified city of Valletta, founded in 1566 by the Knights of Malta, is one of the few Renaissance ideal cities to be built. Planned from the beginning and constructed on virgin ground, it follows a rigid gridiron scheme designed by the Italian architect Francesco Laparelli da Cortona (1521-1570) that is an exemplar of Neoplatonic planning principles of the age of humanism and constitutes a model of modern urban design. Although the founding and development of the city has been well investigated historically, the formal essence of its urban design has not yet been examined satisfactorily from an architectural perspective. While the executed plan has been interpreted until now as an imperfect materialization of one of the four known preparatory drawings by Laparelli, in this article I suggest the possibility that the town built is a further evolutionary step in a fully traceable design process based on a coherent system of trigonometric proportions. In addition to the physical evidence of the city's current digital terrain model, the method of design proposed here is largely supported not only by sixteenth-century architectural theory (as advanced by Albrecht Dürer, Sebastiano Serlio, and Pietro Cataneo) but also historically by a group of new Florentine towns founded in the late Middle Ages that exhibit the same compositional principles.

In this paper we study image recognition tasks in which the images are described by high dimensional feature vectors extracted with deep convolutional neural networks and principal component analysis. In particular, we focus on the problem of high computational complexity of a statistical approach with non-parametric estimates of probability density implemented by the probabilistic neural network. We propose a novel statistical classification method based on the density estimators with orthogonal expansions using trigonometric series. It is shown that this approach makes it possible to overcome the drawbacks of the probabilistic neural network caused by the memory-based approach of instance-based learning. Our experimental study with Caltech-101 and CASIA WebFace datasets demonstrates that the proposed approach reduces the error rate by 1–5 % and increases the computational speed by 1.5 – 6 times when compared to the original probabilistic neural network for small samples of reference images.

Aims Several radiological methods of measuring anteversion of the acetabular component after total hip arthroplasty (THA) have been described. These are limited by low reproducibility, are less accurate than CT 3D reconstruction, and are cumbersome to use. These methods also partly rely on the identification of obscured radiological borders of the component. We propose two novel methods, the Area and Orthogonal methods, which have been designed to maximize use of readily identifiable points while maintaining the same trigonometric principles. Patients and Methods A retrospective study of plain radiographs was conducted on 160 hips of 141 patients who had undergone primary THA. We compared the reliability and accuracy of the Area and Orthogonal methods with two of the current leading methods: those of Widmer and Lewinnek, respectively. Results The 160 anteroposterior pelvis films revealed that the proposed Area method was statistically different from those described by Widmer and Lewinnek (p < 0.001 and p = 0.004, respectively). They gave the highest inter- and intraobserver reliability (0.992 and 0.998, respectively), and took less time (27.50 seconds (sd 3.19); p < 0.001) to complete. In addition, 21 available CT 3D reconstructions revealed the Area method achieved the highest Pearson’s correlation coefficient (r = 0.956; p < 0.001) and least statistical difference (p = 0.704) from CT with a mean within 1° of CT-3D reconstruction between ranges of 1° to 30° of measured radiological anteversion. Conclusion Our results support the proposed Area method to be the most reliable, accurate, and speedy. They did not support any statistical superiority of the proposed Orthogonal method to that of the Widmer or Lewinnek method. Cite this article: Bone Joint J 2019;101-B:1042–1049.

This paper investigates the problem of an optimal sensor placement for better shape deformation sensing of a new antenna structure with embedded or attached Fiber Bragg grating (FBG) strain sensors. In this paper, the deformation shape of the antenna structure is reconstructed using a strain–displacement transformation, according to the measured discrete strain data from limited FBG strain sensors. Moreover, a two-stage sensor placement method is proposed using a derived relative reconstruction error equation. In this method, the initial sensor locations are determined using the principal component analysis based on orthogonal trigonometric (i.e., QR) decomposition, and then a new location is sequentially added into the initial sensor locations one by one by minimizing the relative reconstruction error considering information redundancy. The numerical simulations are conducted, and the comparisons show that the proposed method is advantageous in terms of the sensor distribution and computational cost. Experimental validation is performed using an antenna experimental platform equipped with an optimal FBG strain sensor configuration, and the reconstruction results show good agreements with those measured directly from displacement sensors. The proposed method has a large potential for the strain sensor placement of complex structures, and the proposed antenna structure with FBG strain sensors can be applied to the future wing-skin antenna or flexible space-based antenna.

The development of methodical and mathematical apparatus for formation of a set of diagnostic parameters of complex technical systems, the content of which consists of processing the trajectories of the output processes of the system using the theory of functional spaces, is considered in this paper. The trajectories of the output variables are considered as Lebesgue measurable functions. It ensures a unified approach to obtaining diagnostic parameters regardless a physical nature of these variables and a set of their jump-like changes (finite discontinuities of trajectories). It adequately takes into account a complexity of the construction, a variety of physical principles and algorithms of systems operation. A structure of factor-spaces of measurable square Lebesgue integrable functions, ( spaces) is defined on sets of trajectories. The properties of these spaces allow to decompose the trajectories by the countable set of mutually orthogonal directions and represent them in the form of a convergent series. The choice of a set of diagnostic parameters as an ordered sequence of coefficients of decomposition of trajectories into partial sums of Fourier series is substantiated. The procedure of formation of a set of diagnostic parameters of the system, improved in comparison with the initial variants, when the trajectory is decomposed into a partial sum of Fourier series by an orthonormal Legendre basis, is presented. A method for the numerical determination of the power of such a set is proposed. New aspects of obtaining diagnostic information from the vibration processes of the system are revealed. A structure of spaces of continuous square Riemann integrable functions ( spaces) is defined on the sets of vibrotrajectories. Since they are subspaces in the afore mentioned factor-spaces, the general methodological bases for the transformation of vibrotrajectories remain unchanged. However, the algorithmic component of the choice of diagnostic parameters becomes more specific and observable. It is demonstrated by implementing a numerical procedure for decomposing vibrotrajectories by an orthogonal trigonometric basis, which is contained in spaces. The processing of the results of experimental studies of the vibration process and the setting on this basis of a subset of diagnostic parameters in one of the control points of the system is provided. The materials of the article are a contribution to the theory of obtaining information about the technical condition of complex systems. The applied value of the proposed development is a possibility of their use for the synthesis of algorithmic support of automated diagnostic tools.

This paper studies the free vibration of circular toroidal sectors with circular cross-sections based on the three-dimensional small-strain, linear elasticity theory. A set of orthogonal coordinates, composing the polar coordinate (r,θ) with the origin on the cross-sectional centerline of the sector and the circumferential coordinate φ with the origin at the curvature center of the centerline, is developed to describe the displacements, strains, and stresses in the sector. Each of the displacement components is taken as a product of four functions: a set of Chebyshev polynomials in φ and r coordinates, a set of trigonometric series in θ coordinate, and a boundary function in terms of φ. Frequency parameters and mode shapes have been obtained via a displacement-based extremum energy principle. The upper bound convergence of the first eight frequency parameters accurate up to five figures has been achieved. The present results agree with those from the finite element solutions. The effect of the ratio of curvature radius R to the cross-sectional radius a and the subtended angle φ0 on the frequency parameters of the sectors are discussed in detail. The three-dimensional vibration mode shapes are also plotted.