Search results for: trigonometric splines

Authors: Uzma Bashir, Jamaludin Md. Ali

Abstract:

This study is concerned with the visualization of monotone data using a piece-wise C1 rational trigonometric interpolating scheme. Four positive shape parameters are incorporated in the structure of rational trigonometric spline. Conditions on two of these parameters are derived to attain the monotonicity of monotone data and other two are left-free. Figures are used widely to exhibit that the proposed scheme produces graphically smooth monotone curves.

Keywords: trigonometric splines, monotone data, shape preserving, C1 monotone interpolant

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Authors: Maria Hussin, Malik Zawwar Hussain, Mubashrah Saddiqa

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We present a trigonometric scheme to approximate a circular arc with its two end points and two end tangents/unit tangents. A rational cubic trigonometric Bézier curve is constructed whose end control points are defined by the end points of the circular arc. Weight functions and the remaining control points of the cubic trigonometric Bézier curve are estimated by variational approach to reproduce a circular arc. The radius error is calculated and found less than the existing techniques.

Keywords: control points, rational trigonometric Bézier curves, radius error, shape measure, weight functions

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Authors: Slobodan B. Tričković

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We present the results concerned with trigonometric series that include sine and cosine functions with a parameter appearing in the denominator. We derive two types of closed-form formulas for trigonometric series. At first, for some integer values, as we know that Riemann’s ζ function and Dirichlet η, λ equal zero at negative even integers, whereas Dirichlet’s β function equals zero at negative odd integers, after a certain number of members, the rest of the series vanishes. Thus, a trigonometric series becomes a polynomial with coefficients involving Riemann’s ζ function and Dirichlet η, λ, β functions. On the other hand, in some cases, one cannot immediately replace the parameter with any positive integer because we shall encounter singularities. So it is necessary to take a limit, so in the process, we apply L’Hospital’s rule and, after a series of rearrangements, we bring a trigonometric series to a form suitable for the application of Choi-Srivastava’s theorem dealing with Hurwitz’s zeta function and Harmonic numbers. In this way, we express a trigonometric series as a polynomial over Hurwitz’s zeta function derivative.

Keywords: Dirichlet eta lambda beta functions, Riemann's zeta function, Hurwitz zeta function, Harmonic numbers

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Authors: Jincan Li, Mingyu Gao, Zhiwei He, Yuxiang Yang, Zhongfei Yu, Yuanyuan Liu

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Building an appropriate motion model is crucial for trajectory planning of robots and determines the operational quality directly. An adaptive acceleration and deceleration motion planning based on trigonometric functions for the end-effector of 6-DOF robots in Cartesian coordinate system is proposed in this paper. This method not only achieves the smooth translation motion and rotation motion by constructing a continuous jerk model, but also automatically adjusts the parameters of trigonometric functions according to the variable inputs and the kinematic constraints. The results of computer simulation show that this method is correct and effective to achieve the adaptive motion planning for linear trajectories.

Keywords: kinematic constraints, motion planning, trigonometric function, 6-DOF robots

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Authors: A. A. Soliman

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The Modified Regularized Long Wave (MRLW) equation is solved numerically by giving a new algorithm based on collocation method using quartic B-splines at the mid-knot points as element shape. Also, we use the fourth Runge-Kutta method for solving the system of first order ordinary differential equations instead of finite difference method. Our test problems, including the migration and interaction of solitary waves, are used to validate the algorithm which is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the algorithm. The temporal evaluation of a Maxwellian initial pulse is then studied.

Keywords: collocation method, MRLW equation, Quartic B-splines, solitons

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Authors: Shazalina Mat Zin, Ahmad Abd. Majid, Ahmad Izani Md. Ismail, Muhammad Abbas

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The generalized wave equation models various problems in sciences and engineering. In this paper, a new three-time level implicit approach based on cubic trigonometric B-spline for the approximate solution of wave equation is developed. The usual finite difference approach is used to discretize the time derivative while cubic trigonometric B-spline is applied as an interpolating function in the space dimension. Von Neumann stability analysis is used to analyze the proposed method. Two problems are discussed to exhibit the feasibility and capability of the method. The absolute errors and maximum error are computed to assess the performance of the proposed method. The results were found to be in good agreement with known solutions and with existing schemes in literature.

Keywords: collocation method, cubic trigonometric B-spline, finite difference, wave equation

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Authors: G. Kermarrec, J. Hartmann

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Spline surfaces are a major representation of freeform surfaces in the computer-aided graphic industry and were recently introduced in the field of geodesy for processing point clouds from terrestrial laser scanner (TLS). The surface fitting consists of approximating a trustworthy mathematical surface to a large numbered 3D point cloud. The standard B-spline surfaces lack of local refinement due to the tensor-product construction. The consequences are oscillating geometry, particularly in the transition from low-to-high curvature parts for scattered point clouds with missing data. More economic alternatives in terms of parameters on how to handle point clouds with a huge amount of observations are the recently introduced T-splines. As long as the partition of unity is guaranteed, their computational complexity is low, and they are flexible. T-splines are implemented in a commercial package called Rhino, a 3D modeler which is widely used in computer aided design to create and animate NURBS objects. We have applied T-splines surface fitting to terrestrial laser scanner point clouds from a bridge under load and a sheet pile wall with noisy observations. We will highlight their potential for modelling details with high trustworthiness, paving the way for further applications in terms of deformation analysis.

Keywords: deformation analysis, surface modelling, terrestrial laser scanner, T-splines

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Authors: Sandeep Kaur Chouhan, Jatinderdeep Kaur, S. S. Bhatia

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The existence of sine and cosine series as a Fourier series, their L1-convergence seems to be one of the difficult question in theory of convergence of trigonometric series in L1-metric norm. In the literature so far available, various authors have studied the L1-convergence of cosine and sine trigonometric series with special coefficients. In this paper, we present a modified cosine and sine sums and criterion for L1-convergence of these modified sums is obtained. Also, a necessary and sufficient condition for the L1-convergence of the cosine and sine series is deduced as corollaries.

Keywords: conjugate Dirichlet kernel, Dirichlet kernel, L1-convergence, modified sums

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Authors: Smita Sonker

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Various investigators have determined the degree of approximation of conjugate signals (functions) of functions belonging to different classes Lipα, Lip(α,p), Lip(ξ(t),p), W(Lr,ξ(t), (β ≥ 0)) by matrix summability means, lower triangular matrix operator, product means (i.e. (C,1)(E,1), (C,1)(E,q), (E,q)(C,1) (N,p,q)(E,1), and (E,q)(N,pn) of their conjugate trigonometric Fourier series. In this paper, we shall determine the degree of approximation of 2π-periodic function conjugate functions of f belonging to the function classes Lipα and W(Lr; ξ(t); (β ≥ 0)) by (C1.T) -means of their conjugate trigonometric Fourier series. On the other hand, we shall review above-mentioned work in the light of Lenski.

Keywords: signals, trigonometric fourier approximation, class W(L^r, \xi(t), conjugate fourier series

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Authors: Alberto Hananel

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The aim of this work is to modelize the occlusion of a person with temporomandibular disorders as an evolutionary equation and approach its solution by the construction and characterizing of discrete variational splines. To formulate the problem, certain boundary conditions have been considered. After showing the existence and the uniqueness of the solution of such a problem, a convergence result of a discrete variational evolutionary spline is shown. A stress analysis of the occlusion of a human jaw with temporomandibular disorders by finite elements is carried out in FreeFem++ in order to prove the validity of the presented method.

Keywords: approximation, evolutionary PDE, Finite Element Method, temporomandibular disorders, variational spline

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Authors: K. N. S. Kasi Viswanadham

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Coupled system of second order linear and nonlinear boundary value problems occur in various fields of Science and Engineering. In the formulation of the problem, any one of 81 possible types of boundary conditions may occur. These 81 possible boundary conditions are written as a combination of four boundary conditions. To solve a coupled system of boundary value problem with these converted boundary conditions, a collocation method with cubic B-splines as basis functions has been developed. In the collocation method, the mesh points of the space variable domain have been selected as the collocation points. The basis functions have been redefined into a new set of basis functions which in number match with the number of mesh points in the space variable domain. The solution of a non-linear boundary value problem has been obtained as the limit of a sequence of solutions of linear boundary value problems generated by quasilinearization technique. Several linear and nonlinear boundary value problems are presented to test the efficiency of the proposed method and found that numerical results obtained by the present method are in good agreement with the exact solutions available in the literature.

Keywords: collocation method, coupled system, cubic b-splines, mesh points

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Authors: Semra Turkan

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In this paper, we have examined the factors that affect the productivity of Turkey’s Top 500 Industrial Enterprises in 2014. The labor productivity of enterprises is taken as an indicator of productivity of industrial enterprises. When the relationships between some financial ratios and labor productivity, it is seen that there is a nonparametric relationship between labor productivity and return on sales. In addition, the distribution of labor productivity of enterprises is right-skewed. If the dependent distribution is skewed, the quantile regression is more suitable for this data. Hence, the nonparametric relationship between labor productivity and return on sales by quantile smoothing splines.

Keywords: quantile regression, smoothing spline, labor productivity, financial ratios

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Authors: Xiaogang Li, Jieqiong Miao

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As for the problem of the grey forecasting model prediction accuracy is low, an improved grey prediction model is put forward. Firstly, use trigonometric function transform the original data sequence in order to improve the smoothness of data , this model called SGM( smoothness of grey prediction model), then combine the improved grey model with support vector machine , and put forward the grey support vector machine model (SGM - SVM).Before the establishment of the model, we use trigonometric functions and accumulation generation operation preprocessing data in order to enhance the smoothness of the data and weaken the randomness of the data, then use support vector machine (SVM) to establish a prediction model for pre-processed data and select model parameters using genetic algorithms to obtain the optimum value of the global search. Finally, restore data through the "regressive generate" operation to get forecasting data. In order to prove that the SGM-SVM model is superior to other models, we select the battery life data from calce. The presented model is used to predict life of battery and the predicted result was compared with that of grey model and support vector machines．For a more intuitive comparison of the three models, this paper presents root mean square error of this three different models .The results show that the effect of grey support vector machine (SGM-SVM) to predict life is optimal, and the root mean square error is only 3.18%. Keywords: grey forecasting model, trigonometric function, support vector machine, genetic algorithms, root mean square error

Keywords: Grey prediction model, trigonometric functions, support vector machines, genetic algorithms, root mean square error

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Authors: Fernando Maass, Pablo Martin, Jorge Olivares

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The Bessel function J₀(x) is very important in Electrodynamics and Physics, as well as its zeros. In this work, a method to obtain high accuracy approximation is presented through an application to that function. In most of the applications of this function, the values of the zeros are very important. In this work, analytic approximations for this function have been obtained valid for all positive values of the variable x, which have high accuracy for the function as well as for the zeros. The approximation is determined by the simultaneous used of the power series and asymptotic expansion. The structure of the approximation is a combination of two rational functions with elementary functions as trigonometric and fractional powers. Here us in Pade method, rational functions are used, but now there combined with elementary functions us fractional powers hyperbolic or trigonometric functions, and others. The reason of this is that now power series of the exact function are used, but together with the asymptotic expansion, which usually includes fractional powers trigonometric functions and other type of elementary functions. The approximation must be a bridge between both expansions, and this can not be accomplished using only with rational functions. In the simplest approximation using 4 parameters the maximum absolute error is less than 0.006 at x ∼ 4.9. In this case also the maximum relative error for the zeros is less than 0.003 which is for the second zero, but that value decreases rapidly for the other zeros. The same kind of behaviour happens for the relative error of the maximum and minimum of the functions. Approximations with higher accuracy and more parameters will be also shown. All the approximations are valid for any positive value of x, and they can be calculated easily.

Keywords: analytic approximations, asymptotic approximations, Bessel functions, quasirational approximations

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Authors: P. V. Pramila , V. Mahesh

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Pulmonary Function Tests are important non-invasive diagnostic tests to assess respiratory impairments and provides quantifiable measures of lung function. Spirometry is the most frequently used measure of lung function and plays an essential role in the diagnosis and management of pulmonary diseases. However, the test requires considerable patient effort and cooperation, markedly related to the age of patients esulting in incomplete data sets. This paper presents, a nonlinear model built using Multivariate adaptive regression splines and Random forest regression model to predict the missing spirometric features. Random forest based feature selection is used to enhance both the generalization capability and the model interpretability. In the present study, flow-volume data are recorded for N= 198 subjects. The ranked order of feature importance index calculated by the random forests model shows that the spirometric features FVC, FEF 25, PEF,FEF 25-75, FEF50, and the demographic parameter height are the important descriptors. A comparison of performance assessment of both models prove that, the prediction ability of MARS with the `top two ranked features namely the FVC and FEF 25 is higher, yielding a model fit of R2= 0.96 and R2= 0.99 for normal and abnormal subjects. The Root Mean Square Error analysis of the RF model and the MARS model also shows that the latter is capable of predicting the missing values of FEV1 with a notably lower error value of 0.0191 (normal subjects) and 0.0106 (abnormal subjects). It is concluded that combining feature selection with a prediction model provides a minimum subset of predominant features to train the model, yielding better prediction performance. This analysis can assist clinicians with a intelligence support system in the medical diagnosis and improvement of clinical care.

Keywords: FEV, multivariate adaptive regression splines pulmonary function test, random forest

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Authors: Sajid Hussain

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The complex domain approach for image segmentation based on active contour has been designed, which deforms step by step to partition an image into numerous expedient regions. A novel region-based trigonometric complex pressure force function is proposed, which propagates around the region of interest using image forces. The signed trigonometric force function controls the propagation of the active contour and the active contour stops on the exact edges of the object accurately. The proposed model makes the level set function binary and uses Gaussian smoothing kernel to adjust and escape the re-initialization procedure. The working principle of the proposed model is as follows: The real image data is transformed into complex data by iota (i) times of image data and the average iota (i) times of horizontal and vertical components of the gradient of image data is inserted in the proposed model to catch complex gradient of the image data. A simple finite difference mathematical technique has been used to implement the proposed model. The efficiency and robustness of the proposed model have been verified and compared with other state-of-the-art models.

Keywords: image segmentation, active contour, level set, Mumford and Shah model

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Authors: Davies Obaromi, Qin Yongsong, James Ndege

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To interpolate scattered or regularly distributed data, there are imprecise or exact methods. However, there are some of these methods that could be used for interpolating data in a regular grid and others in an irregular grid. In spatial epidemiology, it is important to examine how a disease prevalence rates are distributed in space, and how they relate with each other within a defined distance and direction. In this study, for the geographic and graphic representation of the disease prevalence, linear and biharmonic spline methods were implemented in MATLAB, and used to identify, localize and compare for smoothing in the distribution patterns of tuberculosis (TB) in Eastern Cape Province. The aim of this study is to produce a more “smooth” graphical disease map for TB prevalence patterns by a 3-D curve fitting techniques, especially the biharmonic splines that can suppress noise easily, by seeking a least-squares fit rather than exact interpolation. The datasets are represented generally as a 3D or XYZ triplets, where X and Y are the spatial coordinates and Z is the variable of interest and in this case, TB counts in the province. This smoothing spline is a method of fitting a smooth curve to a set of noisy observations using a spline function, and it has also become the conventional method for its high precision, simplicity and flexibility. Surface and contour plots are produced for the TB prevalence at the provincial level for 2012 – 2015. From the results, the general outlook of all the fittings showed a systematic pattern in the distribution of TB cases in the province and this is consistent with some spatial statistical analyses carried out in the province. This new method is rarely used in disease mapping applications, but it has a superior advantage to be assessed at subjective locations rather than only on a rectangular grid as seen in most traditional GIS methods of geospatial analyses.

Keywords: linear, biharmonic splines, tuberculosis, South Africa

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Authors: Mahdieh Farzin Asanjan, Erkan Unal Mumcuoglu

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Mitochondria are cytoplasmic organelles of the cell, which have a significant role in the variety of cellular metabolic functions. Mitochondria act as the power plants of the cell and are surrounded by two membranes. Significant morphological alterations are often due to changes in mitochondrial functions. A powerful technique in order to study the three-dimensional (3D) structure of mitochondria and its alterations in disease states is Electron microscope tomography. Detection of mitochondria in electron microscopy images due to the presence of various subcellular structures and imaging artifacts is a challenging problem. Another challenge is that each image typically contains more than one mitochondrion. Hand segmentation of mitochondria is tedious and time-consuming and also special knowledge about the mitochondria is needed. Fully automatic segmentation methods lead to over-segmentation and mitochondria are not segmented properly. Therefore, semi-automatic segmentation methods with minimum manual effort are required to edit the results of fully automatic segmentation methods. Here two editing tools were implemented by applying spline surface dragging and interactive live-wire segmentation tools. These editing tools were applied separately to the results of fully automatic segmentation. 3D extension of these tools was also studied and tested. Dice coefficients of 2D and 3D for surface dragging using splines were 0.93 and 0.92. This metric for 2D and 3D for live-wire method were 0.94 and 0.91 respectively. The root mean square symmetric surface distance values of 2D and 3D for surface dragging was measured as 0.69, 0.93. The same metrics for live-wire tool were 0.60 and 2.11. Comparing the results of these editing tools with the results of automatic segmentation method, it shows that these editing tools, led to better results and these results were more similar to ground truth image but the required time was higher than hand-segmentation time

Keywords: medical image segmentation, semi-automatic methods, transmission electron microscopy, surface dragging using splines, live-wire

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Authors: Abdel-Maksoud Abdel-Kader Soliman

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The General Regularized Long Wave (GRLW) equation is solved numerically by giving a new algorithm based on collocation method using quartic B-splines at the mid-knot points as element shape. Also, we use the Fourth Runge-Kutta method for solving the system of first order ordinary differential equations instead of finite difference method. Our test problems, including the migration and interaction of solitary waves, are used to validate the algorithm which is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the algorithm.

Keywords: generalized RLW equation, solitons, quartic b-spline, nonlinear partial differential equations, difference equations

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Authors: Jinjun Jiang, Lianzhong Chen, Rui Xu

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The accurate of model attitude angel plays an important role on the aerodynamic test results in the wind tunnel test. The original method applies the spherical coordinate system transformation to obtain attitude angel calculation.The model attitude angel is obtained by coordinate transformation and spherical surface mapping applying the nominal attitude angel (the balance attitude angel in the wind tunnel coordinate system) indicated by the mechanism. First, the coordinate transformation of this method is not only complex but also difficult to establish the transformed relationship between the space coordinate systems especially after many steps of coordinate transformation, moreover it cannot realize the iterative calculation of the interference relationship between attitude angels; Second, during the calculate process to solve the problem the arc is approximately used to replace the straight line, the angel for the tangent value, and the inverse trigonometric function is applied. Therefore, in the calculation of attitude angel, the process is complex and inaccurate, which can be solved approximately when calculating small attack angel. However, with the advancing development of modern aerodynamic unsteady research, the aircraft tends to develop high or super large attack angel and unsteadyresearch field.According to engineering practice and vector theory, the concept of vector angel coordinate systemis proposed for the first time, and the vector angel coordinate system of attitude angel is established.With the iterative correction calculation and avoiding the problem of approximate and inverse trigonometric function solution, the model attitude calculation process is carried out in detail, which validates that the calculation accuracy and accuracy of model attitude angels are improved.Based on engineering and theoretical methods, a vector angel coordinate systemis established for the first time, which gives the transformation and angel definition relations between different flight attitude coordinate systems, that can accurately calculate the attitude angel of the corresponding coordinate systemand determine its direction, especially in the channel coupling calculation, the calculation of the attitude angel between the coordinate systems is only related to the angel, and has nothing to do with the change order s of the coordinate system, whichsimplifies the calculation process.

Keywords: attitude angel, angel vector coordinate system, iterative calculation, spherical coordinate system, wind tunnel test

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Authors: Muhammad Younis

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In this article, the novel (G′/G)-expansion method is successfully applied to construct the abundant travelling wave solutions to the modified Burgers’ equation with the aid of computation. The method is reliable and useful, which gives more general exact travelling wave solutions than the existing methods. These obtained solutions are in the form of hyperbolic, trigonometric and rational functions including solitary, singular and periodic solutions which have many potential applications in physical science and engineering. Some of these solutions are new and some have already been constructed. Additionally, the constraint conditions, for the existence of the solutions are also listed.

Keywords: traveling wave solutions, NLPDE, computation, integrability

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Authors: Vernie C. Convicto, Henry P. Aringa, Wilson I. Barredo

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This study presents a conformational model of the helical structures of globular protein particularly ferritin in the framework of white noise path integral formulation by using Associated Legendre functions, Bessel and convolution of Bessel and trigonometric functions as modulating functions. The model incorporates chirality features of proteins and their helix-turn-helix sequence structural motif.

Keywords: globular protein, modulating function, white noise, winding probability

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Authors: Alexander Matrosov, Guriy Shirunov

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A superposition method based on Lame's idea is used to get a general analytical solution to analyze a stress and strain state of a rectangular isotropjc elastic thick plate. The solution is built by using three solutions of the method of initial functions in the form of double trigonometric series. The results of bending of a thick plate under normal stress on its top face with two opposite sides clamped while others free of load are presented and compared with FEM modelling.

Keywords: general solution, method of initial functions, superposition method, thick isotropic plates

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Authors: Ivan Baravy

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Interpolation problem, as it was initially posed in terms of polynomials, is well researched. However, further mathematical developments extended it significantly. Trigonometric interpolation is widely used in Fourier analysis, while its generalized representation as exponential interpolation is applicable to such problem of mathematical physics as modelling of Ziegler-Biersack-Littmark repulsive interatomic potentials. Formulated for finite fields, this problem arises in decoding Reed--Solomon codes. This paper shows the relation between different interpretations of the problem through the class of matrices of special structure - Hankel matrices.

Keywords: Berlekamp-Massey algorithm, exponential interpolation, finite fields, Hankel matrices, Hankel polynomials

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Authors: Jatinderdeep Kaur

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In this paper, the results of Kano from one-dimensional cosine and sine series are extended to two-dimensional cosine and sine series. To extend these results, some classes of coefficient sequences such as the class of semi convexity and class R are extended from one dimension to two dimensions. Under these extended classes, I have checked the function f(x,y) is two dimensional Fourier Cosine and Sine series or equivalently it represents an integrable function. Further, some results are obtained which are the generalization of Moricz's results.

Keywords: conjugate dirichlet kernel, conjugate fejer kernel, fourier series, semi-convexity

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Authors: Negar Riazifar, Nigel G. Stocks

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This paper proposes strategies in level crossing (LC) sampling and reconstruction that provide alias-free high-fidelity signal reconstruction for speech signals without exponentially increasing sample number with increasing bit-depth. We introduce methods in LC sampling that reduce the sampling rate close to the Nyquist frequency even for large bit-depth. The results indicate that larger variation in the sampling intervals leads to an alias-free sampling scheme; this is achieved by either reducing the bit-depth or adding jitter to the system for high bit-depths. In conjunction with windowing, the signal is reconstructed from the LC samples using an efficient Toeplitz reconstruction algorithm.

Keywords: alias-free, level crossing sampling, spectrum, trigonometric polynomial

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Authors: Hee-Choul Kwon, Hyungjin Cho, Heeyong Kwon

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Image rotation is one of main pre-processing step in image processing or image pattern recognition. It is implemented with rotation matrix multiplication. However it requires lots of floating point arithmetic operations and trigonometric function calculations, so it takes long execution time. We propose a new high speed image rotation algorithm without two major time-consuming operations. We compare the proposed algorithm with the conventional rotation one with various size images. Experimental results show that the proposed algorithm is superior to the conventional rotation ones.

Keywords: high speed rotation operation, image processing, image rotation, pattern recognition, transformation matrix

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Authors: Smita Sonker, Uaday Singh

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Various investigators have determined the degree of approximation of functions belonging to the classes W(L r , ξ(t)), Lip(ξ(t), r), Lip(α, r), and Lipα using different summability methods with monotonocity conditions. Recently, Lal has determined the degree of approximation of the functions belonging to Lipα and W(L r , ξ(t)) classes by using Ces`aro-N¨orlund (C 1 .Np)- summability with non-increasing weights {pn}. In this paper, we shall determine the degree of approximation of 2π - periodic functions f belonging to the function classes Lipα and W(L r , ξ(t)) by C 1 .T - means of Fourier series of f. Our theorems generalize the results of Lal and we also improve these results in the light off. From our results, we also derive some corollaries.

Keywords: Lipschitz classes, product matrix operator, signals, trigonometric Fourier approximation

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Authors: Hee-Choul Kwon, Heeyong Kwon

Abstract:

Image rotation is one of main pre-processing steps for image processing or image pattern recognition. It is implemented with a rotation matrix multiplication. It requires a lot of floating point arithmetic operations and trigonometric calculations, so it takes a long time to execute. Therefore, there has been a need for a high speed image rotation algorithm without two major time-consuming operations. However, the rotated image has a drawback, i.e. distortions. We solved the problem using an augmented two-step shear transform. We compare the presented algorithm with the conventional rotation with images of various sizes. Experimental results show that the presented algorithm is superior to the conventional rotation one.

Keywords: high-speed rotation operation, image rotation, transform matrix, image processing, pattern recognition

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Authors: Ranajay Bhowmick

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Analysis of stresses for an infinitesimal tetrahedron leads to a situation where we obtain a cubic equation consisting of three stress invariants. This cubic equation, when solved for a definite condition, gives the principal stresses directly without requiring any cumbersome and time-consuming trial and error methods or iterative numerical procedures. Since the failure criterion of different materials are generally expressed as functions of principal stresses, an attempt has been made in this study to incorporate the solutions of the cubic equation in the form of principal stresses, obtained for a definite condition, into some of the established failure theories to determine their modified descriptions. It has been observed that the failure theories can be represented using the quadratic stress invariant and the orientation of the principal plane.

Keywords: cubic equation, stress invariant, trigonometric, explicit solution, principal stress, failure criterion

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