Detecting and quantifying envelope singularities in the plane

The mathematical envelopes of families of both rigid and non-rigid moving shapes play a fundamental role in a variety of problems from very diverse application domains, from engineering design and manufacturing to computer graphics and computer assisted surgery. Geometric singularities in these envelopes are known to induce malfunctions or unintended system behavior, and the corresponding theoretical and computational difficulties induced by these singularities are not only massive, but also well documented. We describe a new approach to detect and quantify the envelope singularities induced by 2-dimensional shapes of arbitrary complexity moving according to general non-periodic and non-singular planar affine motions. Our approach, which does not require any envelope computations, is reframing the problem in terms of “fold points” and “fold regions” in the neighborhood of geometric singularities, and we show that the existence of these fold points is a necessary condition for the existence of singularities. We establish a mathematically well defined duality between the 2-dimensional Euclidean space in which the motion takes place and a 2+1 spacetime domain. Based on this duality, we recast the problem of detecting and quantifying geometric singularities into inherently parallel tests against the original geometric representation in the 2-dimensional Euclidean space. We conclude by discussing the significance of our results, and the extension of our approach to 3-dimensional moving shapes.     

Detecting real singularities of a space curve from a real rational parametrization

In this paper we give an algorithm that detects real singularities, including singularities at infinity, and counts local branches and multiplicities of real rational curves in the affine n$n$-space without knowing an implicitization. The main idea behind this is a generalization of the D$D$-resultant (see [van den Essen, A., Yu, J.-T., 1997. The D$D$-resultant, singularities and the degree of unfaithfulness. Proc. Amer. Math. Soc. 25 (3), 689–695]) to n$n$ rational functions. This allows us to find all real parameters corresponding to the real singularities between the solutions of a system of polynomials in one variable.     

Fast computation technique of Green's function and its boundary integrals for multilayered medium structures by using Fourier series expansion and method of images

This paper presents a systematic technique to improve the convergence of the Green's function for multilayered medium structure by introducing a three-layered model into the multilayered system. The technique uses a combination of Fourier series expansion and method of images. Numerical analysis demonstrates dramatic improvements of the convergence of Green's function and its boundary integrals. © 1998 John Wiley & Sons, Inc. Int J RF and Microwave CAE 8: 474–483, 1998     

Improving convergence for the approximation of non-periodic functions by Fourier series

Approximation of functions by Fourier series plays an important role in many applied problems of digital signal processing. An effective method is presented for the construction of highly accurate mean-square approximations by Fourier series for nonperiodic functions. This technique employs the subtraction of specially selected functions that enhance the smoothness of the periodic extension of the approximated function. The main advantage of the method is that the function-setting interval is taken as a half-period rather than a whole period. This doubles the smoothness of the periodic extension. The efficiency of the method is illustrated by test functions of one and two variables.     

Identification of discrete Hammerstein systems by the Fourier series regression estimate

The identification of a single-input, single-output (SISO) discrete Hammerstein system is studied. Such a system consists of a non-linear memoryless subsystem followed by a dynamic, linear subsystem. The parameters of the dynamic, linear subsystem are identified by a correlation method and the Newton-Gauss method. The main results concern the identification of the non-linear, memoryless subsystem. No conditions are imposed on the functional form of the non-linear subsystem, recovering the non-linear using the Fourier series regression estimate. The density-free pointwise convergence Of the estimate is proved, that is.algorithm converges for all input densities The rate of pointwise convergence is obtained for smooth input densities and for non-linearities of Lipschitz type.Globle convergence and its rate are also studied for a large class of non-linearities and input densities     

Fourier exponential series matrix of integration

The Fourier exponential operational matrix of integration P is derived which is analogous to that previously derived for other types of orthogonal functions. This matrix P may be used to solve problems such as identification, analysis and optimal control.     

Use of a double Fourier series for three-dimensional shape representation

The representation of three-dimensional star-shaped objects by the double Fourier series (DFS) coefficients of their boundary function is considered. An analogue of the convolution theorem for a DFS on a sphere is developed. It is then used to calculate the moments of an object directly from the DFS coefficients, without an intermediate reconstruction step. The complexity of computing the moments from the DFS coefficients is O(N ² log N), where N is the maximum order of coefficients retained in the expansion, while the complexity of computing the moments from the spherical harmonic representation is O(N ² log ² N). It is shown that under sufficient conditions, the moments and surface area corresponding to the truncated DFS converge to the true moments and area of an object. A new kind of DFS—the double Fourier sine series—is proposed which has better convergence properties than the previously used kinds and spherical harmonics in the case of objects with a sharp point above the pole of the spherical domain.     

Hybrid function projective synchronization of chaotic systems with uncertain time-varying parameters via Fourier series expansion

In this paper, the hybrid function projective synchronization (HFPS) of different chaotic systems with uncertain periodically time-varying parameters is carried out by Fourier series expansion and adaptive bounding technique. Fourier series expansion is used to deal with uncertain periodically time-varying parameters. Adaptive bounding technique is used to compensate the bound of truncation errors. Using the Lyapunov stability theory, an adaptive control law and six parameter updating laws are constructed to make the states of two different chaotic systems asymptotically synchronized. The control strategy does not need to know the parameters thoroughly if the time-varying parameters are periodical functions. Finally, in order to verify the effectiveness of the proposed scheme, the HFPS between Lorenz system and Chen system is completed successfully by using this scheme.     

Identification of non-linear systems using the exponential Fourier series

A new algorithm is presented for the identification of non-linear lumped time-invariant SISO systems. The method is based on differentiation properties of the exponential Fourier series and can be used to identify a wide class of systems. Identification of two specific systems is studied. An illustrative example is provided which shows that the algorithm gives accurate parameter estimates     

A functional presentation of Fourier series convergence

Understanding lengthy mathematical proofs requires strong concentration. Authors must efficiently map whole logical structures into sequential texts. One way to ease such tasks is presenting the logical structure in a functional programming style. In our method, functional proofs are implemented by a real programming language. The behavior of each function appears in the proofs as a building block ready to be visualized with concrete data. This paper contains a case study of the well-known Dirichlet's theorem on the convergence of Fourier series. It shows the relevance of our method in rigorous mathematical presentations that involve - arguments intensively.     

Identification of a class of time-variable linear systems via its output covariance function

The main concern of this paper is to determine the state-space representation of a class of linear time-variable, periodic system, such that when excited by stationary white noise it results in a random process with prescribed covariance function. It is shown that by using a proper transformation on the state covariance matrix of the system it is possible to find a new matrix which has periodicity properties and satisfies a periodic matrix Riceati differential equation; therefore, the time interval of interest, on which the matrix Riceati equation must be solved using previous approaches, will collapse into one period.     

Solution of variational problems using the Fourier series operational matrix

There are many functions having the property of orthogonality. Among them are Walsh functions, block-pulse functions, Chebyshev functions and Legendre functions. Recently, many scholars and researchers made new approaches to solving variational problems via orthogonal functions. This paper suggests a more simple and powerful orthogonal method—the Fourier series operational matrix for integration—which is derived and used to solve a variational problem.     

Convergence acceleration on a general class of power series

In this paper we present several efficient methods for evaluating functions defined by power series expansions. Simple computer codes for two rapid algorithms are given in a companion paper. The convergence rates of the proposed computational schemes are investigated theoretically and the results are illustrated by numerical examples.     

Weighted sharing of a small function by a meromorphic function and its derivative

The paper deals with the problem of meromorphic functions sharing a small function with its derivative and improves the results of Yu [K.W. Yu, On entire and meromorphic functions that share small functions with their derivatives, J. Inequal. Pure Appl. Math. 4 (1) (2003) Art. 21 (Online: http://jipam.vu.edu.au/)], Lahiri–Sarkar [I. Lahiri, A. Sarkar, Uniqueness of meromorphic function and its derivative, J. Inequal. Pure Appl. Math. 5 (1) (2004) Art. 20 (Online: http://jipam.vu.edu.au/)] and a recent result of Zhang [Q.C. Zhang, Meromorphic function that shares one small function with its derivative, J. Inequal. Pure Appl. Math. 6 (4) (2005) Art. 116 (Online: http://jipam.vu.edu.au/)].     

Representation of the Fourier Transform by Fourier Series

The analysis of the mathematical structure of the integral Fourier transform shows that the transform can be split and represented by certain sets of frequencies as coefficients of Fourier series of periodic functions in the interval . In this paper we describe such periodic functions for the one- and two-dimensional Fourier transforms. The approximation of the inverse Fourier transform by periodic functions is described. The application of the new representation is considered for the discrete Fourier transform, when the transform is split into a set of short and separable 1-D transforms, and the discrete signal is represented as a set of short signals. Properties of such representation, which is called the paired representation, are considered and the basis paired functions are described. An effective application of new forms of representation of a two-dimensional image by splitting-signals is described for image enhancement. It is shown that by processing only one splitting-signal, one can achieve an enhancement that may exceed results of traditional methods of image enhancement.     

High-accuracy approximation of the complex probability function by Fourier expansion of exponential multiplier

The real K(x,y) and imaginary L(x,y) parts of the complex probability function are approximated as rapidly convergent series, based on the Fourier expansion of the exponential multiplier. This approach provides rapid and accurate calculations of the Voigt and complex error functions in the most challenging Humlí?ek regions 3 and 4.     

A computer program for approximating a linear operator equation using a generalized Fourier series

Many important engineering problems fall into the category of being linear operators, with supporting conditions. In this paper, an inner-product and norm is used which enables the numerical modeler to approximate such by developing a generalized Fourier series. The resulting approximation is the “best” approximation in that a least-squares (L²) error is minimized simultaneously for fitting both the problem's boundary conditions and satisfying the linear operator relationship (the governing equations) over the problem's domain (both space and time). Because the numerical technique involves a well-defined inner-product, error evaluation is readily available using Bessel's inequality. Minimization of the approximation error is subsequently achieved with respect to a weighting of the inner components, and the addition of basis functions used in the approximation. A computer program source code is provided (see Appendix A) to implement the procedures.     

An efficient algorithm for automatically generating multivariable fuzzy systems by Fourier series method

By exploiting the Fourier series expansion, we have developed a new constructive method of automatically generating a multivariable fuzzy inference system from any given sample set with the resulting multivariable function being constructed within any specified precision to the original sample set. The given sample sets are first decomposed into a cluster of simpler sample sets such that a single input fuzzy system is constructed readily for a sample set extracted directly from the cluster independent of the other variables. Once the relevant fuzzy rules and membership functions are constructed for each of the variables completely independent of the other variables, the resulting decomposed fuzzy rules and membership functions are integrated back into the fuzzy system appropriate for the original sample set requiring only a moderate cost of computation in the required decomposition and composition processes. After proving two basic theorems which we need to ensure the validity of the decomposition and composition processes of the system construction, we have demonstrated a constructive algorithm of a multivariable fuzzy system. Exploiting an implicit error bound analysis available at each of the construction steps, the present Fourier method is capable of implementing a more stable fuzzy system than the power series expansion method of ParNeuFuz and PolyNeuFuz, covering and implementing a wider range of more robust applications.