The Experts below are selected from a list of 267 Experts worldwide ranked by ideXlab platform
Brian J. Mccartin - One of the best experts on this subject based on the ideXlab platform.
-
Spectral structure of the equilateral triangle I: Lamé's formulas
2008Co-Authors: Brian J. MccartinAbstract:Lame's formulas for the eigenvalues and eigenfunctions of the Laplacian on an equilateral triangle with either Dirichlet or Neumann boundary conditions are reviewed. The eigenfunctions are shown to form a complete Orthonormal System. Various properties of the spectrum and nodal/antinodal lines are explored.
-
Plenary lecture III: spectral structure of the equilateral triangle
2008Co-Authors: Brian J. MccartinAbstract:Lame's formulas for the eigenvalues and eigenfunctions of the Laplacian on an equilateral triangle with Dirichlet and Neumann boundary conditions will be reviewed and then extended to the Robin boundary condition. The eigenfunctions will be shown to form a complete Orthonormal System. Various properties of the spectrum and modal functions will be explored.
-
Eigenstructure of the equilateral triangle. Part III. The Robin problem
International Journal of Mathematics and Mathematical Sciences, 2004Co-Authors: Brian J. MccartinAbstract:Lame's formulas for the eigenvalues and eigenfunctions of the Laplacian on an equilateral triangle under Dirichlet and Neumann boundary conditions are herein extended to the Robin boundary condition. They are shown to form a complete Orthonormal System. Various properties of the spectrum and modal functions are explored.
-
Eigenstructure of the Equilateral Triangle, Part I: The Dirichlet Problem
SIAM Review, 2003Co-Authors: Brian J. MccartinAbstract:Lame's formulas for the eigenvalues and eigenfunctions of the Laplacian with Dirichlet boundary conditions on an equilateral triangle are derived using direct elementary mathematical techniques. They are shown to form a complete Orthonormal System. Various properties of the spectrum and nodal lines are explored. Implications for related geometries are considered.
-
Eigenstructure of the Equilateral Triangle, Part I:
2003Co-Authors: Brian J. MccartinAbstract:Lame's formulas for the eigenvalues and eigenfunctions of the Laplacian with Dirichlet boundary conditions on an equilateral triangle are derived using direct elementary math- ematical techniques. They are shown to form a complete Orthonormal System. Various properties of the spectrum and nodal lines are explored. Implications for related geometries are considered.
Juan L Varona - One of the best experts on this subject based on the ideXlab platform.
-
a whittaker shannon kotel nikov sampling theorem related to the dunkl transform
Proceedings of the American Mathematical Society, 2007Co-Authors: Oscar Ciaurri, Juan L VaronaAbstract:A Whittaker-Shannon-Kotel'nikov sampling theorem related to the Dunkl transform on the real line is proved. To this end we state, in terms of Bessel functions, an Orthonormal System which is complete in L 2 (( 1; 1);jxj 2 +1 dx). This Orthonormal System is a generalization of the classical exponential System dening Fourier series.
-
a whittaker shannon kotel nikov sampling theorem related to the dunkl transform on the real line
2000Co-Authors: Oscar Ciaurri, Juan L VaronaAbstract:(the classical Fourier transform corresponds to the case α = −1/2). Here we present a Whittaker-Shannon-Kotel’nikov sampling theorem related to the Dunkl transform Fα. To this end we state, in terms of Bessel functions, an Orthonormal System which is complete in L((−1, 1), dμα). This Orthonormal System is a generalization of the classical exponential System defining Fourier series. We include some pictures that ilustrate the sampling theorem.
Oscar Ciaurri - One of the best experts on this subject based on the ideXlab platform.
-
A Whittaker-Shannon-Kotel’nikov sampling theorem related to the Dunkl transform
2013Co-Authors: Oscar Ciaurri, L. VaronaAbstract:Abstract. A Whittaker-Shannon-Kotel’nikov sampling theorem related to the Dunkl transform on the real line is proved. To this end we state, in terms of Bessel functions, an Orthonormal System which is complete in L 2 ((−1, 1), |x | 2α+1 dx). This Orthonormal System is a generalization of the classical exponential System defining Fourier series. 1
-
a whittaker shannon kotel nikov sampling theorem related to the dunkl transform
Proceedings of the American Mathematical Society, 2007Co-Authors: Oscar Ciaurri, Juan L VaronaAbstract:A Whittaker-Shannon-Kotel'nikov sampling theorem related to the Dunkl transform on the real line is proved. To this end we state, in terms of Bessel functions, an Orthonormal System which is complete in L 2 (( 1; 1);jxj 2 +1 dx). This Orthonormal System is a generalization of the classical exponential System dening Fourier series.
-
a whittaker shannon kotel nikov sampling theorem related to the dunkl transform on the real line
2000Co-Authors: Oscar Ciaurri, Juan L VaronaAbstract:(the classical Fourier transform corresponds to the case α = −1/2). Here we present a Whittaker-Shannon-Kotel’nikov sampling theorem related to the Dunkl transform Fα. To this end we state, in terms of Bessel functions, an Orthonormal System which is complete in L((−1, 1), dμα). This Orthonormal System is a generalization of the classical exponential System defining Fourier series. We include some pictures that ilustrate the sampling theorem.
Margit Pap - One of the best experts on this subject based on the ideXlab platform.
-
Slice Regular Malmquist–Takenaka System in the Quaternionic Hardy Spaces
Analysis Mathematica, 2018Co-Authors: Margit PapAbstract:In this paper the slice regular analogue of the Malmquist–Takenaka System is introduced. It is proved that, under certain restrictions regarding to the parameters of the System, they form a complete Orthonormal System in the quaternionic Hardy spaces of the unit ball. The properties of associated projection operator are also studied.
-
Slice regular Malmquist-Takenaka System in the quaternionic Hardy spaces
arXiv: Complex Variables, 2016Co-Authors: Margit PapAbstract:A slice regular analogue of the Malmquist-Takenaka System is investigated. It is proved that they form a complete Orthonormal System in the quaternionic Hardy spaces of the unit ball. The properties of associated projection operator are studied. Key Words and Phrases: Functions of hypercomplex variables, quaternionic Hardy spaces, series expansions, interpolation, approximation by rational functions, quaternionic Malmquist-Takenaka System
Yoshifumi Ukita - One of the best experts on this subject based on the ideXlab platform.
-
a note on anova in an experimental design model based on an Orthonormal System
Systems Man and Cybernetics, 2012Co-Authors: Yoshifumi Ukita, Toshiyasu Matsushima, Shigeichi HirasawaAbstract:Experiments usually aim to study how changes in various factors affect the response variable of interest. Since the model used most often at present in experimental design is expressed through the effect of each factor, it is easy to understand how each factor affects the response variable. However, since the model contains redundant parameters, a considerable amount of time is often necessary to implement the procedure for estimating the effects. On the other hand, it has recently been shown that the model in experimental design can also be expressed in terms of an Orthonormal System. In this case, the model contains no redundant parameters. Moreover, the theorem with respect to the sum of squares for the 2-factor interaction, needed in the analysis of variance (ANOVA) has been obtained. However, 3-factor interaction is often to be considered in real cases, but the theorem with respect to the sum of squares for the 3-factor interaction has not been obtained up to now. In this paper, we present the theorem with respect to the sum of squares for the 3-factor interaction in a model based on an Orthonormal System. Furthermore, we can also obtain the theorem for interactions with 4 or more factors by the similar proof. Hence, in any real case, we can execute ANOVA in the model based on an Orthonormal System.
-
SMC - A note on ANOVA in an experimental design model based on an Orthonormal System
2012 IEEE International Conference on Systems Man and Cybernetics (SMC), 2012Co-Authors: Yoshifumi Ukita, Toshiyasu Matsushima, Shigeichi HirasawaAbstract:Experiments usually aim to study how changes in various factors affect the response variable of interest. Since the model used most often at present in experimental design is expressed through the effect of each factor, it is easy to understand how each factor affects the response variable. However, since the model contains redundant parameters, a considerable amount of time is often necessary to implement the procedure for estimating the effects. On the other hand, it has recently been shown that the model in experimental design can also be expressed in terms of an Orthonormal System. In this case, the model contains no redundant parameters. Moreover, the theorem with respect to the sum of squares for the 2-factor interaction, needed in the analysis of variance (ANOVA) has been obtained. However, 3-factor interaction is often to be considered in real cases, but the theorem with respect to the sum of squares for the 3-factor interaction has not been obtained up to now. In this paper, we present the theorem with respect to the sum of squares for the 3-factor interaction in a model based on an Orthonormal System. Furthermore, we can also obtain the theorem for interactions with 4 or more factors by the similar proof. Hence, in any real case, we can execute ANOVA in the model based on an Orthonormal System.
-
A Description of Experimental Design on the Basis of an Orthonormal System
Applications of Digital Signal Processing, 2011Co-Authors: Yoshifumi Ukita, Toshiyasu MatsushimaAbstract:The Fourier series representation of a function is a classic representation which is widely used to approximate real functions (Stein & Shakarchi, 2003). In digital signal processing (Oppenheim & Schafer, 1975), the sampling theorem states that any real valued function f can be reconstructed from a sequence of values of f that are discretely sampled with a frequency at least twice as high as the maximum frequency of the spectrum of f . This theorem can also be applied to functions over finite domains (Stankovic & Astola, 2007; Takimoto & Maruoka, 1997). Then, the range of frequencies of f can be expressed in more detail by using a bounded set instead of the maximum frequency. A function whose range of frequencies is confined to a bounded set I is referred to as “bandlimited to I”. Ukita et al. obtained a sampling theorem for bandlimited functions over Boolean (Ukita et al., 2003) and GF(q)n domains (Ukita et al., 2010a), where q is a prime power and GF(q) is Galois field of order q. The sampling theorem can be applied in various fields as well as in digital signal processing, and one of the fields is the experimental design. In most areas of scientific research, experimentation is a major tool for acquiring new knowledge or a better understanding of the target phenomenon. Experiments usually aim to study how changes in various factors affect the response variable of interest (Cochran & Cox, 1992; Toutenburg & Shalabh, 2009). Since the model used most often at present in experimental design is expressed through the effect of each factor, it is easy to understand how each factor affects the response variable. However, since the model contains redundant parameters and is not expressed in terms of an Orthonormal System, a considerable amount of time is often necessary to implement the procedure for estimating the effects. In this chapter, we propose that the model of experimental design be expressed as an Orthonormal System, and show that the model contains no redundant parameters. Then, the model is expressed by using Fourier coefficients instead of the effect of each factor. As there is an abundance of software for calculating the Fourier transform, such a System allows for a straightforward implementation of the procedures for estimating the Fourier coefficients by using Fourier transform. In addition, the effect of each factor can be easily obtained from the Fourier coefficients (Ukita & Matsushima, 2011). Therefore, it is possible to implement easily the estimation procedures as well as to understand how each factor affects the response variable in a model based on an Orthonormal System. Moreover, the analysis of variance can also be performed in a model based on an Orthonormal System (Ukita et al., 2010b). Hence, 18
-
a note on the degrees of freedom in an experimental design model based on an Orthonormal System
Systems Man and Cybernetics, 2011Co-Authors: Yoshifumi Ukita, Toshiyasu Matsushima, Shigeichi HirasawaAbstract:Experiments usually aim to study how changes in various factors affect the response variable of interest. Since the response model used most often at present in experimental design is expressed through the effect of each factor, it is straightforward to ascertain how each factor affects the response variable. However, since the response model contains redundant parameters, we must calculate the degrees of freedom defined by the number of independent parameters in the analysis of variance. In this paper, we show that through a description of experimental design based on an Orthonormal System, the response model can be expressed using only independent parameters. Hence, we do not have to calculate the degrees of freedom defined by the number of independent parameters.
-
ICICS - A note on relation between the Fourier coefficients and the effects in the experimental design
2011 8th International Conference on Information Communications & Signal Processing, 2011Co-Authors: Yoshifumi Ukita, Toshiyasu MatsushimaAbstract:It has recently been shown that the model in experimental design can be expressed in terms of an Orthonormal System. In this case, the model is expressed by using Fourier coefficients instead of the effect of each factor. As there is an abundance of software for calculating the Fourier transform, such a System allows for a straightforward implementation of the procedures for estimating the Fourier coefficients by using Fourier transform. However, Fourier coefficients themselves do not provide a direct representation of the effect of each factor, and the relation between the Fourier coefficients and the effect of each factor has not yet been clarified. In this paper, we present theorems of the relation between the Fourier coefficients and the effect of each factor. By using these theorems, the effect of each factor can be easily obtained from the computed Fourier coefficients. Therefore, with the aid of an Orthonormal System, it is possible to easily implement the estimation procedures as well as to understand how each factor affects the response variable in the model.
