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Peng Lizhong - One of the best experts on this subject based on the ideXlab platform.
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on stein weiss conjugate harmonic Function and octonion Analytic Function
Approximation Theory and Its Applications, 2000Co-Authors: Li Xingmin, Peng LizhongAbstract:It is shown that the Stein-Weiss conjugate harmonic Function is the Quarternion and the Octonion Analytic Function. We find a counter example to show the converse is not ture in the Octonion case, by which we have answered the question proposed in [1].
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on stein weiss conjugate harmonic Function and octonion analysis Function
Approximation Theory and Its Applications, 2000Co-Authors: Li Xingmin, Peng LizhongAbstract:It is shown that the Stein-Weiss conjugate harmonic Function is the Quarternion and the Octonion Analytic Function. We find a counter example to show the converse is not ture in the Octonion case, by which we have answered the question proposed in [1].
David Gottlieb - One of the best experts on this subject based on the ideXlab platform.
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on the gibbs phenomenon iii recovering exponential accuracy in a sub interval from a spectral partial sum of a piecewise Analytic Function
SIAM Journal on Numerical Analysis, 1996Co-Authors: David GottliebAbstract:We continue the investigation of overcoming the Gibbs phenomenon, i.e., obtaining exponential accuracy at all points, including at the discontinuities themselves, from the knowledge of a spectral partial sum of a discontinuous but piecewise Analytic Function. We show that if we are given the first N expansion coefficients of an $L^2 $ Function $f(x)$ in terms of either the trigonometric polynomials or the Legendre polynomials, we can construct an exponentially convergent approximation to the point values of $f(x)$ in any sub-interval in which it is Analytic.
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on the gibbs phenomenon iv recovering exponential accuracy in a subinterval from a gegenbauer partial sum of a piecewise Analytic Function
Mathematics of Computation, 1995Co-Authors: David GottliebAbstract:We continue the investigation of overcoming Gibbs phenomenon, i.e., obtaining exponential accuracy at all points including at the discontinuities themselves, from the knowledge of a spectral partial sum of a discontinuous but piecewise Analytic Function. We show that if we are given the first N expansion coefficients of an L_2 Function f(x) in terms of either the trigonometrical polynomials or the Chebyshev or Legendre polynomials, we can construct an exponentially convergent approximation to the point values of f(x) in any sub-interval in which it is Analytic.
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on the gibbs phenomenon iv recovering exponential accuracy in a subinterval from a gegenbauer partial sum of a piecewise Analytic Function
Mathematics of Computation, 1995Co-Authors: David Gottlieb, Chiwang ShuAbstract:We continue our investigation of overcoming Gibbs phenomenon, i.e., to obtain exponential accuracy at all points (including at the discontinuities themselves), from the knowledge of a spectral partial sum of a discontinuous but piecewise Analytic Function. We show that if we are given the first N Gegenbauer expansion coefficients, based on the Gegenbauer polynomials C(x) with the weight Function (1-x^2)- for any constant 0, of an L_1 Function f(x), we can construct an exponentially convergent approximation to the point values of f(x) in any sub-interval in which the Function is Analytic. The proof covers the cases of Chebyshev or Legendre partial sums, which are most common in applications.
Li Xingmin - One of the best experts on this subject based on the ideXlab platform.
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on stein weiss conjugate harmonic Function and octonion Analytic Function
Approximation Theory and Its Applications, 2000Co-Authors: Li Xingmin, Peng LizhongAbstract:It is shown that the Stein-Weiss conjugate harmonic Function is the Quarternion and the Octonion Analytic Function. We find a counter example to show the converse is not ture in the Octonion case, by which we have answered the question proposed in [1].
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on stein weiss conjugate harmonic Function and octonion analysis Function
Approximation Theory and Its Applications, 2000Co-Authors: Li Xingmin, Peng LizhongAbstract:It is shown that the Stein-Weiss conjugate harmonic Function is the Quarternion and the Octonion Analytic Function. We find a counter example to show the converse is not ture in the Octonion case, by which we have answered the question proposed in [1].
Hansolav Tylli - One of the best experts on this subject based on the ideXlab platform.
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operator weighted composition operators on vector valued Analytic Function spaces
Illinois Journal of Mathematics, 2009Co-Authors: Jussi Laitila, Hansolav TylliAbstract:We study qualitative properties of the operator-\break weighted composition maps ${W_{\psi,\varphi}} : f\mapsto\psi(f\circ\varphi)$ on the vector-valued spaces $H^\infty_v(X)$ of $X$-valued Analytic Functions $f : {\mathbb{D}}\to X$, where ${\mathbb{D}}$ is the unit disk, $X$ is a complex Banach space, $\varphi$ is an Analytic self-map of ${\mathbb{D}}$, $\psi$ is an Analytic operator-valued Function on ${\mathbb{D}}$, and $v$ is a bounded continuous weight on ${\mathbb{D}}$. Boundedness and compactness properties of ${W_{\psi,\varphi}}$ are characterized on $H^\infty_v(X)$ for infinite-dimensional $X$. It turns out that the (weak) compactness of ${W_{\psi,\varphi}}$ also involves properties of the auxiliary operator $T_\psi : x \mapsto\psi(\cdot)x$ from $X$ to $H^\infty_v(X)$, in contrast to the familiar scalar-valued setting $X = \mathbb C$.
Weichen Shi - One of the best experts on this subject based on the ideXlab platform.
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conservation laws from any conformal transformations and the parameters for a sharp v notch in plane elasticity
International Journal of Solids and Structures, 2013Co-Authors: Weichen ShiAbstract:Based on the well known complex Kolosov–Muskhelishvili potentials, two new independent Lagrangian Functions are presented and their variational problems lead to two independent harmonic equations, which are also the Navier’s displacement equations in plane elasticity. By applying Noether’s theorem to these Lagrangian Functions, it is found that their symmetry-transformation in material space is a conformal transformation in planar Euclidean space. Since any Analytic Function is a conformal transformation in planar Euclidean space, the conservation law obtained from this kind of symmetry-transformation possesses universality and leads to a path-independent integral. By adjusting the conformal transformation or Analytic Function, a finite value can be obtained from calculating this kind of path-independent integral around a material point with any order singularity. By applying this path-independent integral to the tip of a sharp V-notch, unlike Rice’s J-integral, the parameters of Mode I and II problems are found, which remain invariant because of path independence for a fixed notch opening angle. That is, these two parameters are equivalent to the notch stress intensity factors (NSIFs), and two examples are presented to show the application.
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conservation integrals in the sense of noether s theorem for an Analytic Function on a physical plane and application
Applied Mathematics and Computation, 2012Co-Authors: Weichen ShiAbstract:Abstract Conservation integrals in the sense of Noether’s theorem for an Analytic Function on a physical plane are investigated. For any physical system described by a continuum field theory, when its general solution under the consideration of plane problem can be expressed in terms of Analytic Functions, there always exist these kinds of conservation integrals. Especially, it is found that any conformal transformation satisfying Cauchy–Riemann equations is the symmetry-transformation for obtaining these kinds of conservation integrals. Since conformal transformations include the translation, rotation and scale change of planner coordinates, the obtained conservation integrals possess universality and diversity. By adjusting the conformal transformation, not only there are countless conserved quantities and conservation integrals indicated by Noether’s theorem in physics, but also these kinds of conservation integrals accord with the imaginary part of Cauchy’s integral theorem mathematically. In order to show some insight into application, the breakdown at the sharp point of a conductor, the steady streaming flow past a spinning cylinder and a crack in plane elasticity are considered. Some conserved quantities and parameters with a kind of physical invariance are presented. Especially, the energy release rate for crack extension and T stress for path selection can be expressed.
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path independent integral for the sharp v notch in longitudinal shear problem
International Journal of Solids and Structures, 2011Co-Authors: Weichen ShiAbstract:Abstract By applying Noether’s theorem to the elastic energy density in longitudinal shear problem, it is shown that its symmetry-transformations of material space can be expressed by the real and imaginary parts of an Analytic Function. This kind of the symmetry-transformations leads to the existence of a conservation law in material space, which does not belong to trivial conservation laws and whose divergence-free expression gives a path-independent integral. It is found that by adjusting the Analytic Function, a finite value can be obtained from this path-independent integral calculated around the material point with any order singularity. For a sharp V-notch placed on the edge of homogenous materials and/or the interface of bi-materials, application shows that the finite value obtained from this path-independent integral is directly related to the notch stress intensity factor (NSIF) and does not depend on the location of integral endpoints chosen respectively along two traction-free surfaces of which form a notch opening angle. Usability is presented in an example to estimate the NSIF of a bi-material plate.