Harmonic differential

Abstract: It is shown that any compact Kahler manifold $M$ gives canonically rise to two strongly homotopy algebras, the first one being associated with the Hodge theory of the de Rham complex and the second one with the Hodge theory of the Dolbeault complex. In these algebras the product of two harmonic differential forms is again harmonic. If $M$ happens to be a Calabi-Yau manifold, there exists a third strongly homotopy algebra closely related to the Barannikov-Kontsevich extended moduli space of complex structures.

Abstract: 1 Holomorphic Functions.- 1.1. Holomorphic Functions.- 1.2. Holomorphic Map.- 2 Complex Manifolds.- 2.1. Complex Manifolds.- 2.2. Compact Complex Manifolds.- 2.3. Complex Analytic Family.- 3 Differential Forms, Vector Bundles, Sheaves.- 3.1. Differential Forms.- 3.2. Vector Bundles.- 3.3. Sheaves and Cohomology.- 3.4. de Rham's Theorem and Dolbeault's Theorem.- 3.5. Harmonic Differential Forms.- 3.6. Complex Line Bundles.- 4 Infinitesimal Deformation.- 4.1. Differentiable Family.- 4.2. Infinitesimal Deformation.- 5 Theorem of Existence.- 5.1. Obstructions.- 5.2. Number of Moduli.- 5.3. Theorem of Existence.- 6 Theorem of Completeness.- 6.1. Theorem of Completeness.- 6.2. Number of Moduli.- 6.3. Later Developments.- 7 Theorem of Stability.- 7.1. Differentiable Family of Strongly Elliptic Differential Operators.- 7.2. Differentiable Family of Compact Complex Manifolds.- Appendix Elliptic Partial Differential Operators on a Manifold.- 1. Distributions on a Torus.- 2. Elliptic Partial Differential Operators on a Torus.- 3. Function Space of Sections of a Vector Bundle.- 4. Elliptic Linear Partial Differential Operators.- 5. The Existence of Weak Solutions of a Strongly Elliptic Partial Differential Equation.- 6. Regularity of Weak Solutions of Elliptic Linear Partial Differential Equations.

Abstract: Numerical solution to linear bending analysis of circular plates is obtained by the method of harmonic differential quadrature (HDQ). In the method of differential quadrature (DQ), partial space derivatives of a function appearing in a differential equation are approximated by means of a polynomial expressed as the weighted linear sum of the function values at a preselected grid of discrete points. The method of HDQ that was used in the paper proposes a very simple algebraic formula to determine the weighting coefficients required by differential quadrature approximation without restricting the choice of mesh grids. Applying this concept to the governing differential equation of circular plate gives a set of linear simultaneous equations. Bending moments, stresses values in radial and tangential directions and vertical deflections are found for two different types of load. In the present study, the axisymmetric bending behavior is considered. Both the clamped and the simply supported edges are considered as boundary conditions. The obtained results are compared with existing solutions available from analytical and other numerical results such as finite elements and finite differences methods. A comparison between the HDQ results and the finite difference solutions for one example plate problem is also made. The method presented gives accurate results and is computationally efficient.