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The purpose of this paper is to work out the theoretical aspects of the vibration problem of rubber wiper blade on convex windshield. Over the past 20 years, some $2$-dimensional spring-mass models were presented in engineering science to simulate the vibration of rubber wiper blade on windshield. In this paper, we will consider the elasticity perspective on this $3$-dimensional vibration problem. Our theoretical analysis suggests that there should exist two classes of vibration frequencies corresponding to “$\ast$-exact deformations (Class I)” and “$\ast$-closed deformations (Class II)”. We prove mathematical theorems on the characterization of deformations of Class I. We also explain how elementary deformations of Class II can be constructed. We then deduce two mathematical formulas, for the vibration problem of rubber wiper blade on convex windshield, from our theoretical analysis. Our theoretical predictions are in almost perfect agreement with experimental data. One of the crucial steps of our analysis is a decomposition theorem motivated by the de Rham Cohomology and the Hodge Theory.

It is shown that for compact Calabi–Yau threefolds, “no strong quantum correction” is equivalent to the condition that, with respect to the Hodge metric, the image Φ ( T ) of the Teichmuller space T under the period map Φ is an open submanifold W of the same dimension as T . In this paper, we extend this result to the case of dimension 4. We also obtained similar results for compact generalized Calabi–Yau manifolds in dimension 3 and 4, under the assumption that their Teichmuller space T exists and is a simply connected complex manifold.

We give sharp lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining an even dimensional nodal hypersurface. This implies the validity of formulas due to M. Saito, L. Wotzlaw and the author for the graded pieces with respect to the Hodge filtration of the top cohomology of the hypersurface complement in many new cases. A classical result by Severi on the position of the singularities of a nodal surface in $\mathbb{P}^3$ is improved and applications to deformation theory of nodal surfaces are given., Comment: v3. Some applications to the deformation theory of nodal surfaces in $\mathbb{P}^3$ are added

The inversion of the Laplace-Beltrami operator and the computation of the Hodge decomposition of a tangential vector field on smooth surfaces arise as computational tasks in many areas of science, from computer graphics to machine learning to computational physics. Here, we present a high-order accurate pseudo-spectral approach, applicable to closed surfaces of genus one in three-dimensional space, with a view toward applications in plasma physics and fluid dynamics. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 941–955, 2017

We provide a self-contained proof of the solvability and regularity of a Hodge-type elliptic system, wherein the divergence and curl of a vector field u are prescribed in an open, bounded, Sobolev-class domain \({\Omega \subseteq \mathbb{R}^{\rm n}}\), and either the normal component \({{\bf u} \cdot {\bf N}}\) or the tangential components of the vector field \({{\bf u} \times {\bf N}}\) are prescribed on the boundary \({\partial \Omega}\). For \({{\rm k} > {\rm n}/2}\), we prove that u is in the Sobolev space \({H^{\rm k+1}(\Omega)}\) if \({\Omega}\) is an \({H^{\rm k+1}}\)-domain, and the divergence, curl, and either the normal or tangential trace of u has sufficient regularity. The proof is based on a regularity theory for vector elliptic equations set on Sobolev-class domains and with Sobolev-class coefficients, and with a rather general set of Dirichlet and Neumann boundary conditions. The resulting regularity theory for the vector u is fundamental in the analysis of free-boundary and moving interface problems in fluid dynamics.

We investigate the performance of various discrete Hodge star operators for discrete exterior calculus (DEC) using circumcentric and barycentric dual meshes. The performance is evaluated through the DEC solution of Darcy and incompressible Navier-Stokes flows over surfaces. While the circumcentric Hodge operators may be favorable due to their diagonal structure, the barycentric (geometric) and the Galerkin Hodge operators have the advantage of admitting arbitrary simplicial meshes. Numerical experiments reveal that the barycentric and the Galerkin Hodge operators retain the numerical convergence order attained through the circumcentric (diagonal) Hodge operators. Furthermore, when the barycentric or the Galerkin Hodge operators are employed, a super-convergence behavior is observed for the incompressible flow solution over unstructured simplicial surface meshes generated by successive subdivision of coarser meshes. Insofar as the computational cost is concerned, the Darcy flow solutions exhibit a moderate increase in the solution time when using the barycentric or the Galerkin Hodge operators due to a modest decrease in the linear system sparsity. On the other hand, for the incompressible flow simulations, both the solution time and the linear system sparsity do not change for either the circumcentric or the barycentric and the Galerkin Hodge operators. Various definitions for discrete Hodge star operators are investigated.Darcy and incompressible Navier-Stokes flows are used for numerical experiments.Barycentric Hodge star reproduces the convergence rate of the circumcentric Hodge.The difference in the computational cost is generally minor.

We study mirror symmetry of complete intersection Calabi-Yau manifolds which have birational automorphisms of infinite order. We observe that movable cones in birational geometry are transformed, under mirror symmetry, to the monodromy nilpotent cones which are naturally glued together.

We show certain symmetry of the dimensions of cohomologies of the funda- mental groups of compact Sasakian manifolds by using the Hodge theory of twisted basic cohomology. As applications, we show that the polycyclic fundamental groups of compact Sasakian manifolds are virtually nilpotent and Sasakian solvmanifolds are finite quotients of Heisenberg nilmanifolds., 6 pages. To appear in Ann. Mat. Pura Appl

On a smoothly stratified space, we identify intersection cohomology of any given perversity with an associated weighted \(L^2\) cohomology for iterated fibred cusp metrics on the smooth stratum.

It is shown that the Hodge theory for metric spaces based on the Alexander Spanier coboundary operator, in the presence of a measure previously developed in [4], holds for the class of compact Alexandrov spaces with curvature bounded from above. In particular, the real cohomology of the space is isomorphic to the corresponding space of harmonic co-chains.

When trying to extend the Hodge theory for elliptic complexes on compact closed manifolds to the case of compact manifolds with boundary one is led to a boundary value problem for the Laplacian of the complex which is usually referred to as Neumann problem. We study the Neumann problem for a larger class of sequences of differential operators on a compact manifold with boundary. These are sequences of small curvature, i.e., bearing the property that the composition of any two neighbouring operators has order less than two.

In this paper we present the rst example of a primitive, totally geodesic subvariety F M g;n with dim(F ) > 1. The variety we consider is a surfaceF M 1;3 dened using the projective geometry of

In this work, we present a Hodge-type decomposition for variable exponent spaces of Clifford-valued functions, where one of the components is the kernel of the parabolic-type Dirac operator.

We study the deformations of twisted harmonic maps \(f\) with respect to the representation \(\rho \). After constructing a continuous “universal” twisted harmonic map, we give a construction of every first order deformation of \(f\) in terms of Hodge theory; we apply this result to the moduli space of reductive representations of a Kahler group, to show that the critical points of the energy functional \(E\) coincide with the monodromy representations of polarized complex variations of Hodge structure. We then proceed to second order deformations, where obstructions arise; we investigate the existence of such deformations, and give a method for constructing them, as well. Applying this to the energy functional as above, we prove (for every finitely presented group) that the energy functional is a potential for the Kahler form of the “Betti” moduli space; assuming furthermore that the group is Kahler, we study the eigenvalues of the Hessian of \(E\) at critical points.

We prove several formulas related to Hodge theory and the Kodaira–Spencer–Kuranishi deformation theory of Kahler manifolds. As applications, we present a construction of globally convergent power series of integrable Beltrami differentials on Calabi–Yau manifolds and also a construction of global canonical family of holomorphic $$(n,0)$$ -forms on the deformation spaces of Calabi–Yau manifolds. Similar constructions are also applied to the deformation spaces of compact Kahler manifolds.

Let M be a compact complex manifold with a complex Finsler metric F. We define a natural projection of complex horizontal Laplacian on M: it is independent of the fiber coordinate. By using Sobolev space theory and spectral resolution theory in Hilbert space, we prove the Hodge theorem for the natural projection of complex horizontal Laplacian on M.

We discuss the variations of mixed Hodge structure for cohomology with compact support of quasi-projective simple normal crossing pairs. We show that they are graded polarizable admissible variations of mixed Hodge structure. Then we prove a generalization of the Fujita–Kawamata semi-positivity theorem.

In this paper we use the vanishing of first cohomology group of a Riemannian manifold (M, g) to find a sufficient condition for a closed vector field ξ on M to be a concircular vector field. Mathematics Subject Classification: 53C55

We prove the existence of global sections trivializing the Hodge bundles on the Hodge metric completion space of the Torelli space of Calabi–Yau manifolds, a global splitting property of these Hodge bundles. We also prove that a compact Calabi–Yau manifold can not be deformed to its complex conjugate. These results answer certain open questions in the subject. A general result about certain period map to be bi-holomorphic from the Hodge metric completion space of the Torelli space of Calabi–Yau type manifolds to their period domains is proved and applied to the cases of K3 surfaces, cubic fourfolds, and hyperkahler manifolds.

In this Note we introduce new definition of Hodge structures and show that R -Hodge structures are determined by R -linear operators that are annihilated by the Weierstrass σ-function.

We prove a representation-theoretic version of Borisov–Batyrev mirror symmetry, and use it to construct infinitely many new pairs of orbifolds with mirror Hodge diamonds, with respect to the usual Hodge structure on singular complex cohomology. We conjecture that the corresponding orbifold Hodge diamonds are also mirror. When X is the Fermat quintic in P 4 , and X ˜ ∗ is a Sym 5 -equivariant, toric resolution of its mirror X ∗ , we deduce that for any subgroup Γ of the alternating group A 5 , the Γ -Hilbert schemes Γ - Hilb ( X ) and Γ - Hilb ( X ˜ ∗ ) are smooth Calabi–Yau threefolds with (explicitly computed) mirror Hodge diamonds.