We demonstrate efficient wireless power transfer between two high Q resonators, especially in a complex electromagnetic environment. In the close proximity of metallic plates, the transfer efficiency stays roughly the same as the free space efficiency with proper designs. The experimental data fits well with a coupled theory model. Resonance frequency matching, alignment of the magnetic field, and impedance matching are shown to be the most important factors for efficient wireless power transfer

We prove the Integral Hodge Conjecture for curve classes on smooth varieties of dimension at least three with nef anticanonical divisor constructed either as an ample hypersurface in a smooth toric Fano variety or as a double cover of a smooth toric Fano variety. In fact, using results of Casagrande and the toric MMP, we prove that in each case, H 2 (X, Z) is generated by classes of rational curves. PreliminariesLet X be a smooth projective variety over C of dimension n. The integral Hodge classes H k−1,k−1 (X, Z) are the classes in H 2k (X, Z) that map to the subspace H k,k (X, C) of the Hodge decomposition of H 2k (X, C) under the natural map The Integral Hodge Conjecture asks if any integral Hodge class is a linear combination of classes of algebraic varieties, in other wordsWe will focus on the Integral Hodge Conjecture for curves, which is the statement that H n−1,n−1 (X, Z) is generated by the classes of algebraic curves contained in X. Recall that on a smooth toric, or more generally rational variety Y , H i (Y, O Y ) = 0 for i > 0, and H 2 (Y, Z) is torsion free. As a consequence,

We prove the integral Hodge conjecture for curve classes on smooth varieties of dimension at least three constructed as a complete intersection of ample hypersurfaces in a smooth projective toric variety, such that the anticanonical divisor is the restriction of a nef divisor. In particular, this includes the case of smooth anticanonical hypersurfaces in toric Fano varieties. In fact, using results of Casagrande and the toric minimal model program, we prove that in each case, 𝐻 2 (𝑋, ℤ) is generated by classes of rational curves.M S C 2 0 2 0 14C25, 14C30 (primary), 4J32, 14J45 (secondary) INTRODUCTIONOn a smooth complex projective variety of dimension 𝑛, the vector space 𝐻 𝑘 (𝑋, ℂ) admits a Hodge decomposition into subspaces 𝐻 𝑝,𝑞 (𝑋, ℂ), with 𝑝 + 𝑞 = 𝑘. The integral Hodge classes 𝐻 𝑘,𝑘 (𝑋, ℤ) are the classes in 𝐻 2𝑘 (𝑋, ℤ) which map to 𝐻 𝑘,𝑘 (𝑋, ℂ) under the natural mapand the class of any algebraic subvariety is an integral Hodge class. The integral Hodge conjecture asks whether the classes of algebraic subvarieties generate the integral Hodge classes as a group.A basic result in this direction is the Lefschetz (1,1)-theorem. This theorem states that the integral Hodge conjecture holds for codimension 1 classes. By the Hard Lefschetz theorem, this also implies that Hodge conjecture holds for degree 2𝑛 − 2 classes, that is, classes of algebraic curves generate 𝐻 𝑛−1,𝑛−1 (𝑋, ℚ) as a vector space.