Hopf Modules and Noncommutative Differential Geometry

We define a new algebra of noncommutative differential forms for any Hopf algebra with an invertible antipode. We prove that there is a one-to-one correspondence between anti-Yetter–Drinfeld modules, which serve as coefficients for the Hopf cyclic (co)homology, and modules which admit a flat connection with respect to our differential calculus. Thus, we show that these coefficient modules can be regarded as “flat bundles” in the sense of Connes’ noncommutative differential geometry.     

The role of matching thickness on the wideband electromagnetic wave suppresser using single layer doped barium ferrite

The effect of Mg²⁺, Co²⁺and Ti⁴⁺ substitution on microwave absorption has been studied for BaMg_0.5Co_0.5Ti_1.0Fe₁₀O₁₉ ferrite-acrylic resin composite in frequency range from 13 to 20 GHz. X-ray diffraction (XRD), scanning electron microscopy (SEM), vector network analysis and vibrating sample magnetometry (VSM) were employed to analyze structure, electromagnetic and microwave absorption properties of prepared ferrite. The obtained results of reflectivity demonstrate that by varying matching thickness along with weight percentage of ferrite to acrylic resin, the bandwidth coupled with reflection loss values of prepared composites can be easily tuned. Based on microwave measurement on reflectivity, it is found that BaMg_0.5Co_0.5Ti_1.0Fe₁₀O₁₉ is a good candidate for wideband electromagnetic compatibility and other practical applications at high frequency.     

Regeneration of surface roughness by the Langevin equation using stochastic analysis on AFM image of a carbon fiber

A new method was developed using AFM images of a fiber surface to regenerate the surface roughness in 3D geometry, such as the cylindrical shape of a “model” fiber. The Langevin equation was used to derive the fluctuations of a carbon fiber surface image. The equation contains two quantities, D⁽¹⁾ (h) and D⁽²⁾ (h) which in physics represent drift and diffusion coefficients. Knowing this coefficient and adding a proper noise function, a similar surface of larger dimension with the same statistical properties of the initial data was created. The generated surface was mapped into cylindrical coordinates, then a mesh generated. The resulting reconstructed surface, input over the geometry of a cylindrical shape, can be implemented for finite element analysis of a single fiber surrounded by matrix and generalized to a many fiber model.