On the volume of a pseudo-effective class and semi-positive properties of the Harder-Narasimhan filtration on a compact Hermitian manifold

This paper divides into two parts. Let $(X,\omega)$ be a compact Hermitian manifold. Firstly, if the Hermitian metric $\omega$ satisfies the assumption that $\partial\overline{\partial}\omega^k=0$ for all $k$, we generalize the volume of the cohomology class in the K\"{a}hler setting to the Hermitian setting, and prove that the volume is always finite and the Grauert-Riemenschneider type criterion holds true, which is a partial answer to a conjecture posed by Boucksom. Secondly, we observe that if the anticanonical bundle $K^{-1}_X$ is nef, then for any $\varepsilon>0$, there is a smooth function $\phi_\varepsilon$ on $X$ such that $\omega_\varepsilon:=\omega+i\partial\overline{\partial}\phi_\varepsilon>0$ and Ricci$(\omega_\varepsilon)\geq-\varepsilon\omega_\varepsilon$. Furthermore, if $\omega$ satisfies the assumption as above, we prove that for a Harder-Narasimhan filtration of $T_X$ with respect to $\omega$, the slopes $\mu_\omega(\mathcal{F}_i/\mathcal{F}_{i-1})\geq 0$ for all $i$, which generalizes a result of Cao which plays a very important role in his studying of the structures of K\"{a}hler manifolds

CR eigenvalue estimate and Kohn-Rossi cohomology

Let $X$ be a compact connected CR manifold with a transversal CR $S^1$-action of dimension $2n-1$, which is only assumed to be weakly pseudoconvex. Let $\Box_b$ be the $\overline{\partial}_b$-Laplacian. Eigenvalue estimate of $\Box_b$ is a fundamental issue both in CR geometry and analysis. In this paper, we are able to obtain a sharp estimate of the number of eigenvalues smaller than or equal to $\lambda$ of $\Box_b$ acting on the $m$-th Fourier components of smooth $(n-1,q)$-forms on $X$, where $m\in \mathbb{Z}_+$ and $q=0,1,\cdots, n-1$. Here the sharp means the growth order with respect to $m$ is sharp. In particular, when $\lambda=0$, we obtain the asymptotic estimate of the growth for $m$-th Fourier components $H^{n-1,q}_{b,m}(X)$ of $H^{n-1,q}_b(X)$ as $m \rightarrow +\infty$. Furthermore, we establish a Serre type duality theorem for Fourier components of Kohn-Rossi cohomology which is of independent interest. As a byproduct, the asymptotic growth of the dimensions of the Fourier components $H^{0,q}_{b,-m}(X)$ for $m\in \mathbb{Z}_+$ is established. Compared with previous results in this field, the estimate for $\lambda=0$ already improves very much the corresponding estimate of Hsiao and Li . We also give appilcations of our main results, including Morse type inequalities, asymptotic Riemann-Roch type theorem, Grauert-Riemenscheider type criterion, and an orbifold version of our main results which answers an open problem.Comment: 39 pages, submitted on January 17, 2018. Comments welcome! arXiv admin note: text overlap with arXiv:1506.06459, arXiv:1502.02365 by other author

Performance of twin-rotor DC homopolar motor

A new concept of dual-rotor motor with counter-rotating propeller increases the efficiency of the ship propulsion used in marine industry. This concept can be realized by twin-rotor permanent magnet homopolar disc motor, which is the object of this thesis. The objectives of this thesis are to determine the motor electromechanical parameters applying a 3-D FEM modeling and to analyze the motor performance in steady-state and dynamic conditions. The objectives of this thesis are to determinate the PM DC homopolar twin-rotor disc motor electromechanical parameters by 3-D FEM modeling. Furthermore, the PM DC Homopolar perform determination in steady-state and dynamic conditions. To reach these objectives, a 3-D model of the twin-rotor type of homopolar PM DC motor is developed using Ansoft Maxwell V12 software package. From modeling the motor magnetic flux and current flow density distribution on the aluminum disc and the electromechanical parameters were determined. Next, applying the electric equivalent circuit and mechanical equivalent system, the steady-state model and dynamic motor mode was developed in Matlab/Simulink software and the performance of the homopolar motor was analyzed. From the analysis, the following conclusions were deduced: - To produce the relatively high electromagnetic torque a very high currents have to be supplied to the rotor aluminum discs what is the major deficiency of the motor - The currents are distributed non-uniformly in the disc what makes the electromagnetic torque smaller. To improve this, more brushes should be applied around the disc periphery. - The proposed construction of the motor allows one rotor disc to rotate in direction opposite to the rotation of other rotor disc. This can be done by reversing the current flow in one of the aluminum discs