On Hard Lefschetz Conjecture on Lawson Homology

Friedlander and Mazur proposed a conjecture of hard Lefschetz type on Lawson homology. We shall relate this conjecture to Suslin conjecture on Lawson homology. For abelian varieties, this conjecture is shown to be equivalent to a vanishing conjecture of Beauville type on Lawson homology. For symmetric products of curves, we show that this conjecture amounts to the vanishing conjecture of Beauville type for the Jacobians of the corresponding curves. As a consequence, Suslin conjecture holds for all symmetric products of curves with genus at most 2.Comment: 7 page

Density Functional Perturbation Theory and Adaptively Compressed Polarizability Operator

Kohn-Sham density functional theory (KSDFT) is by far the most widely used electronic structure theory in condensed matter systems. Density functional perturbation theory (DFPT) studies the response of a quantum system under small perturbation, where the quantum system is described at the level of first principle electronic structure theories like KSDFT. One important application of DFPT is the calculation of vibration properties such as phonons, which can be further used to calculate many physical properties such as infrared spectroscopy, elastic neutron scattering, specific heat, heat conduction, and electron-phonon interaction related behaviors such as superconductivity . DFPT describes vibration proper- ties through a polarizability operator, which characterizes the linear response of the electron density with respect to the perturbation of the external potential. More specifically, in vibration calculations, the polarizability operator needs to be applied to d × NA ∼ O(Ne) perturbation vectors, where d is the spatial dimension (usually d = 3), NA is the number of atoms, and Ne is the number of electrons. In general the complexity for solving KSDFT is O(Ne3), while the complexity for solving DFPT is O(Ne4). It is possible to reduce the computational complexity of DFPT calculations by “linear scaling methods”. Such methods can be successful in reducing the computational cost for systems of large sizes with substantial band gaps, but this can be challenging for medium-sized systems with relatively small band gaps.In the discussion below, we will slightly abuse the term “phonon calculation” to refer to calculation of vibration properties of condensed matter systems as well as isolated molecules. In order to apply the polarizability operator to O(Ne) vectors, we need to solve O(Ne2) coupled Sternheimer equations. On the other hand, when a constant number of degrees of freedom per electron is used, the size of the Hamiltonian matrix is only O(Ne). Hence asymptotically there is room to obtain a set of only O(Ne) “compressed perturbation vectors”, which encodes essentially all the information of the O(Ne2) Sternheimer equations. In this dissertation, we develop a new method called adaptively compressed polarizability operator (ACP) formulation, which successfully reduces the computational complexity of phonon12calculations to O(Ne3) for the first time. The ACP formulation does not rely on exponential decay properties of the density matrix as in linear scaling methods, and its accuracy depends weakly on the size of the band gap. Hence the method can be used for phonon calculations of both insulators and semiconductors with small gaps.There are three key ingredients of the ACP formulation. 1) The Sternheimer equations are equations for shifted Hamiltonians, where each shift corresponds to an energy level of an occupied band. Hence for a general right hand side vector, there are Ne possible energies (shifts). We use a Chebyshev interpolation procedure to disentangle such energy dependence so that there are only constant number of shifts that is independent of Ne. 2) We disentangle the O(Ne2) right hand side vectors using the recently developed interpolative separable density fitting procedure, to compress the right-hand-side vectors. 3) We construct the polarizability by adaptive compression so that the operator remains low rank as well as accurate when applying to a certain set of vectors. This make it possible for fast computation of the matrix inversion using methods like Sherman-Morrison-Woodbury.In particular, the new method does not employ the “nearsightedness” property of electrons for insulating systems with substantial band gaps as in linear scaling methods. Hence our method can be applied to insulators as well as semiconductors with small band gaps.This dissertation also extend the method to deal with nonlocal pseudopotentials as well as systems in finite temperature. Combining all these components together, we obtain an accurate, efficient method to compute the vibrational properties for insulating and metallic systems

A study on Harada Shigeyoshi's Jujireki Chukai (Study of the History of Mathematics 2022)

After being introduced to Japan, the important ancient Chinese calendar, the Shoushi Calendar, was reprinted and disseminated. The Shoushili Yi in Yuan Shi·Li zhi is an important document about the ancient Chinese calendar theory. No one studied it in the Ming and Qing dynasties, but Japanese scholars in the Edo period commented on it, such as Takebe Katahiro (1664-1739), Nishimura Tōsato (1718-1787), and Harada Shigeyoshi (1740-1807), they annotated the Shoushili Yi. The article firstly verifies that the author of the Jujireki Chukai in the library affiliated to Tohoku University is Harada Shigeyoshi, not Takahashi Yoshitoki (1764-1804). Secondly, an investigation was carried out on Harada Shigeyoshi and his writings. The investigation found that there were three manuscripts of Harada Shigeyoshi's Jujireki Chukai, and the contents of the annotations and knowledge sources were verified and sorted out. It is believed that the Jujireki Chukai cited the contents of Tianwen Tujie Fahui (Nakane Genkei), Lisuan Quanshu (Mei Wending) and Juji Kai (Nishimura Tōsato) mostly. Finally, the article analyzes the annotations on “Yanqi (Collect or modify data for the solar terms)” and “Buyong Jinian Rifa (Abolition of the calendar epoch)” in Harada Shigeyoshi's Jujireki Chukai, and thinks that Harada's annotations in “Yanqi” through diagrams are commendable. The “Buyong Jinian Rifa” section is rich in annotations, which supplement the three possible situations that Li Qian and Qi Lvqian proposed to calculate Yanji Shangyuan. The two new situations which do not provide calculation procedures are similar to the methods of Li Qian and Qi Lvqian, and the other two situations are caculated by Seki Takakazu's Jianguan-Method. This method is essentially the same as that of Dayan-Zongshu-Method (Da-yan Rule) [大衍總數術]