Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data

We consider the defocusing, $\dot{H}^1$-critical Hartree equation for the radial data in all dimensions $(n\geq 5)$. We show the global well-posedness and scattering results in the energy space. The new ingredient in this paper is that we first take advantage of the term $\displaystyle - \int_{I}\int_{|x|\leq A|I|^{1/2}}|u|^{2}\Delta \Big(\frac{1}{|x|}\Big)dxdt$ in the localized Morawetz identity to rule out the possibility of energy concentration, instead of the classical Morawetz estimate dependent of the nonlinearity.Comment: 23 pages, 1 figur

Precise Phase Transition of Total Variation Minimization

Characterizing the phase transitions of convex optimizations in recovering structured signals or data is of central importance in compressed sensing, machine learning and statistics. The phase transitions of many convex optimization signal recovery methods such as $\ell_1$ minimization and nuclear norm minimization are well understood through recent years' research. However, rigorously characterizing the phase transition of total variation (TV) minimization in recovering sparse-gradient signal is still open. In this paper, we fully characterize the phase transition curve of the TV minimization. Our proof builds on Donoho, Johnstone and Montanari's conjectured phase transition curve for the TV approximate message passing algorithm (AMP), together with the linkage between the minmax Mean Square Error of a denoising problem and the high-dimensional convex geometry for TV minimization.Comment: 6 page

A brief analysis of introducing curriculum ideological and political elements into "Mapping Practice of Auto Parts"

The article focuses on the new requirements of integrating curriculum ideological and political elements into the professional practice course of "Automotive Parts Mapping Practice", proposes feasible curriculum ideological and political elements, establishes the course teaching concept of ideological and political element integration, and discusses the new teaching mode different from the traditional practice course, striving to integrate professional knowledge education, ideological and political education, and students\u27 practical ability training. The course aims to integrate professional knowledge education, ideological and political education and the cultivation of students\u27 practical skills, and comprehensively promote the construction of the teaching content, curriculum system, teaching methods, teaching means, teaching conditions and teaching staff of Mapping Practice of Auto Parts, so as to lay a solid practical foundation for the revitalization of Chinese automotive industry and the cultivation of China\u27s own manufacturing technology engineers for the first line of automotive production

Blowup dynamics for smooth equivariant solutions to energy critical Landau-Lifschitz flow

In this paper, we study the energy critical equivariant Landau-Lifschitz flow with target manifold $\mathbb{S}^2$. We prove that there exists a codimension one smooth well-localized set of initial data which generates finite-time type-II blowup solutions, and then give a precise description of the corresponding singularity formation. In our proof, both the Schr\"odinger part and the heat part play important roles in the construction of approximate solutions and the mixed Energy-Morawetz arguments. However, the blowup rate is independent of their coefficients.Comment: 84 page

Human Rif1 protein binds aberrant telomeres and aligns along anaphase midzone microtubules

We identified and characterized a human orthologue of Rif1 protein, which in budding yeast interacts in vivo with the major duplex telomeric DNA binding protein Rap1p and negatively regulates telomere length. Depletion of hRif1 by RNA interference in human cancer cells impaired cell growth but had no detectable effect on telomere length, although hRif1 overexpression in S. cerevisiae interfered with telomere length control, in a manner specifically dependent on the presence of yeast Rif1p. No localization of hRif1 on normal human telomeres, or interaction with the human telomeric proteins TRF1, TRF2, or hRap1, was detectable. However, hRif1 efficiently translocated to telomerically located DNA damage foci in response to the synthesis of aberrant telomeres directed by mutant-template telomerase RNA. The hRif1 level rose during late S/G2 but hRif1 was not visible on chromosomes in metaphase and anaphase; however, notably, specifically during early anaphase, hRif1 aligned along a subset of the midzone microtubules between the separating chromosomes. In telophase, hRif1 localized to chromosomes, and in interphase, it was intranuclear. These results define a novel subcellular localization behavior for hRif1 during the cell cycle

Global well-posedness and scattering for the mass-critical Hartree equation with radial data

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Hartree equation $iu_t+\Delta u=\pm(|x|^{-2}*|u|^2)u$ for large spherically symmetric $L^2_x(\Bbb{R}^d)$ initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state.Comment: 38 pages, 1 figur