富里叶级数的收敛速度

对正弦和余弦富立叶级数,通过合并相邻同号项,使其重排成交错级数.讨论了重排形成的交错级数的敛散性.指出根据自变量x的不同取值,该交错级数可能是单调递减或周期递减的级数.按照莱布尼茨判定法提出了不同精度要求的级数项数的计算公式.选取一到三阶收敛的富立叶级数计算了不同比值精度及差值精度要求的级数项数.计算表明,在x的取值为2π的等分点时,富立叶级数的部分和随项数的增加单调地逼近其收敛值.在x的取值为其它点时,富立叶级数的部分和随项数的增加围绕收敛值上下变动,周期地逼近其收敛值.低收敛阶富立叶级数的收敛速度较慢.要达到0.01%的精度,一收敛阶富立叶级数需要数万项,二收敛阶富立叶级数也需要数百项.在不同计算点处,要达到相同的计算精度,需要的级数项数差别较大.     

Low‐Magnetization Magnetic Microcapsules: A Synergistic Theranostic Platform for Remote Cancer Cells Therapy and Imaging

Multifunctional magnetic microcapsules (MMCs) for the combined cancer cells hyperthermia and chemotherapy in addition to MR imaging are successfully developed. A classical layer‐by‐layer technique of oppositely charged polyelectrolytes (poly(allylamine hydrochloride) (PAH) and poly(4‐styrene sulfonate sodium) (PSS)) is used as it affords great controllability over the preparation together with enhanced loading of the chemotherapeutic drug (doxorubicin, DOX) in the microcapsules. Superparamagnetic iron oxide (SPIOs) nanoparticles are layered in the system to afford MMC1 (one SPIOs layer) and MMC2 (two SPIOs layers). Most interestingly, MMC1 and MMC2 show efficient hyperthermia cell death and controlled DOX release although their magnetic saturation value falls below 2.5 emu g^?1, which is lower than the 7–22 emu g^?1 reported to be the minimum value needed for biomedical applications. Moreover, MMCs are pH responsive where a pH 5.5 (often reported for cancer cells) combined with hyperthermia increases DOX release predictably. Both systems prove viable when used as T2 contrast agents for MR imaging in HeLa cells with high biocompatibility. Thus, MMCs hold a great promise to be used commercially as a theranostic platform as they are controllably prepared, reproducibly enhanced, and serve as drug delivery, hyperthermia, and MRI contrast agents at the same time.     

An Unprecedented Two‐Fold Nested Super‐Polyrotaxane: Sulfate‐Directed Hierarchical Polythreading Assembly of Uranyl Polyrotaxane Moieties

The hierarchical assembly of well‐organized submoieties could lead to more complicated superstructures with intriguing properties. We describe herein an unprecedented polyrotaxane polythreading framework containing a two‐fold nested super‐polyrotaxane substructure, which was synthesized through a uranyl‐directed hierarchical polythreading assembly of one‐dimensional polyrotaxane chains and two‐dimensional polyrotaxane networks. This special assembly mode actually affords a new way of supramolecular chemistry instead of covalently linked bulky stoppers to construct stable interlocked rotaxane moieties. An investigation of the synthesis condition shows that sulfate can assume a vital role in mediating the formation of different uranyl species, especially the unique trinuclear uranyl moiety [(UO₂)₃O(OH)₂]²⁺, involving a notable bent [O=U=O] bond with a bond angle of 172.0(9)°. Detailed analysis of the coordination features, the thermal stability as well as a fluorescence, and electrochemical characterization demonstrate that the uniqueness of this super‐polyrotaxane structure is mainly closely related to the trinuclear uranyl moiety, which is confirmed by quantum chemical calculations.     

Identifying a dot-matrix hologram with the fringe-overlapping phenomena of grating dots

Dot-matrix holograms created by two-beam writers contain many grating dots. Because the phase difference between two laser beams for interference cannot be controlled accurately, the fringe positions of grating dots are randomly determined. Therefore, fringe positions are a good kind of tool to identify dot-matrix holograms. In this paper, a number difference between two special fringes of a target grating dot is used to identify a dot-matrix hologram. The two special fringes are determined by three grating dots with parallel fringes. The first special fringe is corresponding to a fringe pair with the best matching for the fringes of the target grating dot and the fringes of the second grating dot. The second special fringe is corresponding to a fringe pair with the best matching for the fringes of the target grating dot and the fringes of the third grating dot. An experiment has proved the proposed method practical and feasible. Because reproducing a grating dot with a specified fringe number difference is difficult, the proposed method is excellent for anti-counterfeiting.     

Enantioselective Self‐Assembly of Triangular Dy3 Clusters with Single‐Molecule Magnet Behavior

Three pairs of enantiopure chiral triangular Ln₃ clusters, [Ln₃L_{RRRRRR/SSSSSS}(μ₃‐OH)₂(H₂O)₂(SCN)₄]?xCH₃OH?yH₂O ( R ‐Dy₃ , Ln=Dy, x=6, y=0; S ‐Dy₃ , Ln=Dy, x=6, y=1; R ‐Ho₃ , Ln=Ho, x=6, y=1; S ‐Ho₃ , Ln=Ho, x=6, y=1; R ‐Er₃ , Ln=Er, x=6, y=0; S ‐Er₃ , Ln=Er, x=6, y=1), have been successfully synthesized by a rational enantioselective synthetic strategy. The core of triangular Ln₃ is bound in the central N₆O₃ of the macrocyclic ligand, and the coordination spheres of Ln ions are completed by four SCN^? anions and two H₂O molecules in axial positions of the macrocycle. The circular dichroism (CD) and vibrational circular dichroism (VCD) spectra of the enantiomers demonstrate that the chirality is successfully transferred from the ligands to the resulting Ln₃ clusters. Ac susceptibility measurements reveal that single‐molecule magnet behavior occurs for both enantiopure clusters of R ‐Dy₃ and S ‐Dy₃ . This work is one of the few examples of the successful design of a pair of triangular Dy₃ clusters showing simultaneously slow magnetic relaxation and optical activity, and this might open up new opportunities to develop novel multifunctional materials.