On the local fractional derivative

We present the necessary conditions for the existence of the Kolwankar-Gangal local fractional derivatives (KG-LFD) and introduce more general but weaker notions of LFDs by using limits of certain integral averages of the difference-quotient. By applying classical results due to Stein and Zygmund (1965) [16] we show that the KG-LFD is almost everywhere zero in any given intervals. We generalize some of our results to higher dimensional cases and use integral approximation formulas obtained to design numerical schemes for detecting fractional dimensional edges in signal processing.     

A two-well structure and intrinsic mountain pass points

Under small dead-load perturbations, and the natural boundary value condition (Neumann problem), we establish the existence of an unstable critical point (mountain pass point) for a variational integral with a two-well structure. The integrands we consider are obtained by the quasiconvex relaxation [18] of the squared distance function and its quasiconvex lower bounds. The models are m otivated by the variational approach to material microstructure when the wells are incompatible. We show that these functions give quasimonotone gradient mappings. We introduce the weak Palais-Smale condition (weak PS) to deal with the lack of compactness in the borderline case where the integrand is . Received March 1, 1999 / Accepted March 29, 2000 / Published online December 8, 2000     

离子束增强沉积制备Cr—N薄膜及其摩擦学性能研究

利用离子束增强沉积（ＩＢＥＤ）技术制备了Ｃｒ－Ｎ薄膜，并对不同能量氮离子轰击所制备的薄膜组成进行了Ｘ射线衍射及光电子能谱分析，测定了涂层的断裂性值，并对涂层的摩擦学性能进行了研究。结果表明：在相同试验条件下，氮离子轰击能量影响Ｃｒ－Ｎ薄膜的相组成及取向，采用低能量氮离子轰击所得到的薄膜具有较高的断裂韧性和优异的耐磨性能。     

Phosphorescent iridium(III) complex with an N^O ligand as a Hg(2+)-selective chemodosimeter and logic gate

Phosphorescent iridium(III) complexes have been attracting increasing attention in applications as luminescent chemosensors. However, no instance of an iridium(III) complex being used as a molecular logic gate has hitherto been reported. In the present study, two iridium(III) complexes, [Ir(ppy)(2)(PBT)] and [Ir(ppy)(2)(PBO)], have been synthesized (PBT, 2-(2-Hydroxyphenyl)-benzothiazole; PBO, 2-(2-hydroxyphenyl)-benzoxazole), and their chemical structures have been characterized by single-crystal X-ray analysis. Theoretical calculations and detailed studies of the photophysical and electrochemical properties of these two complexes have shown that the N^O ligands dominate their luminescence emission properties. Moreover, [Ir(ppy)(2)(PBT)], containing a sulfur atom in the N^O ligand, can serve as a highly selective chemodosimeter for Hg(2+) with ratiometric and naked-eye detection, which is associated with the dissociation of the N^O ligand PBT from the complex. Furthermore, complex [Ir(ppy)(2)(PBT)] has been further developed as an AND and INHIBIT logic gate with Hg(2+) and histidine as inputs.     

30-Membered decanuclear manganese metallacrown

A new macrocyclic decanuclear manganese(III) 30-metallacrowns-10, [Mn₁₀(ipbmshz)₈(dmpmshz)₂(DMF)₁₀]. 4DMF (1) has been prepared by supramolecular self-assembly and characterized by X-ray crystal diffraction, where ipbmshz³⁻ is N-(4-isopropylbenzoyl)-3-methylsalicylhydrazide, dmpmshz³⁻ is N-(2,2-dimethylpropanoyl)-3-methylsalicylhydrazide. The single-crystal structure shows that a novel ring formed by the succession of ten structural moieties of the type [Mn(III)–N–N] through hydrazide N–N groups bridging the ring Mn ions. The ligand enforces the Mn³⁺ ions to form the stereochemistry of a propeller configuration with alternate…ΔΛΔΛ…-type chiral forms depending on the steric repulsions between the tail groups. The decanuclear systems measure ∼2.3 nm in diameter and ∼1.2 nm in thickness. The temperature-dependent magnetic properties have been studied and showed the presence of weakly antiferromagnetic couplings between Mn (III) ions.     

Mountain Pass Solutions for a Double-Well Energy

We establish the existence of a mountain pass solution for a variational integral involving a quasiconvex function with a double-well structure in the geometrically linear elasticity setting. We show that under small dead-load perturbations, the Neumann boundary value problem has at least three solutions, a global minimizer, a local minimizer and a mountain pass solution. We show that our variational integral satisfies a Weak Palais-Smale condition (WPS) hence the mountain pass lemma applies.     

BMO and uniform estimates for multi-well problems

We establish optimal local regularity results for vector-valued extremals and minimizers of variational integrals whose integrand is the squared distance function to a compact set K in matrix space ${{{\mathbb M}^{N \times n}}}$ . The optimality is illustrated by explicit examples showing that, in the nonconvex case, minimizers need not be locally Lipschitz. This is in contrast to the case when the set K is suitably convex, where we show that extremals are locally Lipschitz continuous. The results rely on the special structure of the integrand and elementary Cordes–Nirenberg type estimates for elliptic systems.