Subdifferential representation of homogeneous functions and extension of smoothness in Banach spaces

In this paper, we mainly consider proximal subdifferentials of lower semicontinuous functions defined on real Hilbert space and Clarke＇s subdifferentials of locally Lipschitzian functions defined on Banach space respectively, and obtain the generalized Euler identity of homogenous functions. Then, by introducing a multifunction F, we extend the smoothness of sphere and differentiability of norm function in Banach space.     

A survey on Büchi’s problem: new presentations and open problems

In a commutative ring with a unit, a Büchi sequence is a sequence such that the second difference of the sequence of its squares is the constant sequence (2). Sequences of elements x _n satisfying x _n ² = (x + n)² for some fixed x are Büchi sequences, which we call trivial. Since we want to study sequences whose elements do not belong to certain subrings (e.g., for fields of rational functions F(z) over a field F, we are interested in sequences that are not over F), the concept of trivial sequences may vary. Büchi’s problem for a ring asks whether there exists a positive integer M such that any Büchi sequence of length M or more is trivial.     

Cycle Lemma, Parking Functions and Related Multigraphs

For positive integers a and b, an ${(a, \overline{b})}$ -parking function of length n is a sequence (p ₁, . . . , p _n ) of nonnegative integers whose weakly increasing order q ₁ ≤ . . . ≤ q _n satisfies the condition q _i  < a + (i ? 1)b. In this paper, we give a new proof of the enumeration formula for ${(a, \overline{b})}$ -parking functions by using of the cycle lemma for words, which leads to some enumerative results for the ${(a, \overline{b})}$ -parking functions with some restrictions such as symmetric property and periodic property. Based on a bijection between ${(a, \overline{b})}$ -parking functions and rooted forests, we enumerate combinatorially the ${(a, \overline{b})}$ -parking functions with identical initial terms and symmetric ${(a, \overline{b})}$ -parking functions with respect to the middle term. Moreover, we derive the critical group of a multigraph that is closely related to ${(a, \overline{b})}$ -parking functions.     

Weighted Hardy operators and commutators on Morrey spaces

The operator norms of weighted Hardy operators on Morrey spaces are worked out. The other purpose of this paper is to establish a sufficient and necessary condition on weight functions which ensures the boundedness of the commutators of weighted Hardy operators (with symbols in BMO(ℝ ⁿ )) on Morrey spaces.     

The effective charge effect in partially stripped ion-helium collisions

The double and single ionization cross section ratios of helium by partially stripped carbon, oxygen and fluorine ions are measured for projectile charge states ranging from +1 to +4 and impact energies from 1.5 MeV to 7.5 MeV. The effective charge effect in partially stripped ion-helium collisions is studied. It is found that the effective charge q_eff increases as the impinging energy increases and q _eff shows a modest dependence upon the projectile charge state in the present energy range. The projectile charge state, projectile energy, projectile and target electronic state dependences of the effective charge effect may be explained using orbital interpenetrating.     

Preparation and characterization of ferroelectric SrBi2Ta2O9 thin films on Si using Al2O3 buffer layers

SrBi₂Ta₂O₉ (SBT) thin films were prepared on p-type Si(100) substrates with Al₂O₃ buffer layers. Both the SBT films and the Al₂O₃ buffer layers were deposited by a pulsed laser deposition technique using a KrF excimer laser. An Al prelayer was used to prevent Si surface oxidization in the initial growth stage. It is shown that Al₂O₃ buffer layers effectively prevented interdiffusion between SBT and Si substrates. Furthermore, the capacitance–voltage (C-V) characteristics of the SBT/Al₂O₃/Si heterostructures show a hysteresis loop with a clockwise trace, demonstrating the ferroelectric switching properties of SBT films and showing a memory window of 1.6 V at 1 MHz. Received: 17 July 2000 / Accepted: 16 August 2000 / Published online: 30 November 2000     

Generalized Variational-Like Inequalities with Nonmonotone Set-Valued Mappings

In this paper, we introduce a new class of generalized variational-like inequalities involving a nonmonotone-type set-valued mapping which is more general than those in the known literature. By applying a KKM-type theorem established by the author, some existence theorems for the solutions to the class of generalized variational-like inequalities are proved in Hausdorff locally convex topological vector spaces.     

Microstructures of grain boundaries in (001)-oriented strontium bismuth tantalate thin films grown by pulsed laser deposition

(001)-oriented strontium bismuth tantalate thin films have been grown on Pt/TiO₂/SiO₂/Si (100) substrates by pulsed laser deposition. The room-temperature current–electric field dependence of the films has been investigated, which revealed a space-charge-limited conduction mechanism. The microstructures of grain boundaries and structural defects in these films were also examined by transmission electron microscopy and high-resolution transmission electron microscopy, respectively. The grains of the films deposited at 550 °C exhibited polyhedral morphologies, and the average grain size was about 50 nm in length and 35 nm in width. At a small misorientation angle (8.2°) tilt boundary, a regular array of edge dislocations with about 3-nm periodic distance was observed, and localized strain contrast near the dislocation cores was also observed. The Burgers vector b of the edge dislocation was determined to be [110]. At a high misorientation angle (39.0°) tilt grain boundary lattice strain contrast associated with the distortion of lattice planes was observed, and the mismatching lattice images occurred at about 2 nm along the boundary. The relationship between microstructural defects at grain boundaries and leakage currents of these films is also discussed. Received: 8 September 2000 / Accepted: 18 December 2000 / Published online: 28 February 2001     

Synthesis and Raman scattering of GaN nanorings, nanoribbons and nanowires

Low-dimensional GaN materials, including nanorings, nanoribbons and smooth nanowires have been synthesized by reacting gallium and ammonia using Ag particles as a catalyst on the substrate of MgO single crystals. They were characterized by field emission scanning electron microscopy (FE-SEM), energy dispersive X-ray spectroscopy (EDX) and X-ray diffraction (XRD). EDX, XRD indicated that the low-dimensional nanomaterials were wurtzite GaN. New features are found in Raman scatterings for these low-dimensional GaN materials, which are different from the previous observations of GaN materials. Received: 16 November 2000 / Accepted: 17 November 2000 / Published online: 21 March 2001     

Rupture energy of a pendular liquid bridge

We propose a simple expression for the rupture energy of a pendular liquid bridge between two spheres, taking into account capillary and viscous (lubrication) forces. In the case of capillary forces only, the results are in accordance with curve fitting expressions proposed by Simons et al. [2] and Willett et al. [5]. We performed accurate measurements of the force exerted by liquid bridges between two spheres. Experimental results are found to be close to theoretical values. A reasonable agreement is also found in the presence of viscous forces. Finally, for small bridge volumes, the rupture criterion given by Lian et al. [10] is modified, taking into account additional viscous effects. Received 18 September 2000 and Received in final form 10 June 2001