Cyclic Hypomonotonicity,Cyclic Submonotonicity,and Integration

Rockafellar has shown that the subdifferentials of convex functions are always cyclically monotone operators. Moreover, maximal cyclically monotone operators are necessarily operators of this type, since one can construct explicitly a convex function, which turns out to be unique up to a constant, whose subdifferential gives back the operator. This result is a cornerstone in convex analysis and relates tightly convexity and monotonicity. In this paper, we establish analogous robust results that relate weak convexity notions to corresponding notions of weak monotonicity, provided one deals with locally Lipschitz functions and locally bounded operators. In particular, the subdifferentials of locally Lipschitz functions that are directionally hypomonotone [respectively, directionally submonotone] enjoy also an additional cyclic strengthening of this notion and in fact are maximal under this new property. Moreover, every maximal cyclically hypomonotone [respectively, maximal cyclically submonotone] operator is always the Clarke subdifferential of some directionally weakly convex [respectively, directionally approximately convex] locally Lipschitz function, unique up to a constant, which in finite dimentions is a lower C² function [respectively, a lower C¹ function].     

Peak effect in polycrystalline vortex matter

We report a study of the peak-effect phase diagram of a strongly disordered type-II superconductor V-21 at. %Ti using ac magnetic susceptibility and small-angle neutron scattering (SANS). In this system, the peak effect appears only at fields higher than 3.4 T. The sample is characterized by strong atomic disorder. Vortex states with field-cooled thermal histories show that both deep in the mixed state, as well as close to the peak effect, there exist no long-range orientationally ordered vortex lattices. The SANS scattering radial widths reveal vortex states ordered in the sub-mum scale. We conjecture that the peak effect in this system is a remnant of the Bragg glass disordering transition, but occurs on submicron length scales due to the presence of strong atomic disorder on larger length scales.     

Encapsulation of the corrosion inhibitor 8-hydroxyquinoline into ceria nanocontainers

Ceria nanocontainers were synthesized through a two-step process and then loaded with 8-hydroxyquinoline (8-HQ). The size of the containers was 110 nm as determined by scanning electron microscopy (SEM) and transmission electron microscopy (TEM). X-ray diffraction analysis (XRD) showed that the ceria nanocontainers were of the cerianite crystalline phase. The presence of 8-HQ in the nanocontainers was confirmed with Fourier-transform infrared spectroscopy (FT-IR). The loading of the inhibitor in the nanocontainers was estimated with differential thermal analysis (DTA) and thermogravimetric analysis (TGA). The loading amount of 8-HQ was 4.28% w/w. Based on the size of the nanocontainers and the assumption that they are not broken, we deduced that there were approximately 6.0 × 10⁵ molecules of 8-HQ per container. Furthermore, release of 8-HQ in a corrosive environment was studied by potentiodynamic measurements, showing that the inhibitor is released from the nanocontainers, suppressing the corrosion activities by a strong barrier effect. SEM and dynamic light scattering (DLS) measurements confirmed that the nanocontainers are not significantly agglomerated and maintain their shape after suspension in 0.5 M NaCl solution for more than 72 h.     

Sard theorems for Lipschitz functions and applications in optimization

We establish a “preparatory Sard theorem” for smooth functions with a partially affine structure. By means of this result, we improve a previous result of Rifford [17, 19] concerning the generalized (Clarke) critical values of Lipschitz functions defined as minima of smooth functions. We also establish a nonsmooth Sard theorem for the class of Lipschitz functions from R^d to R^p that can be expressed as finite selections of C^k functions (more generally, continuous selections over a compact countable set). This recovers readily the classical Sard theorem and extends a previous result of Barbet–Daniilidis–Dambrine [1] to the case p > 1. Applications in semi-infinite and Pareto optimization are given.     

Coercivity conditions and variational inequalities

Received November 17, 1997 / Revised version received August 6, 1998?Published online March 16, 1999     

含油岩心显微荧光成像光谱研究

发展了一种显微荧光光谱成像技术,并将其应用于天然岩心进行显微荧光成像光谱研究。利用这种技术同时采集含油岩心表面的荧光光谱信息和空间信息．并对获得的光谱图像给予光谱学和地质学解释。结果表明,不但能显示岩心形貌和组分的大致趋势,而且能揭示其精细细节．为石油地质研究提供了一种新方法,为今后的石油勘探开发工作提供了一种先进的指导手段。     

Approximate convexity and submonotonicity

It is shown that a locally Lipschitz function is approximately convex if, and only if, its Clarke subdifferential is a submonotone operator. Consequently, in finite dimensions, the class of locally Lipschitz approximately convex functions coincides with the class of lower-C¹ functions. Directional approximate convexity is introduced and shown to be a natural extension of the class of lower-C¹ functions in infinite dimensions. The following characterization is established: a multivalued operator is maximal cyclically submonotone if, and only if, it coincides with the Clarke subdifferential of a locally Lipschitz directionally approximately convex function, which is unique up to a constant. Furthermore, it is shown that in Asplund spaces, every regular function is generically approximately convex.     

Some Remarks on the Class of Continuous (Semi-) Strictly Quasiconvex Functions

We introduce the notion of variational (semi-) strict quasimonotonicity for a multivalued operator T  : X⇉X ^* relative to a nonempty subset A of X which is not necessarily included in the domain of T. We use this notion to characterize the subdifferentials of continuous (semi-) strictly quasiconvex functions. The proposed definition is a relaxation of the standard definition of (semi-) strict quasimonotonicity, the latter being appropriate only for operators with nonempty values. Thus, the derived results are extensions to the continuous case of the corresponding results for locally Lipschitz functions.     

The Morse–Sard theorem for Clarke critical values

The Morse–Sard theorem states that the set of critical values of a C^k$C^k$ smooth function defined on a Euclidean space R^d${\mathbb{R}}^d$ has Lebesgue measure zero, provided k≥d$k\ge d$. This result is hereby extended for (generalized) critical values of continuous selections over a compactly indexed countable family of C^k$C^k$ functions: it is shown that these functions are Lipschitz continuous and the set of their Clarke critical values is null.