Invariant discrete flows

In this paper, we investigate the evolution of joint invariants under invariant geometric flows using the theory of equivariant moving frames and the induced invariant discrete variational complex. For certain arc length preserving planar curve flows invariant under the special Euclidean group , the special linear group , and the semidirect group , we find that the induced evolution of the discrete curvature satisfies the differential‐difference mKdV, KdV, and Burgers' equations, respectively. These three equations are completely integrable, and we show that a recursion operator can be constructed by precomposing the characteristic operator of the curvature by a certain invariant difference operator. Finally, we derive the constraint for the integrability of the discrete curvature evolution to lift to the evolution of the discrete curve itself.     

Existence of the uniformly minimum risk equivariant estimators of parameters in a class of normal linear models

In this paper, we study the existence of the uniformly minimum risk equivariant (UMRE) estimators of parameters in a class of normal linear models, which include the normal variance components model, the growth curve model, the extended growth curve model, and the seemingly unrelated regression equations model, and so on. The necessary and sufficient conditions are given for the existence of UMRE estimators of the estimable linear functions of regression coefficients, the covariance matrixV and (trV)^α, where α > 0 is known, in the models under an affine group of transformations for quadratic losses and matrix losses, respectively. Under the (extended) growth curve model and the seemingly unrelated regression equations model, the conclusions given in literature for estimating regression coefficients can be derived by applying the general results in this paper, and the sufficient conditions for non-existence of UMRE estimators ofV and tr(V) are expanded to be necessary and sufficient conditions. In addition, the necessary and sufficient conditions that there exist UMRE estimators of parameters in the variance components model are obtained for the first time.     

Chern and Pontryagin Numbers in Perfect Symmetries of Spheres

The paper presents a procedure for constructing smooth actions of finite perfect groups on spheres with fixed point sets having certain prescribed properties (Theorem A); in particular, having any prescribed configuration of Chern and Pontryagin numbers (Corollary C). The main ingredients used are equivariant thickening and equivariant surgery.     

Orbifolds, Sheaves and Groupoids

We characterize orbifolds in terms of their sheaves, and show that orbifolds correspond exactly to a specific class of smooth groupoids. As an application, we construct fibered products of orbifolds and prove a change-of-base formula for sheaf cohomology.     

Sparse generalized Fourier transforms

Block-diagonalization of sparse equivariant discretization matrices is studied. Such matrices typically arise when partial differential equations that evolve in symmetric geometries are discretized via the finite element method or via finite differences. By considering sparse equivariant matrices as equivariant graphs, we identify a condition for when block-diagonalization via a sparse variant of a generalized Fourier transform (GFT) becomes particularly simple and fast. Characterizations for finite element triangulations of a symmetric domain are given, and formulas for assembling the block-diagonalized matrix directly are presented. It is emphasized that the GFT preserves symmetric (Hermitian) properties of an equivariant matrix. By simulating the heat equation at the surface of a sphere discretized by an icosahedral grid, it is demonstrated that the block-diagonalization is beneficial. The gain is significant for a direct method, and modest for an iterative method. A comparison with a block-diagonalization approach based upon the continuous formulation is made. It is found that the sparse GFT method is an appropriate way to discretize the resulting continuous subsystems, since the spectrum and the symmetry are preserved. AMS subject classification (2000)  43A30, 65T99, 20B25     

Rheological behaviour and shear thickening exhibited by aqueous CTAB micellar solutions

In this experimental work we carefully investigate the rheological behaviour and in particular the shear thickening exhibited by aqueous micellar solutions of CTAB with NaSal as added counterion. We are particularly interested in the evolution of the critical shear rate (at which shear thickening occurs) versus C _D , the surfactant concentration. We show that , at fixed salt concentration C _S , increases with C _D following a power law evolution with a positive exponent of + 5.8. On the other hand we show that if the ratio C _D / C _S is fixed, decreases with C _D with a negative exponent of -2.0. Nevertheless investigations of the zero shear viscosity indicate that in all situations (implying variation of the surfactant concentration C _D , or the salt concentration C _S or the temperature) is a decreasing function of the length of the micelles. All these evolutions are compatible with a gelation mechanism which could possibly be associated with entanglement effects of large interacting flowing structures. Received: 3 March 1998 / Revised: 16 June 1998 / Accepted: 3 July 1998     

疏水缔合型增稠剂聚(丙烯酰胺-丙烯酸高级酯)的合成及增稠性能研究

采用沉淀聚合法合成了聚（丙烯酰胺－丙烯酸十四酯），聚（丙烯酰胺－丙烯酸十六酯）和聚（丙烯酰胺－丙烯酸十八酯），重点讨论了共聚物中疏水基团数量及疏水基团链长对增稠性能的影响。采用凯达尔定氮法和动态热分析（DMTA）测定了共聚物的组成及玻璃化温度。     

Quality by Design Approach for Development of W/O Type Microemulsion-Based Transdermal Systems for Atenolol

The objective of the present investigation was to develop microemulsion-based transdermal systems of highly water soluble drug, Atenolol, by quality by design technique. Atenolol-loaded W/O microemulsions were optimized using D-optimal design with concentrations of oil, surfactants mixture, and water as independent variables, which was converted into microemulsion-based gel (MBG). The results of in vitro permeation of the optimized batch of Atenolol-loaded MBG revealed significant increase in permeability parameters as compared to its convention gel. All results suggested suitability of W/O type MEs as carriers for transdermal delivery of highly water soluble drug, Atenolol.     

Concerted effect of boron and porosity on shear thickening behavior of hybrid mesoporous silica dispersions

In the present work, the influence of porosity and boron on shear thickening behavior of hybrid mesoporous silica has been studied. Three different levels of boron modification were performed by varying the molar composition of boric acid viz., 1.5 mmol, 2.5 mmol, and 3.5 mmol in a co-condensation approach. The incorporation of boron in mesoporous silica network was confirmed by various techniques such as Fourier transform infra-red (FTIR), and ¹¹B solid- state nuclear magnetic resonance (NMR) spectroscopy. The morphology and particle size were confirmed by using scanning and transmission electron microscopy. To evaluate the effect of boron and porosity on the shear thickening behavior, dispersions were prepared from mesoporous boron- modified silica (MSiB), control mesoporous silica (MSi), non-porous boron-modified silica (SiB), and control non-porous silica (Si) in polyethylene glycol. The shear thickening behavior was studied using steady shear rheology. The dispersion prepared from different loadings of synthesized MSiB containing 1.5 mmol boron showed more than 16 times increase in viscosity (657.7 Pa.s) compared to that of MSi (39.2 Pa.s) at a fairly low volume fraction (φ = 0.15) of silica. It is expected that the highly ordered mesoporous architecture of hybrid silica has improved the interaction between the particle and the dispersing medium through hydrogen bonding. The porous morphology of the hybrid mesoporous silica as well as the incorporation of boron in the silica network favors the formation of a frictional contact network, and a transition from continuous shear thickening (CST) to discontinuous shear thickening (DST) behavior was observed. Therefore, silica prepared via incorporation of boron as well as porosity can be material of interest in variety of applications, for example, soft body armors, sporting goods, and shear thickening electrolytes for high impact resistant batteries.