On the rate of change of the sharp constant in the Sobolev–Poincaré inequality

The rate of change of the sharp constant in the Sobolev–Poincaré or Friedrichs inequality is estimated for a Euclidean domain that moves outward. The key ingredients are a Hadamard variation formula and an inequality that reverses the usual Hölder inequality.     

离散外微分与计算电磁学

计算电磁学的核心之一是数值求解Maxwell方程组.适当的离散方式是保证结果能真实反映物理现象的关键.为了在离散的过程中保持该方程组的几何性质,我们建立了基于棱柱网格的系数为R的格点规范理论,其离散曲率满足相应的Bianchi恒等式.通过适当定义离散微分形式之间的内积和棱柱网格上的Hodge星算子,我们由离散变分导出源方程和连续性方程,和Bianchi恒等式一起称为真空中的离散Maxwell方程组.这组方程是内蕴的,并具有规范不变性.     

向量值测度产生的集值测度及性质

由有界变差向量值测度的值域,通过取凸包和闭包,构造了L[0,1],L2[0,1]和C[0,1]空间上的有界变差紧凸集值测度,结果由欧氏空间推广到函数空间.     

Existence for evolutionary problems with linear growth by stability methods

We establish that solutions to the Cauchy–Dirichlet problem

${?}_{t}u?\mathrm{div}(D_{\xi }f(x,Du))=0$

for functionals $f:\mathrm{\Omega }\times {\mathbb{R}}^{N\times n}\to [0,\infty )$ of linear growth can be obtained as limits of solutions to flows with p-growth in the limit $p\downarrow 1$. The result can be interpreted on the one hand as a stability result. On the other hand it provides an existence result for general flows with linear growth.     

Change of variable formulas for non-anticipative functionals on path space

We derive a change of variable formula for non-anticipative functionals defined on the space of Rd-valued right-continuous paths with left limits. The functionals are only required to possess certain directional derivatives, which may be computed pathwise. Our results lead to functional extensions of the Itô formula for a large class of stochastic processes, including semimartingales and Dirichlet processes. In particular, we show the stability of the class of semimartingales under certain functional transformations.     

Topological regular variation: I. Slow variation

Motivated by the Category Embedding Theorem, as applied to convergent automorphisms (Bingham and Ostaszewski (in press) [11]), we unify and extend the multivariate regular variation literature by a reformulation in the language of topological dynamics. Here the natural setting are metric groups, seen as normed groups (mimicking normed vector spaces). We briefly study their properties as a preliminary to establishing that the Uniform Convergence Theorem (UCT) for Baire, group-valued slowly-varying functions has two natural metric generalizations linked by the natural duality between a homogenous space and its group of homeomorphisms. Each is derivable from the other by duality. One of these explicitly extends the (topological) group version of UCT due to Bajšanski and Karamata (1969) [4] from groups to flows on a group. A multiplicative representation of the flow derived in Ostaszewski (2010) [45] demonstrates equivalence of the flow with the earlier group formulation. In companion papers we extend the theory to regularly varying functions: we establish the calculus of regular variation in Bingham and Ostaszewski (2010) [13] and we extend to locally compact, σ-compact groups the fundamental theorems on characterization and representation (Bingham and Ostaszewski (2010) [14]). In Bingham and Ostaszewski (2009) [15], working with topological flows on homogeneous spaces, we identify an index of regular variation, which in a normed-vector space context may be specified using the Riesz representation theorem, and in a locally compact group setting may be connected with Haar measure.     

A fractional partial differential equation based multiscale denoising model for texture image

Traditional integer‐order partial differential equation based image denoising approach can easily lead edge and complex texture detail blur, thus its denoising effect for texture image is always not well. To solve the problem, we propose to implement a fractional partial differential equation (FPDE) based denoising model for texture image by applying a novel mathematical method—fractional calculus to image processing from the view of system evolution. Previous studies show that fractional calculus has some unique properties that it can nonlinearly enhance complex texture detail in digital image processing, which is obvious different with integer‐order differential calculus. The goal of the modeling is to overcome the problems of the existed denoising approaches by utilizing the aforementioned properties of fractional differential calculus. Using classic definition and property of fractional differential calculus, we extend integer‐order steepest descent approach to fractional field to implement fractional steepest descent approach. Then, based on the earlier fractional formulas, a FPDE based multiscale denoising model for texture image is proposed and further analyze optimal parameters value for FPDE based denoising model. The experimental results prove that the ability for preserving high‐frequency edge and complex texture information of the proposed fractional denoising model are obviously superior to traditional integral based algorithms, as for texture detail rich images. Copyright © 2013 John Wiley & Sons, Ltd.