Maximal operators and differentiation theorems for sparse sets

We study maximal averages associated with singular measures on \rr. Our main result is a construction of singular Cantor-type measures supported on sets of Hausdorff dimension $1 - \epsilon$, $0 \leq \epsilon < {1/3}$ for which the corresponding maximal operators are bounded on $L^p(\mathbb R)$ for $p > (1 + \epsilon)/(1 - \epsilon)$. As a consequence, we are able to answer a question of Aversa and Preiss on density and differentiation theorems in one dimension. Our proof combines probabilistic techniques with the methods developed in multidimensional Euclidean harmonic analysis, in particular there are strong similarities to Bourgain's proof of the circular maximal theorem in two dimensions. Updates: Andreas Seeger has provided an argument to the effect that our global maximal operators are in fact bounded on L^p(R) for all p>1; in particular, it follows that our differentiation theorems are also valid for all p>1. Furthermore, David Preiss has proved that no such differentiation theorems (let alone maximal estimates) can hold with p=1. These arguments are included in the new version. We have also improved the exposition in a number of places.Comment: Revised version. The final version will appear in Duke Math.

Viscosity scaling of fingering instability in finite slices with Korteweg stress

We perform linear stability analyses (LSA) and direct numerical simulations (DNS) to investigate the influence of the dynamic viscosity on viscous fingering (VF) instability in miscible slices. Selecting the characteristic scales appropriately the importance of the magnitude of the dynamic viscosity of individual fluids on VF in miscible slice has been shown in the context of the transient interfacial tension. Further, we have confirmed this result for immiscible fluids and manifest the similarities between VF in immiscible and miscible slices with transient interfacial tension. In a more general setting, the findings of this letter will be very useful for multiphase viscous flow, in which the momentum balance equation contains an additional stress term free from the dynamic viscosity.Comment: 16 pages, 8 figure

Multi-Sensor Image Fusion Based on Moment Calculation

An image fusion method based on salient features is proposed in this paper. In this work, we have concentrated on salient features of the image for fusion in order to preserve all relevant information contained in the input images and tried to enhance the contrast in fused image and also suppressed noise to a maximum extent. In our system, first we have applied a mask on two input images in order to conserve the high frequency information along with some low frequency information and stifle noise to a maximum extent. Thereafter, for identification of salience features from sources images, a local moment is computed in the neighborhood of a coefficient. Finally, a decision map is generated based on local moment in order to get the fused image. To verify our proposed algorithm, we have tested it on 120 sensor image pairs collected from Manchester University UK database. The experimental results show that the proposed method can provide superior fused image in terms of several quantitative fusion evaluation index.Comment: 5 pages, International Conferenc