A comment on a controversial issue: A generalized fractional derivative cannot have a regular kernel

The problem whether a given pair of functions can be used as the kernels of a generalized fractional derivative and the associated generalized fractional integral is reduced to the problem of existence of a solution to the Sonine equation. It is shown for some selected classes of functions that a necessary condition for a function to be the kernel of a fractional derivative is an integrable singularity at 0. It is shown that locally integrable completely monotone functions satisfy the Sonine equation if and only if they are singular at 0.

Ideal structure and factorization properties of the regular kernel operators

We consider the structure of the lattice of (order and algebra) ideals of the band of regular kernel operators on $$L^p$$ L p -spaces. We show, in particular, that for any $$L^p(\mu )$$ L p ( μ ) space, with $$\mu $$ μ $$\sigma $$ σ -finite and $$1<p<\infty $$ 1 < p < ∞ , the norm-closure of the ideal of finite-rank operators on $$L^p(\mu )$$ L p ( μ ) , is the only non-trivial proper closed (order and algebra) ideal of this band. Key to our results in the $$L^p$$ L p setting is the fact that every regular kernel operator on an $$L^p(\mu )$$ L p ( μ ) space ($$\mu $$ μ and p as before) factors with regular factors through $$\ell _p$$ ℓ p . We show that a similar but weaker factorization property, where $$\ell _p$$ ℓ p is replaced by some reflexive purely atomic Banach lattice, characterizes the regular kernel operators from a reflexive Banach lattice with weak order unit to a KB-space with weak order unit.

An Unsupervised Method to Detect Rock Glacier Activity by Using Sentinel-1 SAR Interferometric Coherence: A Regional-Scale Study in the Eastern European Alps

Rock glaciers are widespread periglacial landforms in mountain regions like the European Alps. Depending on their ice content, they are characterized by slow downslope displacement due to permafrost creep. These landforms are usually mapped within inventories, but understand their activity is a very difficult task, which is frequently accomplished using geomorphological field evidences, direct measurements, or remote sensing approaches. In this work, a powerful method to analyze the rock glaciers’ activity was developed exploiting the synthetic aperture radar (SAR) satellite data. In detail, the interferometric coherence estimated from Sentinel-1 data was used as key indicator of displacement, developing an unsupervised classification method to distinguish moving (i.e., characterized by detectable displacement) from no-moving (i.e., without detectable displacement) rock glaciers. The original application of interferometric coherence, estimated here using the rock glacier outlines as boundaries instead of regular kernel windows, allows describing the activity of rock glaciers at a regional-scale. The method was developed and tested over a large mountainous area located in the Eastern European Alps (South Tyrol and western part of Trentino, Italy) and takes into account all the factors that may limit the effectiveness of the coherence in describing the rock glaciers’ activity. The activity status of more than 1600 rock glaciers was classified by our method, identifying more than 290 rock glaciers as moving. The method was validated using an independent set of rock glaciers whose activity is well-known, obtaining an accuracy of 88%. Our method is replicable over any large mountainous area where rock glaciers are already mapped and makes it possible to compensate for the drawbacks of time-consuming and subjective analysis based on geomorphological evidences or other SAR approaches.

ESTIMASI DENSITAS KERNEL ADJUSTED: STUDI SIMULASI

Kernel adjusted density estimation is a modification of the regular kernel density estimation. The modification is applied to a kernel function. This kernel function is derived from the location-scale transformation. Simulation study shows that this estimation have better results than the regular estimation because it has smaller MSE value. In addition, if normal kernel is used as a kernel function then the curve estimation will be smoother than other kernel function such as uniform kernel and Epachenikov kernel.