GPM DPR Profile Classification Algorithm: Enhancement from V6X to V7

<p>The profile classification module in GPM DPR level-2 algorithm outputs various products  such as rain type classification, melting layer  detection and  identification of  surface snowfall , as well as presence of graupel and hail. Extensive evaluation and validation activities have been performed on these products and have illustrated excellent performance. The latest version of these products is 6X.  With increasing interests  on severe weather  such as hail and  extreme precipitation, in  the next version (version 7), we development a flag to identify hail along the vertical profile using  precipitation type index (PTI).</p><p>Precipitation type index (PTI) plays an important role in a couple of algorithms in the profile classification module. PTI is a value calculated for each dual-frequency profile with precipitation observed by GPM DPR.   DFRm slope, the maximum value of the Zm(Ku) , and  storm top height  are used in calculating PTI. PTI is effective in separating snow and Graupel/Hail  profiles. In version 7, we zoom in further into PTI for  Graupel/ hail profiles and separate  them into graupel and hail profiles with different PTI thresholds. A new Boolean product of “flagHail” is a hail only identifier for each vertical profile.  This hail product will be validated with ground radar products and other DPR products from Trigger module of DPR level-2 algorithm.   In version 7, we make improvements of the surface snowfall algorithm. An adjustment is made accounting for global variability of storm top profiles.. A storm top normalization is introduced to obtain a smooth transition of surface snowfall identification algorithm along varying latitudes globally.</p>

Boolean Product Polynomials, Schur Positivity, and Chern Plethysm

Let $k \leq n$ be positive integers, and let $X_n = (x_1, \dots , x_n)$ be a list of $n$ variables. The Boolean product polynomial $B_{n,k}(X_n)$ is the product of the linear forms $\sum _{i \in S} x_i$, where $S$ ranges over all $k$-element subsets of $\{1, 2, \dots , n\}$. We prove that Boolean product polynomials are Schur positive. We do this via a new method of proving Schur positivity using vector bundles and a symmetric function operation we call Chern plethysm. This gives a geometric method for producing a vast array of Schur positive polynomials whose Schur positivity lacks (at present) a combinatorial or representation theoretic proof. We relate the polynomials $B_{n,k}(X_n)$ for certain $k$ to other combinatorial objects including derangements, positroids, alternating sign matrices, and reverse flagged fillings of a partition shape. We also relate $B_{n,n-1}(X_n)$ to a bigraded action of the symmetric group ${\mathfrak{S}}_n$ on a divergence free quotient of superspace.

Boolean like algebras

Using Vaggione's concept of central element in a double pointed algebra, weintroduce the notion of Boolean like variety as a generalisation ofBoolean algebras to an arbitrary similarity type. Appropriately relaxing therequirement that every element be central in any member of the variety, weobtain the more general class of semi-Boolean like varieties, whichstill retain many of the pleasing properties of Boolean algebras. We provethat a double pointed variety is discriminator iff it is semi-Boolean like,idempotent, and 0-regular. This theorem yields a new Maltsev-stylecharacterisation of double pointed discriminator varieties.Moreover, we point out the exact relationship between semi-Boolean-likevarieties and the quasi-discriminator varieties, and we providesemi-Boolean-like algebras with an explicit weak Boolean productrepresentation with directly indecomposable factors. Finally, we discuss idempotentsemi-Boolean-like algebras. We consider a noncommutative generalisation of Boolean algebras and prove - along the lines of similar results available for pointed discriminator varieties or for varieties with a commutative ternary deduction term that every idempotent semi-Boolean-like variety is term equivalent to a variety of noncommutative Boolean algebras with additional operations.

Classification and GNS-construction for general universal products

It is known that there are exactly five natural products, which are universal products fulfilling two normalization conditions simultaneously. We classify universal products without these extra conditions. We find a two-parameter deformation of the Boolean product, which we call (r, s)-products. Our main result states that, besides degnerate cases, these are the only new universal products. Furthermore, we introduce a GNS-construction for not necessarily positive linear functionals on algebras and study the GNS-construction for (r, s)-product functionals.

THE FIVE INDEPENDENCES AS NATURAL PRODUCTS

Let [Formula: see text] be the class of all algebraic probability spaces. A "natural product" is, by definition, a map [Formula: see text] which is required to satisfy all the canonical axioms of Ben Ghorbal and Schürmann for "universal product" except for the commutativity axiom. We show that there exist only five natural products, namely tensor product, free product, Boolean product, monotone product and anti-monotone product. This means that, in a sense, there exist only five universal notions of stochastic independence in noncommutative probability theory.