scholarly journals Boolean product representations of algebras via binary polynomials

2018 ◽  
pp. 297-321
Author(s):  
Antonino Salibra ◽  
Antonio Ledda ◽  
Francesco Paoli
Keyword(s):  
Representations Of Algebras ◽  
Boolean Product

THE FIVE INDEPENDENCES AS NATURAL PRODUCTS

2003 ◽  
Vol 06 (03) ◽  
pp. 337-371 ◽  
Author(s):  
NAOFUMI MURAKI
Keyword(s):  
Probability Theory ◽  
Natural Products ◽  
Tensor Product ◽  
Natural Product ◽  
Free Product ◽  
Noncommutative Probability ◽  
Stochastic Independence ◽  
Boolean Product ◽  
Probability Spaces ◽  
Noncommutative Probability Theory

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.

GPM DPR Profile Classification Algorithm: Enhancement from V6X to V7

2021 ◽  
Author(s):  
Chandrasekar V Chandra ◽  
Minda Le
Keyword(s):  
Vertical Profile ◽  
Smooth Transition ◽  
Identification Algorithm ◽  
Dual Frequency ◽  
A Value ◽  
Extensive Evaluation ◽  
Precipitation Type ◽  
Boolean Product ◽  
Level 2 ◽  
Type Classification

<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>

Positive Forms on Nuclear *-Algebras and Their Integral Representations

1990 ◽  
Vol 42 (3) ◽  
pp. 410-469 ◽  
Author(s):  
Alain Bélanger ◽  
Erik G. F. Thomas
Keyword(s):  
Integral Representation ◽  
Positive Cone ◽  
Approximate Identity ◽  
Integral Representations ◽  
Regularity Conditions ◽  
Direct Integral ◽  
Positive Form ◽  
Representations Of Algebras ◽  
Almost All ◽  
Order Intervals

Abstract.The main result of this paper establishes the existence and uniqueness of integral representations of KMS functionals on nuclear *- algebras. Our first result is about representations of *-algebras by means of operators having a common dense domain in a Hilbert space. We show, under certain regularity conditions, that (Powers) self-adjoint representations of a nuclear *-algebra, which admit a direct integral decomposition, disintegrate into representations which are almost all self-adjoint. We then define and study the class of self-derivative algebras. All algebras with an identity are in this class and many other examples are given. We show that if is a self-derivative algebra with an equicontinuous approximate identity, the cone of all positive forms on is isomorphic to the cone of all positive invariant kernels on These in turn correspond bijectively to the invariant Hilbert subspaces of the dual space This shows that if is a nuclear -space, the positive cone of has bounded order intervals, which implies that each positive form on has an integral representation in terms of the extreme generators of the cone. Given a continuous exponentially bounded one-parameter group of *-automorphisms of we can define the subcone of all invariant positive forms satisfying the KMS condition. Central functionals can be viewed as KMS functionals with respect to a trivial group action. Assuming that is a self-derivative algebra with an equicontinuous approximate identity, we show that the face generated by a self-adjoint KMS functional is a lattice. If is moreover a nuclear *-algebra the previous results together imply that each self-adjoint KMS functional has a unique integral representation by means of extreme KMS functionals almost all of which are self-adjoint.

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