Γ∗-Relation on Fuzzy Hyperrings and Fundamental Ring

Fundamental relation performs an important role on fuzzy algebraic hyperstructure and is considered as the smallest equivalence relation such that the quotient is a universal algebra. In this paper, we introduce a new fuzzy strongly regular equivalence on fuzzy hyperrings such that the set of the quotient is a ring that is non-commutative. Also, we introduce the concept of a complete part of a fuzzy hyperring and study its principal traits. At last, we convey the relevance between the fundamental relation and complete parts of a fuzzy hyperring.

Use of Silica-Encapsulated Pseudomonas sp. Strain NCIB 9816-4 in Biodegradation of Novel Hydrocarbon Ring Structures Found in Hydraulic Fracturing Waters

ABSTRACTThe most problematic hydrocarbons in hydraulic fracturing (fracking) wastewaters consist of fused, isolated, bridged, and spiro ring systems, and ring systems have been poorly studied with respect to biodegradation, prompting the testing here of six major ring structural subclasses using a well-characterized bacterium and a silica encapsulation system previously shown to enhance biodegradation. The direct biological oxygenation of spiro ring compounds was demonstrated here. These and other hydrocarbon ring compounds have previously been shown to be present in flow-back waters and waters produced from hydraulic fracturing operations.Pseudomonassp. strain NCIB 9816-4, containing naphthalene dioxygenase, was selected for its broad substrate specificity, and it was demonstrated here to oxidize fundamental ring structures that are common in shale-derived waters but not previously investigated with this or related enzymes.Pseudomonassp. NCIB 9816-4 was tested here in the presence of a silica encasement, a protocol that has previously been shown to protect bacteria against the extremes of salinity present in fracking wastewaters. These studies demonstrate the degradation of highly hydrophobic compounds by a silica-encapsulated model bacterium, demonstrate what it may not degrade, and contribute to knowledge of the full range of hydrocarbon ring compounds that can be oxidized usingPseudomonassp. NCIB 9816-4.

LOCAL ANALYSIS OF GRAUERT–REMMERT-TYPE NORMALIZATION ALGORITHMS

Normalization is a fundamental ring-theoretic operation; geometrically it resolves singularities in codimension one. Existing algorithmic methods for computing the normalization rely on a common recipe: successively enlarge the given ring in form of an endomorphism ring of a certain (fractional) ideal until the process becomes stationary. While Vasconcelos' method uses the dual Jacobian ideal, Grauert–Remmert-type algorithms rely on so-called test ideals. For algebraic varieties, one can apply such normalization algorithms globally, locally, or formal analytically at all points of the variety. In this paper, we relate the number of iterations for global Grauert–Remmert-type normalization algorithms to that of its local descendants. We complement our results by a study of ADE singularities. All intermediate singularities occurring in the normalization process are determined explicitly. Besides ADE singularities the process yields simple space curve singularities from the list of Frühbis-Krüger.

Computing in quotients of rings of integers

We develop algorithms to turn quotients of rings of integers into effective Euclidean rings by giving polynomial algorithms for all fundamental ring operations. In addition, we study normal forms for modules over such rings and their behavior under certain quotients. We illustrate the power of our ideas in a new modular normal form algorithm for modules over rings of integers, vastly outperforming classical algorithms.