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In this article, we study the zero-divisor graphsΓ(Zn) of rings of integers modulo η as information systems L(Γ(Zn)) using equivalence classes and rough sets. Equivalence classes are referred as granules and partitions are referred as indiscernible partitions. We define an indiscernibility relation on the vertex set and identify different sets of attributes that induce the same indiscernibility partition. A reduct is a minimal subset of attributes which yields the same partition as the original attribute set. We compute all reducts of the defined information system and classify them in to two types including: (i) the set P of all prime divisors of η and (ii) the set consisting of P/pi, prime powers of P/pi in the prime factorization of η and the elements of the form P/pi pj, where pj ε P/pi . Moreover, we give the structures and the cardinalities of the attribute subsets whose removal yields a different indiscernibility partition than that of the set of all attributes, referred as essential sets. We prove that the essential sets of L(Γ(Zn)) either consist of two prime divisors of η or one prime divisor P/pi combined with prime powers of P/pi . Further, we determine the lower and upper approximations of various vertex subsets to study the properties of the zero-divisor graphs. We also study properties of the rough membership function forΓ(Zn). Furthermore, we introduce the class-based discernibility matrix induced by indiscernibility classes of zero divisors and determine general form of its entries. We also prove that the minimal entries of the class-based discernibility matrix coincide with the essential sets ofΓ(Zn). Based on these results, we determine information-granularity measures corresponding to the notable partitions of Γ(Zn) and provide an example to establish consistency of the proved results with these well-known measures. Thus, starting from introducing an information system, we investigate the components of granular computing and utilize our findings to compute information granularity measures, contributing to a deeper understanding of the zero divisor graphs via rough set theory. [ABSTRACT FROM AUTHOR]

In this article, we study the zero-divisor graphsΓ(Zn) of rings of integers modulo η as information systems L(Γ(Zn)) using equivalence classes and rough sets. Equivalence classes are referred as granules and partitions are referred as indiscernible partitions. We define an indiscernibility relation on the vertex set and identify different sets of attributes that induce the same indiscernibility partition. A reduct is a minimal subset of attributes which yields the same partition as the original attribute set. We compute all reducts of the defined information system and classify them in to two types including: (i) the set P of all prime divisors of η and (ii) the set consisting of P/pi, prime powers of P/pi in the prime factorization of η and the elements of the form P/pi pj, where pj ε P/pi . Moreover, we give the structures and the cardinalities of the attribute subsets whose removal yields a different indiscernibility partition than that of the set of all attributes, referred as essential sets. We prove that the essential sets of L(Γ(Zn)) either consist of two prime divisors of η or one prime divisor P/pi combined with prime powers of P/pi . Further, we determine the lower and upper approximations of various vertex subsets to study the properties of the zero-divisor graphs. We also study properties of the rough membership function forΓ(Zn). Furthermore, we introduce the class-based discernibility matrix induced by indiscernibility classes of zero divisors and determine general form of its entries. We also prove that the minimal entries of the class-based discernibility matrix coincide with the essential sets ofΓ(Zn). Based on these results, we determine information-granularity measures corresponding to the notable partitions of Γ(Zn) and provide an example to establish consistency of the proved results with these well-known measures. Thus, starting from introducing an information system, we investigate the components of granular computing and utilize our findings to compute information granularity measures, contributing to a deeper understanding of the zero divisor graphs via rough set theory. [ABSTRACT FROM AUTHOR]

Objective: To determine the usefulness and safety of transthoracic echocardiography (TTE) imaging for real time guidance in transcatheter device closure of atrial septal defect (ASD) secundum. Study Design: Cross-sectional study. Setting: Department of Paediatric Cardiology, CPEIC, Multan, Pakistan. Period: January 2022 to December 2023. Material & Methods: Patients of both genders aged above 3 years with clinical and echocardiographic evidence of ASD secundum with significant left to right shunt were analysed. All cases had morphologically suitable margins of defect confirmed on TTE. Usefulness and safety of intra-procedure and immediate post-procedure transcatheter device closure of secundum ASD were noted. Results: In a total of 34 patients, 20 (58.8%) were female and 14 (41.2%) male. The mean age was 21.32±17.21 years. The TTE findings revealed that the mean ASD side was 17.47±4.32 mm. Aortic margins were deficient among 28 (82.4%) cases. The mean fluoroscopy and procedure time were 9.29±6.98 minutes and 49.91±39.76. Residual leak was noted in one (2.9%) cases while none of the cases were observed to have superior vena cava, inferior vena cava or pulmonary vein obstructions. Post-procedure, transient supraventricular tachycardia (SVT) arrhythmia was the most common type of complication reported among 2 (5.9%) cases. None of the cases needed any kind of emergency surgical exploration. No cases of pericardial effusion were found. No mortality was reported. Conclusion: Transcatheter device closure of secundum atrial septal defect was found to be effective and safe using TTE. [ABSTRACT FROM AUTHOR]

One of the most recent developments in the field of graph theory is the analysis of networks such as Butterfly networks, Benes networks, Interconnection networks, and David-derived networks using graph theoretic parameters. The topological indices (TIs) have been widely used as graph invariants among various graph theoretic tools. Quantitative structure activity relationships (QSAR) and quantitative structure property relationships (QSPR) need the use of TIs. Different structure-based parameters, such as the degree and distance of vertices in graphs, contribute to the determination of the values of TIs. Among other recently introduced novelties, the classes of ev-degree and ve-degree dependent TIs have been extensively explored for various graph families. The current research focuses on the development of formulae for different ev-degree and ve-degree dependent TIs for s-dimensional Benes network and certain networks derived from it. In the end, a comparison between the values of the TIs for these networks has been presented through graphical tools. [ABSTRACT FROM AUTHOR]