Cubic pythagorean fuzzy linear spaces

The first author is supported by the “CSIR” Council of Scientific and Industrial Research.Pythagorean fuzzy sets assist in handling more uncertain and vague data than fuzzy sets and intuitionistic fuzzy sets. The notion of cubic pythagorean fuzzy sets is defined by combining interval valued pythagorean fuzzy sets and pythagorean fuzzy sets In this paper, based on the notion of cubic Pythagorean fuzzy sets we initiate a new theory called cubic pythagorean fuzzy linear spaces. Inspired by the notion of Cubic linear spaces we also present P(resp. R)-union and P(resp. R) intersection of cubic pythagorean fuzzy linear spaces. The concept of internal(resp. external) cubic pythagorean fuzzy linear spaces and its properties are examined.Publisher's Versio

Some Results of Double Sequences in 2-Normed and n-Normed Spaces

The primary purpose of this paper is to introduce the notion of double sequences in 2-normed space. We provide a simple way to derive a norm from the standard 2-norm by using double sequences when a 2-normed space is given. Equivalence relation between derived norm and the usual norm are established. Using this derived norm, we examine the completeness property of a 2-normed space and we extend the results to n-normed spaces

A Whole-Chromosome Analysis of Meiotic Recombination in Drosophila melanogaster

Although traditional genetic assays have characterized the pattern of crossing over across the genome in Drosophila melanogaster, these assays could not precisely define the location of crossovers. Even less is known about the frequency and distribution of noncrossover gene conversion events. To assess the specific number and positions of both meiotic gene conversion and crossover events, we sequenced the genomes of male progeny from females heterozygous for 93,538 X chromosomal single-nucleotide and InDel polymorphisms. From the analysis of the 30 F1 hemizygous X chromosomes, we detected 15 crossover and 5 noncrossover gene conversion events. Taking into account the nonuniform distribution of polymorphism along the chromosome arm, we estimate that most oocytes experience 1 crossover event and 1.6 gene conversion events per X chromosome pair per meiosis. An extrapolation to the entire genome would predict approximately 5 crossover events and 8.6 conversion events per meiosis. Mean gene conversion tract lengths were estimated to be 476 base pairs, yielding a per nucleotide conversion rate of 0.86 × 10−5 per meiosis. Both of these values are consistent with estimates of conversion frequency and tract length obtained from studies of rosy, the only gene for which gene conversion has been studied extensively in Drosophila. Motif-enrichment analysis revealed a GTGGAAA motif that was enriched near crossovers but not near gene conversions. The low-complexity and frequent occurrence of this motif may in part explain why, in contrast to mammalian systems, no meiotic crossover hotspots have been found in Drosophila