Multiset topology via DNA and RNA mutation

Abstract

Topology is the most important branch of modern mathematics, which plays an important role in applications. In this paper, we use the concept of the topology, based on the concept of multiset to solve an important problems in life (DNA and RNA mutation) to detect diseases and help biologists in the treatment of diseases. Also, we introduce a new theory that explains if there is an existence of a mutation or not, and we have a set of metric functions through which we examine the congruence and similarity and dissimilarity between "types," which may be a strings of bits, vectors, DNA or RNA sequences, … , etc. Finally, we will introduce a theory that which can be used to know the existence and place of mutation.In this section, a brief survey of results and notations as introduced in literature 7-15 is presented. 5820 Definition 2.1. A mset M drawn from the set X is represented by a mapping C M (x) defined as C M (x) ∶ X → N, where N is the set of positive integers. C M (x) is the number of occurrences of the element x in the mset M. We present the mset M drawn from the set X = {x 1 , x 2 , … , x n } as M = {m 1 ∕x 1 , m 2 ∕x 2 , … , m n ∕x n } where m i is the number of occurrences of the element x i , i = 1, 2, … , n, in the mset M . Clearly, a set is a special case of an mset.Let M and N be 2 msets drawn from a set X. Then, the following are defined. 6,13,[16][17][18] The cardinality of an mset M is symbolized by Card (M) or |M| and is given byDefinition 2.2. (Girish and John 12 and Jena et al 19 ; whole submsets) A submset N of M is a whole submset of M with each element in N having full multiplicity as in M, ie, C N (x) = C M (x) for every x in N. Definition 2.3. (Girish and John 12 and Jena et al 19 ; partial whole submsets) A submset N of M is a partial whole submset of M with at least one element in N having full multiplicity as in M, ie, C N (x) = C M (x) for some x in N. Definition 2.4. (Girish and John 12 and Jena et al 19 ; full submsets) A submset N of M is a full submset of M if M and N having the same support set with C N (x) ≤ C M (x) for every x in N, ie, M * = N * with C N (x) ≤ C M (x) for every x in N. Example 2.5. (Girish and John 12 ) Let M = {3∕a, 4∕b, 6∕c} be an mset. The following are some of submsets of M that are whole submsets, partial whole submsets, and full submsets: (a) A submset {3∕a, 4∕b} is a whole submset and partial whole submset of M. (b) A submset {2∕a, 4∕b, 3∕c} is a partial whole submset and full submset of M. (c) A submset {2∕a, 4∕b} is a partial whole submset of M. Definition 2.6. (Previous studies 9-12 ) Let M 1 and M 2 be 2 msets drawn from a set Y; then the Cartesian product of M 1 and M 2 is defined as M 1 × M 2 = {(m∕x, n∕ )∕mn ∶ x∈ m M 1 , ∈ n M 2 }. Definition 2.7. (Girish and John 9 and Girish and Sunil 10 ) A submset R of M × M is called an mset relation on M if every member (m∕x, n∕y) of R has a count and product of C 1 (x, y) and C 2 (x, y), We denote m∕x related to n∕y by m∕xRn∕y. Definition 2.8. (Girish and John 9 and Girish and Sunil 10 ) (i) An mset relation...

Decomposition of some kinds of mixed multiset continuity on multiset topological spaces

“…We must acknowledge that this subject has been studied by biologists as well as others. EI-Sharkasy and Badr [3], EI-Sharkasy et al [4] have shown that the notion of multiset topology plays a profound role to find the mutations of deoxyribonucleic acid (DNA) and ribonucleic acid (RNA). These articles demonstrate how a multiset topology can be used to identify diseases and assist biologists with disease diagnosis.…”

“…Consider two M-topologies τ 1 and τ 2 on M as τ 1 = {M, ∅, {1/a}, {2/b}, {1/a, 2/b}} and τ 2 = {M, ∅}.Let N = {3/b}, then it is mixed (1, 2)-semi-open, but not a τ 1 -open mset. (ii) Let X = {a, b, c} and M = {2/a, 4/b, 1/c} ∈ [X]4 . Consider two M-topologies τ 1 and τ 2 on M as τ 1 = {M, ∅, {2/a}, {3/b}, {2/a, 3/b}} andτ 2 = {M, ∅, {1/b, 1/c}}.…”

Modeling of COVID-19 in View of Rough Topology

“…Recently, graphs, multisets, and rough sets have been used to represent structures such as the human heart [9] and medicine [10][11][12], which are useful in medicine, physics, and biology. Previously, we have used topology to study the similarity of DNA sequences and to identify mutations in genes, chromosomes, their locations, and amino acids that have changed [13]. We also used topology to study the recombination of DNA and to form a mathematical model for the recombination process [14].…”