Nano δI-closed Sets in Nano Ideal Topological Spaces

This article investigates the notion of N-δI-closed sets and its properties in ideal nano topological space using the closure operator NclδI(S)={xÎՍ/ Nint(Ncl*(W))∩ S ≠ ∅, for each WÎWN(x)}. We establish NclδI(S) is the intersection of all N-δI-closed supersets of S and NintδI(S) is the union of all N-δI-open subsets of S in this space. Apart from that N-δI-closed sets occurs between the class of Nδ-closed sets and N-closed sets. Moreover, we explore the concepts of N-δI-interior, N-δI-Exterior, N-δI-Border, and N-δI-Frontier and discuss its properties

Gaussian Twin Neighborhood Prime Labeling on Fan Digraphs

Gaussian integers are complex numbers of the form \gamma=x+iy where x and y are integers and i^2=-1. The set of Gaussian integers is usually denoted by \mathbb{Z}[i]. A Gaussian integer \gamma=a+ib\in\mathbb{Z}[i] is prime if and only if either \gamma=\pm(1\pm i),N(\gamma)= a^2+b^2 is a prime integer congruent to 1(mod4), or \gamma=p+0i or =0+pi where p is a prime integer and |p|\equiv3(mod4). Let D=(V,A) be a digraph with |V|=n. An injective function f:V(D)\rightarrow\left[\gamma_n\right] is said to be a Gaussian twin neighborhood prime labeling of D, if it is both Gaussian in and out neighborhood prime labeling. A digraph which admits a Gaussian twin neighborhood prime labeling is called a Gaussian twin neighborhood prime digraph. In this paper, we introduce some definitions of fan digraphs. Further, we establish the Gaussian twin neighborhood prime labeling in fan digraphs using Gaussian integers

Evaluation of Bode Index as a Predictor of Severity and Its Correlation with Pulmonary Hypertension in COPD Patients

INTRODUCTION: Chronic obstructive pulmonary disease (COPD) is a major cause of mortality and morbidity and health care costs worldwide. The COPD severity is generally assessed on the basis of a single parameter i.e forced expiratory volume in one second (FEV1). However, the systemic manifestations of COPD were not reflected by the FEV1. Hence, a multidimensional grading system which assesses the respiratory and systemic expressions of COPD is designed to predict outcome in these patients. These variables were used to construct the multidimensional BODE index which is a 10-point scale in which higher scores indicate the more severe nature of the disease and higher risk of complications and death. AIMS AND OBJECTIVES: 1. To compare the severity of COPD using BODE index than primary lung function test alone. 2. To determine whether higher BODE index in chronic obstructive pulmonary disease correlates with more years of cigarette smoking and more days of hospitalisation. 3. To determine whether higher BODE index is associated with more severe systemic compilcations in the form of cardiac involvement (PHT). METHODOLOGY: COPD patients who attended our outpatient clinic / admitted as inpatient at the Govt.Kilpauk Medical College & hospital in the department of General Medicine were enrolled into the study. The BODE index was calculated for each patient using the body mass index, the threshold value of FEV1, the distance walked in 6 min, and the score on the Modified Medical Research Council (MMRC) dyspnea scale. The patients received points ranging from 0 (lowest value) to 3 (maximal value). For Body mass index the values were 0 (> 21) or 1 ( 350 ms), 1 (250 – 350 ms), 2 (150 – 249 ms) and 3 (< 150 ms). The MMRC dyspnea class 0 and I were given 0 points, class II – 1 point, class III – 2 points and class IV – 3 points. The BODE score of 0 – 2 was taken as mild COPD. Scores between 3-5 was considered as moderate disease and those more than or equal to 6 was considered as severe COPD. RESULTS: A total of 81 patients with COPD were enrolled in the study. All 81 patients were males. Among the patients with COPD, 21 (25.9%), 27 (33.3%), 28 (34.6%) and 5 (62%) belongs to age 41 – 50 years, 51 – 60 years, 61 – 70 years and 71 – 80 years respectively. Most patients are in the age group of 51 – 70 years. Of the total 81 patients enrolled in the study, 42 (51.9%) comes under Mild COPD, 15 (18.5%) comes under Moderate COPD and 24 (29.6%) comes under Severe COPD categories. On analysing the distribution of number of exacerbations in relation to the severity of COPD categories, it was observed that the mean number of exacerbations were 0.62±1.15, 5.53±1.96 and 14.38±1.95 in mild, moderate and severe category of COPD. Among 81 patients enrolled in the study, 53 (65.4%) patients had no changes in ECG but only 28 (34.6%) patients had ECG changes i.e ‘P’ Pulmonale and Right axis deviation (RAD). Among the 81 patients included in the study, 30 (37%) had Mild PHT, 14 (17.3%) had Moderate PHT, 26 (32.1%) had Severe PHT and 11(13.6%) had no evidence of PHT. On analysing the distribution of BODE scores in relation to the PHT categories, it was observed that the mean BODE Scores were 1.00±1.29, 3.50±1.79 and 7.62±1.68 in mild, moderate and severe category of PHT. On analysing the distribution of BODE score in relation to the severity of COPD categories, it was observed that the mean BODE Scores were 0.57±0.63, 4.13±0.92 and 8.00±1.25 in mild, moderate and severe category of COPD. CONCLUSION: BODE Index can be used as a very useful and reliable index to assess the severity of Chronic Obstructive Pulmonary Disease (COPD). BODE index is directly correlated with the number of exacerbations. As the Smoking intensity and the Smoking Pack Years increases the BODE Index also increases. PHT increases with the severity of COPD as assessed by BODE index

Reach Energy of Digraphs

A Digraph D consists of two finite sets ), where  denotes the vertex set and denotes the arc set. For vertices  if there exists a directed path from  to  then  is said to be reachable from  and vice versa. The Reachability matrix of D is the  matrix , where  if  is reachable from and  otherwise. The eigen values corresponding to the reachability matrix are called reach eigen values. The reach energy of a digraph is defined by where  is the eigen value of the reachability matrix. In this paper we introduce the reach spectrum of a digraph and study its properties and bounds. Moreover, we compute reachspectrum for some digraphs

Product Signed Domination in Graphs

Let  be a simple graph. The closed neighborhood of , denoted by , is the set . A function  is a product signed dominating function, if for every vertex where . The weight of , denoted by , is the sum of the function values of all the vertices in . . The product signed domination number of  is the minimum positive weight of a product signed dominating function. In this paper, we establish bounds on the product signed domination number and estimate product signed domination number for some standard graph

Quotient Energy of Zero Divisor Graphs And Identity Graphs

إذا اعتبرنا ان (p,q) هورسم البياني المتصل البسيط    مجموع القيم المطلقة لسلسلة مصفوفة الحاصل في الرسم البياني تشكل الرسم البياني لطاقة الحاصل. الهدف من هذه الدراسة هو فحص طاقة حاصل الرسوم البيانية للمطابقة والرسوم البيانية القاسم الصفري للحلقات التبادلية باستخدام نظرية المجموعة ونظرية الرسم البياني، والتطبيقات. في هذه الدراسة، يتم فحص الرسوم البيانية للمطابقة المشتقة من المجموعة وفئات قليلة من الرسوم البيانية لقاسم الصفر للحلقة التبادلية R.Consider the (p,q) simple connected graph . The sum absolute values of the spectrum of quotient matrix of a graph  make up the graph's quotient energy. The objective of this study is to examine the quotient energy of identity graphs and zero-divisor graphs  of commutative rings using group theory, graph theory, and applications. In this study, the identity graphs  derived from the group  and a few classes of zero-divisor graphs  of the commutative ring R are examined