The aim of this paper is to introduce strongest relation, cosets and middle cosets of anti \(Q\)-fuzzy subgroups of \(G\) with respect to \(t\)-conorm \(C.\) We investigate equivalent characterizations of them and we construct a new group induced by them, and give the homomorphism theorem between them.

New Hadamard type inequalities for a class of \(s\)-Godunova-Levin functions of the second kind for fractional integrals are obtained. These new estimates extend and generalize some existing results for the \(Q\)-class and \(P\)-class functions. The generalized case for the Katugampola fractional integrals are also given.

For any non-square integer multiplier \(k\), there is an infinity of triangular numbers multiple of other triangular numbers. We analyze the congruence properties of indices \(\xi\) of triangular numbers multiple of triangular numbers. Remainders in congruence relations \(\xi\) modulo \(k\) come always in pairs whose sum always equal \((k-1)\), always include 0 and \((k-1)\), and only 0 and \((k-1)\) if \(k\) is prime, or an odd power of a prime, or an even square plus one or an odd square minus one or minus two. If the multiplier \(k\) is twice the triangular number of \(n\), the set of remainders includes also \(n\) and \((n^{2}-1)\) and if \(k\) has integer factors, the set of remainders include multiples of a factor following certain rules. Algebraic expressions are found for remainders in function of \(k\) and its factors, with several exceptions. This approach eliminates those \(\xi\) values not providing solutions.

Aim: To assess risk factors for development of type II diabetes. Materials & Methods: 80 patients of type 2 diabetes mellitus >40 years of age were put in group 1 group 2 were healthy subjects irrespective of gender. Factors such as family history, physical activity, blood pressure, alcohol consumption and BMI was recorded. Results: Alcohol consumption was present in 45 in group 1 and 20 un group 2, family history was positive in 65 group 1 and 12 in group 2, sedentary life was seen in 52 group 1 and 25 in group 2, BMI was underweight seen in 14 in group 1 and 5 in group 2, normal 12 in group 1 and 46 in group 2, overweight 30 BMI was underweight seen in 14 in group 1 and 5 in group 2, normal12 in group 1 and 14 in group 2 and obese 22 and 15 46 in group 2. Blood pressure was normal seen 16 in group 1 and 42 in group 2, pre- hypertension 24 in group 1 and 26 in group 2, hypertension stage 1 in 30 in group 1 and 10 in group 2 and hypertension stage 2 seen in 10 in group 1 and 2 in group 2. A significant difference was observed (P< 0.05). Conclusion: Common risk factors in diabetes was overweight, hypertension, lack of physical activity and alcohol consumption.

Aim: To evaluate anesthetic management in renal transplant patients. Methodology: Ninety- two renal transplant patients were part of the study. Parameters such as type of transplant, reason for chronic kidney disease, preoperative data, history of dialysis, preoperative anesthesia management, monitoring details and the outcome were recorded. Results: Chronic glomerulonephritis (CGN) in 28 (30.4%), chronic interstitial nephritis (CIN) in 20 (21.7%), polycystic kidney disease (PCKD) in 11 (11.9%), obstructive nephropathy (Ob. N) in 4 (4.3%), diabetic nephropathy (DN) in 8 (8.7%), hereditary nephropathy in 3 (3.2%), reflux nephropathy in 12 (13%) and membranoproliferative glomerulonephritis (MPGN) in 6 (6.5%). In 32 (35.6%) patients, isoflurane was inhalational agent and recovery time was 25.1 minute, in 40 (43.4%), desflurane was inhalational agent and recovery time was 22.7 minutes and in 20 (21.7%), sevoflurane was inhalational agent and recovery time was 32.1 minutes. A significant difference was observed (P< 0.05). Conclusion: Anesthesia management has made renal transplantation safe and predictable. Postoperative maintenance of renal transplant patients have contributed to the success of renal transplant programme.

Aim: To compare tension band wire and circumferential wiring for patellar fractures. Materials & Methods: One hundred twenty adult patients in age ranged 18- 50 years of either gender was randomly divided into groups viz. group 1 treated with tension band wire and group 2 with circumferential wiring for patellar fractures. Reich and Rosenberg scale, rage of motion and complications were recorded in both groups and compared. Results: At 4 weeks in group 1 and group 2, restriction of last \(10^{0}- 20^{0}\) was seen in 24 and 27, restriction of \(20^0-50^0\) was seen in 14 and 13, restriction \(>50^0\) was observed in 22 and 20. At 8 weeks, no restriction was seen in 8 and 9, restriction of last \(10^0- 20^0\) was seen in 38 and 39, restriction of \(20^0-50^0\) was seen in 10 and 9, restriction >500 was seen in 4 and 3. At 12 weeks, no restriction was seen in 15 and 20, restriction of last \(10^0- 20^0\) was seen in 35 and 32, restriction of \(20^0-50^0\) was seen in 8 and 7 and restriction \(>50^0\) was seen in 2 and 1 in group 1 and 2 patients respectively. A significant difference was observed (P< 0.05). Conclusion: Both techniques for the management of patellar fractures were equally effective in achieving functional outcome.

This paper aims to compare 0.1% topical triamcinolone acetonide, oral methotrexate, and a combination of 0.1% topical triamcinolone acetonide and oral methotrexate in the management of oral lichen planus. 60 histologically confirmed cases of oral lichen planus were divided into 3 groups. Group T was given 0.1% topical triamcinolone acetonide, group M was given topical methotrexate and group C was given combination of both 0.1% topical triamcinolone acetonide and oral methotrexate. Clinical severity score and VAS was compared. The mean CSS at baseline was 5.4 in group T, 4.2 in group M and 4.1 in group C and at 4 months was 2.6 in group T, 2.1 in group M and 0.82 in group C. Baseline VAS was 6.5 in group T, 6.2 in group M and 7.1 in group C and at 4 months was 2.5 in group T, 1.3 in group M and 0.25 in group C. Group T had 3.2 years, group M had 3.1 years and group C had 3.3 years of duration of symptoms. It is concluded that the combination of triamcinolone and methotrexate exhibited maximum relieve of symptoms in patients with oral lichen planus.

The Cauchy problem for a higher order modification of the nonlinear Schrödinger equation (MNLS) on the line is shown to be well-posed in Sobolev spaces with exponent \(s > \frac{1}{4}\). This result is achieved by demonstrating that the associated integral operator is a contraction on a Bourgain space that has been adapted to the particular linear symbol present in the equation. The contraction is proved by using microlocal analysis and a trilinear estimate that is shown via the \([k; Z]\)-multiplier norm method developed by Terence Tao.

In this paper, we give characterizations of orthogonality conditions in certain classes of normed spaces. We first consider Range-Kernel orthogonality in norm-attainable classes then we characterize orthogonality conditions for Jordan elementary operators.

Copulas played a key role in numerous areas of statistics over the last few decades. In this paper, we offer a new kind of trigonometric bivariate copula based on power and cosine functions. We present it via analytical and graphical approaches. We show that it may be used to create a new bivariate normal distribution with interesting shapes. Subsequently, the simplest version of the suggested copula is highlighted. We discuss some of its relationships with the Farlie-Gumbel-Morgensten and simple polynomial-sine copulas, establish that it is a member of a well-known semi-parametric family of copulas, investigate its dependence domains, and show that it has no tail dependence.