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A Study of Fuzzy Ag-Subgroups

Thesis Info

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Author

Amanullah

Program

PhD

Institute

University of Malakand

City

Malakand

Province

KPK

Country

Pakistan

Thesis Completing Year

2016

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/10333/1/AmanUllah_Maths_2016_HSR_UoM_Malakand_20.02.2017.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676725411972

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The idea of fuzzy sets has opened new door of research in the world of contemporary Mathematics. The concept of fuzzy sets provided a new approach to model imprecision and uncertainty present in phenomena without sharp boundaries. The fuzzi cation of algebraic structures, play a dynamic role in Mathematics with diverse applications in many other branches such as computer arithmetic''s, control engineering, error correcting codes and formal languages and many more. Moreover, during the course of the last decade, non-associative algebraic structures have gained popularity among the researchers. In this background, many researchers initiated the notion of AG-groupoids, its newly introduced subclasses and its fuzzi cation. The present research is among the very few where non-associative algebraic structures are investigated and fuzzi ed. In this thesis various constructions of AG-groups over theeld Zn are introduced, some related results and example of AG-groups are provided. Further, the structural properties of fuzzy AG-subgroup are introduced and various notions of fuzzy AG-subgroups are investigated, e.g. conjugate of a fuzzy AG-subgroup, fuzzy normal AG-subgroups, relations between fuzzy normal AG-subgroup and commutators in AG-groups and equal-height elements in fuzzy AG-subgroups. Moreover, the notion of fuzzy AG-subgroups is further extended and a fuzzy coset in AG-subgroups is introduced. It is worth mentioning that if A is any fuzzy AGsubgroup of G, thenA(xy) =A(yx) for all x; y 2 G, i.e. each fuzzy left coset is fuzzy right coset and vice versa. Also, fuzzy coset in AG-subgroup could be empty contrary to coset in group theory. However, order of the nonempty fuzzy coset is the same as the index number [G : A] where H is an AG-subgroup of an AG-group G. The notion of fuzzy quotient AG-subgroup, fuzzy AG-subgroup of the quotient (factor) AG-subgroup, fuzzy homomorphism of AG-group and fuzzy Lagrange''s Theorem ofnite AG-group is introduced. Finally, cubic AG-subgroups and its properties are explored.
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علامہ یوسف علی

علامہ یوسف علی
دوسرا حادثہ علامہ یوسف علی کی وفات کا ہے، وہ جدید تعلیم کی بہترین پیداوار بہت پرانے اور بڑے فاضل آئی سی ایس تھے، انگریزی پر ان کو اہل زبان جیسی قدرت حاصل تھی، وہ متعدد بلندپایہ کتابوں کے مصنف تھے، ان کی ایک کتاب کا ترجمہ ’’ہندوستان قرونِ وسطیٰ میں‘‘ کے نام سے اردو میں بھی شائع ہوچکا ہے، یہ ان کے ان لکچروں کا مجموعہ ہے جو انھوں نے ہندوستانی اکیڈمی الہ آباد میں دیے تھے، اگرچہ ان کا قالب مغربی تھا، مگر ان کا قلب ہمیشہ مومن اور ان کا قلم ابتداء سے اسلام کی خدمت میں مصروف رہا، ان کا سب سے بڑا علمی و دینی کارنامہ قرآن مجید کا انگریزی ترجمہ ہے، جو ادبی خوبیوں کے لحاظ سے انگریزی زبان کا شاہکار سمجھا جاتا ہے، اس کی ایک خوبی یہ بھی ہے کہ مترجم نے ان جدید مفسرین کے برعکس جو محض یورپ کی مرعوبیت میں آیاتِ قرآنی میں خود ساختہ تاویلیں کرتے ہیں، اپنے حواشی میں کوئی ایسی تاویل نہیں کی ہے جو سلف کے عقائد کے خلاف ہو، اس لئے دینی طبقہ میں بھی اس ترجمہ کو پسندیدگی کی نظر سے دیکھا گیا، اﷲ تعالیٰ اپنی کتاب عزیز کے اس خادم کو اس کی خدمت کا صلہ عطا فرمائے۔ (شاہ معین الدین ندوی، جنوری ۱۹۵۴ء)

 

اردو تراجم و تفاسیر میں تفسیر مرادیہ کا مقام و مرتبہ

This is a historical fact that along with Arabs, rather morethan Arabs, the Quranic and Islamic sciences were dealt by the nonArabs. After Arabic, the Persian language attained the status of anIslamic language, and great books were written in Islamic literaturein Persian. After Persian, Urdu succeeded to hold the title of Islamiclanguage. A great many works of Islamic sciences and translationand exegesis of the Qur’ān were rendered into Urdu by the scholarsof the subcontinent and others. It is said that Urdu tafsīr began in the 12th century from theHijrah. As Jamīl Naqī says that the first Tafsīr was "Basā’ir alQur’ān" by Nikhal Shāh Jahānpūrī (114 A. H/1231AD), he points outthat Ḥakīm Muḥammad Ashraf Khān was the first one whotranslated the Qur’ān into Urdu with some comments. Shāh ‘AbdulQādir (1230 AH/1815AD) and Shah Rafi’udddīn followed him. However, Urdu translation and exegesis of the Quran byMurād’ullāh Anṣārī Sanbhalī, a disciple of Mirzā Maẓhar Jan-eJānān, is rightly said to be the earlier work than those of Shāh‘Abdul Qādir and Shāh Rafī’uddīn. However, the first completetranslations were of course of both of them. The Author of this research article, explores and discussesTafsīr-e-Murādiyah and highlights its scholarly merits, whichdetermine its status among the exegetical literature of the Quran.

Bioevaluation of Cabbage Based Designer Food to Modulate Oxidative Stress Markers

Nowadays, plants bioactive moieties are gaining attention amongst the masses to mitigate lifestyle related dysfunctions owing to their safe nature and functional properties. Considering phytochemistry and cost-effectiveness of cabbage, the current project was designed to probe the antioxidant capacity of locally grown green and red cabbage with special emphasis on modulation of oxidative stress biomarkers in response to hypercholesterolemic diet. Purposely, green and red cabbage samples were analyzed for compositional analyses that depicted high proportion of potassium and vitamin C in both samples. The red cabbage showed higher amount of total polyphenols & flavonoids (224.37±6.96 & 219.15±10.30 mg/100g F.W.) than green cabbage (58.41±3.01 & 34.04±1.06 mg/100g F.W.) along with the existence of anthocyanins (69.86±4.12 mg/100g F.W.) in red cabbage only. Comparative HPLC analysis regarding antioxidant moieties showed significant proportion of kempferol (171.10±5.99 mg/100g F.W.) and vitamin C (139.07±2.23 mg/100g F.W.) in red cabbage however, vitamin C (121.46±3.28 mg/100g F.W.) was found as the major antioxidant in green cabbage. The red cabbage depicted higher free radical quenching and reducing ability in contrast to green cabbage using DPPH (1, 1-diphenyl-2- picrylhydrazyl), ABTS [2, 2''-azino-bis (3-ethylbenzothiazoline-6-sulphonic acid)], FRAP (ferric reducing antioxidant power) and PFRAP (potassium ferricyanide reducing antioxidant power) reagents. The corresponding assays were ranged from 31.22±1.65 to 87.79±3.69%, 1.51±0.07 to 6.04±0.21 μM Trolox/g F.W., 0.95±0.04 to 1.16±0.04 μM Fe2+/g F.W., 41.16±2.10 to 59.32±2.14% and 48.03±1.68 to 63.45±3.05%. In the product development phase, four types of croquettes based on green and red cabbage leaves & their respective aqueous extracts along with control were stored under frozen conditions and analyzed at three different intervals; 0, 15 & 30 days, employing two types of cooking procedures; baking and frying. The resultant prototypes were assessed for physicochemical analyses (caloric count, color, texture, water activity & antioxidant assays) as well as sensory response. The current study portrayed that designer croquettes remained freezer friendly for one month without employing any preservative. Comparative assessment of cooking methods inferred higher antioxidant potential and taste scores for fried prototypes. Based on antioxidant assays and overall acceptability, red cabbage and its respective aqueous extract were screened for biological evaluation. In animal study (12 weeks), male white New Zealand rabbits were divided into six groups; three groups under normal dietary pattern; normal diet, normal diet+cabbage leaves & normal diet+cabbage extract and next three under hypercholesterolemic dietary regimen; hypercholesterolemic diet, hypercholesterolemic diet+cabbage leaves & high calorie+aqueous cabbage extract. At termination, overnight fasted rabbits were sacrificed to assess serum lipid profile. The percent reduction in serum cholesterol and LDL by red cabbage & its extract was reported up to 10.79 & 15.19 and 12.24 & 18.09 in hypercholesterolemic rabbits, respectively. Furthermore, serum specific and tissues (liver, heart & kidney) oxidative stress biomarkers alongside, their somatic index and histopathology were studied. The red cabbage supplementation suppressed the leakage of liver functioning enzymes in sera up to 15.63% (ALT), 13.88% (ALP), 12.96% (ɤ-GT) and 10.77% (total bilirubin) besides reduction in hepatic lipid peroxidation up to 29.60% thus improved endogenous antioxidant enzymes. Likewise, red cabbage showed protection against cardiac oxidative stress. Additionally, the red cabbage extract lowered renal MDA (lipid peroxidation) up to 24.07% in oxidative stressed rabbits that ultimately restored renal SOD (11.38%) and CAT (16.72%), accordingly. The hepatosomatic index (HSI) and nephrosomatic index (NSI) expounded significant impact of treatments on hypercholesterolemic diet fed rabbits except cardiosomatic index (CSI). In hypercholesterolemic rabbits, mild degree of fibrotic and necrotic changes in hepatic parenchyma were rectified by red cabbage supplementation, whereas red cabbage extract based diet showed slower rate of amelioration. Likewise, mild degree of necrotic changes and congestion in cardiac fenestrations were modulated by red cabbage feeding. Moreover, severe necrotic & pyknotic alterations in renal parenchyma along with mild to moderate degree of congestion in hypercholesterolemic rabbits were effectively reversed via extract supplemented group in contrast to cabbage fed group. In the nutshell, dietary inclusions based on red cabbage has proven restorative against hepatic and cardiac compromised conditions, whereas altered renal fenestrations were also effectively restored by red cabbage extract enriched diet.