Complex symmetry analysis (CSA) establishes a connection between a class of n and 2n, dimen- sional complex and real systems of ordinary/partial differential equations (ODEs/PDEs), respec- tively. Similarly, another class of systems of PDEs is extractable from the base complex systems of ODEs by CSA. The equivalence of the base complex systems under an invertible transforma- tion of the dependent and independent variables has been exploited to study the corresponding real systems. Of particular interest is the extension of Lie groups of transformations developed to solve scalar and systems of non-linear second order ODEs, to two and four dimensional systems of the same order, respectively. For scalar equations, they may be used to either reduce them to the free particle equation (linearize) or integrate them. The former requires an 8-dimensional algebra while a 2-dimensional solvable algebra is sufficient to apply the latter. In this thesis, those two and four dimensional systems are characterized that arise from linearizable or integrable scalar and two dimensional systems of second order ODEs, respectively. Further, invariance of systems of PDEs under equivalence transformations is studied with CSA by employing the invariance properties of the base complex PDEs. Three linearizable classes of two dimensional systems of cubically semi-linear (in the first deriva- tive) second order ODEs has appeared so far in the literature. A comparison of two of the classes, obtainable from geometric methods and CSA, is presented. Both these classes are transformable to the system of free particle equations subject to certain linearization conditions, even though their general (cubic) semi-linear forms are proved to be inequivalent under point transformations. There are five equivalence classes of two dimensional linearizable systems of second order ODEs namely, those with 5, 6, 7, 8 and 15-dimensional Lie algebras. For those systems that arise from a scalar complex linearizable second order ODE, treated as a pair of real ODEs, a reduced optimal canonical form is established. Of the five only three equivalence classes with 6, 7 or 15-dimensional algebras are recovered by this procedure. Both the equations of these systems are found to sat- isfy Cauchy-Riemann (CR) equations with respect to the dependent variables. Therefore, here as elsewhere in this thesis, such systems are called CR-structured systems. A class of non-linearizable two dimensional CR-structured systems of second order ODEs is presented to show that the linearizability of the scalar complex equations is not sufficient to map ii iii the emerging systems to linear forms. A general system of n second order ODEs with 2n symmetry generators may not be amenable to quadratures by real symmetry analysis. However, it is shown that the CR-structured systems may be solvable by a procedure called complex-linearization even if they have fewer symmetries than required to linearize or integrate them. A symmetry generator of the base complex ODE associates a pair of Lie-like operators with the CR-structured systems. It is proved that all such operators are not necessarily real symmetries of the emerging system. A criterion has been developed which shows when and how the real symme- try generators of the CR-structured systems of two second order ODEs are extractable from the associated complex Lie symmetries of the base ODEs. The most general complex-linearizable form and the complex-linearization criteria for four di- mensional systems of second order ODEs are derived by extending the geometric linearization criteria presented for two dimensional systems of cubically semi-linear second order ODEs. Two canonical forms of such systems have been derived by employing CSA on a system of dimension two once and a scalar equation twice. A specific form of the complex linearizing transformations associated with the base two dimensional systems is shown to furnish the reduction of the corresponding four dimensional complex-linearizable systems to the free particle Newtonian systems. Semi-invariants for a class of systems of two linear parabolic type PDEs in two independent variables under equivalence transformations of the dependent variables have been deduced. This class of systems of two linear parabolic type PDEs and the real transformations that map such systems into themselves with different coefficients in general, are shown to correspond to complex scalar linear parabolic equations and associated complex transformations, respectively. Moreover, the semi-invariants for such systems of PDEs also correspond to complex Ibragimov invariants of the complex scalar linear parabolic PDEs. Particular cases of systems of parabolic type equations, i.e., when they are uncoupled or coupled in a special manner, have been studied with CSA.
آہ مولانا عبدالسلام قدوائی! آنکھیں اشکبار ہیں، دل اندوہ و غم کا جوئبار ہے، جب یہ قلم فگار ہوکر لکھ رہا ہے کہ مولانا عبداسلام قدوائی جو دارالمصنفین کی علمی اور نجی مجلسوں کی رونق، عزت اور آبرو تھے، ہم سب کو چھوڑ کر یکایک آغوش رحمت الٰہی میں چلے گئے۔ وہ ۱۹۷۵ء میں دارالمصنفین اس وقت آئے جب جناب شاہ معین الدین احمد ندوی سابق ناظم دارالمصنفین کی رحلت سے یہاں کا پتہ پتہ، بوٹا بوٹا سوگوار اور بے رونق ہورہا تھا، وہ یہاں آئے تو اپنے جلو میں علامہ شبلیؒ کی بے پناہ عظمت، اپنے استاد مولانا سید سلیمان ندویؒ کی غیر معمولی عقیدت، جناب شاہ معین الدین احمد ندوی مرحوم کی شخصیت سے اپنی مخلصانہ محبت اور خود اپنی ذات کی لینت، مروت اور ملاطفت کا لشکر ساتھ لائے اور اس راقم سے ایسے گھل مل گئے کہ ہم دونوں کے درمیان شیروشہد کی کوثر بہتی نظر آنے لگی، ان کی آمد سے دارالمصنفین کی سرگرمیوں میں شادابی، اس کی امیدوں کے پھولوں میں رعنائی اور اس کی تمناؤں کے مرغزاروں میں دل فریبی پیدا ہونے لگی، مگر معلوم نہیں مصلحتِ خداوندی کیا تھی کہ دارالمصنفین کے رفقاء ان کی علمی بصیرت اور بزرگانہ الفت سے ہر طرح کا استفادہ کررہے تھے کہ وہ اچانک دائمی طور پر ان سے جدا ہوگئے، وہ ۲۷؍ رمضان المبارک کو تراویح پڑھ کر اور تہجد اور فجر کی نماز ادا کرکے چار بجے صبح اعظم گڑھ سے اپنے وطن تُھلنیڈی ضلع رائے بریلی عید منانے روانہ ہوئے، وہاں پہنچنے کے دوسرے روز سحری کے وقت اٹھے، یکایک بیہوش ہوئے اور جمعہ کے روز گیارہ بجے دن کو اﷲ کو پیارے ہوگئے، دوسرے دن عید کی نماز کے بعد ان کی طالب علمی کے محبوب اور شفیق ترین ساتھی اور اسلامی ممالک کے فاضل اجل مولانا سیدابوالحسن علی...
During the former era of the State of Bahawalpur the Nawabs were in perpetual conflict with their relatives. The Kehlwar family of Sindh and during the modern era remained under influence of the British. So we can say that the State of Bahawalpur remained under constant foreign influence and the Nawabs did not have chance to rule with liberty and ease. In spite of these facts, the government of the State had many Islamic qualities and there are clear effects of Fatwa on judicial system in both eras.
The project “Chronic effects of metals on their accumulation, DNA damage and oxidative stress in the fish” was conducted in four phases i.e. (i) acute toxicity of metals to the fish (ii) chronic exposure based accumulation of metals in the fish (iii) DNA damage caused by chronic exposure of metals (iv) effects of chronic exposure of metals on oxidative stress in the fish. The acute toxicities (96-h LC50 and lethal concentration) of cadmium, chromium, copper, lead and metals mixture (Cd+Cr+Cu+Pb) were determined for 120 days old three species of fish viz. Catla catla, Labeo rohita and Cirrhina mrigala, separately (n=10), under controlled laboratory conditions. After determination of 96-h LC50 of selected metals and MM, all the three species of fish were exposed to 2/3rd, 1/3rd, 1/4th and 1/5th of their respective 96-h LC50 for 14, 28, 42, 56, 70 and 84 days, separately, and after each period of exposure the accumulation of metals in their organs and the extent of DNA damage in the peripheral erythrocytes and oxidative stress in terms of superoxide dismutase (SOD), catalase (CAT) and peroxidase (POD) activities in the fish tissues were determined. The sensitivity of three species of fish, determined in terms of 96-h LC50, for all the individual metals and MM varied significantly, showing species-specific sensitivity towards each metals/MM. C. mrigala were significantly (p<0.05) less sensitive to all the individual metals and MM according to 96-h LC50. Exposure of metals mixture caused significantly higher toxicity to all the three species of fish, followed by Pb, Cu, Cd and Cr exposures. The accumulation of all metals in the fish exhibited significantly positive correlation with both dose and duration of exposure. Chronic exposure of metals caused their significantly higher accumulation in C. mrigala, followed by L. rohita and C. catla. However, the overall amassing of metals in the fish, during exposure of individual metal and MM followed the order: Cr > Cd > Cu > Pb. Fish gills, liver, kidney, muscle and brain exhibited significant variability for the accumulation of all the individual metals and MM. Fish liver and gills showed significantly higher ability to concentrate all metals while fish muscles showed significantly lowest ability to amass metals. The overall accumulation of metals in the organs of three species of fish followed the order: liver > gills > kidney > brain > muscles. The accumulation of all metals in the fish body showed significantly direct correlation with the concentration and duration of exposure. The chronic exposure of Cd, Cr, Cu, Pb and MM caused significantly variable damage to the DNA of peripheral blood erythrocytes of all the three fish species under investigation. However, C. mrigala exhibited significantly higher frequency of damaged nuclei, followed by L. rohita and C. catla. The genetic damage index (GDI) values for three species of fish, exposed to metals, varied significantly also. However, both C. mrigala and L. rohita showed significantly higher GDI values. The fish exposed to MM showed significantly higher mean GDI value while it was significantly lower due to Cd exposure. The overall nuclear damage in the fish, caused by individual metals and MM exposures, followed the order: MM > Cu > Pb > Cr > Cd. C. mrigala developed significantly longer tail length of comets while it was significantly lower in C. catla. There existed significant differences among individual metals and MM to induce comet tail lengths in the nuclei of fish erythrocytes. However, exposure of MM induced significantly longer tails, followed by Cu, Pb, Cr and Cd exposures. Therefore, both GDI and CTL of comets may be used to predict genotoxic potentials of various toxicants for the fish.