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دشتِ افسوس میں اک پھول کھلا ہو جیسے
دشتِ افسوس میں اک پھول کھلا ہو جیسے
تو خزاں زاد شجر سے ہی ملا ہو جیسے
ایک مدت سے بیابانی تھی دل میں میرے
تو مرے غم مرے ہر دکھ کا صلہ ہو جیسے
روح مجروح تھی اور ادھڑے تھے ٹانکے دل کے
تیرے آنے سے ہر اک زخم سلا ہو جیسے
تم سے بچھڑیں تو کسی طور بھی ہم جی نہ سکیں
سانس تو سانس ہے تم دل کی جلا ہو جیسے
تم فضاؔ چھائی ہو اک ابرِ کرم کی صورت
ہم نہ مل پائے یہی خود سے گلہ ہو جیسے
أحكام الحرابة و إختطاف الطائرات
This research article consist unique study regarding constitution of Human being character building in the thoughts of eminent philosopher Shah Wali Ullah (1703-1764). In present critique the focus has been made to explore how individual characters build in the specific environments? How surrounding effects on the character building? Moreover linkage of Islamic ‘IB└DA and its positive impact on the Muslim society has been explored. In interpretation of Shah Wali Ullah, All ‘IB└DA are like tools which lead to generate four basic ethics i.e purity and transparency capitulation, gainful and abstinence. These are the basic moral code which are the ultimate result of the four kind of ‘IB└DA i.e prayer, fasting, zakat and hajj. Muslim has inestimable inner power in the form of six lat┐’ef )اطلفئ, )which ultimately resulted upon the change of behavior. Character building are etiquettes, noble practices, decentness and good morality. It is generally refers to a code of conduct, that an individual group or society hold as authoritative in distinguishing right from wrong. Ethics are phenomenon values and can develop up to reasonable universal standards. Conduct in Islam governs all aspects of life and specifically addresses such principles as truthfulness, honesty, trust, sincerity, brotherhood and justice, while Islam forbid false, conspiracy, dodge, rude, irascibility, corruption. To materialize the virtues and disgrace the fake a role model prophet Muhammad (S.A.W) were deputed from Allah to guide the human being. So In present article character building in the theory of Shah Wali Ullah especially while in other Muslims scholars in general has diagnosed.Homomorphic Images of Generalized Triangle Subgroups of Psl 2, Z
The modular group generated by two linear fractional transformations, u : z 7! 1 z and v : z 7! z1 z , satisfying the relations u2 = v3 = 1 [46]. The linear transformation t : z 7! 1 z inverts u and v, i,e, t2 = (vt)2 = (ut)2 = 1 and extends PSL(2; Z) to PGL(2;Z). In [72] a condition for the existence of t is explained. G. Higman introduced coset diagrams for.PSL(2; Z) and PGL(2;Z): Since then, these have been used in several ways, particularly for nding the subgroups which arise as homomorphic images or quotients of PGL(2;Z). The coset diagrams of the action of PSL(2;Z) represent permutation representations of homomorphic images. In these coset diagrams the three cycles of the homomorphic image of v, say v, are represented by small triangles whose vertices are permuted counter-clockwise, any two vertices which are interchanged by homomorphic image of u, say u, are joined by an edge, and t is denoted by symmetry along the vertical line. The xed points of u and v, if they exist are denoted by heavy dots. The xed points of t lies on the vertical line of symmetry. A real quadratic irrational eld is denoted by Qpd, where d is a square free positive integer. If =a1 + b1pdc1 is an element of Qpd, where a1;b1;c1;d; are integers, then has a unique representation such that a1;a2 1 dc1 and c1 are relatively prime. It is possible that ; and and its algebraic conjugate = a1 pdc1 have opposite signs. In this case is called an ambiguous number by Q. Mushtaq in [69]. The coset diagrams of the action of PSL(2;Z) on Qpddepict interesting results. It is shown in [69] that for a xed value of d, there is only one circuit in the coset diagram of the orbit, corresponding to each .Any homomorphism 1 : PGL(2;Z) ! PGL(2;q) give rise to an action on PL(Fq): We denote the generators ()1; ()1 and (t)1 by ; and t: If neither of the generators , and t lies in the kernel of 1; so that , and t are of order 2, 3 and 2 respectively, then 1 is said to be a non-degenerate homomorphism: In addition to these relations, if another relation ( )k = 1 is satised by it, then it has been proved in [74] that the conjugacy classes of non-degenerate homomorphisms of PGL(2;Z) into PGL(2;q) correspond into one to one way with the conjugacy classes of 1 and an element of Fq: That is, the actions of PGL(2;Z) on PL(Fq) are parametrized by the elements of Fq: This further means that there is a unique coset diagram, for each conjugacy class corresponding to 2 Fq. Finally, by assigning a parameter 2 Fq to the conjugacy class of 1, there exists a polynomial f() such that for each root i of this polynomial, a triplet ; ; t 2 PGL(2;q) satises the relations of the triangle group (2;3;k) =D ; ; t : 2 = 3 = ( t)2 = ( )k = ( t)2 = ( t)2 = 1E: Hence, we can obtain the triangle groups (2;3;k) through the process of parametrization. Thegeneralizedtrianglegrouphasthepresentationu;v : ur;vs;Wk;where r; s; k are integers greater than 1, and W = u1v1:::ukvk, where k > 1;0 < i < r and 0 < i < s for all i. These groups are obtained by natural generalization of (r;s;k) dened by the presentationsDu;v : ur = vs = (uv)k = 1E, where r;s and k are integers greater than one. It was shown in [37] that G is innite if 1 r + 1 s + 1 k 1 provided r 3 or k 3 and s 6, or (r; s;k) = (4;5;2): This was generalized in [4], where it was shown that G is innite whenever 1 r + 1 s + 1 k 1 . A proof of this last fact can be seen in [101].A generalized triangle group may be innite when 1 r + 1 s + 1 k > 1. The complete classication of nite generalized triangle groups is given in 1995 by J. Howie in [39] and later by L. Levai, G. Rosenberger, and B. Souvignier in [57] which are fourteen in number. As there are fourteen, generalized triangle groups classied as nite [39], our area of interest is the set of groups which are homomorphic images or quotients of PSL(2;Z). Out of these fourteen only eight groups are quotients of the modular group. In this study, we have extended parametrization of the action of PSL(2;Z) on PL(Fp), where p is a prime number, to obtain the nite generalized triangle groupsD 2 = 3 = 23 = 1E by this parametrization. By parametrization of action of PGL(2;Z) on PL(Fp) we have obtained the coset diagrams of D 2 = 3 = 23 = 1E for all 2 Fp. This thesis is comprised of six chapters. The rst chapter consists of some basic denitions and concepts along with examples. We have given brief introduction of linear groups, the modular and the extended modular group, real quadratic irrational elds, nite elds, coset diagrams, triangle groups, and generalized triangle groups. In the second chapter, we show that entries of a matrix representing the element g =(v)m1v2m2l where l 1 of PSL(2;Z) =;v : 2 = v3 = 1are denominators of the convergents of the continued fractions related to the circuits of type (m1;m2); for all m1;m2 2N: We also investigate xed points of a particular class of circuits of type (m1;m2) and identify location of the Pisot numbers in a circuit of a coset diagram of the action of PSL(2;Z) on Qpd[f1g, where d is a non-square positive integer.In the third chapter we attempt to classify all those subgroups of the homomor phic image of PSL(2;Z) which are depicted by coset diagrams containing circuits of the type (m1; m2). In the fourth chapter we devise a special parametrization of the action of modular group PSL(2;Z) on PL(Fp), where p is prime, to obtain the generalized triangle groups D 2 = 3 = 2k = 1E and by parametrization we obtain the coset diagrams of D 2 = 3 = 2k = 1E for all 2 Fp. In the fth chapter we investigate the action of PSL(2;Z) on PL(F7n) for di⁄erent values of n, where n 2 N, which yields PSL(2;7). The coset diagrams for this action are obtained, by which the transitivity of the action is inspected in detail by nding all the orbits of the action. The orbits of the coset diagrams and the structure of prototypical D168 Schwarzite [48], are closely related to each other. So, we investigate in detail the relation of these coset diagram with the carbon allotrope structures with negative curvature D168 Schwarzite. Their relation reveals that the diagrammatic structure of these orbits is similar to the structure of hypothetical carbon allotrope D56 Protoschwarzite which has a C56 unit cell. In the last chapter, we investigate the actions of the modular group PSL(2;Z) on PL(F11m) for di⁄erent values of m; where m 2 N and draw coset diagrams for various orbits and prove some interesting results regarding the number of orbits that occur.Journals by Discipline
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