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Algorithms for Primary Decomposition

Thesis Info

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Author

Idrees, Nazeran

Program

PhD

Institute

Government College University

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2006

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/handle/123456789/1029

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676725452343

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The primary decomposition methods of Eisenbud, Huneke and Vasconcelos are anal- ysed in detail providing proofs of important theorems and all the corresponding al- gorithms are programmed in the language of Singular. MOreover, we investigated the parallelization of two modular algorithms. In fact, we consider the modular com- putation of Gr ̈obner bases (resp. standard bases) and the modular computation of the associated primes of a zero–dimensional ideal and describe their parallel imple- mentation in Singular. The algorithms of Shimoyama and Yokoyama for primary decomposition of ideals are generalized to submodules of a free module over the polynomial ring in several variables with coefficients in a field. The algorithms are implemented in Singular.
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وحید الدین سلیم ؔپانی پتی

مولوی وحیدالدین سلیمؔ پانی پتی
مولانا وحیدالدین سلیم پانی پتی، عربی اور اردو کے ادیب تھے، وہ مولانا فیض الحسن صاحب سہارنپوری کے شاگرد تھے، لاہور کے مشرقی شعبہ میں تعلیم پائی تھی اور وہیں سے تحریر و انشاء اور ترجمہ و تالیف کا شوق اپنے ساتھ لائے تھے، ۱۸۹۰؁ء کے بعد سے غالباً سرسید مرحوم کے علمی مددگار مقرر ہوئے، یعنی سرسید کی تصنیفات اور مضامین کے لئے عربی کتابوں سے معلومات فراہم کیا کرتے تھے، پھر معارف نام کا ایک علمی رسالہ انہوں نے علی گڑھ سے نکالا، جس نے اہل علم میں بڑی عزت حاصل کی، چند سال نکل کر یہ بند ہوگیا، پھر ۱۹۰۴؁ء کے قرب میں وہ علی گڑھ گزٹ کے ایڈیٹر ہوئے اور بالآخر اس سے بھی الگ ہوکر خانہ نشین ہوگئے، ۱۹۱۰؁ء میں جب لکھنؤ سے مسلم گزٹ نکلا، جس نے مسلمانوں کی اس نئی سیاسی بیداری میں خاصہ حصہ لیا، تو مولانا شبلی مرحوم کے مشورہ سے وہی اس کے اڈیٹر مقرر ہوئے اور حق یہ ہے کہ انہوں نے بڑی خوبی سے اس فرض کو انجام دیا، مسلم گزٹ کے بند ہونے کے بعد وہ پھر خانہ نشین ہوگئے اور آخر غالباً ۱۹۱۶؁ء میں یا اس کے گردوپیش زمانہ میں وہ حیدرآباد گئے اور جامعہ عثمانیہ میں اردو کے پروفیسر مقرر ہوئے اور اسی منصب پر اس مہینہ میں انہوں نے ملیح آباد (ضلع لکھنؤ) میں وفات پائی، مرحوم کی عمر ستر سال کے قریب ہوگی۔
مرحوم نے چھوٹے بڑے مضامین بے شمار لکھے، ان کی خاص خصوصیت زود نویسی تھی، وہ قلم برداشتہ لکھتے تھے اور بڑے بڑے ہفتہ وار اخبار کو ایک رات میں بیٹھ کر پورا کرلیتے تھے اور ان کی کوئی مستقل تصنیف، ’’وضعِ اصطلاحات علمیہ‘‘ کے سوا دوسری نہیں، نئے الفاظ کے تراشنے اور وضع کرنے میں ان کو پوری مہارت تھی، علی...

حدیثِ لولاک کا تحقیقی و تجزیاتی مطالعہ A Research Review of the Hadith e Lawlāk

The importance and authenticity of the hadiths of the Prophet (PBUH) in the Muhammadan Ummah is an undisputed fact. A large number of Muḥaddithīn and imams spent their lives in collecting and editing the hadiths of the Prophet (PBUH) and compiled a collection of hadiths. These hadiths were classified according to their health and weakness. Due to the efforts of the imams, there are hadiths narrated from the Messenger of Allah on all sciences and knowledge and issues and rulings. Among them are Hadith Qudsī, Ma‘rūf, Mawkūf, Maktū‘, Saḥiḥ, Ḥasan Za‘īf and all other types of Hadith. The same Hadith e Qudsīyyah also includes a Hadith Lawlāk "لولاک" refers to those hadiths in which the Prophet ﷺ has been described as the cause of the creation of the universe and the origin of the universe. That is the hadith in which Allah Almighty has related the Prophet (PBUH) as the reason for creating the universe and its beings. However, the words of "Hadith e Lawlāk" The authenticity of the hadith has been made a subject of discussion. Some imams have declared this hadith to be weak and some have also said that it is valid. However, a large number of scholars have agreed on the authenticity of this hadith and the hadith has been declared correct in terms of meaning and has been recorded in the books of hadith, biographies and interpretations and other subject books. In the article under review, the sayings of Imams on the health and weakness of Hadith Lawlāk have been presented and research analysis of the said Hadith has been done through simulation and rational arguments.

Effect of Doping of Various Metal Cations on Structural, Electrical and Magnetic Properties of Nano Cobalt Ferrites

Cobalt ferrite, having an inverse spinel structure and the inherent properties of high coercivity, moderate saturation magnetization and high electrical resistivity, is a potential candidate for magnetic storage devices and high frequency applications. In the present study, cobalt ferrite has been doped with various dopants like Cr, rare earths (Sm, Ho, Er, Dy and Pr) and Zr co-doped with Mg, Mn and Ni, in order to improve the electrical and magnetic properties while maintaining a spinel structure and moderate saturation magnetization values a micro-emulsion method of preparation in which a cheap surfactant, namely polyethylene glycol, has been used. The formation of spinel phase occurs between 573 and 673K as indicated by the thermal analyses (TG/DTA), but a well crystalline and stable spinel phase is achieved at 1073K as evident from the powder X-ray diffraction studies of the synthesized samples. All the doped cobalt ferrite samples are in single spinel phase as confirmed by XRD and magnetic susceptibility measurements. The average crystallite sizes of the doped cobalt ferrite samples are in the range of 13 nm to 70 nm. The elemental composition of doped cobalt ferrites is confirmed by energy dispersive X-ray fluorescence analysis which shows an agreement between the theoretical and experimental compositions of the prepared samples. Electrical resistivity as measured at 293K the by two point probe method is found to have a value of 1.25 × 106 Ωm for un-doped cobalt ferrite which is enhanced by doping with Cr, Zr-Mg, Zr-Mn and Zr- Ni. For small contents of rare earth metal cations introduced into spinel lattice of cobalt ferrite the electrical resistivity (at 293K) increases to a larger extent due to insulating nature of rare earth oxides. The variation in electrical resistivity with composition and temperature has been discussed on the basis of hopping model of electron conduction in ferrites. The activation energy of hopping and drift mobility of the charge carrier is calculated from the resistivity data. The dielectric properties are measured by inductance capacitance resistance (LCR) meter in the frequency range of 100Hz – 1MHz and dielectric constant (έ), dielectric loss angle (tanδ) and dielectric loss factor ( ε ′′ ) are calculated from the capacitance data. These dielectric parameters are found to decrease with increasing frequency. This behaviour is typical of ferrites as explained by Koop’s model. The dielectric constant (έ) and the dielectric losses of the un-doped cobalt ferrite have been reduced by doping in the present work. Curie temperature has been determined from the low field AC-magnetic susceptibility measurements and was found to increase for specific contents of dopants as compared to the un-doped cobalt ferrite while for others a lower value of Curie temperature was observed. Saturation magnetization has been increased by doping with Cr up to x = 0.5, Zr-Mn content x = 0.1, Sm content x = 0.04 and Er content x = 0.08 while for the rest of compositions the saturation magnetization has been decreased as compared to the un-doped cobalt ferrite sample. Beside this, the coercivity of the materials prepared in the present study has been increased by doping rare earth metal cations, while it has been reduced by Cr and Zr co-doped with Mg, Mn and Ni, in cobalt ferrites. The reduction in magnetization has been discussed in terms of dilution of magnetization, crystallite size effects and the spin canting introduced by the dopants at octahedral sites.