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Means of Cauchy’S Type

Thesis Info

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Author

Matloob Anwar

Program

PhD

Institute

Government College University

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2008

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/handle/123456789/584

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726624246

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Deciding where to begin is a major step. One procedure is to lay out all necessary preliminary material, introduce the major ideas in their most general setting, prove the theorems and then specialize to obtain classical results and various applications. We experience convexity all the times and in many ways. The most prosaic example is our upright position, which is secured as long as the vertical projection of our center of gravity lies inside the convex envelope of our feet. Also convexity has a great impact on our every day life through numerous applications in industry, business, medicine and art. So do the problems of optimum allocation of resources and equilibrium of non cooperative games. The theory of convex functions is a part of the general subject of convexity, since a convex function is one whose epigraph is a convex set. Nonetheless it is an important theory, which touches almost all branches of mathematics. In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal (parallel) to the ”average” derivative of the section. It is used to prove theorems that make global conclusions about a function on an interval starting from local hypotheses about derivatives at points of the interval. This theorem can be understood concretely by applying it to motion: if a car travels one hundred miles in one hour, so that its average speed during that time was 100 miles per hour, then at some time its instantaneous speed must have been exactly 100 miles per hour. An early version of this theorem was first described by Parameshvara (1370-1460) from the Kerala school of astronomy and mathematics in his commentaries on Govin- dasvami and Bhaskara II. The mean value theorem in its modern form was later stated viiviii by Augustin Louis Cauchy (1789-1857). It is one of the most important results in differential calculus, as well as one of the most important theorems in mathematical analysis, and is essential in proving the fundamental theorem of calculus. The mean value theorem can be used to prove Taylor’s theorem, of which it is a special case. We use this Mean value theorem and its other generalized version to define new Cauchy’s means. In the first chapter some basic notions and results from the theory of means and convex functions are being introduced along with classical results of convex functions. In the second chapter we define some further results about logarithmic convexity of differences of of power means for positive linear functionals as well as some related results. In the third chapter we define new means of Cauchy’s type. We prove that this mean is monotonic. Also we give some applications of this means. In the fourth chapter we give Cauchy’s means of Boas type for non positive measure. We show that these Cauchy’s means are monotonic. In the fifth chapter, we give definition of Cauchy means of Mercer’s type. Also, we show that these means are monotonic. In the sixth chapter, we define the generalization of results given by S. Simi ́c, for log- convexity for differences of mixed symmetric means. We also present related Cauchy’s means. In the last chapter we give an improvement and reversion of well known Ky-Fan inequality. Also we introduce in this chapter Levinson means of Cauchy’s type. We prove that these means are monotonic.
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سید عدید ؔ

سید عدید ؔ(۱۹۶۵ء پ) کا اصل نام تنویر حسین شاہد ہے۔ آپ کھروٹہ سیداں سیالکوٹ میں پیدا ہوئے۔ ایم۔اے اردو گورنمنٹ مرے کالج سیالکوٹ سے کیا۔ ۱۹۸۰ء میں مرے کالج میں آپ حلقہ اربابِ ذوق کے جائنٹ سیکرٹری تھے۔ آپ نے شاعری میں یوسف نیر اور اصغر سودائی سے ابتدائی راہنمائی لی۔ سب سے پہلے مرے کالج کے ادبی رسالے ’’مفکر‘‘ میں آپ کا کلام شائع ہوا۔ ’’بالتحقیق ‘‘ سیالکوٹ اور ’’اخبار جہاں‘‘ لاہور میں بھی ان کا ابتدائی کلام چھپتا رہا۔ (۱۱۴۹) ’’وقت ‘‘ سید عدید کا پہلا شعری مجموعہ ہے۔ جو سیالکوٹ سے ۱۹۸۸ء کو شائع ہوا۔ دوسرا شعری مجموعہ ’’تلاش‘‘۱۹۹۴ء میں شائع ہوا۔ ’’ہم نفس‘‘ تیسرا شعری مجموعہ ہے جو ۱۹۹۵ء میں شائع ہوا۔ آپ کا چوتھا شعری مجموعہ ’’فریب دے کر چلا گیا ہے‘‘ ہے جسے ادیب پبلی کیشنز لاہور نے ۱۹۹۶ء میں شائع کیا۔ ’’محبتوں میں حساب کیا‘‘عدید کا پانچواں شعری مجموعہ ہے جسے الحمد پبلی کیشنز لاہور نے ۱۹۹۸ء میں شائع کیا۔ چھٹا شعری مجموعہ ’’پیاربے اختیار ہوتا ہے‘‘ جسے الحمد پبلی کیشنز لاہور نے ۲۰۰۰ء میں شائع کیا۔ ’’ساتھ تمہار ا اگر ملے‘‘ ساتواں شعری مجموعہ ہے۔ جسے القلم پبلی کیشنز لاہور نے ۲۰۰۶ء میں شائع کیا۔ آٹھواں شعری مجموعہ ’’تیرے بن زندگی‘‘ ہے جسے مراد پبلی کیشنز لاہور نے ۲۰۱۰ء میں شائع کیا۔ اس کے علاوہ ’’وفائیں ساتھ رہتی ہیں‘‘ ،’’گردش‘‘ ،’’تمنادل میں رہتی ہے‘‘، ’’درد کے سمندر میں‘‘،عدید کے زیر طبع کتابوں کے نام ہیں جو جلد شائع ہونے والی ہیں۔کافی مسودے ایسے بھی ہیں جن کے نام ابھی تک تجویز نہیں کیے گئے ہیں۔

                عشق مجازی سید عدیدؔ کی شاعری کا بڑا موضوع ہے۔ ان کے ہاں نسوانی عشق کے ساتھ ساتھ جنون بھی ملتا ہے۔ وہ محبوب کی بات بھی کرتے ہیں اور ا س کے ظلم و ستم کا ذکر بھی کرتے ہیں۔ انھیں اپنے محبوب سے سچی...

عقائد باطلہ كا رد ، شیخ احمد سرہندی کا کردار اورعصری استفادہ

Shaikh Ahmed Sirhindi, known as Mujaddid Alf Sani (R.A) was not only a well-known Muslim mystic but also a great reformer. He brought reformative changes in political system of subcontinent and introduced social set up of Islamic society,  spiritual patterns of Islamic mystics and religious scholars of Islamic sciences. His services regarding the revival of Islam and purification of Islamic beliefs are versatile and multidimensional. Mughal emperor Akbar tried to reconcile the differences of both religions, and introduced a new faith called the Din-i-Ilahi, which incorporated both Islam and Hinduism which stressed the Muslim community a lot. This article is a study of the role of Shaikh Ahmad Sirhandi in purification believes. Main tiopics of this study are oneness of Ram and Raheem, wahdat-ul-wajood and wahdat-ul-Shahood, finality, eternality, sustainability and universality of Prophethood, Meraj-ul-Nabi, Aqeeda Tanasukh and Aqeeda Halool with reference to the life hereinafter. Article also mentions the methodology of this pious personality in purification of believers and its utility in contemporary period. This ends with findings of the research paper and no doubt it provides important guidance for researchers of Islam to face the challenges for the betterment of Muslim Ummah.

High Throughput Molecular Screening of Citrus Tristeza Virus from Trhe Citrus Germplasm of Punjab and Production of Virus-Free Citrus Plants Via Shoot-Tip Grafting.

Citrus is the prominent tree fruit crop consumed world wide because of its wide nutritional and health values. Pakistan is also famous for its citrus production that plays an important role in establishing the country’s economy. Tactlessly, various diseases including many viruses, virus-like, bacterial and fungal diseases are damaging the citrus crop in the world including Pakistan. Citrus tristeza virus (CTV) is the causal agent of the most deadly virus disease of citrus leaving a disastrous impact on citrus production. It is a phloem-limited and graft-transmissible virus that belongs to Closteroviridae family transferred by using the already infected propagative material or by vector, causing a wide range of symptoms depending on the isolate and the host. The current research was designed to study the prevalence of CTV in Pakistan screening the two major regions of Punjab i.e. Sargodha and Silanwali using PCR techniques and the ultimate goal was the establishment of virus-free citrus plants via in vitro shoot-tip grafting (STG). A total of 400 citrus samples representing major citrus groups including sweet oranges, mandarins, grapefruits, kumquats, tangelos, lemons and limes, rootstocks and others were collected. Symptoms in the field ranged from asymptomatic or the symptomatic plants bearing pin holing on the bark, seedling yellow, vein clearing, leaf and bark flecking, necrotic lesions on the leaf edges and twigs, stunted growth and quick decline like symptoms were noticed. RNA extraction was done using two methods i.e. Trizol and SDS-Triton. The quality of RNA from both the extraction methods was checked using 37% formaldehyde denaturing agarose gel electrophoresis as well as using UV-VIS Nanodrop spectrophotometer. Molecular screening of CTV using PCR techniques was used to confirm the presence or absence of CTV. Initial detection was done using conventional PCR with cDNA as a template in which 12 samples were positive for CTV with an infection rate of only 3%. A total of 400 RNA samples were checked for their integrity, out of which 392 samples were positive for NAD5 and COX assay and were used for further CTV screening. Cq values for NAD5 ranged from 15.40 to 28.50 with an average of 19.84 for 392 samples. COX Cq values ranged from 12.33 to 27.00 with an average of 16.00 for 392 samples. The detection of CTV using highthroughput qRT-PCR was done using reported primers and TaqMan probe. Twenty-five samples were found positive for the CTV using both the multiplex (CTV, CPsV and CLBV) as well as singleplex (CTV) assay. The Cq values for the CTV positive samples in a multiplex and singleplex assay ranged from 11.4-32.0 and 11.413-31.749 respectively. The estimated infection rate for CTV was 7% in both the regions ix of Punjab. The most infected group found was sweet oranges (22/252), followed by mandarins (2/52) and lemons and limes (1/25). The infection rate between the two regions Sargodha and Silanwali was 6.7% and 2.6% respectively. A one-step RT-PCR was used to further characterizes the CTV positive samples. As one-step RT-PCR is far less sensitive from qPCR, out of 25, only 15 samples reacted with universal coat protein primers for the detection of CTV at the desired amplicon size of 672bp. The fifteen CTV positive samples from one-step RT-PCR for the conserved generic coat protein unit (CPU) were characterized for the presence of common CTV genotypes as VT, T30. T36, T3, RB and a newly found S1 using a multiple molecular marker (MMM) method based on the one-step RT-PCR amplification of sequence specific PCR products with sets of primers derived from the most conserved sites within the genomes of VT, T30, T36, T3, RB and a newly discovered S1 isolates. Out 15 CTV-CPU positive samples 10 samples reacted with VT marker, 2 reacted with T30 and RB marker and 1 reacted for T36, T3 and S1 marker respectively. Many of the genotypes were found in mix infection such as VT+T30, VT+T36 and VT+T30+S1 etc. The CTV coat protein gene, VT, T30, T36, T3, RB and S1 isolates were sequenced for the phylogenetic analysis with isolates present worldwide. The results of the nucleotide sequence comparison of individual isolate were done using MEGA 7.0. phylogenetically. The Pakistani isolates showed highly similar homology with American, Italian, Indian and Chinese isolates. The control of CTV and many other graft-transmissible diseases is now possible by using the disease free nursery plants, controlling the vectors and the eradication of diseased plants. Availability of virus free planting material is the utmost necessity of the citrus industry in Pakistan for which in vitro shoot-tip grafting has already proved successful in establishing the clean plant material. In vitro shoot-tip grafting technique was introduced for the production of pathogen tested citrus plants. Nine commercially important citrus cultivars were STGed and were indexed at various steps. The most significant aspect of the present study was to learn and develop the procedure for in vitro shoot-tip grafting for establishing the virus-free citrus plants that will help the citrus growers of Pakistan in providing the clean plant material. The procedure involved rootstock preparation, scion preparation, in vitro shoot-tip grafting, acclimatization, pre- and post- indexing of the successful grafts.