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On Means and Inequalities for C0-Semigroup of Operators

Thesis Info

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Author

Gul I Hina Aslam

Program

PhD

Institute

National University of Sciences & Technology

City

Islamabad

Province

Islamabad

Country

Pakistan

Thesis Completing Year

2016

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/9920/1/Gul_I_Hina_Aslam_Mathematics_2016_HSR_NUST_31.08.2016.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726799119

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The strongly continuous semigroups of operators are of great significance due to the fact that these have numerous applications in various areas of science. The qualitative understanding of a complicated deterministic system can be acquired by analyzing the solutions of a related differential equations in terms of these operators. Moreover, the theory of convexity, means, Cauchy means and inequalities has a huge impact in everyday science. Therefore, the pronounced nature of the means and inequalities defined on semigroups of operators, can not be contradicted. This dissertation is a staunch effort to generalize the theory of means and inequalities to the operator semigroups. A new theory of power means is introduced on a C0-group of continuous linear operators. A mean value theorem is proved. Moreover, the Cauchy-type power means on a C0-group of continuous linear operators, are obtained systematically. A Jessen’s type inequality for normalized positive C0-semigroups is obtained. An adjoint of Jessen’s type inequality has also been derived for the corresponding adjoint semigroup, which does not give the analogous results but the behavior is still interesting. Moreover, it is followed by some results regarding exponential convexity of complex structures involving operators from a semigroup. Few applications of Jessen’s type inequality are also presented, yielding the Hölder’s type and Minkowski’s type inequalities for corresponding semigroup. Moreover, a Dresher’s type inequality for two-parameter family of means, is also proved. A Jensen’s and Hermite-Hadamard’s type inequalities are also obtained for a semigroup of positive linear operators and a superquadratic mapping defined on a Banach lattice algebra. The corresponding mean value theorems conduct us to find a new sets of Cauchy’s type means.
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بھولتا ہی نہیں وہ خواب مجھے

بھولتا ہی نہیں وہ خواب مجھے
وہ ملا جس میں بے نقاب مجھے

کرچیاں اتنی ہو گئیں دل کی
رکھنا مشکل ہے یہ حساب مجھے

میرا دشمن نصیب ہے میرا
ہونے دے گا نہ کامیاب مجھے

اشک بہتے ہیں، تارے گنتا ہوں
دے گیا ہجر کا نصاب مجھے

جو گزرتے ہیں تیرے بن تائب
لمحے لگتے ہیں وہ عذاب مجھے

تمسکات میں سرمایہ کاری اور اس کی شرعی حیثیت تحقیقی وتجزیاتی مطالعہ

Islam is a complete code of life which provides guidance in political, social and economic affairs. Economics deals with very importnt sphere of human life that involves struggle for survival. This struggle is always appreciated because Allah Almighty Himself motivates for it. The basic aim of this is to eradicate poverty and huger and to bring happiness and satisfaction in society but the condition is that all the economic activities should be done with within the limits of sharia. Otherwise the efforts of human beings in this world as well as hereafter will never be successful. In contemporary economic trends, investment in securities is well known and popular. Government and private institutions issue bonds, shares, debentures etc to provide economic security to the people but in various types of sceurities Shar฀‘ah laws are not taken care of. Where, for a Muslim, injunctions of Shar฀‘ah are everything. There are tidings of rewards on obeying these injunctions and warning of punishment on their violation boath in this world and hereafter. Economic experts suggest to invest but People remain uncertain in these schemes. Regarding this objective, in this article few types of securities (Shares, Debentures and Prize Bonds) have been discussed in Shar฀‘ah perspectives and prize bonds were given special attention because of difference of openion of scholars about it. Some alternate solutions which may provide an insight into Islamic fiscal monetary system have been provided at the end.

Competitive Analysis of Pakistan Higher Education and Strategic Response A Case Study of Khyber Pakhtunkhwa Private and Public Sector Institutions

This research work includes the exploration of ethnobotanical, pharmacognostic, physicochemical and pharmacological aspects of Monotheca buxifolia (Falc.) A. DC. of the dicotyledonous family, Sapotaceae. It is locally known as ―Gurgura‖ and is used as a source of fuel, fodder, agricultural tools and for Honey bee farming in various areas of Khyber Pakhtunkhwa. Medicinally the fruits are used as a digestive, purgative, laxative, in urinary disorders, diuretic, vermifuge, refrigerant, tonic and in antiseptic pastes. Morphological, anatomical and organoleptic features of different parts of the plant (fruit, seed, leaf, stem bark and root bark) were worked out in detail. M. buxifolia had a palisade ratio of 6.75± 0.5, vein islets number 37.4±2.88 per mm2, vein termination number 35.4±1.8 per mm2, stomatal number of lower epidermis 77.4±3.57 per mm2 and 7.75± 0.97 on upper epidermis while stomatal index of lower epidermis 10.53±0.40 and upper epidermis 5.39±0.33. Stomata on the upper epidermis were of actinocytic type while on the lower epidermis actinostephanocytic type of stomata were present. The powder drug studies of the fruit, seed, leaf, stem bark and root bark showed characteristic fragments.The qualitative preliminary phytochemical screening of different parts of M. buxifolia gave positive indications for presence of aminoacids, proteins, reducing and nonreducing sugars, fixed oils, fats, glycosides, alkaloids, tannins, phenolic compounds, saponins, anthocyanins, Triterpenoids, phytosterols and Flavonoids in both aqueous and methanol extracts. The florescence behaviour, moisture contents and ash values were also worked out. The fruit pulp, seeds and leaves contained 8.33%, 10.62% and 0.56% fixed oil respectively. Linolenic acid, Oleic acid, Palmitic acid, Myristic acid and Stearic acid were the major component fatty acids. Elemental analysis of various parts of M. buxifolia revealed presence of good quantities of Nitrogen, potassium and phosphorus. Trace elements were found to be within the WHO permissible limits except for lead (22.48±0.33) and Cobalt iv (10.7±0.01), which exceeded the permissible limits in barks of stem and root respectively.The methanol extract of M. buxifolia fruit and leaf were tested at 500, 1000 and 2000 mg/kg body weight doses for any toxicological effects. They were found to be safe at all the tested doses.The methanol extracts of different parts of M. buxifolia were evaluated for their cytotoxic potentials against Artemia salina larvae. The fruit pulp had no visible cytotoxic effects. The seed extract produced most significant cytotoxic effects, giving an LD50 value of 4.668 (µg/ml) followed by root bark (31.265 µg/ml), leaf (97.59 µg/ml) and stem bark (199.65 µg/ml). In the in vitro spasmolytic bioassay the crude methanol extract of M. buxifolia fruit produced a significant inhibition of jejunal contractions through cholinergic pathway and voltage gated calcium channel blockade, similar to verapomil.The 250, 500 and 750 mg/kg doses of M. buxifolia fruit and leaf extracts produced significant antidiarrheal effects against Castor oil induced diarrhea in mice, in a dose dependent manner. The fruit extract caused 38.8%, 61% and 61.2% while the leaf extract caused 67.2%, 74.6% and 91.1% inhibition of faecal droppings, respectively, as compared to negative control. The 250 and 500 mg/kg doses of crude methanol extract of M. buxifolia fruit produced remarkable hepatoprotective activity against paracetamol induced hepatic damage in mice. the post-damage treatment was more effective and comparable to the standard hepatoprotective drug, Silymarin. The extract significantly (P ≤ 0.05) lowered the elevated levels of biochemical markers (Total Bilirubin, Direct Bilirubin, SGPT, alkaline phosphatase and Gamma‐GT) towards normal. Histopathological studies further confirmed hepatoprotective potentials of the plant. This study provides useful pharmacognostic standards for M. buxifolia and elaborates its pharmacological significance.