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On the Averages of Convex Functions

Thesis Info

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Author

Roqia, Gholam

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/bitstream/123456789/2054/1/1732S.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726805563

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“Behind every theorem lies an inequality”. Mathematical inequalities play an impor- tant role in almost all branches of mathematics as well as in other areas of science. The basic work ”Inequalities” by Hardy, Littlewood and Polya appeared 1934 [37]and the books ”Inequalities” by Beckenbach and Bellman published in 1961 [9] and ”An- alytic inequalities” by Mitronovic published in 1970 made considerable contribution to this field and supplied motivation, ideas, techniques and applications. This theory in recent years has attached the attention of large number of researchers, stimulated new research directions and influenced various aspect of mathematical analysis and applications. Since 1934 an enormous amount of effort has been devoted to the dis- covery of new types of inequalities and the application of inequalities in many part of analysis. The usefulness of Mathematical inequalities is felt from the very be- ginning and is now widely acknowledged as one of the major deriving forces behind the development of modern real analysis. This dissertation deals with the inequali- ties for Jensen inqualites involving average of convex functions, Hermite-Hadamard inequalities. Chapter 1 offers an overview of the basic results contains a survey of basic concepts, indications and results from theory of convex functions and theory of inequalities used in subsequent chapters to which we refer as the known facts. Chapter 2 we give proofs of convexity and Schur convexity of the generalized inte- gral and weighted integral quasi-arithmetic mean. An overview of assorted proofs of schur convexity of integral arithmetic mean is discussed. In a detailed proof, discrete Jensen inequality for integral arithmetic mean is derived. Also integral version of Jensen inequality for integral arithmetic mean is proved. Motivated by discrete and viiviii integral Jensen inequalities functionals are defined. Two different method is given for constructing new examples of exponentially convex functions from non trivial gen- erating families of functions. Mean value theorem are proved. Different classes of monotonically increasing Cauchy means are created. Chapter 3 gives us convexity and Schur convexity of functions connected to Hermite- Hadamrd inequality as well as Schur convexity of differences of Hermite-Hadamrd inequality and Hammar-Bullen inequality by different proofs. Applying assorted gen- eralizations of Hermite-Hadamard inequality and Hammer-Bullen inequality on some special families of functions from varied classes, n-exponentially convex functions are generated by quite new method. Lyponuve, Dresher and Gramm’s type inequalities are developed. Pretty different Stolarsky type means are derives preserving inherited monotonically increasing property. Chapter 4 deals with inequalities of higher order convexity and divided difference. Two of them use majorization results and others are related to Jensen inequalities and Hermite-Hadamrd inequality. Integral Jensen inequality for divided difference is proved. Applications of averages of 3-convex functions as first order divided difference of convex functions are acquired. Method of producing n-exponentially convex func- tions is applied using divided differences. Produced functions are used in studying Stolarsky type means In the fifth chapter results about averages values of convex func- tions with variable limits and average values of composition functions is given. Study functionals for inequalities proved by D.E. Wulbert ( call them Wulbert’s inequalities for convenience) for convex and three convex functions. Extensions, improvements are accomplished. Variety of Stolarsky type means of a concave (convex) functions are obtained.
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ORGANIZATIONAL ORGANIZATIONAL CULTURE OF EDUCATIONAL INSTITUTION IN INDONESIA ORGANIZATIONAL CULTURE OF EDUCATIONAL INSTITUTION IN INDONESIA

Organizational culture is defined as the underlying beliefs, assumption, values and ways of interacting that contribute to the unique social and psychological environment of an organization. There are two types of culture, i.e. Material culture and non-material culture, which had been organized and maintained according to the needs of human being. Source of the culture is originated from the product of human being’s thought. Human being itself is a unique individual whose behaviour influenced by many variations and shapes of culture in society. As for the object of organizational culture in educational institution, it included quality, achievement, and professionalism which can be  developed by educational institution.

In Vitro and in Vivo Evaluation of Traditional Herbs for the Control of Staphylococcus Aureus, a Causative Agent of Sore Throat

Staphylococcus aureus normally colonizes about 30 percent healthy population asymptomatically, but it has the potential to causes deep-seated infections. Life threatening diseases like septicemia, Pneumonia and endocarditis. S. aureus is also involved in many toxicoses like food poisoning. Moreover, it is considered as one of the most common causes of nosocomial infection worldwide and its increasing resistance to antibiotic makes it difficult to control spread within the community. The aims of this work was (a) to increase the knowledge about virulence and pathogenesis of S. aureus isolated from sore throat patients (b) validation of traditional herbs as an effective antibacterial remedy that can be used against methicillin resistantS. aureus (c) the effectiveness of these herbs in in vivosetting and finally (d) the toxicological assessment of active herbs. The first part of this thesis comprised of identification and characterization of S. aureus. Isolated from a sore throat patients. The strains were identified using conventional microbiological and molecular methods. The agar disc diffusion method was used to evaluate the antibiotic susceptibility profiling. Virulence-associated genes were detected using PCR while Logistic regression was used to test the likelihood of strains carrying combinations of genes involved in toxin production and/or host immune evasion. Highest resistance was observed against beta-lactam group followed by cephalosporin, lincosamide, tetracycline, macrolides and aminoglycosides. No resistance was observed against vancomycin and linezolid. Among genes involved in host immune evasion, Staphylococcus protein A (spa) was identified most frequently (81%) and proportions of capsular polysaccharides (CP8), clumping factor A (clfA) and intracellular adhesion A (ica A) were 78%, 68.5% and 40% respectively. Intracellular adhesion D (ica D) and capsular polysaccharide 5 (CP5) could not be amplified from any isolate. Toxin genes were present in 43.5% isolates. Staphylococcus enterotoxins genes (SEs) (69 %) were most frequently detected among toxin gene containing isolates, followed by enterotoxin (ETs) (24.09 %) and toxic shock syndrome toxin (TSST-1) (15 %). More than one toxin genes were present in the 32.53% isolates. Host immune evasion and toxin genes were not associated with each other. Coaand spa gene polymorphism and band association analysis revealed that spa negative isolates possess Coa 1200 and 900bp, whereas spa positive isolates contain coa of 650bp and 750bp. The spa, CP8 and sea may be considered molecular targets in designing treatment and control strategies. In the second part in vitro, antibacterial activities of 29 traditional medicinal plants used in respiratory ailments were assessed on multidrug-resistant Gram-positive and Gram-negative bacteria isolated from the sore throat patients and two reference strains. The methanolic, n-hexane and aqueous extracts were screened by the agar well diffusion assay. Bioactive fractions of effective extracts were identified on TLC coupled with bio-autography, while their toxicity was determined using hemolytic assay against human erythrocytes. Qualitative and quantitative phytochemical analysis of effective extracts was also performed. Methanolic extract of 18 plants showed antimicrobial activity against test strains. Adhatoda vasica (ZI = 17–21 mm, MIC: 7.12–62.5 ?g/ml), Althaea officinalis (ZI = 16–20 mm, MIC: 15.62–31.25 ?g/ml), Cordia latifolia (ZI = 16– 20 mm, MIC: 12.62–62.5 ?g/ml), Origanum vulgare(ZI = 20–22 mm, MIC: 3–15.62 ?g/ml), Thymus vulgaris (ZI = 21–25 mm, MIC: 7.81–31.25 ?g/ml) and Ziziphus jujuba (ZI = 14–20 mm, MIC: 7.81–31.25 ?g/ml) showed significant antibacterial activity. Alkaloid fractions of Adhatoda vasica, Cordia latifolia and Origanum vulgare and flavonoid fraction of the Althaea officinalis, Origanum vulgare, Thymus vulgaris and Ziziphus jujuba exhibited antimicrobial activity. Effective plant extracts show 0.93–4.1 % erythrocyte haemolysis. The results obtained from this study provide a scientific rationale for the traditional use of these herbs and emphasized on the in vivo validation of in vitro evaluation. The third part of study comprised of experiments on in vivo validation of in vitro finding. Six plants extract having MIC ≤ 100 μg/ml were selected for their in vivo validation against anti-MRSA activity in an animal model. Efficacy of plant extracts on the course of sore throat was checked by analyzing selected parameters and general health. Two experiments were conducted in this regard. A pilot study was conducted to determine the dose of clinical isolate required to induce clinical infection in mice. The dose at which infection established was used for the main experiment to check the effect of treatments on the progress of infections. In the main trial, after the establishment of infection treatment with plant extracts was given up to 7th-day after wards. Re-isolation, gross lesion scoring and bacterial load in tissues were measured. In addition, hematological parameters were recorded. The quantitative evaluation of bacterial load and blood parameters proved A. vasica and O. vulgare to be the best active plant even in in vivo setting. The fourth part of study comprises of the toxicological assessment of the A. vasica and O. vulgarea recommended dose in human (200 mg/ml/Kg) and double the recommended dose (400 mg/ml/Kg). Extracts were forcibly fed to the Mus musculus for 07 days consecutively and on 08 day blood sample was drawn serum were analyzed for known markers of toxicity including (ALT, AST, ALP, T. bilirubin), (Urea and Creatinine) test. Liver and kidney section were also studied through histology for any toxic effect of both extracts. High dose of both plants induced toxicity which was also supported by histological findings. It is concluded that A. vasica and O. vulgare plants showed and used to isolate broad spectrum antimicrobial compounds that will serve as a novel agent for controlling resistant and highly virulent strains of S. aureus. The plant contains a lot of secondary metabolites that can be explored for the treatment infections condition in place of antibiotics. It may prevent the development of resistance to life saving drugs.