On a Review of the Lebesgue Integrals with Application to Probability Theory

There are two primary motivations for the study of Lebesgue measure theory and they are

a) It is desirable to measure the length of any subset of the real line

b) It is desirable to have a theory of the integral in which the syllogism

Holds in greatest possible generality. It turns out that both these desidereta are too ambiguous. In fact (a) is impossible. In order to have a feasible and useful theory of measuring sets, we must restrict attention to a particular class of sets. As for (b), we can certainly construct a theory of the integral in which (I) is easy and natural. But there is no ”optimal” theory.

The Lebesgue integral addresses both of the above issues very nicely. We shall invest a few pages in this work to providing a brief introduction to the pertinent ideas. We will not be able to prove all the results, but can state them all precisely and provide some elucidating examples. In this work we will discuss the application of some of the important results of the Lebesgue measure to probability theory. Note that the notion of length that we shall develop here is called a ”measure” However, probability dates to the days of B. Pascal (1633-1662) and even before, when gamlers wanted to anticipate the results of certain bets. The subject did not develop space, and was fraught with paradoxes and conundrums. It was not until 1933, when A.N. Komogorov (1903-1987) realized that measure theory was the correct language for formulating probabilistic statements, that the subject could be set on rigorous footing.

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On the Uniform Bounded ness Principle with Applications to Space of Polynomials and Fourier Series

The uniform bounded ness principle is of great importance as throughout analysis as there are many instances of results related to this theorem. The development of this theorem greatly centres on Baire’s category theorem which is contained in section one of this work. Section two is the detail of the theorem in full while section three centres on its applications. Finally, the major objective of this work is to classically present the Uniform Bounded ness Principle as a vital ingredient in the theory of Hilbert Space Analysis.

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The Role of Gender Diversity on the Firms Performance a Case of India

The paper examines the role of gender diversity in the financial performance of the listed companies in India. In the literature, the role of gender diversity is extensively studied, but little attention has been paid to developing countries like India. The results found that there is a significant positive relationship between gender diversity and firms performance.

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The Corporate Governance Emergence in India

This study gives an overview of the corporate governance in India. The corporate governance issue has been emerged all around the world and still not fully developed in emerging economies like India. From the previous study in the last decade, it has been found India makes the impressive progress towards not only the implementation but also on the improvement in the system.

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ESSENTIAL OIL PHYSICOCHEMICAL COMPOSITION OF THE Thymus fontanesii COLLECTED FROM TIARET A SEMI-ARID REGION OF ALGERIA

The aim of this work was to determine the exact composition of the essential oil extracted from Thymus fontanesii collected from Tiaret at the western region of Algeria by gas chromatography method. The fresh aerial parts of T. fontanesii were collected at the flowering stage on May 2017 from Tiaret, at the western region of Algeria, characterised by a semi-arid climate. The essential oil extracted was dried with anhydrous sodium sulfate and kept at a cool temperature of +4°C in glass bottles away from light. The essential oil from leaves of Thymus fontanesii was analysed by gas chromatography (GC-FID) and gas chromatography–mass spectrometry (GC/MS) realized by Pyrenessences Analysis, France. The oil was obtained from the aerial parts of T. fontanesii with an average yield of 2,256±0,094%. The chromatographic analyses identified ninety-one (91) compounds which represents approximately 99,92% of the total detected oil components mainly represented by carvacrol with 62,25% being by far the main component the other appreciable constituents identified were γ-terpinene with 12,62%, β-cymene with 7,43%, linalool with 4,26%, β-myrcene with 1,29% and α-terpinene with 1,31%. and thymol represented only 1,76% of the composition.

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Effect of Different Concentration of (Indole 3-Butyric Acid) on Stem Cutting of Jamrul (Syzygium samarangense) at Patuakhali District under Dumki Upazila

The experiment was carried out at the Agriculture research field, Patuakhali Science and Technology University. Data on rooting of Wax Jambu were collected after 5 weeks of setting the cuttings inside the propagator. This research work was describes the scope of clonal propagation of Jamrul (Syzygium samarangense) by stem cutting. It was laid out completely randomized block design (RCBD) with 4 replications. The treatments were different concentrations of IBA (Indole 3-Butyric Acid) solution as T0 (0% IBA or Control), T1 (0.2% IBA), T2 (0.4% IBA) and T3 (0.8% IBA). Cuttings were treated with IBA and placed inside the non-mist propagation system. The rooting ability of cuttings was significantly influenced by the application of IBA. The highest rooting percentage (85.75) and root diameter (1.53) were obtained from the cuttings treated with 0.4% IBA solution followed by 0.2% IBA. Maximum root number (21.6) was obtained from the cuttings treated with 0.4% IBA solution followed by 0.8% IBA. The longest root length (14.07 cm) was recorded with 0.4% IBA followed by 0.2% IBA (8.65 cm). Rooted cuttings were transplanted in the polythene bags filled with soil and cow dung mixed at the ratio of 3:1 (by volume) for further growth and steckling production. The objective of the present investigation was to evaluate rooting ability from the cutting of small leafy branch with or without rooting hormone IBA in low cost farmers friendly non-mist propagator and clone good quality syzygium samarangense species and propagate through stem cutting. As a Results revealed that Syzygium samarangense was highly amenable for clonal propagation by stem cuttings and treatment of leafy stem cuttings with 0.4% IBA is recommended considering performance in all studied and vegetative propagation through stem cutting can be a better and helpful method of plant. So clonal propagation technique, increase the yield and quality fruit.

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STUDIES ON ASSESSING THE DIVERSITY OF BIRDS, PHYTOPLANKTONS AND ZOOPLANKTONS IN PONDS IN MUTHUPETTAI BLOCK, THIRUVARUR DISTRICT, TAMIL NADU

The study was aimed to make an assessment on birds, phytoplanktons and zooplanktons in the ponds of Muthupet block, Thiruvarur and samples were collected from five ponds. Plankton samples were collected from the surface water at predecided locations and were identified. Nearly 22 phytoplanktons and 11 zooplanktons were identified from the ponds. Spot counting was used for recording the number of water birds and counting was done during early morning hours. The identification was done using an identification manual of Indian sub Continental birds by Grimmett and nearly 22 water birds were recorded. The pond water remains green in color in all the seasons. The presence of cyanophyceae members like Anacystis sp. and Spirulina sp. may be responsible for green color of the pond water. Migratory birds like gull billed tern, yellow wagtail and wooded sand piper were recorded in the month of February. The water quality of the pond is in eutrophic condition but the presence of rich bank vegetation attracts the birds for roosting. Further, the thick vegetation of Eicchornia sp. provides hiding place for the water birds.

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SEASONAL VARIATIONS OF THE REPRODUCTIVE ACTIVITY IN LOCAL BREED’S EWES RAISED IN THE SOUTH-WEST OF ALGERIA

The objective of this study was to determine the characteristics of variations in sexual activity of ewes taken in a saharian environment in the southwest of Algeria. Two experiments were conducted in this study. In the first one, a monthly monitoring was carried out on a population of 1430 ewes dominated by the Ouled Djellal breed, aged between 1 and 7 years, and belonging to 12 livestock farms distributed in different zones of the study area. In the second experiment, 63 healthy Ouled Djellal ewes, assumed non-pregnant and aged from 1 to 5 years, were involved to determine the hormonal profiles of progesterone, oestradiol, FSH, and LH during the four seasons (autumn, winter, spring, and summer). Hormonal assays were measured by using the chemiluminescent immunoenzymatic technique (CLIA). The monthly distribution of lambing in ewes has allowed us to deduce that sexual activity is continuous throughout the year with a maximum of lambing in autumn. The results of the hormone assays also revealed the existence of a hormonal activity during the four seasons with a decrease in winter and spring for progesterone and in summer for LH. This permit us to conclude that Ouled Djellal ewes raised under difficult climatic conditions of the south-west of Algeria are relatively non-seasonal with sometimes only a slight decrease in reproductive activity.

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