Die Integration von organischen und anorganischen Düngemitteln verbessert die Nährstoffversorgung und Bodenqualität, was zu höherer Reisproduktivität führt. Organische Dünger wie FYM erhöhen die Effizienz der Düngemittel und fördern die Bodenfruchtbarkeit sowie die Gesundheit, indem sie Mikronährstoffe bereitstellen und das Bodenverhalten verändern.
Nonelementary Integrals are traditioanlly known as Nonelementary Functions but it doesn't given clear explanation that what was the reason behind it to call such function as Nonelementary Functions, whereas the name Nonelementary Integrals are the clear explanation that it has been originated from integration and there must be relation between elementary function and integration, when we call an integral nonelementary. In fact nonelementary integrals are those antiderivatives of elementary functions, which cann't be expressed in terms of elementary functions. This book contains the basic theorems and properties related to integration and these are not found in general textbooks. I collected these material from different resources, which have been given as references in the last chapter in the book.
An Application of Binomial Theorem in Vedic Mathematics
The article explores the Vedic mathematics subsutra "anurupyena vidhi," extending its application to calculate the nth power of integers and rational numbers using the binomial theorem. It aims to bridge Vedic and modern mathematics, uncovering hidden logic within the sutra. While the arithmetic operations may initially seem challenging, the author encourages readers to engage with the material through practice, ultimately leading to a deeper appreciation of the unconventional methods presented.
Focusing on teaching methodologies in higher education, the article examines the common structure shared by all subjects, including mathematics, which consists of assumptions, properties, and applications. It proposes an idealistic method for research-oriented students and a realistic inductive method as effective approaches for graduate and postgraduate learners. The discussion highlights the absence of a universally best teaching method while emphasizing the importance of adapting techniques to suit different educational levels, concluding with suggestions for further research in the field.
The book presents innovative mathematical concepts, introducing dominating functions, sequential functions, and dominating sequential functions that encompass nearly all classical elementary functions. Each new function is categorized into four distinct types: trigonometric, hyperbolic, exponential, and logarithmic. The research highlights that classical elementary functions are specific instances of these newly defined functions, expanding the understanding of mathematical function classification and relationships.
The book explores the concept of arithmetic progressions, extending it into a new framework called Multi-dimensional Arithmetic Progression with Multiplicity. It begins with traditional one-dimensional progressions and progresses through two and three-dimensional cases, detailing various multiplicities. The final chapter introduces rth dimensional progressions, encompassing all previous forms. Additionally, it provides a foundation for further research, encouraging new scholars to explore the topic. The work is backed by references and the author's prior research contributions.
The book explores the extension of geometric progressions, introducing the concept of Multi-dimensional geometric Progression with Multiplicity. It emphasizes the significance of geometric progressions and series across various research fields, highlighting their applications and relevance in advanced mathematical analysis. The research presents a thorough examination of related sub-topics, aiming to deepen the understanding of these mathematical constructs.
An Application of strong Liouville's theorem
Focusing on the proof of a specific conjecture related to indefinite nonintegrable functions, this research paper explores Yadav's six conjectures. It employs strong Liouville's theorem and its special cases, alongside examples of well-known nonelementary functions. The work also references properties discussed by Marchisotto and Zakeri, contributing to the understanding of these complex mathematical concepts.