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This textbook provides a succinct introduction to algebraic topology. It follows a modern categorical approach from the beginning and gives ample motivation throughout so that students will find this an ideal first encounter to the field. Topics are treated in a self-contained manner, making this a convenient resource for instructors searching for a comprehensive overview of the area.It begins with an outline of category theory, establishing the concepts of functors, natural transformations, adjunction, limits, and colimits. As a first application, van Kampen's theorem is proven in the groupoid version. Following this, an excursion to cofibrations and homotopy pushouts yields an alternative formulation of the theorem that puts the computation of fundamental groups of attaching spaces on firm ground. Simplicial homology is then defined, motivating the Eilenberg-Steenrod axioms, and the simplicial approximation theorem is proven. After verifying the axiomsfor singular homology, various versions of the Mayer-Vietoris sequence are derived and it is shown that homotopy classes of self-maps of spheres are classified by degree.The final chapter discusses cellular homology of CW complexes, culminating in the uniqueness theorem for ordinary homology.Introduction to Algebraic Topology is suitable for a single-semester graduate course on algebraic topology. It can also be used for self-study, with numerous examples, exercises, and motivating remarks included.

This book presents the basic theoretical concepts of dynamical systems with applications in population dynamics. Existence, uniqueness and stability of solutions, global attractors, bifurcations, center manifold and normal form theories are discussed with cutting-edge applications, including a Holling's predator-prey model with handling and searching predators and projecting the epidemic forward with varying level of public health interventions for COVID-19.As an interdisciplinary text, this book aims at bridging the gap between mathematics, biology and medicine by integrating relevant concepts from these subject areas, making it self-sufficient for the reader. It will be a valuable resource to graduate and advance undergraduate students for interdisciplinary research in the area of mathematics and population dynamics.

This volume explores the nature, philosophies and genres of indigenous African popular music, focusing on how indigenous African popular music artistes are seen as prophets and philosophers, and how indigenous African popular music depicts the world. Indigenous African popular music has long been under-appreciated in communication scholarship. However, understanding the nature and philosophies of indigenous African popular music reveals an untapped diversity which only be unraveled by knowledge of the myriad cultural backgrounds from which its genres originate. Indigenous African popular musicians have become repositories of indigenous cultural traditions and cosmologies.With a particular focus on scholarship from Nigeria, Zimbabwe and South Africa, this volume explores the work of these pioneering artists and their protégés who are resiliently sustaining, recreating and popularising indigenous popular music in their respective African communities, and at the same time propagatingthe communal views about African philosophies and the temporal and spiritual worlds in which they exist. ¿

In this newly revised and expanded edition the reader finds both an introduction to igneous petrology and an exposition of the quantitative use of phase diagrams in understanding the origin and crystallization history of basic magmas. The book provides a step-by- step analysis of crystallization and melting in a limited number of geologically significant phase diagrams. The limiting processes of fractional and equilibrium melting and fractional and equilibrium crystallization are rigorously examined. Examples of the use of phase diagrams are drawn from the literature of layered intrusions, redox equilibria in iron-bearing systems, and the high pressure equilibria of melting in the mantle.