Maran Illustrated Effortless Algebra (repost)

In spite of some recent applications of ultraproducts in algebra, they remain largely unknown to commutative algebraists, in part because they do not preserve basic properties such as Noetherianity. This work wants to make a strong case against these prejudices. More precisely, it studies ultraproducts of Noetherian local rings from a purely algebraic perspective, as well as how they can be used to transfer results between the positive and zero characteristics, to derive uniform bounds, to define tight closure in characteristic zero, and to prove asymptotic versions of homological conjectures in mixed characteristic. Some of these results are obtained using variants called chromatic products, which are often even Noetherian. This book, neither assuming nor using any logical formalism, is intended for algebraists and geometers, in the hope of popularizing ultraproducts and their applications in algebra.

This unique textbook focuses on the structure of fields and is intended for a second course in abstract algebra. Besides providing proofs of the transcendance of pi and e, the book includes material on differential Galois groups and a proof of Hilbert's irreducibility theorem. The reader will hear about equations, both polynomial and differential, and about the algebraic structure of their solutions. In explaining these concepts, the author also provides comments on their historical development and leads the reader along many interesting paths.

Radiology Illustrated: Gastrointestinal Tract is the second of two volumes designed to provide clear and practical guidance on the diagnostic imaging of abdominal diseases. The book presents approximately 300 cases with 1500 carefully selected and categorized illustrations of gastrointestinal tract diseases, along with key text messages and tables that will help the reader easily to recall the relevant images as an aid to differential diagnosis., Essential points are summarized at the end of each text message to facilitate rapid review and learning. Additionally, brief descriptions of each clinical problem are provided, followed by case studies of both common and uncommon pathologies that illustrate the roles of the different imaging modalities, including ultrasound, radiography, computed tomography, and magnetic resonance imaging.

Produced by the award-winning maranGraphics group, Maran Illustrated Guide to Piano is a valuable resource to a wide range of readers-from people who want to play a few songs for their family in the living room to those who aspire to become serious musicians. Maran Illustrated Guide to Piano shows the reader the best way to perform each task, while the full-color photographs, music examples and clear, step-by-step instructions walk the reader through each task from beginning to end.