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Galois Theory, the theory of polynomial equations and their solutions, is one of the most fascinating and beautiful subjects of pure mathematics. Using group theory and field theory, it provides a complete answer to the problem of the solubility of polynomial equations by radicals: that is, determining when and how a polynomial equation can be solved by repeatedly extracting roots using elementary algebraic operations. This textbook contains a fully detailed account of Galois Theory and the algebra that it needs and is suitable both for those following a course of lectures and the independent reader (who is assumed to have no previous knowledge of Galois Theory). The second edition has been significantly revised and re-ordered; the first part develops the basic algebra that is needed, and the second a comprehensive account of Galois Theory. There are applications to ruler-and- compass constructions, and to the solution of classical mathematical problems of ancient times. There are new exercises throughout, and carefully-selected examples will help the reader develop a clear understanding of the mathematical theory.
Ecosystem modeling, a pillar of the systems ecology paradigm (SEP), addresses questions such as, how much carbon and nitrogen are cycled within ecological sites, landscapes, or indeed the earth system? Or how are human activities modifying these flows? Modeling, when coupled with field and laboratory studies, represents the essence of the SEP in that they embody accumulated knowledge and generate hypotheses to test understanding of ecosystem processes and behavior. Initially, ecosystem models were primarily used to improve our understanding about how biophysical aspects of ecosystems operate. However, current ecosystem models are widely used to make accurate predictions about how large-scale phenomena such as climate change and management practices impact ecosystem dynamics and assess potential effects of these changes on economic activity and policy making. In sum, ecosystem models embedded in the SEP remain our best mechanism to integrate diverse types of knowledge regarding how the earth system functions and to make quantitative predictions that can be confronted with observations of reality. Modeling efforts discussed are the Century ecosystem model, DayCent ecosystem model, Grassland Ecosystem Model ELM, food web models, Savanna model, agent-based and coupled systems modeling, and Bayesian modeling.
There is a paucity of Asian-based data regarding the diagnostic yield of computed tomography imaging in the initial assessment of idiopathic unilateral vocal fold palsy.
To investigate the diagnostic yield of computed tomography in idiopathic unilateral vocal fold palsy cases in an Asian tertiary hospital, and to determine the causative pathologies and positive predictive factors.
A retrospective chart review was conducted of patients (between 2010 and 2018) with a clinical diagnosis of idiopathic unilateral vocal fold palsy who underwent contrast-enhanced computed tomography of the neck and chest at Tan Tock Seng Hospital, Singapore.
The overall computed tomography diagnostic yield was 21 per cent, with malignancy accounting for 63.6 per cent of diagnoses. Degree of vocal fold weakness was the only significant predictor of positive computed tomography findings (11.5 per cent in vocal fold paresis vs 29.1 per cent in vocal fold paralysis, p = 0.025). None of the patients with negative computed tomography findings went on to develop disease after a mean follow up of 14.3 months.
Computed tomography is a useful initial investigation for idiopathic unilateral vocal fold palsy, particularly in cases with vocal fold paralysis.