They are associated essentially to a closed smooth manifold equipped with a (co)Euler structure and a Riemannian metric in the first case, a smooth triangulation in the second case, and a smooth flow of type described in Section 2 in the third case. The functions we consider are the complex-valued Ray-Singer torsion, the Milnor-Turaev torsion, and the dynamical torsion.
Riemannian Geometry, Topology and Dynamics permit to introduce partially defined holomorphic functions on the variety of representations of the fundamental group of a manifold. The paper, which is published in an international mathematics journal, presents a resolution to this serious problem, which is important, as that would be conducive to peace and harmony. It would be difficult and might be impossible to reason with and convince someone with a closed mind-set, someone who has made up the mind not to be convinced, or even someone who is not intelligent enough to be convinced. The paper also delves into the problems encountered in logical reasoning as well as some modes of reasoning. Scenarios from mathematics, which probably represents the most rigorous form of logical reasoning, and science are described as well. It describes the scenario of people frequently involved in fruitless arguments and debates, and shows why the application of logic, for example, in logical argument or debate, could not often achieve the desired outcomes, much of the time ending up with frustration, unhappiness, bad feelings and poor relationships. This paper points out the frequent inefficacy of logical presentations, arguments and debates per se in bringing about the correct and wonted outcomes. Logical reasoning in any form is an important aspect of life it is persuading or convincing others with logic through writing or speech, for example, scientists, politicians, businessmen, financiers, solicitors and many others do this.
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