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Concept# Duality (mathematics)

Summary

In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry.
In mathematical contexts, duality has numerous meanings. It has been described as "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics".
Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some family of scalars. For instance, linear algebra dual

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Ileana Ozana Jelescu, Nicolas Kunz

A two-compartment model of diffusion in white matter, which accounts for intra- and extra-axonal spaces, is associated with two plausible mathematical scenarios: either the intra-axonal axial diffusivity Da,‖ is higher than the extra-axonal De,‖ (Branch 1), or the opposite, i.e. Da,‖

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Emerging reconfigurable nanotechnologies allow the implementation of self-dual functions with a fewer number of transistors as compared to traditional CMOS technologies. To achieve better area results for Reconfigurable Field-Effect Transistors (RFET)-based circuits, a large portion of a logic representation must be mapped to self-dual logic gates. This, in turn, depends upon how self-duality is preserved in the logic representation during logic optimization and technology mapping. In the present work, we develop Boolean size-optimization methods– a rewriting and a resubstitution algorithms using Xor-Majority Graphs(XMGs) as a logic representation aiming at better preserving self-duality during logic optimization. XMGs are more compact for both unate and binate logic functions as compared to conventional logic representations such as And-Inverter Graphs(AIGs) or Majority-Inverter Graphs (MIGs). We evaluate the proposed algorithm over crafted benchmarks (with various levels of self-duality), and cryptographic benchmarks. For cryptographic benchmarks with a high self-duality ratio, the XMG-based logic optimisation flow can achieve an area reduction of up to17% when compared to AIG-based optimization flows implemented in the academic logic synthesis tool ABC.

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