ARCHIMEDEAN DUAL Meaning and
Definition
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The term "Archimedean dual" refers to a mathematical concept related to lattices, convex sets, and optimization problems. Specifically, it pertains to a specific type of transformation applied to a lattice or a convex set in a vector space.
In mathematics, a lattice is a discrete subgroup of a real vector space. A convex set, on the other hand, is a region that includes the line segment connecting any two of its points. The transformation known as the Archimedean dual is applied to a lattice or a convex set to yield another lattice or convex set.
To elaborate, if the original lattice is in N-dimensional space, its Archimedean dual will also be in the same N-dimensional space. The Archimedean dual of a convex set, which can be described as the set of all linear functionals that are at most one on the convex set, can be defined similarly.
This transformation has practical applications in optimization problems, particularly in the realm of linear programming. By applying the Archimedean dual transformation to a given problem, one can reformulate it into an equivalent problem that is easier to solve or has desired properties for optimization purposes.
In summary, the term "Archimedean dual" refers to a transformation applied to a lattice or a convex set, yielding another lattice or convex set in the same vector space. It is commonly utilized in the field of optimization, allowing for reformulation and simplification of various mathematical problems.
Etymology of ARCHIMEDEAN DUAL
The term "Archimedean dual" is a combination of two key words: "Archimedean" and "dual", each with their own etymology.
1. Archimedean: The word "Archimedean" derives from the name of the ancient Greek mathematician and inventor, Archimedes. Archimedes made significant contributions to the fields of mathematics, physics, and engineering. He is famously known for his principle of buoyancy, his work on the lever, and his development of several mathematical concepts, including the method of exhaustion. The term "Archimedean" is used to honor his name and his influential work.
2. Dual: The term "dual" has Latin origins and comes from the Latin word "duelis", meaning "two-fold" or "double".
