Design of Polar Code Lattices of Finite Dimension

The Standard Metric

10.23943/princeton/9780691166902.003.0026 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Affine Transformation ◽

Convex Sets ◽

Finite Dimension ◽

Affine Subspace ◽

Affine Transformations ◽

Real Vector ◽

Euclidean Metric ◽

Affine Hyperplane ◽

Tits Building ◽

Convex Closure

This chapter presents some results about groups generated by reflections and the standard metric on a Bruhat-Tits building. It begins with definitions relating to an affine subspace, an affine hyperplane, an affine span, an affine map, and an affine transformation. It then considers a notation stating that the convex closure of a subset a of X is the intersection of all convex sets containing a and another notation that denotes by AGL(X) the group of all affine transformations of X and by Trans(X) the set of all translations of X. It also describes Euclidean spaces and assumes that the real vector space X is of finite dimension n and that d is a Euclidean metric on X. Finally, it discusses Euclidean representations and the standard metric.

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POLAR Code Calculations of the Shuttle Orbiter Plasma Wake and Direct Comparisons with Spacelab II Measurements.

10.21236/ada181160 ◽

1986 ◽

Author(s):

I. Katz ◽

G. A. Jongeward ◽

J. R. Lilley ◽

Mandell Jr. ◽

M. J.

Keyword(s):

Polar Code

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Finite dimensional locally convex cones

Filomat ◽

10.2298/fil1716111a ◽

2017 ◽

Vol 31 (16) ◽

pp. 5111-5116

Author(s):

Davood Ayaseha

Keyword(s):

Finite Dimension ◽

Convex Cones ◽

Vector Spaces ◽

Topological Vector Spaces ◽

Euclidean Topology ◽

Finite Dimensional ◽

Dimensional Cone ◽

Locally Convex Cones ◽

Locally Convex ◽

Special Case

We study the locally convex cones which have finite dimension. We introduce the Euclidean convex quasiuniform structure on a finite dimensional cone. In special case of finite dimensional locally convex topological vector spaces, the symmetric topology induced by the Euclidean convex quasiuniform structure reduces to the known concept of Euclidean topology. We prove that the dual of a finite dimensional cone endowed with the Euclidean convex quasiuniform structure is identical with it?s algebraic dual.

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