| 词条 | Orthant |
| 释义 |
In geometry, an orthant[1] or hyperoctant[2] is the analogue in n-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions. In general an orthant in n-dimensions can be considered the intersection of n mutually orthogonal half-spaces. By independent selections of half-space signs, there are 2n orthants in n-dimensional space. More specifically, a closed orthant in Rn is a subset defined by constraining each Cartesian coordinate to be nonnegative or nonpositive. Such a subset is defined by a system of inequalities: ε1x1 ≥ 0 ε2x2 ≥ 0 · · · εnxn ≥ 0, where each εi is +1 or −1. Similarly, an open orthant in Rn is a subset defined by a system of strict inequalities ε1x1 > 0 ε2x2 > 0 · · · εnxn > 0, where each εi is +1 or −1. By dimension:
The nonnegative orthant is the generalization of the first quadrant to n dimensions and is important in many constrained optimization problems. See also
Notes1. ^[https://books.google.com/books?id=FV_s8W58D4UC&pg=PA394&lpg=PA394&dq=positive+orthant+definition&source=bl&ots=LV7f7KQZIP&sig=445iHTVnZB4MJ6-BNOoAeHpz-1A&ei=ajp8TPeUFs2nnQfmraSdCw&sa=X&oi=book_result&ct=result&resnum=10&ved=0CEkQ6AEwCQ#v=onepage&q=positive%20orthant%20definition&f=false Advanced linear algebra By Steven Roman, Chapter 15] 2. ^{{MathWorld|title=Hyperoctant|urlname=Hyperoctant}} 3. ^J. H. Conway, N. J. A. Sloane, The Cell Structures of Certain Lattices (1991)
[[Mohammad Amin Ahmadi] 2 : Euclidean geometry|Linear algebra |
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