|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
algebra I-Math |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
theory I-Math |
|
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|
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|
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|
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|
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|
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|
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|
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|
algebra B-Math |
|
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|
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|
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|
combinatorics B-Math |
|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
matrices B-Math |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
numerical B-Math |
|
analysis I-Math |
|
|
|
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|
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|
vector B-Math |
|
spaces I-Math |
|
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|
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|
structure O |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
hilbert B-Math |
|
spaces I-Math |
|
|
|
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|
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|
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|
the O |
|
trivial B-Math |
|
subspaces I-Math |
|
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|
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|
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|
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|
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|
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|
vector B-Math |
|
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|
|
|
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|
vector B-Math |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
tensors B-Math |
|
|
|
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|
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|
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|
calculus B-Math |
|
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|
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|
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|
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|
concepts O |
|
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|
techniques O |
|
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|
analysi O |
|
analysis B-Math |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
space I-Math |
|
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|
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|
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|
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|
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|
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|
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|
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|
|
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|
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|
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|
combinations I-Math |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
classical B-Math |
|
geometry I-Math |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
space I-Math |
|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
normed O |
|
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|
hilbert B-Math |
|
spaces I-Math |
|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
|
real O |
|
vector O |
|
space O |
|
and O |
|
complex O |
|
vector O |
|
space O |
|
are O |
|
kinds O |
|
of O |
|
vector B-Math |
|
spaces I-Math |
|
ased O |
|
on O |
|
different O |
|
kinds O |
|
of O |
|
scalars O |
|
real O |
|
coordinate O |
|
space O |
|
or O |
|
complex O |
|
coordinate O |
|
spac O |
|
ased O |
|
on O |
|
different O |
|
kinds O |
|
of O |
|
scalars O |
|
real B-Math |
|
coordinate I-Math |
|
space I-Math |
|
r O |
|
complex O |
|
coordinate O |
|
spac O |
|
r O |
|
complex B-Math |
|
coordinate I-Math |
|
space I-Math |
|
|
|
most O |
|
geometric B-Math |
|
transformation I-Math |
|
such O |
|
as O |
|
translations O |
|
rotations O |
|
reflections O |
|
rigid O |
|
motions O |
|
isometries O |
|
and O |
|
projections O |
|
transform O |
|
lines O |
|
into O |
|
line O |
|
such O |
|
as O |
|
translations B-Attributes |
|
rotations O |
|
reflections O |
|
rigid O |
|
motions O |
|
isometries O |
|
and O |
|
projections O |
|
transform O |
|
lines O |
|
into O |
|
line O |
|
rotations B-Attributes |
|
reflections O |
|
rigid O |
|
motions O |
|
isometries O |
|
and O |
|
projections O |
|
transform O |
|
lines O |
|
into O |
|
line O |
|
reflections B-Attributes |
|
rigid O |
|
motions O |
|
isometries O |
|
and O |
|
projections O |
|
transform O |
|
lines O |
|
into O |
|
line O |
|
rigid B-Attributes |
|
motions I-Attributes |
|
isometries O |
|
and O |
|
projections O |
|
transform O |
|
lines O |
|
into O |
|
line O |
|
isometries B-Attributes |
|
and O |
|
projections O |
|
transform O |
|
lines O |
|
into O |
|
line O |
|
and O |
|
projections B-Attributes |
|
transform I-Attributes |
|
lines O |
|
into O |
|
line O |
|
|
|
moreover O |
|
it O |
|
maps O |
|
linear B-Math |
|
subspaces I-Math |
|
in O |
|
onto O |
|
linear O |
|
subspaces O |
|
in O |
|
possibly O |
|
of O |
|
a O |
|
lower O |
|
dimension O |
|
for O |
|
example O |
|
it O |
|
maps O |
|
a O |
|
plane O |
|
through O |
|
the O |
|
origin O |
|
in O |
|
to O |
|
either O |
|
a O |
|
plane O |
|
through O |
|
the O |
|
origin O |
|
in O |
|
a O |
|
line O |
|
through O |
|
the O |
|
origin O |
|
in O |
|
or O |
|
just O |
|
the O |
|
origin O |
|
i O |
|
in O |
|
onto O |
|
linear O |
|
subspaces O |
|
in O |
|
possibly O |
|
of O |
|
a O |
|
lower O |
|
dimension B-Math |
|
for O |
|
example O |
|
it O |
|
maps O |
|
a O |
|
plane O |
|
through O |
|
the O |
|
origin O |
|
in O |
|
to O |
|
either O |
|
a O |
|
plane O |
|
through O |
|
the O |
|
origin O |
|
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|
a O |
|
line O |
|
through O |
|
the O |
|
origin O |
|
in O |
|
or O |
|
just O |
|
the O |
|
origin O |
|
i O |
|
|
|
a O |
|
system O |
|
of O |
|
linear B-Math |
|
equations I-Math |
|
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|
differently O |
|
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|
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|
general O |
|
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|
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|
the O |
|
equations O |
|
are O |
|
linearly O |
|
dependent O |
|
or O |
|
if O |
|
it O |
|
is O |
|
inconsistent O |
|
and O |
|
has O |
|
no O |
|
more O |
|
equations O |
|
than O |
|
unknown O |
|
behave O |
|
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|
from O |
|
the O |
|
general O |
|
case O |
|
if O |
|
the O |
|
equations O |
|
are O |
|
linearly B-Math |
|
dependent I-Math |
|
or O |
|
if O |
|
it O |
|
is O |
|
inconsistent O |
|
and O |
|
has O |
|
no O |
|
more O |
|
equations O |
|
than O |
|
unknown O |
|
|
|
the O |
|
binary O |
|
operation O |
|
called O |
|
vector O |
|
addition O |
|
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|
simply O |
|
addition O |
|
assigns O |
|
to O |
|
any O |
|
two O |
|
vectors B-Math |
|
v O |
|
and O |
|
w O |
|
in O |
|
v O |
|
a O |
|
third O |
|
vector O |
|
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|
v O |
|
which O |
|
is O |
|
commonly O |
|
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|
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|
v O |
|
w O |
|
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|
called O |
|
the O |
|
sum O |
|
of O |
|
these O |
|
two O |
|
vector O |
|
|
|
this O |
|
point O |
|
of O |
|
view O |
|
turned O |
|
out O |
|
to O |
|
be O |
|
particularly O |
|
useful O |
|
for O |
|
the O |
|
study O |
|
of O |
|
differential B-Math |
|
and I-Math |
|
integral I-Math |
|
equations I-Math |
|
|
|
if O |
|
such O |
|
a O |
|
linear B-Math |
|
combination I-Math |
|
exists O |
|
then O |
|
the O |
|
vectors O |
|
are O |
|
said O |
|
to O |
|
be O |
|
linearly O |
|
dependen O |
|
exists O |
|
then O |
|
the O |
|
vectors O |
|
are O |
|
said O |
|
to O |
|
be O |
|
linearly B-Math |
|
dependent I-Math |
|
|
|
this O |
|
is O |
|
the O |
|
case O |
|
with O |
|
mechanics O |
|
and O |
|
robotics O |
|
for O |
|
describing O |
|
rigid B-Attributes |
|
body I-Attributes |
|
dynamics O |
|
geodesy O |
|
for O |
|
describing O |
|
earth O |
|
shape O |
|
perspectivity O |
|
computer O |
|
vision O |
|
and O |
|
computer O |
|
graphics O |
|
for O |
|
describing O |
|
the O |
|
relationship O |
|
between O |
|
a O |
|
scene O |
|
and O |
|
its O |
|
plane O |
|
representation O |
|
and O |
|
many O |
|
other O |
|
scientific O |
|
domain O |
|
|
|
many O |
|
other O |
|
physical O |
|
quantities O |
|
can O |
|
be O |
|
usefully O |
|
thought O |
|
of O |
|
as O |
|
vectors B-Math |
|
|
|
all O |
|
these O |
|
questions O |
|
can O |
|
be O |
|
solved O |
|
by O |
|
using O |
|
gaussian B-Math |
|
elimination I-Math |
|
or O |
|
some O |
|
variant O |
|
of O |
|
this O |
|
algorith O |
|
|
|
most O |
|
physical O |
|
phenomena O |
|
are O |
|
modeled O |
|
by O |
|
partial B-Math |
|
differential I-Math |
|
equations I-Math |
|
|
|
the O |
|
binary O |
|
function O |
|
called O |
|
scalar O |
|
multiplicationassigns B-Math |
|
to O |
|
any O |
|
scalar O |
|
a O |
|
in O |
|
f O |
|
and O |
|
any O |
|
vector O |
|
v O |
|
in O |
|
v O |
|
another O |
|
vector O |
|
in O |
|
v O |
|
which O |
|
is O |
|
denoted O |
|
a O |
|
to O |
|
any O |
|
scalar B-Math |
|
a O |
|
in O |
|
f O |
|
and O |
|
any O |
|
vector O |
|
v O |
|
in O |
|
v O |
|
another O |
|
vector O |
|
in O |
|
v O |
|
which O |
|
is O |
|
denoted O |
|
a O |
|
a O |
|
in O |
|
f O |
|
and O |
|
any O |
|
vector B-Math |
|
v O |
|
in O |
|
v O |
|
another O |
|
vector O |
|
in O |
|
v O |
|
which O |
|
is O |
|
denoted O |
|
a O |
|
|
|
finitedimensional O |
|
vector B-Math |
|
spaces I-Math |
|
occur O |
|
naturally O |
|
in O |
|
geometry O |
|
and O |
|
related O |
|
area O |
|
occur O |
|
naturally O |
|
in O |
|
geometry B-Math |
|
and O |
|
related O |
|
area O |
|
finitedimensional B-Attributes |
|
vector O |
|
spaces O |
|
occur O |
|
naturally O |
|
in O |
|
geometry O |
|
and O |
|
related O |
|
area O |
|
|
|
these O |
|
equations O |
|
often O |
|
complex O |
|
and O |
|
nonlinear O |
|
can O |
|
be O |
|
linearized O |
|
using O |
|
linear B-Math |
|
algebra I-Math |
|
methods O |
|
allowing O |
|
for O |
|
simpler O |
|
solutions O |
|
and O |
|
analyse O |
|
|
|
however O |
|
the O |
|
general O |
|
concept O |
|
of O |
|
a O |
|
functional B-Math |
|
had O |
|
previously O |
|
been O |
|
introduced O |
|
in O |
|
by O |
|
the O |
|
italian O |
|
mathematician O |
|
and O |
|
physicist O |
|
vito O |
|
volterr O |
|
|
|
this O |
|
means O |
|
that O |
|
for O |
|
two O |
|
vector B-Math |
|
spaces I-Math |
|
over O |
|
a O |
|
given O |
|
field O |
|
and O |
|
with O |
|
the O |
|
same O |
|
dimension O |
|
the O |
|
properties O |
|
that O |
|
depend O |
|
only O |
|
on O |
|
the O |
|
vectorspace O |
|
structure O |
|
are O |
|
exactly O |
|
the O |
|
same O |
|
technically O |
|
the O |
|
vector O |
|
spaces O |
|
are O |
|
isomorphi O |
|
over O |
|
a O |
|
given O |
|
field O |
|
and O |
|
with O |
|
the O |
|
same O |
|
dimension O |
|
the O |
|
properties O |
|
that O |
|
depend O |
|
only O |
|
on O |
|
the O |
|
vectorspace O |
|
structure O |
|
are O |
|
exactly O |
|
the O |
|
same O |
|
technically O |
|
the O |
|
vector O |
|
spaces O |
|
are O |
|
isomorphic B-Attributes |
|
dimension B-Math |
|
he O |
|
properties O |
|
that O |
|
depend O |
|
only O |
|
on O |
|
the O |
|
vectorspace O |
|
structure O |
|
are O |
|
exactly O |
|
the O |
|
same O |
|
technically O |
|
the O |
|
vector O |
|
spaces O |
|
are O |
|
isomorphi O |
|
|
|
vector B-Math |
|
spaces I-Math |
|
generalize O |
|
euclidean O |
|
vectors O |
|
which O |
|
allow O |
|
modeling O |
|
of O |
|
physical O |
|
quantities O |
|
such O |
|
as O |
|
forces O |
|
and O |
|
velocity O |
|
that O |
|
have O |
|
not O |
|
only O |
|
a O |
|
magnitude O |
|
but O |
|
also O |
|
a O |
|
directio O |
|
generalize O |
|
euclidean O |
|
vectors O |
|
which O |
|
allow O |
|
modeling O |
|
of O |
|
physical O |
|
quantities O |
|
such O |
|
as O |
|
forces O |
|
and O |
|
velocity O |
|
that O |
|
have O |
|
not O |
|
only O |
|
a O |
|
magnitude B-Attributes |
|
but O |
|
also O |
|
a O |
|
directio O |
|
but O |
|
also O |
|
a O |
|
direction B-Attributes |
|
vectors B-Math |
|
which O |
|
allow O |
|
modeling O |
|
of O |
|
physical O |
|
quantities O |
|
such O |
|
as O |
|
forces O |
|
and O |
|
velocity O |
|
that O |
|
have O |
|
not O |
|
only O |
|
a O |
|
magnitude O |
|
but O |
|
also O |
|
a O |
|
directio O |
|
|
|
this O |
|
is O |
|
important O |
|
because O |
|
if O |
|
we O |
|
have O |
|
m O |
|
independent B-Math |
|
vectors I-Math |
|
a O |
|
solution O |
|
is O |
|
guaranteed O |
|
regardless O |
|
of O |
|
the O |
|
righthand O |
|
side O |
|
and O |
|
otherwise O |
|
not O |
|
guarantee O |
|
|
|
many O |
|
algebraic B-Math |
|
operations I-Math |
|
on O |
|
real O |
|
numbers O |
|
such O |
|
as O |
|
addition O |
|
subtraction O |
|
multiplication O |
|
and O |
|
negation O |
|
have O |
|
close O |
|
analogues O |
|
for O |
|
vectors O |
|
operations O |
|
which O |
|
obey O |
|
the O |
|
familiar O |
|
algebraic O |
|
laws O |
|
of O |
|
commutativity O |
|
associativity O |
|
and O |
|
distributivit O |
|
on O |
|
real O |
|
numbers O |
|
such O |
|
as O |
|
addition O |
|
subtraction B-Attributes |
|
multiplication O |
|
and O |
|
negation O |
|
have O |
|
close O |
|
analogues O |
|
for O |
|
vectors O |
|
operations O |
|
which O |
|
obey O |
|
the O |
|
familiar O |
|
algebraic O |
|
laws O |
|
of O |
|
commutativity O |
|
associativity O |
|
and O |
|
distributivit O |
|
addition B-Attributes |
|
subtraction O |
|
multiplication O |
|
and O |
|
negation O |
|
have O |
|
close O |
|
analogues O |
|
for O |
|
vectors O |
|
operations O |
|
which O |
|
obey O |
|
the O |
|
familiar O |
|
algebraic O |
|
laws O |
|
of O |
|
commutativity O |
|
associativity O |
|
and O |
|
distributivit O |
|
subtraction O |
|
multiplication B-Attributes |
|
and O |
|
negation O |
|
have O |
|
close O |
|
analogues O |
|
for O |
|
vectors O |
|
operations O |
|
which O |
|
obey O |
|
the O |
|
familiar O |
|
algebraic O |
|
laws O |
|
of O |
|
commutativity O |
|
associativity O |
|
and O |
|
distributivit O |
|
and O |
|
negation B-Attributes |
|
have O |
|
close O |
|
analogues O |
|
for O |
|
vectors O |
|
operations O |
|
which O |
|
obey O |
|
the O |
|
familiar O |
|
algebraic O |
|
laws O |
|
of O |
|
commutativity O |
|
associativity O |
|
and O |
|
distributivit O |
|
have O |
|
close O |
|
analogues O |
|
for O |
|
vectors O |
|
operations O |
|
which O |
|
obey O |
|
the O |
|
familiar O |
|
algebraic O |
|
laws O |
|
of O |
|
commutativity B-Attributes |
|
associativity O |
|
and O |
|
distributivit O |
|
associativity B-Attributes |
|
and O |
|
distributivit O |
|
and O |
|
distributivity B-Attributes |
|
|
|
the O |
|
magnitude O |
|
of O |
|
the O |
|
vector B-Math |
|
is O |
|
the O |
|
distance O |
|
between O |
|
the O |
|
two O |
|
points O |
|
and O |
|
the O |
|
direction O |
|
refers O |
|
to O |
|
the O |
|
direction O |
|
of O |
|
displacement O |
|
from O |
|
a O |
|
to O |
|
is O |
|
the O |
|
distance B-Attributes |
|
between O |
|
the O |
|
two O |
|
points O |
|
and O |
|
the O |
|
direction O |
|
refers O |
|
to O |
|
the O |
|
direction O |
|
of O |
|
displacement O |
|
from O |
|
a O |
|
to O |
|
between O |
|
the O |
|
two O |
|
points O |
|
and O |
|
the O |
|
direction B-Attributes |
|
refers O |
|
to O |
|
the O |
|
direction O |
|
of O |
|
displacement O |
|
from O |
|
a O |
|
to O |
|
|
|
in O |
|
mathematics B-Math |
|
a O |
|
matrix O |
|
p O |
|
a O |
|
matrix B-Math |
|
p O |
|
|
|
the O |
|
concept O |
|
of O |
|
vector B-Math |
|
spaces I-Math |
|
is O |
|
fundamental O |
|
for O |
|
linear O |
|
algebra O |
|
together O |
|
with O |
|
the O |
|
concept O |
|
of O |
|
matrices O |
|
which O |
|
allows O |
|
computing O |
|
in O |
|
vector O |
|
space O |
|
is O |
|
fundamental O |
|
for O |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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homogeneous B-Math |
|
equation I-Math |
|
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|
|
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|
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|
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|
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|
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|
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|
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|
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|
space I-Math |
|
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|
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|
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|
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|
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|
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|
|
|
linear B-Math |
|
algebra I-Math |
|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
equation I-Math |
|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
linear B-Math |
|
maps I-Math |
|
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often O |
|
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|
represented O |
|
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|
matrices O |
|
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|
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|
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|
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|
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|
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|
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|
linear O |
|
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|
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|
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|
matrices B-Math |
|
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|
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|
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|
rotation O |
|
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|
reflection O |
|
linear O |
|
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|
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|
simple O |
|
examples O |
|
include O |
|
rotation B-Attributes |
|
and O |
|
reflection O |
|
linear O |
|
transformation O |
|
and O |
|
reflection O |
|
linear B-Math |
|
transformations I-Math |
|
|
|
presently O |
|
most O |
|
textbooks O |
|
introduce O |
|
geometric O |
|
spaces O |
|
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|
linear B-Math |
|
algebra I-Math |
|
and O |
|
geometry O |
|
is O |
|
often O |
|
presented O |
|
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|
elementary O |
|
level O |
|
as O |
|
a O |
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subfield O |
|
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|
linear O |
|
algebr O |
|
geometric B-Math |
|
spaces I-Math |
|
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linear O |
|
algebra O |
|
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|
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|
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|
often O |
|
presented O |
|
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|
elementary O |
|
level O |
|
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|
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|
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|
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|
algebr O |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
solution B-Math |
|
|
|
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|
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|
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|
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|
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|
define O |
|
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|
spaces I-Math |
|
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|
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|
vector O |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
space I-Math |
|
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|
|
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|
theory I-Math |
|
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|
nonlinear I-Math |
|
functionals I-Math |
|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
focuses O |
|
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|
matrices B-Math |
|
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|
linear O |
|
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|
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|
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|
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|
specified O |
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matrices O |
|
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|
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|
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|
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|
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viewed O |
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|
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|
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|
algebra I-Math |
|
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|
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|
represent O |
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|
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|
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|
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|
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|
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|
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|
matrices B-Math |
|
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|
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|
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|
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|
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|
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|
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|
maps I-Math |
|
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|
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|
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|
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|
|
|
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|
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|
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|
multiplicative O |
|
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|
in O |
|
fields O |
|
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|
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|
involved O |
|
in O |
|
the O |
|
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|
defining O |
|
a O |
|
vector B-Math |
|
space I-Math |
|
|
|
matrices B-Math |
|
is O |
|
a O |
|
rectangular O |
|
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|
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|
table O |
|
of O |
|
numbers O |
|
symbols O |
|
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|
expressions O |
|
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|
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|
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|
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|
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|
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|
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|
used O |
|
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|
represent O |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
these O |
|
applications O |
|
synthetic B-Math |
|
geometry I-Math |
|
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|
often O |
|
used O |
|
for O |
|
general O |
|
descriptions O |
|
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|
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|
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|
approach O |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
mathematics B-Math |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
linear B-Math |
|
algebra I-Math |
|
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|
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|
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|
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linear B-Math |
|
subspace I-Math |
|
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|
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|
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|
algebra I-Math |
|
|
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nonlinear O |
|
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|
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|
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|
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|
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|
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|
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|
often O |
|
used O |
|
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|
dealing O |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
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|
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|
linear B-Math |
|
maps I-Math |
|
|
|
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|
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|
field O |
|
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|
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|
dynamics O |
|
linear B-Math |
|
algebra I-Math |
|
finds O |
|
its O |
|
application O |
|
in O |
|
computational O |
|
fluid O |
|
dynamics O |
|
cfd O |
|
a O |
|
branch O |
|
that O |
|
uses O |
|
numerical O |
|
analysis O |
|
and O |
|
data O |
|
structures O |
|
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|
solve O |
|
and O |
|
analyze O |
|
problems O |
|
involving O |
|
fluid O |
|
flow O |
|
|
|
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|
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|
system O |
|
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|
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|
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|
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|
namely O |
|
all O |
|
of O |
|
the O |
|
points B-Math |
|
on O |
|
the O |
|
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|
lin O |
|
solutions B-Math |
|
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|
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|
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|
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|
points O |
|
on O |
|
the O |
|
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|
lin O |
|
|
|
weather O |
|
forecasting O |
|
or O |
|
more O |
|
specifically O |
|
parametrization O |
|
for O |
|
atmospheric O |
|
modeling B-Attributes |
|
is O |
|
a O |
|
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|
example O |
|
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|
a O |
|
realworld O |
|
application O |
|
where O |
|
the O |
|
whole O |
|
earth O |
|
atmosphere O |
|
is O |
|
divided O |
|
into O |
|
cells O |
|
of O |
|
say O |
|
km O |
|
of O |
|
width O |
|
and O |
|
km O |
|
of O |
|
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|
|
|
a O |
|
vector B-Math |
|
space I-Math |
|
is O |
|
finitedimensional O |
|
if O |
|
its O |
|
dimension O |
|
is O |
|
a O |
|
natural O |
|
numbe O |
|
is O |
|
finitedimensional B-Attributes |
|
if O |
|
its O |
|
dimension O |
|
is O |
|
a O |
|
natural O |
|
numbe O |
|
|
|
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|
solve O |
|
them O |
|
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|
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|
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|
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|
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|
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|
which O |
|
the O |
|
solutions B-Math |
|
are O |
|
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|
into O |
|
small O |
|
mutually O |
|
interacting O |
|
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|
|
|
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|
example O |
|
as O |
|
three O |
|
parallel O |
|
planes O |
|
do O |
|
not O |
|
have O |
|
a O |
|
common O |
|
point O |
|
the O |
|
solution O |
|
set O |
|
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|
their O |
|
equations O |
|
is O |
|
empty O |
|
the O |
|
solution O |
|
set O |
|
of O |
|
the O |
|
equations O |
|
of O |
|
three O |
|
planes O |
|
intersecting O |
|
at O |
|
a O |
|
point O |
|
is O |
|
single O |
|
point O |
|
if O |
|
three O |
|
planes O |
|
pass O |
|
through O |
|
two O |
|
points O |
|
their O |
|
equations O |
|
have O |
|
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|
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|
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|
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|
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|
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|
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|
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|
solution B-Math |
|
set I-Math |
|
is O |
|
infinite O |
|
and O |
|
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|
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|
all O |
|
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|
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|
passing O |
|
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|
these O |
|
point O |
|
|
|
in O |
|
mathematics O |
|
differential B-Math |
|
refers O |
|
to O |
|
several O |
|
related O |
|
notions O |
|
derived O |
|
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|
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|
early O |
|
days O |
|
of O |
|
calculus O |
|
put O |
|
on O |
|
a O |
|
rigorous O |
|
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|
such O |
|
as O |
|
infinitesimal O |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
days O |
|
of O |
|
calculus O |
|
put O |
|
on O |
|
a O |
|
rigorous O |
|
footing O |
|
such O |
|
as O |
|
infinitesimal B-Attributes |
|
differences I-Attributes |
|
and O |
|
the O |
|
derivatives O |
|
of O |
|
function O |
|
and O |
|
the O |
|
derivatives B-Math |
|
of O |
|
function O |
|
|
|
in O |
|
contrast O |
|
linear B-Math |
|
algebra I-Math |
|
deals O |
|
mostly O |
|
with O |
|
finitedimensional O |
|
spaces O |
|
and O |
|
does O |
|
not O |
|
use O |
|
topolog O |
|
deals O |
|
mostly O |
|
with O |
|
finitedimensional B-Math |
|
spaces I-Math |
|
and O |
|
does O |
|
not O |
|
use O |
|
topolog O |
|
and O |
|
does O |
|
not O |
|
use O |
|
topology B-Math |
|
|
|
this O |
|
is O |
|
in O |
|
particular O |
|
the O |
|
case O |
|
in O |
|
graph B-Math |
|
theory I-Math |
|
f O |
|
incidence O |
|
matrices O |
|
and O |
|
adjacency O |
|
matrice O |
|
f O |
|
incidence O |
|
matrices B-Math |
|
and O |
|
adjacency O |
|
matrice O |
|
|
|
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|
every O |
|
vector B-Math |
|
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|
that O |
|
span O |
|
has O |
|
exactly O |
|
one O |
|
expression O |
|
as O |
|
a O |
|
linear O |
|
combination O |
|
of O |
|
the O |
|
given O |
|
lefthand O |
|
vectors O |
|
then O |
|
any O |
|
solution O |
|
is O |
|
uniqu O |
|
within O |
|
that O |
|
span O |
|
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|
exactly O |
|
one O |
|
expression O |
|
as O |
|
a O |
|
linear B-Math |
|
combination I-Math |
|
of O |
|
the O |
|
given O |
|
lefthand O |
|
vectors O |
|
then O |
|
any O |
|
solution O |
|
is O |
|
uniqu O |
|
|
|
these O |
|
concepts O |
|
are O |
|
central O |
|
to O |
|
the O |
|
definition O |
|
of O |
|
dimension B-Math |
|
|
|
in O |
|
mathematics O |
|
a O |
|
set O |
|
b O |
|
of O |
|
vectors O |
|
in O |
|
a O |
|
vector O |
|
space O |
|
v O |
|
is O |
|
called O |
|
a O |
|
basis B-Attributes |
|
p O |
|
vector B-Math |
|
space I-Math |
|
v O |
|
is O |
|
called O |
|
a O |
|
basis O |
|
p O |
|
vectors B-Math |
|
in O |
|
a O |
|
vector O |
|
space O |
|
v O |
|
is O |
|
called O |
|
a O |
|
basis O |
|
p O |
|
|
|
with O |
|
respect O |
|
to O |
|
general O |
|
linear B-Math |
|
maps I-Math |
|
linear O |
|
endomorphisms O |
|
and O |
|
square O |
|
matrices O |
|
have O |
|
some O |
|
specific O |
|
properties O |
|
that O |
|
make O |
|
their O |
|
study O |
|
an O |
|
important O |
|
part O |
|
of O |
|
linear O |
|
algebra O |
|
which O |
|
is O |
|
used O |
|
in O |
|
many O |
|
parts O |
|
of O |
|
mathematics O |
|
including O |
|
geometric O |
|
transformations O |
|
coordinate O |
|
changes O |
|
quadratic O |
|
forms O |
|
and O |
|
many O |
|
other O |
|
part O |
|
of O |
|
mathematic O |
|
linear B-Math |
|
endomorphisms I-Math |
|
and O |
|
square O |
|
matrices O |
|
have O |
|
some O |
|
specific O |
|
properties O |
|
that O |
|
make O |
|
their O |
|
study O |
|
an O |
|
important O |
|
part O |
|
of O |
|
linear O |
|
algebra O |
|
which O |
|
is O |
|
used O |
|
in O |
|
many O |
|
parts O |
|
of O |
|
mathematics O |
|
including O |
|
geometric O |
|
transformations O |
|
coordinate O |
|
changes O |
|
quadratic O |
|
forms O |
|
and O |
|
many O |
|
other O |
|
part O |
|
of O |
|
mathematic O |
|
and O |
|
square B-Math |
|
matrices I-Math |
|
have O |
|
some O |
|
specific O |
|
properties O |
|
that O |
|
make O |
|
their O |
|
study O |
|
an O |
|
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|
part O |
|
of O |
|
linear O |
|
algebra O |
|
which O |
|
is O |
|
used O |
|
in O |
|
many O |
|
parts O |
|
of O |
|
mathematics O |
|
including O |
|
geometric O |
|
transformations O |
|
coordinate O |
|
changes O |
|
quadratic O |
|
forms O |
|
and O |
|
many O |
|
other O |
|
part O |
|
of O |
|
mathematic O |
|
have O |
|
some O |
|
specific O |
|
properties O |
|
that O |
|
make O |
|
their O |
|
study O |
|
an O |
|
important O |
|
part O |
|
of O |
|
linear O |
|
algebra O |
|
which O |
|
is O |
|
used O |
|
in O |
|
many O |
|
parts O |
|
of O |
|
mathematics O |
|
including O |
|
geometric O |
|
transformations O |
|
coordinate B-Math |
|
changes I-Math |
|
quadratic O |
|
forms O |
|
and O |
|
many O |
|
other O |
|
part O |
|
of O |
|
mathematic O |
|
quadratic B-Math |
|
forms I-Math |
|
and O |
|
many O |
|
other O |
|
part O |
|
of O |
|
mathematic O |
|
geometric B-Math |
|
transformations I-Math |
|
coordinate O |
|
changes O |
|
quadratic O |
|
forms O |
|
and O |
|
many O |
|
other O |
|
part O |
|
of O |
|
mathematic O |
|
|
|
sciences O |
|
concerned O |
|
with O |
|
this O |
|
space O |
|
use O |
|
geometry B-Math |
|
widel O |
|
sciences B-Attributes |
|
concerned O |
|
with O |
|
this O |
|
space O |
|
use O |
|
geometry O |
|
widel O |
|
|
|
in O |
|
both O |
|
cases O |
|
very O |
|
large O |
|
matrices B-Math |
|
are O |
|
generally O |
|
involve O |
|
|
|
the O |
|
following O |
|
pictures O |
|
illustrate O |
|
this O |
|
trichotomy O |
|
in O |
|
the O |
|
case O |
|
of O |
|
two O |
|
variables B-Math |
|
|
|
vectors B-Math |
|
play O |
|
an O |
|
important O |
|
role O |
|
in O |
|
physics O |
|
the O |
|
velocity O |
|
and O |
|
acceleration O |
|
of O |
|
a O |
|
moving O |
|
object O |
|
and O |
|
the O |
|
forces O |
|
acting O |
|
on O |
|
it O |
|
can O |
|
all O |
|
be O |
|
described O |
|
with O |
|
vector O |
|
play O |
|
an O |
|
important O |
|
role O |
|
in O |
|
physics O |
|
the O |
|
velocity O |
|
and O |
|
acceleration O |
|
of O |
|
a O |
|
moving B-Attributes |
|
object O |
|
and O |
|
the O |
|
forces O |
|
acting O |
|
on O |
|
it O |
|
can O |
|
all O |
|
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|
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|
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|
vector O |
|
object O |
|
and O |
|
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|
forces B-Attributes |
|
acting O |
|
on O |
|
it O |
|
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|
all O |
|
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|
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|
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|
vector O |
|
|
|
it O |
|
assists O |
|
in O |
|
the O |
|
modeling O |
|
and O |
|
simulation O |
|
of O |
|
fluid O |
|
flow O |
|
providing O |
|
essential O |
|
tools O |
|
for O |
|
the O |
|
analysis O |
|
of O |
|
fluid B-Attributes |
|
dynamics I-Attributes |
|
problems I-Attributes |
|
|
|
the O |
|
term O |
|
is O |
|
used O |
|
in O |
|
various O |
|
branches O |
|
of O |
|
mathematics O |
|
such O |
|
as O |
|
calculus B-Math |
|
differential I-Math |
|
geometry I-Math |
|
algebraic O |
|
geometry O |
|
and O |
|
algebraic O |
|
topolog O |
|
algebraic B-Math |
|
geometry I-Math |
|
and O |
|
algebraic O |
|
topolog O |
|
and O |
|
algebraic B-Math |
|
topology I-Math |
|
|
|
a O |
|
linear B-Math |
|
combination I-Math |
|
of O |
|
x O |
|
and O |
|
y O |
|
would O |
|
be O |
|
any O |
|
expression O |
|
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|
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|
form O |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
y O |
|
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|
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|
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|
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|
of O |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
linear B-Math |
|
span I-Math |
|
of O |
|
a O |
|
subset O |
|
s O |
|
one O |
|
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|
uses O |
|
the O |
|
following O |
|
phraseseither O |
|
s O |
|
spans O |
|
v O |
|
s O |
|
is O |
|
a O |
|
spanning O |
|
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|
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|
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|
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|
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|
spannedgenerated O |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
vector B-Math |
|
space I-Math |
|
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|
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|
a O |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
phraseseither O |
|
s O |
|
spans O |
|
v O |
|
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|
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|
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|
spanning O |
|
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|
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|
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|
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|
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|
spannedgenerated O |
|
by O |
|
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|
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|
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|
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|
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|
generator O |
|
or O |
|
generator O |
|
set O |
|
of O |
|
|
|
spans O |
|
can O |
|
be O |
|
generalized O |
|
to O |
|
matroids B-Attributes |
|
and O |
|
module O |
|
and O |
|
modules B-Attributes |
|
spans B-Math |
|
can O |
|
be O |
|
generalized O |
|
to O |
|
matroids O |
|
and O |
|
module O |
|
|
|
the O |
|
elements O |
|
of O |
|
a O |
|
basis B-Attributes |
|
are O |
|
called O |
|
basis O |
|
vector O |
|
are O |
|
called O |
|
basis B-Math |
|
vectors I-Math |
|
|
|
f O |
|
a O |
|
linear B-Math |
|
map I-Math |
|
is O |
|
a O |
|
bijection O |
|
then O |
|
it O |
|
is O |
|
called O |
|
a O |
|
linear O |
|
isomorphis O |
|
is O |
|
a O |
|
bijection O |
|
then O |
|
it O |
|
is O |
|
called O |
|
a O |
|
linear B-Math |
|
isomorphism I-Math |
|
bijection B-Attributes |
|
then O |
|
it O |
|
is O |
|
called O |
|
a O |
|
linear O |
|
isomorphis O |
|
|
|
in O |
|
mathematics B-Math |
|
and O |
|
physics O |
|
a O |
|
vector O |
|
space O |
|
also O |
|
called O |
|
a O |
|
linear O |
|
space O |
|
is O |
|
a O |
|
set O |
|
whose O |
|
elements O |
|
often O |
|
called O |
|
vectors O |
|
may O |
|
be O |
|
added O |
|
together O |
|
and O |
|
multiplied O |
|
scaled O |
|
by O |
|
numbers O |
|
called O |
|
scalar O |
|
and O |
|
physics O |
|
a O |
|
vector B-Math |
|
space I-Math |
|
also O |
|
called O |
|
a O |
|
linear O |
|
space O |
|
is O |
|
a O |
|
set O |
|
whose O |
|
elements O |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
digital O |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
mechanics O |
|
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|
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|
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|
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|
differential I-Math |
|
equations I-Math |
|
digital O |
|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
space I-Math |
|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
linear B-Math |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
mechanics O |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
mathematics O |
|
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|
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|
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|
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|
mechanics I-Attributes |
|
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|
dynamics O |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
geometr O |
|
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|
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|
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|
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|
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|
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|
lines O |
|
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|
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|
synthetic B-Math |
|
geometry I-Math |
|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
complex O |
|
numbers O |
|
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|
functions B-Math |
|
|
|
functional B-Math |
|
analysis I-Math |
|
studies O |
|
function O |
|
space O |
|
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|
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|
spaces I-Math |
|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
analysis I-Math |
|
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|
basis I-Math |
|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
matrices O |
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
various O |
|
application O |
|
|
|
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|
algebraic O |
|
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|
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|
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|
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|
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|
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|
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|
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|
are O |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
problems O |
|
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|
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|
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|
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|
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|
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|
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|
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|
performance O |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
example O |
|
rotations B-Attributes |
|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
fundamental O |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
lines B-Math |
|
planes I-Math |
|
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|
rotation O |
|
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|
rotations B-Math |
|
|
|
linear B-Math |
|
algebra I-Math |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
functions I-Math |
|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
theory I-Math |
|
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|
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|
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|
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|
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|
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|
vector O |
|
space O |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
technique O |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
system I-Math |
|
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|
linearization O |
|
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|
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|
technique O |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
linear B-Math |
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
number O |
|
of O |
|
elements O |
|
called O |
|
the O |
|
dimension O |
|
of O |
|
the O |
|
vector O |
|
spac O |
|
however O |
|
all O |
|
the O |
|
bases O |
|
have O |
|
the O |
|
same O |
|
number O |
|
of O |
|
elements O |
|
called O |
|
the O |
|
dimension B-Math |
|
of O |
|
the O |
|
vector O |
|
spac O |
|
|
|
for O |
|
a O |
|
system O |
|
involving O |
|
two O |
|
variables O |
|
x O |
|
and O |
|
y O |
|
each O |
|
linear B-Math |
|
equation I-Math |
|
determines O |
|
a O |
|
line O |
|
on O |
|
the O |
|
xyplan O |
|
|
|
tropical B-Math |
|
geometry I-Math |
|
is O |
|
another O |
|
example O |
|
of O |
|
linear O |
|
algebra O |
|
in O |
|
a O |
|
more O |
|
exotic O |
|
structur O |
|
is O |
|
another O |
|
example O |
|
of O |
|
linear B-Math |
|
algebra I-Math |
|
in O |
|
a O |
|
more O |
|
exotic O |
|
structur O |
|
|
|
in O |
|
numerical B-Math |
|
analysis I-Math |
|
many O |
|
computational O |
|
problems O |
|
are O |
|
solved O |
|
by O |
|
reducing O |
|
them O |
|
to O |
|
a O |
|
matrix O |
|
computation O |
|
and O |
|
this O |
|
often O |
|
involves O |
|
computing O |
|
with O |
|
matrices O |
|
of O |
|
huge O |
|
dimensio O |
|
many O |
|
computational O |
|
problems O |
|
are O |
|
solved O |
|
by O |
|
reducing O |
|
them O |
|
to O |
|
a O |
|
matrix B-Math |
|
omputation O |
|
and O |
|
this O |
|
often O |
|
involves O |
|
computing O |
|
with O |
|
matrices O |
|
of O |
|
huge O |
|
dimensio O |
|
|
|
in O |
|
general O |
|
the O |
|
behavior O |
|
of O |
|
a O |
|
linear B-Math |
|
system I-Math |
|
is O |
|
determined O |
|
by O |
|
the O |
|
relationship O |
|
between O |
|
the O |
|
number O |
|
of O |
|
equations O |
|
and O |
|
the O |
|
number O |
|
of O |
|
unknown O |
|
|
|
geometric B-Math |
|
interpretatio O |
|
|
|
a B-Math |
|
first I-Math |
|
order I-Math |
|
differential I-Math |
|
equation I-Math |
|
is O |
|
said O |
|
to O |
|
be O |
|
homogeneous O |
|
if O |
|
it O |
|
may O |
|
be O |
|
writte O |
|
is O |
|
said O |
|
to O |
|
be O |
|
homogeneous B-Math |
|
if O |
|
it O |
|
may O |
|
be O |
|
writte O |
|
|
|
in O |
|
mathematics B-Math |
|
a O |
|
system O |
|
of O |
|
linear O |
|
equations O |
|
or O |
|
linear O |
|
system O |
|
is O |
|
a O |
|
collection O |
|
of O |
|
one O |
|
or O |
|
more O |
|
linear O |
|
equations O |
|
involving O |
|
the O |
|
same O |
|
variable O |
|
a O |
|
system O |
|
of O |
|
linear B-Math |
|
equations I-Math |
|
or O |
|
linear O |
|
system O |
|
is O |
|
a O |
|
collection O |
|
of O |
|
one O |
|
or O |
|
more O |
|
linear O |
|
equations O |
|
involving O |
|
the O |
|
same O |
|
variable O |
|
or O |
|
linear B-Math |
|
system I-Math |
|
is O |
|
a O |
|
collection O |
|
of O |
|
one O |
|
or O |
|
more O |
|
linear O |
|
equations O |
|
involving O |
|
the O |
|
same O |
|
variable O |
|
|
|
for O |
|
solutions O |
|
in O |
|
an O |
|
integral O |
|
domain O |
|
like O |
|
the O |
|
ring O |
|
of O |
|
the O |
|
integers O |
|
or O |
|
in O |
|
other O |
|
algebraic O |
|
structures O |
|
other O |
|
theories O |
|
have O |
|
been O |
|
developed O |
|
see O |
|
linear B-Math |
|
equation I-Math |
|
over O |
|
a O |
|
rin O |
|
|
|
modules B-Math |
|
over O |
|
the O |
|
integers O |
|
can O |
|
be O |
|
identified O |
|
with O |
|
abelian O |
|
groups O |
|
since O |
|
the O |
|
multiplication O |
|
by O |
|
an O |
|
integer O |
|
may O |
|
be O |
|
identified O |
|
to O |
|
a O |
|
repeated O |
|
additio O |
|
over O |
|
the O |
|
integers O |
|
can O |
|
be O |
|
identified O |
|
with O |
|
abelian B-Math |
|
groups I-Math |
|
since O |
|
the O |
|
multiplication O |
|
by O |
|
an O |
|
integer O |
|
may O |
|
be O |
|
identified O |
|
to O |
|
a O |
|
repeated O |
|
additio O |
|
since O |
|
the O |
|
multiplication B-Attributes |
|
by O |
|
an O |
|
integer O |
|
may O |
|
be O |
|
identified O |
|
to O |
|
a O |
|
repeated O |
|
additio O |
|
by O |
|
an O |
|
integer O |
|
may O |
|
be O |
|
identified O |
|
to O |
|
a O |
|
repeated O |
|
addition B-Attributes |
|
integers B-Math |
|
can O |
|
be O |
|
identified O |
|
with O |
|
abelian O |
|
groups O |
|
since O |
|
the O |
|
multiplication O |
|
by O |
|
an O |
|
integer O |
|
may O |
|
be O |
|
identified O |
|
to O |
|
a O |
|
repeated O |
|
additio O |
|
|
|
thus O |
|
computing O |
|
intersections O |
|
of O |
|
lines O |
|
and O |
|
planes O |
|
amounts O |
|
to O |
|
solving O |
|
systems O |
|
of O |
|
linear B-Math |
|
equations I-Math |
|
lines B-Math |
|
and O |
|
planes O |
|
amounts O |
|
to O |
|
solving O |
|
systems O |
|
of O |
|
linear O |
|
equation O |
|
and O |
|
planes B-Math |
|
amounts O |
|
to O |
|
solving O |
|
systems O |
|
of O |
|
linear O |
|
equation O |
|
|
|
the O |
|
concepts O |
|
of O |
|
linear B-Math |
|
independence I-Math |
|
span I-Math |
|
basis I-Math |
|
and O |
|
linear O |
|
maps O |
|
also O |
|
called O |
|
module O |
|
homomorphisms O |
|
are O |
|
defined O |
|
for O |
|
modules O |
|
exactly O |
|
as O |
|
for O |
|
vector O |
|
spaces O |
|
with O |
|
the O |
|
essential O |
|
difference O |
|
that O |
|
if O |
|
r O |
|
is O |
|
not O |
|
a O |
|
field O |
|
there O |
|
are O |
|
modules O |
|
that O |
|
do O |
|
not O |
|
have O |
|
any O |
|
basi O |
|
and O |
|
linear B-Math |
|
maps I-Math |
|
also O |
|
called O |
|
module O |
|
homomorphisms O |
|
are O |
|
defined O |
|
for O |
|
modules O |
|
exactly O |
|
as O |
|
for O |
|
vector O |
|
spaces O |
|
with O |
|
the O |
|
essential O |
|
difference O |
|
that O |
|
if O |
|
r O |
|
is O |
|
not O |
|
a O |
|
field O |
|
there O |
|
are O |
|
modules O |
|
that O |
|
do O |
|
not O |
|
have O |
|
any O |
|
basi O |
|
also O |
|
called O |
|
module B-Math |
|
homomorphisms I-Math |
|
are O |
|
defined O |
|
for O |
|
modules O |
|
exactly O |
|
as O |
|
for O |
|
vector O |
|
spaces O |
|
with O |
|
the O |
|
essential O |
|
difference O |
|
that O |
|
if O |
|
r O |
|
is O |
|
not O |
|
a O |
|
field O |
|
there O |
|
are O |
|
modules O |
|
that O |
|
do O |
|
not O |
|
have O |
|
any O |
|
basi O |
|
|
|
in O |
|
any O |
|
event O |
|
the O |
|
span O |
|
has O |
|
a O |
|
basis O |
|
of O |
|
linearly O |
|
independent O |
|
vectors O |
|
that O |
|
do O |
|
guarantee O |
|
exactly O |
|
one O |
|
expression O |
|
and O |
|
the O |
|
number O |
|
of O |
|
vectors B-Math |
|
in O |
|
that O |
|
basis O |
|
its O |
|
dimension O |
|
can O |
|
not O |
|
be O |
|
larger O |
|
than O |
|
m O |
|
or O |
|
n O |
|
but O |
|
it O |
|
can O |
|
be O |
|
smalle O |
|
|
|
consequently O |
|
linear B-Math |
|
algebra I-Math |
|
algorithms O |
|
have O |
|
been O |
|
highly O |
|
optimize O |
|
|
|
these O |
|
operations O |
|
and O |
|
associated O |
|
laws O |
|
qualify O |
|
euclidean O |
|
vectors O |
|
as O |
|
an O |
|
example O |
|
of O |
|
the O |
|
more O |
|
generalized O |
|
concept O |
|
of O |
|
vectors O |
|
defined O |
|
simply O |
|
as O |
|
elements O |
|
of O |
|
a O |
|
vector B-Math |
|
space I-Math |
|
|
|
a O |
|
linear B-Math |
|
map I-Math |
|
from O |
|
to O |
|
always O |
|
maps O |
|
the O |
|
origin O |
|
of O |
|
to O |
|
the O |
|
origin O |
|
o O |
|
|
|
in O |
|
modern O |
|
introductory O |
|
texts O |
|
on O |
|
functional B-Math |
|
analysis I-Math |
|
the O |
|
subject O |
|
is O |
|
seen O |
|
as O |
|
the O |
|
study O |
|
of O |
|
vector O |
|
spaces O |
|
endowed O |
|
with O |
|
a O |
|
topology O |
|
in O |
|
particular O |
|
infinitedimensional O |
|
space O |
|
the O |
|
subject O |
|
is O |
|
seen O |
|
as O |
|
the O |
|
study O |
|
of O |
|
vector B-Math |
|
spaces I-Math |
|
endowed O |
|
with O |
|
a O |
|
topology O |
|
in O |
|
particular O |
|
infinitedimensional O |
|
space O |
|
endowed O |
|
with O |
|
a O |
|
topology B-Math |
|
in O |
|
particular O |
|
infinitedimensional O |
|
space O |
|
|
|
this O |
|
article O |
|
deals O |
|
mainly O |
|
with O |
|
finitedimensional O |
|
vector B-Math |
|
spaces I-Math |
|
|
|
matrix B-Math |
|
theory I-Math |
|
is O |
|
the O |
|
branch O |
|
of O |
|
mathematics O |
|
that O |
|
focuses O |
|
on O |
|
the O |
|
study O |
|
of O |
|
matrice O |
|
is O |
|
the O |
|
branch O |
|
of O |
|
mathematics B-Math |
|
that O |
|
focuses O |
|
on O |
|
the O |
|
study O |
|
of O |
|
matrice O |
|
that O |
|
focuses O |
|
on O |
|
the O |
|
study O |
|
of O |
|
matrices B-Math |
|
|
|
a O |
|
solution B-Math |
|
to O |
|
a O |
|
linear O |
|
system O |
|
is O |
|
an O |
|
assignment O |
|
of O |
|
values O |
|
to O |
|
the O |
|
variables O |
|
such O |
|
that O |
|
all O |
|
the O |
|
equations O |
|
are O |
|
simultaneously O |
|
satisfie O |
|
to O |
|
a O |
|
linear O |
|
system O |
|
is O |
|
an O |
|
assignment O |
|
of O |
|
values O |
|
to O |
|
the O |
|
variables B-Math |
|
uch O |
|
that O |
|
all O |
|
the O |
|
equations O |
|
are O |
|
simultaneously O |
|
satisfie O |
|
uch O |
|
that O |
|
all O |
|
the O |
|
equations O |
|
are O |
|
simultaneously B-Math |
|
satisfie O |
|
|
|
in O |
|
multilinear B-Math |
|
algebra I-Math |
|
one O |
|
considers O |
|
multivariable O |
|
linear O |
|
transformations O |
|
that O |
|
is O |
|
mappings O |
|
that O |
|
are O |
|
linear O |
|
in O |
|
each O |
|
of O |
|
a O |
|
number O |
|
of O |
|
different O |
|
variable O |
|
one O |
|
considers O |
|
multivariable O |
|
linear B-Math |
|
transformations I-Math |
|
that O |
|
is O |
|
mappings O |
|
that O |
|
are O |
|
linear O |
|
in O |
|
each O |
|
of O |
|
a O |
|
number O |
|
of O |
|
different O |
|
variable O |
|
|
|
otherwise O |
|
it O |
|
is O |
|
infinitedimensional B-Attributes |
|
and O |
|
its O |
|
dimension O |
|
is O |
|
an O |
|
infinite O |
|
cardina O |
|
|
|
a O |
|
normed B-Math |
|
vector I-Math |
|
space I-Math |
|
is O |
|
a O |
|
vector O |
|
space O |
|
along O |
|
with O |
|
a O |
|
function O |
|
called O |
|
a O |
|
norm O |
|
which O |
|
measures O |
|
the O |
|
size O |
|
of O |
|
element O |
|
is O |
|
a O |
|
vector B-Math |
|
space I-Math |
|
along O |
|
with O |
|
a O |
|
function O |
|
called O |
|
a O |
|
norm O |
|
which O |
|
measures O |
|
the O |
|
size O |
|
of O |
|
element O |
|
|
|
such O |
|
a O |
|
system O |
|
is O |
|
known O |
|
as O |
|
an O |
|
underdetermined B-Math |
|
system I-Math |
|
|
|
if O |
|
in O |
|
addition O |
|
to O |
|
vector B-Math |
|
addition I-Math |
|
and O |
|
scalar O |
|
multiplication O |
|
there O |
|
is O |
|
a O |
|
bilinear O |
|
vector O |
|
product O |
|
v O |
|
v O |
|
v O |
|
the O |
|
vector O |
|
space O |
|
is O |
|
called O |
|
an O |
|
algebra O |
|
for O |
|
instance O |
|
associative O |
|
algebras O |
|
are O |
|
algebras O |
|
with O |
|
an O |
|
associate O |
|
vector O |
|
product O |
|
like O |
|
the O |
|
algebra O |
|
of O |
|
square O |
|
matrices O |
|
or O |
|
the O |
|
algebra O |
|
of O |
|
polynomial O |
|
and O |
|
scalar B-Math |
|
multiplication I-Math |
|
there O |
|
is O |
|
a O |
|
bilinear O |
|
vector O |
|
product O |
|
v O |
|
v O |
|
v O |
|
the O |
|
vector O |
|
space O |
|
is O |
|
called O |
|
an O |
|
algebra O |
|
for O |
|
instance O |
|
associative O |
|
algebras O |
|
are O |
|
algebras O |
|
with O |
|
an O |
|
associate O |
|
vector O |
|
product O |
|
like O |
|
the O |
|
algebra O |
|
of O |
|
square O |
|
matrices O |
|
or O |
|
the O |
|
algebra O |
|
of O |
|
polynomial O |
|
there O |
|
is O |
|
a O |
|
bilinear B-Math |
|
vector I-Math |
|
product I-Math |
|
v O |
|
v O |
|
v O |
|
the O |
|
vector O |
|
space O |
|
is O |
|
called O |
|
an O |
|
algebra O |
|
for O |
|
instance O |
|
associative O |
|
algebras O |
|
are O |
|
algebras O |
|
with O |
|
an O |
|
associate O |
|
vector O |
|
product O |
|
like O |
|
the O |
|
algebra O |
|
of O |
|
square O |
|
matrices O |
|
or O |
|
the O |
|
algebra O |
|
of O |
|
polynomial O |
|
v O |
|
v O |
|
v O |
|
the O |
|
vector B-Math |
|
space I-Math |
|
is O |
|
called O |
|
an O |
|
algebra O |
|
for O |
|
instance O |
|
associative O |
|
algebras O |
|
are O |
|
algebras O |
|
with O |
|
an O |
|
associate O |
|
vector O |
|
product O |
|
like O |
|
the O |
|
algebra O |
|
of O |
|
square O |
|
matrices O |
|
or O |
|
the O |
|
algebra O |
|
of O |
|
polynomial O |
|
|
|
analysis B-Math |
|
is O |
|
the O |
|
branch O |
|
of O |
|
mathematics O |
|
dealing O |
|
with O |
|
continuous O |
|
functions O |
|
limits O |
|
and O |
|
related O |
|
theories O |
|
such O |
|
as O |
|
differentiation O |
|
integration O |
|
measure O |
|
infinite O |
|
sequences O |
|
series O |
|
and O |
|
analytic O |
|
function O |
|
is O |
|
the O |
|
branch O |
|
of O |
|
mathematics O |
|
dealing O |
|
with O |
|
continuous B-Math |
|
functions I-Math |
|
limits I-Math |
|
and O |
|
related O |
|
theories O |
|
such O |
|
as O |
|
differentiation O |
|
integration O |
|
measure O |
|
infinite O |
|
sequences O |
|
series O |
|
and O |
|
analytic O |
|
function O |
|
and O |
|
related O |
|
theories O |
|
such O |
|
as O |
|
differentiation O |
|
integration O |
|
measure O |
|
infinite O |
|
sequences O |
|
series O |
|
and O |
|
analytic B-Math |
|
functions I-Math |
|
|
|
vectors O |
|
can O |
|
be O |
|
added O |
|
to O |
|
other O |
|
vectors O |
|
according O |
|
to O |
|
vector B-Math |
|
algebra I-Math |
|
vectors B-Math |
|
can O |
|
be O |
|
added O |
|
to O |
|
other O |
|
vectors O |
|
according O |
|
to O |
|
vector O |
|
algebr O |