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Mathematics
Linear and Abstract Algebra
Question regarding fundamental region of a lattice
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[QUOTE="Peter_Newman, post: 6861157, member: 656585"] Hey [USER=53426]@Office_Shredder[/USER], thanks for your answer! I would also proceed similarly. See below: We would have to show uniqueness and assume that there is an overlap, then let ##x_1 = P(B^*) + \Lambda(B)## and ##x_2 = P(B^*) + \Lambda(B)##. If there is an overlap, then it means that ##x_1 = x_2##, this is noting else then $$x_1 = P(B^*) + \Lambda(B) = a_1 b_1^* + ... + a_n b_n^* + a_1'b_1 + ... + a_n'b_n$$ and $$x_2 = P(B^*) + \Lambda(B) = c_1 b_1^* + ... + c_n b_n^* + c_1'b_1 + ... + c_n'b_n$$ with ##a_i,c_i \in## [0,1) and ##a_i', c_i' \in \mathbf{Z}##. The problem now is that once here we have the Gram Schmidt vectors and the basis vectors. You could try to rewrite the basis vectors, but I think that makes things even more difficult. Even if I am sure that this is the way for the proof. But what happens next? This can be shown relatively easy with the idea from above, this is easier since we only have one "sort" of vectors. The calculation of uniqueness works out excellently here! [/QUOTE]
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Question regarding fundamental region of a lattice
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