User alphan sahin - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T18:02:06Z http://mathoverflow.net/feeds/user/30966 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120056/injective-mapping Injective Mapping Alphan Sahin 2013-01-27T21:37:13Z 2013-01-27T22:36:19Z <p>Let y=Ax. A is a matrix n by m and m>n. Also, x gets its values from a finite alphabet. Elements of A and x can be complex numbers. How can i show if the mapping from x to y is injective for given A and alphabet (beside a search method)? </p> <p>For example, let A and the alphabet be</p> <p>[1 0 1/sqrt2 1/sqrt2] [0 1 1/sqrt2 -1/sqrt2]</p> <p>and </p> <p>{1, -1}</p> <p>respectively. For this alphabet, x can be [1 1 -1 -1]'. Actually, there are 2^m=16 possible options for x, considering all permutations. Using this permutation set and A matrix, the operation will be one-to-one even if we map larger space to smaller space! Indeed, it is possible to test the injectivity with this scale. However, when we have larger matrices, search algorithm (i.e., testing all mappings) would take very long. If i can, i would like to verify it without a search algorithm.</p> http://mathoverflow.net/questions/122804/yax-with-finite-alphabet Comment by Alphan Sahin Alphan Sahin 2013-02-25T02:10:48Z 2013-02-25T02:10:48Z Thank you very much. I have updated the question. http://mathoverflow.net/questions/122804/yax-with-finite-alphabet Comment by Alphan Sahin Alphan Sahin 2013-02-24T22:56:22Z 2013-02-24T22:56:22Z I would also like to note that A is not square matrix. http://mathoverflow.net/questions/122804/yax-with-finite-alphabet Comment by Alphan Sahin Alphan Sahin 2013-02-24T22:51:02Z 2013-02-24T22:51:02Z Maybe, i didn't use the correct terminology. Basically, i am trying to understand the injection between x and y for the equation of y=Ax, when x gets its elements from a finite alphabet. I don't know which topic can help me to figure this issue out. http://mathoverflow.net/questions/120056/injective-mapping Comment by Alphan Sahin Alphan Sahin 2013-01-29T16:37:46Z 2013-01-29T16:37:46Z Do you think that it is helpful for this question if i constraint my alphabet as the roots of polynomials with the integer coefficients? http://mathoverflow.net/questions/120056/injective-mapping Comment by Alphan Sahin Alphan Sahin 2013-01-27T23:03:22Z 2013-01-27T23:03:22Z I think, we can say that this question is also related with cryptology. How can we sure that a crypto algorithm results in one-to-one transformation? We cannot test all inputs.