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p(x) is a polynomial
p_k(x) = product of (x - product(i)) for i in the k-th degree cartesian product of the roots of p
For example, if p(x) = (x - 1) * (x - 2), then p_3(x) = (x-1*1*1)*(x-1*1*2)*(x-1*2*1)*(x-1*2*2)*(x-2*1*1)*(x-2*1*2)*(x-2*2*1)*(x-2*2*2) = 4096 - 13824*x + 19328*x^2 - 14688*x^3 + 6648*x^4 - 1836*x^5 + 302*x^6 - 27*x^7 + x^8.
My question is, is there a way to calculate the coefficients of p_k without knowing the roots of p exactly (i.e., when p has a large degree), and when using approximations doesn't give enough precision? Will p_k always have integer coefficients? 70.190.182.236 ( talk) 04:06, 15 January 2015 (UTC)
Hi guys,
I'm reading the http://en.wikipedia.org/wiki/One-way_function article.
References seem to be clear that mathematically there is no proof that one-way functions (hashes) actually exist. Does this mean that there is absolutely no bounds on computational complexity that can be proved at all? For example, it could be that O(n) where n is exactly seven operations, for any hash? --- but that's clearly insane. We can know 7 operations aren't enough to produce an arbitrary hash value because we can just brute-force all 7-operation "algorithms" and prove that 7 operations won't get you a plaintext that hashes to x for any x in the space.
so there has to be some bounds, right? (obviously brute-forcing is a formal proof but an expensive one, so we can formally prove that 7 operations can't reverse sha-256)
But if there is some bounds, then what is the sense that "one-way functions don't exist." ... That we have formally proved minimum bounds (such as 7), they're just not high enough?
or what is the key insight I'm missing. thanks. 212.96.61.236 ( talk) 05:20, 15 January 2015 (UTC)
Hello :-)
I followed the instructions of Principal_component_regression#Details_of_the_method (which was very exciting). I did this with statistic software R. I got final PCR-estimates, e.g.
[1,] 27.005477 [2,] 28.531195 [3,] 4.031036 [4,] 29.464202 [5,] 18.974255 [6,] 47.658639 [7,] 24.125975 [8,] 30.831690 [9,] 32.111585 [10,] 21.811584 [11,] 34.054133 [12,] 28.901388 [13,] 37.990794 [14,] 9.021954 [15,] -66.069150 [16,] 74.483241 [17,] -3.654576 [18,] -6.004836 [19,] 11.401041
I can calculate the R^2 (and adj.) and the F-statistic. But is it possible to calculate SE and p-values to check the significance? What else can I do to check whether and to what extent my PCR-Model is better than the OLS-version (original data was strongly multicollinear). I checked Principal_component_regression#Two_basic_properties and for k=p=19 I got same reg-coefficients with OLS and PCR. I also can calculate MSE-values, as in Principal_component_regression#Efficiency it says MSE of OLS should be bigger (e.g. worser) than PCR. For k=p=19 it is the same, for less components PCR-MSE gots smaller. But what else can I see from my results? Would be very thankful for any idea :-) Thanks a lot! -- WissensDürster ( talk) 11:49, 15 January 2015 (UTC)
The proposition "(B if C) and (C if B)", is tautologically equivalent to "B iff C", in which B is not mentioned twice.
My question is about whether, the following proposition is - tautologically equiavalent to any proposition - in which B is not mentioned twice (by using as many connectives as we wish and no matter what connectives are used, provided that all connectives are binary or unary):
"(B if C), and (C if (B and A))".
77.126.32.139 ( talk) 23:10, 15 January 2015 (UTC)
Mathematics desk | ||
---|---|---|
< January 14 | << Dec | January | Feb >> | January 16 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
p(x) is a polynomial
p_k(x) = product of (x - product(i)) for i in the k-th degree cartesian product of the roots of p
For example, if p(x) = (x - 1) * (x - 2), then p_3(x) = (x-1*1*1)*(x-1*1*2)*(x-1*2*1)*(x-1*2*2)*(x-2*1*1)*(x-2*1*2)*(x-2*2*1)*(x-2*2*2) = 4096 - 13824*x + 19328*x^2 - 14688*x^3 + 6648*x^4 - 1836*x^5 + 302*x^6 - 27*x^7 + x^8.
My question is, is there a way to calculate the coefficients of p_k without knowing the roots of p exactly (i.e., when p has a large degree), and when using approximations doesn't give enough precision? Will p_k always have integer coefficients? 70.190.182.236 ( talk) 04:06, 15 January 2015 (UTC)
Hi guys,
I'm reading the http://en.wikipedia.org/wiki/One-way_function article.
References seem to be clear that mathematically there is no proof that one-way functions (hashes) actually exist. Does this mean that there is absolutely no bounds on computational complexity that can be proved at all? For example, it could be that O(n) where n is exactly seven operations, for any hash? --- but that's clearly insane. We can know 7 operations aren't enough to produce an arbitrary hash value because we can just brute-force all 7-operation "algorithms" and prove that 7 operations won't get you a plaintext that hashes to x for any x in the space.
so there has to be some bounds, right? (obviously brute-forcing is a formal proof but an expensive one, so we can formally prove that 7 operations can't reverse sha-256)
But if there is some bounds, then what is the sense that "one-way functions don't exist." ... That we have formally proved minimum bounds (such as 7), they're just not high enough?
or what is the key insight I'm missing. thanks. 212.96.61.236 ( talk) 05:20, 15 January 2015 (UTC)
Hello :-)
I followed the instructions of Principal_component_regression#Details_of_the_method (which was very exciting). I did this with statistic software R. I got final PCR-estimates, e.g.
[1,] 27.005477 [2,] 28.531195 [3,] 4.031036 [4,] 29.464202 [5,] 18.974255 [6,] 47.658639 [7,] 24.125975 [8,] 30.831690 [9,] 32.111585 [10,] 21.811584 [11,] 34.054133 [12,] 28.901388 [13,] 37.990794 [14,] 9.021954 [15,] -66.069150 [16,] 74.483241 [17,] -3.654576 [18,] -6.004836 [19,] 11.401041
I can calculate the R^2 (and adj.) and the F-statistic. But is it possible to calculate SE and p-values to check the significance? What else can I do to check whether and to what extent my PCR-Model is better than the OLS-version (original data was strongly multicollinear). I checked Principal_component_regression#Two_basic_properties and for k=p=19 I got same reg-coefficients with OLS and PCR. I also can calculate MSE-values, as in Principal_component_regression#Efficiency it says MSE of OLS should be bigger (e.g. worser) than PCR. For k=p=19 it is the same, for less components PCR-MSE gots smaller. But what else can I see from my results? Would be very thankful for any idea :-) Thanks a lot! -- WissensDürster ( talk) 11:49, 15 January 2015 (UTC)
The proposition "(B if C) and (C if B)", is tautologically equivalent to "B iff C", in which B is not mentioned twice.
My question is about whether, the following proposition is - tautologically equiavalent to any proposition - in which B is not mentioned twice (by using as many connectives as we wish and no matter what connectives are used, provided that all connectives are binary or unary):
"(B if C), and (C if (B and A))".
77.126.32.139 ( talk) 23:10, 15 January 2015 (UTC)