Math 121A Linear Algebra
... (x) = amxm + am-1xm-1 + … + a1x + ao g(x) = bnxn + bn-1xn-1 + … + b1x + bo Without loss of generality, if n m, bm, bm-1, …, bn+1 = 0. g(x) = bmxm + … + bnxn + … + b1x + bo (x) + g(x) = (am + bm)xm + … + (an + bn)xn + … + (a1 + b1)x + (ao + bo) Scalar Multiplication: Let t F and be a polynomi ...
... (x) = amxm + am-1xm-1 + … + a1x + ao g(x) = bnxn + bn-1xn-1 + … + b1x + bo Without loss of generality, if n m, bm, bm-1, …, bn+1 = 0. g(x) = bmxm + … + bnxn + … + b1x + bo (x) + g(x) = (am + bm)xm + … + (an + bn)xn + … + (a1 + b1)x + (ao + bo) Scalar Multiplication: Let t F and be a polynomi ...
Deterministic factorization of sums and differences of powers
... But this implies (ab−1 )dj 6≡ 1 mod p for 1 ≤ j < i, and we conclude that o = di . Since the order of any element is a divisor of the group order p − 1, we derive p ≡ 1 mod di and the claim follows. We now consider the runtime of the algorithm. The cost for Step 1 is in O(1) and negligible. We are l ...
... But this implies (ab−1 )dj 6≡ 1 mod p for 1 ≤ j < i, and we conclude that o = di . Since the order of any element is a divisor of the group order p − 1, we derive p ≡ 1 mod di and the claim follows. We now consider the runtime of the algorithm. The cost for Step 1 is in O(1) and negligible. We are l ...
functors of artin ringso
... then F xG H -> H is smooth. Proof, (i) This is more or less well known (see [3, Theorem 3.1]), but we give a proof for the sake of completeness. Suppose hs ->■hR is smooth. Let r (resp. s) be the maximal ideal in R (resp. S), and pick xx,..., xn in S which induce a basis of t*iR=s/(s2+rS). If we set ...
... then F xG H -> H is smooth. Proof, (i) This is more or less well known (see [3, Theorem 3.1]), but we give a proof for the sake of completeness. Suppose hs ->■hR is smooth. Let r (resp. s) be the maximal ideal in R (resp. S), and pick xx,..., xn in S which induce a basis of t*iR=s/(s2+rS). If we set ...
"Associated class functions and characteristic polynomial on the
... Of course, the strong law of large numbers and the central limit theorem are only one of many similar universality-type results now known in probability theory. In this thesis we establish universality-type results for two classes of random objects: random matrices and stochastic processes. In the f ...
... Of course, the strong law of large numbers and the central limit theorem are only one of many similar universality-type results now known in probability theory. In this thesis we establish universality-type results for two classes of random objects: random matrices and stochastic processes. In the f ...
