Table I must be further corrected. In row 1, replace "s^T ⋅ t" with "s ⋅ t" and "H^T ⋅ s" with "H ⋅ s"

Algorithm 1 looks fine.

A great number of changes must be made to restore "★" to everything that had "⋅" notation and was removed in the first proof. A list follows.

 - On line 72, replace "denoted rs" to "denoted r ★ s"
 - On line 75, replace "\mu(rA)" to "\mu(r ★ A)" 
 - In Table I row 1, replace "0s" and "1s" to "0 ★ s" and "1 ★ s", respectively
 - In Table I row 2, replace "Ms" to "M ★ s"
 - In Table I row 3, replace "(i_1, …, i_q)s" with "(i_1, …, i_q) ★ s"
 - In Table I row 4, replace "r_m s" and "s_m s" to "r_m ★ s" and "s_m ★ s", respectively
 - In Table I row 5, replace "r_m s" and "s_m s" to "r_m ★ s" and "s_m ★ s", respectively
 - On line 93, replace "rs" and "rt" with "r ★ s" and "r ★ t", respectively
 - In Algorithm I step 2, replace "Z(s, rs)" with "Z(s, r ★ s)"
 - In equation (4), replace "rs_m" to "r ★ s_m"
 - In Algorithm I step 3(b)iii, replace "rs_m" with "r ★ s_m"
 - On line 143, replace "rs" with "r ★ s"
 - On line 151, replace "Z(rs', s') = Z(rs, s)" with "Z(r ★ s', s') = Z(r ★ s, s)"
 - In equation (7), replace "s_0^{-1}s_i" with "s_0^{-1} ★ s_i"
 - In equation (8), replace "s^{-1}t" and "t^{-1}s" with "s^{-1} ★ t" and "t^{-1} ★ s", respectively
 - In equation (9) line 1, replace "\tilde Z(rs_0, rs) = B[(rs_0)^{-1}(rs)]" with "\tilde Z(rs_0, r ★ s) = B[(rs_0)^{-1} ★ (r ★ s)]"
 - In equation (9) line 2, replace "s_0^{-1}s" with "s_0^{-1} ★ s"
 - On line 323, replace "r_n m" and "s_n m" with "r_n ★ m" and "s_n ★ m", respectively
 - On line 336, replace "r_i j" and "s_i j" with "r_i ★ j" and "s_i ★ j", respectively
 - In equation (13), replace "r s_m" with "r ★ s_m"