Equating the two and rearranging, we obtain which is the spherical sine rule. Again, in the infinitesimal limit we obtain the familiar Euclidean sine rule As a variant of the above analysis, we have from (5) again that is equal to As before, the first expression simplifies to ...
Rearranging, we conclude that and hence giving the claim. Thus for instance: For the norm, one can take to be the family of linear exponential phases with and , and obtain a linear lower bound of for the -entropy, thus matching the upper bound of up to constants when is fixed. ...
Mathematical logic is the application of mathemat- ical techniques to logic. What is logic? I believe I am following the ancient Greek philosopher Aristotle when I say that logic is the (correct) rearranging of facts to find the information that we want. Logic has two aspects: formal and...
is equal to The first expression simplifies by (1) and properties of the inner product to which by (2), (3) simplifies further to . Similarly, the second expression simplifies to which by (2), (3) simplifies to . Equating the two and rearranging, we obtain which is the spherical...
Equating the two and rearranging, we obtain which is the spherical sine rule. Again, in the infinitesimal limit we obtain the familiar Euclidean sine rule As a variant of the above analysis, we have from (5) again that is equal to As before, the first expression simplifies to ...
is the order divisor function. The sum (1) then corresponds to the case . For , , and a routine application of the Dirichlet hyperbola method (or Perron’s formula) gives the asymptotic or more accurately where is a certain explicit polynomial of degree with leading coefficient ; see e.g...
Let’s first prove the weak form of the divisor bound (3), which is already good enough for many applications (because a loss of is often negligible if, say, the final goal is to extract some polynomial factor in n in one’s eventual estimates). By rearranging a bit, our task is to...
although this is now an antihomomorphism rather than a homomorphism: . One can then split up a quaternion into its real part and imaginary part by the familiar formulae (though we now leave the imaginary part purely imaginary, as opposed to dividing by in the complex case). The inner ...