From the symbolic manipulation viewpoint, # has much to recommend it. Some more properties of #: # is commutative and associative: A#B = B#A, A#(B#C) = (A#B)#C. # distributes over multiplication: (A*B)#C = (A#C)*(B#C) and A#(B*C) = (A#B)*(A#C). 1 is an absorber: A#1 = 1#A = 1. Arguably, A#0 = 0. If the logs are base 2, 2 is an identity: A#2 = A; and A#(2B) = A*(A#B). One argument for using base sqrt2 for the logs is so 2#2 = 4. Over the positive reals (except 1), we have inverses: A' = 2 ^ (1/log2(A)). A # A' = A' # A = 2; (A#B)#B' = A. If either of A or B is a power of 2, then A#B is an integer. One drawback is that A#B is usually not an integer; I think expressions like 3^log_2(3) are transcendental, but I don't know the present state of knowledge. The mod P and finite field versions are also interesting: the operation tables have lots of (obvious) patterns. I use the smallest positive primitive root for the log base. Rich ----------------- Quoting Mike Stay <metaweta@gmail.com>:
The Diffie-Hellman operation a # b = exp(ln(a) ln(b)). This is clearly commutative, unlike exponentiation, but it still distributes over multiplication. This can be extended to an infinite series of operations in both directions:
. . . a $ b = ln(exp(a) @ exp(b)) a @ b = ln(exp(a) + exp(b)) = exp(ln(a) $ ln(b)) a + b = ln(exp(a) * exp(b)) = exp(ln(a) @ ln(b)) a * b = ln(exp(a) # exp(b)) = exp(ln(a) + ln(b)) a # b = exp(ln(a) * ln(b)) . . .
The operation @ is roughly max (Maslov dequantization can deform it into max). Each operation distributes over the operation above it. What is this sequence called? -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
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