Does anyone know what "\ll" (latex) or "<<" means?
Thanks.
+3
A:
It means "Much less than". Precise interpretation depends on how accurate you want your conclusions to be. For example, sometimes the result is A only if B << C, then the error with the result is likely to depend on the ratio B/C.
Victor Liu
2010-06-26 17:49:41
Victor, precisely, does "B<<C" mean B=o(C) or lim B/C =0?
2010-06-26 17:53:49
It means B/C < epsilon; the latter one. The limit that you describe depends on the situation. For an algorithm which does some constant time T1=O(1) preprocessing, but then does some linear time T2=O(n) work, the run time is O(n) if A << B.
Victor Liu
2010-06-26 17:58:07
Victor, if you look into http://en.wikipedia.org/wiki/Table_of_mathematical_symbols, you can see that "f ≪ g means the growth of f is asymptotically bounded by g". Thus B<<C could mean B/C=1, according to that. So I am confused again.
2010-06-26 18:08:37
As you can see, the Wikipedia page has several definitions for the symbol, depending on the field/context you are in. I am an engineer, so to me, if I want 1% error in my results, I read B << C as B < 0.01 C. Other people should chime in on what it means to them. The asymptotic bounding is probably only used in the area of theory of computation.
Victor Liu
2010-06-26 21:29:21
A:
Sounds like you're asking about the mathematical meaning of the symbol <<, not about how to produce it... which is off-topic for StackOverflow.
But it means "much less than."
David Zaslavsky
2010-06-26 17:49:44
David, precisely, does "B<<C" mean B=o(C) or lim B/C =0? as people use it in their CS or math papers without telling its exact meaning.
2010-06-26 17:55:52
Most often I do not think that any exact meaning is included in the notation.
Andreas Rejbrand
2010-06-26 18:00:13
`B=O(C)` is something totally different - it suggests that B is of the same order as C. A limit as B/C goes to zero is closer to the meaning you're looking for. But as Andreas implied, it does sort of depend on context. You'd have to ask about a specific example (probably on a math forum) to get more detailed information.
David Zaslavsky
2010-06-26 18:05:40
+3
A:
In a programming context, it usually means a bit shift left. In a more general math context, it means "much less than".
Jan Gorzny
2010-06-26 17:50:20
A:
If it's a math notation question, I'd say a << b means a very much smaller than b, or a << : very small a.
Mau
2010-06-26 17:51:25
+2
A:
a ≪ b means that "a is much less than b". It is often used in mathematics and physics when justifying approximations. For instance, if d is the distance between two particles, one with positive charge and one with negative charge, and D is the distance between the center of mass of those and a fixed point P, then if d ≪ D, you can make a few approximations and treat the system as an electric dipole.
A more concrete example: If 0 < A ≪ B, then A / B is very close to zero, and so cos(A / B) can be replaced by 1. (If the situation allows this. Of course, sometimes the deviation from unity might be the part of interest...)
By the way, you could easily have figured this out yourself: Using a Unicode editor, or Windows charmap.exe, you would have found that ≪ is U+226A: "MUCH LESS-THAN".
Andreas Rejbrand
2010-06-26 17:53:37
Andreas, if you look into en.wikipedia.org/wiki/…, you can see that "f ≪ g means the growth of f is asymptotically bounded by g". Thus B<<C could mean B/C=1, according to that. So I am confused again.
2010-06-26 18:10:42
@user376089: In most cases "≪" simply means "much less than". The second meaning, which probably is rather uncommon, was new to me.
Andreas Rejbrand
2010-06-26 18:19:12
@Andreas Rejbrand yep. Usually in physics it is used before justifying keeping only the first non null order in a series expansion on A/B ... which is the only way to solve the Hamiltonian equations :D
belisarius
2010-06-27 00:30:15