premjan

Does entropy depends on the cosmological model? I think the big bang universe is one where entropy continuously increases, but there are other (less favored) cosmological models where low-grade energy may recycle back as mass.

doesn't linux have some sort of indirection of file blocks so that the blocks for a file are not likely to be adjacent on the hdd anyway?

This comment was probably a joke.

Probably for no good reason. Somehow the idea of matrix singularity bugged me, but while we can take some reasonable steps to eliminate it, 0*0=0 seems too much like an obvious thing to kludge. I guess some matrices (and numbers) had best just remain singular (or uninvertible).

Yeah, oo (and 0) are kind of multiplicative information sinks. oo * 1 = oo oo * 0 = ?? (shall we say this is 0 or undefined?) 0*a = 0 I was just thinking that we might somehow be able to guarantee an invertible matrix inverse function but oo * 0 is still a problem. Why not just say that oo * 0 = 1 (by definition)? Here are possible truth tables for {Z_2,oo} + 0 1 oo 0 0 1 oo 1 1 0 oo oo oo oo oo * 0 1 oo 0 0 0 1 1 0 1 oo oo 1 oo oo Basically multiplication by infinity is the problem. It is not invertible, as oo*oo = oo*1. Also 0*0 = 0*1. I'm guessing there is no way to eliminate these problems? Perhaps we could put oo * oo = 0 (kind of return to home do not pass go situation) whereas oo * 1 = oo. And 0*0 could plausibly be put as oo (it even looks like two zeros already). This would at least remove the information sink properties of 0 and oo. Would this help in any non Z_2 setting is the question. But I guess it would make matrix inversion in Z_2 (only) invertible, in some sense.

now doesn't that translate into the analytic setting somehow though? I bet if we introduced an explicit infinity element it would make the whole shebang work with closed division and matrix inverse. What is a "bog" standard? If we had an infinity element, we would have 0*infinity = 1 and 1*infinity = infinity. Then it would be Z_2 augmented with infinity. 0+oo = oo and 1*oo = oo. Then singular matrices would have inverses, would they not?

wow I like the way you delicately said "shed" instead of s***. I get it. The whole thing doesn't make as much sense if constructed without the usual behavior of 0 and 1. If we changed it, it would complicate all our formulas and equation solving.

I seem to have heard that over time, intermarriage between fair skinned and dark skinned people will end up in dark skin (but of course this may be anecdotal and influenced more by environmental factors than pure genetics).

Is there some way to eliminate "singularity" in Z_2 matrix inverse while retaining other properties (possibly by redefining + or *)? Basically I was wondering if it is possible to make the matrix inverse work for all matrices including the 0 matrix (groupify the inverse operation I guess)? (probably not, I guess). Is there some way to gain intuition why this is not possible? Is it possible to achieve something like a field that is closed under inverse? I guess the multiplicative property of 0 would have to be sacrificed. Could this be done by introducing an explicit "max" or "infinity" element? Maybe if I use ^ (XOR) and !^ (XNOR) for * and + respectively?

Mod 2 operations ought to do fine I suppose. Will matrix inversion be injective in Z mod 2? I suppose it ought to be. Do you have a link to "sliding" somewhere? Thanks.

Imagine an m x n matrix which has only 0 and 1 as its elements. I couldn't think of a better term for it, so I'm calling it a boolean matrix. 1) suppose m = n. I want to calculate the inverse of a given boolean matrix. Is this faster than computing it for a regular matrix (with elements other than only 0 and 1)? Is there an algorithm (analogous to, say, Gaussian elimination) which always preserves the boolean property at each step? Basically if you can maintain the boolean property, you can do everything using bit operations so it ought to be more efficient (besides there not being any numerical instability problems). 2) suppose m != n. I would like to compute the left and right inverse of a given boolean matrix. Is it likely that the left inverse would be equal to the right inverse? Is it easy to tell when this would be the case? Could we force it to be the case somehow? 3) is it easy to tell which boolean matrices are nonsingular?

Isn't white skin recessive?

bubbles (maybe of water vapor or dissolved gases coming out of solution). it also happens with fast running water (e.g. coming out of the tap, or, I suppose sea spray).

off generally doesn't go off immediately, the phosphors take some time to stop emitting.

Doron claims that Dedekind cuts do not work because the limit point of a sequence (e.g. sequence of rationals) is always separate by a finite interval from its "limit" irrational (and due to the self-similar scaling of the real line, any gap is a "1" gap. Here's his latest set of posts: http://www.iidb.org/vbb/showthread.php?p=2001576#post2001576

but the diagonal argument does this only for Z and R.

well universal quantifiers show up even in the definition of a limit (for all epsilon, there is a delta or N for which...) so in a sense, under constructive math, it is not sound to speak of such a definition for a limit. Strictly speaking, infinite decimals ought not to be allowed at all since they are not constructible, so Cantor's proof would fall apart if we were forced to use only constructible numbers. Thing, is, strict constructability is more of computer science than math, and that is why even Doron doesn't push the argument home, since he is clearly coming more from a logic than a CS angle.

Kon Tiki, and extremely muscular Tongans indicates that it is possible but difficult.

epsilon-delta?

my photo is a few years old (I have a few more pounds on me now).

so your notation has nothing to do with set cardinality at all? I guess it does not strictly need to. Then again, {__} is apparently not related to the cardinality of {. in N}?

I guess transfinite cardinals are a bit like infinitesimals (dx,dy) used by Newton which are also nonsensical, strictly speaking, or like renormalization in QED which is also mathematically nonsensical. However, some limited manipualation with these entities does work. In your model aleph0 is basically unmanipulable?? In your system, how does one arrive at the fact that aleph0 is the cardinality of the naturals? Is it possible to say this?

it is all in ratios so it is bound to be very easy to solve (if you count all the exponents correctly).

OK I see you would like to do away with the limit concept, but the limit is a property of the sequence, by the epsilon, n proof, isn't it? What are "information forms"?