We can still find ways to bump ourselves off out there.
Yes. That is true. On occasion people may meet again. And we have still not solved the problem of them killing each other when they do. More importantly we have not solved the problem of both parties being annihilated in the fight, and also the problem of some 'accident' wiping out some part of humanity. However, on the level of the whole of mankind, expanding into space stops either of these from causing the extinction of our species. The reason being that, although disasters may occur, it becomes likely that no single disaster (or simultaneous disasters) ever wipes out all of mankind in one go (so long as we can keep expanding). This is like the principle of not having all our eggs in one basket/planet/solar-system/galaxy.
This is illustrated by a thought experiment in statistics (curiously enough we use it to make species of disease extinct which is the opposite of what we want to do to ourselves). Consider that there is a fixed chance each year of the annihilation of the earth. Then the probability of humanity surviving forever is zero. However, if we expand to other planets fast enough and each planet has the same probability per year, then the probability of humanity surviving forever is non-zero. This means that our species only becomes doomed again once we fill the universe and can no longer expand at this rate. Hence we have a lot more time to enjoy ourselves.
You're probably right that it won't give you an oracle, but my comment is about the properties of a Turing machine (or similarly any computer based on the same algorithmic logic) as regards runtime orders. There are many applications (unlike NP) in which one can prove useful lower bounds for runtime. However, I don't think it goes against causality or thermodynamics to beat these by getting some funky physical process to come up with the answer in a way that avoids the necessary computation by 'outsourcing' it to the 'processing' that is the underlying cause of the laws of physics. Okay, so this is a bit wacky, but what the hell. I am a mathematician...
We do not apriori know that the laws of physics cannot be (ab)used to cause a computation to happen in a way which is strictly better than the way a Turing machine (read 'pretty much any computer you can think of') works. Though this apparatus requires a large number of photons it is an exciting result towards what could be a real paradigm shift in computing. For similar reasons quantum computing is interesting to us, but it too has its drawbacks. Alternatively one could hope for an (IMHO unlikely) proof of P=NP, which would say that a Turing maching can after all achieve similar efficiency.
I hope that the parent is not a sensible person with a good understanding of scientific reason just making a sarcastic joke to make creationists look bad, because that would be unfair and terribly unkind. On the other hand if the parent is being serious, then it has done a great service to the argument it opposes (purely by its own lack of reason). Either way it was an entertaining read.
Let me get this right? You would have had your diplomas withheld for not going to the prayer session and you are complaining about the law that, if followed would have saved those who did not wish to go to it. Wow. You people sadden me.
Aside from the obvious remark about random recombination of genes (one cause of things 'skipping' one or more generations), you are correct. However, chances are the genes that caused people to be 'black' or 'white' would still be there and should it ever be advantageous (to survival or reproduction) to be one colour or other, that is the colour more people would become in the space of a few generations. Nothing is lost, unlike in the case of cultures and languages dying out.
I assume you are referring to the US public schooling system. In which case you are quite right and it worries me to see that in the US children are increasingly taught that scientific reasoning must depend on such assumptions. This turns the whole debate away from what it should be. This is why, when I have visited the US I have never dared wear my "Just Say No To Evolution" T-shirt for fear that people might think it anything other than a cynical joke about this debate. Over here in Britain we are only just beginning to feel the effects of the Christian Nonsense Lobby in our state schools and it has yet to provoke the sort of outrage that leads people to publicly make idiotic arguments against it for lack of understanding of the perfectly good ones.
"It's just that it's a theory based on the NON-existence of a creator."
Not at all. Me thinks you do not understand the concept of scientific reasoning as well as one might hope. It is a theory 'not based on the existence of a creator', which is a far cry from 'a theory based on the NON-existence of a creator'. Not assuming the existence of a 'creator' (whatever one may choose to mean with that) one does ones best to understand and explain observed phenomenon in a rational manner. While one cannot yet prove that the flying spaghetti monster does or does not exist through repeatable experimentation (and people should feel free to contribute their research in this area to the scientific community as a whole), one can make a very good description of the functioning of the world around us without having to tackle the issue of the influence of his omnipotent noodly appendages.
The parent makes several glaring mistakes in his reasoning.
Most importantly, it is the correct combination of genes that makes a successful organism as well as individual genes. 'Mixing' of groups of people is hugely advantageous for this reason.
Secondly, genes do not become lost when they combine with genes from another person to make a child. There is just a new combination of genes which can contribute to the whole genetic diversity of mankind. For example, we could take the idea that races should not interbreed a little further and say that people should not breed outside of their immediate family. The problem with this would be that genetic diversity could hardly ever increase, and by attrition mankind would be doomed. By separating races one creates several smaller separate gene-pools each of which is smaller than the original whole and hence more vulnerable.
Thirdly, by separating the societies it would become genetically/evolutionarily advantageous for one race to think of or treat the others as subhumans. By this argument I claim that you have implicitly invoked Godwin's law.
Also, I wish you luck procreating with your sister...
Damn those Ents and their slow decision making. First they nearly refused to act to stop the downfall of middle Earth and now, even worse, they are causing problems with Steve's divine creation. Personally I think we should ban them from having I phones if they are going to do this.
Thus, connected smooth subsets of real algebraic hypersurfaces are orientable.
That sounds convincing (that you need more than three dimensions).
I am not sure this works: the Boy surface has a curve of self intersection (three loops at mutual right angles) so it is not quite obvious that you can remove that and still have enough left of the topology to have a Möbius band...
Trace out a simple closed curve on the surface whose tangent bundle is non-orientable. This is possible since we may take a simple closed curve on the domain RP^2 and move it away from the three triple points, then whenever it touches two corresponding points in the double point locus we move it to the side a little near one of them. Now the restriction of Boys' map to a small enough neighbourhood of this curve is an embedding. And the complement of this neighbourhood in the domain RP^2 is a disk.
"You cannot embed RP^2 in R^3 at all, not even continuously."
You're right. So one needs to find a map (at least as nasty as the Boys surface) of RP^2 into R^3 such that the removal of the neighbourhood of a point in the domain of this map leaves an embedded Moebius strip. Smoothly this is possible, but I imagine it is hard or impossible algebraically.
So, not really being an algebraic geometer myself, I just asked the nearest one (Pelham Wilson). He says that real algebraic varieties do not have to be orientable. But he also does not know how one would realise the Moebius strip as an embedded part of a real algebraic surface in 3D.
Not so, I think. At least in the projective case one should be able to make the real locus of complex algebraic varieties. So RP^2 for example. So my guess is that the problem comes when you try to algebraically get that RP^2 into R^3. Of course, once you've done that somewhat nicely, you allow yourself an algebraic inequality and cut it down to a Moebius strip. No?
You are right that you only get one object, not two as one might expect, but it is not a Moebius strip. Here's how to verify this fact (and teach the kids even more cool geometry while you're at it):
a) Trace a line around the surface of a Moebius strip, you will find yourself drawing on both sides before getting back to where you started, hence the strip only really has one side.
b) Now do the same with the object you made in step 2. You will find this behaves far more normally as it has two sides (i.e. you can't get to the other side without taking the pen off the paper).
Indeed, the article is about something quite different from what it claims.
Actually the Wikipedia equations are trigonometric, not algebraic. 'Algebraic' would mean that they exhibit the Moebius strip as the common zero locus of a set of polynomials. In this case real polynomials in three real variables. I'm disappointed that (in spite of what the article claims) they don't do that as it would be really cool for us geometers.
To a topologist these small differences do not matter, so any loop is a circle and any half-twisted flat loop is a Moebius strip. And the Moebius strip is specifically a (smooth) topological object.
If they really had found an ALGEBRAIC equation for the Moebius strip, as the post wrongfully claims, it would, as I understand it, be a significant advance in real algebraic geometry (study of spaces arising as the zero set of real polynomial equations). As it is we can only approach the Moebius strip in algebraic geometry as an object living in higher than 3 dimensions.
The parent's link leads to is a parametrisation of the moebius strip (which turns out to require trig functions, and to be quite easy to come up with). I maintain, the actual Moebius strip problem (to find algebraic equations) is also not solved here.
This is an integral (hence analytic) equation if you read the article. An algebraic equation would be much more interesting as it would be a lot easier to study and maybe gain geometric insight from.
So all of this DRM is horribly implemented and cracked quickly. Does anyone know at least theoretically if there is a way to implement DRM which is inherently secure (at least up to the cracking of, say, RSA encryption)? i.e. what might we be up against?
Slashdot Mirror
This is illustrated by a thought experiment in statistics (curiously enough we use it to make species of disease extinct which is the opposite of what we want to do to ourselves). Consider that there is a fixed chance each year of the annihilation of the earth. Then the probability of humanity surviving forever is zero. However, if we expand to other planets fast enough and each planet has the same probability per year, then the probability of humanity surviving forever is non-zero. This means that our species only becomes doomed again once we fill the universe and can no longer expand at this rate. Hence we have a lot more time to enjoy ourselves.
You're probably right that it won't give you an oracle, but my comment is about the properties of a Turing machine (or similarly any computer based on the same algorithmic logic) as regards runtime orders. There are many applications (unlike NP) in which one can prove useful lower bounds for runtime. However, I don't think it goes against causality or thermodynamics to beat these by getting some funky physical process to come up with the answer in a way that avoids the necessary computation by 'outsourcing' it to the 'processing' that is the underlying cause of the laws of physics. Okay, so this is a bit wacky, but what the hell. I am a mathematician...
We do not apriori know that the laws of physics cannot be (ab)used to cause a computation to happen in a way which is strictly better than the way a Turing machine (read 'pretty much any computer you can think of') works. Though this apparatus requires a large number of photons it is an exciting result towards what could be a real paradigm shift in computing. For similar reasons quantum computing is interesting to us, but it too has its drawbacks. Alternatively one could hope for an (IMHO unlikely) proof of P=NP, which would say that a Turing maching can after all achieve similar efficiency.
I hope that the parent is not a sensible person with a good understanding of scientific reason just making a sarcastic joke to make creationists look bad, because that would be unfair and terribly unkind. On the other hand if the parent is being serious, then it has done a great service to the argument it opposes (purely by its own lack of reason). Either way it was an entertaining read.
My apologies then. I misunderstood your post.
Let me get this right? You would have had your diplomas withheld for not going to the prayer session and you are complaining about the law that, if followed would have saved those who did not wish to go to it. Wow. You people sadden me.
Aside from the obvious remark about random recombination of genes (one cause of things 'skipping' one or more generations), you are correct. However, chances are the genes that caused people to be 'black' or 'white' would still be there and should it ever be advantageous (to survival or reproduction) to be one colour or other, that is the colour more people would become in the space of a few generations. Nothing is lost, unlike in the case of cultures and languages dying out.
I assume you are referring to the US public schooling system. In which case you are quite right and it worries me to see that in the US children are increasingly taught that scientific reasoning must depend on such assumptions. This turns the whole debate away from what it should be. This is why, when I have visited the US I have never dared wear my "Just Say No To Evolution" T-shirt for fear that people might think it anything other than a cynical joke about this debate. Over here in Britain we are only just beginning to feel the effects of the Christian Nonsense Lobby in our state schools and it has yet to provoke the sort of outrage that leads people to publicly make idiotic arguments against it for lack of understanding of the perfectly good ones.
Not at all. Me thinks you do not understand the concept of scientific reasoning as well as one might hope. It is a theory 'not based on the existence of a creator', which is a far cry from 'a theory based on the NON-existence of a creator'. Not assuming the existence of a 'creator' (whatever one may choose to mean with that) one does ones best to understand and explain observed phenomenon in a rational manner. While one cannot yet prove that the flying spaghetti monster does or does not exist through repeatable experimentation (and people should feel free to contribute their research in this area to the scientific community as a whole), one can make a very good description of the functioning of the world around us without having to tackle the issue of the influence of his omnipotent noodly appendages.
Most importantly, it is the correct combination of genes that makes a successful organism as well as individual genes. 'Mixing' of groups of people is hugely advantageous for this reason.
Secondly, genes do not become lost when they combine with genes from another person to make a child. There is just a new combination of genes which can contribute to the whole genetic diversity of mankind. For example, we could take the idea that races should not interbreed a little further and say that people should not breed outside of their immediate family. The problem with this would be that genetic diversity could hardly ever increase, and by attrition mankind would be doomed. By separating races one creates several smaller separate gene-pools each of which is smaller than the original whole and hence more vulnerable.
Thirdly, by separating the societies it would become genetically/evolutionarily advantageous for one race to think of or treat the others as subhumans. By this argument I claim that you have implicitly invoked Godwin's law.
Also, I wish you luck procreating with your sister...
I recognise the grandparent's mistake. I bet he/she/it was using a Dvorak keyboard layout. It happons to mo all the timo.
Damn those Ents and their slow decision making. First they nearly refused to act to stop the downfall of middle Earth and now, even worse, they are causing problems with Steve's divine creation. Personally I think we should ban them from having I phones if they are going to do this.
You're right. So one needs to find a map (at least as nasty as the Boys surface) of RP^2 into R^3 such that the removal of the neighbourhood of a point in the domain of this map leaves an embedded Moebius strip. Smoothly this is possible, but I imagine it is hard or impossible algebraically.
So, not really being an algebraic geometer myself, I just asked the nearest one (Pelham Wilson). He says that real algebraic varieties do not have to be orientable. But he also does not know how one would realise the Moebius strip as an embedded part of a real algebraic surface in 3D.
Not so, I think. At least in the projective case one should be able to make the real locus of complex algebraic varieties. So RP^2 for example. So my guess is that the problem comes when you try to algebraically get that RP^2 into R^3. Of course, once you've done that somewhat nicely, you allow yourself an algebraic inequality and cut it down to a Moebius strip. No?
What you've written is about the nicest way of embedding the Möbius strip in 3D, but it uses trig functions, not just polynomials (see my post again).
a) Trace a line around the surface of a Moebius strip, you will find yourself drawing on both sides before getting back to where you started, hence the strip only really has one side.
b) Now do the same with the object you made in step 2. You will find this behaves far more normally as it has two sides (i.e. you can't get to the other side without taking the pen off the paper).
Enjoy.
Actually the Wikipedia equations are trigonometric, not algebraic. 'Algebraic' would mean that they exhibit the Moebius strip as the common zero locus of a set of polynomials. In this case real polynomials in three real variables. I'm disappointed that (in spite of what the article claims) they don't do that as it would be really cool for us geometers.
To a topologist these small differences do not matter, so any loop is a circle and any half-twisted flat loop is a Moebius strip. And the Moebius strip is specifically a (smooth) topological object.
If they really had found an ALGEBRAIC equation for the Moebius strip, as the post wrongfully claims, it would, as I understand it, be a significant advance in real algebraic geometry (study of spaces arising as the zero set of real polynomial equations). As it is we can only approach the Moebius strip in algebraic geometry as an object living in higher than 3 dimensions.
The parent's link leads to is a parametrisation of the moebius strip (which turns out to require trig functions, and to be quite easy to come up with). I maintain, the actual Moebius strip problem (to find algebraic equations) is also not solved here.
This is an integral (hence analytic) equation if you read the article. An algebraic equation would be much more interesting as it would be a lot easier to study and maybe gain geometric insight from.
You make me smile in you're own, very American, way.
So all of this DRM is horribly implemented and cracked quickly. Does anyone know at least theoretically if there is a way to implement DRM which is inherently secure (at least up to the cracking of, say, RSA encryption)? i.e. what might we be up against?