MandrakeRoot

Probably in any reasoning format fullness and emptyness would be related. So the concepts you are trying to define will be dependant in any situation. Why trying to reason in another format that current mathematical reasoning ? This reasoning would work excellent in society also i guess. Can you explain better why our society would die out if we continue to reason using mathematical reasoning ? Mandrake

I've never quite believed in this stuff. It is nice for movies and so but i have difficulty believing that it could exist, though i am far from an expert in this field Probably the EEG and all this stuff would show the same thing regardless of whatever personality is manifested. I guess that this sort of evidence would have otherwise been made known to the large public already dont you think ? Is there someone that beliefs this stuff actually exists and can argument why ? Or someone that can argument why psy's and stuff talk of this ? Mandrake

Well you will need an axiom to define the emptyset. Basically something like "there is a set such that every element is not in it". So then you have what you call emptyness. I am not sure what you mean with fullness though ? A point is a synonym for element (of a set) i would say. With the axioms of set theory you can create the natural numbers, once you have these you can create and define perfectly all other numbers and prove statements such as "in between any two rational numbers there is an irrational number" (also in between any two irrational numbers there is a rational number). An interval is just a set : the notation [a,b] is just short hand for the expression : [math] [a,b] := \{x \in \mathbb{R} \; : a \leq x \leq b\}[/math], Does that answer your question ? Mandrake

what is that supposed to mean ? Anyway i dont care. I remind you that the car was originally seen as something that could help to reduce city pollution (i.e., horse shit !) Nobody ever imagined at that time that someday millions and millions of cars would be used every day creating tons of pollution. Thinking that you can perfectly manipulate genes without any side effects is rather arrogant and arrogance is surely not something science needs ! Mandrake

I dont quite understand what you want to do, but it looks like you are trying to reinvent some parts of set theory here. In set theory the so called finite cardinals can be identified with the natural numbers, but they are also sets : e.g., 0 = emptyset 1 = {emptyset} 2 = {emptyset,{emptyset}} so 0 is an element of 1 which is an element of 2 etc... (allowing you to create an order, a <= b iff a element of b) Mandrake

A Master in applied mathematics is a very usefull diploma. There are many job opportunities having such a diploma. You can think of working for banks,insurance companies, doing research, working in some R&D department, etc... Mandrake

I think that the so called genetic manipulation you are suggestion will be this shotgun turned upon ourselves ! I dont think that it is possible to create some sort of superhuman by genetic manipulation without any indesirable consequences (on the long term, maybe originally neglected). Mandrake

I encountered this stuff in the first year of my math studies at the university and it was in some course material written by the professor. Though any book on introduction to functional analysis would be surely contain all this. You could try to catch a book called "introduction to functional analysis" of J.B. Conway or "functional analysis" by W. Rudin (this is an excellent book but maybe a little hard to begin with) Measure theory is sort of seperate but becomes all teh more powerfull when you combine metric spaces and measure theory ! The L_p's are just some examples of banach spaces, with rather pleasant properties, since banach spaces can be rather strange (in the sense that there exists spaces with really counter-intuitive properties). Mandrake

I have a phd in math. One of the most interesting fields is functional analysis i would say. Mandrake

Most banach space and hilbert space theory is developped by analyst, and functional analyst and not statisticians or in probability theory. In fact Hilbert and banach spaces are generalisations of spaces with a distance on it. Indeed hilbert space theory and operator theory isused all the time in quantum mechanics, but these spaces come back all the time in mathematics too. It is just very convenient to work with them, since they can be anything from the [math]\mathbb{R}^n[\math] to the space of all continuous functions from some space with values in [math]\mathbb{R}[\math], or whatever else. So in proving some result for a banach space, you have immediately your result for all these spaces allowing great flexability of your theorems and lemmas. In short a metric space is a vector space with a distance on it, a mapping assigning a positive number to any pair of elements with some trivial properties d(x,y) = 0 iff x = y d(x,y) >= 0 for all x,y in the space d(x,y) = d(y,x) and finally d(x,y) <= d(x,z) + d(z,y) such a space is called complete if every sequence that eventually sticks to itself has a limit. A banach space is a vector space with a norm on it, thati s complete, a norm is a mapping assigning to every element a positive number : norm(x) = 0 iff x = 0 norm(alpha x) = abs(alpha) norm(x) for every scalar alpha (complex or real depending if iti s a complex or real valued vector space) norm(x + y) <= norm(x) + norm(y) It is easily seen that d(x,y) = norm(x-y) will be a metric on this space, so every banach space is a complete metric space Now a hilbert space is again more specific then that, it has an inner product, and indeed each hilbert space is specially a banach space. If you guys want i could put the definitions more formally and more complete ? Mandrake

Yeah it is that. There are some algorithms with O(exp(m)) behaviour, where m is the size of the problem, but in practice some of them are not that bad and actually terminate. It is just that the theoretical worst case behaviour is disastrous for such algorithms, though for some "real life" problems that might be a lot better. Mandrake

Banach space theory is very interesting and usefull indeed. [math]l_1[/math] is indeed a proper subset of [math]l_2[/math], this is easily seen as "every absolutely summable sequence" has to be convergent to zero and hence there exists some N such that [math]|a_n|^2 \leq |a_n| \; \forall n \geq N[/math]. However they are not equal as [math]a = \{\frac{1}{n}\}_{n=1}^{\infty}[/math] is an element of [math]l_2[/math] that is not in [math]l_1[/math]. For all [math]l_p[/math], [math]l_p \subseteq c_0[/math] ([math]p \geq 1[/math] in this post). In formula for real valued l_p's : [math]l_p = \{ a = \{a_n\}_{n=1}^\infty \subseteq \mathbb{R} \; : \sum_{n=1}^{\infty} |a_n|^p < \infty \}[/math] This result is not true for the function space [math]L_p[/math] though. Mandrake

No you are wrong the big "O" notation is also used in mathematics and is used there to indicate the behaviour of errors in function of the "step of discretization" for example. I was using this notation in that sense, so O(1/n) is the behaviour of the error the algorithm makes in function of the number of iterations ! (When we do more iterations, the error will become smaller), which means nothing on the number of iterations. I will write it down even more clear because otherwise you might misunderstand : Suppose the algorithm is supposed to calculate the quantity X, but in fact calculates X_bar, and we know by analysing the algorithm that |X - X_bar| = O(1/n), where n is the number of iterations. Then if we desire a precision of approx. 1/100, we will need to do something like 100 iterations ! Depending on the size of the problem m, the algorithm could easily have time complexity O(m^5). The goal of the O notation is to majorate some function of something, by another more easy function in order to be sure to have some knowledge of "worst" case behaviour. If your algorithm has time complexity O(m), (m size of problem), then that does not mean that for every instance of your problem the algorithm wil terminate in this time, it might as well terminate before, but the "worst" instance will generate this behaviour ! Mandrake

I was talking about the precision of the algorithm not the time complexity (n being the number of iterations or something like that and not the size of the problem). Time complexities could ofcourse never converge to zero for obvious reasons, like you say. Still say having an O(n) (this time n = size of problem) alg. and one O(n^2), then both would be just as "infinite" since it would suffice to put an n^2 size problem to the first and an n-size problem to the second. Mandrake

I do not agree with you Daphtar. The big O notation is something indicating the speed at which something tends to zero or infinity or whatever. Say for instance you have two algorithms to solve something with error resp. O(1/n) and O(1/n^2), the second would be preferable since it converges more quickly, but that has nothing to do with infinity. The only reference i see to different degrees of infinity is the different (infinite) cardinals. You can create sets which are all infinite and each time one is strictly bigger than another. (two sets are equal in size if you can create a bijection in between them). In the original question of jordan infinity is not an element, but a notation. Mandrake

It is pretty simple actually. [math][0,\infty)[/math] is just a notation to say that your function can take on any positive real value. In this sense infinity is just a notation and nothing more. The lesser and greater number of infinity is more something that arises in set theory. For exemple it is possible to show that [math]\mathbb{N}[/math] and [math]\mathbb{Q}[/math] contain just as much elements (though both contain an infinite number of them. On the contrairy [math]\mathbb{R}[/math] contains strictly more elements than either of these two sets mentioned above, and the set consisting of all subsets of [math]\mathbb{R}[/math] contains again strictly more elements and so on and so on, but that has nothing to do with numbers or calculating. You cannot calculate with infinite "as a number" except when you impose trivial rules as : [math]\infty + a = \infty \forall a \in \mathbb{R}[/math] and so on. Here [math]\infty - \infty[/math] will always remain undefined, because otherwise you can arrive to contradictions. So to make short the argument, infinity is a usefull concept or in the use [math][0,\infty)[/math] just a simple notation to say "all positive real numbers greater or equal to zero". I hope that makes it more clear now ? Mandrake

I dont think it is very hard to show that the sum is convergent, since it is bounded by above (clearly a_n is less than the number of digits in 2^n, which seems to grow as O(n) ) and its partial sums are clearly increasing, hence convergent. Mandrake

conquaring ecology would be "nice and warm jacket on the antartic with a loaded shotgun to kill everything that moves !"

Yeah i agree with you, cheating on natural selection is not possible in the long run. But i surely don't agree on the point where you say humans dominate ecology with the technology, destroy would be more likely (a slow autodestruction maybe ??). Mandrake

I disagree that asthma is non relevant to natural selection. In the time where h. sapiens was a hunter/gatherer, clearly someone with asthma would have had a disadvantage. I know there is a difference between having having "undesirable genes" and them actually being active, but having them could allow you to pass them on to your children that might have these genes active (i.e., show signs of the disease or whatever). I think most people will take the chance, even if they know their children might be sick. I even know some people that knew their child would be seriously handicapped (drown syndrom + other physical complications) and didnt choose an abortion ! It is clearly true that natural selection is something that has effect after many years, but can also go rather suddenly. A famous example is this butterfly species (white) in england of which normally only a few specimens are black and would have been selected against, (since they were to easy to remark) but due to industrial revolution and pollution it was the black specimens that became predominant. Maybe the correct formulation of the issue is : Can we cheat natural selection by use of technology ? (on the long run). Mandrake

With degrading the gene pool i meant that an increasing number of "flawed genes" is present,i.e., genes with genetic defauts. There are families, where the parents have serious health problems, the children the same plus more etc..., so that is what initialised this debate (though like i said i am not convinced either, it is a debate , an hypothesis that needs testing, disproving etc....) The idea is the following : Someone that has an heriditary kidney problem for instance, (would have died in the 3 world countries and hence not reproduced), but we keep him alive by technology, kidney transplant whatever, so he can reproduce and hence passes over the genes of this hereditary kidney failure. So in "natural" circumstances these genes would have disappeared from the pool , whereas in the technologic circumstances they are allowed to stay in it ! (if the couple has even more children then 2, then maybe they will even be more frequent ?) Mandrake

Yes true, but even taking that into account, the number of children suffering from allergies increases. But that is not the most important, the debate i wanted to iniaet is about the following : Will our technologic influence (keeping people that are sick alive, and thus allowing (some) to reproduce whereas they never could have, where we always to live in "natural" circumstances) ultimately degrade the DNA pool ? Mandrake

When looking to some stats, there are an ever increasing number of children with allergies or other illnesses. Since we are sort of trying to cheat evolution in using technology to keep people alive that would have otherwise died or help a couple having difficulty to have children to have children etc... So my question/debate issue is : Does our DNA degrade ? with this i mean that, will our population contain an ever increasing number of individuals with illnesses (including allergies etc...), that could ultimately (or not) lead to the destruction of the species ? I know this is a pretty wild statement, and i am not convinced it is true, i find it just remarkable that the number of children with allergies/illnesses has increased a lot, making me think of some genetic defauts (maybe) ? So what do you guys/girls think of it ? Mandrake

Not all christians are against gay weddings, so surely there is no need for generalization here ! The netherlands is a christian country at the foundation, though ofcourse church and state are seperated, but the christian community is still relatively strong and gay weddings are allowed since some years now. I think it is clear that gay couples should get the same rights and obligations as married couples. It could always be called something else than a marriage if people would prefer so.

I have a nice problem for you guys, just a funny problem to stimulate the mind. For each non-negative integer [math]n[/math], let [math]a_n[/math] be the number of digits in the decimal expansion of [math]2^n[/math] that are at least 5. For example [math]a_{16} = 4,since 2^{16}=65536[/math] has four digits that are 5 or higher. What is the sum [math]\sum_{n=0}^\infty \frac{a_n}{2^n}[/math] ? I evaluated machinally the first 299 (see attachment) terms a of the sum. The sum converges to 0.086617 numerically, what is the exact value ? What is the growth rate of a_n ? Mandrake a.txt