Bignose

Really? The evidence against one of the most verified law of physics ever is based on supposed extraterrestrial contacts? If the experiment can't be repeated, it is suspect at best. Certainly nothing I'd consider enough to overthrow an exceptionally well verified law.

I must have missed this. All I've seen is seemingly random posting. Can you please, in explicit detail, show what you have calculated that is 'impossible' in current mathematics?

Are you going to have a meaningful conversation? Demonstrate some way in which this new math can be used? Address some of the questions raised? I only ask because you are basically treating this like a personal blog, and if that is your intention, you can start your own blog/website and post whatever you want without using this forum's resources.

It is kind of a clever way of attempting to pass the Turing test. The software has a database of all the phrases that have been typed into it. And then, based on what was just immediately typed into it, it searches that db for current keywords and replies with an old phrase that was input into it. That's why it seems to easily slip into accusing the user into being a bot... it has a healthy db of phrases input into it accusing it of being a bot. Years ago, you could actually trick it into digging certain responses back out of it by asking it about really unusual stuff. That is, the input phrase would have a few rare keywords the bot searched its database for an appropriate response. Because there were only a few responses that came back in the search, you could get a known reply to the input. But I wonder how true that is now... this was 4-5 years ago, and who knows how many more inputs it has gotten since then. (I also don't remember those inputs...)

Does making it rain bring the low pressure in the center of the storm closer to normal? Because that is a major source of energy of the storms. That is, if you just limit the raining, I am not sure that necessarily helps the wind and storm surge. Not to mention the weather conditions are usually favorable to just re-evaporate more moisture into the air and rain again (e.g. the normally fairly warm ocean water). In short, while it may help, I think a lot more study is needed before an answer could be given either way.

Define resistance. Because if you use a fairly liberal definition of resistance, I daresay every paper experiences it. Very rarely is a paper accepted as-is the very first time. Very rarely is a paper presented at a conference and no questions are asked. No idea is accepted without a critique and evaluation of the evidence supporting it. All of these could be seen as 'resistances'. But, all are also reasons why science is exceptionally strong today, compared to most any other time in human history. Probably not perfect, but far, far superior to the vast majority of human history.

This should be an excellent start... http://relativity.livingreviews.org/Articles/lrr-2001-4/ It is a journal article solely on the topic of discussing the experiments that support GR.

No [math] \frac{\displaystyle\sum_{k=1}^2 x^k}{\displaystyle\sum_{k=1}^2 x^{2k}} = \frac{x+x^2}{x^2 + x^4} \ne \frac{x}{x^2} + \frac{x^2}{x^4} = \displaystyle\sum_{k=1}^2 \frac{x^k}{x^{2k}} [/math] It is certainly not true in general, and as I wrote above, it really is only going to be special cases where you are going to be able to turn it into a single sum.

perhaps something like cellular automata are what you are looking for? Stephen Wolfram's books A New Kind Of Science covers them in depth. While my opinion is that Wolfram goes a little overboard with his enthusiasm for the subject, there are some interesting things that happen with CA rules.

nduman, we tend not to just solve or do work for other people here. Please post what you have done so far, and where you have gotten stuck and we'll try to guide you from there. you may want to look into using this forum's LaTeX capabilities, too, to make your post much easier to read. http://www.scienceforums.net/topic/3751-quick-latex-tutorial/

They don't know if it is impossible or not. That is exactly what the problem is. If someone can definitely prove one way or the other, they win the $.

No, I provided you the link to show that in some cases, the question of possibility or not, is really THE key question.

So, you don't care that the question you posed doesn't have an answer in a very large majority of the cases? How very odd. I thought that that was rather relevant and important, really. Sometimes, the toughest thing with a problem is to show that it does or doesn't have an answer. For example, if you can prove that the Navier-Stokes equations have unique solutions, the Clay Mathematics Institute will give you $1 million. http://theconversation.edu.au/millennium-prize-the-navier-stokes-existence-and-uniqueness-problem-4244

I think we're finally on the same page, since this is what I've been trying to say form the very first post.

It really depends on the form of the sums. You might be able to, and you may not be able to. Do you have a specific example in mind?

It is always a possibility. To really look into this, you need to propose a good mechanism for the interaction, and from that explain why we 'only use a fraction of the ones available'. And, again, as above, what this really needs is a quantitative prediction, not just a qualitative phrase like 'a fraction'.

This is exactly what all the numerical integration techniques do. But, except in some (rare) special cases, the sum is going to going to be from i = 1 to infinity. As I wrote above, this would take an infinite amount of time to compute, and hence you will never actually know its "exact value". The most straightforward numerical technique, Euler's method, is just the literal computation of a Reimann sum. Improvements on that method allow the computation to consist of fewer terms and/or greater accuracy to the approximation. And, yes I'm going to post it again, it is really only in very, very rare circumstances does the numerical approximation equal the actual exact evaluation of the integral. This is how they test numerical techniques, compare the numerical approximations with known solutions. And, it is also rare that a summation would be amenable to analysis to be convergent to an exact or easily calculable value. Or to a summation that wouldn't require a large number of terms. Again, I say rare simply because it is almost wholly a function of the form the integrand takes. One can do the Reimann sum, and get this summation you are looking for, for nicely behaved integrand functions, like polynomials. But again, polynomials are exceptionally rare in the entire range of all functions. In other words, in the vast majority of cases, forming the Reimann sum will give you a summation formula that won't be easily analyzed. If the Reimann sum even exists, hence ajb's comment about Lesbegue measures.

And this is not taken to be snide or rude at all, but I would similarly lobby that the question you have asked has also been not clear. You didn't. I was just trying to factitiously show how unclear that last question is. And it does fulfill your requirement of a perfectly accurately evaluated integral. But, back to your question, with no specifics at all, I am not sure how anyone can help much more. You can either evaluate an integral using various integration techniques (again, integration by parts, trigonometric integration, etc.) or numerical techniques (most of which will approximate the value to a precision that is dependent upon the specific technique applied). Beyond that, I really am unsure what specific question is being asked. And no, it didn't take much time at all to write up that integral. This forum's LaTeX capabilities are very easy to use.

The 'cannot' refers to the implied 'in every case' in your question. Your question was stated very generally, and the answer to the general case is 'no'. But there are special cases. And this isn't a thread about grammar, is it? I concede I may not be the best writer, but no one else who is posting seems to have much of a problem understanding what I am trying to get across. Now, this next followup question about 25/3 is also unclear. Without more specifics, how can anyone actually answer this? How can we be expected to read your mind and know what your arbitrary function that was integrated to 25/3 is? [math] \int^{\sqrt[3]{5}}_{0} x^2 \, \mathrm{d} x = \frac{25}{3} [/math] ?

I have answered it, since I have tried to explain to you many times, what you are asking cannot occur. It only occurs in special cases. So, the answer to your general question (paraphrasing here) "can you show me a process by which you can evaluate any integral to any precision?" is "no." Whether you accept it, or not. This is why I tried to show you many times, that what you are asking for happens rarely. Because there is NO general answer to your question. Perhaps I should have been much more explicit. I really thought my post #15 was rather clear, however. Now, if you want to modify your question, and talk about those you can evaluate more accurately, or limit yourself to talking about certain kinds of functions and integrals, we can. But, just because you don't like the answer, doesn't mean you can shout at me in all-caps. I did put a little more detail into my answer in a different thread in this section. http://www.scienceforums.net/topic/69457-probability-in-an-infinite-set/page__view__findpost__p__707277 In short, when dealing with a set with infinite choices, it doesn't make sense to talk about choosing any specific element. You have to talk about selecting elements in a subset of the set (or a range).

you do realize that measure theory and integrals are intimately related, yes? Most people are far more familiar with integrals, so I think it is generally more useful to try to explain these concepts using that more familiar framework.

It only makes sense to talk about ranges in an infinite set. I'll use the same example from the other thread. Consider a uniform distribution from 0 to 1. That means [math] f(x) = 1 [/math] for 0 < x < 1. And we'll agree that there are infinite number of numbers between 0 and 1. The definition of the probability of a single sample X located between a and b is [math]P(A < x < B) = \int_A^B f(x) \, \mathrm{d} x[/math] And, you can see if A = B... that integral is 0. So, the probability of choosing any individual element in the infinite set is 0. You only get non-zero answers when talking about ranges withing the infinite set. Now, the above definition is true for any f(x). And there are functions that have more weight on places that in others, such as the above mentioned bell curve a.k.a. the normal or Gaussian distribution. Again, you have to talk about ranges. If you take a range with a fixed width about the mean of the normal distribution, the above integral will evaluate to a higher number than if take that same fixed width on any range on the curve that doesn't include the mean. Again, any single number that is sampled will have probability zero because that integral evaluates to zero. But the probability density curve, f(x), can take on most any form, within a few restrictions. Namely, not-negative, and the integral over the entire range should evaluate to exactly 1.

I hope 'more' means actually defining what symbols in equations mean and posting graphs that are actually titled and have axes that are labelled.

But, you cannot be infinite accurate with it. Because you don't know it to infinite decimal points. This is your original question, that despite how many times you don't care for my answer, I have been trying to answer as best I can. Because the set of non-polynomial functions is uncountably infinite, whereas the set of polynomial functions are countably infinite. Just like there are more numbers between 0 and 100 than there are between 0 and 1. Wouldn't you say that it is fair, that if you selected numbers at random across all of the reals, that it is more rare to select one between 0 and 1 than it is to select one between 0 and 100? Yet, there are infinite numbers between each of them. This is how you can have something be infinite, and yet be rare. The analogy is exactly the same for polynomials out of the set of all functions. And exactly the same for integrals that are evaluatable to infinite precision out of the set of all polynomials. Yes, there are an infinite number of integrals that can be evaluated exactly. But, they are like the reals between 0 and 1 when picking from the entire set of all real numbers. They are rare. I don't know how else to explain it; so please ask more questions. Please tell me where I lose you. I do want to help you understand, if you are willing to learn. ACUV, on a continuous distribution, the probability of picking any single individual number is 0. Because over the range of a distribution, there are infinite choices. This again is the uncountably infinite-ness of the real numbers. What you can talk about, however, it he probability of pulling a number between two numbers. For example, consider the uniform distribution over 0 to 1. The probability of selecting exactly 0.72358723875365329853232875354876 is 0. But, the probability of selection a number between 0.7 and 0.8 is 10%. It seems counterintuitive, but this is what happens over sets with an infinite number of elements. The way you think about probability with sets with finite counts of elements aren't always straightforward to extend to sets with infinite counts.

No, there are different levels of infinite. Here's an easy example. There are an infinite number of numbers between 0 and 1. There are also an infinite number of numbers between 0 and 100. Both an infinite number, but there are still more between 0 and 100. Now, going with my example in the previous post, there are more irrational numbers than rational. This is because the rational numbers are countably infinite. That means, you can use logic and put them in an order, with a direct 1 to 1 correspondence with the natural numbers. http://en.wikipedia.org/wiki/Countable_set No such logic exists for irrational numbers, and hence they are considered uncountably infinite. http://en.wikipedia.org/wiki/Uncountable_set This also implies that there are more irrationals than rationals. In terms of its rareness... if you really had a true random number generator that picked any and all numbers equally between 0 and 1, you would pick an irrational number far more often than a rational one. In the same way, the polynomials (per the traditional definition of non-negative integer exponents) are also countable. Yes, there are an infinite number of them, but they are still countable. However, the set of all functions is uncountably infinite. Again, uncountable implies there are more than any countable set, even if both are infinite. Again, if you had a true random function picker, something that generated a random function from all possible functions. The chances of it picking a polynomial function are really quite rate. I think you do not have a very good appreciation of the true definition of a function, nor an understanding that the very tiny number that we (mankind) have named are only a very, very, very small amount of all the function that are out there. Yes, there are an infinite number of Bessel Functions of the First Kind, and an infinite number of polynomials.... but that is such a small slice of all the available functions out there. I will stick by my assertion that polynomials are very rare in the set of all functions. As well as stick with my assertion that the number of integrals that can be evaluated with infinite precision is also rare. To wit, even if an integral is over a polynomial, the limits still have to be rational as well. [math]\int^{\sqrt{3}}_{\sqrt{2}} 1 \, \mathrm{d} x[/math] isn't knowable to infinite accuracy, for example, despite being a very simple polynomial integrand. Because you can't know the two irrational endpoints to infinite accuracy. Lastly, there is no reason to be rude and use derogatory phrases like "I would think that a math expert..." and then follow that up with a snide remark. I invited you to take up any personal issue you have with me via the mods or via the PM system. It also obviously wasn't a lead in to ask me to clarify -- which I obviously was willing to do per the paragraphs before this one. I didn't just make that statement willy-nilly or create it out of thin air. There are formal definitions and concepts which back it up. I am happy to discuss these, but the attitude I am getting from your posts in this thread really make me think you aren't open to discussion. You have an attitude that you already know everything, and anyone who says something different isn't worth your time. If I am misreading this, I do apologize, but I would ask that you look over your posting style and review, because the impression you are giving off seems very aggressive and smug.