Keen

My understanding of relativity is very limited, so I'd appreciate if someone could help me understand this concept. When it comes to an object, I can move it in space by applying forces change direction etc... I can even make an object stop moving altogether in a certain frame of reference, but when it comes to time, it seems to me that there is little control over how an object moves in time. It always goes forward, I can't make the object stop etc... so how is it helpful to think of time as a dimension when it behaves very separately from the others?

If you do not want to use Wolfram, you're going to have a difficult time calculating this integral. The general method for calculating these kinds of integrals is using following formulas: [math]sin(x)=\frac{2tan(\frac{x}{2})}{1+tan(\frac{x}{2})^2}[/math] and [math]cos(x)=\frac{1-tan(\frac{x}{2})^2}{1+tan(\frac{x}{2})^2} [/math] You then change the integration variable by putting [math]u=tan(\frac{x}{2})[/math] and you should obtain a rational function in u. You then decompose it into partial fractions and integrate these. However considering how many different constants there are, it is likely to take enormous time and I definitely am not motivated to do this kind of calculations.

This is typically a statement that is best proven by contraposition. Assume that the discriminant is positive and then prove that your polynomial cannot be non negative in all points. When you have a positive discriminant, your polynomial has exactly two roots and depending on the sign of a, it is either positive between these roots or negative. If it is negative, you're done. If not then it must be positive between the largest of the two roots and infinity and between minus infinity and the smallest of your two roots. I guess you could also prove it directly, but then you'd probably have to waste time with who knows how many different cases. Hope it is clear enough.

The best hint I can give you is that for any interval I and any function y positive on I: dy/dx=y (d ln(y)/dx) with ln being the natural logarithm. With that, you should be able to solve it.

Look. You could call me an advocate of mathematics for the sake of mathematics and I am not criticizing the fact that you are exposing theories that require a higher level of abstraction. What I am criticizing is your approach at exposing them. For me to consider some theory interesting, it must either have some sort of problems or solve other problems from other fields. Also usually I find a much better approach when explaining abstract mathematics: to give a lots of examples to make your audience grasp what kind of objects you are dealing with. Also what does your abstraction generalize? Does this generalization help to tackle some problems? And so on... I once took courses in Arakelov geometry, which were unfortunately pretty useless to me, because our teacher made something similar to what you are doing right now. He started with some very abstract notions like Grothendieck topology or some very general functorial inductive limits (not the usual ones with a directed set), without ever motivating these notions, so most students got lost in these abstractions pretty quickly. It's a pity that you are over dramatizing people's criticism and lack of enthusiasm, because I am sure there are interesting things to be learned from your posts. When it comes to category theory I usually prefer to use the Godel-Neumann class theory. It's most likely just a cosmetic change to talk about the class of all sets instead of Grothendieck universe, but in theory it should be larger since it encompasses all the sets, while Grothendieck universe only encompasses the sets we "care" about. I doubt it changes much in practice, but it's a thing I wanted to point out.

We might not have the same notion of fun.... I think it would be much more interesting if you showed some problems that naturally arise from what you just showed or at least some reason why this deserves any attention. Perhaps you could show some concrete vector spaces and give an example of a specific pull-back or push-forward? Otherwise it just seems (at least to me) like some pointless abstract symbol manipulation.

The most disappointing thing about new Star Wars to me was the villain. I understand what they were going for: a great build up only to make you realize then that he's nothing but a brat, but I just think they went way too far with this. If the villain isn't a least bit scary (and he stopped to be as soon as he took off his mask and made a ton of serious mistakes), then how can the story be interesting? They also recycled quite a lot of things from the original series: like father and son conflict, destroying a giant machine of doom, stormtroopers that never hit anything, secret plans sent by a droid. The worst part is that not only they basically recycled what we have already seen, but somehow managed to make it worse in some cases. Take for example the scene where Kylo Ren kills his father. It is reminiscent of the scene from the episode 5, so I will make a comparison. In episode 5, you had Luke totally beaten and Vader tries to persuade him to join the dark side. Then you get the shocking reveal that Vader is Luke's father. Not only it is quite a shock, but also it puts Luke into a very desperate situation. That is an example of what I would consider a good writing. On the other hand in the scene where Solo tries to persuade Ren to join the light side, we already know he's his father, so no big reveal. Also being the only notable villain in the movie it was quite predictable what he was going to do. He either had to kill Solo or suddenly become a good guy out of nowhere and the movie would be suddenly without a villain therefore it was obvious he was going to kill Solo, therefore all the build up they made in this scene felt kind of wasted and the resulting drama much more inefficient. Also in my opinion the actor could also have done a better job portraying his inner conflict. I didn't feel very convinced by his acting. Considering all the hype I really hoped we would get a far better movie than we got. In the end for me it turned out to be an "OK" movie. There were two things that saved it for me. -Scenery: I really loved the wreckages of the Imperial machines in the desert: those scenes always felt majestic. It also felt good to see the imperial warmachines like TIEs or star destroyers with that old nostalgic feel and yet quite polished with new effects. -Cast from the original series: I think that Carrie Fisher and Harrison Ford made a good job. I could still see Han Solo and princess Leia in them. Older and with new experiences, but I did not have any doubts that those two are Solo and Leia. Their acting to me was the most believable thing in this movie. Maybe it's simply because those two are such iconic characters, but those two performances were my favorite.

I suppose yeah. As soon as the client does not see much difference or the security measures become too technical, credit card companies will probably not want to invest in it. Being an amateur enthusiast in cryptography I just find it disappointing to see a system where you send the same secret data to verify identity being used while there are in my opinion much more secured ways. There are some banks, which send a one time password by sms, which is already quite a good security measure, but I'd still like to see a system with something like zero knowledge proofs or digital signatures implemented simply because I don't like much the idea of giving my card number to strangers.

I'm not sure whether to put this in the computer science section or applied mathematics, but I've always considered cryptography to be applied mathematics, so I will post it here. I don't unfortunately know all the details behind how online transactions work, but as far as I know, you send to the merchant your credit card number and cryptogram and he uses those informations to validate the transaction. This seems to me a bit insecure, because you have to trust the merchant as you are giving him all the necessary information that can be used to pay anywhere and if someone for example steals it from him, he could reuse it somewhere else. (Yes I am aware that the communication is encrypted, so stealing that number isn't that easy). An alternative, that to me sounds much more secure is to use the credit card number as a private key in some digital signature algorithm like for example DSA. That way, the merchant sends you all the necessary information for the transaction like for instance some identification, price to pay, date etc.. you digitally sign it with your credit card and then send it back to the merchant. That information can be then validated by the payment server and cannot be reused by anyone else, since it is a digital signature of only one specific transaction. If needed, this could most likely be as well adapted for monthly payments. You would simply send a monthly payment order signed by your digital signature to a company like netflix instead of your credit card. Maybe I'm not getting something, but that to me seems much more secure, than sending a simple unchanging information over the internet.

Or you could encrypt the private key by some symetric encryption algorithm like AES or blowfish and decrypt it in your RAM only when needed. That's how for example PGP works if I'm not mistaken.

Very interesting idea, but unless I did not get something, I believe your first hypothesis to be false. You claim that numbers which differ by one have no more than two brackets of difference between them in their bracket notation. I tried among the Fermat's primes as they grow rather quickly and it's easy to decompose the preceeding number. I have found [math]257=2^8+1[/math]. 257 is 65th prime, so according to your notiation it should have 65 opening brackets and 65 closing brackets. The preceeding number [math]256=2^8[/math] has eight twos in it's decomposition therefore if I get it correctly it should have 8 opening and 8 closing brackets, which is quite a difference with the next number. I'd personally speculate that there is no bound for the difference between brackets of consecutive numbers simply because when you multiply small prime numbers among them the result tends to grow rather quickly and if the consecutive number is a prime then it would be a rather large prime with lots of brackets, while the preceding number won't have that many brackets, since it's composed of a relatively small number of small primes.

That would probably be for the best. I don't want you to discourage from mathematics, but if you really wanted to write a serious paper on such a very difficult topic as continuum hypothesis, it would take years of study in mathematics and then even more hard work in mathematical logic. Either way as far as I know, continuum hypothesis is independent of ZFC axioms, which is a standard framework for modern mathematics, so you can't neither prove nor disprove it: you can only make a theory based on it or its negation. I am sure there are lots of fascinating results in this kind of non standard mathematics, but I am unfortunately not knowledgeable enough in mathematical logic to discuss this topic further.

Ok, I thought you were considering this type of numbers. I just wasn't sure. It actually does make perfect sense. You just consider the infinite sequence (un) of 0 and 1 and to this sequence you can associate a unique number which is the limit of the infinite series [math] \Sigma [/math] un 2-n. On the other hand your limit does not make much sense. You claim, that by making n going to the cardinal of aleph0, you obtain the result, that the cardinal of numbers with finitely many digits is aleph1. However you still do not explain in what sense you take your limit. It surely isn't the classical limit of sequences, since the limit of 2n in the classical sense is simply infinity, which does not say then anything on the cardinality of your set [math]B_F[/math], besides that it is infinite, but that's not really what you are trying to prove. Also I would like to add, that the notation of the powerset [math]2^{\mathbb{N}} [/math] is a mere notation and isn't anything particularly special, which makes your use of limit even weirder and if I am to be honest, I do not think there is a way to write your limit formally in a way that would be compatible with modern mathematics as it is a very well known result, that a set can't have the same cardinal as its powerset. The proof of this statement is actually not that difficult and is reminiscent of Bertrand Russel's paradox, which I recommend you to check out. If there was a set A with the same cardinal as its powerset, there would be a one to one correspondence between A and [math] 2^A [/math], let's call it f. You can think of one to one correspondence as a label. To each member x of A, you associate a unique subset of A denoted f(x). One to one simply means, that each subset of A has a unique label. Now we consider a very particular subset of A, which has all the elements x, such that x is not a member of f(x). We call this subset B. Since we know that f is one to one, it means that there is y in A such that B=f(y) and now the question is whether y is in B or not. First let's suppose it is. Then by definition of B y is not in f(y). but f(y) is precisely B. If on the other hand it isn't in B, that means that y is in f(y) by definition of B, but f(y) is B, so y is in B, which is a contradiction. No matter how you put it you obtain something contradictory, which means that the set B does not exist and therefore neither the one to one correspondence f does.

Well honestly I am kind of lost already in your first paragraph. When it mentions "We let n go to the cardinal number of the set of natural numbers", what does that thing even mean? From what I understand, what is called here a binary fractional number, is pretty much a binary number with finite decimals, which actually is simply a rational number. Claiming, that this set has the cardinal of the powerset of natural numbers is a very bold assertion, since it challenges already well established results, so this magical use of limit "We let n go to the cardinal number of the set of natural numbers." had better be very well explained. The problem is that with limits you can make pretty much anything if you don't use them carefully and here, I do not think there is a sufficient justification behind that use of limit. If on the other hand the set of binary fractional numbers is the set of all numbers in [0,1[ written in binary, then it seems to me there is a misrepresentation, because the process that is used to list those numbers only deals with numbers with finite decimals. So yeah it's all quite messy and needs rewriting and I am almost 100% sure, that if those arguments were written correctly a flaw would quickly appear, because one does not simply contradict Cantor's diagonal argument.

Hi. I am not sure whether to put this topic in physics or mathematics, since it's kind of both. I used to take physics courses back when I was an undergraduate student and unfortunately I didn't like them much. Mainly because of how the mathematical models were treated 'poorly'. By that I mean that we lacked rigorous definitions and I wasn't even sure of for instance how regular the functions that we employed were. I once stumbled upon a book whose name I have unfortunately forgotten, but I remember it was about special relativity and it was really written for people with my mindset. To give you an example, it defined a material point as a couple (gamma, m) with gamma an infinitely differentiable curve in a Minkowsky space and m a positive real number. I know that physics is supposed to represent real world and I don't mind that. I just think that as soon as physicists use mathematical representations, they should define their objects well... mathematically. I wanted to ask if you know of other books which take this kind of formal approach to physics? I am mainly interested in classical and relativistic mechanics, electromagnetism and thermodynamics. Again I don't mind references to experiments and physical explanations: it's physics after all, but I would just like that all the formal mathematical part is treated "correctly".

And just to answer the actual question: the set of real numbers is not a group under the usual multiplication law, but not because there is no neutral element, because 1 is a neutral element under multiplication, but not everyone is invertible, because 0 isn't. If you put e=0 in your original argument Backes, it still works: 1*0=0*1=0, so 1 is neutral even for 0. The real numbers are on the other hand a group under the addition law.

Algebra being my favorite subject in mathematics, I think I can give you some suggestions concerning what you can study. It's not very easy to get into abstract algebra, because if you only consider structures like fields groups rings vector spaces by themselves, you usually don't get far in their study as they are in some sense closely related, so I do not think there is a perfect order in which one should learn about all these structures. But since one has to start somewhere, I would personally recommend to learn in this order: Start with some elementary group theory. Nothing complicated, just get the hang of notions like order, symmetric group, group action etc... Then learn about finite dimensional vector spaces and linear maps and how they relate to matrices and systems of equations. Try to think about different examples: it seems you already got the hang of polynomial functions and that's great. Linear algebra is an extremely powerful tool in mathematics and I think it is important to familiarize oneself with as many linear structures as possible. Once you are familiar enough with basic notions in linear algebra, you can study things like scalar product, determinant and eigenvalues. Then I think you will be ready to study the ring structure. The most important here is to learn what are ideals, quotient rings and ring of polynomials. Once you understand it, you can study fields and general linear algebra over abstract fields. I come from a French academic background, so I am not sure I can recommend you some good books on those subjects in English, but you can try to start for example with J.S Milne Group Theory Chapters 1-6 Then for example continue with this book on linear algebra https://www.math.ucdavis.edu/~linear/linear-guest.pdfand you can end with http://www.maths.usyd.edu.au/u/bobh/UoS/rfwhole.pdf Those are what I consider to be the basic notions in algebra.

In math, a dimension higher than 3 is usually used, because geometric analogies help to solve a lot of problems. If you want to solve for example equations with lots of variables, you view those equations as some geometrical entity in a higher dimensional space and sometimes, you can find interesting results concerning these equations The definitions stay the same for all dimensions. In basic geometry you can define a dimension as the number of coordinates in a coordinate system. In dimension two, you only need two coordinates to find a point (x,y), in dimension 3 you need three coordinates (x,y,z), in dimension 4, you need four coordinates (x,y,z,t) and so on... It's of course not obvious to imagine something that has more than 3 dimensions, but what usually helps is to work with such spaces and explore their properties on paper. You find out, that lots of things which are true in dimension 3 are also true in higher dimension as well, so there are some analogies that can be found. If you want to acquire some intuition concerning higher dimensional spaces, I'd suggest to wait for the release of a quite clever video game called Miegakura in which you have to solve puzzles in 4 dimensions.

That's actually a good point: and a reason I do not really consider it a real paradox. If we want to use notions such as mass or volume, we can only work with sets that are measurable, because anything else doesn't really make sense and the sets involved in this paradox aren't measurable. With that being said, you can't really physically obtain such sets, but that's probably irrelevant, because as far as I know, on microscopic levels you have so many strange things that are going on, that measurability it's the least of your concerns. The other way around for me is quite difficult as I have little to none experience with physics and engineering, so I can just think about some strange stuff I've seen in mathematics and tell myself "That kind of thing can't exist in real world". With that being said mathematics is a model and just a formal way to describe the laws you can observe, so as soon as in physics you require some formal approach that doesn't have a mathematical description yet, you can be pretty sure that a mathematician creates a new field in mathematics. Lots of mathematical notions actually originated in problems in physics.

What exactly is an algebra of vertices to you? It seems to me that what you are looking for is simply an affine space.

Look for Banach Tarski paradox. In summary, you can cut a ball of volume 1 into some pieces in such way that after some rearrangement of those pieces you obtain two identical balls of volume 1. That kind of paradox made mathematicians question, whether or not the axiom of choice is legitimate since using it creates paradoxes completely contradictory to nature and physics. (conservation of mass) There are also functions which are continuous and yet not differentiable in any point. You are quite unlikely to encounter them in nature, because at least from my limited knowledge in physics I am pretty convinced that most of the functions you encounter are infinitely differentiable. (by parts at least) In the same category you also have Devil's staircase. It's a function which you can differentiate almost anywhere and it's derivative will by always 0 and it's increasing. In some sense you move up with 0 speed. That's all that comes to my mind right now.

I'm not exactly an expert on chess, but I like to play it and sometimes I watch some videos analyzing chess games. I like tactical players like Kasparov or Tal, as that kind of play fits much more my mindset. Tal is actually my favorite, because he always made those crazy kind of sacrifices incomprehensible to mere mortals.

That's exactly what I was looking for: actually even better as it's not a simulation, but real images. Thanks a lot!

I come from pure mathematics background and my philosophy when it comes to using computers in mathematics is to avoid them as much as possible, so I believe I can provide you with some downside of computers. When it comes to pure calculations, computer is an invaluable tool, but it should just be a means to gain time. I never put calculations into a computer, that I would not be able to do myself, because precomputed functions are kind of a black box and it may become problematic when you simply trust your computer without knowing what it is doing. If I understand well all the algorithms implied, then I am not against using computers to gain time in calculations. However I refuse to use computers when doing some rigorous mathematical proofs. The problem is: computer does the calculations just fine, but does not help you to understand what is happening. When a proof is done by some brute force calculations I consider it incomplete in a sense, that it does not make people any wiser. I don't know what your mathematical background is, so it's hard to give specific examples.

When I see videos (mostly movies) about space shuttle take off, the part I think I would find quite fascinating is skipped: The transition between when the Earth looks flat from above and when you are in a sufficient distance to see it is round and the transition between a blue sky and depths of space. Is there any simulation, where you could see the Earth from above while getting further away from it?