It makes it hard when the textbook is terrible. :( I always feel bad having to ask, but when we ask our prof he says surely we can ask someone else. It's like... We're paying you to teach us and you're grading our assignments with no feedback. Literally he puts a check mark or an X. That's it. sigh
Well you'll have to come up with some specific questions and I can see what I can do... I'm decent at sciences like Chemistry, but my majors were psychology and sociology. :P
This problem is actually an intermediary step in DUI (wikipedia's first example). The partial derivative of the expression wrt $a$ isn't too nice, I doubt it will help much. Also, that would leave 2 integrals at the end to mop up
I remember when I first learned the quadratic equation I stopped paying attention in class trying to find a cubic equation and a general equation for polynomials of degree n.
Give an example of a polynomial in the variable t, such that it is fourth degree, in descending powers of the variable, with exactly five terms, and having a positive coefficient for its quadratic term.
Here is a lovely limit I'd like to share. I work on it and try to find a beautiful way of doing things $$\lim_{n\to\infty}\frac{\sqrt[n^2]{(4^n+1)(4^n+2)\cdots (4^n+n)}}{2^n}$$
I read Brun's proof of Brun's theorem here : http://gallica.bnf.fr/ark:/12148/bpt6k486270d/f138.image
(and the following pages)
But I was unsatisfied. I did not understand it and I did not even get the notation he used. It seems like " some statistical arguments " because of the infinite product...
how long is the proof? is there a readable version? or an english translation? asking for a step-by-step walkthrough of a historical proof may or may not be asking too much of MSE depending on these sorts of bits.
@Shayna how you punch things into your calculator depends entirely on what calculator you have. is there a reason you don't want to punch them into website calculators?
LOL I need to know how to use mine for the exam... I know how on a graphing calculator I think, but we're only allowed to use scientific ones in the exam. @anon
@Shayna familiarize yourself with the buttons. do you know how parentheses and order of operations work? you could go with e.g. 7/2-(1/4)√290 or 7/2-(1/4)(290^(1/2)) or 7/2-(290^(1/2))/4. (frankly, these things should be taught in gradeschool.)
No , but mathematicians are the highest lifeform on the planet and they often have skills outside math , in fact math is used in all science. Hence my sensitity and defense/attack towards the subject @anon @Shayna
No answers yet to my question , but a star though.
@Shayna I hope your ok with my comments , not trying to upset you.
But im not the type of teen you can walk over :p
Let go a step further. math has applications , well at least indirectly in other sciences for sure. What is an application of sociology that really works / worked ?
@Chris'swisesister my thoughts, where I marked the suspicious parts in red: $$\lim_{n\to\infty}\frac{\sqrt[n^2]{(4^n+1)(4^n+2)\cdots (4^n+n)}}{2^n}\\=\lim_{n\to\infty}2^{-n}\prod_{k=1}^{n}4^n\left(1+\frac{k}{4^n}\right)^{1/n^2}\\=\lim_{n\to\infty}2^{-n}\prod_{k=1}^{n}\sum_{j\geqslant 0}{n^{-2}\choose j}\left(\frac{k}{4^n}\right)^j\\ \color{Red}{=\lim_{n\to\infty}2^{-n}\prod_{k=1}^{n}\left[1+\frac{k}{n^2 4^n}\right]}=L.$$
@mick Obviously you don't even know what Sociology is. I hope you weren't speaking for all mathematicians when you said you were more intelligent... Because that would be quite a rude fallacy to project upon to the others. ;)
@Shayna If you can read , i said im just 14 and not a mathematician yet , so I do not speak for all and btw never claimed so. Further SOCIOLOGY IS ABOUT HUMAN SOCIAL ACTIVITY hence it implies relationships !!
Here's a nice problem: find the set of unit vectors $\mathbf{v_1},\mathbf{v_2},...,\mathbf{v_{5}}$ whose bases are at the origin such that $\sum_{1 \le i <j \le 5} \theta _{i.j}$ is a maximum, with $\theta_{1,2}$ being the angle between $\mathbf{v_1}$ and $\mathbf{v_2}$.
@Chris'swisesister I used the binomial theorem and collected only the first and second terms: $$\sum_{j\geqslant 0}{n^{-2}\choose j}\left(\frac{k}{4^n}\right)^j\\=1+\frac{(1/n^2)(k/4^n)}{1!}+\frac{(1/n^2)(1/n^2-1)(k/4^n)^2}{2!}+\cdots$$
one could equally well speak of open boxes, open polyhedra, open ellipsoids, etc. In euclidean space, an open set is any set X such that every point x in X has a ball around it contained in X.
@exitingcorpse I had a feeling of deja vu there :)
it's a basic fact that norms on R^n are all equivalent. but some of them are more finitary than others: suppose the unit norm ball (which specifies the norm) is a polytope with coordinates lying in Q^n (a rational polytope)
just think about R^2 at first. imagine you have an oracle to compute the norm, and you know the unit norm ball is a rational polytope a priori. can you come up with an algorithm to determine the vertices in finite time?
(the application is the thurston norm in 3-manifold topology. it has a lot of information, and we know it is always a rational polytopal norm. of course, the issue is that we don't know how to compute it exactly... even bounds can be tough)
@mick In fact, the way in which you referred to Sociology implied you were speaking of Psychology... I'm sorry, but you still have much to learn at 14.
@Shayna I'm quite off when it comes to Sociology, but what would be an "open problem" in that field? I mean, can you cite some nice "Sociology problems"?
@IanMateus World hunger, social politics, global waste & recycling, natural resources... Basically anything humans interact with environment wise... If humans are interacting with one another in most cases it becomes Psychology (it will have little depth in Sociology anyway). Sociology is more qualitative and Psychology is more quantitative.
@Shayna ok, these are famous problems. I mean, what connections specialists in that field know/see further than the mere mortals? I have seen some work in geopolitics, and it is awesome!
@IanMateus Without the Sociologists you wouldn't have nearly as much information on those topics... They wouldn't be as widely known. Being that their research is more qualitative, they discover things first hand. They don't often conduct the quantitative statistics themselves, but instead pass them on to other disciplines. Sociologists are masters of interviewing and surveying techniques... They also use participant observation quite a bit.
@mick You can't, really... It still doesn't mean you're right though. There is a point at which Anthropology, Sociology and Psychology all meet and diverge. Sociology refers more to the social interactions with bureaucracy and thus overlaps with many disciplines. Psychology is raw human interaction with other humans. Anthropology is the study of social interaction and bureaucracy in times past by the study of ancient ways or ways little known in today's societies (such as of the Roma Gypsies).
LOL @mick THE MEDIA... HA! They get their info from researchers. Without researchers they would have no leads.
@mick I sincerely don't know what you are trying to prove. If the media tells you something, it is because someone thought it before, they are Sociologists.
@Ian @mick Not in all cases. Sociologists often inform political scientists of what needs to be studied as sociologists are those who specialize in interviewing and surveying techniques.
@IanMateus I've been lucky enough to take courses in many of the Humanities... Majored in Psychology and Sociology... Plus I was the president of the Sociology-Anthropology club and went on some field school trips for Anthropology (super cool!).
@mick LOL IQ tests are the least valid tests psychologists use. Pick up a book on psychological testing and assessment. Do us a favor and learn something before you shoot your mouth off.
@Shayna Alright. Studies on the effects of alcohol on the brain whilst performing motor functions. Studies on how drugs effect individuals. Studies on how sleep vs wakefulness effects individuals, REM sleep, circadian rhythms. Pick up a book and read. I'm not your human book. Seriously.
@mick how would you know that unless someone studied them and told you they were bad for you? how would you know which neurotransmitters they effect? also, nice... you only had one come back to my whole spiel. i believe that you, sir, are done. now shush.
@PeterTamaroff I feel like I've spent too long typing information for someone who is ignorant enough not to absorb what is being offered up... Thus a spiel. :p
Is it a pretty safe bet that if a question is asking for the unique smallest topology containing the all collections of some set of topologies it is refering to $\subset$
@PeterTamaroff Well, I'm doing math for teachers... Going into a BEd program so I had to pick up a math other than stats LOL. I want to be a high school counselor, but where I live they require you to have 5 years teaching experience first. :/
@Bageer indeed, the topologies on a given set are partially ordered under inclusion, and that's what we're referring to when we use words like "coarse" vs. "fine"
@PeterTamaroff The full question is: Let $\{\mathcal{T}_\alpha \}$ be a family of topologies on $X$. Show that there is a unique smallest topology on $X$ containing all the collections of $\mathcal{T}_\alpha$. There is a similar one for largest topolgy.
