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I understand how derivatives work based on the definition, and the fact that my professor explained it step by step until the point where I can derive it myself.
However when it comes to the area under a curve for some reason when you break it up into an infinite amount of rectangles, magically it turns into the ant... |
Why is the area under a curve the intergral? |
Suppose $\kappa$ is an uncountable cardinal. Then $S\subseteq\kappa$ is club (CLosed UnBounded) if $S$ is unbounded in $\kappa$ and is a closed subset of $\kappa$ under the order topology. My question is the following: suppose $S\subseteq\kappa$ is club, and $S=S_1\cup S_2$. Is there necessarily some club set $C$ such ... |
Why is the area under a curve the integral? |
First: the integral is *defined* to be the (net signed) area under the curve. The definition in terms of Riemann sums is precisely designed to accomplish this. The integral is a limit, a number. There is, a priori, no connection whatsoever with derivatives. (That is one of the things that makes the Fundamental Theorems... |
First: the integral is *defined* to be the (net signed) area under the curve. The definition in terms of Riemann sums is precisely designed to accomplish this. The integral is a limit, a number. There is, a priori, no connection whatsoever with derivatives. (That is one of the things that makes the Fundamental Theorems... |
First: the integral is *defined* to be the (net signed) area under the curve. The definition in terms of Riemann sums is precisely designed to accomplish this. The integral is a limit, a number. There is, a priori, no connection whatsoever with derivatives. (That is one of the things that makes the Fundamental Theorems... |
First: the integral is *defined* to be the (net signed) area under the curve. The definition in terms of Riemann sums is precisely designed to accomplish this. The integral is a limit, a number. There is, a priori, no connection whatsoever with derivatives. (That is one of the things that makes the Fundamental Theorems... |
First: the integral is *defined* to be the (net signed) area under the curve. The definition in terms of Riemann sums is precisely designed to accomplish this. The integral is a limit, a number. There is, a priori, no connection whatsoever with derivatives. (That is one of the things that makes the Fundamental Theorems... |
First: the integral is *defined* to be the (net signed) area under the curve. The definition in terms of Riemann sums is precisely designed to accomplish this. The integral is a limit, a number. There is, a priori, no connection whatsoever with derivatives. (That is one of the things that makes the Fundamental Theorems... |
For an university assignment I have to visualize some curves in 3 dimensional space.
Until now I've implemented Bézier, helix and conical spiral.
Could you give me some advice about some famous curves in geometry history? |
famous space curves in geometry history? |
Suppose you have a double sequence $\displaystyle a_{nm}$. What are sufficient conditions for you to be able to say that $\displaystyle \lim_{n\to \infty}\,\lim_{m\to \infty}{a_{nm}} = \lim_{m\to \infty}\,\lim_{n\to \infty}{a_{nm}}$? Bonus points for necessary and sufficient conditions.
For an example of a sequenc... |
OK, I know that Matiyasevich's solution to Hilbert's 10th problem shows that there is no algorithm to decide whether or not a polynomial $p(x_1,\ldots,p_n)$ with integer coefficients has a solution over the integers. I have no doubts about the truth of that. However, I have been unable to figure out why the following... |
I got an assignment to prove certain things about right triangles in Lobachevsky geometry, but so far I don't know where to start. What model is the best for studying these objects? What is the general approach to this kind of tasks? |
How to analize triangles in Lobachevsky geometry? |
Everything is in $\mathbb{Z}$. Let $v_1 < v_2 < ... < v_n = k$, and $v_1 = 1$ for $k >> n$. Let $ P = \Pi_{i < j} (v_j - v_i)$. How can I show that $P \le k^{n^2}$?
There are $n + (n-1) + ... + 1 = \frac{n(n+1)}{2}$ terms in the product. Starting from $v_n - v_{n-1} = 1$, etc. Clearly, $P = 1(1*2)(1*2*3) ...(k-1)! =... |
i have n points that determine a poligon. I need to color the interior of this poligon with a color . How can i do this? |
The eigenvalues of the eigenvectors of a rotation matrix are complex numbers. I understand that they cannot be real numbers because when you rotate something no direction stays the same.
**My question**<br>
What is the intuition that the eigenvectors are complex? Why do they exist at all for rotation matrices? I me... |
I have n points that determine a polygon. I need to color the interior of this polygon with a color . How can I do this? |
The eigenvalues of a rotation matrix are complex numbers. I understand that they cannot be real numbers because when you rotate something no direction stays the same.
**My question**<br>
What is the intuition that the eigenvalues are complex? Why do they exist at all for rotation matrices? I mean it is not so that ... |
I got an assignment to prove certain things about right triangles in Lobachevsky geometry, but so far I don't know where to start. What model is the best for studying these objects? What is the general approach to this kind of task? |
How to analyze triangles in Lobachevsky geometry? |
Given the Right ascension in h m s, Declination in deg ' " and the Trigonometric parallax How can I get the cartesian (x,y,z) coordinates of a star? I'm guessing I need 3 separate formulas to get each x, y and z values. |
there is a prove that there is no $2^n that is divisible by 5? |
there is a prove that there is no $2^n$ that is divisible by 5? |
I´m trying to develop a architecture hardware to make a implementation of an algorithm that can be descompose in terms of sums, multiplications, substrations and exponential functions. I,m trying to modelling, exp(-x) through Taylor series. The domain of my function is bounded between 0 and 1500, but I want to use a pa... |
For improper single integrals with positive integrands of the first, second or mixed type there are the comparison and the limit tests to determine their convergence or divergence. There is also the absolute convergence theorem.
For multiple integrals I know that the comparison test can be used as well.
**Qu... |
Suppose there is a birthday party to which all of the girls (and none of the boys) in a small town are invited. If you run into a mother who has dropped off a kid at this birthday party and who has two children, the chance that she has two girls is $1/3$. Why? $3/4$ of the mothers with two children will be at the birth... |
I think this question is generally confusing because there's a lack of context -- I'll try to supply that.
Suppose there is a birthday party to which all of the girls (and none of the boys) in a small town are invited. If you run into a mother who has dropped off a kid at this birthday party and who has two children... |
I think this question confuses a lot of people because there's a lack of intuitive context -- I'll try to supply that.
Suppose there is a birthday party to which all of the girls (and none of the boys) in a small town are invited. If you run into a mother who has dropped off a kid at this birthday party and who has ... |
It is well known how to use algebraic geometry (differentials, divisors, and Riemann-Roch) in order to prove the finite dimensionality of the vector space of modular forms of some fixed weight and level.
Is it possible to prove the finite dimensionality of modular forms using representation theory? If so, any refere... |
Proving finite dimensionality of modular forms using representation theory? |
If we start with $\int\frac{v}{g-2v} dv$ how would we go about integrating it? (g is a constant)
The mark scheme tells me to rearrange it to $\frac{1}{2}\int ( -1 + \frac{g}{g-2v} ) dv$ which integrates to $\frac{1}{2}v-\frac{1}{4}gln(g-2v) + c$
I can see that the rearrangement works 'in reverse', but I'm not sur... |
Is there a good explanation of a manifold on the web somewhere? The wikipedia article isn't really working for me. I was actually hoping for a whiteboard lecture on youtube, but can't find one.
My math experience is calculus through differential equations, twenty five years ago. I also have some computer science ... |
My question is ; How can I solve the following integral question?
$ \int{(1 + cosx)^3 dx} $
Thanks in advance,
|
How can I find $ \int{(1 + cosx)^3 dx} $ ? |
In the definition of the complexity class NP, we allow the certificate to be polynomial in the input size. But do we ever need a certificate that is longer than the input?
In all the NP problems I could think of, we don't. The "worst" problem was factorization<s>, where the obvious incoding is $n+O(\lg(n))$ (where ... |
Is $$\int_1^N e^{ix^\alpha} dx=O(N^\alpha)?$$
where $0\lt \alpha \lt 1$. |
Is this integral of order $N^{alpha}$? |
My question is ; How can I solve the following integral question?
$$\int{(1 + \cos x)^3\mathrm dx}$$
Thanks in advance,
|
How can I find $ \int{(1 + \cos x)^3\mathrm dx} $ ? |
I'm sorry that I'm putting this in an answer rather than a comment, but I'm too new to comment on your question. My thoughts for proving this process converges (question 1.) were along the lines of:
1. $A\,\&\,B$ both SPD $\implies AB$ and $A+B$ are also SPD
2. If $A$ and $B$ are symmetric $n\times n$ matrices, $... |
Is this integral of order $N^{\alpha}$? |
If we start with $\int\frac{v}{g-2v} dv$ how would we go about integrating it? (g is a constant)
The mark scheme tells me to rearrange it to $\frac{1}{2}\int ( -1 + \frac{g}{g-2v} ) dv$ which integrates to $\frac{1}{2}v-\frac{1}{4}g\ln(g-2v) + c$
I can see that the rearrangement works 'in reverse', but I'm not su... |
I think that the reason that these puzzles are so often confusing is that they rely on the limitations of the English language rather than on any mathematical difficulties. Of course this is not unique to English, and I think you should be able to find similar similar puzzles in pretty much any natural language.
... |
If we start with $\int\frac{v}{g-2v} dv$ how would we go about integrating it? (g is a constant)
The answers to the past exam paper I have tell me to rearrange it to $\frac{1}{2}\int ( -1 + \frac{g}{g-2v} ) dv$ which integrates to $\frac{1}{2}v-\frac{1}{4}g\ln(g-2v) + c$
I can see that the rearrangement works 'in... |
(I'll keep this one short)
Given a Dirichlet series
$$g(s)=\sum_{k=1}^\infty\frac{c_k}{k^s}$$
where $c_k\in\mathbb R$ and $c_k \neq 0$ (i.e., the coefficients are a sequence of arbitrary nonzero real numbers), and assuming that $g(s)$ can be analytically continued, does it follow that $g(s)$ possesses a critic... |
I think it's an interesting coincidence that the [locker puzzle][1] and this [puzzle about duplicate array entries (see problem 6b)][2] have such similar solutions. _Spoiler alert!_ Don't read on if you want to solve these puzzles yourself first (they're two of the best puzzles I've ever seen).
In both solutions we... |
I think it's an interesting coincidence that the [locker puzzle][1] and this [puzzle about duplicate array entries (see problem 6b)][2] have such similar solutions. _Spoiler alert!_ Don't read on if you want to solve these puzzles yourself first (they're two of the best puzzles I've ever seen).
In both solutions we... |
Is $$\int_1^N e^{ix^\alpha} dx=O(N^\alpha)?$$
where $0\lt \alpha \lt 1$ and $i=\sqrt{-1}$.
{** Edit**.} As Willie Wong shows below, the correct equation is
$$\int_1^N e^{ix^\alpha} dx=O(N^{1-\alpha}).$$ |
Is $$\int_1^N e^{ix^\alpha} dx=O(N^\alpha)?$$
where $0\lt \alpha \lt 1$ and $i=\sqrt{-1}$.
${\bf Edit.}$ As Willie Wong shows below, the correct equation is
$$\int_1^N e^{ix^\alpha} dx=O(N^{1-\alpha}).$$ |
How one should go about proving
$\displaystyle\frac{\Gamma'(s)}{\Gamma(s)}=O(\log|s|)$
directly? By a direct proof, I mean not to go through the usual Stirling's formula with the exact error term. The use of a rough form of Stirling's formula is welcome. |
How may one go about proving
$\displaystyle\frac{\Gamma'(s)}{\Gamma(s)}=O(\log|s|)$
directly? By a direct proof, I mean not to go through the usual Stirling's formula with the exact error term. The use of a rough form of Stirling's formula is welcome. |
I'm working with percentage splits and I'm trying to add extra players into the split but I'm coming unstuck...
Scenario:
I have agreements with a number of parties that set out what type of split we get from a pot of cash. With some it's 50/50, with others it may be 60/40 in favour of either party or perhaps ev... |
How can I factor in an extra player on a percentage split? |
The [integral test of convergence](http://en.wikipedia.org/wiki/Integral_test_for_convergence) states that, if $f:[1,+\infty)\to[0,+\infty)$ is a monotonically decreasing nonnegative function, then the series $\sum_1^\infty f(n)$ converges iff $\int_1^\infty f(n) dn$ is finite.
Is the high-dimensional generalization... |
Does the integral test work on higher dimensions? |
**HINT** $\ $ By [Gauss's algorithm][0] for computing inverses modulo a prime $\rm\: 2\:n+1\:$ (here $5$) we have:
$\rm\quad\ \ \displaystyle \frac{1}2\ \equiv\ \frac{n+1}{2\:(n+1)}\ \equiv\ n+1\ \ (mod\:\ 2\:n+1)\:.\quad $ Hence $\rm\ 2^n \equiv 0\ $ times $\ (1/2)^n\ $ yields $\rm\ 1\equiv 0\ \Rightarrow\Leftarrow... |
Worth having a look at is "Harmonic analysis on finite groups" for applications to probability theory (one of your stated strengths). That way you'll be going from what you already know to the new stuff. Google books link is
http://books.google.com/books?id=3trud4weQ3AC&printsec=frontcover&dq=inauthor:%22Tullio+Cecch... |
The [integral test of convergence](http://en.wikipedia.org/wiki/Integral_test_for_convergence) states that, if $f:[1,+\infty)\to[0,+\infty)$ is a monotonically decreasing nonnegative function, then the series $\sum_1^\infty f(n)$ converges iff $\int_1^\infty f(n) dn$ is finite.
Is the high-dimensional generalization... |
I'm doing a physics course in my spare time and the problem set is:
> Ball A is dropped from rest from a building of height H exactly as ball B is thrown up vertically from the ground. When they collide A has twice the speed of B. If the collision occurs at height h, what is h/H?
It also gives a hint: Write equat... |
Newtonian Physics: ball A is dropped, ball B is thrown - where do they coincide? |
I'm doing a physics course in my spare time and the problem set is:
> Ball A is dropped from rest from a building of height H exactly as ball B is thrown up vertically from the ground. When they collide A has twice the speed of B. If the collision occurs at height h, what is h/H?
It also gives a hint: *Write equa... |
I was reading a comment on MathOverflow and did not understand what was being suggested. I also don't understand Anton's response:
http://mathoverflow.net/questions/17732/difference-between-measures-and-distributions/18062#18062
I initially thought Regenbogen's comment was being applied to the derivative of the d... |
Is $$\left |\int_1^N e^{ix^\alpha} \ \text{dx} \right| = \mathcal{O}(N^\alpha)?$$
where $\displaystyle 0\lt \alpha \lt 1$ and $\displaystyle i=\sqrt{-1}$.
${\bf Edit.}$ As Willie Wong shows below, the correct equation is
$$\left |\int_1^N e^{ix^\alpha} \ \text{dx} \right| = \mathcal{O}(N^{1-\alpha}).$$ |
Couldn't one just do a u-substitution with u=g-2v? |
I have found the following ellipse representation $(x,y)=(x_0\cos(\theta+d/2),y_0\cos(\theta-d/2))$, $\theta \in [0,2\pi]$. This is a countour of bivariate normal distribution with uneven variances and correlation $\rho=\cos(d)$. I know that this is a rotated ellipse with centre $(0,0)$. How to find the lengths of majo... |
How may one go about proving
$\displaystyle\frac{\Gamma'(s)}{\Gamma(s)}=O(\log|s|)$
directly? By a direct proof, I mean not to go through the usual Stirling formula with the exact error term. The use of a rough form of Stirling's formula is welcome. |
How may one go about proving
$\displaystyle\frac{\Gamma'(s)}{\Gamma(s)}=O(\log|s|)$,
(away from the poles) directly? By a direct proof, I mean not to go through the usual Stirling formula with the exact error term. The use of a rough form of Stirling's formula is welcome. |
**HINT** $\ $ Show that $\rm (a^{n+1}-1)/(a-1)\ $ and $\rm a^n+a^{n-1}+\:\cdots\:+1\ $ are both solutions of the recurrence $\rm\ f(n+1)\ = (a+1)\ f(n-1) - a\ f(n-2)\ $ with identical initial conditions $\rm f(1),\ f(0)\:$. Now use strong induction to prove the **uniqueness theorem** for solutions of such difference eq... |
**HINT** $\ $ Show that $\rm (a^{n+1}-1)/(a-1)\ $ and $\rm a^n+a^{n-1}+\:\cdots\:+1\ $ are both solutions of the recurrence $\rm\ f(n+1)\ = (a+1)\ f(n-1) - a\ f(n-2)\ $ with identical initial conditions $\rm f(1),\ f(0)\:$. Now use strong induction to prove the **uniqueness theorem** for solutions of such difference eq... |
**HINT** $\ \ $ Show that $\rm\ (a^{n+1}-1)/(a-1)\ $ and $\rm\ a^n+a^{n-1}+\:\cdots\:+1\ $ are both solutions of the recurrence $\rm\ f(n+2)\ = (a+1)\ f(n+1) - a\ f(n)\ $ with identical initial conditions $\rm\ f(1),\ f(0)\:.\ \ \ $
Now use strong induction to prove the **uniqueness theorem** for solutions of such d... |
I just started real analysis. I don't have a background in proofs or logic, simply calculus. So I'm trying to learn more about proofs--so forgive the basic question, please.
How do you go about proving this theorem: If a and b are any numbers, then there is one and only one number x such that a + x=b. This number... |
For improper single integrals with positive integrands of the first, second or mixed type there are the comparison and the limit tests to determine their convergence or divergence. There is also the absolute convergence theorem.
For multiple integrals I know that the comparison test can be used as well.
**Qu... |
Let $\kappa$ be a measurable cardinal, we say that $\mathcal{D}$ is a normal ultrafilter iff whenever $g\in\kappa^\kappa$ such that $g<_\mathcal{D} Id$, we have some $\alpha<\kappa$ such that $\{\gamma<\kappa | g(\gamma) = \alpha\}\in\mathcal{D}$.
Given $\kappa$ a measurable cardinal and some $M$ which is a transiti... |
Edit: Here is a more verbose proof. The original proof is retained below.
<h3>Introduction</h3>
Let $\Sigma$ be some alphabet with $n$ symbols. Let $L$ be the language of words in which each symbol appears an even number of times. Suppose $G$ is a grammar for $L$. Our goal is to given an exponential lower bound on ... |
I'm looking for an algorithm for the transitive reduction of a digraph (a DAG in fact).
The [wikipedia article](http://en.wikipedia.org/wiki/Transitive_reduction#Graph_algorithms_for_transitive_reduction) displays a formula R<sup>-</sup> = R - (R × R<sup>+</sup>) — where R<sup>-</sup> is the transitive re... |
While combing around my notes looking for other possible examples for [this question][1], I chanced upon another one of my unsolved problems:
Cycloidal curves are curves generated by a circle rolling upon a plane or space curve. It's not too hard to derive the required parametric equations if the fixed curve is a pl... |
A helical cycloid? |
How can I add random "fuzz" to this formula, so that the randomness range diminishes as values increase? |
How do we numerically integrate a rapidly decaying exponential function? A simple Gauss quadrature which is based on approximating the function by polynomial, I think will not work, since rapidly decaying exponential function will require more terms in the series. Is there a series approximation for the function? |
as show above, how do i show an isomorphism ? the method suggested by the book is to defined a map $k$: Hom$(Z,G)$ -> $G$ by $f -> f(1)$. i am stuck on how exactly to show that it is a homomorphism. Could anyone shed some light into this? is f(1) an element of G? thanks.
(For those who wanted to know, this is not h... |
Prove that there is a largest $n$ such that $n \le x$ for any rational $x$.
What about any real number $x$? |
How do I show that $\text{Hom}(\mathbb{Z},G)$ and $G$ are isomorphic?
The method suggested by the book is to define a map $k$: $\text(\mathbb{Z},G) \to G$ by $f \to f(1)$.
I am stuck on how exactly to show that it is an isomorphism. Could anyone shed some light into this? Is $f(1)$ an element of G?
(For those... |
(Sorry to post this as an answer, but I can't comment yet). Do you mean that given a fixed rational number $x$, the set $\lbrace n: n\leq x, n \in \mathbb Z\rbrace$ has a maximum? (Hint: Use the well-ordering principle) |
If we define $x$ = displacement, $v$ = velocity and $a$ = acceleration then I am used to the ideas that $a= \frac{dv}{dt} = \frac{d^2x}{dt^2}$
However I also understand $a=v \frac{dv}{dx}$. Can someone explain to me why this is?
|
Given set of points in 3D ( X = (x1, x2, x3), Y = (y1, y2, y3) ), how can I fit transformation from X to Y?
As far as I know this is called projective transformation.
Here is example of X and Y.
Blue and red lines in X are parallel, but they are not parallel in Y.
![alt text][1]
![alt text][2]
[1]: ht... |
<i>I've revised my question a bit in response to the (very helpful) advice so far--</i>
I have an engineering background but am interested in learning abstract harmonic analysis. My interest is rather unstructured; the Fourier transform sparked my interest in mathematics, and I've been following since I first learne... |
I have this function, $e^{-x}$ bounded between 0 and 1500 and I have an approximation by Taylor Series of the same function bounded between 0 and 0.5. I would like to express my function $e^{-x}$ bounded between 0 and 15oo in terms of the last approximation I described, what is bounded between 0 and 0.5.
Does anyone k... |
Say we have $N$ [$3 \le n \le 100,000$] spheres indexed as $1,2,3,\cdots N$,all of them have identical weight apart from one.We have to determine which sphere it is (index) by using only the pair of scales.
We could solve this problem by weighing repeatedly buy I am interested on weighing as minimum as possible so w... |
Say we have $N$ [$3 \le N \le 100,000$] spheres indexed as $1,2,3,\cdots N$,all of them have identical weight apart from one.We have to determine which sphere it is (index) by using only the pair of scales.
We could solve this problem by weighing repeatedly buy I am interested on weighing as minimum as possible so w... |
In the comments section, the following Dirichlet series came up: the Riemann $\zeta$-function, Dirichlet $L$-functions, and Ramanujans series $\sum_{n \geq 1}\tau(n) n^{-s}$, where $\tau(n)$ is the coefficient of $q^n$ in $\Delta(q) = q\prod_{n=1}^{\infty} (1-q^n)^{24}$.
First note that the $\zeta$-function is a spe... |
For an abelian variety $A_{/\mathbb{Q}}$, its volume $vol(A(\mathbb{R}))$ appears in the conjectured Birch Swinnerton-Dyer formula for the L-series at 1.
I am having trouble in understanding the size of this volume, as a function of the (minimal) equations defining the variety. Hence the question:
Say we have a h... |
***Say we have $N$ [$3 \le N \le 100,000$] spheres indexed as $1,2,3,\cdots N$,all of them have identical weight apart from one.We have to determine which sphere it is (index) by using only the pair of scales.***
We could solve this problem by weighing repeatedly,but I am obviously interested weighing as minimum as ... |
In the comments section to Willie Wong's answer, the following Dirichlet series came up: the Riemann $\zeta$-function, Dirichlet $L$-functions, and Ramanujans series $\sum_{n \geq 1}\tau(n) n^{-s}$, where $\tau(n)$ is the coefficient of $q^n$ in $\Delta(q) = q\prod_{n=1}^{\infty} (1-q^n)^{24}$.
First note that the $... |
Situation A: Once only, I toss 2 identical fair coins and don't look at the outcomes. A truthful observer looks at one of the coins and tells me that at least one of the coins is a head.
Situation B: Once only, I toss 2 identical fair coins and don't look at the outcomes. A truthful observer looks at both of the co... |
2 slightly different situations in which 2 coins are tossed. Does the knowledge of an observer effect the probabilities of the outcomes? |
While combing around my notes looking for other possible examples for [this question][1], I chanced upon another one of my unsolved problems:
Cycloidal curves are curves generated by a circle rolling upon a plane or space curve. It's not too hard to derive the required parametric equations if the fixed curve is a pl... |
Situation A: Once only, I toss 2 identical fair coins and don't look at the outcomes. A truthful observer looks at one of the coins and tells me that at least one of the coins is a head.
Situation B: Once only, I toss 2 identical fair coins and don't look at the outcomes. A truthful observer looks at both of the co... |
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