qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,822,336 | <p>My friend asked me what the roots of $y=x^3+x^2-2x-1$ was.</p>
<p>I didn't really know and when I graphed it, it had no integer solutions. So I asked him what the answer was, and he said that the $3$ roots were $2\cos\left(\frac {2\pi}{7}\right), 2\cos\left(\frac {4\pi}{7}\right)$ and $2\cos\left(\frac {8\pi}{7}\ri... | egreg | 62,967 | <p>Set $x=t+t^{-1}$. Then the equation becomes
$$
t^3+3t+3t^{-1}+t^{-3}+t^2+2+t^{-2}-2t-2t^{-1}-1=0
$$
and, multiplying by $t^3$,
$$
t^6+t^5+t^4+t^3+t^2+t+1=0
$$
and it should be now clear what the solutions are. For each root there's another one giving the same solution in $x$.</p>
|
3,001,335 | <p>I know little formal math terminology and don't understand much of anything about complex analysis. Also, if this isn't a good starting point for complex integration feel free to say (I'm learning about it partly for Cauchy's residue theorem). </p>
<p>My first and intuitive idea of residue has to do with remainder,... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$a\in\mathbb C$</span>, <span class="math-container">$r\in(0,\infty)$</span>, <span class="math-container">$f\colon B_r(a)\setminus\{a\}\longrightarrow\mathbb C$</span> is an analytic function, and <span class="math-container">$\gamma\colon[a,b]\longrightarrow B_r(a)\setminus\{a\}$</s... |
2,734,338 | <p>I would like to show that
$$\forall n\in\mathbb{N}^*, \quad \sqrt{\frac{n}{n+1}}\notin \mathbb{Q}$$</p>
<p>I'm interested in more ways of proofing this.</p>
<p>My method :</p>
<p>suppose that $\sqrt{\frac{n}{n+1}}\in \mathbb{Q}$ then there exist $(p,q)\in\mathbb{Z}\times \mathbb{N}^*$ such that $\sqrt{\frac{n}... | rtybase | 22,583 | <p>From <a href="https://en.wikipedia.org/wiki/Rational_root_theorem" rel="nofollow noreferrer">RRT</a> for $P(x)=(n+1)x^2-n=0$, if $x=\frac{p}{q}, \gcd(p,q)=1$ is a solution for P(x), then $\color{green}{p\mid n}$ and $\color{red}{q\mid n+1}$. In this particular case it's even $\color{green}{p^2\mid n}$ and $\color{re... |
4,508,796 | <p>How to find the integral
<span class="math-container">$$\int_0^1 x\sqrt{\frac{1-x}{1+x}}dx$$</span></p>
<p>I tried by substituting <span class="math-container">$x=\cos a$</span>. But it's leading to a form <span class="math-container">$\sin2a\cdot\tan a/2$</span> which I can't integrate further.</p>
| MathDona | 1,084,986 | <p><span class="math-container">$$I=\int_{0}^{1} x \sqrt{\frac{1-x}{1+x}}dx$$</span>
Let <span class="math-container">$x=\cos 2t$</span>, then
<span class="math-container">$$I=-2\int_{\pi/4}^{0} \cos 2t \sin 2t \tan t dt=2\int_{0}^{\pi/4} \left[ \cos 2t-\cos^2 2t\right] dt$$</span> <span class="math-container">$$=2\int... |
4,508,796 | <p>How to find the integral
<span class="math-container">$$\int_0^1 x\sqrt{\frac{1-x}{1+x}}dx$$</span></p>
<p>I tried by substituting <span class="math-container">$x=\cos a$</span>. But it's leading to a form <span class="math-container">$\sin2a\cdot\tan a/2$</span> which I can't integrate further.</p>
| dan_fulea | 550,003 | <p>Observe that the given integral is an integral over <span class="math-container">$\sqrt{1-x^2}$</span> times a rational function of <span class="math-container">$x$</span>, so any of the three <a href="https://en.wikipedia.org/wiki/Euler_substitution" rel="nofollow noreferrer">Euler substitutions</a>
leads to an int... |
227,873 | <p>I am looking for robust generalizations of matrix rank.</p>
<p>Think of the the following problem: A big matrix of low rank is perturbed by random noise, such that it becomes a full-rank matrix. Is there a generalization of matrix rank that still 'sees' that the perturbed matrix is close to a low-rank matrix?</p>
| Vahan | 28,157 | <p>You can perform Principle Component Analysis and consider the rank of the resulting matrix. This is similar to what Sebastian Goette suggested.</p>
|
3,991,072 | <p>Consider, two planar vectors:</p>
<p><span class="math-container">$$V= a \hat{x} + b \hat{y}$$</span></p>
<p>And
<span class="math-container">$$ U = a' \hat{x} + b' \hat{y}$$</span></p>
<p>These are analogous to the complex numbers:</p>
<p><span class="math-container">$$ v = a + bi$$</span></p>
<p>and,</p>
<p><span... | Mustafa Said | 90,927 | <p>For planar vectors we can indeed multiply and divide.</p>
<p>First identify <span class="math-container">$V = a \hat{x} + b \hat{y}$</span> with <span class="math-container">$(a, b)$</span> and</p>
<p><span class="math-container">$U = c \hat{x} + d \hat{y} $</span> with <span class="math-container">$(c, d)$</span></... |
3,445,768 | <p>I am trying to solve a nonlinear differential equation of the first order that comes from a geometric problem ; <span class="math-container">$$x(2x-1)y'^2-(2x-1)(2y-1)y'+y(2y-1)=0.$$</span></p>
<p>edit1 <strong><em>I am looking for human methods to solve the equation</em></strong> </p>
<p>edit2 the geometric pr... | Robert Israel | 8,508 | <p>Maple finds the solutions</p>
<p><span class="math-container">$$y(x) = \frac12,\ y(x) =\frac12-x,\ y(x) = \sqrt {(c-c^2 )(2\,x-1)}-cx+c $$</span></p>
|
3,445,768 | <p>I am trying to solve a nonlinear differential equation of the first order that comes from a geometric problem ; <span class="math-container">$$x(2x-1)y'^2-(2x-1)(2y-1)y'+y(2y-1)=0.$$</span></p>
<p>edit1 <strong><em>I am looking for human methods to solve the equation</em></strong> </p>
<p>edit2 the geometric pr... | JJacquelin | 108,514 | <p>This is not an answer to the question but a complement to the Robert Israel's answer. It was not possible to edit it in the comments section. </p>
<p><span class="math-container">$$y(x) = \sqrt {(c-c^2 )(2\,x-1)}-cx+c \tag 1$$</span>
is not the complete set of solutions. One must not forget
<span class="math-con... |
972,530 | <blockquote>
<p>If $S_1 = \sqrt{2}$, and</p>
<p>$S_{n+1} = \sqrt{2 + \sqrt{S_n}}$ (n = 1,2,3....),</p>
<p>prove that $\{S_n\}$ converges, and that $S_n < 2$ for all $n \in \Bbb{N}$</p>
<p>This is one the questions from Principles of Mathematical Analysis by Rudin. I am not sure how to proceed with t... | Paul | 17,980 | <p>Hints:</p>
<p><strong>1)</strong> $S_n < 2.$ Clearly, $S_1<2$. Suppose that $S_k <2$. Then $S_{k+1}=\sqrt{2 + \sqrt{S_k}} <\sqrt{2 + 2}=2.$</p>
<p><strong>2)</strong> $S_n \le S_{n+1}$. Clearly, $S_1<S_2$. Suppose that $S_{k-1} <S_k$. Then $$S_{k}=\sqrt{2 + \sqrt{S_{k-1}}} <\sqrt{2 +\sqrt{S_{... |
381,067 | <p>I am asked this question:</p>
<blockquote>
<p>Prove that $x^5 - 5x + 1$ has no double roots in $\mathbb C[x]$. </p>
</blockquote>
<p>Now here's what I said:</p>
<blockquote>
<p>$p(x) \in \mathbb C[x]$ has no double roots if and only if $gcd(p,p') = 1$. Now we need to prove that. So:
$p(x) = x^5 - 5x + 1$ a... | Community | -1 | <p>In the Euclidean algorithm, you divide the larger by the smaller. A cubic polynomial is larger than a linear polynomial.</p>
<p>Computing gcds by finding the prime factorization is often a feasible alternative. In this setting, prime factorization means "factor", and is almost synonymous with finding its roots.</p>... |
2,003 | <p>I use some custom shortcut keys in <code>KeyEventTranslations.tr</code>. One is for the <code>Delete All Output</code> function: </p>
<pre><code>Item[KeyEvent["w", Modifiers -> {Control}],
FrontEnd`FrontEndExecute[FrontEnd`FrontEndToken["DeleteGeneratedCells"]]]
</code></pre>
<p>or simply:</p>
<pre><code>... | István Zachar | 89 | <p>Under version 10 at least for the <strong>Delete All Output</strong> menuoption one doesn't have to hit <kbd>Enter</kbd> any more to make it effective. This is not a full answer but it certainly makes my life one keystroke easier.</p>
<p>This now works without putting up a confirmation dialog:</p>
<pre><code>Front... |
629,989 | <p>Given function sequence $\{f_n(x)\}^\infty$ defined as $f_n(x) = \frac{nx}{2 + n + x}. (0 \le x \le 1)$</p>
<p>I need to find the limit function and whether it converges uniformly or not uniformly.</p>
<p>I found that the limit is:</p>
<p>$$\lim_{n \to \infty} \frac{nx}{2 + n + x} = \lim_{n \to \infty} \frac{x}{... | Haha | 94,689 | <p>$f_n$ are differentiable . Then use the fact that $f_n(x)\to x$ uniformly iff $\|f_n(x)-x\|_{\infty}\to 0$.You can find the maximum of $f_n(x)-x$ in terms of $n$ and show that the maximum goes to $0$ for $n\to \infty$.</p>
|
439,745 | <blockquote>
<p>Prove:$|x-1|+|x-2|+|x-3|+\cdots+|x-n|\geq n-1$</p>
</blockquote>
<p>example1: $|x-1|+|x-2|\geq 1$</p>
<p>my solution：(substitution)</p>
<p>$x-1=t,x-2=t-1,|t|+|t-1|\geq 1,|t-1|\geq 1-|t|,$</p>
<p>square,</p>
<p>$t^2-2t+1\geq 1-2|t|+t^2,\text{Since} -t\leq -|t|,$</p>
<p>so proved.</p>
<p><em>ques... | Elias Costa | 19,266 | <p>After the answer of Thomas Andrews my answer appears to be dispensable. But I'll leave it here as a poor alternative. Here a proof valid for any $ x $ since $ n $ is even. For the case $ n $ odd simply discard the parcel | x-n |.
\begin{align}
|x-1|+|x-2|+|x-3|+\ldots+|x-n|
=
&
|x-1|+|2-x|+|x-3|+
\\
+
&
\ld... |
1,311,466 | <p>My concept of real no. Is not very clear. Please also tell the logic behind the question.
The expression is true for 19, is it true for all the multiples? </p>
| DeepSea | 101,504 | <p>Observe that : there are $n$ numbers which are both divisible by $2$ and $3$ that appear in the product of $(3n)! = 1\cdot 2\cdot 3\cdots (3n)$ are: $2,4,6,...,2n$, and $3,6,9,...,3n$. Thus the product is divisible by both $3^n$ and $2^n$, and since $(2^n,3^n) = 1$, we have $6^n \mid (3n)!$</p>
|
1,569,728 | <p>Find one $z\in \mathbb{C}$ in the inequality $|z-25i|\le 15$ that has the largest argument ($\arg (z)$)</p>
<p>The inequality is equivalent to $x^2+(y-25)^2\le 15^2$ that represents the set of points in the circle of radius $15$ and center coordinate $C(0,25)$.</p>
<p>In this set, how to find one complex number wh... | Piquito | 219,998 | <p>It is clear the largest argument corresponds to the red tangent below. The involved triangle is $5$ times the pythagorean one of sides $(3,4,5)$ i.e. of sides $(15,20,25)$. Thus $$z=-12+16i$$</p>
<p><a href="https://i.stack.imgur.com/lSzkg.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/lSzkg.png... |
4,090,259 | <p>I began watching Gilbert Strang's lectures on Linear Algebra and soon realized that I lacked an intuitive understanding of matrices, especially as to why certain operations (e.g. matrix multiplication) are defined the way they are. Someone suggested to me 3Blue1Brown's video series (<a href="https://youtube.com/play... | CoveredInChocolate | 901,866 | <p>Not sure if I am addressing what you really are after, but I always like considering simple examples if there is something I don't understand. Using the same matrix in two different contexts. Here is an example of a linear transformation.
<span class="math-container">$$
A =
\begin{bmatrix}
2 & 0 \\
0 & 1
\e... |
1,014,476 | <p>I pick 6 cards from a set of 13 (ace-king). If ace = 1 and jack,queen,king = 10 what is the probability of the sum of the cards being a multiple of 6? </p>
<p><strong>Tried so far:</strong>
I split the numbers into sets with values:
6n, 6n+1, 6n+2, 6n+3
like so:</p>
<p>{6}{1,7}{2,8}{3,9}{4,10,j,q,k}{5}</p>
<p>and... | Ross Millikan | 1,827 | <p>Your combinations are $4\cdot 4+1\cdot 0+1\cdot 2, 2\cdot 1+2\cdot2+2\cdot3, 4 \cdot 4+2\cdot 1,+$ something that doesn't make sense because there are three places you choose from $1$ and only $0,5$ qualify. The left number is the number of cards of that value $\pmod 6$ . You could also have $3\cdot 4+1 \cdot 5+1 ... |
1,014,476 | <p>I pick 6 cards from a set of 13 (ace-king). If ace = 1 and jack,queen,king = 10 what is the probability of the sum of the cards being a multiple of 6? </p>
<p><strong>Tried so far:</strong>
I split the numbers into sets with values:
6n, 6n+1, 6n+2, 6n+3
like so:</p>
<p>{6}{1,7}{2,8}{3,9}{4,10,j,q,k}{5}</p>
<p>and... | Satish Ramanathan | 99,745 | <p>Answer:<img src="https://i.stack.imgur.com/lcFvj.png" alt="enter image description here"></p>
<p>Repetition of 10 is not the same as they represent different cards. I have done it through brute force. Hopefully I have captured everything.</p>
<p>Good luck</p>
<p>Satish</p>
|
2,417,356 | <p>I found the following very nice post yesterday which presented the conditional expectation in a way which I found intuitive;</p>
<p><a href="https://math.stackexchange.com/questions/1492306/conditional-expectation-with-respect-to-a-sigma-algebra?noredirect=1&lq=1">Conditional expectation with respect to a $\sig... | Sangchul Lee | 9,340 | <p>Assume that $X \geq 0$. Then $\nu(E) = \int_{E} X \, d\mathbb{P}$ defined a measure on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Of course, its Radon-Nikodym derivative is simply $d\nu/d\mathbb{P} = X$.</p>
<p>Now we restrict $\nu$ to the space space $(\Omega, \mathcal{G}, \mathbb{P}|_{\mathcal{G}}... |
1,068,719 | <p>The problem is to prove that the quintic
$$x^5+10x^4+15x^3+15x^2-10x+1$$
is irreducible in the rationals. </p>
<p>I don't have much knowledge in group theory, and certainly not in Galois theory, and I'm pretty sure this problem can be solved without those tools. </p>
<p>I know about Eisenstein's criterion, but it ... | Kaj Hansen | 138,538 | <p><strong>Hint</strong>:</p>
<p>First, prove that $f(x)$ is irreducible over a field $F$ $\iff$ $f(x+c)$ is also irreducible over $F$ for any $c \in F$.</p>
<p>Given this result, note that $f(x-1) = x^5 + 5x^4 - 15x^3 + 20x^2 - 30x + 20$.</p>
|
2,419,753 | <p>How would you approach to solve questions like:</p>
<p>$\sin(z)=\sin(2)$, $z$ is an arbitrary complex number.</p>
| Robert Israel | 8,508 | <p>By the laws of exponents
$$x^{10n} x^5 \left(x^{-5}\right)^n = x^{10 n + 5 - 5 n} = x^{5 + 5 n}$$
so you want
$$ x^{5+5n} = x^{-10}$$
Assuming $x$ is not $0$ or $\pm 1$, $x^t$ is a one-to-one function of $t$, so this will
imply $$5 + 5 n = -10$$</p>
|
4,638,393 | <p><a href="https://i.stack.imgur.com/SgVlq.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/SgVlq.png" alt="enter image description here" /></a></p>
<p>How does the author obtain formula (4)? From formula (2), I only get that <span class="math-container">$u\left(\frac{k}{n}+\frac{1}{n},\cdot\right)=\... | Abolfazl Chaman motlagh | 938,462 | <p>for one step backward it is what you said:
<span class="math-container">$$
u(t + \frac{1}{n},.) \approx (1-\frac{c}{n} \partial_x)u(t,.)
$$</span>
so if you start from <span class="math-container">$t=0$</span> and use this approximation <span class="math-container">$k$</span> times each time with step size <span cla... |
36,735 | <p>In Peter J. Cameron's book "Permutation Groups" I found the following quote</p>
<blockquote>
<p>It is a slogan of modern enumeration theory that the ability to count a set is closely related to the ability to pick a random element from that set (with all elements equally likely).</p>
</blockquote>
<p>Indeed, one... | Robin Kothari | 8,075 | <p>Yes, there is formal way of saying this using complexity theory. I think the statement is something like: For all self-reducible relations, the problems of approximate sampling and approximate counting are equivalent (with polynomial time reductions). More specifically, for such problems, the existence of an FPRAS (... |
3,192,068 | <p><span class="math-container">\begin{matrix}
1 & 2 & 0 & 1 \\
2 & 4 & 1 & 4 \\
3 & 6 & 3 & 9 \\
\end{matrix}</span>
I have tried to transpose it and then reduce it by row echelon form and i get zeros on the last two rows. But i can't grasp if i should be doing that ... | Deepak | 151,732 | <p>Just to explain both my comment and Kavi Rama Murthy's,</p>
<p>The distance is equal to the arc length of the curve. I hope you can see this. Just sketch out any arbitrary curve and pretend that it's a particle (or a bumblebee!) moving around on the Cartesian plane. The arc length of the curve is the distance trave... |
1,809,022 | <p>I've been working on some very basic differential equations, but I came to a
problem where I need to figure out the behavior of $y(t)$ as $t \rightarrow
\infty$ Given that
$$\frac{dy}{dt} = \frac{3t}{1+2e^{y}}.$$
In this case, it was very apparent to me that I would not be able to solve for
a simple solution of $y(t... | Doug M | 317,162 | <p>since $y'>0$ when $t>0.$ $y(t)$ is constantly increasing for all $t>0$</p>
<p>If $y(t)$ were bounded $y'(t)$ would either have to oscillate around $0,$ or the limit as $t$ goes to infinity of $y(t)$ would equal $0.$</p>
|
1,809,022 | <p>I've been working on some very basic differential equations, but I came to a
problem where I need to figure out the behavior of $y(t)$ as $t \rightarrow
\infty$ Given that
$$\frac{dy}{dt} = \frac{3t}{1+2e^{y}}.$$
In this case, it was very apparent to me that I would not be able to solve for
a simple solution of $y(t... | Pipicito | 93,689 | <p>I will give you a precise proof describing the behaviour of $y(t)$ when $t$ goes to $+\infty$. Observe that no matter what your initial condition at $t=0$ is, you will have $y'(t)>0$ for $t>0$. You get this information by checking the sign of the right hand side of $$\frac{dy}{dt} = \frac{3t}{1+2e^{y}}$$</p>
... |
3,547,384 | <p>I saw this equation<span class="math-container">$$S(q)=\int_a^bL(t,q(t),\dot q(t))dt$$</span>
in <a href="https://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation" rel="nofollow noreferrer">wikipedia</a>.</p>
<p>So I would think that <span class="math-container">$f(x,y)$</span> must be equal to <span class="ma... | Arthur | 15,500 | <blockquote>
<p>Because it is always correct that <span class="math-container">$\cfrac{df}{dt}=\cfrac{\partial f}{\partial t}$</span>,</p>
</blockquote>
<p>No, that's not correct, at all. At least not in this context.</p>
<p><span class="math-container">$\dfrac{\partial f}{\partial t}$</span> means "the partial der... |
2,646,251 | <p>Given the linear system in $\mathbb{Z_3}$:</p>
<p>$$
\left\{
\begin{array}{c}
a+b+c+d=1 \\
b+c+e=2 \\
a+2e=0
\end{array}
\right.
$$</p>
<p>I used the row reduction with matrices and I got:</p>
<p>$$
\left\{
\begin{array}{c}
a+b+c+d=1 \\
b+c+e=2 \\
d=2
\end{array}
\right.
$$</p>
<p>But now I don't know ho... | Bernard | 202,857 | <p>I'd use the symbols $0,1,-1$ for the elements of $\mathbf Z/3\mathbf Z$. Here's how to put the augmented matrix in reduced row echelon form:
\begin{align}
&\begin{bmatrix}
1&1&1&1&0&\hspace{-0.6em}|\phantom{-}~1\\
0&1&1&0&1&\hspace{-0.6em}|\:{-}1 \\1&0&0&0&... |
1,662,090 | <p>First of all, hi. I am new here.</p>
<p>Let$$X_1,\dots, X_n$$
be i.i.d. exponential random variables.</p>
<p>$$
Pr({\max X_n}>{(\sum X_n-\max X_n ) }) = ?
$$</p>
<p>I think we should take integrals on exponential distribution functions over corresponding intervals but I could not make it work.</p>
<p>Thanks.<... | stity | 285,341 | <p>$75583$ in base $9$ is $50052=2^2*3*43*97$ in base 10</p>
<p>The unknown base $b$ is at least $5$ since $402$ start with a $4$</p>
<p>If $b>10$, $402_b*302_2 \geq (11*11*4+2)*(11*11*3+2)=177390 > 50052$ so $b\leq 10$</p>
<p>$402_b =2 \mod(b)$ and $302_b=2 \mod(b)$ so $75583_9=50052_{10}=4\mod(b)$</p>
<p>$$... |
450,785 | <p>I want to obtain the formula for binomial coefficients in the following way: elementary ring theory shows that $(X+1)^n\in\mathbb Z[X]$ is a degree $n$ polynomial, for all $n\geq0$, so we can write</p>
<p>$$(X+1)^n=\sum_{k=0}^na_{n,k}X^k\,,\ \style{font-family:inherit;}{\text{with}}\ \ a_{n,k}\in\mathbb Z\,.$$</p>
... | Marc van Leeuwen | 18,880 | <p>If you define $a_{n,k}=\binom nk$ to be the coefficient of $X^k$ in $(1+X)^n$ (with no combinatorial meaning attached), then you easily find $\binom nk=\binom {n-1}{k-1}+\binom{n-1}k$ for all $n,k\geq1$ (as you indicated in the question). Also $\binom n0=1$ for all $n\geq0$, and $\binom0k=0$ for all $k>0$. Expand... |
1,135,045 | <p>I need to compute
\begin{align}
S = \sum_{k=-\infty}^j \sum_{m=-1}^2 w_{k,m} f_{k+m-1}
\end{align}
but I only want to access the elements of $f$ once, so I would prefer something like
\begin{align}
\sum_k f_k \sum_m ...
\end{align}
Here is what I did: substitute $l=m-1+k$ to get
\begin{align}
S &= \sum_{k=-\inf... | PdotWang | 212,686 | <p>Let us assume that:
$$S_l=\sum_{l=-\infty}^{j+1}N_{l} \cdot f_{l}$$
Then we need to figure out what the $N_l$ is.
$$N_l=\sum_{k=-\infty}^{j}\sum_{m=-1}^{2} \omega_{k,m}=\sum_{m=-1}^{2}\sum_{k=-\infty}^{j} \omega_{k,m}$$
Under the condition of
$$l=k+m-1$$
Therefor,
$$k=l-m+1$$
$$N_l=\sum_{m=-1}^{2}\omega_{l-m+1,m}$... |
2,634,791 | <blockquote>
<p>How can I show that the map $f: GL_n(\mathbb R)\to GL_n(\mathbb R)$ defined by $f(A)=A^{-1}$ is continuous?</p>
</blockquote>
<p>The space $GL_n(\mathbb R)$ is given the operator norm and so I want to show for all $\epsilon$ there exists $\delta$ such that $\|A-B\|<\delta \implies \|A^{-1}-B^{-1}\... | caffeinemachine | 88,582 | <p>One can use the Cayley-Hamilton Theorem, which states that if $$p(x)=\det(xI-M)= x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$$ is the characteristic polynomial of $M$, then </p>
<p>$$M^n+a_{n-1}M^{n-1}+\cdots+a_1M+a_0I=0$$.</p>
<p>If $M\in GL_n(\mathbf R)$, then note that $a_0$ above cannot be $0$, for otherwise the determ... |
4,517,429 | <p>Let <span class="math-container">$ {\textstyle \{X_{1},\ldots ,X_{n},\ldots \}}$</span> be a sequence of independent random variables, each of those random variable follow a Gamma distribution.</p>
<p>For the summation of those random variable:</p>
<p><span class="math-container">$ {\displaystyle {\bar {X}}_{n}\equi... | Harry | 676,771 | <p>Well, CLT doesn't tell sum is Normal. It tells the normalized average <span class="math-container">$(\bar X_n-n\mu)/\sqrt{n}\,\sigma$</span> has an asymptotic distribution that is normal. The sum may be Gamma or not depending on the parameters.</p>
|
2,232,095 | <blockquote>
<p>Let $a, b, c, p, q$ be real numbers. Suppose $\{α, β\}$ are the roots
of the equation $x^2 + 2px+ q = 0$ and $\{α,\frac{1}{β}\}$ are the
roots of the equation $ax^2 + 2bx+ c = 0$, where $β \notin \{−1, 0, 1\}$.</p>
</blockquote>
<p>STATEMENT-1 : $(p^2 − q)(b^2 − ac) ≥ 0$</p>
<p>STATEMENT 2: $b \... | Miguel | 259,671 | <p><strong>Statement 1</strong></p>
<p>Note that you can write a second order equation with its roots as:
$$x^2+2px+q=(x-\alpha)(x-\beta)$$
so that $2p=-\alpha-\beta$ and $q=\alpha\beta$. Doing the same with the second equation:
$$a x^2+2bx+c=a(x-\alpha)(x-\frac{1}{\beta})$$
yields $c=a\frac{\alpha}{\beta}$ and $2b=a(... |
13,843 | <p>We have a natural number $n>1$. We want to determine whether there exist
natural numbers $a, k>1$ such that $n = a^k$. </p>
<p>Please suggest a polynomial-time algorithm.</p>
| Community | -1 | <p>In order to test whether or not a natural number <span class="math-container">$n$</span> is a perfect power, we can conduct a binary search of the integers {1,2,...,n} for a number <span class="math-container">$m$</span> such that <span class="math-container">$n = m^b$</span> for some <span class="math-container">$b... |
13,843 | <p>We have a natural number $n>1$. We want to determine whether there exist
natural numbers $a, k>1$ such that $n = a^k$. </p>
<p>Please suggest a polynomial-time algorithm.</p>
| Stefan Kohl | 28,104 | <p>The computer algebra system <a href="http://www.gap-system.org" rel="nofollow">GAP</a> performs this test and determines a
smallest root $a$ of a given integer $n$ quite efficiently.
The following is copied directly from its source code (file gap4r6/lib/integer.gi),
and should be self-explaining:</p>
<pre><code>###... |
3,055,365 | <p>While doing my research, I came across this integral and don't know how to solve for this:
<span class="math-container">$$\int_{0}^{\infty}x^2\exp\{ax-be^{ax}\}dx,\text{where $a,b>0$}.$$</span>
My attempt:
<span class="math-container">\begin{align}
\int_{0}^{\infty}x^2\exp\{ax-be^{ax}\}dx &\overset{x = \ln ... | Ininterrompue | 622,553 | <p>The integral you call <span class="math-container">$A$</span> can be split into two integrals.</p>
<p><span class="math-container">$$ A = \frac{1}{a^{3}}\int_{1}^{\infty}\ln^{2}t\,e^{-bt}\,\mathrm{d}t = \frac{1}{a^{3}}\left[\int_{0}^{\infty}\ln^{2}t\,e^{-bt}\,\mathrm{d}t - \int_{0}^{1}\ln^{2}t\, e^{-bt}\,\mathrm{d}... |
3,055,365 | <p>While doing my research, I came across this integral and don't know how to solve for this:
<span class="math-container">$$\int_{0}^{\infty}x^2\exp\{ax-be^{ax}\}dx,\text{where $a,b>0$}.$$</span>
My attempt:
<span class="math-container">\begin{align}
\int_{0}^{\infty}x^2\exp\{ax-be^{ax}\}dx &\overset{x = \ln ... | Zachary | 433,146 | <p>Starting with <span class="math-container">$A$</span>, and knowing that the constant of proportionality is <span class="math-container">$a^{-3}$</span>, we can see that
<span class="math-container">\begin{align}
A&=a^{-3}\int_1^\infty \log^2 x\, e^{-bx}\,dx \\
&= a^{-3}\frac{\partial^2}{\partial \mu^2} \int_... |
1,212,336 | <p>Let R be the set of all real numbers. Is $\{\mathbb R^+,\mathbb R^−,\{0\}\}$ a partition of $\mathbb R$? Explain your answer.</p>
<p>My answer is no because of $\{0\}$. I am confused with $\{0\}$. please help.</p>
| robjohn | 13,854 | <p>Consider that $\log(x)$ is an increasing function and that $\log(1)=0$, then look at the following plot:</p>
<p><img src="https://i.stack.imgur.com/rozpp.png" alt="enter image description here"></p>
<p>Normally, if $f(x)$ is an increasing function, then, for $a,b\in\mathbb{Z}$,
$$
\sum_{k=a+1}^bf(k)\ge\int_a^bf(x)... |
656,701 | <p>Suppose we have:</p>
<p>$A = \{(x,v,w):x+v=w\}$</p>
<p>$B = \{(x,v):x=v\}$</p>
<p>$C = \{(w,u):\exists x 2x=w\}$</p>
<p>Can we say that $C = A \cup B$?</p>
| André Nicolas | 6,312 | <p>The question has been modified a bit from the previous one. The replacement of $-$ by $=$ makes sense. However, in the definition of $C$, you had
$C=\{w: \exists x(2x=w)\}$, and that should have been kept. </p>
<p>Presumably the set that our variables range over is the set of natural numbers, or the set of integer... |
74,347 | <blockquote>
<p>Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$.</p>
</blockquote>
<p>This question is just after the definition of differentiation and the theorem that if $f$ is finitely derivable at $c$, then $f$ is also continuous at $c$. Please help, my textbook does not h... | user7530 | 7,530 | <p>$|x|$ is continuous, and differentiable everywhere except at 0. Can you see why?</p>
<p>From this we can build up the functions you need: $|x-2| + |x-3| + |x-4|$ is continuous (why?) and differentiable everywhere except at 2, 3, and 4.</p>
|
290,050 | <p>Are there good lower/upper bounds for
$ \sum\limits_{i = 0}^k {\left( \begin{array}{l} n \\ i \\ \end{array} \right)x^i } $ where $0<x<1$, $k \ll n$?</p>
| Brendan McKay | 9,025 | <p>Write $x=p/(1-p)$ and then
$$ \sum_{i=0}^k \binom ni x^i = (1-p)^{-n}\sum_{i=0}^k \binom ni p^k(1-p)^{n-k}.$$
The last sum is the cumulative binomial distribution, which has no exact formula (except as a special function) but a large literature on bounds. It is quite a common topic on Mathoverflow, see these for ex... |
1,290,516 | <p>Find the values of $m$ if the line $y=mx+2$ is a tangent to the curve $x^2-2y^2=1$.</p>
<p>My working:</p>
<p>First we differentiate $x^2-2y^2=1$ with respect to $y$ to get the gradient. We get $y^2=\frac{1}{2}x^2-\frac{1}{2}\implies y=\pm\sqrt{\frac{1}{2}x^2-\frac{1}{2}}$.</p>
<p>We take the positive one for dem... | Community | -1 | <p>Let the given line be tangent to the curve at the point $(x_0, y_0).$ Then we have $$y_0 = m x_0 + 2, \\ x_0^2 = 1 + 2y_0^2.$$ Using the fist equation in the second, we have $$x_0^2 = 1 + 2 (m x_0^2 + 4 m x_0 + 4),$$ which is $$m = \frac {x_0^2 - 9} {2 x_0^2 + 8 x_0}.$$ We also know from the second equation that $$y... |
1,131,622 | <p>The question itself is a very easy one:<br/></p>
<blockquote>
<p>Somebody has got two kids, one of whom is a girl. Then what's the probability that he's got <strong>at least</strong> one boy?</p>
</blockquote>
<p>My answer is that, since he's already got a girl, then "he's got at least one boy" amounts to "the o... | KSmarts | 192,747 | <p>All of the answers so far explain why you are wrong. However, strictly speaking, you are <strong>not</strong> wrong. You are simply making different assumptions and interpreting the question differently than other people.</p>
<p>The question tells us that someone has (exactly) two children, (at least) one of whom i... |
29,115 | <p>I just read a proof and, after struggling some time with a mental leap, I think that it uses tacitly the following:</p>
<p>Let $\kappa$ be a regular cardinal, $\theta > \kappa$ a regular cardinal too then:
$ S \subset \kappa$ is stationary if and only if
$\forall \mathcal{A} = (H(\theta), \in, <,..) \exists... | Philip Welch | 6,942 | <p>(I first wanted to give an answer, but I was not quick enough. I then wanted to add a small comment and found out after 20 minutes that I had insufficient reputation.)</p>
<p>The comment was regarding 2) of oktan's original query: having $H(\theta)$
in the structure is overkill: it suffices to have $( \kappa, <,... |
1,392,576 | <p>How can the followin question be solved algebraically?</p>
<p>A certain dealership has a total of 100 vehicles consisting of cars and trucks. 1/2 of the cars are used and 1/3 of the trucks are used. If there are 42 used vehicles used altogether, how many trucks are there?</p>
| Anurag A | 68,092 | <p>$$\frac{x^2+2\sqrt{x^2}}{x}=x+\frac{2|x|}{x}$$
Now as $x\to 0$. The first term goes to $0$ but the second term goes to $\pm 2$, depending on which side you approach $0$.</p>
|
1,265,801 | <p>Let $p$ be an odd prime and $a, b \in \Bbb Z$ with $p$ doesn't divide $a$ and $a$ doesn't divide $b$. Prove that among the congruence's $x^2 \equiv a \mod p$, $\ x^2 \equiv b \mod p$, and $x^2 \equiv ab \mod p$, either all three are solvable or exactly one.</p>
<p>Please help I'm trying to study for final in number... | Rolf Hoyer | 228,612 | <p>What you should prove directly is the following:</p>
<ul>
<li>The product of two quadratic residues is a quadratic residue.</li>
<li>The product of a quadratic residue and a quadratic non-residue is a quadratic non-residue.</li>
</ul>
<p>A counting argument then yields additionally:</p>
<ul>
<li>The product of tw... |
73,238 | <p>How can I calculate the solid angle that a sphere of radius R subtends at a point P? I would expect the result to be a function of the radius and the distance (which I'll call d) between the center of the sphere and P. I would also expect this angle to be 4π when d < R, and 2π when d = R, and less than 2π when d ... | Neil G | 774 | <p>I think an easy way to visualize this is to see it as a bunch of transformation matrices.</p>
<p>A circle is just $SRx$ where</p>
<p>$x = \left[\matrix{1\\0\\0}\right],$</p>
<p>$S = rI$ is a scale matrix, and</p>
<p>$R = \left[\matrix{\cos\theta & \sin\theta & 0 \\ \sin\theta & -\cos\theta & 0 \\... |
2,059,604 | <p>Let's say I have a two periodic functions f(x) and g(x) each with the same period of p. Is it always the case that the sum of these two functions will also have the period of p? Is there any counter example?</p>
| Community | -1 | <p>For sums and products: Let $p$ be a period of $f(x)$ and let $q$ be a period of $g(x)$. Suppose that there are positive integers $a$ and $b$ such that $ap=bq=r$. Then $r$ is a period of $f(x)+g(x)$, and also of $f(x)g(x)$. </p>
<p>So, if $f(x)$ has $5\pi$ as a period, and $g(x)$ has $3\pi$ as a period, then $f(x)+... |
1,601,970 | <p>I want to use proof by contradiction.</p>
<p>Suppose that real numbers are bounded, then according to the axiom of continuity, there exists a least upper bound $b$.</p>
<p>But if $x\in \Bbb R$, then $x+1\in \Bbb R$ because of the inclusion property of real numbers.</p>
<p>But $x+1\in \Bbb R\Longrightarrow x+1\leq... | frosh | 211,697 | <p>From Calculus by Apostol:</p>
<blockquote>
<p><strong>Theorem #1:</strong> The set <strong>P</strong> of positive integers <span class="math-container">$1, 2, 3,...$</span> is unbounded above.</p>
<p><strong>Proof #1:</strong> Assume <strong>P</strong> is bounded above. We shall show that this leads to a contradicti... |
1,119,563 | <p>Why is $\sec^{-1}(2/\sqrt{2}) = \sec^{-1}(\sqrt{2})$ true?</p>
| Daniel W. Farlow | 191,378 | <p>As Workaholic pointed out (here with more explanation):
\begin{align}
\frac{2}{\sqrt{2}} &= \frac{2}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}}\tag{$\frac{\sqrt{2}}{\sqrt{2}}=1$}\\[1em]
&= \frac{2\sqrt{2}}{2}\\[1em]
&= \sqrt{2}.
\end{align}
Since $\dfrac{2}{\sqrt{2}}=\... |
490,641 | <p>In Niels Lauritzen, <em>Concrete Abstract Algebra</em>, I'm having trouble showing the following:</p>
<p>The problem starts out like this:</p>
<p>$f(X)=a_nX^n+\cdots+a_1X+a_0, a_i \in \mathbb Z, n \in \mathbb N$ </p>
<p>Part (i) which I think I've done right:</p>
<p>i) Show $X-a \mid X^n-a^n$:
$X^n - a^n = (X-a)... | PITTALUGA | 94,471 | <p>Of course $(x-a)g(x)$ vanishes at $x=a$, so $a$ is a root also for this polynomial. If the degree of $f$ is $n$ modulo $N$, then $N$ does not divide $a_n$, which consequently must be the leading coefficient of $g(x)$.
As a consequence, the degree of $g(x)$ must be exactly $n-1$, otherwise $(x-a)g(x) \neq f(x)$ mod $... |
2,466,949 | <p>Room coordinates are following my walls, to use the guidance system I build the position from various other sensors & built a GPS position from it.</p>
<p>As I also need the a "fake" compass I'm trying to interface a moving robot with a sensor I made.</p>
<p>Robot expect compass to send him the values of a 3-a... | N. F. Taussig | 173,070 | <p>Your instructor is counting distinguishable arrangements of the word <em>anagram</em>.</p>
<p>The word <em>anagram</em> has seven letters, so we have seven positions to fill with $3$ <em>a</em>s, $1$ <em>g</em>, $1$ <em>m</em>, and $1$ <em>r</em>. We can fill three of these seven positions with <em>a</em>s in $\bi... |
10,974 | <p>Is the following true: If two chain complexes of free abelian groups have isomorphic homology modules then they are chain homotopy equivalent.</p>
| Leonid Positselski | 2,106 | <p>Yes, this is true, and it does not matter whether the complexes are bounded from any side (nor of course does it matter whether the homology is finitely generated). This is so because:</p>
<ol>
<li>The homotopy category of free abelian groups is equivalent to the derived category of abelian groups. This holds eve... |
2,231,092 | <p>I am reading <a href="http://people.ucalgary.ca/~rzach/static/open-logic/open-logic-complete.pdf" rel="nofollow noreferrer">Open Logic TextBook</a>. In which there is a proposition about Extensionality of first order sentences (6.12) It goes like this, </p>
<p>Let $\phi$ be a sentence, and $M$ and $M'$
be structure... | Dr. Sonnhard Graubner | 175,066 | <p>we have the equation in cartesian coordinates as $$y=\frac{1}{7}x-\frac{19}{7}$$
now we have $$\cos(\theta)=\frac{x}{r}$$ and $$\sin(\theta)=\frac{y}{r}$$
can you finish?</p>
|
241,998 | <p>Consider a list of even length, for example <code>list={1,2,3,4,5,6,7,8}</code></p>
<p>what is the fastest way to accomplish both these operations ?</p>
<p><strong>Operation 1</strong>: two by two element inversion, the output is:</p>
<pre><code>{2,1,4,3,6,5,8,7}
</code></pre>
<p>A code that work is:</p>
<pre><code>... | Henrik Schumacher | 38,178 | <p>If you have the do that very often, store your lists in a packed matrix like <code>a</code> below (that's a good idea anyways!), use your current code to generate a permutation, and then use <code>Part</code> to apply the permutations on the columns of <code>a</code></p>
<pre><code>n = 16;
a = RandomInteger[{1, 100}... |
3,772,399 | <p>I need help with the following question:</p>
<p>Let <span class="math-container">$X_i$</span> be independent, non-negative random variables, <span class="math-container">$i \in \{1,...,n\}$</span>. I want to show that for all <span class="math-container">$t > 0$</span>, <span class="math-container">$$P(S_n > ... | Adina Goldberg | 250,127 | <p>This post about <a href="https://math.stackexchange.com/questions/3180034/commuting-matrices-up-to-a-scalar">which matrices commute up to a scalar</a> may be helpful, as you are looking for <span class="math-container">$M$</span> and <span class="math-container">$\lambda$</span> such that <span class="math-container... |
959,525 | <p>Could someone tell me what i've done wrong?</p>
<p>I tried to find out the derivative of $3^(2x)-2x+1$ but I got it wrong.
What I did was derivate $3^a-2x+1$ where a = 2x then multiply those two.</p>
<p>$(ln3*3^a - 2)*2$ = $2ln3*3^(2x)-4$</p>
<p>Ps. x = 2 so the answer is supposed to be 176.</p>
| copper.hat | 27,978 | <p>Here is a more complex answer:</p>
<p>Suppose $z \in \mathbb{C}$, then $|1+z|+|1-z| \ge \sqrt{|1+z|^2 + |1-z|^2} = \sqrt{2+ |z|^2} \ge 2$. Furthermore, we have equality <strong>iff</strong> $z=0$.</p>
<p>Now suppose $z \neq 0$ and let $w \in \mathbb{C}$, then $|z+w|+|z-w| = |z| (|1+{ w \over z}| + |1-{w \over z}|)... |
1,874,914 | <p>in order to find $e^{AT}$ We can't just take the exponential of A as we would do in its diagonalized form. We need to diagonalize $A=S^{-1}e^{\delta(t)}S$ in order to find $e^{AT}$ why is this the case? I know we can't take the exponential of the matrix right away, do we need to take the exponential of the diagonal ... | Andrew D. Hwang | 86,418 | <p>It's not that you can't write down the series
$$
\exp(tA) = \sum_{k=0}^{\infty} \frac{t^{k} A^{k}}{k!},
\tag{1}
$$
it's that the entries of $A^{k}$ usually aren't easy to express in terms of the entries of $A$ (try it yourself for a $2 \times 2$ matrix!), so (1) isn't an explicit description of the entries of $\exp(... |
943,048 | <p><strong>Question:</strong></p>
<blockquote>
<p>let $x_{i}=1$ or $-1$,$i=1,2,\cdots,1990$, show that
$$x_{1}+2x_{2}+\cdots+1990x_{1990}\neq 0$$</p>
</blockquote>
<p>this problem it seem is easy,But I think is not easy. </p>
<p>I think note
$$1+2+3+\cdots+1990\equiv \pmod { 1990}?$$</p>
| Kim Jong Un | 136,641 | <p>You are summing up $\frac{1990\times 1991}{2}=1981045$ numbers each of which is odd. The result must be odd and in particularly cannot be $0$.</p>
|
33,153 | <p>Here is one definition of a differential equation:</p>
<blockquote>
<p>"An equation containing the derivatives of one or more dependent variables, with respect to one of more independent variables, is said to be a differential equation (DE)" <em>(Zill - A First Course in Differential Equations)</em></p>
</... | Pacciu | 8,553 | <p>When I was a student I was taught the following definition:</p>
<blockquote>
<p>Let <span class="math-container">$N\in \mathbb{N}$</span>, <span class="math-container">$U\subseteq \mathbb{R}^{N+2}$</span> and <span class="math-container">$F:U\to \mathbb{R}$</span>.</p>
<p>Then the <em><span class="math-conta... |
378,966 | <p>$$A_t-A_{xx} = \sin(\pi x)$$
$$A(0,t)=A(1,t)=0$$
$$A(x,t=0)=0$$
Find $A$.</p>
<p>I know I need to find the homogeneous and particular solutions. Im just not sure on this PDE.</p>
| Daryl | 36,034 | <p>Since the non-homogeneity depends only on $x$, we can assume a solution of the form $A(x,t)=u(x,t)+\phi(x)$.</p>
<p>Substituting this into the PDE gives
$$u_t-u_{xx}-\phi_{xx}=\sin(\pi x).$$
Choosing $\phi(x)$ such that $-\phi_{xx}=\sin(\pi x)$, means that $u$ only needs to satisfy a homogeneous PDE. </p>
<p>Note ... |
3,613,854 | <p>Let <span class="math-container">$$A=\begin{bmatrix}
3 & 2 \\
2 & 3
\end{bmatrix}.$$</span>
Find the spectral decomposition of <span class="math-container">$A$</span>. This is <span class="math-container">$$A=VDV^{-1}=\begin{bmatrix}
-1 & 1 \\
1 & 1
\end{bmatrix}\begin{bmatrix}
1 & 0 \\
0 & 5... | SMA.D | 255,648 | <p>For a diagonalizable matrix <span class="math-container">$A=VDV^{-1}$</span>:
<span class="math-container">$$f(A) = V\begin{bmatrix}f(d_1)&0 \\ 0&f(d_2)\end{bmatrix}V^{-1}$$</span>
Hence
<span class="math-container">$$ 2^A = V\begin{bmatrix}2^1&0 \\ 0&2^5\end{bmatrix}V^{-1}$$</span></p>
<p>See <a hr... |
1,355,684 | <p>How to find the lower and upper focus? Hyperbola </p>
<p>I started with this
$$ 9x^2 + 54x - y^2 + 10y + 81 = 0 $$</p>
<p>and broke it down to</p>
<p>$$ \frac{9(x+3)^2}{25} - \frac{(y-5)^2}{25} = -1 $$</p>
<p>center = (-3,5)
Lower Vertex = (-3,0)
Upper Vertex = (-3,10)</p>
<p>How to get the foci? </p>
<p>foci... | Bassball Batman | 169,579 | <p>The cool thing about 37 and 111 is this: the three identical digits divided by their sum equals 37. Another fun fact is that 3 × 7 × 37 = 777, which times 13 equals 10,101 (and thus, 151,515 and 474,747)! The product of 7 × 13 (that is, 91) always goes evenly into multiples of 10,101 because...</p>
<ul>
<li>91 × 1... |
163,640 | <p>Early in a course in Algebra the result that every group can be embedded as a subgroup<br>
of a symmetric group is introduced. One can further work on it to embed it as a subgroup of a suitable (higher degree) alternating group.</p>
<p>Inverting the view point we can say that the family of simple groups $A_n, n\... | YCor | 14,094 | <p>There are plenty of ways of characterizing bounded rank for finite simple groups. Let $G$ be a finite group. Define </p>
<ul>
<li>$r_n(G)$ as the largest $k$ such that $(\mathbf{Z}/n\mathbf{Z})^k$ embeds into $G$</li>
<li>$\mathrm{nc}(G)$ the largest $k$ such that there exist non-abelian subgroups $H_1\dots,H_k$ su... |
3,910,739 | <p>I am trying to find a pdf for a random variable <span class="math-container">$X$</span> where <span class="math-container">$X=-2Y+1$</span> and <span class="math-container">$Y$</span> is given by <span class="math-container">$N(4,9)$</span></p>
<p>Here is my attempt:</p>
<p>we know <span class="math-container">$\mu=... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$-2Y+1 <x$</span> is not equivalent to <span class="math-container">$Y <-\frac {x-1} 2$</span>.</p>
<p><span class="math-container">$P(X\leq x)=P(-2Y+1 \leq x)=P(Y \geq \frac {1-x} 2)=\int_{\frac {1 -x} 2}^{\infty} \phi (t)dt$</span> where <span class="math-container">$\phi$</span... |
4,032,983 | <p>I would like to know math websites that are useful for students, PhD students and researchers (useful in the sense most of the students or researchers—of a particular area—are using it). Maybe you can share which math websites you sometime use and why you use it.</p>
<p>Let me give my websites and why I use them:</p... | RavenclawPrefect | 214,490 | <p>On /r/math, there is <a href="https://www.reddit.com/r/math/comments/8ewuzv/a_compilation_of_useful_free_online_math_resources/" rel="nofollow noreferrer">A Compilation of Useful, Free, Online Math Resources</a>. It is somewhat geared towards students in scope, but references many tools used by research mathematicia... |
3,234,217 | <p>Let <span class="math-container">$a,b,c \in \mathbb{R},$</span> <span class="math-container">$\vec{v_1}=\begin{pmatrix}1\\4\\1\\-2 \end{pmatrix},$</span> <span class="math-container">$\vec{v_2}=\begin{pmatrix}-1\\a\\b\\2 \end{pmatrix},$</span> and <span class="math-container">$\vec{v_1}=\begin{pmatrix}1\\1\\1\\c \en... | PierreCarre | 639,238 | <p>Just build a matrix with these vectors as rows and perform row reduce. The vectors will be linearly dependent if at least one row is made of zeros. The idea is that the rank of a matrix is the maximum number of linearly independent rows (or columns), hence, the rows will be linearly dependent if and only if <span cl... |
3,234,217 | <p>Let <span class="math-container">$a,b,c \in \mathbb{R},$</span> <span class="math-container">$\vec{v_1}=\begin{pmatrix}1\\4\\1\\-2 \end{pmatrix},$</span> <span class="math-container">$\vec{v_2}=\begin{pmatrix}-1\\a\\b\\2 \end{pmatrix},$</span> and <span class="math-container">$\vec{v_1}=\begin{pmatrix}1\\1\\1\\c \en... | amd | 265,466 | <p>Fleshing out a comment by Bernard, the rank of a matrix is equal to the order of its largest nonzero minor. Writing the three vectors as rows to save vertical space, for the three vectors to be linearly dependent we want the matrix <span class="math-container">$$A=\begin{bmatrix}1&4&1&-2 \\ -1&a&... |
1,474,123 | <p>I have tried to use u-substitution but for some reason am not doing it right and thus not getting the correct answer. I want to know the most obvious/ intuitive way to solve this integral.</p>
| David Vallis | 279,109 | <p>$x=z$, $\mathrm{d}x=z\sec^{2}(k) \mathrm{d}k$, $(a^2+x^2)^{\frac32}=z^4$</p>
|
3,098,587 | <p>I have no answers to refer to, hence, would be great if someone could check up if my procedure to solve the following problem is correct. Also, I am struggling to solve for event B from part b) - any tips would be much much appreciated! I'm preparing for an exam, hence, it is of vital importance. Thank you! </p>
<p... | Asatur Khurshudyan | 261,161 | <p>It seems that rotation indeed involves Euclids fifth postulate. Section 8 of <a href="http://www.michaelbeeson.com/research/papers/ConstructiveGeometryAndTheParallelPostulate.pdf" rel="nofollow noreferrer">this paper</a> is all about that.</p>
|
3,072,720 | <p>Let <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> be independent random variables with uniform distribution between <span class="math-container">$0$</span> and <span class="math-container">$1$</span>, that is, have joint density <span class="math-container">$f_{xy}(x, y) = 1$</s... | Maxim | 491,644 | <p>The conditional pdf is given by
<span class="math-container">$$f_{X | W = w}(x) =
\frac {f_{X, W}(x, w)} {f_W(w)}.$$</span>
<span class="math-container">$f_W$</span> is the pdf of a sum of two independent uniformly distributed r.v.:
<span class="math-container">$$f_W(w) = 4 w \left[0 < w \leq \frac 1 2 \right] +
... |
4,586,527 | <p>Consider the following problem.</p>
<p><a href="https://i.stack.imgur.com/uzgHX.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/uzgHX.png" alt="enter image description here" /></a></p>
<p>I'm fairly new to probability theory, but have some experience with combinatorics. For that reason, after fail... | user469053 | 1,027,291 | <p>For this problem, your probabilistic approach is correct. But I would use the following alternative: The sample space consists of <span class="math-container">$\binom{4}{2}=6$</span> outcomes. You have four components (in order) and there are <span class="math-container">$\binom{4}{2}=6$</span> ways to choose a set ... |
986,754 | <p>So I'm kind of stuck on this question and I don't exactly know how to describe this on the title header and I apologize... </p>
<blockquote>
<p>For some values of $x$, the assignment statement $y := 1-\cos(x)$ involves a difficulty. What is the difficulty? What values of $x$ are involved? What remedy do you propo... | Spencer | 71,045 | <p>Another remedy, </p>
<blockquote class="spoiler">
<p> $$1-\cos(x) = 2\sin^2(x/2)$$</p>
</blockquote>
|
4,246,719 | <p>Consider two random variables <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>, both distributed as a <a href="https://en.wikipedia.org/wiki/Gumbel_distribution" rel="nofollow noreferrer">Gumbel</a> with location 0 and scale 1.</p>
<p>Let <span class="math-container">$Z\equiv X-Y$<... | greg | 357,854 | <p><span class="math-container">$
\def\a{\alpha}\def\b{\beta}
\def\o{{\tt1}}\def\p{\partial}
\def\E{{\cal E}}\def\F{{\cal F}}\def\G{{\cal G}}
\def\L{\left}\def\R{\right}\def\LR#1{\L(#1\R)}
\def\vec#1{\operatorname{vec}\LR{#1}}
\def\trace#1{\operatorname{Tr}\LR{#1}}
\def\grad#1#2{\frac{\p #1}{\p #2}}
\def\c#1{\color{red... |
2,421,771 | <p>I’m attempting to explain curvature in layman’s terms to my class before explaining the formula. I like to do this first to give my students an idea of what we are finding. </p>
<p>Some people explain curvature as a “measure of how fast a curve is changing direction at a given point.” But this seems misleading t... | operatorerror | 210,391 | <p>Time can be an ok way to talk about it. Take a particle travelling along a curve, maybe defined by $x(t)$ or maybe it happens to be the graph of some function of time. Then, curvature is measuring how the velocity is changing, encoding the sign (think concavity/convexity) of this change as well.</p>
|
15,237 | <p><a href="https://matheducators.stackexchange.com/questions/176/knowing-mathematics-does-not-translate-to-knowing-to-teach-mathematics-why">A question</a> has been asked about why great mathematicians are not necessarily great teachers. On the other hand, I am wondering if knowing more mathematics actually helps with... | KCd | 893 | <p>The title of the question asks how knowing "more about math" can help in teaching calculus, while the question itself asks specifically about the math learned towards a masters degree or a doctorate. I am going to focus on the "more about math" part, regardless of the stage at which it was acquired, because the math... |
1,860,134 | <p>In <em>The logic of provability</em>, by G. Boolos, there is a remark in chapter 7 saying that $\diamond^{m} \top\implies \diamond^{n} \top$ is false if $m<n$ <strong>(unless $PA$ is 1-inconsistent)</strong>.</p>
<p>Now, it seems to me that the parenthetical expression is not necessary, since earlier in the chap... | user21820 | 21,820 | <p>Hmm your reasoning seems weird. Take any theory $T$ that satisfies the <a href="http://plato.stanford.edu/entries/logic-provability/" rel="nofollow">provability conditions</a>. Take any $m,n \in \mathbb{N}$ such that $m<n$. Then by Lob's theorem, if $T \vdash \square^n \bot \to \square^{n-1} \bot$ then $T \vdash ... |
141,346 | <p>I am not sure where one looks up this type of fact. Google was not very helpful.</p>
| sheesh | 39,611 | <p>Appealing to Grothendieck's theorem is pretty much over the top. It's completely elementary. Suppose $\cal A$ is an abelian (or exact) category with enough injectives.</p>
<blockquote>
<p>A complex is injective if and only if it is (a retract of a) split exact complex with injective components.</p>
</blockquote>
... |
3,395,910 | <p>I understand how to apply the trapezoidal rule to approximate the area under a curve.</p>
<p>But I'm not sure how to apply it when approximating areas between two functions. </p>
<ul>
<li>Do you use the formula like how you normally would, except apply it to the <strong>first function - the second function</strong... | mathcounterexamples.net | 187,663 | <p><span class="math-container">$T= \{x \in \mathbb R \mid (f-g)(x)=0\} = (f-g)^{-1}(\{0\})$</span>, i.e. <span class="math-container">$T$</span> is the inverse image under <span class="math-container">$f-g$</span> of the singleton <span class="math-container">$\{0\}$</span>. As <span class="math-container">$\{0\}$</sp... |
56,162 | <p>I'm trying to understand the Cartan decomposition of a semisimple Lie algebra, $\mathfrak g=\mathfrak k \oplus \mathfrak p$, where $[\mathfrak k,\mathfrak p] \subseteq \mathfrak p$, cf. the wikipedia article on <a href="http://en.wikipedia.org/wiki/Cartan_decomposition" rel="noreferrer">Cartan decomposition</a>.</p>... | Peter Woit | 11,670 | <p>As Konrad Waldorf noted, in this case G-bundles are trivializable (since $\pi_2(G)$ is trivial). So gauge transformations are just maps
$$\phi:M\rightarrow G$$</p>
<p>and these have a homotopy invariant that can be non-trivial, the degree of the map. One way to compute this is as
$$\int_M \phi^*\omega_3$$</p>
<p... |
2,740,349 | <blockquote>
<p>A triangle has the side lengths of $3$, $5$, and $7$. Express $\cos(y)+\sin(y)$, where $y$ is the largest angle in the triangle.</p>
</blockquote>
<p>I have tried to apply pythagoras theorm, trying to express the other two angles in some way, split the triangle into smaller triangles, but all without... | Gibbs | 498,844 | <p>If $n = 2$, you can see that $\det(X,Y) = \sigma_1 \wedge \sigma_2 (X,Y)$. Generalize to any $n$.</p>
|
1,521,518 | <p>Determine if the given vectors span $\mathbb{R}^4$</p>
<p>${(1, 1, 1, 1), (0, 1, 1, 1), (0, 0, 1, 1), (0, 0, 0, 1)}$.</p>
<p>I'm completely confused on this question. My textbook gives a different problem but in $\mathbb{R}^3$. How would i go about this?</p>
| z100 | 259,327 | <p>Not needed any sophisticated knowledge, just a simple understanding: the most right vector gives 1 dimension,
2nd rightmost 2nd dimension, 2nd left 3rd dimension and first one 4th dimension.
If you do not believe, subtract two consetive vectors - and get the standard basis exactly.</p>
|
1,666,977 | <p><strong>Background</strong></p>
<p>This is purely a "sate my curiosity" type question.</p>
<p>I was thinking of building a piece of software for calculating missing properties of 2D geometric shapes given certain other properties, and I got to thinking of how to failsafe it in case a user wants to calculate the ar... | Ng Chung Tak | 299,599 | <p>You may define a digon on $S^{2}$. By the ways, there're $\{ \frac{n}{k} \}$ star polygons. In particular $\{ \frac{n}{1} \}$ or $\{ \frac{n}{n-1} \}$ is ordinary polygon.</p>
|
3,995,272 | <p>I am a master student in mathematical physics. I study soliton and traveling wave solutions of the differential equations.</p>
<p>Let's consider the following ODE:
<span class="math-container">$$Q^{\prime}(\xi)=ln(A)(\alpha+\beta Q(\xi)+\sigma Q^2(\xi))$$</span>
where
<span class="math-container">$A \neq 0,1.$</span... | Kavi Rama Murthy | 142,385 | <p>Hint: Since <span class="math-container">$\ln (1+s) \sim s$</span> as <span class="math-container">$s \to 0$</span> we get <span class="math-container">$\lim_{s\to 0} \frac {\ln (1+s)} {1-(1+s)^{2}}=-\frac 1 2$</span>. Now put <span class="math-container">$s=\sin x-1$</span> in this.</p>
|
62,177 | <p>One of the most mind boggling results in my opinion is, with the axiom of choice/well-ordering principle, there exist such things as uncountable well-ordered sets $(A,\leq)$. </p>
<p>With this is mind, does there exist some well ordered set $(B,\leq)$ with some special element $b$ such that the set of all elements ... | Ted | 15,012 | <p>Let $b$ be the first <a href="http://en.wikipedia.org/wiki/First_uncountable_ordinal" rel="nofollow">uncountable ordinal</a>, and $B$ be the set of all ordinals up to and including $b$.</p>
|
1,291,511 | <p>This may seem like a silly question, but I just wanted to check. I know there are proofs that if $f(x)=f'(x)$ then $f(x)=Ae^x$. But can we 'invent' another function that obeys $f(x)=f'(x)$ which is <strong>non-trivial</strong>?</p>
| k1.M | 132,351 | <p>Observe that if $ f(x)=f'(x) $ then
$$
\left(\frac{f(x)}{e^x}\right)'=\frac{f'(x)-f(x)}{e^x}=0
$$
Hence $\dfrac{f(x)}{e^x}$ is constant...</p>
|
3,374,248 | <p>I haven't worked out all the details yet, but it seems to be true for the following functions:</p>
<ul>
<li><span class="math-container">$f(k) = 1$</span></li>
<li><span class="math-container">$f(k) = 1/k!$</span></li>
<li><span class="math-container">$f(k) = a^k$</span></li>
<li><span class="math-container">$f(k) ... | reuns | 276,986 | <p><span class="math-container">$\sum_{k=0}^n f(k)f(n-k) = O\Big(n f^2(\frac{n}{2})\Big)$</span> doesn't hold when <span class="math-container">$f(0) \ne 0$</span> and <span class="math-container">$f(n)$</span> grows much faster than <span class="math-container">$f(n/2)^2$</span> for example with <span class="math-con... |
2,860,321 | <p>Suppose $L/K$ is a Galois extension of local fields with Galois group $G = \operatorname{Gal}(L/K)$. Let $K'$ be the maximal unramified Extension of $K$ in $L$.</p>
<p>The Definition of the <strong>inertia group</strong> of $L/K$ is given by $I = I_{L/K} = \operatorname{Gal}(L/K')$ which I understand.</p>
<p>In so... | Tomo | 62,940 | <p>The inertia group is usually defined as the kernel of the homomorphism
<span class="math-container">$$\varepsilon:D\twoheadrightarrow G(\overline L/\overline K),$$</span>
where here <span class="math-container">$D$</span> denotes the decomposition group (in the case of a Galois extension of local fields as in your q... |
129,439 | <p>This code in Mathematica 9 returns two graphs that are NOT complementary.</p>
<pre><code>{GraphData[{7, 172}], GraphData[{7, 172}, "ComplementGraph"]}
</code></pre>
| Feyre | 7,312 | <p>In's an unfortunate rendering I think because the connection between what according to the <code>AdjacencyMatrix[]</code> are <code>6</code> and <code>7</code> isn't obvious.</p>
<p>But please consider the following mapping:</p>
<p><a href="https://i.stack.imgur.com/1ubsN.png" rel="nofollow noreferrer"><img src="h... |
1,579,781 | <blockquote>
<p>If $x+y+z=6$ and $xyz=2$, then find the value of $$\cfrac{1}{xy}
+\cfrac{1}{yz}+\cfrac{1}{zx}$$</p>
</blockquote>
<p>I've started by simply looking for a form which involves the given known quantities ,so:</p>
<p>$$\cfrac{1}{xy} +\cfrac{1}{yz} +\cfrac{1}{zx}=\cfrac{yz\cdot zx +xy \cdot zx +xy \cdot... | Harish Chandra Rajpoot | 210,295 | <p>Notice, the method is straight forward simply take L.C.M. & substitute known values as follows $$\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}=\frac{x+y+z}{xyz}=\frac{6}{2}=\mathbb{\color{red}{3}}$$</p>
|
81,982 | <p>I am beggining to do some work with cubical sets and thought that I should have an understanding of various extra structures that one may put on cubical sets (for purposes of this question, connections). I know that cubical sets behave more nicely when one has an extra set of degeneracies called connections. The que... | Peter Arndt | 733 | <p>A very concrete instance where you can see the meaning and usefulness of connections is <a href="http://www.tac.mta.ca/tac/volumes/1999/n7/5-07abs.html" rel="nofollow">this</a> article by Brown and Mosa: They show that double categories (which do have an underlying (truncated) cubical set) with connections are the s... |
258,226 | <p>As an algebraist, I have some strong intuitions about what it means for an algebraic result to be true. In particular, my intuition would lead me to believe that if I cannot construct a counter-example to a claim, then the claim must be true. This is what motivated <a href="https://mathoverflow.net/questions/25813... | Panu Raatikainen | 102,468 | <p>No. It works with T0 and Π01 sentences, as PA refutes any false Π01 sentence. But not so with T1 and Π02 sentences: there are false Π02 sentences which are independent of T1. </p>
|
2,237,441 | <p>Let $n$ be a natural number.</p>
<p>I need to prove that $9 \mid 4^n-3n-1$</p>
<p>Could anyone give me some hints how to prove it without using induction.</p>
| Ayoub Falah | 337,716 | <p>Let $u_n=4^n-3n-1$</p>
<p>so $$u_{n+1}=4^{n+1}-3(n+1)-1$$</p>
<p>$$\Leftrightarrow u_{n+1}=4(4^n-1)-3n$$</p>
<p>$$\Leftrightarrow u_{n+1}=4(4^n-1)- 4\cdot3n + 9n$$</p>
<p>$$\Leftrightarrow u_{n+1}=4(4^n - 3n - 1) + 9n$$</p>
<p>$$\Leftrightarrow u_{n+1}=4u_n + 9n$$</p>
<p>$$\Leftrightarrow u_{n+1}=4(4u_{n-1} + ... |
658,758 | <p>How do you plot $$f(x,y) = \frac{x}{1-y} \text{with}~ x^2+y^2<1$$ in Mathematica or Maple?</p>
| Mikasa | 8,581 | <p>In Maple, you can use the following codes:</p>
<pre><code>[> with(plots):
plot3d(x/(1-y), x = -1 .. 1, y = -sqrt(1-x^2) .. sqrt(1-x^2), axes = boxed, filled = true,numpoints = 1000,color=green);
</code></pre>
<p><img src="https://i.stack.imgur.com/ysKzx.png" alt="enter image description here"></p>
|
1,397,190 | <p>Find the sum of following series:</p>
<p>$$1 + \cos \theta + \frac{1}{2!}\cos 2\theta + \cdots$$</p>
<p>where $\theta \in \mathbb R$.</p>
<p>My attempt: I need hint to start.</p>
| Chinny84 | 92,628 | <p>Hint to start
$$
1+\cos \theta + \frac{1}{2!}\cos(2\theta)+\cdots = \mathcal{Re}\sum_{k=0}^n\frac{\mathrm{e}^{ik\theta}}{k!} = \mathcal{Re}\sum_{k=0}^n\frac{\left(\mathrm{e}^{i\theta}\right)^k}{k!}
$$
Also remember that in the $\lim_{n\to\infty}$ we have
$$
\lim_{n\to\infty}\sum_{k=0}^n\frac{x^k}{k!}=\mathrm{e}^x
$$... |
1,397,190 | <p>Find the sum of following series:</p>
<p>$$1 + \cos \theta + \frac{1}{2!}\cos 2\theta + \cdots$$</p>
<p>where $\theta \in \mathbb R$.</p>
<p>My attempt: I need hint to start.</p>
| GeorgSaliba | 142,772 | <p><strong>HINTS</strong>: You have $$\sum_{k=0}^\infty\frac{\cos(k\theta)}{k!}$$</p>
<p>Remember that:
$$e^x=\sum_{k=0}^\infty\frac{x^k}{k!}$$
And:
$$\cos(\theta)=\Re{e^{i\theta}}$$</p>
|
1,983,614 | <p>Consider a measurable space $(\Omega, \mathcal{F})$ and let $I$ be an arbitrary index set. </p>
<p>Is the following true?</p>
<blockquote>
<p>If $\left( A_i \right)_{i \in I}$ is a chain in $\mathcal{F}$ – that is, $\forall i \in I$, $A_i \in \mathcal{F}$ and for all $i, j \in I$, we have $A_i \subseteq A_... | Mitchell Spector | 350,214 | <p>This isn't true in general (assuming the axiom of choice).</p>
<p>Let $\kappa$ be the least cardinal of a non-measurable set of reals; let $f$ map $\kappa$ 1-1 onto such a non-measurable set.</p>
<p>Then $\{f\!"\!\alpha \mid \alpha\lt\kappa\}$ (where $f\!"\!\alpha$ denotes the range of $f$ on $\alpha)$ is a chain ... |
213,102 | <p>Is it true than an aperiodic, ergodic, minimal and equicontinuous dynamical system on a compact metric space is totally ergodic ?</p>
<p>According to some results I found in some books, a rotation on a compact metric group is equicontinous, and it is minimal <del>and totally ergodic</del> whenever it is ergodic.</p... | Ian Morris | 1,840 | <p>Just knowing that a transformation $T$ is minimal is no guarantee that $T^n$ is also minimal. For example, let $T_1$ be the non-identity homeomorphism of a two-point metric space and let let $T_2$ be an aperiodic, minimal, equicontinuous, uniquely ergodic transformation of a compact metric space (for example, an irr... |
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