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 |
|---|---|---|---|---|
3,208,613 | <p>We have a fair <span class="math-container">$3$</span> sided die <span class="math-container">$(a,b,c)$</span>, and we perform the following experiment:</p>
<p>Roll the die until we have seen <span class="math-container">$10$</span> of any of the sides, let <span class="math-container">$X$</span> be the number of t... | angryavian | 43,949 | <p>The roots of the two equations are
<span class="math-container">$$\frac{1}{2}\left(-(k-1) \pm \sqrt{(k-1)^2 + 8 (k+1)}\right) = \frac{1}{2}(-(k-1) \pm (k+3)) = \{2, -(k+1)\}$$</span>
<span class="math-container">$$\frac{1}{2(k-1)} \left(-k \pm \sqrt{k^2 - 4(k-1)}\right) = \frac{1}{2(k-1)}(-k \pm (k-2)) = \{-\frac{1}... |
4,336,706 | <p>Let <span class="math-container">$\mathbb {K}$</span> be a field. Let <span class="math-container">$f: \mathbb {K}^2 \rightarrow \mathbb {K}^2; x \mapsto Ax+b$</span> be an affine transformation. Suppose <span class="math-container">$f$</span> has a fixed point line (i.e. a line such that every point on that line is... | Kavi Rama Murthy | 142,385 | <p>Counter-example <span class="math-container">$f(x)=\frac 1x$</span> when <span class="math-container">$f$</span> is only <span class="math-container">$C^{\infty}$</span> on <span class="math-container">$(0,1)$</span>.</p>
<p>If <span class="math-container">$f$</span> is also continuous on <span class="math-containe... |
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... | Joffan | 206,402 | <p>Since $p$ is a prime, it has at least one primitive root $g$. </p>
<ul>
<li>Every quadratic residue $q$ (coprime to $p$) can be expressed as $g^{2k}\equiv q \bmod p$. </li>
<li>Every quadratic non-residue $n$ can be expressed as $g^{2k+1}\equiv n \bmod p$.</li>
</ul>
<p>Therefore your assertion is equivalent to th... |
2,402,429 | <p>Let $P(x) = x^3 + 2x^2+3x+4$ and $a$ be the root of equation $x^4+x^3+x^2+x+1=0$.</p>
<p>Find the value of $P(a)P(a^2)P(a^3)P(a^4)$</p>
<p>Is my answer correct ?</p>
<p>Since root of equation $x^4+x^3+x^2+x+1=0$ is the $5^{th}$ primitive root of 1,</p>
<p>so $a, a^2, a^3, a^4$ are roots of $x^4+x^3+x^2+x+1=0$ </... | Mark Bennet | 2,906 | <p>Here is another way. Call the product that you want Q.</p>
<p>Note that $xP(x)-P(x)=x^4+x^3+x^2+x-4$</p>
<p>As you correctly observe the values you are substituting are all roots of $x^4+x^3+x^2+x+1=0$ and hence for these values you get $(x-1)P(x)=-5$ whence $$(a-1)(a^2-1)(a^3-1)(a^4-1)Q=(-5)^4=625$$</p>
<p>Now p... |
408,590 | <p>I'm looking for references (books/lecture notes) for :</p>
<ul>
<li>Cardinality without choice, Scott's trick;</li>
<li>Cardinal arithmetic without choice.</li>
</ul>
<p>Any suggestions? Thanks in advance.</p>
| Cameron Buie | 28,900 | <p>Azriel Lévy's "Basic Set Theory" discusses Scott's trick, and does some discussion of choiceless arithmetic.</p>
|
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)... | Calvin Lin | 54,563 | <p><strong>Hint:</strong> Ignoring modulo $N$, what would be the coefficient of $x^{n-1} $ in $g(x)$.</p>
<p><strong>Hint:</strong> If this coefficient is 0 modulo $N$, what can we say about the coefficient of $x^{n}$ in $f$?</p>
|
323,128 | <p>Show that in every (not necessarily connected) graph there is a path from every vertex $u$ of odd degree to some other vertex $v$ ($u \neq v$), also of odd degree.</p>
| Brian M. Scott | 12,042 | <p>HINT: Let $u$ be vertex of odd degree. Start at $u$ and walk from vertex to vertex, never repeating an edge, until you can’t proceed any further.</p>
|
2,263,759 | <p>Let $E$ be a vector space and $\varphi: E \to E$ be a linear map. Let $x, y \in E \setminus \{0\}$ and $\lambda, \mu \in F$ such that
$\varphi(x) = \lambda x$ and $\varphi(y) = \mu y$. Prove that if $\lambda \neq \mu$ then $\{x, y\}$ is linearly independent.</p>
<p>This proof seems like it should be on the simpler ... | egreg | 62,967 | <p>You may want to see a proof that can be generalized to more than two vectors.</p>
<p>Consider $\alpha x+\beta y=0$. Then
\begin{align}
\varphi(\alpha x+\beta y)&=0 \\
\mu(\alpha x+\beta y)&=0
\end{align}
which become
\begin{align}
\alpha\lambda x+\beta\mu y&=0 \\
\alpha\mu x+\beta\mu y&=0
\end{align... |
920,429 | <p>Given that $x(t)=(c_1+c_2 t + c_3 t^2)e^t$ is the general solution to a differential equation, how do you work backwards to find the differential equation? </p>
| mvw | 86,776 | <p>The general solution
$$
x(t) = (c_1 + c_2 t + c_3 t^2)\, e^t
$$
has three parameters $c_i$, so one would need three integrations to reintroduce them from a differential equation, or
three differentiations to eleminate them:
$$
\begin{align}
x(t) &= (c_1 + c_2 t + c_3 t^2)\, e^t \Rightarrow \\
\dot{x}(t) &... |
920,429 | <p>Given that $x(t)=(c_1+c_2 t + c_3 t^2)e^t$ is the general solution to a differential equation, how do you work backwards to find the differential equation? </p>
| Ana M | 167,359 | <p>Given a homogeneous linear differential equation of order $n$, we can get the solutions by writing the characteristic polynomial and equating to zero.</p>
<p>$c_nx^{(n)}(t)+c_{n-1}x^{(n-1)}(t)+...+c_{1}x^{(1)}(t)+c_0x(t)=0$</p>
<p>$x(t)=\sum_iA_ie^{\lambda_i t}$, where $\lambda_i$ is a $\lambda$ such that solves t... |
366,844 | <p>Using the infinite product of $\sin(\pi z)$, one can find the Hadamard product for $e^z-1$:</p>
<p>$$e^z-1 =2ie^{z/2}\sin(-iz/2)= 2i e^{z/2} (-iz/2) \prod_n \left(1+\frac{z^2}{4\pi n^2}\right)\\= e^{z/2} z \prod_n \left(1+\frac{z^2}{4\pi n^2}\right).$$</p>
<p>I don't see a way to find the product for $\cos\pi z$.... | Clayton | 43,239 | <p><strong>Hint:</strong> Use $\sin(2z)=2\sin(z)\cos(z)$ so that $$\cos(z)=\frac{\sin(2z)}{2\sin(z)}.$$ If you're careful about how you write it, you will see that all of the 'even terms' cancel nicely. I do not have time right now, but if you haven't been able to solve it within a few hours, I will return and post my ... |
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... | hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>Your line has the following cartesian equation </p>
<p>$$y=\frac {1}{7}x+b $$</p>
<p>with $$b=-3+\frac {2}{7}=-\frac {19}{7} $$</p>
<p>thus</p>
<p>$$r=\sqrt {x^2+y^2}=f (x) $$</p>
<p>and $$\tan (\theta)=\frac {y}{x} =g (x)$$
or $$x=h (\theta) $$</p>
<p>thus, your polar equation wi... |
2,576,344 | <p>This problem is about expected value, and it's a real world problem.</p>
<p>I know so far that $f$ is strictly increasing, if that makes the proof more concise (but if you can also prove it without this assumption, that would be awesome). Find all solutions for $f$ when $f(P_a \cdot a+P_b \cdot b)=P_a \cdot f(a)+P_... | angryavian | 43,949 | <p>By taking $P_b=0$, you have $f(P_a \cdot a) = P_a \cdot f(a)$ for any $a$ and $P_a$.</p>
<p>Taking $a=1$ shows that $f(P_a) = P_a \cdot f(1)$ for any $P_a$.</p>
<p>For purely cosmetic reasons you can rewrite the last result as $f(x) = x \cdot f(1)$ for all $x$.</p>
<p>Letting $m:=f(1)$ shows that $f(x) = mx$.</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>... | Daniel Huber | 46,318 | <p>We can achieve a speed up of approx. 2 by using <code>Riffle</code>:</p>
<p>list = {1, 2, 3, 4, 5, 6, 7, 8};</p>
<p>For the first problem:</p>
<pre><code>Riffle[list[[2 ;; ;; 2]], list[[1 ;; ;; 2]]]
</code></pre>
<p>For the second problem:</p>
<pre><code>Riffle[list[[-2 ;; 1 ;; -2]], list[[-1 ;; 2 ;; -2]]]
</code></... |
1,969,903 | <blockquote>
<p>a) Evaluate the one-dimensional Gaussian integral</p>
<p><span class="math-container">$I(a)$</span> = <span class="math-container">$\int_R exp(-ax^2)dx$</span>, <span class="math-container">$a>0$</span></p>
<p>b) evaluate the two-dimensional Gaussian integral using a)</p>
<p><span class="math-contai... | Batman | 127,428 | <p>Note that $e^{-ax^2-by^2} = e^{-a x^2}e^{-b y^2}$.</p>
<p>Then, note that $\iint_{\mathbb{R}^2} e^{-ax^2-by^2} dy dx = \iint_{\mathbb{R}^2} e^{-a x^2}e^{-b y^2} dy dx = \int_{\mathbb{R}} e^{-a x^2} dx \int_{\mathbb{R}} e^{-b y^2} dy$, and apply your result from part (a). </p>
<p>Also, note $\int_{\mathbb{R}} \frac... |
3,984,230 | <blockquote>
<p><span class="math-container">$2^x=4x$</span></p>
</blockquote>
<p>I cant seem to solve this equation. The furthest I have been able to come is
<span class="math-container">$x-\log_2(x)=2$</span>, but I can't figure how to solve. When I graph <span class="math-container">$2^x$</span> and <span class="mat... | Community | -1 | <p>The equation has indeed the easy integer solution <span class="math-container">$x=4$</span> which can be found by inspection. Graphing reveals a second solution (and it is possible to formally prove that there are no other real roots).</p>
<p>If you heard of the Taylor development, as the second root is not that lar... |
40,572 | <p>Dummit and Foote, p. 204</p>
<p>They suppose that $G$ is simple with a subgroup of index $k = p$ or $p+1$ (for a prime $p$), and embed $G$ into $S_k$ by the action on the cosets of the subgroup. Then they say</p>
<p>"Since now Sylow $p$-subgroups of $S_k$ are precisely the groups generated by a $p$-cycle, and dist... | André Nicolas | 6,312 | <p>Don't really like to give general rules or tricks, since then looking for them can interfere with the analysis of a problem. But the presence of <strong>or</strong> means we are trying to count the <strong>union</strong> of two sets $A$ and $B$. And sometimes the best way to count a union is to count $A$, count $B... |
269,242 | <p>The number of primes in each of the $\phi(n)$ residue classes relatively prime to $n$ are known to occur with asymptotically equal frequency (following from the proof of the Prime Number Theorem). Does the same result hold on pairs of consecutive primes on the $\phi(n)^2$ pairs of congruence classes?</p>
<p>To wit:... | user54998 | 54,998 | <p>I think it would be extremely unlikely that, at this point in time, one could prove anything of this kind. Even the much weaker problem (for general $n$) of whether there are infinitely many such pairs (if $\gcd(ab,n) = 1$) seems extremely difficult as soon as $\phi(n) > 2$. </p>
<p>Consider the very special ca... |
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 ... | SC Maree | 357,023 | <p>Definitions with matrices are not that obvious as you might expect. Consider for example the matrix inverse. If $x$ is a number, its inverse $x^{-1}$ should satisfy $x x^{-1} = 1$. A simple computation gives of course $x^{-1} = \frac{1}{ x}$. With a matrix $A$, this is not as simple, We require $A A^{-1} = I$, where... |
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>
| Kaster | 49,333 | <p>You have to guess particular solution first.
$$
A^p = B\sin \pi x \\
-A^p_{xx} = B\pi^2\sin \pi x = \sin \pi x \\
B = \frac 1{\pi^2}
$$
so
$$
A^p = \frac 1{\pi^2} \sin \pi x
$$
General solution of inhomogeneous problem is a sum of general solution of homogeneous problem and particular solution. So
$$
A = A^h + A^p
$... |
2,706,776 | <p>In solving the wave equation
$$u_{tt} - c^2 u_{xx} = 0$$
it is commonly 'factored'</p>
<p>$$u_{tt} - c^2 u_{xx} =
\bigg( \frac{\partial }{\partial t} - c \frac{\partial }{\partial x} \bigg)
\bigg( \frac{\partial }{\partial t} + c \frac{\partial }{\partial x} \bigg)
u = 0$$</p>
<p>to get
$$u(x,t) = f(x+ct) + g(x-c... | knzhou | 247,403 | <p>Yes, this logic is totally legitimate. In the language of linear algebra, we have two operators $A$ and $B$ and the wave equation is
$$AB \psi = 0, \quad A = \partial_t - c \partial_x, \quad B = \partial_t + c \partial_x.$$
Since $A$ and $B$ commute, they may be simultaneously diagonalized, i.e. there is a basis $\p... |
2,416,446 | <p>I need to prove the following inequality: </p>
<blockquote>
<p>$$\sum_{n=m+1}^\infty \frac{1}{n^2}\leq \frac1m$$</p>
</blockquote>
<p>But I'm stuck with it. I found online geometric justifications for this but I'd really appreciate to see actual proof. Any hints? </p>
| Kenny Lau | 328,173 | <p>$$\begin{array}{rcl}
\displaystyle \sum_{n=m+1}^\infty \frac{1}{n^2}
&\le& \displaystyle \sum_{n=m+1}^\infty \frac{1}{(n-1)n} \\
&=& \displaystyle \sum_{n=m+1}^\infty \left(\frac{1}{n-1} - \frac1n \right) \\
&=& \displaystyle \left(\frac{1}m - \frac1{m+1} \right) + \left(\frac{1}{m+1} - \frac... |
2,416,446 | <p>I need to prove the following inequality: </p>
<blockquote>
<p>$$\sum_{n=m+1}^\infty \frac{1}{n^2}\leq \frac1m$$</p>
</blockquote>
<p>But I'm stuck with it. I found online geometric justifications for this but I'd really appreciate to see actual proof. Any hints? </p>
| Bernard | 202,857 | <p><strong>Hint:</strong></p>
<p>For each $n$ you have
$$\frac1{n^2}\le\frac1{n(n-1)}=\frac1{n-1}-\frac1n.$$</p>
|
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... | Some Guy | 730,299 | <p>(<a href="https://artofproblemsolving.com" rel="nofollow noreferrer">https://artofproblemsolving.com</a>) is a very good math site. It has artofporblemsolving books and it also has a lot of math games to practice your math skills. Alcumus is the best game on there (<a href="https://artofproblemsolving.com/alcumus" r... |
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... | AD - Stop Putin - | 1,154 | <p>Your start is fine. </p>
<p>Consider in the equation <span class="math-container">$x_1v_1+x_2v_2 + x_3v_3 =0$</span>. </p>
<p>We have two choices:</p>
<ol>
<li><span class="math-container">$x_3=0$</span>.</li>
</ol>
<p>(Can you see the similarity between <span class="math-container">$v_1$</span> and <span class=... |
147,095 | <p>I was wondering if there is any stationary distribution for bipartite graph? Can we apply random walks on bipartite graph? since we know the stationary distribution can be found from Markov chain, but we have two different islands in bipartite graph and connections occur between nodes from different groups. </p>
| JP McCarthy | 35,482 | <p>$\newcommand{\Raw}{\Rightarrow}\newcommand{\raw}{\rightarrow}\newcommand{\N}{\mathbb{N}}$
I just want to echo a few of the other answers and add one point to RW's: I understand that the graph is bipartite if and only if $-1$ is an eigenvalue of the stochastic operator. There is a bit more info than the OP asked for ... |
3,389,659 | <p>if <span class="math-container">$S_n={a_1,a_2,a_3,...,a_{2n}}$</span>} where <span class="math-container">$a_1,a_2,a_3,...,a_{2n}$</span> are all distinct integers.Denote by <span class="math-container">$T$</span> the product
<span class="math-container">$$T=\prod_{i,j\epsilon S_n,i<j}{(a_i-a_j})$$</span> Prove t... | antkam | 546,005 | <p>HINT:</p>
<p>You're not quite correct for the powers of <span class="math-container">$2$</span>. There are <span class="math-container">$2n$</span> numbers total, so there can in fact be exactly <span class="math-container">$n$</span> odds and <span class="math-container">$n$</span> evens (i.e. neither has to be <... |
986,754 | <p>So I'm kind of stuck on this question and I don't exactly know how to describe this on the title header and I apologize... </p>
<blockquote>
<p>For some values of $x$, the assignment statement $y := 1-\cos(x)$ involves a difficulty. What is the difficulty? What values of $x$ are involved? What remedy do you propo... | lhf | 589 | <p>The point is that when $x$ is small, $\cos(x) \approx 1$ and so you can expect loss of precision in $y$.</p>
<p>One remedy is this:</p>
<blockquote class="spoiler">
<p> $$1-\cos(x) = (1-\cos(x)) \dfrac{1+\cos(x)}{1+\cos(x)} = \dfrac{1-\cos^2(x)}{1+\cos(x)}=\dfrac{\sin^2(x)}{1+\cos(x)}$$<br>
For $x$ really smal... |
413,108 | <p>Given a commutative ring <span class="math-container">$ R $</span> and a multiplicatively closed subset <span class="math-container">$ S $</span> of <span class="math-container">$ R $</span>, there are two ways to consturct <span class="math-container">$ S^{-1}R $</span>:</p>
<ol>
<li><p>define an equivalence relati... | Johannes Huisman | 85,592 | <p>Maybe these are the arguments you're looking for: First reduce to the case of <span class="math-container">$S$</span> being finitely generated as a multiplicative subset of <span class="math-container">$R$</span>. Next, reduce to the case <span class="math-container">$S$</span> where <span class="math-container">$S$... |
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$<... | Ben Grossmann | 81,360 | <p>What you have computed is really <span class="math-container">$\frac{\delta \operatorname{vec}(Y^TY)}{\delta \operatorname{vec}(Y)}$</span>; I will assume this is what you're really after. I will also assume that you are using the <a href="https://en.wikipedia.org/wiki/Row-_and_column-major_order" rel="nofollow nore... |
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... | Nick C | 470 | <p><em>I believe that a good student with bachelor's degree in mathematics should have sufficient knowledge to teach calculus.</em></p>
<p>I agree with this. I work with a few colleagues who came to teach at the community college after working in the high school, and their credential is a Master's of Science for Teach... |
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... | Benoît Kloeckner | 187 | <p>An aspect that does not show up much in other's answer is: exercise & test design. To construct an exercise, it is very often quite useful to have more advanced knowledge, either to be able to choose the values correctly, or to know what will echo later courses.</p>
<p>Some example:</p>
<ul>
<li>knowing finite f... |
285,227 | <p>I am trying to prove $\exp(x+y) = \exp(x) \exp(y)$.</p>
<p>I may use that $$\exp(x) = \sum_{n=0}^\infty \frac {x^n}{n!}$$
I further know how to multiply two power series in one point, i.e. if $f(x) = \sum_{n=0}^\infty c_n(x-a)^n$ and $g(x) = \sum_{k=0}^\infty d_n(x-a)^n$ then
$$
f(x)g(x) = \sum_{n=0}^\infty e_n(x-a... | salaku | 388,854 | <p>$$
\begin{align*}
& \exp(x+y)=\sum_{n=0}^{\infty}\frac{1}{n!}(x+y)^n = \sum_{n=0}^{\infty}\frac{1}{n!}\sum_{k=0}^{n}\frac{n!}{(n-k)!k!}
x^ky^{n-k} \\
=& \sum_{n=0}^{\infty}\sum_{k=0}^{n}\frac{x^k}{k!}\frac{y^{n-k}}{(n-k)!} = \lim_{N \to \infty}\sum_{n=0}^{N}\sum_{k=0}^{n} \frac{x^k}{k!}\frac{y^{n-k}}{(n-k)!}... |
599,602 | <p>Please help with this calculus question. I'm asked to solve
$$(1+y^2) \,\mathrm{d}x = (\tan^{-1}y - x)\,\mathrm{d}y.$$</p>
| mainak | 114,270 | <p><strong>strong text</strong>
we can write the given equation as, </p>
<pre><code> dx/dy = {tan^(-1)y -x} / {1 + y^(2)}
dx/dy + x/{1 + y^(2)} = tan^(-1)y /{1 + y^(2)}
</code></pre>
<p>the above one is a linear differential equation in x.
therefore, Integrating factor = exp^[ ∫ 1/{1 + y^(2)} dy]
... |
141,346 | <p>I am not sure where one looks up this type of fact. Google was not very helpful.</p>
| Daniel Schäppi | 1,649 | <p>Yes, any Grothendieck abelian category has enough injectives. I believe this goes back to Grothendieck's Tohoku paper. The category in question is Grothendieck abelian since it is equivalent to a category of additive presheaves. The domain category has objects the integers, and morphisms generated by $d_n: n \righta... |
1,850,069 | <p>Let the incircle (with center $I$) of $\triangle{ABC}$ touch the side $BC$ at $X$, and let $A'$ be the midpoint of this side. Then prove that line $A'I$ (extended) bisects $AX$.<a href="https://i.stack.imgur.com/pd7Di.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pd7Di.png" alt="enter image desc... | Jack D'Aurizio | 44,121 | <p>That is an almost trivial exercise in <a href="http://mathworld.wolfram.com/BarycentricCoordinates.html" rel="nofollow noreferrer">barycentric coordinates</a>: you just have to check that
<span class="math-container">$$ \det\left(\frac{A+X}{2};I;A'\right) = 0 \tag{1} $$</span>
given <span class="math-container">$I=\... |
3,080,005 | <p>So, I was trying to understand the "Group action" theory. I read the definition of <span class="math-container">$Stab_G$</span> and I was trying to solve some basic questions.</p>
<p>I came across with the following question: </p>
<blockquote>
<p>Let <span class="math-container">$S_7$</span> be a group that on i... | user289143 | 289,143 | <p>A group action on a set <span class="math-container">$X$</span> is defined as <span class="math-container">$\varphi : G\ \mathrm{x}\ X \rightarrow X$</span> such that <span class="math-container">$g \cdot x=\varphi (g,x)$</span> In this case <span class="math-container">$X=G$</span> and <span class="math-container">... |
342,064 | <p>We know that if an estimator, say $\widehat{\theta}$, is an unbiased estimator of $\theta$ and if its variance tends to 0 as n tends to infinity then it is a consistent estimator for $\theta$. But this is a sufficient and not a necessary condition. I am looking for an example of an estimator which is consistent but ... | gerw | 58,577 | <p>Just take an estimator $\hat\theta_n$ which has just two values:
\begin{align*}P(\hat\theta_n = \theta) &= p_n\\P(\hat\theta_n = \delta_n) &= 1-p_n\end{align*}
Now, choose sequences $\delta_n$ and $p_n$ appropriate and you should get a consistent estimator ($p_n \to 0$), whose variance does not converge to z... |
4,374,307 | <p>Problem:<br />
Suppose there are <span class="math-container">$7$</span> chairs in a row. There are <span class="math-container">$6$</span> people that are going to randomly
sit in the chairs. There are <span class="math-container">$3$</span> females and <span class="math-container">$3$</span> males. What is the pro... | Thomas Andrews | 7,933 | <p>An alternative approach is that you have the outer seats both occupied with probability <span class="math-container">$\frac57$</span> and the conditional probability that they both have a woman is <span class="math-container">$$\frac{\binom32}{\binom62}=\frac15.$$</span></p>
<p>So the probability is <span class="mat... |
4,374,307 | <p>Problem:<br />
Suppose there are <span class="math-container">$7$</span> chairs in a row. There are <span class="math-container">$6$</span> people that are going to randomly
sit in the chairs. There are <span class="math-container">$3$</span> females and <span class="math-container">$3$</span> males. What is the pro... | Toby Mak | 285,313 | <p>Here is an approach that is closest to your thought process. The three females and the three males are all <em>indistinguishable</em>, so we use combinations instead of permutations. Picking any one sex first gives:</p>
<p><span class="math-container">$${7 \choose 3} \cdot {4 \choose 3}$$</span></p>
<p>total possibi... |
3,011,080 | <p>I was reading:</p>
<blockquote>
<p>Take <span class="math-container">$N \in \mathbf N$</span> such that <span class="math-container">$a_i \leq N$</span> for all <span class="math-container">$i \leq n$</span> and N
is a multiple of every prime number ≤ n. We claim that then 1 + N, 1 +
2N, . . . , 1 + nN, 1 + (... | Will Jagy | 10,400 | <p>All you need is the conclusion that <span class="math-container">$p$</span> does not divide <span class="math-container">$N.$</span> It is enough to show that <span class="math-container">$\gcd(p,N) = 1.$</span> Now, as there is an integer <span class="math-container">$t$</span> such that <span class="math-container... |
3,011,080 | <p>I was reading:</p>
<blockquote>
<p>Take <span class="math-container">$N \in \mathbf N$</span> such that <span class="math-container">$a_i \leq N$</span> for all <span class="math-container">$i \leq n$</span> and N
is a multiple of every prime number ≤ n. We claim that then 1 + N, 1 +
2N, . . . , 1 + nN, 1 + (... | Charlie Parker | 118,359 | <p>We want to show:</p>
<p><span class="math-container">$$ 1+N, 1+2N, \dots, 1+(n+1)N$$</span></p>
<p>is coprime (i.e. share no common factors). It's sufficient to show they don't share a common prime factor (since all other factors are made of primes, since if its a factor of a specific number then that number can b... |
2,127,679 | <p>I need to find $\frac{a}{b} \mod c$.<br>
This is equal to $(a\cdot b^{\phi(c)-1}\mod c)$, when $b,c$ are co-prime. But what if that's not the case?<br>
To be more clear, I need $$\frac{10^{a\cdot b}-1}{10^b-1}\mod P$$ </p>
| Bill Dubuque | 242 | <p>Update: you edited your question to include a specific example. Here there is some ambiguity depending on whether the division is intended in the integers, or in the integers mod $m$. Let's consider a simpler case, the fraction $\ x\equiv {6}/2\pmod{\!10}.\,$ If this denotes division in the integers then $\,6/2\,$ d... |
2,127,679 | <p>I need to find $\frac{a}{b} \mod c$.<br>
This is equal to $(a\cdot b^{\phi(c)-1}\mod c)$, when $b,c$ are co-prime. But what if that's not the case?<br>
To be more clear, I need $$\frac{10^{a\cdot b}-1}{10^b-1}\mod P$$ </p>
| Matt B | 111,938 | <p>In general, it doesn't make sense if $b$ is not coprime to $c$. However, in your case, the fraction is actually an integer so we can make sense of it.</p>
<p>Not that $\dfrac{x^n-1}{x-1}=x^{n-1}+x^{n-2}+ \cdots +1$ so setting $x=10^b$ and $n=a$, you get $\dfrac{10^{ab}-1}{10^b-1}=10^{a-1}+x^{a-2}+ \cdots +1$ and yo... |
235,430 | <p>Suppose that a bounded sequence of real numbers $s_i$ ($i\in\omega$) has a limit $\alpha$ along some ultrafilter $\mu_1\in \beta{\Bbb N}\setminus{\Bbb N}$. Then given another ultrafilter $\mu_2\in \beta{\Bbb N}\setminus{\Bbb N}$, surely there exists some rearrangement $s_{r(i)}$ of $s_i$ that has the same limit $\al... | Pushpendre | 29,887 | <p>If Y is any diagonal matrix then the lower bound is achieved so the lower bound is tight for diagonal matrices. There can't be any sharper bound defined in terms of eigen values since there exists a diagonal matrix with those eigenvalues. </p>
<p>Update after edit to the question: </p>
<p>If a bound of the type $... |
202,719 | <p><code>Reduce</code> often provides a much fuller solution than <code>Solve</code>. But it's always in the form of a true statement rather than functions or replacement rules, e.g.</p>
<p>Input:</p>
<pre><code>Reduce[Sin[x^2] + Cos[a] == 0 && -π/2 <= x <= π/2, x]
</code></pre>
<p>Output:</p>
<pre><c... | J. M.'s persistent exhaustion | 50 | <p>At least for the OP's specific case, here is one possibility:</p>
<pre><code>Piecewise[Append[
KeyValueMap[{#2, #1} &,
GroupBy[Cases[
BooleanConvert[Reduce[Sin[x^2] + Cos[a] == 0 &&
-π/2 <= x <= π/2,... |
769,504 | <p>It is mentioned in <a href="http://ac.els-cdn.com/0166864182900657/1-s2.0-0166864182900657-main.pdf?_tid=2ecebd88-ccce-11e3-ae74-00000aab0f6c&acdnat=1398467347_2b1c578dc3ae8c1a9107e7444203edb6" rel="nofollow noreferrer">this</a> article, that, the one point compactification of an uncountable discrete space, is A... | topsi | 60,164 | <p>I think that if we take, $\mathbb R$ with the discrete topology and
$X = \mathbb R \{ \infty \}$ to be the Alexandroff one point compactification of $\mathbb R$, we get a space which is not first counable in which ONE has a winning strategy in the game: $G_{np}(\infty,X)$.</p>
<p><strong>Proof:</strong> Suppose we... |
440,439 | <p>I was working more on the topic on my previous question when I have to know whether the following statement is true to circumvent the "exception" caused by division by singular matrices; again, long story short, the statement follows:</p>
<p>If two singular matrices <span class="math-container">$A, B$</spa... | Alexandre Eremenko | 25,510 | <p>Every bounded analytic function <span class="math-container">$h$</span> in the disk has the representation
<span class="math-container">$$h(z)=B(z)\exp(-P(z)),$$</span>
where <span class="math-container">$B$</span> is a Blaschke product and <span class="math-container">$P$</span> has positive imaginary part. Applyin... |
62,177 | <p>One of the most mind boggling results in my opinion is, with the axiom of choice/well-ordering principle, there exist such things as uncountable well-ordered sets $(A,\leq)$. </p>
<p>With this is mind, does there exist some well ordered set $(B,\leq)$ with some special element $b$ such that the set of all elements ... | Andrés E. Caicedo | 462 | <p>There is no need to use the axiom of choice here. Suppose $X$ is an infinite well-orderable set. We argue that there is a well-ordered set $(Y,<)$ with $Y$ of strictly larger cardinality than $X$. </p>
<p>For this, consider the set $A$ of all binary relations $R\subseteq X\times X$ such that $R$ is a well-orderi... |
863,364 | <p><img src="https://i.stack.imgur.com/w6R2g.jpg" alt="enter image description here"></p>
<p>I am missing the 3D graph for the equation $x^2+2z^2=1$.</p>
| angryavian | 43,949 | <p>I think that as long as everything is well defined (for example, being in the <a href="http://en.wikipedia.org/wiki/Schwartz_space" rel="nofollow">Schwartz space</a>), then the result is what you think it should be. For example, $$F\left(\frac{\partial^2}{\partial x \partial y}\right) u(x,y,t) = (ix)(iy) \hat{u}(x,y... |
1,175,297 | <p>Note: The following definitions from my book, Discrete Mathematics and Its Applications [7th ed, 598].</p>
<p>This is my book's definition for a reflexive relation
<img src="https://i.stack.imgur.com/og5wE.png" alt="enter image description here"></p>
<p>This is my book's definition for a anti symmetric relation
<i... | Graham Kemp | 135,106 | <p>Reflexivity <em>requires</em> all elements to be both way related with themselves. However, this does <em>not</em> prohibit non-equal elements from being both way related with each other.</p>
<p>That is: $\qquad\forall a\in A: (a,a)\in R$ from which we can prove: $$\forall a\in A\;\forall b\in A:\Big(a=b \;\to\; ... |
868,935 | <p>I saw a proof that
$$
\lim_{x\to 0} \ln|x|\cdot x = 0
$$
where is is argued that for $x \in (0,1)$ we have
$$
| \ln(x) x | = \left| \int_1^x x/t ~\mathrm d t \right| = \left| \int_x^1 x/t ~\mathrm d t \right| \le \left|\int_x^1 1 dt\right| = |1 - x| \le 1
$$
and therefore the result follows, but why should the f... | Matt B. | 164,029 | <p>$x \ln(x)$ is monotonous as well (decreasing around 0), so it must have a limit.</p>
|
2,860,321 | <p>Suppose $L/K$ is a Galois extension of local fields with Galois group $G = \operatorname{Gal}(L/K)$. Let $K'$ be the maximal unramified Extension of $K$ in $L$.</p>
<p>The Definition of the <strong>inertia group</strong> of $L/K$ is given by $I = I_{L/K} = \operatorname{Gal}(L/K')$ which I understand.</p>
<p>In so... | D_S | 28,556 | <p>Assume $L/K$ are local fields. Let $\mathfrak p$ (resp. $\mathfrak P$) be the unique maximal ideal of $\mathcal O_K$ (resp. $\mathcal O_L$). Let $\kappa(\mathfrak p) = \mathcal O_K/\mathfrak p$ and $\kappa(\mathfrak P) = \mathcal O_L/\mathfrak P$. The inclusion of $\mathcal O_K$ into $\mathcal O_L$ induces an inc... |
2,860,321 | <p>Suppose $L/K$ is a Galois extension of local fields with Galois group $G = \operatorname{Gal}(L/K)$. Let $K'$ be the maximal unramified Extension of $K$ in $L$.</p>
<p>The Definition of the <strong>inertia group</strong> of $L/K$ is given by $I = I_{L/K} = \operatorname{Gal}(L/K')$ which I understand.</p>
<p>In so... | nguyen quang do | 300,700 | <p>Your question makes no sense if you don't specify what your fields $K, L$ are. Since you introduce the maximal unramified subextension $K'/K$ of $L/K$, there is only one possible interpretation: $K$ is a local field, i.e. it is complete w.r.t. a non archimedean discrete valuation $v$. The ring of integers of $K$ is ... |
4,072,769 | <blockquote>
<p>How to evaluate this?
<span class="math-container">$$\prod_{k=1}^m \tan \frac{k\pi}{2m+1}$$</span></p>
</blockquote>
<p>My work</p>
<p>I couldn't figure out a method to solve this product.
I thought that this identity could help.
<span class="math-container">$$\frac{e^{i\theta}-1}{e^{i\theta}+1}=i\tan \... | mathlover123 | 761,688 | <p>So you can start by breaking the product to get:
<span class="math-container">$$\prod_{k=1}^{2m}\tan{(\frac{k\pi}{2m+1})}=\prod_{k=1}^{2m}\sin{(\frac{k\pi}{2m+1})}\prod_{k=1}^{2m}\frac{1}{\cos{(\frac{k\pi}{2m+1}})}$$</span>
Now :
<span class="math-container">$$\prod_{k=1}^{2m}\cos{(\frac{k\pi}{2m+1})}=\frac{(-1)^{m}... |
1,115,545 | <p>In my lecture notes we have the following:</p>
<p>The set <span class="math-container">$$\mathbb{P}^2(K)=\{[x, y, z] | (x, y, z) \in (K^3)^{\star}\}$$</span> is called projective plane over <span class="math-container">$K$</span>.</p>
<p>There are the following cases:</p>
<ul>
<li><p><span class="math-container">$z ... | sciona | 195,458 | <p>Use Cauchy-Schwarz Inequality: $$(a+b+b+c+c+d+d+a)\left(\frac{a^2}{a+b}+\frac{d^2}{a+d}+\frac{b^2}{b+c}+\frac{c^2}{c+d}\right) \ge (a+b+c+d)^2$$</p>
<p>Alternative approach: $$\sum\limits_{cyc} \frac{a^2}{a+b} = \sum\limits_{cyc} \left(\frac{a^2}{a+b} - (a-b)\right) = \sum\limits_{cyc} \frac{b^2}{a+b}$$</p>
<p>The... |
1,115,545 | <p>In my lecture notes we have the following:</p>
<p>The set <span class="math-container">$$\mathbb{P}^2(K)=\{[x, y, z] | (x, y, z) \in (K^3)^{\star}\}$$</span> is called projective plane over <span class="math-container">$K$</span>.</p>
<p>There are the following cases:</p>
<ul>
<li><p><span class="math-container">$z ... | user2345215 | 131,872 | <p>Another approach using $4ab\le(a+b)^2$:
\begin{align*}\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+d}+\frac{d^2}{d+a}&=\left(a-\frac{ab}{a+b}\right)+\left(b-\frac{bc}{b+c}\right)+\left(c-\frac{cd}{c+d}\right)+\left(d-\frac{da}{d+a}\right)\\
&=1-\left(\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{cd}{c+d}+\frac{da}{d+a... |
4,492,930 | <p>Let <span class="math-container">$\mathfrak{c}$</span> denote the cardinality of the continuum. I sketch an intuitive but non-rigorous argument that <span class="math-container">$|\mathbb{R}^\mathbb{N}| = \mathfrak{c}$</span>, with the question:</p>
<p><strong>Question</strong>: can this argument be made rigorous?</... | Sourav Ghosh | 977,780 | <p>Yes. As inner product is additive in each component. But you have to be careful while dealing with homogeneity as inner product is conjugate linear. But in real inner product it's linear in both components i.e a real inner product is a bilinear map.</p>
<p>Here we only need additivity, so it's fine for any inner pr... |
1,579,781 | <blockquote>
<p>If $x+y+z=6$ and $xyz=2$, then find the value of $$\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... | Pieter Rousseau | 286,128 | <p>$$x+y+z=6$$
Divide both sides with $xyz$
$$\frac{x+y+z}{xyz}=\frac{6}{xyz}$$
$$\frac{x}{xyz}+\frac{y}{xyz}+\frac{z}{xyz}=\frac{6}{2}$$
$$\frac{1}{yz}+\frac{1}{xz}+\frac{1}{xy}=3$$</p>
|
90,876 | <p>$$2x-\dfrac{x+1}{2} + \dfrac{1}{3}(x+3)= \dfrac{7}{3}$$</p>
<p>When I solve this I always end up with 11x = 5, which is wrong, no matter which way I solve it. Does anyone know how to solve it? Steps? (Because I know the answer should be x=1)</p>
| The Chaz 2.0 | 7,850 | <p>Multiply by $6$ to clear fractions:</p>
<p>$12x - 3(x +1) +2(x +3) = 14$</p>
<p>Eventually you'll get</p>
<p>$11x = 11$</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... | Dany Majard | 5,375 | <p>As far as higher cubical categories are concerned, a connection will allow you to literally rotate a face, i.e. turn a face of one type into a face of another type, in an invertible way.
In short it materializes an equivalence between the different types of faces into special degenerate cubes.</p>
<p>The 2d case fo... |
3,023,726 | <p>I'm trying to solve a problem that I can't seem to work out.</p>
<blockquote>
<p><span class="math-container">$f$</span> is an entire function. Prove that <span class="math-container">$|f^{(n)}(0)|< n!n^n$</span> for at least 1 <span class="math-container">$n$</span>. </p>
</blockquote>
<p>I've been thinking ... | Community | -1 | <p><strong>Hint:</strong> Try proving it by contradiction. Suppose <span class="math-container">$|f^{(n)}(0)| \geq n! n^n$</span> for all <span class="math-container">$n \in \Bbb{N}$</span>. You are given that <span class="math-container">$f$</span> is entire. Can you say anything about the radius of convergence of <sp... |
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>
| Guy | 206,544 | <p>Note that
$$4^3=64\equiv 1\mod 9$$</p>
<p>Now divide to three cases, depending on $\left(n\mod 3\right)$.</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>
| Maximal Ideal | 352,912 | <p>Another approach: consider the fact that
$$(4^{n}-1)-3n = (4-1)(1+4+\cdots+4^{n-1})-3n = 3[1+4+\cdots+4^{n-1} - n].$$</p>
|
2,403,201 | <p>How do I solve for $x$:</p>
<p>$$\log\left(\frac{1.07^x}{1050-2.5x}\right)=\log\left(\frac{1.2}{828}\right)$$</p>
<p>If I raise both sides to the power of $10$, I get:
$\dfrac{1.07^x}{1050-2.5x}=\frac{1}{690}$</p>
<p>Then I'm stuck. How do I solve this ?</p>
<p>As suggest by @Kevin, I have decided to add my take... | Community | -1 | <p>After seeing the plot of the function, the orders of magnitude are such that in a first approximation the $x$ at the denominator can be ignored, and you get</p>
<p>$$x\approx\log_{1.07}\frac{1050}{690}=6.2054\cdots.$$</p>
<p>As said by Claude, next approximations are given by Newton, and two or three iterations sh... |
3,454,514 | <blockquote>
<p>How to change the integration order in the given integral?
<span class="math-container">$$
\int\limits_0^1dx\int\limits_0^1dy\int\limits_0^{x^2+y^2}fdz\rightarrow
\int\limits_?^?dz\int\limits_?^?dy\int\limits_?^?fdx
$$</span></p>
</blockquote>
<p>I tried to make a graphic interpretation, but it see... | Hans Lundmark | 1,242 | <p>The iterated integral on the left equals the triple integral
<span class="math-container">$$
I = \iiint_D f \, dxdydz
,
$$</span>
where
<span class="math-container">$$
D = \{ (x,y,z) \in \mathbf{R}^3 : 0 \le x \le 1, \, 0 \le y \le 1, \, 0 \le z \le x^2+y^2 \}
.
$$</span>
To write this as an iterated integral with <... |
103,540 | <p>Suppose you have a triangular chessboard of size $n$, whose "squares" are ordered triples $(x,y,z)$ of nonnegative integers that add up to $n$. A rook can move to any other point that agrees with it in one coordinate -- for example, if you are on $(3,1,4)$ then you can move to $(2,2,4)$ or to $(6,1,1)$, but not to ... | Cristi Stoica | 10,095 | <p>Here is a paper about this problem: <a href="http://www.cin.ufpe.br/~pcp/nonattacking_queens.pdf" rel="noreferrer">"Non-attacking queens on a triangle"</a>.</p>
<p>And here's another one <a href="http://arxiv.org/abs/0910.4325" rel="noreferrer">"Putting Dots in Triangles"</a></p>
|
103,540 | <p>Suppose you have a triangular chessboard of size $n$, whose "squares" are ordered triples $(x,y,z)$ of nonnegative integers that add up to $n$. A rook can move to any other point that agrees with it in one coordinate -- for example, if you are on $(3,1,4)$ then you can move to $(2,2,4)$ or to $(6,1,1)$, but not to ... | Gerhard Paseman | 3,402 | <p>This has a connection with additive permutations, asked about here:
<a href="https://mathoverflow.net/q/211276">Are there enough additive permutations?</a> .</p>
<p><a href="http://oeis.org/A002047" rel="nofollow noreferrer">http://oeis.org/A002047</a> has references to a similar problem involving a hexagonal board... |
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>
| Michael Galuza | 240,002 | <p>Hint:
$$
1+\cos x + \frac{1}{2!}\cos 2x + \ldots = \Re(e^{0ix} + e^{1ix} + \frac{1}{2!}e^{2ix} + \ldots)=\Re e^{e^{ix}}
$$</p>
<p>$$
e^{ix}=\cos x+i\sin x\\\Longrightarrow e^{e^{ix}} = e^{\cos x}e^{i\sin x}=e^{\cos x}(\cos(\sin x)+i\sin(\sin x))
$$
Your sum is
$$
e^{\cos x}\cos(\sin x)
$$</p>
|
3,917,912 | <p>I am reading an article where the author seems to use a known relationship between the sum of a finite sequence of real positive numbers <span class="math-container">$a_1 +a_2 +... +a_n = m$</span> and the sum of their reciprocals. In particular, I suspect that
<span class="math-container">\begin{equation}
\sum_{i=1... | TurlocTheRed | 397,318 | <p>This follows form Lagrange Multipliers.</p>
<p>Let <span class="math-container">$S_1=\sum_{i=0}^n a_i=m$</span>. Minimize <span class="math-container">$S_2=\sum_{i=0}^n\frac{1}{a_i}$</span>.</p>
<p>Treat each <span class="math-container">$a_i$</span> as an independent variable.</p>
<p><span class="math-container">$\... |
3,981,983 | <p>I had an interesting math problem presented to me some time ago by a friend (he stated it in non-mathematical terms). At what angle would you launch a projectile from a spaceship/satellite such that it left that object and went on to hit another orbiting object? Then as a supplemental question he asked at what angle... | Brian M. Scott | 12,042 | <p>If <span class="math-container">$x\in X\setminus A$</span>, then <span class="math-container">$X\setminus A$</span> is an open nbhd of <span class="math-container">$x$</span> disjoint from <span class="math-container">$A$</span>. If <span class="math-container">$a\in A$</span>, then <span class="math-container">$a$<... |
18,413 | <p>The thought came from the following problem:</p>
<p>Let $V$ be a Euclidean space. Let $T$ be an inner product on $V$. Let $f$ be a linear transformation $f:V \to V$ such that $T(x,f(y))=T(f(x),y)$ for $x,y\in V$. Let $v_1,\dots,v_n$ be an orthonormal basis, and let $A=(a_{ij})$ be the matrix of $f$ with respect ... | Chris Card | 1,470 | <p>It's not true in general, e.g. </p>
<p>$A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$
$B = \begin{pmatrix} 1 & 1 \\ 2 & 1 \end{pmatrix}$</p>
<p>$AB = \begin{pmatrix} 4 & 3 \\ 3 & 2 \end{pmatrix}$</p>
<p>(with thanks to Rahul for formatting help)</p>
|
2,515,765 | <p>The following question is from an intermediate calculus book I am going through: </p>
<p>Find two sets in $\mathbb R^2$ that have the same interior, but whose complements have different interiors. </p>
<p>This seems like the kind of question that should be fairly straightforward, but I just can't think of an answe... | Carlos Jiménez | 356,536 | <p>Consider $A=\mathbb{Q}\times\mathbb{Q}$ and $B=\emptyset$. Clearly, $\text{Int}(A)=\text{Int}(B)=\emptyset$ but $\text{Int}(\mathbb{R}^2\setminus A)=\emptyset$ and $\text{Int}(\mathbb{R}^2\setminus B)=\text{Int}(\mathbb{R}^2)=\mathbb{R}^2$ </p>
|
3,000,862 | <p>I can name at least 4 different ways of representing <span class="math-container">$\exp$</span> function:</p>
<ol>
<li>Taylor series: For <span class="math-container">$x \in \mathbb{R}, \exp(x) = \sum_{k=0}^{\infty} \frac{x^k}{k!}$</span>.</li>
<li>Differential equation: <span class="math-container">$f: \mathbb{R} ... | Oscar Lanzi | 248,217 | <p>You may have a sign error. I render <span class="math-container">$s^3$</span> as <span class="math-container">$-(1-i\sqrt{3})/16$</span> but you seem to have <span class="math-container">$+(1-i\sqrt{3})/16$</span>. But even with the sign correction (which probably does get you a good answer), your approach is not ... |
668,291 | <p>If $h$ and $k$ are any two distinct integers, then $h^n-k^n$ is divisible by $h-k$.</p>
<p>Let's start with the basis. Let $n=1$, then
$h^1-k^1 = h-k$</p>
<p>Now for the induction, I can't use $k$ because I don't want to be confused. So let $P(r)$ for $h^n-k^n$ and that's $h^r-k^r$</p>
<p>$h^r-k^r = h-k$</p>
<... | Pedro | 23,350 | <p>Another option is to write $$h^{r+1}-k^{r+1}=hh^{r}-kk^{r}=(h+k)(h^r-k^r)-kh(h^{r-1}-k^{r-1})$$</p>
<p>in a similar fashion to what we do to prove that if $\alpha,\beta$ are the roots of $f[X]\in\Bbb Z[X]$, $\alpha^n+\beta^n\in\Bbb Z$ for every $n\in\Bbb Z$. Indeed, $\alpha+\beta,\alpha\beta\in\Bbb Z$ by Vieta and<... |
1,710,523 | <p>Im asked to find the arc length of :</p>
<p>$$
\int_{-2}^{x}\sqrt{3w^4-1}dw
$$
where x is between -2 and -1.</p>
<p>Do I find the integral just as I would normally find it then find the arc length of that? Im a little confused on the notation I guess.</p>
| Jan Eerland | 226,665 | <p>Use the formula for the arc length:</p>
<p>$$\int_{-2}^{-1}\sqrt{1+\frac{\partial}{\partial x}\left[\int_{-2}^{x}\sqrt{3w^4-1}\space\text{d}w\right]^2}\space\text{d}x=$$</p>
<hr>
<p>Using the fundamental theorem of calculus:</p>
<p>$$\frac{\partial}{\partial x}\left[\int_{-2}^{x}\sqrt{3w^4-1}\space\text{d}w\righ... |
19,305 | <p>If I compute the eigenvalues and eigenvectors using <code>numpy.linalg.eig</code> (from Python), the eigenvalues returned seem to be all over the place. Using, for example, <a href="http://archive.ics.uci.edu/ml/machine-learning-databases/iris/iris.data" rel="nofollow">the Iris dataset</a>, the normalized Eigenvalue... | user72308 | 72,308 | <p>I replicate the dataset in your question.and got the normalized Eigenvalues are [2.9108 0.9212 0.1474 0.0206].</p>
<p>first to get this answer your need to normalize the data :1) subtract the mean for each column (each variable over time),2)divided demeaned column by its variance (variance after demean) 3) then yo... |
3,441,346 | <p>I was asked to prove that a set <span class="math-container">$X$</span> is closed if and only if it contains all its limit points. I proceeded like so:</p>
<p>Let <span class="math-container">$X^\dagger=\partial X \cap X´$</span> and <span class="math-container">$X^\ast=\partial X \backslash X´$</span> with <span c... | copper.hat | 27,978 | <p>There is no need to drag the boundary and derived sets into the proof.</p>
<p>Suppose <span class="math-container">$X$</span> is closed and <span class="math-container">$p$</span> is a limit point of <span class="math-container">$X$</span>. Suppose <span class="math-container">$p \notin X$</span>. Since <span class... |
2,853,668 | <blockquote>
<p>Show that $$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{x^n}$$ converges for every $x>1$.</p>
</blockquote>
<p>let $a(x)$ be the sum of the series. does $a$ continious at $x=2$? differentiable?</p>
<p>I guess the first part is with leibniz but I am not sure about it.</p>
| Mostafa Ayaz | 518,023 | <p>Hint:</p>
<p>What if you use the ratio test?$$\lim_{n\to\infty}|\dfrac{a_{n+1}}{a_n}|$$</p>
|
2,484,004 | <p><span class="math-container">$M^*(a,b)=b-a$</span> we know that this fact but how we can prove closed intervals are Lebesgue measurable. I tried to prove by using <span class="math-container">$\cap ((a-\frac1n),(b+\frac1n))$</span> But ı totaly stucked :( please help me guys</p>
| Dr. Sonnhard Graubner | 175,066 | <p>HINT: plug in $$y^2=1-4x^2$$ in your function, then it containes only the variable $x$</p>
|
15,871 | <p>I would like to state something about the existence of solutions $x_1,x_2,\dots,x_n \in \mathbb{R}$ to the set of equations</p>
<p>$\sum_{j=1}^n x_j^k = np_k$, $k=1,2,\dots,m$</p>
<p>for suitable constants $p_k$. By "suitable", I mean that there are some basic requirements that the $p_k$ clearly need to satisfy... | Joel David Hamkins | 1,946 | <p>First, let me mention that one must be careful when asserting that an implication fails. Taken literally, the assertion "D(x) does not imply P(x)" is logically equivalent to the assertion that D(x) is true and P(x) is false. This meaning of material implication that is used in mathematics is not the same as the nat... |
1,291,107 | <p>Let $X$ be random variable and $f$ it's density. How can one calculate $E(X\vert X<a)$?</p>
<p>From definition we have:</p>
<p>$$E(X\vert X<a)=\frac{E\left(X \mathbb{1}_{\{X<a\}}\right)}{P(X<a)}$$</p>
<p>Is this equal to:</p>
<p>$$\frac{\int_{\{X<a\}}xf(x)dx}{P(X<a)}$$</p>
<p>? If yes, then ho... | Martigan | 146,393 | <p>Euh... I think you overcomplicated things here...</p>
<p>$(1-x)(x-5)^3=x-1$ is equivalent to $(1-x)[(x-5)^3+1]=0$</p>
<p>Either $x=1$ or $(x-5)^3=-1$...</p>
|
135,675 | <p>Let $D$ be the Dirac-Operator on $\mathbb{R}^n$ or more generally the Dirac spinor bundle $\mathcal{S}\to M$ of a (semi-)Riemannian spin manifold $M$. Then we consider $D$ as an unbouded Operator on $\mathcal{H}=L^2(\mathbb{R}^n)$ with domain $C^\infty_c(\mathbb{R}^n,\mathbb{C}^N)$. Then it is said that the operator... | Paul Siegel | 4,362 | <p>As discussed in the comments, the statement probably needs to be modified in order for $\langle D \rangle^{-n}$ to be defined. I'm guessing that the correct statement should fit into the following framework:</p>
<hr>
<p>Proposition: Let $D$ be an essentially self-adjoint first order elliptic operator on a possibl... |
1,176,615 | <p>I am invited to calculate the minimum of the following set:</p>
<p>$\big\{ \lfloor xy + \frac{1}{xy} \rfloor \,\Big|\, (x+1)(y+1)=2 ,\, 0<x,y \in \mathbb{R} \big\}$.</p>
<p>Is there any idea?</p>
<p>(The question changed because there is no maximum for the set (as proved in the following answers) and I assume ... | Christian Blatter | 1,303 | <p>From $xy+x+y=1$ and $x>0$, $y>0$ it follows that $xy<1$. Since $t\mapsto t+{1\over t}$ is decreasing when $t<1$ we conclude that we have to make
$xy$ is as large as possible. Let $x+y=:s$. Then
$$1-s=xy\leq{s^2\over4}\ .$$
The largest possible $xy$ goes with the smallest admissible $s>0$, and the la... |
75,880 | <p>Say $f:X\rightarrow Y$ and $g:Y\rightarrow X$ are functions where $g\circ f:X\rightarrow X$ is the identity. Which of $f$ and $g$ is onto, and which is one-to-one?</p>
| karmic_mishap | 17,529 | <p>I think $f$ should be one-to-one and $g$ should be onto, since $g$ has to cover all of $X$ in its range and $f$ has to make $X$ correspond in a one-to-one fashion with $Y$. It seems that $g$ could be not one-to-one if it is an inverse of $f$ that discards some of the information that being a member of Y conveys. For... |
2,362,790 | <p>I'm interested in the problem linked with <a href="https://math.stackexchange.com/questions/770117/determinant-of-circulant-matrix/2362754#2362754">this answer</a>. </p>
<p>Let $ f(x) = a_n + a_1 x + \dots + a_{n-1} x^{n-1} $ be polynomial with distinct $a_i$ which are <strong>primes</strong>. </p>
<p>(Po... | Gerry Myerson | 8,269 | <p>For $n=4$, $x^4-1$ and $13+11x+17x^2+19x^3$ both have the root $x=-1$. Any number of similar examples can be produced. </p>
|
879,640 | <p>Does a matrix have only one inverse matrix (like the inverse of an element in a field)? If so, does this mean that</p>
<p>$A,B \text{ have the same inverse matrix} \iff A=B$?</p>
| Disintegrating By Parts | 112,478 | <p>If $A$, $B$ are square matrices with same inverse $C$, then $AC=CA=I$ and $BC=CB=I$. Therefore,
$$
A =AI= A(CB)= (AC)B = IB = B.
$$
The odd thing about matrices is this: If $A$, $B$ are $n\times n$ matrices over a field, then $AB=I$ iff $BA=I$. This is a direct consequence of the fact that the $N\times N$ ma... |
3,243,503 | <p>If <span class="math-container">$x + y = 2c$</span>, find minimum value of
<span class="math-container">$ \sec x +\sec y $</span> if <span class="math-container">$x,y\in(0,\pi/2)$</span>, in terms of <span class="math-container">$c$</span>.</p>
<p>I was able to solve by differentiating the equation and got the ans... | Ma Joad | 516,814 | <p><span class="math-container">$$\frac{d}{dx}(\sec x+\sec y)=\frac{\sin x}{\cos^2x}+\frac{\sin y}{\cos^2y}\frac{dy}{dx}=0,\\
\frac{dy}{dx}=-1,\\
\frac{\sin x}{\cos^2x}=\frac{\sin y}{\cos^2y}\\
\Rightarrow x=y=c.$$</span>
Minimum value: <span class="math-container">$2\sec c.$</span></p>
|
885,778 | <ol>
<li><p>Is there any group of order 36 with no subgroup of order 6?</p></li>
<li><p>Is there any group of order $p^2q^2$ with no subgroup of order $pq$?</p></li>
<li><p>Is there any group of order $p^{2m}q^{2m}$ with no subgroup of order $p^mq^m$?</p></li>
<li><p>Is there any group of order $p^{2m}q^{2n}$ with no s... | Geoff Robinson | 13,147 | <p>For question 2, I believe the answer is yes, which also settles question $3$ and $4.$ Take $p = 149$ and $q = 5.$ Then a cyclic group of order $25$ acts irreducibly and faithfully on an elementary Abelian $p$-group of order $p^{2}.$ Furthermore, in the resulting semidirect product, each element of order $5$ acts i... |
1,450,476 | <p>I'm in number theory and I currently have these problems assigned as homework. I've looked through the sections containing these problems and I've solved/proved most of the other problems, but I can't figure these ones out.</p>
<ol>
<li>For $n>1$, show that every prime divisor of $n!+1$ is an odd integer that is... | Eric Auld | 76,333 | <p>It is certainly not bijective. Showing it is a monomorphism is trivial (why?) Showing it is an epimorphism is almost as easy; think about two functions $f,g: \mathbb{Q} \to R$ that disagree on some value. Can it be that $f \circ i = g \circ i$?</p>
|
1,949,874 | <blockquote>
<p><span class="math-container">$P_2(R)$</span> is the set of polynomials of degree two or lower.</p>
<p>Show that there is a unique basis <span class="math-container">$\{p_1, p_2, p_3\}$</span> of <span class="math-container">$P_2(R)$</span> with the property that <span class="math-container">$p_1(0) = 1,... | Paul Sinclair | 258,282 | <p>You are asked to prove two things:</p>
<ul>
<li>Existance: there is such a basis, and</li>
<li>Uniqueness: there is at most one such basis.</li>
</ul>
<p>The proof you describe is only for proving uniqueness. Uniqueness is almost always easier to prove than existance.</p>
<p>To prove such a basis exists, you need... |
1,949,874 | <blockquote>
<p><span class="math-container">$P_2(R)$</span> is the set of polynomials of degree two or lower.</p>
<p>Show that there is a unique basis <span class="math-container">$\{p_1, p_2, p_3\}$</span> of <span class="math-container">$P_2(R)$</span> with the property that <span class="math-container">$p_1(0) = 1,... | Gerry Myerson | 8,269 | <p>Paul Sinclair's solution is fine, but if you want to find $p_1,p_2,p_3$, you can do it this way: </p>
<p>First, you know that if $p(x)$ is a polynomial and $p(a)=0$ then $x-a$ is a factor. So $p_1$ must be $m_1(x-1)(x-2)$ for some constant $m_1$, and you can evaluate that constant by considering $p_1(0)$. Similarly... |
1,507,526 | <p>Let $S$ be the portion of the sphere $x^2+y^2+z^2=9$, where $1\leq x^2+y^2\leq4$ and $z\geq0$. Calculate the surface area of $S$</p>
<p>Ok i'm really confused with this one. I know i have to apply the surface area formula but and possibly spherical coordinates but i can't seem how to get the integral out.</p>
<p>T... | Simon | 282,296 | <p>Cylindrical coordinates are the way to go!
Recall that if you have a surface S the surface integral is equal to
\begin{equation}
\int\int f(x,y,z) dS
\end{equation}
Well, you can represent z as a function of x and y.
Also recall that dS stands for the "arc-length" at that point.
Therefore, you can rewrite the int... |
1,598,545 | <p>Maybe I am not well versed with the actual definition of mean, but I have a doubt. On most resources, people say that arithmetic mean is the sum of $n$ observations divided by n. So my first question: </p>
<blockquote>
<p>How does this formula work? Is there any derivation to it? If not,
then while creating thi... | Ross Millikan | 1,827 | <p>The arithmetic mean formula is a definition of the term, so there is no derivation. The mean does not involve the range that the numbers might be over, like your $1$ to $9$ example. It only involves the actual numbers. You can define the term central value to take a set of numbers and return half the sum of the m... |
1,598,545 | <p>Maybe I am not well versed with the actual definition of mean, but I have a doubt. On most resources, people say that arithmetic mean is the sum of $n$ observations divided by n. So my first question: </p>
<blockquote>
<p>How does this formula work? Is there any derivation to it? If not,
then while creating thi... | Ian | 83,396 | <p>I really don't think that people came up with the formula for the arithmetic mean by starting with a nice property and deriving the formula to obtain that property. However, one useful characterization of the mean is that it is the unique minimizer of the function $f(m)=\sum_{i=1}^n (x_i-m)^2$. That is, it is the nu... |
761,286 | <p>let $G$ be an infinite group of the form $G_1 \oplus G_2 \oplus \dots \oplus G_n$ where each $G_i$ is a <strong>non trivial</strong> group and $n>1$. Prove that $G$ is not cyclic.</p>
<p><strong>Attempt</strong> : Let $G = G_1 \oplus G_2 \oplus \dots \oplus G_n$ be cyclic.</p>
<p>then $\exists ~g =(g_1,g_2,....... | ajotatxe | 132,456 | <p>It seems that the worst (and the best) scenario will give $n^2\sqrt n$ times.</p>
|
1,726,187 | <p>Recently in class our teacher told us about the evaluating of the sum of reciprocals of squares, that is $\sum_{n=1}^{\infty}\frac{1}{n^2}$. We began with proving that $\sum_{n=1}^{\infty}\frac{1}{n^2}<2$ by induction. However, we actually proved a stronger result, namely that $\sum_{n=1}^{\infty}\frac{1}{n^2}<... | Jack D'Aurizio | 44,121 | <p>You may Euler's acceleration method or just an iterated trick like my $(1)$ <a href="https://math.stackexchange.com/a/1409131/44121">here</a> to get:
$$ \zeta(2) = \sum_{n\geq 1}\frac{1}{n^2} = \color{red}{\sum_{n\geq 1}\frac{3}{n^2\binom{2n}{n}}}\tag{A}$$
and the last series converges pretty fast. Then you may noti... |
1,859,810 | <p>Consider the function
$$
f(x)=\sum_{n=1}^{\infty}\frac{\mathrm{sign}\left(\sin(nx)\right)}{n}\, .
$$
This is a bizarre and fascinating function. A few properties of this function that SEEM to be true:</p>
<p>1) $f(x)$ is $2\pi$-periodic and odd around $\pi$.</p>
<p>2) $\lim_{x\rightarrow \pi_-} f(x) = \ln 2$. (Can... | Marco Cantarini | 171,547 | <p>We can prove the convergence almost everywhere using the following </p>
<blockquote>
<p><strong>Theorem:</strong> Let $\varphi
$ a function such that $$\varphi\left(x\right)\in L^{2}\left(-\pi,\pi\right),\,\varphi\left(x+2\pi\right)=\varphi\left(x\right),\,\int_{0}^{2\pi}\varphi\left(x\right)dx=0.$$ Assume that ... |
1,859,810 | <p>Consider the function
$$
f(x)=\sum_{n=1}^{\infty}\frac{\mathrm{sign}\left(\sin(nx)\right)}{n}\, .
$$
This is a bizarre and fascinating function. A few properties of this function that SEEM to be true:</p>
<p>1) $f(x)$ is $2\pi$-periodic and odd around $\pi$.</p>
<p>2) $\lim_{x\rightarrow \pi_-} f(x) = \ln 2$. (Can... | Mark McClure | 21,361 | <p>The series certainly converges when $x$ is a rational multiple of $\pi$, say $x=p\pi/q$ where $p/q$ is in lowest terms. Here is an outline of a proof of this using <a href="https://en.wikipedia.org/wiki/Dirichlet's_test" rel="noreferrer">Dirichlet's test</a>.</p>
<p>The numbers
$$\alpha_k = k\,\pi\frac{p}{q} ... |
535,757 | <p>The exercise is to give an example for two sets $M$ and $N$, and functions $f$ and $g$, for which $f \circ g = id_M$, but $g \circ f \ne id_N$.</p>
<p>My idea is a bit based on my computer programming background, where <code>(x/2)*2</code> is <code>0</code> for integers. Here it is:</p>
<p>$$M=N=\mathbb{N_0}.$$
$$... | Peter LeFanu Lumsdaine | 2,439 | <p><strong>Yes, your example is valid, and your overall argument for it is good.</strong> You have a few minor errors in details and notation, though.
$\newcommand{\id}{\mathit{id}} \newcommand{\N}{\mathbb{N}}$</p>
<ul>
<li><p>In your line “$(g \circ f)(x) = \cdots = x = \id_\mathbb{N}$”, the last item should be not ... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.