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 |
|---|---|---|---|---|
4,544,787 | <p>These sums showed up in a probability problem I was working on. They're not quite the Stirling numbers of the first kind since it's possible to have e.g. <span class="math-container">$i_1 = i_2$</span>. Denoting the sum by <span class="math-container">$(k\mid n)$</span> we have the recurrence relation</p>
<p><span c... | Parcly Taxel | 357,390 | <p>Here I will show by induction that the Stirling numbers of the second kind give the general answer for the multisubset sum: with <span class="math-container">$Q_{n,k}$</span> being the set of all <span class="math-container">$k$</span>-multisubsets of <span class="math-container">$[n]=\{1,\dots,n\}$</span>,
<span cl... |
4,544,787 | <p>These sums showed up in a probability problem I was working on. They're not quite the Stirling numbers of the first kind since it's possible to have e.g. <span class="math-container">$i_1 = i_2$</span>. Denoting the sum by <span class="math-container">$(k\mid n)$</span> we have the recurrence relation</p>
<p><span c... | Mike Earnest | 177,399 | <p>Parcly has given a proof that
<span class="math-container">$$
\sum_{1\le i_1\le \dots \le i_k\le n}i_1\cdots i_k={n+k \brace n}\tag{$\star$}
$$</span>
Surely, such a beautiful equation must have a combinatorial proof? Indeed it does!</p>
<p>Given a partition of <span class="math-container">$\{1,2,\dots,n+k\}$</span>... |
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>Or you can continue on your method, just make the range of $l$ large enough:</p>
<p>Let $l \in (- \infty, + \infty)$, </p>
<p>$$S=\sum_{k=-\infty}^{j}\sum_{l=- \infty}^{+ \infty} \Omega_{k,l-k+1} \cdot {f_l}=\sum_{l=- \infty}^{+ \infty} \sum_{k=-\infty}^{j}\Omega_{k,l-k+1} \cdot {f_l}$$
where:
$$
\Omega=\begin{cas... |
1,271,942 | <p>I am a little bit confused with the definition of finitely presented modules. In Lang's <em>Algebra</em> he defines a module <span class="math-container">$M$</span> to be finitely presented if and only if there is a exact sequence <span class="math-container">$F'\to F\to M \to 0$</span> such that both <span class="m... | user26857 | 121,097 | <p>In Lang's <em>Algebra</em> he defines a module <span class="math-container">$M$</span> to be finitely presented if and only if there is an exact sequence <span class="math-container">$F'\to F\to M \to 0$</span> such that both <span class="math-container">$F', F$</span> are free <em>of finite rank</em>, and this is t... |
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}\... | D_S | 28,556 | <p>Use the fact that $A^{-1} = \frac{1}{\textrm{det(A)}}\textrm{adj}(A)$ and the following general principles:</p>
<p>1 . The topology on $\textrm{GL}_n(\mathbb{R})$, with the operator norm, is the subspace topology coming from $\mathbb{R}^{n^2}$, the $n^2$-fold product of copies of $\mathbb{R}$. In general, norm t... |
3,566,603 | <p>I need a simple way to show <span class="math-container">$\mathbb R^2$</span> is not isomorphic <span class="math-container">$\mathbb{R}[x]/(x^2)$</span>. Both are not integral domains, and both are not fields, so I’m not sure how to go about it.</p>
| Marc Olschok | 19,950 | <p>One possibility (among many others) is to combine the following observations:</p>
<p>(1) <span class="math-container">$\epsilon = x + (x^2) \in \mathbb{R}[x]/(x^2)$</span> satisfies
<span class="math-container">$\epsilon^2 = 0$</span>.</p>
<p>(2) any isomorphism must map nonzero nilpotent elements to nonzero nilp... |
3,695,868 | <p>In right triangle <span class="math-container">$ABC,$</span> <span class="math-container">$\angle C = 90^\circ.$</span> Let <span class="math-container">$P$</span> and <span class="math-container">$Q$</span> be points on <span class="math-container">$\overline{AC}$</span> so that <span class="math-container">$AP = P... | Jan Eerland | 226,665 | <p>Well, we can use Pythagoras twice (as @Narasimham pointed out):</p>
<p><span class="math-container">$$
\begin{cases}
\text{BP}^2=\text{CP}^2+\text{BC}^2\\
\\
\text{BQ}^2=\text{CQ}^2+\text{BC}^2\\
\\
\text{CP}=\text{CQ}+\text{PQ}\\
\\
\text{CQ}=\text{PQ}\color{\white}{+}\\
\\
\text{BP}=76\\
\\
\text{BQ}=67
\end{case... |
3,695,868 | <p>In right triangle <span class="math-container">$ABC,$</span> <span class="math-container">$\angle C = 90^\circ.$</span> Let <span class="math-container">$P$</span> and <span class="math-container">$Q$</span> be points on <span class="math-container">$\overline{AC}$</span> so that <span class="math-container">$AP = P... | Quanto | 686,284 | <p>Let AP = PQ = QC = <span class="math-container">$x$</span>. Note that PB and QB are the medians of the triangles AQB and PCB respectively. Per the Apollonius's Theorem for medians</p>
<p><span class="math-container">$$AB^2 + QB^2 = 2x^2 + 2PB^2\tag1 $$</span>
<span class="math-container">$$CB^2 +PB^2 = 2x^2 + 2QB^... |
13,843 | <p>We have a natural number $n>1$. We want to determine whether there exist
natural numbers $a, k>1$ such that $n = a^k$. </p>
<p>Please suggest a polynomial-time algorithm.</p>
| Felipe Voloch | 2,290 | <p>For each $k \le \log n/\log 2$, compute an approximation to the positive real $k$-th root of $n$ using Newton's method to enough precision to check if it is an integer. Alternatively, use $p$-adic roots for a suitable $p$, with Newton turning into Hensel.</p>
|
3,959,178 | <p>Let <span class="math-container">$G$</span> be a group and <span class="math-container">$\mathbb k$</span> a field. Then, let <span class="math-container">$\mathfrak 1$</span> be the trivial <span class="math-container">$\mathbb k G$</span>-module.</p>
<p>According to Lorenz's A Tour of Representation Theory, he sta... | Qiaochu Yuan | 232 | <p>I'll write <span class="math-container">$k$</span> for the trivial module. There's a natural map <span class="math-container">$\varepsilon : k[G] \to k$</span> of <span class="math-container">$k[G]$</span>-modules given by sending every <span class="math-container">$g \in G$</span> to <span class="math-container">$1... |
907,893 | <p>I wanted to know about this convention :</p>
<p>By rate of growth of R, we normally mean : (change in R) / (change in Time)</p>
<p>But Rate of growth of a geometric sequence "a(1+r)^n" is r, which is strange i feel</p>
<p>I am kind of confused, can anyone clear it </p>
| Did | 6,179 | <blockquote>
<p>By rate of growth of R, we normally mean : (change in R) / (change in Time)</p>
</blockquote>
<p>Actually no, the rate of growth of some quantity $R$ over the time interval $(t_1,t_2)$ is not $$\frac{R(t_2)-R(t_1)}{t_2-t_1}$$ but $$\frac{R(t_2)-R(t_1)}{R(t_1)}$$ and, when the time is continuous, the... |
521,740 | <p>$x$,$y$ are real numbers satisfying $(x-1)^{2}+4y^{2}=4$<br>
find the maximum of $xy$ and justify it without calculus.<br>
Does there exist a tricky solution using elementary inequalities (AM-GM or Cauchy-Schwarz) ?</p>
<p>I tried and got it's when $x=\dfrac{3+\sqrt{33}}{4}$</p>
| Elmar Zander | 10,076 | <p>If you need to <em>justify only the solution</em> without calculus, you can do it the following way: The solution you have is
$$
x=\frac{1}{4} \left(3+\sqrt{33}\right) \qquad y=\frac{1}{4} \sqrt{\frac{1}{2} \left(15+\sqrt{33}\right)}
$$
If you go along the ellipse from that $(x,y)$ by a very small amount it will be... |
1,701,176 | <p>The problem I'm having is with the logs. I go:</p>
<p>$$\lim_{n \to \infty} \Big( \frac{\log{(n+1)}}{\log{(n)}} \cdot \frac{n-2}{n-1} \Big)$$</p>
<p>$$=\lim_{n \to \infty} \Big( \frac{\log{(n+1)}}{\log{(n)}}\Big) \cdot \lim_{n \to \infty} \Big(\frac{n-2}{n-1} \Big)$$</p>
<p>and here I know that $$\lim_{n \to \inf... | quasicoherent_drunk | 195,334 | <p>We can write $\log{(n+1)}$ as $$\log{(n(1+\frac{1}{n}))}=\log{n}+\log\left(1+\frac{1}{n}\right).$$</p>
<p>Now $\log\left(1+\frac{1}{n}\right)$ is bounded, so is insignificant compared to $\log{n}.$ So the limit of $$\frac{\log(n+1)}{\log(n)}=\frac{\log{(n)}}{\log{(n)}}+\frac{\log{\left(1+\frac{1}{n}\right)}}{\log{(... |
4,196,109 | <p>While studying about inequalities, I came across the following definition (<span class="math-container">$\forall a > 0)$</span>:</p>
<p><span class="math-container">$$
\begin{alignat}{1}
& |x| > a \iff \{ x \mid x < -a \text{ or } x > a \} \\
& |x| < a \iff \{ x \mid -a < x < a \}
\e... | user97357329 | 630,243 | <p>A framework proposed by Cornel (<strong>answer to the main integral</strong>)</p>
<p>The skeleton of the solution may be immediately obtained by using the strategy given for the generalization in Sect. <span class="math-container">$1.24$</span>, page <span class="math-container">$14$</span>, <strong>(Almost) Impossi... |
2,867,207 | <p>This question is a (perhaps naive) 'simplification' of a result in a paper, so the answer could be negative.</p>
<p>Define the cone $\Sigma(\theta)$ for $\theta\in(0,\pi/2]$,
$$\Sigma(\theta) = \left\{ z = x+iy : x>0, |y|<(\tan\theta)x\right\}, $$
and define the norm $\|f\|_\theta$ for functions $f$ analytic ... | Ben West | 37,097 | <p>Writing $(n,m)$ for the GCD of $m$ and $n$, immediately one has $(n,m)=(n,m-kn)$ for any $k\in\mathbb{Z}$. Then
$$
(21n+4,14n+3)=(7n+1,14n+3)=(7n+1,1)=1.
$$</p>
<p>Thus $21n+4$ and $14n+3$ are coprime, so their ratio is in lowest terms.</p>
|
1,320,874 | <p>I am trying to answer the following: Does the congruence $x^2 \equiv -1$ (mod $p$) have any solutions if $p \equiv 3$ (mod $4$)? If so, how many incongruent solutions does it have? If not, why not?</p>
<p>I know from the previous part of the question that if $p$ is a prime and $p \equiv 1$ (mod $4$), then the congr... | Zev Chonoles | 264 | <p>André Nicolas has an approach that uses Wilson's theorem, as requested by the OP, but I guess I'll leave my answer up.</p>
<hr>
<p>No, it has no solutions. The first step is to observe that, for an odd prime $p$,
$$x^2\equiv -1\bmod p\iff x\text{ has order 4 in }(\mathbb{Z}/p\mathbb{Z})^\times$$
But the group $(\... |
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... | lhf | 589 | <p>You can <a href="https://en.wikipedia.org/wiki/Euler%27s_criterion" rel="nofollow">Euler's criterion</a>.
Let
$$
A=a^{\frac{p-1}{2}} \bmod p,
\quad
B=b^{\frac{p-1}{2}} \bmod p,
\quad
C=(ab)^{\frac{p-1}{2}} \bmod p
$$
Then $C=AB$. This implies that if two of the equations are solvable, then so is the third.</p>
|
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 ... | Zarrax | 3,035 | <p>Let $(a_1,a_2,a_3)$ and $(b_1,b_2,b_3)$ be two unit vectors perpendicular to the direction of the axis and each other, and let $(c_1,c_2,c_3)$ be any point on the axis. (If ${\bf v} = (v_1,v_2,v_3)$ is a unit vector in the direction of the axis, you can choose ${\bf a} = (a_1,a_2,a_3)$ by solving ${\bf a} \cdot {\bf... |
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>
| Antoine | 73,561 | <p>If you define period of a function $h$ as the number $p$, such that $h(x + p) = h(x)$, for all $x$, then yes (try it).</p>
<p>If you define period as the smallest positive number, such that $h(x + p) = h(x)$, then no, for example: $g = \sin$ and $f = -\sin$ will give you $h(x) = \sin(x) - \sin (x) = 0$.</p>
|
19,842 | <p>Seeing <a href="https://stackoverflow.com/help/self-answer">this</a> I though this thing was promoted, and for avoiding for the question becoming boring, I didn't answer it suddenly and waited and I did mentioned that I knew the answer, maybe it's just misunderstanding that I don't know the answer. Anyways, what's t... | apnorton | 23,353 | <p>Self-answering a question means that you actually post an answer to your question, not that you had already solved the question in the past. Thus, your second link isn't a "self-answered question" by any measure. </p>
<p>Instead of editing the question to include an answer, it is best to <em>post an answer</em> t... |
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)... | Martin Brandenburg | 1,650 | <p>Somehow the statement is unnatural und unnecessarily complicated.</p>
<p>If $R$ is an arbitrary commutative ring, and $r \in R$ is a root of a polynomial $f \in R[X]$, then $X-r$ divides $f$. Proof: Using the transformation $X \mapsto X-r$ it suffices to check the case $r=0$. But then $f$ has no constant term and t... |
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... | Allawonder | 145,126 | <p>I think what you notice is that the process reminds you of the number of permutations (not selections, as you wrote) of <span class="math-container">$4$</span> out of <span class="math-container">$7$</span> objects. This is indeed the case.</p>
<p>Now, first note that to count the number of arrangements of <span cl... |
858,494 | <p>Where does the definition of the $L_\infty$ norm come from?</p>
<p>$$\|x\|_\infty=\max \{|x_1|,\dots,|x_k|\}$$</p>
| Cameron Williams | 22,551 | <p>Here's the way I like to think about it (which is not too rigorous but can be made rigorous). We have, more generally, the $L^p$ norms ($1\le p < \infty$):</p>
<p>$$(\|x\|_p)^p:=\sum_{i=1}^n |x_i|^p.$$</p>
<p>The Euclidean norm is a special case of this (take $p = 2$); the taxicab norm is also a special case (t... |
3,026,097 | <p>I was studying an article where I encountered <span class="math-container">$\mathbb{R}^E_{\gt 0}$</span>. I couldn't find out what does this notation mean exactly.
I'm sorry if my question is basic, I searched this community but I didn't find the answer to my question.</p>
<p>Here is a part of that article:</p>
<b... | YiFan | 496,634 | <p>Hint: break the integral up into
<span class="math-container">$$\int_0^\infty=\int_0^1+\int_1^2+\int_2^3+\cdots.$$</span>
On each of these intervals the denominator takes a constant value, so you can bring it out of the integral sign and evaluate each. After that just sum all the terms you get.</p>
|
3,757,763 | <p>Let <span class="math-container">$T: V\rightarrow V$</span> be a linear operator of the vector space <span class="math-container">$V$</span>.</p>
<p>We write <span class="math-container">$V=U\oplus W$</span>, for subspaces <span class="math-container">$U,W$</span> of <span class="math-container">$V$</span>, if <span... | Tsemo Aristide | 280,301 | <p>Consider the shift operator <span class="math-container">$s$</span>, defined on <span class="math-container">$\text{Vect}(e_i, i\in\mathbb{N})$</span>, where <span class="math-container">$s(e_n)=e_{n+1}$</span> for <span class="math-container">$n\in\mathbb{N}$</span>. Note that <span class="math-container">$\ker(s)... |
917,302 | <p>If $p(x)$ is a polynomial of degree 4 such that $p(2)=p(-2)=p(-3)=-1$ and $p(1)=p(-1)=1$, then find $p(0)$.</p>
| Community | -1 | <p>This is a case of Lagrangian interpolation.</p>
<p>$$P(x)=\sum_i\left(y_i\prod_{j\ne i}\frac{x-x_j}{x_i-x_j}\right).$$</p>
<p>It is most efficiently computed using <a href="http://en.wikipedia.org/wiki/Neville%27s_algorithm" rel="nofollow noreferrer">Neville's scheme</a>.</p>
<p>Using the abscissas in the order $... |
589,309 | <p>Finding all sets of primes $p$ and $q$ such that $p$ divides $q^2 -4$ and $q$ divides $p^2-1$.</p>
| Robert Israel | 8,508 | <p>Hint: $q^2-4$ and $p^2-1$ can be factored.</p>
|
589,309 | <p>Finding all sets of primes $p$ and $q$ such that $p$ divides $q^2 -4$ and $q$ divides $p^2-1$.</p>
| Superguy | 510,499 | <p>Four cases would be formed and only two would be left .</p>
<p>$p+1 \equiv 0 \pmod q$</p>
<p>$q+2 \equiv 0 \pmod p$</p>
<p>After some estimation it would be found that only $p=5$ and $q=3$ can be the solution.</p>
|
844,355 | <p>Let $T\colon V \rightarrow W$ a linear transformation between the real vector spaces $V$ and $W$ both with finite dimension.</p>
<p>How can i prove that $\dim(V) = \dim T(V) + \dim T^{-1}(0)$.</p>
<p>I can't understand this problem and how to solve it , if you can help me please.</p>
| mookid | 131,738 | <p>Consider a space $F$ such as $V = F\oplus T^{-1}(0)$. Consider the restriction $T'$ of $T$ from
$F$ to $T(V)$.</p>
<ol>
<li>Let $x,y\in F.$</li>
</ol>
<p>$$
T'(x) = T'(y)\implies T(x-y) = T(x)-T(y) = 0\implies
x-y\in T^{-1}(0)
$$but as $x,y\in F$:
$$
x-y\in T^{-1}(0)\cap F= \{0\}\implies x=y.
$$2. Let $y\in T(V)... |
2,531,714 | <p>Given is: </p>
<p>$(1-x^2)\dfrac{d^2y(x)}{dx^2} + 2x\dfrac{dy(x)}{dx} - 2y(x) = 0 $</p>
<p>The Solution is: </p>
<p>$y(x) =C_1x + C_2(x^2+1)$ </p>
<p>How do I factor the $x$ out in order to get it into a normal "linear" form that contains only coefficients to show that the solution is valid? </p>
<p>Edit: The e... | Jack D'Aurizio | 44,121 | <p>We have that
$$ f(x)=\sum_{n\geq 0}a_n x^n\quad\Longleftrightarrow\quad \frac{f(x)}{1-x}=\sum_{n\geq 0}A_n x^n $$
where $A_n = a_0+a_1+\ldots+a_n$. In order to find the OGF of $H_{n}^2$ it is enough to find the OGF of
$$ H_{n+1}^2-H_{n}^2 = \left(H_{n+1}-H_n\right)\left(H_{n+1}+H_n\right)=\frac{2H_n}{n+1}+\frac{1}{(... |
3,884,581 | <p>Please don't just throw an answer at me, please explain how you arrived at it cause I've been fiddling with this for the past 30min...</p>
| Community | -1 | <p><strong>Hint:</strong></p>
<p><span class="math-container">$$a^2+b^2\pm2ab=(a\pm b)^2.$$</span></p>
<hr />
<p>The standard way is to try and eliminate one of the unknonws, for instance getting <span class="math-container">$b$</span> from the second equation and plugging in the first. Then solve the equation in a sin... |
3,884,581 | <p>Please don't just throw an answer at me, please explain how you arrived at it cause I've been fiddling with this for the past 30min...</p>
| Spectre | 799,646 | <p>A larger hint :</p>
<p><span class="math-container">$a^2 + b^2 = 6, ab = 4$</span></p>
<p><span class="math-container">$(a+b)^2 = a^2 + b^2 + 2ab = 6 + 2 \times 4 = 14 \implies a + b = \pm \sqrt{14}\longrightarrow(1)$</span>
<span class="math-container">$(a-b)^2 = a^2+b^2-2ab = 6 - 2\times 4 = -2 \implies a - b = \s... |
3,103,991 | <p>Suppose <span class="math-container">$X$</span> is a topological space, <span class="math-container">$C$</span> is a closed subset of <span class="math-container">$X$</span>, <span class="math-container">$U$</span> is an open subset of <span class="math-container">$X$</span> and <span class="math-container">$U$</spa... | tomasz | 30,222 | <p>This is true if and only if <span class="math-container">$C$</span> is the closure of its interior. Such sets are called <em>regular closed sets</em>.</p>
<p>Indeed, if <span class="math-container">$C=\overline{V}$</span> for some open <span class="math-container">$V$</span>, then <span class="math-container">$U\ca... |
2,429,769 | <p>I watched a <a href="https://www.youtube.com/watch?v=lXkRj6MKbZs" rel="nofollow noreferrer">great youtube video</a> about how to prove a limit of a multivariable function exists. It explained that one method is by substitution. For example, we can solve $$lim_{(x, y) \to 0,0} \frac{xy}{\sqrt{x^2 + y^2}}$$</p>
<p>By... | Raphael Rafatpanah | 61,853 | <p>The formula to sum an arithmetic sequence is:</p>
<p><a href="https://i.stack.imgur.com/NQt1T.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/NQt1T.gif" alt="enter image description here"></a></p>
<p><code>n</code> = <code>iterations</code> = <code>initial velocity</code> / <code>drag</code></p>... |
2,231,092 | <p>I am reading <a href="http://people.ucalgary.ca/~rzach/static/open-logic/open-logic-complete.pdf" rel="nofollow noreferrer">Open Logic TextBook</a>. In which there is a proposition about Extensionality of first order sentences (6.12) It goes like this, </p>
<p>Let $\phi$ be a sentence, and $M$ and $M'$
be structure... | Emilio Novati | 187,568 | <p>Hint:</p>
<p>You can write the equation in cartesian coordinate as $y=mx+q$, than substitute
$y=\rho \sin \theta$ and $x=\rho \sin \theta$. From this it is easy to obtain $\rho$ as a function of $\theta$.</p>
|
622,090 | <p>We are asked to solve the following linear system</p>
<p>$$x_1-3x_2+x_3=1$$
$$2x_1-x_2-2x_3=2$$
$$x_1+2x_2-3x_3=-1$$</p>
<p>by using gauss-jordan elimination method. The augmented matrix of the linear system is $$\left(\begin{array}{ccc|c}1 & -3 & 1 & 1 \\2 & -1 & -2 & 2 \\1 & 2 & -... | Zhoe | 99,231 | <p>Yes, you can stop there and conclude that the system is inconsistent as $0\ne-2$. If you were to continue to reduce the matrix to reduced-row echelon form, row $3$'s inconsistency would remain unaffected.
$$\left(\begin{array}{ccc|c}1 & 0 & -\frac{7}{5} & 0 \\0 & 1 & -\frac{4}{5} & 0 \\0 &... |
4,564,882 | <p>Suppose there are two types of weathers. Sunny and Rainy. <br />
The probability that a sunny day is followed by a sunny day is 70% and followed by a rainy day is 30%. <br />
The probability that a rainy day is followed by a rainy day is 60% and followed by a sunny day is 40%. <br />
In a year (365 days), how many... | true blue anil | 22,388 | <p>Taking that the steady state probability exists, let these probabilitie be <strong>s</strong> for sunny and <strong>r</strong> for rainy, then one more iteration won't change these probabiliies, hence</p>
<p><span class="math-container">$s*0.7 + r*0.4 = s$</span></p>
<p><span class="math-container">$s*0.3 + r*0.6 = ... |
959,525 | <p>Could someone tell me what i've done wrong?</p>
<p>I tried to find out the derivative of $3^(2x)-2x+1$ but I got it wrong.
What I did was derivate $3^a-2x+1$ where a = 2x then multiply those two.</p>
<p>$(ln3*3^a - 2)*2$ = $2ln3*3^(2x)-4$</p>
<p>Ps. x = 2 so the answer is supposed to be 176.</p>
| Arodi007 | 135,954 | <p>Find value of C <br/><br/></p>
<p>from Left-side <br/>
$C=2\sqrt{a^2-m^2}$</p>
<p>From right side
$m^2=\sqrt{b^2+a^2}/(2)$</p>
<p>You might continue the solution from these...</p>
|
2,750,783 | <p>There is a problem I am having trouble understanding. </p>
<p>We are asked to evaluate the definite integral:</p>
<p>$$\int_0^2\sin(e^x+x^2)(e^x+2x)\,dx$$ </p>
<p>If anyone would be so kind as to walk me through it, I would be extremely grateful. </p>
| giobrach | 332,594 | <p>I do not understand how you can postulate the existence of $y \notin A^c$ such that $y$ is a limit point of $A^c$ just from the fact that $A^c$ is open (i.e., just by using your definition of openness). I think you may be implicitly using the fact that $A$ is closed, and as such it contains all its border points, wh... |
2,750,783 | <p>There is a problem I am having trouble understanding. </p>
<p>We are asked to evaluate the definite integral:</p>
<p>$$\int_0^2\sin(e^x+x^2)(e^x+2x)\,dx$$ </p>
<p>If anyone would be so kind as to walk me through it, I would be extremely grateful. </p>
| Nuntractatuses Amável | 537,135 | <p>If $A^C$ is open, then for each $x \in A^C$ there is an $\epsilon>0$ such that $B(x; \epsilon) = \{ y \in \mathbb{R}^2 \colon |x - y| < \epsilon\} \subseteq A^C$. </p>
<p>If $x \in \bar{A}$, then, for each $\epsilon > 0$, $B(x; \epsilon) \cap A \neq \emptyset$.</p>
<p>These are your definitions, I guess. ... |
2,750,783 | <p>There is a problem I am having trouble understanding. </p>
<p>We are asked to evaluate the definite integral:</p>
<p>$$\int_0^2\sin(e^x+x^2)(e^x+2x)\,dx$$ </p>
<p>If anyone would be so kind as to walk me through it, I would be extremely grateful. </p>
| aschepler | 2,236 | <p>There are a number of problems with your proof. Taking it one bit at a time,</p>
<blockquote>
<p>We want to show if $A^c$ is open, the $A$ is closed. Let $X=\mathbb{R}^2$ and $A\subseteq X$. Then, denote $A^c$ as the complement of A such that $A^c=\{x\in X|x\notin A\}$.</p>
</blockquote>
<p>So far, just restatin... |
299,170 | <p>Given the following integer programming formulation, how can I specify that the variables are unique and none of them has the same value as the other one. basically <code>x1</code>, <code>x2</code>, <code>x3</code> , and <code>x4</code> need to get only one unique value from 1, 2, 3 or 4. and same applies to <code>y... | Is7aq | 61,660 | <p>I found a workaround by adding the following constraints, to ensure that the sum of every 2 numbers, every 3 number, and every 4 numbers is at the least their minumum sum if they are to be distinct. For the above problem, the following additional constraints ensured that values are distinct.</p>
<pre><code>+x1 +x2 ... |
943,048 | <p><strong>Question:</strong></p>
<blockquote>
<p>let $x_{i}=1$ or $-1$,$i=1,2,\cdots,1990$, show that
$$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}?$$</p>
| extremeaxe5 | 174,546 | <p>Consider the equation modulo 2. Regardless of whether $x_i = 1,or -1$, $x_i\equiv 1\pmod{2}$.</p>
<p>Thus $\sum_{i=1}^{1990}{ix_i}\equiv 1+0+1+\cdots +0\pmod{2}$, where there are $\dfrac{1990}{2}=995$ $1$'s in the summation. </p>
<p>We conclude that $\sum_{i=1}^{1990}{ix_i}\equiv 1\pmod{2}$, so it definitely canno... |
1,355,684 | <p>How to find the lower and upper focus? Hyperbola </p>
<p>I started with this
$$ 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 $$</p>
<p>center = (-3,5)
Lower Vertex = (-3,0)
Upper Vertex = (-3,10)</p>
<p>How to get the foci? </p>
<p>foci... | Daniel Griscom | 253,093 | <p>Forming a number by "repeating a two digit number three times" is the equivalent of "multiplying a two digit number by 10101". And, as @achuille-hui said, $10101 = 3 \times 7 \times 17 \times 37$. So, any number in your form will be a multiple of 3, 7, 17 and 37.</p>
|
1,996,290 | <p>I don't know how to solve the following integral:</p>
<p>$$ \frac{2\pi}R\int_{r_1}^{r_2}r(r+R-|r-R|)dr $$</p>
<p>I have solved it when $R \le r_1$ and $R \ge r_2$ but I need the answer for $ r_1\lt R \lt r_2$. <br/>
R is a constant as well as $r_1$ and $r_2$.</p>
<p>I appreciate your help.</p>
| Alexis Olson | 11,246 | <p>If $r_1 < R < r_2$, then</p>
<p>\begin{eqnarray}
\int_{r_1}^{r_2}r(r+R-|r-R|)dr
&=& \int_{r_1}^{R}r(r+R+(r-R))dr + \int_{R}^{r_2}r(r+R-(r-R))dr\\
&=& \int_{r_1}^{R}2r^2dr + \int_{R}^{r_2}2rR\,dr
\end{eqnarray}</p>
|
4,115,069 | <p>I understand 'functionals' as functions of functions, for example:</p>
<p><span class="math-container">$$ S[y(x)]= \int_{t_1}^{t_2} \sqrt{1+(y')^2} dx$$</span></p>
<p>Which is the famous arc length integral</p>
<p>Now, in a similar way, a limit we can write as:</p>
<p><span class="math-container">$$L(a, [y(x)] ) = \... | peek-a-boo | 568,204 | <p>A functional is just a function whose domain and target space are of a specific type. Typically the domain is assumed to be a space like <span class="math-container">$L^p, C^k$</span>, or some other Hilbert/Banach/Frechet space (or maybe you may not even want all this structure, and just assume the domain is some re... |
1,665,443 | <p>How do we show the ring homomorphism for </p>
<p>$\phi :\mathbb F_p(\alpha) \rightarrow\mathbb F_p(\alpha)$ which is defined as $ \phi(\alpha)=\alpha +1$.</p>
<p>This is a very basic fact but I am unable to prove it by the definition of ring homomorphism. Same thing happens for ring homomorphism over $\phi :K[x] \... | DonAntonio | 31,254 | <p>Ok, so $\;\Bbb F_p(\alpha)\;$ is a finite, and thus algebraic, extension of $\;\Bbb F_p\;$ , but also</p>
<p>$$\alpha^p-\alpha+a=0\implies (\alpha+1)^p-(\alpha+1)+a=\alpha^p+1-\alpha-a+a=0$$</p>
<p>so both $\;\alpha,\,\alpha+1\;$ are roots of the same polynomial irreducible polynomial over $\;\Bbb F_p[x]\;$ and th... |
1,821,248 | <p>Which of the following are true?</p>
<ol>
<li><p>$\sigma \circ \sigma(j)=j~\forall j, 1 \leq j \leq 5$.</p></li>
<li><p>$\sigma^{-1}(j)=\sigma(j)~\forall j, 1 \leq j \leq 5$.</p></li>
<li><p>The set $ \{k: \sigma(k) \neq k \}$ has even number of elements.</p></li>
<li><p>The set $\{k:\sigma(k) =k \}$ has an odd num... | Brian Cheung | 248,555 | <p>$\sigma^{-1}(j)\leq\sigma(j)\forall 1\leq j \leq 5$<br>
since $\sigma$ is a permutation,<br>
$j\leq\sigma^2(j)\forall 1\leq j\leq 5$<br>
since $\sigma^2$ is also a permutation, we have $j=\sigma^2(j)\forall 1\leq j\leq 5$<br>
So the longest cycle in $\sigma$ is at most of length $2$.<br>
That is, an element is eithe... |
122,776 | <p>Let $\Gamma$ be a lattice in a (real or p-adic) Lie group.
Is it true that for a given natural number $n$ there exists a finite index subgroup $\Sigma\subset\Gamma$ such that each $\sigma\in\Sigma$ is an $n$-th power of some element of $\Gamma$?</p>
<p>In other words, is it true that for given $\sigma\in\Sigma$ the... | Ian Agol | 1,345 | <p>This is false for uniform lattices in rank one semi simple Lie groups and large $n$ by a result of <a href="http://www.ams.org/journals/tran/1996-348-06/S0002-9947-96-01510-3/home.html" rel="nofollow">Ivanov and Olshanskii</a>, which implies that the normal subgroup generated by $n$th powers is infinite index for ce... |
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... | James Arathoon | 448,397 | <p>The standard definition for the curvature at a particular point on a curved path is just $\frac{1}{R}$, where R is the radius of curvature back to some notional associated point (valid for that particular point on the curved path only). The definition need not involve time, but most applications involve time and mot... |
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... | ChocolateAndCheese | 148,590 | <p>I think about it as related to the radius of a circle or ball that will juuust lie tangent to the curve at that point. Any larger radius and the circle will be too large to touch at that point, and so will necessarily touch the curve in 2 other points.</p>
<p>Of course this isn't exactly rigorous, but I found it ... |
118,763 | <p>Hello,</p>
<p>Let $X'X$ be a positive definite matrix and let $\mathbf{1}$ denote the vector of ones. </p>
<p>I'm hoping to construct a positive, diagonal matrix $W$ such that
$$(W X'X W) \mathbf{1} = \mathbf{1}$$</p>
<p>$X$ and $W$ are all assumed to have real-valued entries, and $X'$ denotes the transpose of $X... | Will Sawin | 18,060 | <p>Consider the simplex of nonzero diagonal matrices W with nonnegative entries up to scaling, and the simplex of nonzero vectors V with nonnegative entries up to scaling.</p>
<p>There is a map, $V=\max(WX′XW\mathbf 1,0)$, from the first simplex to the second, with $\max(a,0)$ interpreted entrywise. This is well-defin... |
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... | Tommi | 2,083 | <p>There are four ways an advanced knowledge of mathematics can help, in rough order of importance: Mastery, understanding student thinking, context and content and tricks. However, general teaching skill also matters, and I would guess that it does not reliably increase along with mathematics studies, or at least that... |
481,313 | <p>Show that in an abelian group the product of two elements of finite order is itself an element of finite order.</p>
<p>I need some hint to start with, I am familiar with the basic</p>
| Dan Rust | 29,059 | <p>If $a$ has order $l$ and $b$ has order $m$, can you find an $n\geq 1$, in terms of $l$ and $m$, such that $a^n=b^n=e$?</p>
|
2,462,722 | <p><a href="https://i.stack.imgur.com/jtWGA.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/jtWGA.jpg" alt="enter image description here"></a></p>
<p>I got this from QFT Demystified in the author attempt to derive the Euler Lagrange equation. But isn't the Taylor expansion for $f(x+a)$ supposed to b... | alexjo | 103,399 | <p>$$
\frac{\mathrm d V(x)}{\mathrm d x}=\lim_{\varepsilon\to 0}\frac{V(x+\varepsilon)-V(x)}{\varepsilon}
$$
so for $\varepsilon\ll 1$ we can write</p>
<p>$$
\frac{\mathrm d V(x)}{\mathrm d x}\approx \frac{V(x+\varepsilon)-V(x)}{\varepsilon}
$$
that is
$$
V(x+\varepsilon)\approx V(x) + \varepsilon \frac{\mathrm d V(x... |
3,395,910 | <p>I understand how to apply the trapezoidal rule to approximate the area under a curve.</p>
<p>But I'm not sure how to apply it when approximating areas between two functions. </p>
<ul>
<li>Do you use the formula like how you normally would, except apply it to the <strong>first function - the second function</strong... | cqfd | 588,038 | <p>Suppose <span class="math-container">$y\in[a,b]\setminus T$</span>. Then <span class="math-container">$a:=|f(y)-g(y)|\gt 0$</span>. As <span class="math-container">$f$</span> and <span class="math-container">$g$</span> are continuous, we can find a <span class="math-container">$\delta\gt0$</span> such that <span cl... |
56,162 | <p>I'm trying to understand the Cartan decomposition of a semisimple Lie algebra, $\mathfrak g=\mathfrak k \oplus \mathfrak p$, where $[\mathfrak k,\mathfrak p] \subseteq \mathfrak p$, cf. the wikipedia article on <a href="http://en.wikipedia.org/wiki/Cartan_decomposition" rel="noreferrer">Cartan decomposition</a>.</p>... | Konrad Waldorf | 3,473 | <p>I think the important information is that $H^3(G,\mathbb{R}) \neq 0$. </p>
<p>By the way, every $G$-bundle over $M$ is trivializable, under the conditions you have mentioned. That's why a gauge transformation can be regarded as a (smooth) map $g: M \to G$. </p>
<p>Now you look at the behaviour of the Chern-Simons ... |
3,756,649 | <p>If <span class="math-container">$f:\mathbb{R}\to \mathbb{R}$</span> such that <span class="math-container">$\lim\limits_{x \to \infty}xf(x)=L$</span>. Then <span class="math-container">$\lim\limits_{x \to \infty}f(x)=0$</span>.</p>
<p>My proof is as follows:</p>
<p>Let <span class="math-container">$g(x)=xf(x)$</span... | Community | -1 | <p>Unfortunately,</p>
<p><span class="math-container">$$|f(x)|\le\frac{L+\epsilon}\delta$$</span> is not conclusive. Nothing says that <span class="math-container">$\delta$</span> is even large.</p>
<hr />
<p>A simple proof is</p>
<p><span class="math-container">$$\lim_{x\to\infty}f(x)=\lim_{x\to\infty}\left(xf(x)\cdot... |
7,268 | <p>I'm a private tutor working with a 7th grader who is struggling with solving equations. Given a simple equation, he is able to solve it using a formulaic procedure, but it is very obvious that he has no idea what the solution really means. Hence, if he gets a problem that's slightly different from ones he's solved b... | Tamisha Thompson | 4,702 | <p>I would use Algebra tiles (like <a href="http://illuminations.nctm.org/Activity.aspx?id=3482" rel="nofollow">these</a>). I've found that as students see the difference between an x tile and a unit tile, for example, they tend to make fewer mistakes with combining like terms. When they have to physically remove the... |
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... | Misha Lavrov | 383,078 | <p>We can think of the seating assignment as a random permutation of <span class="math-container">$7$</span> items (the three males, the three females, and the empty seat). This random permutation puts a female in the first chair with probability <span class="math-container">$\frac37$</span>. Conditional on having done... |
2,679,393 | <p>If $X=\{\emptyset,\{\emptyset\},\{\{\emptyset\}\}\}$ and $Y=X\setminus\{\{\emptyset\}\}$ </p>
<p>then what element is excluded from $X$? Is it $\{\{\emptyset\}\}$, or $\{\emptyset\}$?</p>
<p>In a similar vein, if $Z=\{a, b, c\}$, does it make sense to say $Z\setminus a$?</p>
<p>Thanks</p>
| Pietro Paparella | 414,530 | <p>For your first question, if $A$ and $B$ are sets, then $A \setminus B := \{ x \mid x \in A \wedge x \not \in B\}$. Thus, $X \setminus \{\{\emptyset\}\} = \{\emptyset,\{\{\emptyset\}\}\}$.</p>
<p>For the second question, if you follow the previous definition strictly, then it doesn't make sense to write $Z \setminus... |
2,679,393 | <p>If $X=\{\emptyset,\{\emptyset\},\{\{\emptyset\}\}\}$ and $Y=X\setminus\{\{\emptyset\}\}$ </p>
<p>then what element is excluded from $X$? Is it $\{\{\emptyset\}\}$, or $\{\emptyset\}$?</p>
<p>In a similar vein, if $Z=\{a, b, c\}$, does it make sense to say $Z\setminus a$?</p>
<p>Thanks</p>
| Mohammad Riazi-Kermani | 514,496 | <p>$$X=\{\emptyset,\{\emptyset\},\{\{\emptyset\}\}\}$$</p>
<p>and $$ Y=X\setminus\{\{\emptyset\}\}= \{\emptyset,\{\{\emptyset\}\}\} $$ because $\{\emptyset\}$ is removed from your $X$.</p>
<p>For your next question regarding $$ Z=\{a, b, c\}$$ $Z\setminus a$ does not make sense unless $a$ is a set.</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... | Bobson Dugnutt | 259,085 | <p>For regular $n$-gons with side-length $1$, the area is given as $$\frac{1}{4}n \cot \frac{\pi}{n}$$</p>
<p>Here are some values of the formula for negative values of $n$:</p>
<p>\begin{array}{c|c}
n & \cot\frac{\pi}{n} \\
\hline -1 & \text{complex } \infty \\
-2 & 0\\
-3 & -\frac{1}{\sqrt{3}} \\
n ... |
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... | zyx | 14,120 | <p>For regular n-gons inscribed in a given oriented circle, a regular $(-n)$-gon can be defined to be regular n-gon with the opposite orientation. </p>
<p>This is not a definition I have ever seen in a publication, but it is consistent with most of the standard conventions.</p>
<p>Extending the idea a bit, one can t... |
3,367,672 | <p>Assume <span class="math-container">$a_n$</span> is a non-negative sequence, that is, <span class="math-container">$a_n \geq 0$</span> for every <span class="math-container">$n \in \mathbb{N}$</span>. Prove that if <span class="math-container">$a_n$</span> converges then the limit is non-negative. Clue: prove it by ... | msm | 340,064 | <p>One way to do this is by contrapositive: if <span class="math-container">$L \in \mathbb{R}$</span> is negative, show that the sequence does not converge to <span class="math-container">$L$</span>. Take <span class="math-container">$L \in \mathbb{R}$</span> with <span class="math-container">$s<0$</span>. Then you ... |
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... | Aatmaj | 769,348 | <p>I have a lenthier method, put sinx=<span class="math-container">$e^t$</span> As x→π/2,sinx→1, so the limit t tends to 0. (<span class="math-container">$e^0=1$</span>)</p>
<p>we get</p>
<blockquote>
<p><span class="math-container">$lim_{t\rightarrow 0} \frac{t}{1-e^{2t}}$</span>=</p>
</blockquote>
<blockquote>
<p><sp... |
1,879,076 | <p>How to show that $(1+x/n)^n\geq (1+x/10)^{10}$? (for $n\geq 10$)</p>
<p>I see that if I consider $n\to\infty$, the LHS approaches $e^x$, but that's all I could really see.</p>
<p>Please assume that $x\in[0,\infty)$</p>
| H. H. Rugh | 355,946 | <p>$$(1+\frac{x}{n})^n = \sum_{k=0}^n \frac{1 \times (1-\frac1n) ... (1-\frac{k-1}{n})}{k!} x^k \geq \sum_{k=0}^{10} \frac{1 \times (1-\frac{1}{10}) ... (1-\frac{k-1}{10})}{k!} x^k = (1+ \frac{x}{10})^{10}$$</p>
<p>In fact, same proof shows that $(1+x/n)^n$ is increasing in $n$ (fixed $x>0$).</p>
|
2,762,230 | <blockquote>
<p>Let $I:=[a,b]$ a perfect interval and $\gamma\in C(I,\Bbb R^n)$ an injective path such that $\Gamma:=\gamma(I)$ is rectifiable. Show that $\dim_H(\Gamma)=1$.</p>
</blockquote>
<p>Here $\dim_H$ is the Hausdorff dimension. My work so far: </p>
<p>Note that the canonical projections $\pi_k$ are Lipschi... | Jonas | 382,832 | <p>Let $H^1$ be the 1-dimensional outer Hausdorff measure on $\mathbb{R}^n$. Since the curve is injective, you can show that $H^1(\Gamma) =$ "Length of $\gamma$". See for instance <a href="https://math.stackexchange.com/questions/1186574/hausdorff-measure-of-rectifiable-curve-equal-to-its-length">here</a>. </p>
<p>Sin... |
137,595 | <p>I was using the <code>StreamPlot</code> function to plot the direction field of a system of two first order differential equations. Is there any way I could add solution curves to my direction field with this function? Or is there another function that could do that for me? I looked around, but I couldn't find anyth... | Nasser | 70 | <blockquote>
<p>Is there any way I could add solution curves to my direction field
with this function</p>
</blockquote>
<h2>First method</h2>
<p>One direct way, is to use <code>Show</code> and simply add the solution to the Stream plot. Here is a quick example (since you did not give one)</p>
<pre><code>f[x_, y_... |
269,450 | <p><strong>Bug introduced in 9.0 or earlier and persisting through 13.0.1 or later</strong></p>
<p>I noticed that initially valid syntax becomes invalid within nested <code>RootSum</code> objects. Consider for example the expression</p>
<pre><code>test=RootSum1[
Function[{x},2 x^7 Log[5805]-(25 Log[5805]^6 RootSum[58... | cvgmt | 72,111 | <pre><code>styles = Directive[Black, Dotted, Thickness[0.003], Opacity[1]];
Plot[x^2, {x, 0, 5}, GridLines -> {{{3, styles}, {4, styles}}},
PlotRange -> {5, 30}, PlotStyle -> {Black, Dotted, Thickness[0.003]}]
</code></pre>
<p><a href="https://i.stack.imgur.com/MV8P4.png" rel="nofollow noreferrer"><img src="... |
269,450 | <p><strong>Bug introduced in 9.0 or earlier and persisting through 13.0.1 or later</strong></p>
<p>I noticed that initially valid syntax becomes invalid within nested <code>RootSum</code> objects. Consider for example the expression</p>
<pre><code>test=RootSum1[
Function[{x},2 x^7 Log[5805]-(25 Log[5805]^6 RootSum[58... | LouisB | 22,158 | <p>Another way (without <code>GridLines</code>) is use <code>Epilog</code> or <code>Prolog</code>. An advantage of this is grid lines can be added to the plot with the simplest <code>GridLines->Automatic</code> option without affecting the the vertical lines. Here is an example that uses color, thickness and dashi... |
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... | 5xum | 112,884 | <p>By itself, what you have shown does not imply that the limit is zero. After all, the value of $\frac1x\cdot x$ is also bounded by $1$ but does not have a limit of $0$.</p>
<p>If your book says that the result follows soley from the bounds you wrote, it is wrong. I suggest, however, that you reread the entire argume... |
1,639,081 | <p>I have been unable to solve the following question, </p>
<p>If $$\sin(2x) - \tan(x) = 0$$</p>
<p>Find $x$ , $-\pi\le x\le \pi$</p>
<p>So far my workings have been
Use following identity: </p>
<p>$$\sin(2x) = 2\sin(x)\cos(x)\\2\sin(x)\cos(x) - \tan(x) = 0\\2\sin(x)\cos(x) - \frac{\sin(x)}{\cos(x)} = 0\\
2\frac{\s... | Harish Chandra Rajpoot | 210,295 | <p>Notice, $$\sin 2x-\tan x=0$$
$$\frac{2\tan x}{1+\tan^2x}-\tan x=0$$
$$\tan x\left(\frac{1-\tan^2 x}{1+\tan^2x}\right)=0$$
$$\color{blue}{\tan x\cos 2x=0}$$
Now, solving for $x$,<br>
$$\tan x=0\iff x=n\cdot 180^\circ$$
where $n$ is any integer </p>
<p>For given interval $[-180^\circ, 180^\circ]$, setting $n=-1, 0, ... |
204,218 | <p>I can't bear an expression containing radicals of imaginary numbers,
in case it can be expressed as in terms of radicals of real numbers only.</p>
<p>For example, I can't bear the expression</p>
<pre><code>Sqrt[2 + I]
</code></pre>
<p>because
it can be expressed as </p>
<pre><code>Sqrt[1/2 (2 + Sqrt[5])] + Sqrt[... | J. M.'s persistent exhaustion | 50 | <p>Yet another possibility:</p>
<pre><code>ComplexExpand[Sqrt[2 + I], TargetFunctions -> {Re, Im}] // FunctionExpand // Simplify
(5^(1/4) ((2 + I) + Sqrt[5]))/Sqrt[10 + 4 Sqrt[5]]
</code></pre>
|
4,461,327 | <p>To show that they are equal, I need to show</p>
<p><span class="math-container">$\bigcap_{n=1}^{\infty}[0,1+1/n) \subset [0,1]$</span> and <span class="math-container">$[0,1] \subset \bigcap_{n=1}^{\infty}[0,1+1/n)$</span></p>
<p>My attempt is: let <span class="math-container">$x \in [0,1] \Rightarrow 0 \leq x \leq ... | xpaul | 66,420 | <p>Showing
<span class="math-container">$$[0,1] \subset \bigcap_{n=1}^{\infty}[0,1+1/n)$$</span>
is equivalent to showing
<span class="math-container">$$ \overline{[0,1]} \supset \overline{\bigcap_{n=1}^{\infty}[0,1+1/n)}. \tag1$$</span>
Since
<span class="math-container">$$ \overline{[0,1]}=(-\infty, 0) \cup (1, \inft... |
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 ... | Redundant Aunt | 109,899 | <p>Use Cauchy-Schwarz:
$$ \frac{a^2}{a+b}+\frac{d^2}{a+d}+\frac{b^2}{b+c}+\frac{c^2}{c+d}=\left(\frac{a^2}{a+b}+\frac{d^2}{a+d}+\frac{b^2}{b+c}+\frac{c^2}{c+d}\right)\cdot\left((a+b)+(a+d)+(b+c)+(c+d)\right)\cdot 0.5\geq (a+b+c+d)^2\cdot 0.5= 0.5 $$</p>
|
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>
| Community | -1 | <p>In addition to the mapping presented in <a href="https://mathematica.stackexchange.com/a/129442/31159">Feyre's answer</a>, it is possible to check that those two graphs are complementary by generating the complement of the first graph, and verifying that it is isomorphic to the second graph.</p>
<pre><code>{graph1,... |
3,946,591 | <blockquote>
<p>The sides of a triangle are on the lines <span class="math-container">$2x+3y+4=0$</span>, <span class="math-container">$ \ \ x-y+3=0$</span>, and <span class="math-container">$5x+4y-20=0$</span>. Find the equations of the altitudes of the triangle.</p>
</blockquote>
<p>Should I find the vertices first? ... | user2661923 | 464,411 | <p>Hint</p>
<p><span class="math-container">$\sum_{k=1}^n [f(k) \times g(k)].$</span></p>
<p><span class="math-container">$f(k)$</span> is the # of possible sequences of length <span class="math-container">$k$</span>, where the very first occurrence of the letter <span class="math-container">$b$</span> is on the <span ... |
3,231,271 | <blockquote>
<p>Suppose <span class="math-container">$X$</span> is Banach and <span class="math-container">$T\in B(X)$</span> (i.e. <span class="math-container">$T$</span> is a linear and continuous map and <span class="math-container">$T:X \to X$</span>). Also, suppose <span class="math-container">$\exists c > 0$... | Martin Argerami | 22,857 | <p>Suppose that <span class="math-container">$X$</span> is infinite-dimensional. Then its unit ball is not compact, so there exists a sequence <span class="math-container">$\{x_n\}$</span> in the unit ball that admits no convergent subsequence; by replacing with a subsequence if necessary, we may assume that there exis... |
3,814,199 | <p>On a Hilbert space <span class="math-container">$\mathcal{H}$</span>, it is well known that trace class operators have a finite trace. However, there are operators which are not trace class but have a finite trace, e.g. the <a href="https://math.stackexchange.com/a/3039962/138382">unilateral shift</a>. Then, conside... | Peter | 463,556 | <p>Your second approach seems right. Every assignment of lectures to teachers corresponds to a mapping from the set of lectures <span class="math-container">$S_M$</span> to the set of teachers <span class="math-container">$S_T$</span>. The number of ways to assign lectures to teachers such that every teacher has at lea... |
3,814,199 | <p>On a Hilbert space <span class="math-container">$\mathcal{H}$</span>, it is well known that trace class operators have a finite trace. However, there are operators which are not trace class but have a finite trace, e.g. the <a href="https://math.stackexchange.com/a/3039962/138382">unilateral shift</a>. Then, conside... | Math Lover | 801,574 | <p>While neither of your answers seem to be correct, your starting approach is fine. The problem is that you get duplicates and they need to be removed.<br />
For example, say, there are <span class="math-container">$4$</span> lectures <span class="math-container">$L1, L2, L3, L4$</span> and <span class="math-container... |
1,190,345 | <p>If $f$ is Riemann integrable on $[a,b]$ , is $|f|$ Riemann integrable on $[a,b]$ ?
(The metric is $\mathbb R$ usual)</p>
<p>The other is question is $f$ is Riemann integrable on $[a,b]$ , can I claim $f$ is bounded on $[a,b]$ ? (I think the answer can be either yes or no that depend on considerating generalized fun... | RRL | 148,510 | <p>Using the reverse triangle inequality we have</p>
<p>$$\begin{align}\sup_{x \in [x_{j-1},x_j]}|f|(x) - \inf_{x \in [x_{j-1},x_j]}|f|(x) \\&=\sup_{x,y \in [x_{j-1},x_j]}| |f(x)|-|f(y)|| \\ &\leqslant \sup_{x,y \in [x_{j-1},x_j]}| f(x)-f(y)| \\ & = \sup_{x \in [x_{j-1},x_j]}f(x) - \inf_{x \in [x_{j-1},x_j... |
1,301,937 | <p>A curve $C$ is said to be <em>trigonal</em> if it admits a rational function $f: C \to \mathbb{CP}^1$ of degree $3$. Is any nonsingular plane curve of degree four trigonal? Can the map be chosen so that it has exactly $10$ ramification points?</p>
| user149792 | 149,792 | <p>The answer to both of the questions is yes.</p>
<p>Construct an $($almost everywhere$)$ $3$-sheeted covering map by choosing a point on our curve $C$ and a hyperplane not passing through it, and projecting from the point to the hyperplane. If we worked out the coordinates, this would clearly be given by a rational ... |
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 ... | Joseph O'Rourke | 6,094 | <p>A visualization aid, $n=10$:
<br />
<img src="https://i.stack.imgur.com/bBQwy.jpg" alt="HexChess n=10"><br /></p>
<p>And now here are Will Swain's rook placements:
<br /><img src="https://i.stack.imgur.com/mHq1b.jpg" alt="HexRooks n=6,5"><img src="https://... |
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 ... | Noam D. Elkies | 14,830 | <p>Nice question!</p>
<p>For the maximum number of pairwise non-defending rooks,
Will Sawin proved an upper bound of $(2n/3) + 1$
in his comment to the original question. This bound is attained,
at least to within $O(1)$, by two rows of $n/3 - O(1)$ rooks each,
starting from around $(2n/3,n/3,0)$ and $(n/3,2n/3,1)$
a... |
2,710,200 | <p>I need to find the norm of an operator from $l^2 \to l^1$, but I'm struggling because of the different norms on $l^2$ and $l^1$. </p>
<p>The operator is defined by $T:l^2 \to l^1, x_i \mapsto 2^{-i}x_i$. </p>
<p>Using the canonical basis, I have that $||T||\geq 1/2$, but I have a feeling this is not a very good lo... | Arian | 172,588 | <p>By Cauchy-Schwartz
$$\Big(\sum_{i=1}^{n}\frac{|x_i|}{2^i}\Big)^2\leqslant\Big(\sum_{i=1}^{n}\frac{1}{2^{2i}}\Big)\Big(\sum_{i=1}^nx^2_i\Big)$$
Taking limit $n\to\infty$
$$\lim_n\Big(\sum_{i=1}^{n}\frac{|x_i|}{2^i}\Big)^2\leqslant\lim_n\Big(\sum_{i=1}^{n}\frac{1}{2^{2i}}\Big)\lim_n\Big(\sum_{i=1}^nx^2_i\Big)$$
which... |
668,291 | <p>If $h$ and $k$ are any two distinct integers, then $h^n-k^n$ is divisible by $h-k$.</p>
<p>Let's start with the basis. Let $n=1$, then
$h^1-k^1 = h-k$</p>
<p>Now for the induction, I can't use $k$ because I don't want to be confused. So let $P(r)$ for $h^n-k^n$ and that's $h^r-k^r$</p>
<p>$h^r-k^r = h-k$</p>
<... | Gurvan | 127,051 | <p>Notice that you have
$$
\begin{align}
&a^2-b^2 = (a-b)(a+b) \\
&a^3-b^3 = (a-b)(a^2+ab+b^2) \\
&a^4-b^4 = (a-b)(a^3+a^2b+ab^2+b^3) \\
etc
\end{align}
$$
In fact for every integer $n \ge 1$ you have
$$
a^n-b^n = (a-b) \cdot \sum_{k=0}^{n-1}a^kb^{n-1-k}
$$
Of course you can prove this formula by induct... |
213,102 | <p>Is it true than an aperiodic, ergodic, minimal and equicontinuous dynamical system on a compact metric space is totally ergodic ?</p>
<p>According to some results I found in some books, a rotation on a compact metric group is equicontinous, and it is minimal <del>and totally ergodic</del> whenever it is ergodic.</p... | V. Delecroix | 15,342 | <p>The answer is no in this generality! If you consider the classical odometer (i.e. addition by 1 on 2-adic integers) then its second power (addition by 2) is not minimal. This second power preserves the numbers starting with 0 (the "even numbers") and the numbers starting with 1 (the "odd numbers"). But of course thi... |
2,960,734 | <p>So basically, I am given the following to prove:</p>
<blockquote>
<p>Let <span class="math-container">$+\gamma$</span> be a positively oriented smooth Jordan arc, and let <span class="math-container">$\omega$</span> denote the interior of <span class="math-container">$+\gamma$</span>. Recall that if <span class="mat... | Gibbs | 498,844 | <p>Choose <span class="math-container">$F(x,y) = (-y+y(a),x-x(a))$</span> and define a <span class="math-container">$1$</span>-form <span class="math-container">$\alpha := (x-x(a))\mathrm{d}y+(y(a)-y)\mathrm{d}x = F_2(x,y)\mathrm{d}y+F_1(x,y)\mathrm{d}x$</span>. Then
<span class="math-container">$$\mathrm{d}\alpha = 2\... |
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>
| Plutoro | 108,709 | <p>The function $$f(x)=\int_{-2}^x\sqrt{3w^4-1}\,dw$$ is a curve in the plane. The formula for arc-length of a curve is $$L=\int_a^b\sqrt{(f'(x))^2+1}\,dx.$$ The fundamental theorem of calculus tells us that $$f'(x)=\sqrt{2x^4-1}.$$ Therefore, you need to solve the integral $$L=\int_{-2}^{-1}\sqrt{(\sqrt{3x^4-1})^2+1}\... |
1,816,109 | <p>Does anyone know how to deal with the integral
$$\int_0^{\pi /2}\cot^n(x)dx$$</p>
<p>with $n\in (-1,1)$?</p>
<p>Apparently it is a well known identity: it is listed in the <a href="http://mathworld.wolfram.com/Cotangent.html" rel="nofollow">page of wolfram for the cotangent</a>, where it says that it equals $2^{-1... | Mark Viola | 218,419 | <p>For $|n|<1$, we can write</p>
<p>$$\begin{align}
\int_0^{\pi/2}\cot^n(x)\,dx&=\frac12 B\left(\frac{1-n}{2},\frac{1+n}{2}\right)\\\\
&=\frac12 \Gamma\left(\frac{1-n}{2}\right)\Gamma\left(\frac{1+n}{2}\right)\\\\
&=\frac{\pi}{2\sin\left(\pi \frac{1+n}{2}\right)}\\\\
&=\frac{\pi}{2\cos(n\pi/2)}
\end... |
4,496,736 | <p>Question: Use the variation of parameter method to find the general solution of the following differential equation
<span class="math-container">$$(\cos x) y''+(2\sin x) y'-(\cos x) y =0\;\;\;\;,\;\;\;\;0<x<1$$</span></p>
<p><strong>My Try:</strong></p>
<p>I think the question is wrong, since the right hand si... | Doug | 1,064,129 | <p>By trial and error, you can show that one of the fundamental solutions is <span class="math-container">$y_1(x) = c\sin(x)$</span>. Now, Abel's Theorem tells us that the Wronskian of solutions is given by
<span class="math-container">$$W(x) = c\text{exp}\left(-\int 2\tan(x) dx\right) = c\cos^2(x).$$</span>
But, <spa... |
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... | Douglas S. Stones | 2,264 | <p>A specific (and rather well-known) example exists for prime numbers. There are necessary conditions (such as Fermat's Little Theorem) which is satisfied by all primes, but sometimes satisfied by composites as well. These composites are called <em><a href="http://en.wikipedia.org/wiki/Pseudoprime" rel="nofollow">ps... |
3,433,249 | <p>I need to find the number of <span class="math-container">$7$</span>s if we write all the numbers from <span class="math-container">$1$</span> to <span class="math-container">$1000000$</span>(so <span class="math-container">$77$</span>, for example, counts as two <span class="math-container">$7$</span>s and not one)... | kingW3 | 130,953 | <p>You're counting <span class="math-container">$7$</span> digit numbers but you only need <span class="math-container">$6$</span>.</p>
|
3,134,991 | <p>If nine coins are tossed, what is the probability that the number of heads is even?</p>
<p>So there can either be 0 heads, 2 heads, 4 heads, 6 heads, or 8 heads.</p>
<p>We have <span class="math-container">$n = 9$</span> trials, find the probability of each <span class="math-container">$k$</span> for <span class="... | Asinomás | 33,907 | <p>The probability is <span class="math-container">$\frac{1}{2}$</span> because the last flip determines it.</p>
|
3,134,991 | <p>If nine coins are tossed, what is the probability that the number of heads is even?</p>
<p>So there can either be 0 heads, 2 heads, 4 heads, 6 heads, or 8 heads.</p>
<p>We have <span class="math-container">$n = 9$</span> trials, find the probability of each <span class="math-container">$k$</span> for <span class="... | Arthur | 15,500 | <p>All the possible sequences of tosses (which are all equally likely) may be divided into the two categories "even number of heads" and "odd number of heads". You can then pair up each sequence in one category with the "flipped" version (flip every coin in the sequence around) in the other category. This shows that th... |
3,134,991 | <p>If nine coins are tossed, what is the probability that the number of heads is even?</p>
<p>So there can either be 0 heads, 2 heads, 4 heads, 6 heads, or 8 heads.</p>
<p>We have <span class="math-container">$n = 9$</span> trials, find the probability of each <span class="math-container">$k$</span> for <span class="... | Frxstrem | 1,556 | <p>There are two cases here:</p>
<ul>
<li>There's an even number of heads: 0, 2, 4, 6 or 8 heads</li>
<li>There's an odd number of heads: 1, 3, 5, 7 or 9 heads</li>
</ul>
<p>But notice that having an odd number of heads means having an even number of tails (e.g. 5 heads means 4 tails), so the second case is the same ... |
2,406,061 | <p>I am also confused about whether these are symbols or have some meaning of their own.
PS- I know that <span class="math-container">$\operatorname{d}y\over\operatorname{d}x$</span> geometrically represents the slope. But, I've come across <span class="math-container">$\operatorname{d}x\over\operatorname{d}y$</span> t... | Community | -1 | <p><a href="https://i.stack.imgur.com/XWjZz.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XWjZz.png" alt="enter image description here"></a></p>
<p>$dy$ represents the $y$ increment along the tangent. It is strictly proportional to $dx$.</p>
<p>$\Delta y$ represents the $y$ increment along the cu... |
1,458,405 | <p>(p ∨ q) → r ≡ (p → q) ∨ (p → r) could be valid or invalid</p>
<p>I need to prove it using logical equivalences (can't use truth table)</p>
<p>This is how far I've gotten by working with the right side:</p>
<p><-> p→(q v r)</p>
<p><-> ¬p v (q v r) </p>
<p>then commutative law</p>
<p><-> (q v r) v ¬p </... | Brian M. Scott | 12,042 | <p>When you get to $\neg p\lor(q\lor r)$, I would apply associativity:</p>
<p>$$\begin{align*}
(p\to q)\lor(p\to r)&\equiv p\to(q\lor r)\\
&\equiv\neg p\lor(q\lor r)\\
&\equiv(\neg p\lor q)\lor r\;.
\end{align*}$$</p>
<p>Your target, after all, is $(p\lor q)\to r$, which you know is equivalent to $\neg(p\... |
2,002,201 | <p>simplify <span class="math-container">$\sqrt{(45+4\sqrt{41})^3}-\sqrt{(45-4\sqrt{41})^3}$</span>.</p>
<blockquote>
<p>1.<span class="math-container">$90^{\frac{3}{2}}$</span></p>
<p>2.<span class="math-container">$106\sqrt{41}$</span></p>
<p>3.<span class="math-container">$4\sqrt{41}$</span></p>
<p>4.<span class="ma... | Bernard | 202,857 | <p><strong>Hint:</strong>
$$45\pm4\sqrt{41}=(2\pm\sqrt{41})^2.$$</p>
|
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