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 |
|---|---|---|---|---|
299,698 | <p>I am having trouble with the following exercise.
We are considering the following equation: </p>
<p>$m\frac{dv}{dt}=gm-Kv^2$</p>
<p>I need to solve the above equation knowing that $v(0)=32; m=128;g=10;K=5$</p>
<p>We have: $128\frac{dv}{dt}=1280-5v^2$</p>
<p>$\frac{dv}{dt}=10-\frac{5}{128}v^2$</p>
<p>$\frac{dv}{... | André Nicolas | 6,312 | <p><strong>Hint:</strong> Use <strong>partial fractions</strong> on the left. We have
$$\frac{1}{v^2-256}=\frac{1}{32}\left(\frac{1}{v-16}-\frac{1}{v+16}\right).$$</p>
|
299,698 | <p>I am having trouble with the following exercise.
We are considering the following equation: </p>
<p>$m\frac{dv}{dt}=gm-Kv^2$</p>
<p>I need to solve the above equation knowing that $v(0)=32; m=128;g=10;K=5$</p>
<p>We have: $128\frac{dv}{dt}=1280-5v^2$</p>
<p>$\frac{dv}{dt}=10-\frac{5}{128}v^2$</p>
<p>$\frac{dv}{... | Ron Gordon | 53,268 | <p>Following @Andre</p>
<p>$$\int \frac{dv}{v^2-256} = \frac{1}{32} \int dv \left ( \frac{1}{v-16} - \frac{1}{v+16} \right ) = \frac{1}{32} \ln{\left ( \frac{v-16}{v+16} \right ) }+ C$$</p>
|
3,185,927 | <p><strong>Proposition:</strong> Let <span class="math-container">$X$</span> be a compact Hausdorff space. Suppose there are countable real valued continuous functions <span class="math-container">$\{f_n\}_{n \in \mathbb{Z}_+}$</span> separating <span class="math-container">$X$</span> i.e. for all <span class="math-con... | Henno Brandsma | 4,280 | <p>I'll denote <span class="math-container">$\Bbb Z^+$</span> by <span class="math-container">$\omega$</span>.</p>
<p><span class="math-container">$\mathbb{R}^{\omega}$</span> (in the product topology) is metrisable by the metric <span class="math-container">$$D((x_n), (y_n))=\sum_{n \in \omega} \frac{\min(|x_n-y_n|, ... |
3,117,139 | <p>I am learning to calculate the arc length by reading a textbook, and there is a question</p>
<p><a href="https://i.stack.imgur.com/Zigqv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Zigqv.png" alt="enter image description here"></a></p>
<p>However, I get stuck at calculating</p>
<p><span cla... | caden Hong | 517,671 | <p>The example is inappropriate as this does not have an elementary solution. I think you actually mean <span class="math-container">$\int e^{x^2} x dx$</span>, substitution allows you see <span class="math-container">$1/2 \int e^u du$</span> then you can use the established formula to obtain the elementary solution.</... |
3,511,565 | <p>I am in the middle of an exercise and I am stuck in a final step in which I want to show that <span class="math-container">$y-x^2 \notin \langle x(x-1), (x-1)y\rangle$</span> in <span class="math-container">$\mathbb{C}[x,y]$</span>. My first thought was that <span class="math-container">$x-1$</span> does not divide ... | Blue | 409 | <p>There's an interesting generalization of this result to three dimensions.</p>
<p>Take <span class="math-container">$\triangle ABC$</span> (with angles <span class="math-container">$\alpha$</span>, <span class="math-container">$\beta$</span>, <span class="math-container">$\gamma$</span> and circumradius <span class=... |
4,215,226 | <p>Let <span class="math-container">$f$</span> be an integrable function on <span class="math-container">$[a,b]$</span>. Then I have to show that there exists <span class="math-container">$x$</span> in <span class="math-container">$[a,b]$</span> such that <span class="math-container">$\int_a^x f=\int_x^b f$</span>.
I u... | paw88789 | 147,810 | <p>The <span class="math-container">$48$</span> non-ace cards go into five (some possibly empty) runs of non-ace cards (before the first ace; between the first and second aces; ...; after the fourth ace).</p>
<p>So the expected length of each of these non-ace runs is <span class="math-container">$\frac{48}{5}=9.6$</spa... |
2,982,368 | <blockquote>
<p>Let <span class="math-container">$f:\mathbb R\to\mathbb R$</span> differentiable such that <span class="math-container">$f(0)=0$</span> and <span class="math-container">$f'(x)=[f(x)]^2$</span> for all <span class="math-container">$x\in\mathbb R$</span>. Prove that <span class="math-container">$f(x)=0$... | Kavi Rama Murthy | 142,385 | <p>On the open set <span class="math-container">$\{x:f(x) \neq 0\}$</span> we get <span class="math-container">$(x+c)f(x)=-1$</span> for some constant <span class="math-container">$c$</span>. This implies that the continuous function <span class="math-container">$(x+c)f(x)$</span> takes only the values <span class="mat... |
451,326 | <p>I am having trouble understanding a certain part of the proof on why a function cannot approach two different limits near $a$, so I will just list the relevant parts. If this is not enough/ambiguous then please tell me and I will type out the whole proof.</p>
<p>So, suppose we now have:</p>
<p>$$ \text{if } 0<|... | Etienne | 80,469 | <p>The function $t\mapsto \vert B(y,t)\vert$ is positive and $\mathcal C^1$ (in fact, it is $c\, t^n$ for some constant $c$); so you just need to show that $\varphi(t)=\int_{B(y,t)} u(x)\, dx$ is $\mathcal C^1$. If you put $x=y+t\xi$, then the change of variable formula gives
$\varphi (t)=t^n\int_{B(0,1)} u(y+t\xi )\,... |
204,003 | <p>I have a sequence of numbers
like 1,7,22,45,12,96,21,45,65,36,85,14,51,16,18,17,16....65...</p>
<blockquote>
<p>IS there any formula to check whether the sequence is random or not ?</p>
</blockquote>
<p>In my case</p>
<ol>
<li>odd numbers are not random since previous+2 </li>
<li>even numbers are not random <... | Gerry Myerson | 8,269 | <p>There is a very good discussion of this question in Seminumerical Algorithms, which is Volume 2 of Knuth's The Art Of Computer Programming. </p>
|
474,260 | <p>We know that irrational number has not periodic digits of finite number as rational number.<br>
All this means that we can find out which digit exist in any position of rational number.<br>
But what about non-rational or irrational numbers?<br>
For example:<br>
How to find out which digit exists in Fortieth position... | Mark Bennet | 2,906 | <p>You can use <a href="http://en.wikipedia.org/wiki/Continued_fraction" rel="nofollow">continued fraction</a> approximations to find rational numbers arbitrarily close to any irrational number.</p>
<p>For $\sqrt 2$ this is equivalent to the chain of approximations $\frac 11, \frac 32, \frac 75, \frac {12}{17} \dots$ ... |
2,869,189 | <p>Is the method of calculating determinant of $3\times 3$ matrix by diagonals,
apply also on $4\times 4$ matrix?</p>
<p>for example:</p>
<p>$$\begin{matrix}2&2&1&3|\\1&4&4&5|\\5&1&1&6|\\7&1&4&5|\end{matrix}\begin{matrix}2&2&1\\1&4&4\\5&1&1\\7&am... | saulspatz | 235,128 | <p>No, it isn't valid. When you calculate the determinant of an $n\times n$ matrix by minors, you compute $n!$ products. When $n=3, n!=6$ and there are $6$ broken diagonals, and the "spaghetti rule" still involves $6$ products. But when $n=4, n!=24,$ and there are only $8$ broken diagonals, so there isn't much hope.... |
2,869,189 | <p>Is the method of calculating determinant of $3\times 3$ matrix by diagonals,
apply also on $4\times 4$ matrix?</p>
<p>for example:</p>
<p>$$\begin{matrix}2&2&1&3|\\1&4&4&5|\\5&1&1&6|\\7&1&4&5|\end{matrix}\begin{matrix}2&2&1\\1&4&4\\5&1&1\\7&am... | Bernard | 202,857 | <p>The fastest method is Gaussian elimination to obtain a triangular matrix – in this case the determinant is the product of the diagonal elements:
\begin{align}
&\begin{vmatrix}
2&2&1&3\\1&4&4&5\\5&1&1&6\\7&1&4&5
\end{vmatrix}
=-\begin{vmatrix}
1&4&4&5\\... |
1,903,495 | <p>Suppose,$y_1=y_1(x_1,x_2)$ and $y_2=y_2(x_1,x_2)$,
such that,</p>
<p>$$dy_1=\frac{\partial y_1}{\partial x_1}dx_1+\frac{\partial y_1}{\partial x_2} \, dx_2$$</p>
<p>$$dy_2=\frac{\partial y_2}{\partial x_1}dx_1+\frac{\partial y_2}{\partial x_1} \, dx_2$$</p>
<p>Then,I've taken the product of the above two,but unab... | Doug M | 317,162 | <p>(5 red, 0 white) -- ${15\choose 5}$</p>
<p>(4 red, 1 white) -- ${15\choose 4}{10\choose1}$</p>
<p>(3 red, 2 white) -- ${15\choose 3}{10\choose2}$</p>
|
4,068,493 | <p><a href="https://i.stack.imgur.com/Rzj9N.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Rzj9N.jpg" alt="triangle diagram" /></a></p>
<p>Given that:</p>
<ol>
<li>[BM) is the bisector of the angle ABC.</li>
<li>(BM) and (AN) are parallel straight lines.</li>
</ol>
<p>I am trying to prove that the t... | dodoturkoz | 804,253 | <p>Look for the alternate interior angles and corresponding angles in the figure: <span class="math-container">$\measuredangle MBA=\measuredangle BAN$</span> and <span class="math-container">$\measuredangle CBM=\measuredangle BNA$</span>, so the triangle is isosceles.</p>
<hr />
<p>If you insist to use Thales':</p>
<p>... |
14,167 | <p>I randomly pick a natural number <em>n</em>. Assuming that I would have picked each number with the same probability, what was the probability for me to pick <em>n</em> before I did it?</p>
| AplanisTophet | 437,710 | <p>Some clarification should be added to the above answers.</p>
<p>By partitioning the reals into non-measurable sets, one can devise a way to relate a real selected uniformly at random from an interval such as [0, 1) to a natural number using a method by which all natural numbers have an "equal," however undefined, p... |
2,006,870 | <p>Is the following inequality true?</p>
<p>$\left( \sum \limits_{i=1}^\infty \sum \limits_{j=1}^\infty \sum \limits_{k=1}^\infty \sum \limits_{l=1}^\infty a_{ij}\,a_{ik}\,a_{jl}\,a_{kl} \right) \leq \left( \sum \limits_{i=1}^\infty \sum \limits_{j=1}^\infty a_{ij}^2 \right)^{1/2}\left( \sum \limits_{i=1}^\infty \sum ... | barak manos | 131,263 | <p>If $x,y\in[0,999999999]$, then you can use $f(x,y)=1000000000x+y$.</p>
|
3,059,150 | <p>Does there exist any open linear (vector) subspace of a Hilbert space? I could not think of any example.</p>
<p>Actually, I was reading the book by Simmons, there almost in every theorem it assumed that "If M is a closed linear subspace".It seemed natural to me to think about subspaces which are not closed. I have ... | pitariver | 614,358 | <p>If <span class="math-container">$M \leq \mathcal{H}$</span> a subspace of a Hilbert space (or generally any normed space) is open, then it contains a ball around the origin <span class="math-container">$0 \in B_r(0) \subset M$</span>, but for every (none-zero) vector <span class="math-container">$v \in \mathcal{H}$<... |
257,839 | <p>I was trying to solve the ODE</p>
<p>\begin{equation}
\ddot{r} r = \alpha(\dot{r}^2-1)
\end{equation}
where $\alpha$ is an arbitrary constant. There are some simple cases when $\alpha = -1 $ then you can use separation of variables to find the solution. For the initial condition when $r'(0)=1$ it also simplifies, $... | Mathlover | 22,430 | <p>HINT:</p>
<p>\begin{equation}
\ddot{r} r = \alpha(\dot{r}^2-1)
\end{equation}</p>
<p>$$\frac{\dot{r}\ddot{r}}{\dot{r}^2-1}=\alpha\frac{\dot{r}}{r}$$</p>
<p>$$\frac{\dot{r}\ddot{r}}{(\dot{r}+1)(\dot{r}-1)}=\alpha\frac{\dot{r}}{r}$$</p>
<p>$$\frac{\ddot{r}}{\dot{r}+1} +\frac{\ddot{r}}{\dot{r}-1}=2\alpha\frac{\dot{... |
257,839 | <p>I was trying to solve the ODE</p>
<p>\begin{equation}
\ddot{r} r = \alpha(\dot{r}^2-1)
\end{equation}
where $\alpha$ is an arbitrary constant. There are some simple cases when $\alpha = -1 $ then you can use separation of variables to find the solution. For the initial condition when $r'(0)=1$ it also simplifies, $... | doraemonpaul | 30,938 | <p>Case $1$ : $\alpha=0$</p>
<p>Then $\ddot{r}r=0$</p>
<p>$r=0$ or $\ddot{r}=0$</p>
<p>$r=0$ or $r=C_1t+C_2$</p>
<p>$\therefore r=C_1t+C_2$</p>
<p>Case $2$ : $\alpha\neq0$</p>
<p>Then $\ddot{r}r=\alpha(\dot{r}^2-1)$</p>
<p>$r\dfrac{d^2r}{dt^2}=\alpha\left(\left(\dfrac{dr}{dt}\right)^2-1\right)$</p>
<p>Let $u=\d... |
2,696,320 | <p>If $\gcd(m,15)= \gcd(n,15)=1$, show that either $15|(m^4 + n^4)$ or $15|(m^4 - n^4)$. </p>
<p>I'm really stuck on this proof. This is what I know: </p>
<p>Since the $\gcd(m,15)= 1$ we can write it as $mx+15y=1$ where $x,y$ are integers. Also, since $\gcd(n,15)=1$ it can also be written as $nu + 15v = 1$ where $u,v... | DonAntonio | 31,254 | <p>The conditions mean that $\;m,\,n\;$ aren't divisible by $\;3\;$ and neither by $\;5\;$ , yet $\;m^4\,,\,n^4\;$ are both $\;=1\pmod 3\;$ , whereas by Fermat's Little Theorem both $\;m^4,\,n^4=1\pmod 5\;$ ...</p>
|
307,466 | <p>Is there any simple way to construct an entiere function $f$ such that :
$$\forall p \in {\mathbb N} \quad f(2^p)=(-1)^p$$</p>
| Community | -1 | <p>Typically, one uses both the <a href="https://en.wikipedia.org/wiki/Weierstrass_factorization_theorem" rel="noreferrer">Weierstrass factorization</a> and <a href="https://en.wikipedia.org/wiki/Mittag-Leffler%27s_theorem" rel="noreferrer">Mittag-Leffler</a> theorem to prove the existence of an entire function $f$ suc... |
606,656 | <p>So I heard this a long time ago and I recently started thinking about it again. So I was told that the complex function $f(z)=1/z$ maps everything inside a circle to points outside the circle (the remaining part of the complex plane). Why is this?</p>
<p>I realize we can write $$f(z)=\frac{1}{z}=\frac{\overline{z}}... | Daniel Robert-Nicoud | 60,713 | <p>A point is inside the (unit) circle if $|z|<1$, thus
$$|f(z)|=\frac{1}{|z|}>1$$
is outside the circle if $z$ was inside the circle, and vice-versa.</p>
|
3,197,301 | <p>Let <span class="math-container">$OABC$</span> be a tetrahedron such that <span class="math-container">$|OA|=|OB|=|OC|$</span>. Denote by <span class="math-container">$D$</span> and <span class="math-container">$E$</span> the midpoints of segments <span class="math-container">$AB$</span> and <span class="math-contai... | Martin Argerami | 22,857 | <p>This works in any metric space. So you have a distance <span class="math-container">$d$</span> and the ball of radius <span class="math-container">$1$</span> around <span class="math-container">$x_0$</span>. </p>
<p>Let <span class="math-container">$r=1-d(x_1,x_0)$</span>. If <span class="math-container">$x$</span>... |
146,394 | <p>Here is a nice figure</p>
<p><a href="https://i.stack.imgur.com/e6Kuf.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/e6Kuf.jpg" alt="enter image description here"></a></p>
<p>Now let's create a grid</p>
<pre><code>plot = Show[GraphicsGrid[{{im1, im1, im1},
{SpanFromLeft, im1, im1}... | Peter | 63,945 | <p>I had the same issue. You have to find first the hidden directory. Go to the search icon in task bar and </p>
<p>1)type file explorer,</p>
<p>2) view hidden systems files and hidden system data something like that(I have a greek version of windows 10 so it's in greek)</p>
<p>3)select the option for programmers a... |
1,988,021 | <p>Consider $n$ objects each with an associated probability $p_i$, $i\in\{1,\dots,n\}$. If I sample objects $k$ times independently with replacement according to the probability distribution defined by the $p_i$, how does one compute the expected number of times you sample an object you have sampled before?</p>
<p>W... | Kiki | 281,305 | <p>Try the hyperbolic substitution $x=sinh(y)$</p>
|
4,010,070 | <p>The question was:</p>
<ul>
<li>Find <span class="math-container">$n$</span> where <span class="math-container">$GCD(a_{n}, 14) = 7$</span> where <span class="math-container">$n$</span> is natural if you knew that <span class="math-container">$a_{n} = n + 3$</span></li>
</ul>
<p>The book solved it by saying that mean... | Son Gohan | 865,323 | <p>Your last line comes straight from the definition of <span class="math-container">$\varphi(f)$</span> and the definition of continuity. Since <span class="math-container">$\varphi(f)$</span> is continuous, for every <em>small number</em> (here it is 1), we can find a neighborhood <span class="math-container">$V$</sp... |
4,010,070 | <p>The question was:</p>
<ul>
<li>Find <span class="math-container">$n$</span> where <span class="math-container">$GCD(a_{n}, 14) = 7$</span> where <span class="math-container">$n$</span> is natural if you knew that <span class="math-container">$a_{n} = n + 3$</span></li>
</ul>
<p>The book solved it by saying that mean... | Evangelopoulos Phoevos | 739,818 | <p>Some background: Suppose X is a vector space and <span class="math-container">$\Phi $</span> is a family of linear functionals on <span class="math-container">$X$</span>. This family defines a topology on <span class="math-container">$X$</span>, denoted by <span class="math-container">$\mathcal T_\Phi$</span> such ... |
184,142 | <p>I am currently an undergraduate math student. (In fact, freshmen.)</p>
<p>I know that usually abstract algebra is taught somehow late in the undergraduate course, and curious how studies of abstract algebra at graduate level differ from studies at undergraduate level.</p>
<p>So, things like what gets new treatment... | Mathemagician1234 | 7,012 | <p>What constitutes a "graduate algebra' course in the United States has undergone a lengthy evolution. As MTurgon and rschwieb said earlier, it's highly subjective from teacher to teacher. But I think the evolution of the subject gives some insight of what can be expected at most programs today for a year long graduat... |
279,238 | <p>I want to create a regular polygon from the initial two points <span class="math-container">$A$</span>, <span class="math-container">$B$</span> and number of vertices <span class="math-container">$n$</span>,<br />
<a href="https://i.stack.imgur.com/dKr9A.png" rel="noreferrer"><img src="https://i.stack.imgur.com/dKr9... | flinty | 72,682 | <p>You <em>can</em> use <code>CirclePoints</code>. You just need to transform them:</p>
<pre><code>{x1, y1} = {3, 6};
{x2, y2} = {-4, 2};
circ = N@CirclePoints[5];
tran = Last@FindGeometricTransform[{{x1, y1}, {x2, y2}}, circ[[;;2]]];
With[{tcirc = tran[circ]},
ListLinePlot[Append[tcirc, First@tcirc], AspectRatio ->... |
3,462,868 | <p>What is the minimal size of a set <span class="math-container">$\mathfrak S$</span> of <span class="math-container">$k$</span>-element subsets of <span class="math-container">$\{1,...,n\}$</span> such that for any <span class="math-container">$k$</span>-element subset <span class="math-container">$S$</span> of <span... | Matt Samuel | 187,867 | <p>Try taking the derivative again. The derivative of the constant is zero, so adding the constant gives you the same derivative. It turns out the converse is true: if two functions have the same derivative, then they differ by a constant. In order to get all antiderivatives, you have to include the constant. </p>
|
3,462,868 | <p>What is the minimal size of a set <span class="math-container">$\mathfrak S$</span> of <span class="math-container">$k$</span>-element subsets of <span class="math-container">$\{1,...,n\}$</span> such that for any <span class="math-container">$k$</span>-element subset <span class="math-container">$S$</span> of <span... | Gibbs | 498,844 | <p>Your wording is far from being formal. The operator <span class="math-container">$d/dt$</span> should not be interpreted as a fraction, rather as a differential operator, a derivation, or a vector field depending on the context. On the other hand, the differential <span class="math-container">$dt$</span> is its "dua... |
549,245 | <p>Apparently, it is a common mistake to write $\forall x[R(x) \rightarrow P(x)]$ instead of $\forall x[R(x)\wedge P(x)]$ in some cases, however I can't seem to find the difference between the two. </p>
<p>I did some research in ProofWiki and a Discrete Mathematics book, but I couldn't find this specific difference ex... | DonAntonio | 31,254 | <p>Try with the following basic, intuitive definitions:</p>
<p>$$x\in\Bbb Z=\text{the set of integer numbers}\;,\;\;R(x):=x\;\text{is a prime number}\;,\;$$</p>
<p>$$P(x):=x\;\text{not divisible by}\;4$$</p>
<p>Then you try to explain the difference between "For any integer $\;x\;$ , if $\;x\;$ is a prime number the... |
549,245 | <p>Apparently, it is a common mistake to write $\forall x[R(x) \rightarrow P(x)]$ instead of $\forall x[R(x)\wedge P(x)]$ in some cases, however I can't seem to find the difference between the two. </p>
<p>I did some research in ProofWiki and a Discrete Mathematics book, but I couldn't find this specific difference ex... | Trevor Wilson | 39,378 | <p>It's the difference between</p>
<ol>
<li><p>"anyone who gets a perfect score on the quiz will receive a cookie," and</p></li>
<li><p>"everyone gets a perfect score on the quiz and will receive a cookie."</p></li>
</ol>
<p>It's a bit tricky to translate number 1 from colloquial language into first-order logic, but ... |
3,020,825 | <p>let A be a subset of <span class="math-container">$\mathbb{R}^P$</span> and <span class="math-container">$ x \in \mathbb{R}^P$</span> denotes <span class="math-container">$d(x,A) = \inf \{ d(x,y) : y \in A\}$</span>.There exists a point <span class="math-container">$y_0 \in A $</span> with <span class="math-c... | Patrick Stevens | 259,262 | <ul>
<li>Show that "<span class="math-container">$n$</span> is square" is primitive recursive (more formally, that <span class="math-container">$n \mapsto 1$</span> if <span class="math-container">$n$</span> is square, and <span class="math-container">$0$</span> otherwise).</li>
<li>Show that the function <span class="... |
281,386 | <p>I have asked this question on stackexchange and have not received any answers or comments after 2 days of it being there.</p>
<p>I read somewhere that the following statement is correct. A proof or any hint as to how to prove it would be helpful.</p>
<p>Let $G$ be a connected reductive group defined over $k$(mayno... | nfdc23 | 81,332 | <p>First, some easy reduction steps to pass to the case where $G$ is semisimple. Let $T$ be a maximal $k$-torus in $G$, and pass to a finite separable extension of $k$ so that $T$ is split. We know that $H$ is the almost-direct product $\mathscr{D}(H) \cdot Z$ for a subtorus $Z \subset T_{\overline{k}}$ that is the ma... |
1,017,320 | <p>Assume ZFC. Let $B\subseteq\mathbb R$ be a set that is not Borel-measurable. Clearly, $B$ must be uncountable, since countable sets are always Borel being a countable union of measurable singletons.</p>
<p><strong>Question:</strong> can one conclude that $B$ necessarily has the cardinality of the continuum <em>with... | Ramiro de la Vega | 94,514 | <p>The situation is quite the opposite. Every Borel set is either countable or has size continuum. So if $B \subseteq \mathbb{R}$ is an uncountable set of size less than the continuum, it is not Borel-measurable.</p>
|
15,702 | <p>One way to prove that a field $K$ has no ideals except the entire field and the trivial ideal is to note the fact that every element $x$ has an inverse. By the definition of an ideal, if $x$ is in the ideal then $x^{-1}x$ is because $x^{-1} \in K$. But now we have that 1 is in the ideal, and so again by the definiti... | Alon Amit | 308 | <p>Ideals are not "unitless subsets", in the sense that there are many unitless subsets that aren't ideals. Proper ideals, indeed, contain no units, but they are also closed under addition, subtraction and multiplication by arbitrary ring elements.</p>
<p>I'm not sure why you're confused about the case of "multiples o... |
4,271,166 | <p>I have been studying probability and there are many results regarding the sum of random variables.</p>
<p>For example the sum of iid bernouli random variables is a binomial distributed
The sum of Geometric random variables is Negative binomial distributed
Sum of exponential random variables is Gamma distributed. Al... | Gwendolyn Anderson | 964,863 | <p>The most complete resource you may find online might be Wikipedia's List of Probability Distributions:
<a href="https://en.wikipedia.org/wiki/List_of_probability_distributions" rel="nofollow noreferrer">Wikipedia: List of Probability Distributions</a>.
You can click on each distribution listed to go to its own page,... |
4,056,627 | <p>Equation to minimize using Boolean Algebra Laws:
<span class="math-container">$A+\bar{A} B \bar{C}$</span></p>
<p>I have tried doing this but i am unsure of the answer:
<span class="math-container">$$
\begin{array}{l}
\text { Let } K=B \bar{C} \\
A+\bar{A} K=A+K=A+B \bar{C}
\end{array}
$$</span></p>
| Community | -1 | <p><span class="math-container">$G$</span> acts by left multiplication on the left quotient <span class="math-container">$G/H$</span>. If <span class="math-container">$G$</span> is simple, then <span class="math-container">$G$</span> embeds into <span class="math-container">$S_{[G:H]}$</span> and hence <span class="mat... |
1,523,313 | <p>Let $n$ be a positive integer. I want to prove one of these:</p>
<p>1.$\mathbb{R}^n$ cannot be expressed as $A\cup B$ where $A,B$ are algebraic varieties $V(I),V(J)$ and $A,B\neq\mathbb{R}^n$.</p>
<p>2.$\mathbb{R}^n$ is not Haussdorff under Zariski's topology.</p>
<p>3.If $V(J)=\mathbb{R}^n$ then $J=0$. </p>
<p>... | Servaes | 30,382 | <p>No; if $v$ is an eigenvector of a matrix $A$ for distinct eigenvalues $\lambda$ and $\mu$, then
$$\lambda v=Av=\mu v,$$
and hence $(\lambda-\mu)v=0$, which implies that $v=0$, contradicting the fact that $v$ is an eigenvector.</p>
|
1,523,313 | <p>Let $n$ be a positive integer. I want to prove one of these:</p>
<p>1.$\mathbb{R}^n$ cannot be expressed as $A\cup B$ where $A,B$ are algebraic varieties $V(I),V(J)$ and $A,B\neq\mathbb{R}^n$.</p>
<p>2.$\mathbb{R}^n$ is not Haussdorff under Zariski's topology.</p>
<p>3.If $V(J)=\mathbb{R}^n$ then $J=0$. </p>
<p>... | Alekos Robotis | 252,284 | <p>Suppose $v$ is an eigenvector of $A$ corresponding to two distinct eigenvalues $\lambda_1,\lambda_2$. This gives us $Tv=\lambda_1v$, $Tv=\lambda_2 v\implies \lambda_1v=\lambda_2v$, contrary to $\lambda_1\ne\lambda_2$ (because $v$ is nonzero).</p>
|
2,846,369 | <p>As the title suggests, I would like to find the eigenfunctions, $y_n(x)$ of a self-adjoint linear differential operator $\dfrac{d^2y}{dx^2}$ that satisfy the boundary conditions $y_n(0)=y_n(\pi)=0$. Thereafter, I would like to construct its Green's function, $G(x, \xi).$. </p>
<p>I know that it is common to define ... | Lutz Lehmann | 115,115 | <p>You find the general solution to $y''=λy$ and try to satisfy the boundary conditions. For positive $λ$ and almost all negative ones the only solution should be the zero solution. Cases with non-trivial solutions are the eigenvalues/eigenfunctions.</p>
<p>For the Green function compute IVP solutions $y_1,y_2$ with $... |
2,846,369 | <p>As the title suggests, I would like to find the eigenfunctions, $y_n(x)$ of a self-adjoint linear differential operator $\dfrac{d^2y}{dx^2}$ that satisfy the boundary conditions $y_n(0)=y_n(\pi)=0$. Thereafter, I would like to construct its Green's function, $G(x, \xi).$. </p>
<p>I know that it is common to define ... | Disintegrating By Parts | 112,478 | <p>The problem in solving the eigenfunction equation is that solutions are unique only up to non-zero multiplicative constants. That is, any non-trivial solution of
$$
y''=\lambda y , \;\;\; y(0)=y(\pi)=0
$$
may be scaled by any non-zero constant, and the new function is also a solution. Define $y_l$ a... |
1,540,635 | <p>Given $$\min\limits_x \|x\|_2^2$$ $$\text{s.t.} Ax = b$$ <strong>show $x^* = A^T(AA^T)^{-1}b$ where $A \in \mathbb{R}^{m \times n}, m < n$</strong></p>
<p>This is projection $x$ onto the hyperplane $Ax - b = 0$</p>
<p>Multiplying $A^T$ on both sides</p>
<p>$A^TAx - A^Tb = 0$</p>
<p>So we have $\langle A^TAx ... | Michael Grant | 52,878 | <p>It's not clear to me under what principle we could have expected "multiplying $A^T$ on both sides" to lead to a proper answer. For one thing, since $m<n$, $A^TA$ is not invertible. (As it is, the correct answer assumes that $A$ has full row rank so that $AA^T$ is invertible; this is of course not always true.)</p... |
2,859,411 | <p>Let $y(t)$ be a real valued function defined on the real line such that
$y'= y(1 − y)$, with $y(0) \in [0, 1]$. Then $\lim_{t\to\infty} y(t) = 1$.</p>
<p>The solution is given as false .But i have no idea about that. I try some counterexample but it won't work.</p>
<p>How can i find solution with 3 minutes?</p>
| José Carlos Santos | 446,262 | <p>Take $y(t)=0$. Then $y(0)\in[0,1]$ and $y'=0=y(1-y)$. But $\lim_{t\to\infty}y(t)=0\neq1$.</p>
|
161,766 | <p>What is the best way to write a quick function which takes in a vertex (of some undirected graph), and <strong>gives the value of the k-Core's in which that vertex is in?</strong></p>
<p>The obvious modification of <code>kCoreComponents[Graph,k]</code> using <code>Intersection</code> is either taking ages, since it... | Acus | 18,792 | <p>Here is how I used it in version 6. On my recent 10.3 it still works:
You can execute lines below directly in kernel (in text console, i. e. run math) or put it in .m file and use as script. Graphics server needs to be installed on the server.</p>
<pre><code>whereIam="/home/acus/temp";
$Path=Append[$Path,whereIam]... |
3,109,226 | <p>Express the following expression <span class="math-container">$$E=(x^3-y^3)(y^3-z^3)(z^3-x^3)$$</span> in terms of <span class="math-container">$a, b$</span> where <span class="math-container">$a,b \in \mathbb R$</span> and
<span class="math-container">$$a=x^2y+y^2z+z^2x$$</span> <span class="math-container">$$b=xy... | nonuser | 463,553 | <p>Start: <span class="math-container">$$ a-b = xy(x-y)+yz (y-z)+zx(z-x) $$</span>
<span class="math-container">$$= xy(x-y)+yz (y-z)+zx(z-\color{red}y)+zx(\color{red}y-x) $$</span>
<span class="math-container">$$= (xy-zx)(x-y)+(yz -zx)(y-z)$$</span></p>
<p><span class="math-container">$$ =x(y-z)(x-y)+z(y-x)(y-z) $$</s... |
3,109,226 | <p>Express the following expression <span class="math-container">$$E=(x^3-y^3)(y^3-z^3)(z^3-x^3)$$</span> in terms of <span class="math-container">$a, b$</span> where <span class="math-container">$a,b \in \mathbb R$</span> and
<span class="math-container">$$a=x^2y+y^2z+z^2x$$</span> <span class="math-container">$$b=xy... | Parallelism Alert | 639,984 | <p>Firstly, we use <span class="math-container">$x^3-y^3=(x-y)(x^2+xy+y^2)$</span> for <span class="math-container">$(x,y)=(a,b),(b,c),(a,c)$</span> and we obtain
<span class="math-container">$$E=(x-y)(y-z)(z-x)(x^2+xy+y^2)(y^2+yz+z^2)(x^2+xz+z^2)$$</span>
<span class="math-container">$$E=(x^2z+y^2x+z^2y-x^2y-y^2z-z^2x... |
3,173,125 | <p>Let <span class="math-container">$f(x) = x^3 + a x^2 + b x + c$</span> and <span class="math-container">$g(x) = x^3 + b x^2 + c x + a\,$</span> where <span class="math-container">$a, b, c$</span> are integers and <span class="math-container">$c\neq 0\,$</span>. Suppose that the following conditions hold:</p>
<ol>
... | Hanno | 316,749 | <p>The conditions 1 and 2 imply that</p>
<ul>
<li><span class="math-container">$\;c\,$</span> must be <span class="math-container">$\,0\,$</span> or <span class="math-container">$\,-1$</span>,</li>
<li><span class="math-container">$\;a= -c^2$</span>, and <span class="math-container">$\:b= c^2-c-1\,$</span>.</li>
</ul>... |
1,125,891 | <p>Let $U=<\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0\end{pmatrix}, \begin{pmatrix}0 & 0 & -1 \\ 0 & 1 & 0\end{pmatrix}, \begin{pmatrix}2 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}>$ and $V= <\begin{pmatrix}2& 0 & 0 \\ 0 & 2 & 0 \end{pmatrix}, \begin{pmatrix}... | Bernard | 202,857 | <p>Let's denote the generators of $U$: $u_1$, $u_2$ and $u_3$ (in your very order) and those of $V$: $v_1$ and $v_2$. Note $f(u_1)=v_1+v_2$ and $f(u_2)$ and $f(u_3)$ are colinear with $f(u_1)$. With this information you can deduce $\operatorname{im} f$ and you can write the matrix of $f$ relative to the bases $(u_1,u_... |
204,087 | <p>What would be an isomorphism between
$\mathcal{F}(S;V / U)$ and $\mathcal{F}(S;V)/ \mathcal{F}(S;U)$, where
$S$ is a set, $V$ a vector space and $U$ a subspace of $V$. $\mathcal{F}(A,B)$ denotes the set of functions from $A$ to $B$ and $V/U$ is the quotient space. </p>
| Gerry Myerson | 8,269 | <p>You could take $\lambda_i={n\choose i}n^i$. Then your expression would be $$(a+abn)^n+(b+abn)^n-(abn)^n$$</p>
|
2,921,390 | <blockquote>
<p>Aashna and Radhika see the integers $1$ to $211$ written on a blackboard. They alternate turns and in every step each of them wipes out any $11$ numbers until only $2$ numbers are left on the blackboard. If the difference of these $2$ numbers (by subtracting the smaller from the larger) is $\geq 111$,... | user1234567890 | 492,496 | <p>If you chose to play first you would have 10 turns. -> 10 x 11 = 110</p>
<p>If you chose to play second you would have 9 turns. -> 9 x 11 = 99 </p>
<p>If you are second and remove every high number starting with 211 while the first removes the low numbers you would stop at 211 - 99 = 112.</p>
<p>The last left num... |
1,318,880 | <p>I'm trying to prove that $\operatorname{lcm}(n,m) = nm/\gcd(n,m)$
I showed that both $n,m$ divides $nm/\gcd(n,m)$
but I can't prove that it is the smallest number.
Any help will be appreciated.</p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> <span class="math-container">$\,\ n,m\mid j \!\iff\! nm\mid nj,mj\!$</span> <span class="math-container">$\overset{\ \rm\color{darkorange}U}\iff\! nm\mid (nj,mj) \overset{\ \rm \color{#0a0}D_{\phantom |}}= (n,m)j\!$</span> <span class="math-container">$\iff\! nm/(n,m)\mid j$</span></p>
<p>where... |
3,068,270 | <p>Given that <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are real constants and that the equation <span class="math-container">$x^4+ax^3+2x^2+bx+1=0$</span> has at least one real root, find the minimum possible value of <span class="math-container">$a^2+b^2$</span>.</p>
<p>I be... | vadim123 | 73,324 | <p><span class="math-container">$$x^4+ax^3+2x^2+bx+1=$$</span>
<span class="math-container">$$=(x^2+\frac{a}{2}x)^2-\frac{a^2}{4}x^2+2x^2+bx+1=$$</span>
<span class="math-container">$$=(x^2+\frac{a}{2}x)^2-\frac{a^2}{4}x^2+2x^2-\frac{b^2}{4}x^2+(\frac{b}{2}x+1)^2=$$</span>
<span class="math-container">$$=(x^2+\frac{a}{... |
2,337,357 | <p><a href="https://i.stack.imgur.com/iWuCv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/iWuCv.png" alt="enter image description here"></a></p>
<p>The derivative of $\frac1x$ is $\frac{-1}{x^2}$.</p>
<p>How do I find the $c$ if there is no zero in the derivative of the function?</p>
<p>I starte... | user164144 | 458,090 | <p>Assuming that the rolls are independent, the event X of rolling 3, 20-sided dice may be considered as a binomial distributed variable with n=3 and p=1/20. You are looking for P(X>1) which is the probability that you will get at least 2 outcomes the same. This is 0.00725. However, since there are 20 possible outcomes... |
1,549,506 | <p>To Prove $$\lim_{x \to 0}\frac{x^2+2\cos x-2}{x\sin^3 x}=\frac{1}{12}$$
I tried with L'Hospital rule but in vain.</p>
| Olivier Oloa | 118,798 | <p><strong>Hint</strong>. You may use Taylor expansions, as $x \to 0$,
$$
\cos x =1-\frac{x^2}{2}+\frac{x^4}{24}+O(x^6)
$$
$$
\sin x =x+O(x^3)
$$ giving
$$
x^2+2\cos x-2=\frac{x^4}{12}+O(x^6)
$$ $$
x\sin^3 x =x^4+O(x^6)
$$ and</p>
<blockquote>
<p>$$
\frac{x^2+2\cos x-2}{x\sin^3 x}=\frac{\frac{x^4}{12}+O(x^6)}{x^4+O(... |
1,549,506 | <p>To Prove $$\lim_{x \to 0}\frac{x^2+2\cos x-2}{x\sin^3 x}=\frac{1}{12}$$
I tried with L'Hospital rule but in vain.</p>
| nicole | 165,997 | <p>L'Hopital's rule may be used repeatedly until you find an answer:</p>
<p>$$\lim_{x \to 0}\frac{x^2+2\cos x-2}{x\sin^3 x}=\lim_{x \to 0}\frac{2x-2\sin x}{\sin^2 x(\sin x+3x\cos x)}$$</p>
<p>$$= \lim_{x \to 0}\frac{2-2\cos x}{ 6\sin ^2(x)\cos (x) - 3x(\sin^3(x)-2\sin(x)\cos^2(x))}$$</p>
<p>$$= \lim_{x \to 0}\frac{2... |
3,725,638 | <p>The exterior derivative of a scalar function is</p>
<p><span class="math-container">$d f(x,y,z) = (
\frac{\partial f}{\partial x} dx
+ \frac{\partial f}{\partial y} dy
+ \frac{\partial f}{\partial z} dz
)$</span></p>
<p>Am I correct in assuming then that</p>
<p><span class="math-container">$d\left( F_x(x,y,z) e_x ... | azif00 | 680,927 | <p><span class="math-container">$$d(F_xdx + F_ydy + F_zdz) = dF_x \wedge dx + dF_y \wedge dy + dF_z \wedge dz$$</span> and use that <span class="math-container">$$dF = \frac{\partial F}{\partial x}dx + \frac{\partial F}{\partial y}dy + \frac{\partial F}{\partial z}dz$$</span> according to your first claim.</p>
|
483,173 | <p>A non-zero matrix $A$ is said to be nilpotent for some positive integer $k\geq2$. If $A$ is nilpotent then is $I+A$ invertible?? Where $I$ is the identity matrix.</p>
| DonAntonio | 31,254 | <p>Hint: Suppose $\,A^k=0\;$ , then:</p>
<p>$$(A-I)(A^{k-1}+A^{k-2}+...+A+I)=A^K-1\ldots$$</p>
<p>...but you have $\,A\color{red}+I\;$ , so what to do...? :)</p>
|
3,836,662 | <p>I understand basic group theory. I would say that I've seen most of the standard stuff up to, say, the quotient group.</p>
<p>I feel like I've seen in more than one place the suggestion that group theory is the study of symmetries, or actions that leave something (approximately) unchanged. Unfortunately I can only f... | Community | -1 | <p>In the case that<span class="math-container">$H\triangleleft G$</span>, this follows immediately from the second isomorphism theorem.</p>
<p>But, actually your result is well known, and called the <em>product formula</em>. Neither <span class="math-container">$H$</span> nor <span class="math-container">$K$</span> i... |
3,974,768 | <p>Suppose <span class="math-container">$a_n \rightarrow a$</span>, <span class="math-container">$b_n \rightarrow b$</span> and <span class="math-container">$ a < b$</span>.
Is it true that there is <span class="math-container">$ N \in \mathbb{N} $</span> such that for any <span class="math-container">$ m > N $</... | Martin R | 42,969 | <p>Assume that <span class="math-container">$z$</span> is a root of the equation with <span class="math-container">$|z| \ge 2$</span>. Then
<span class="math-container">$$
1 = \left|\frac{7z^4+4z+1}{z^7} \right|
\le \left|\frac{7}{z^3}\right| + \left|\frac{4}{z^6}\right| +\left| \frac{1}{z^7}
\right| \\
\le \frac{7}{... |
3,974,768 | <p>Suppose <span class="math-container">$a_n \rightarrow a$</span>, <span class="math-container">$b_n \rightarrow b$</span> and <span class="math-container">$ a < b$</span>.
Is it true that there is <span class="math-container">$ N \in \mathbb{N} $</span> such that for any <span class="math-container">$ m > N $</... | offline | 556,883 | <p>This question can be solved using Roché's Theorem. Consider the holomorphic function</p>
<p><span class="math-container">\begin{align}
h(z) := f(z) + g(z) = z^7 + 7z^4 + 4z + 1
\end{align}</span>
with <span class="math-container">$f(z) = z^7$</span> and <span class="math-container">$g(z) = 7z^4+4z+1$</span>. Conside... |
20,470 | <p>I couldn't find a more descriptive title, but I guess an example will explain my problem.</p>
<p>I set up some customized <code>Grid</code> function including some additional functionalities which I control with custom Options. Additionally, I would like to change some of the standard <code>Grid</code> Options, e.... | Leonid Shifrin | 81 | <p>The problem in this particular case comes from the default setting of the <code>Dividers</code> option, which overrides the <code>Frame</code> option settings. This does not strike me as a right behavior, or at least these options are not as orthogonal as they should be. This appears to fix the problem:</p>
<pre><c... |
366,589 | <p>how do I solve this one? </p>
<p>$$\lim_{q \to 0}\int_0^1{1\over{qx^3+1}} \, \operatorname{d}\!x$$</p>
<p>I tried substituting $t=qx^3+1$ which didn't work, and re-writing it as $1-{qx^3\over{qx^3+1}}$ and then substituting, but I didn't manage to get on. </p>
<p>Thanks in advance! </p>
| Wolphram jonny | 43,048 | <p>Try substituting $t=q^{1/3}x$. Or, you can put the limit inside the definite integral first (the easy way)</p>
|
4,557,275 | <p>I am studying measure theory from Cohn's Measure theory textbook. If <span class="math-container">$(X, \scr A, \mu)$</span> is a measure space then a subset <span class="math-container">$N$</span> of <span class="math-container">$X$</span> is said to be locally null if for every <span class="math-container">$A \in \... | geetha290krm | 1,064,504 | <p>You cannot prove this. <span class="math-container">$\int |f|d\mu=0$</span> <strong>if and only</strong> if <span class="math-container">$f=0$</span> a.e. Take a locally null set <span class="math-container">$A$</span> which is not a null set and take <span class="math-container">$f=\chi_A$</span>. This is a counter... |
4,557,275 | <p>I am studying measure theory from Cohn's Measure theory textbook. If <span class="math-container">$(X, \scr A, \mu)$</span> is a measure space then a subset <span class="math-container">$N$</span> of <span class="math-container">$X$</span> is said to be locally null if for every <span class="math-container">$A \in \... | Oliver Díaz | 121,671 | <p>The statement in the OP's title is not quite correct.
Here is a a counter example: consider the set <span class="math-container">$[0,1]$</span> equipped with the Borel <span class="math-container">$\sigma$</span>-algebra, and define the measure <span class="math-container">$\mu$</span> as <span class="math-container... |
1,091,549 | <p>How to compute $\int_{0}^{+\infty} \frac{dx}{e^{x+1} + e^{3-x}}$?</p>
<p>My partial solution:
$$
\int_{0}^{+\infty} \frac{dx}{e^{x+1} + e^{3-x}} = \int_{0}^{+\infty} \frac{dx}{e^{3-x}(1 + e^{2x-2})} \\
= \int_{0}^{+\infty} \frac{e^{x-3}dx}{1 + e^{2x-2}}.
$$
Thank you very much.</p>
| DeepSea | 101,504 | <p><strong>Hint:</strong> Put $t = e^{x-1} \to e^{2x-2} = t^2\to I = \dfrac{1}{e^2}\cdot \displaystyle \int_{e^{-1}}^\infty \dfrac{dt}{1+t^2}$</p>
|
3,126,080 | <p>Is there any particular equation which doesn't work on the real plane of numbers but works on other planes?</p>
| trancelocation | 467,003 | <p><span class="math-container">$1 + 1 = 0$</span> in <span class="math-container">$\mathbb{F}_2$</span></p>
<p>(<span class="math-container">$\mathbb{F}_2$</span> is the field only consisting of <span class="math-container">$0$</span> and <span class="math-container">$1$</span> where <span class="math-container">$1$<... |
2,651,486 | <p>I have an exercise here that is asking me to write the MacLaurin formula of orders II, III, IV for a multivariable function. Example: $ f(x,y)=\cos x \cos y$</p>
<p>Can anyone tell me what the formula looks like for a multivariable function and maybe guide me through this example? Would be much appreciated!</p>
| user | 505,767 | <p><strong>HINT</strong></p>
<p>You can expand separately</p>
<ul>
<li>$\cos x=1-\frac{x^2}2+\frac{x^4}{4!}+o(x^4)$</li>
<li>$\cos y=1-\frac{y^2}2+\frac{y^4}{4!}+o(y^4)$</li>
</ul>
<p>and then multilply taking the terms to the desidered order.</p>
<p>That is for order IV</p>
<p>$$f(x,y)= \cos x \cdot \cos y =\left... |
3,648,315 | <p><a href="https://i.stack.imgur.com/IKnlD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/IKnlD.png" alt="enter image description here"></a></p>
<p>I have an item, let's say sword. </p>
<p>As described above, <br/>
If my sword is at state 1, I can try upgrading it. There are 3 possibilities. <br/... | Oliver Clarke | 373,486 | <p>Let's assume we have a solution <span class="math-container">$(x,y)$</span>. First we make a substitution <span class="math-container">$y = x + a$</span> for some integer <span class="math-container">$a$</span>. Substituting gives <span class="math-container">$2x^2 +2x(a+1) + (a - 83) = 0$</span> which we can think ... |
3,648,315 | <p><a href="https://i.stack.imgur.com/IKnlD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/IKnlD.png" alt="enter image description here"></a></p>
<p>I have an item, let's say sword. </p>
<p>As described above, <br/>
If my sword is at state 1, I can try upgrading it. There are 3 possibilities. <br/... | user21820 | 21,820 | <p>Will Jagy's approach is obviously the correct one. However, there is another way that does not rely on finding the algebraic factorization:</p>
<p><span class="math-container">$y·(2x+1) = 83-x$</span>.</p>
<p><span class="math-container">$2x+1 \mid 83-x$</span>.</p>
<p><span class="math-container">$2x+1 \mid 2·(8... |
471,710 | <p>Why do small angle approximations only hold in radians? All the books I have say this is so but don't explain why.</p>
| Andrew | 946,023 | <p>There are several correct ways to answer this that illuminate different aspects of what is going on (and I wouldn't be surprised if the answer is present on this site somewhere EDIT: indeed the other responses to this question are different ways of looking at it). Here is a geometrical answer.</p>
<p>The basic form... |
2,698,555 | <p><strong>Question</strong></p>
<p>Find all the rational values of $x$ at which $y=\sqrt{x^2+x+3}$</p>
<p><strong>My attempt</strong></p>
<p>Since we only have to find the rational values of $x$ and $y$, we can assume that
$$ x \in Q$$
$$ y \in Q$$
$$ y-x \in Q $$
Let$$ d = y-x$$
$$d=\sqrt{x^2+x+3}-x$$
$$d+x=\sqrt... | mbomb007 | 206,300 | <p>The red area $R$ is the area of the big quarter circle minus the internal white and blue areas. The radius of the larger circle is 2*r. So the area of the big quarter circle is:</p>
<p>$$ Q = \frac {\pi(2r)^2}{4} = \pi r^2 $$</p>
<p>Visually, this is equivalent to a quarter of each small circle (together, these ar... |
1,639,521 | <p>Let <span class="math-container">$f:[0,\infty)\to\mathbb{R}$</span> differentiable and suppose that <span class="math-container">$$\lim_{x\to\infty}f'(x)=L.$$</span> How can I prove that <span class="math-container">$$\lim_{x\to\infty}\frac{f(x)}{x} = L\;?$$</span></p>
<p>I have solved some similar problems using t... | Clement C. | 75,808 | <p><em>From the basic definition of convergence, with $\varepsilon$'s:</em></p>
<p>Fix $\varepsilon > 0$. By definition, there exists $a \geq 0$ such that, for all $ x\geq a$, $L-\varepsilon \leq f^\prime(x) \leq L+\varepsilon$.</p>
<p>For $x\geq a$, write
$$
f(x) - f(a) = \int_a^x f^\prime
$$
which gives
$$
(L-\v... |
569,484 | <p>A and B are events in a sample space with $p(A) > 0$ and $p(B) > 0$. Write $p(A|B)$ for the conditional probability of $A$ given that $B$ has occurred.</p>
<p>1) If $p(A|B) < p(A)$, show that $p(B|A) < p(B)$ </p>
<p>2) Show that $p(A|B) ≥ \frac{p(A)+p(B)-1}{p(B)}$</p>
<p>For the first part:
I subbed... | MasterOfBinary | 101,063 | <p>For 1, Start with what you know: $P(A|B) < P(A)$ and $P(A|B) = \frac{P(A \cap B)}{P(B)}$.</p>
<p>So do a bit of substitution and see what you get:</p>
<p>$$\frac{P(A \cap B)}{P(B)} < P(A)$$</p>
<p>You might notice something there that you can work with. Remember the formula for $P(B|A)$.</p>
|
128,533 | <p>For the description, I have a simplified problem like the following:</p>
<pre><code>MapAt[f[1, #1], {a, b, c, d}, #2] & @@@ {{1, 2}, {3, 4}}
</code></pre>
<p>will give</p>
<blockquote>
<pre><code>{{a, f[1, 1][b], c, d}, {a, b, c, f[1, 3][d]}}
</code></pre>
</blockquote>
<p>But actually <code>{a, f[1, 1][b], ... | Ali Hashmi | 27,331 | <p><strong>This may be the fastest way in my analysis so far and may compete with <code>Fold</code> or perhaps perform better</strong></p>
<pre><code>Module[{m = Range@10},
SetAttributes[func, HoldFirst];
func[x_, {}] := x;
func[x_, y_] := (x[[y[[1, 2]] ]] = BitSet[x[[y[[1, 2]]]], y[[1, 1]]];
func[x, Rest@y]);
func[... |
3,080,566 | <p>please tell me how I can solve the following equation. </p>
<p><span class="math-container">$$z^3+\frac{(\sqrt2+\sqrt2i)^7}{i^{11}(-6+2\sqrt3i)^{13}}=0$$</span></p>
<p>What formula should I use? If possible, tell me how to solve this equation or write where I can find a formula for solving such an equation. I sear... | Dr. Sonnhard Graubner | 175,066 | <p>Simplify at first, <span class="math-container">$$(\sqrt{2}+\sqrt{2}i)^7=64\sqrt{2}(1-i)$$</span>
to prove this write <span class="math-container">$$\sqrt{2}^7(1+i)^7$$</span> Calculate <span class="math-container">$$(1+i)^3\cdot (1+i)^3\cdot (1+i)$$</span></p>
|
2,729,364 | <p>What are direct methods for proving that a ring is a UFD in general without proving that it's a PID/Euclidean domain/field and using the fact that all those things are UFDs?</p>
<p>As an example, we can take <span class="math-container">$\mathbb{Z}[i]$</span> or <span class="math-container">$\mathbb{Z}[\sqrt{-2}]$<... | lhf | 589 | <p>Here are some thoughts:</p>
<ul>
<li><p><em>Exsitence</em> is easy to prove using induction on the norm.</p></li>
<li><p><em>Uniqueness</em> is the hard part, especially since it fails for most rings of the form <span class="math-container">$\mathbb Z[\sqrt d]$</span>. For the rings you've mentioned, it can be prov... |
593,746 | <p>In general, if a random process is ergodic, does it imply that it is also stationary in any sense?</p>
| Fred S | 349,583 | <p>Yes, ergodicity implies stationarity.</p>
<p>Consider an ensemble of realizations generated by a random process. Ergodicity states that the time-average is equal to the ensemble average. The time-average is obtained by taking the average of a single realization, giving you a particular number. To obtain the ensembl... |
4,627,133 | <p>We have <span class="math-container">$A$</span> <span class="math-container">$(3×3)$</span> matrix with real entries. We know that A is orthogonal and <span class="math-container">$\operatorname{trace}(A)>1$</span>. Show that matrix <span class="math-container">$A+I_{3}$</span> is invertible.</p>
<p>We can see th... | P. Lawrence | 545,558 | <p>All thw eigenvalues of <span class="math-container">$A$</span> are on the unit circle. There exists a real invertible <span class="math-container">$3 \times 3$</span> matrix <span class="math-container">$P$</span> such that <span class="math-container">$P^{-1}AP$</span> is <span class="math-container">$I$</span> or ... |
1,429,937 | <p>From <a href="http://www.gottfriedville.net/mathprob/comb-subrect.html" rel="nofollow">this</a> page it can be shown that the number of possible rectangles in and m*n grid can be found by first choosing $2$ lines from a possible $m+1$ to form the vertical sides of the rectangle, and then $2$ from the $n+1$ horizonta... | Yuriy S | 269,624 | <p>Yes, it's possible with complex numbers. If you are allowed to use complex numbers, then use Euler's identity: $\Bbb e ^{\Bbb ix} = \cos x + \Bbb i \sin x$.</p>
<p>Then you get</p>
<p>$$
\sin ax = \frac {\Bbb e ^{\Bbb i ax} - \Bbb e ^{-\Bbb i ax}} {2 \Bbb i} = \frac {(\Bbb e^ {\Bbb i bx})^\frac a b - (\Bbb e ^{-\B... |
1,818,281 | <p>Suppose I have a pair of 2 non-linear differential equations of the form:
$$\begin{matrix}
\frac{dy}{dt}=f(x,y)\\
\frac{dx}{dt}=g(x,y)
\end{matrix}$$
Equilibrium points are where the trajectory ends up on, when plotted on the $x-y$ plane.</p>
<p>What are the qualitative differences between a 'stable node' and a 's... | Rodrigo de Azevedo | 339,790 | <p>If we have two coupled nonlinear ODEs</p>
<p>$$\dot{x}_1 = f_1 (x_1, x_2) \qquad \qquad \dot{x}_2 = f_2 (x_1, x_2)$$</p>
<p>and the origin is an equilibrium point, i.e., $f_1 (0,0) = f_2 (0,0) = 0$, then let the <strong>linearization</strong> of vector field $f$ near the origin be</p>
<p>$$\dot{\mathrm{x}} = \mat... |
2,391,769 | <p>I'm concerned with the total number of ones, and the total number of runs, but not with the size of any of the runs.</p>
<p>For example, $N=8$, $R=3$, $C=5$ includes 11101010, 01101011 among the 24 total possible strings.</p>
<p>I can compute these for small $N$ easily enough, but I am specifically interested in t... | Marko Riedel | 44,883 | <p>Perhaps there is some value in solving this with generating functions.
In the present case we have the marked generating function using $z$
for ones and $w$ for zeros and $y$ for runs of ones</p>
<p>$$(1+y(z+z^2+z^3+\cdots))
\\ \times \left(\sum_{q\ge 0} (w+w^2+w^3+\cdots)^q
y^q (z+z^2+z^3+\cdots)^q\right)
\\ \ti... |
1,105,056 | <p>There's something I've never understood about polynomials.</p>
<p>Suppose $p(x) \in \mathbb{R}[x]$ is a real polynomial. Then obviously,</p>
<p>$$(x-a) \mid p(x)\, \longrightarrow\, p(a) = 0.$$</p>
<p>The converse of this statement was used throughout high school, but I never really understood why it was true. I ... | user 1 | 133,030 | <p><strong>Hint</strong>: <em>using Division Algorithm</em>, you have:<br>
$$p(x)=(x-a)q(x)+r $$
$$0=p(a)=r$$ </p>
<hr>
<p>Edition: (by comment of Marc van Leeuwen). definition from "<em>Cox D., Little J., O'Shea D. Ideals, Varieties, and Algorithms</em> "
<img src="https://i.stack.imgur.com/gQNEL.png" alt="enter... |
199,235 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/9505/xy-yx-for-integers-x-and-y">$x^y = y^x$ for integers $x$ and $y$</a> </p>
</blockquote>
<p>Determine the number of solutions of the equation
$n^m = m^n$
where both m and n are integers.</p>
| Berci | 41,488 | <p>I remember only the result, but not the proof (anyway, probably not too hard):</p>
<p>Either $n=m$ or $\{n,m\}=\{2,4\}$.</p>
|
1,567,473 | <p>$-\frac12·\frac{iz-2z}{z^2}+\frac{-1-i}{2i-2z}=
\frac{\frac{-3}4-\frac12i}{z}$</p>
<p>if $z=a+bi$,
How to find $a$ and $b$?
Thank you.</p>
| gerw | 58,577 | <p>Hint:
You could choose $M = b^2 + 1$. Then, try to find $a,b$ which maximizes $h = \min\{a, b/(b^2+1)\}$.</p>
|
38,597 | <p>Or equivalently, if $G$ is a group, do the projective and injective dimension of $Z$ (viewed as a $ZG$-module) agree?</p>
<p>Thanks! </p>
| Xiaolei Wu | 4,760 | <p>The cohomological and homological dimension of a group do not agree in general. For example, the homological dimension of the group <span class="math-container">$Z[\frac{1}{2}]$</span> is one, while its cohomological dimension is 2. However, if |G| is countable, then cohomological dimension is either equal to the h... |
1,226,162 | <p>\begin{align}
x' &= -x^3 + x^5 + (x^4)(y^5)\\[.7em]
y' &= -8y^3 + y^5 - 10(y^4)(x^5)
\end{align}
$(0,0)$ is obviously a critical point of the system, and we are given that it is asymptotically stable, but have to show it. </p>
<p>I have tried to make a Lyapunov function $V(x,y) = ax^2 + cy^2$, with ... | Robert Israel | 8,508 | <p>Hint: choose $a$ and $c$ so the terms in $x^5 y^5$ cancel. Note that for $x$ near $0$, lower powers of $x$ dominate higher powers, and similarly for $y$.</p>
|
115,657 | <p>In topology the spheres <span class="math-container">$S^n$</span> are the "simplest" closed manifolds, and they are like "Dirac's delta at <span class="math-container">$n$</span>" for (reduced) cohomology groups. Furthermore they are boundaries of the simplest compact manifolds-with-boundary, i.e... | Tom Goodwillie | 6,666 | <p>To expand on Will Sawin's comment in a vague sort of way (which is the best I can do):</p>
<p>The cofiber of the pair $(\mathbb P^n,\mathbb P^{n-1})$ (which doesn't exist as a scheme but does exist if you widen your scope to a suitable category of (pre)sheaves) can be called a $2n$-sphere, the motivic $2n$-sphere.<... |
115,657 | <p>In topology the spheres <span class="math-container">$S^n$</span> are the "simplest" closed manifolds, and they are like "Dirac's delta at <span class="math-container">$n$</span>" for (reduced) cohomology groups. Furthermore they are boundaries of the simplest compact manifolds-with-boundary, i.e... | Konrad Voelkel | 956 | <p>To expand on Tom Goodwillie's answer:</p>
<p>a precise definition of "motivic sphere" would be
$$S^{p,q} = \big( \Delta^{p-q} / \partial \Delta^{p-q} \big) \wedge \big( \bigwedge^q \mathbb{G}_m\big)$$
which you can interpret as the $q$-fold smash product of the multiplicative group $\mathbb{G}_m$ smashed with a $(p... |
4,482,600 | <p>Prove
<span class="math-container">$$\int_{0}^{\frac{\pi}{2}} \sqrt[n]{\tan x} \,dx = \frac{\pi}{2} \sec \left(\frac{\pi}{2n}\right)$$</span></p>
<p>for all natural numbers <span class="math-container">$n \ge 2$</span>.</p>
<p>There are several answers (<a href="https://math.stackexchange.com/questions/4385116/how-t... | robjohn | 13,854 | <p><span class="math-container">$$
\begin{align}
\int_0^{\frac\pi2}\sqrt[n]{\tan(x)}\,\mathrm{d}x\
&=\int_0^\infty\frac{u^{1/n}\,\mathrm{d}u}{1+u^2}\tag{1a}\\
&=\frac12\int_0^\infty\frac{v^{\frac{1-n}{2n}}}{1+v}\,\mathrm{d}v\tag{1b}\\
&=\frac\pi2\csc\left(\pi\frac{n+1}{2n}\right)\tag{1c}\\[6pt]
&=\frac\... |
4,482,600 | <p>Prove
<span class="math-container">$$\int_{0}^{\frac{\pi}{2}} \sqrt[n]{\tan x} \,dx = \frac{\pi}{2} \sec \left(\frac{\pi}{2n}\right)$$</span></p>
<p>for all natural numbers <span class="math-container">$n \ge 2$</span>.</p>
<p>There are several answers (<a href="https://math.stackexchange.com/questions/4385116/how-t... | Quanto | 686,284 | <p>Here is to integrate with partial fractions. Utilize the factorization<br />
<span class="math-container">$ 1+y^{2n} = \prod^n_{k=1} (y^2-2y\cos a_k +1 )$</span>, with <span class="math-container">$a_k=\frac{(2k-1)\pi}{2n}$</span>
<span class="math-container">\begin{align}
&\int_{0}^{\frac{\pi}{2}} \sqrt[n]{\tan... |
18,691 | <p>Let's say I have a folowing set of data:</p>
<ul>
<li>k = 1 : list of values </li>
<li>k = 3 : list of values </li>
<li>k = 10 : list of values</li>
</ul>
<p>I know that to make a <code>BoxWhiskerChart</code> I have to give it a list of listen of these values as data and ks as labels. </p>
<p>How do I force the o... | kglr | 125 | <p>From <a href="http://reference.wolfram.com/mathematica/ref/BoxWhiskerChart.html#" rel="nofollow noreferrer">docs > BoxWhiskerChart >Scope > Data and Wrappers</a>:</p>
<blockquote>
<p>Nonreal data is taken to be missing and typically yields a gap in the box-and-whisker chart</p>
</blockquote>
<p>So one can use th... |
2,028,646 | <p>Let two random variables: $$x_1 \sim Bin(100, 0.5) \\ x_2 \sim Bin(100, 0.6)$$</p>
<p>Now, we define a third random variable, $x_{12}$ which it's distribution is the aggregated distributions of $x_1$ and $x_2$, so it's <strong>not</strong> quite like $x_1 + x_2$ even though empirically the variance seems like the s... | BruceET | 221,800 | <p>From a theoretical point of view it is exactly the distribution of $X_1 + X_2.$ But it is <em>not</em> binomial because the two Success probabilities are not the same.
$$E(X_1 + X_2) = n_1p_1 + n_2p_2 = 50 + 60.$$
Provided $X_1$ and $X_2$ are <em>independent</em>, you also have
$$Var(X_1 + X_2) = n_1p_1(1-p_1) + n_2... |
116,392 | <p>As <a href="https://mathematica.stackexchange.com/questions/110808/a-custom-function-is-too-slow-when-i-first-time-to-run-it">my this previous post</a>.I get a custom function which can help me got some function contain a certain option.And I have put it in my "init.m" file.</p>
<pre><code>LookupOptionFunc... | Mr.Wizard | 121 | <p>The behavior you are encountering is the time taken by <em>Mathematica</em> evaluate all the Symbols in the System context, including definitions (and Options) that are only loaded on first use. (For one of my own encounters with this delayed loading please see <a href="https://stackoverflow.com/q/5649379">Why do I... |
3,800,833 | <p><strong>Question: Bob invests a certain sum of money in a scheme with a return of 22% p.a . After one year, he withdraws the entire amount (including the interest earned) and invests it in a new scheme with returns of 50% p.a (compounded annually) for the next two years. What is the compounded annual return on his ... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
2,006,927 | <p>I want to prove that $\cap_{i=1}^{\infty}\left(A_i \cap \left(\cup_{j=1}^{\infty}B_j\right)\right)=\cup_{j=1}^{\infty}\left(\left(\cap_{i=1}^{\infty}A_i\right) \cap B_j\right)$ for sets $A_i,B_j$ and natural numbers $i,j$. If an element $x$ belongs to the left hand side, then $x\in A_1$ and $x\in$ some of $B_j$ and ... | Jared N | 385,410 | <p>If $x$ is in the set on the left, then $x\in A_i$ for <em>all</em> indices $i$, and $x\in B_j$ for <em>some</em> index $j$. That would imply that
$$x\in B_j \cap \bigcap_{i=1}^\infty A_i,$$
(again, just for a <em>single</em> index $j$.)
Fortunately, we only need it to be true for a single $j$, because that implies ... |
98,317 | <p>This comes from Artin Second Edition, page 219. Artin defined $G = \langle x,y\mid x^3, y^3, yxyxy\rangle$, and uses the Todd-Coxeter Algorithm to show that the subgroup $H = \langle y\rangle$ has index 1, and therefore $G = H$ is the cyclic group of order 3.</p>
<p>That being the case, $x$ cannot be either $y$ or ... | Srivatsan | 13,425 | <p>Here's one way -- quite a bit ad hoc. </p>
<p>The basic idea is write all the elements in terms of $z := yx$. From the third relation, we can see that $z^2 y = 1$, or $z^2 = y^{-1} = y^2$. Therefore, $y = y^4 = z^4$. Now, we can also write $x$ in terms of $z$:
$$x = y^{-1} z = y^2 \cdot z = z^8 \cdot z = z^9 .$$... |
4,494,199 | <p>This is a problem from a past qualifying exam in complex analysis. I'm working through these to study for my own upcoming qual. For this question, I think my proof is fairly straightforward, but I'd like to know whether or not it is correct and complete. I'm also interested in other ways of answer the question. Than... | Z Ahmed | 671,540 | <p>Let <span class="math-container">$0\le a , b \le 1$</span></p>
<p>We prove that <span class="math-container">$$0\le \frac a{1+b}+\frac b{1+a} \le 1$$</span></p>
<p>Let <span class="math-container">$$F=\frac a{1+b}+\frac b{1+a}$$</span>
Then <span class="math-container">$F\ge 0$</span>, note that
<span class="math-co... |
158,978 | <p><a href="https://i.stack.imgur.com/5nKVV.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5nKVV.jpg" alt="Explore the Fourier Series for a square wave"></a></p>
<pre><code>f (t) = (4/pi) Sum[(1/n) sin (2 pi (f (t))), {n, 1, Infinity}]
</code></pre>
| José Antonio Díaz Navas | 1,309 | <p>Once you have the function and its Fourier representation, you can try with this:</p>
<pre><code>f[t_] = (4/Pi) Sum[(1/n) Sin[2 Pi n t], {n, 1, \[Infinity], 2}];
fserf[t_,nmax_] := (4/Pi) Sum[(1/n) Sin[2 Pi n t], {n, 1, 2*nmax-1, 2}];
GraphicsGrid[
Partition[
Plot[{f[t], fserf[t, #]}, {t, 0, 3}, PlotLabel ->... |
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