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 |
|---|---|---|---|---|
101,929 | <p>I am trying to do the following :-</p>
<pre><code>f[x_] := {x, 2};
Do[f[x_] :=Append[f[x], {x, 4}],{3}] % runs into recursion limit
</code></pre>
<p>because I want to grow my function (which is a list of functions) in a loop ? What is the right way to do this ?</p>
| djphd | 27,205 | <p>This can be done pretty neatly with <code>Nest</code> in which case everything is built before assigning it to your function.</p>
<pre><code>f[x_,n1_,n2_,n3_]:=Nest[Append[#,{x,n2}]&,{x,n1},n3]
f[x, 2, 4, 3]
{x, 2, {x, 4}, {x, 4}, {x, 4}}
</code></pre>
<p>Now you can set the constants and number of repetitions... |
1,901,170 | <blockquote>
<p>Consider the region $D$ bounded by the positive $y$-axis, the line $y=8$ and the line $y=x^3$. Evaluate the following integral between $x=0$ and $x=2$.
$$\iint_D \frac{x^3y^2}4\ dx\ dy$$</p>
</blockquote>
<p>What are the limits and what is the solution? I've asked people and they all give conflicti... | Dave L. Renfro | 13,130 | <p>Andres Mejia has recommended a couple of books (<strong>Div, Grad, Curl</strong> is especially nice), but his books are for vector <strong>calculus</strong> and from your question it sounds like what you want (now, at least) are books for vector <strong>geometry</strong>.</p>
<p>Consider <a href="http://rads.stacko... |
2,919,096 | <p>I am trying to solve a system of two second-order ODEs. After separating them, I obtained a fourth-order independent ODE as illustrated below. I wonder if there is a specific technique to solve it.</p>
<p>$$y^{(4)}+\frac{a_1}{x} y^{(3)}+\frac{a_2}{x^2}y^{(2)}+a_3y^{(2)}+\frac{a_4}{x}y^{(1)}+a_5y=0$$</p>
| Przemo | 99,778 | <p>This is a follow-up to the answer given by doraemonpaul above.Let us express the unknown function through its Laplace transform as follows:
<span class="math-container">\begin{equation}
y(x)=\int\limits_0^\infty e^{-x s} K(s) ds
\end{equation}</span>
By inserting the above into the ODE and then by integrating by par... |
910,552 | <blockquote>
<p>$\sin(nx)$ does not contain Cauchy subsequence in $L^p([0,2\pi]) $ for $1\leq p < \infty$</p>
</blockquote>
<p><strong>My attempt:</strong> </p>
<p>Set $f_n(x) = \sin(nx)$.</p>
<p>Argue by contradiction, suppose there exists a Cauchy subsequence $f_{n_k}$ in $L^p$ for some fixed $p$, then $f_{n_... | Davide Giraudo | 9,849 | <p>The given arguments are good. I agree with Golbez, the fact that if $(f_{n_k})_k$ converge weakly to something, then it converges to $0$ has to be detailed, for example noticing that $\int_{[0,2\pi]} f_{n_k}g\mathrm d\lambda\to 0$ for each continuous function $g$ and concluding by a density argument. </p>
|
4,159,337 | <p>How would you show
<span class="math-container">$$\log_2 3 + \log_3 4 + \log_4 5 + \log_5 6 > 5?$$</span></p>
<p>After trying to represent the mentioned expression in the same base, a messy expression is created.
The hint mentioned that the proof can take help of quadratic equations.</p>
<p>Could you provide some... | Boshu | 257,404 | <p><span class="math-container">$\log_2 3 + \log_3 4 + \log_4 5 + \log_5 6 > 4\times\sqrt[4]{\frac{\log 3}{\log 2}\times\frac{\log 4}{\log 3}\times\frac{\log 5}{\log 4}\times\frac{\log 6}{\log 5}}=4\times\sqrt[4]{\log_2 6}.$</span></p>
<p>Now, <span class="math-container">$5/4=1.25$</span>. So we are done if we can ... |
626,920 | <p>$a_n=\sum_{k=1}^{n} \frac{1}{n+k}=\frac{1}{n+1}+\frac{1}{n+2}+\dots+\frac{1}{2n}$</p>
<p>How to find $\lim a_n$?</p>
| WLOG | 21,024 | <p>With Eulero-Mascheroni : $$\sum_{k = 1}^{n}\frac{1}{k} - \log{n} \rightarrow \gamma$$ $$\sum_{k = 1}^{2n}\frac{1}{k} - \log{2n} \rightarrow \gamma$$ so $$\sum_{k = n+1}^{2n}\frac{1}{k} - \log{2n} +\log{n} \rightarrow 0$$ and then$$\sum_{k = n+1}^{2n}\frac{1}{k} \rightarrow \log{2}$$</p>
|
3,805,452 | <p>Given a stick and break it randomly at two places, what is the probability that you can form a triangle from the pieces?</p>
<p>Here is my attempt and the answer does not match, so I am confused what went wrong with this argument.</p>
<p>I first denote the two randomly chosen positions by <span class="math-container... | Draditate | 819,566 | <p>Here is my view of this problem, not a debug of your procedure(sorry)</p>
<p>Actually I figure this 'forming triangle' problem by 'forming triangles'.</p>
<p><a href="https://i.stack.imgur.com/LxTB7.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/LxTB7.png" alt="enter image description here" /></a... |
3,816,140 | <p>The points A, B, C, D have position vectors <strong>a</strong>, <strong>b</strong>, <strong>c</strong> and <strong>d</strong> respectively. Prove that the lines joining the midpoints of opposite edges of the tetrahedron ABCD bisect each other and give the position vector of the point of intersection.</p>
<p>I have s... | 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... |
3,713,206 | <p>Spivak (3rd edition) proposes solving the integral <span class="math-container">$$\int \frac{1+e^x}{1-e^x} dx$$</span> by letting <span class="math-container">$u=e^x$</span>, <span class="math-container">$x=\ln(u)$</span>, and <span class="math-container">$dx=\frac{1}{u}du$</span>. This results in the integral <span... | Joe | 623,665 | <p>Believe it or not,
<span class="math-container">$$
\int f(g(x)) \, dx
$$</span>
is still of the correct form to admit substitution, provided that <span class="math-container">$g$</span> is invertible. To begin with, consider what we are aiming for: we know that any integral of the form
<span class="math-container">$... |
764,437 | <p>I am trying to prove that $\mathbb{Z}[i]\cong \mathbb{Z}[x]/(x^2+1)$.</p>
<p>My initial plan was to use the first isomorphism theorem. I showed that there is a map $\phi: \mathbb{Z}[x] \rightarrow \mathbb{Z}[i]$, given by $\phi(f)=f(i).$ This map is onto and homorphic. The part I have a question on is showing th... | Christoph | 86,801 | <p>Let me elaborate on Daniel Fischers comment. You have a ring homomorphism $\phi: \mathbb Z[x]\to\mathbb Z[i]$ given by $x\mapsto i$. Take $f \in \ker \phi$. By the division algorithm,
$$ f = q\cdot(x^2+1) + r,$$
where $\deg r < 2$ and $q,r\in\mathbb Z[x]$, since $x^2+1$ is monic. Applying $\phi$ to this equation ... |
3,984,322 | <p>I have the 2nd order homogeneous ODE:
<span class="math-container">$$y(x)''+\frac{2}{x}y'(x)-By=0 $$</span>
Where <span class="math-container">$B$</span> is a constant. Since I am a bit rusty with analytical solutions, I plugged it in MatLab's symbolic solver which found the following solution.
<span class="math-con... | user577215664 | 475,762 | <p><span class="math-container">$$y''+\frac{2}{x}y'-\beta y=0$$</span>
<span class="math-container">$$xy''+{2}y'-\beta xy=0$$</span>
Substitute <span class="math-container">$z=\sqrt \beta x ( \beta >0)$</span>:
<span class="math-container">$$x{\beta} y''(z)+{2}{\sqrt {\beta}}y'(z)-\beta xy(z)=0$$</span>
<span class... |
99,617 | <p>How can I animate a point on a polar curve? I have used <code>Animate</code> and <code>Show</code> together before in order to get the curve and the moving point together on the same plot, but combining the polar plot and the point doesn't seem to be working because point only works with Cartesian coordinates.</p>
... | m_goldberg | 3,066 | <p>It is not necessary to resort to <code>Show</code>. Your animation can easily be accomplished with <code>Epilog</code>.</p>
<pre><code>Animate[
PolarPlot[2 Sin[4 θ], {θ, 0, 2 Pi},
Epilog ->
{Red, PointSize[Large], Point[{2 Sin[4 θ] Cos[θ], 2 Sin[4*θ] Sin[θ]}]}],
{θ, 0, 2 Pi},
AnimationRunning -... |
99,617 | <p>How can I animate a point on a polar curve? I have used <code>Animate</code> and <code>Show</code> together before in order to get the curve and the moving point together on the same plot, but combining the polar plot and the point doesn't seem to be working because point only works with Cartesian coordinates.</p>
... | MarcoB | 27,951 | <p>Conceptually you are there already; you only had a small syntax error in your code, i.e. <strong>a missing set of braces</strong> around the items in the <code>Graphics</code> expression:</p>
<pre><code>Animate[
Show[
PolarPlot[2 Sin[4*t], {t, 0, 2 Pi}],
Graphics[{Red, PointSize[0.02], Point[{2 Cos[θ] Sin[4 θ]... |
3,786,797 | <p>I have two variables <span class="math-container">$x$</span> and <span class="math-container">$y$</span> which are measured at two time steps <span class="math-container">$0$</span> and <span class="math-container">$1$</span>. Available to me is the percent change from the first timestep to the second, relative to t... | N. F. Taussig | 173,070 | <p><strong>Method 1:</strong> Arrange the five women in a row. This can be done in <span class="math-container">$5!$</span> ways. Doing so creates six spaces in which we can place the men, four between successive pairs of women and two at the ends of the row.</p>
<p><span class="math-container">$$\square W \square W... |
3,585,111 | <p>Given that <span class="math-container">$x^{\frac{2}{3}}=({x^2})^{\frac{1}{3}}$</span>, I thought that <span class="math-container">$f(x)$</span> is continuous in <span class="math-container">$\mathbb{R}$</span> but when I plot this function, I get something like this in wolfram.</p>
<p><a href="https://i.stack.img... | Deepak | 151,732 | <p>Not sure if this is the issue but, allowing for complex answers, there are actually three cube roots of every real number (except <span class="math-container">$0$</span>). For negative real numbers, the principal cube root (one with the smallest argument) is a complex number with non-zero imaginary part. Some comput... |
33,125 | <p>For some fixed $n$ define the quadratic form $$Q(x,y) = x^2 + n y^2.$$</p>
<p>I think that if $Q$ represents $m$ in two different ways then $m$ is composite.</p>
<p>I can prove this for $n$ prime. I was hoping someone could give me a hint towards proving this result for general $n$? Also would be interested in gen... | Gerry Myerson | 8,269 | <p>John Brillhart published a paper about this in the American Mathematical Monthly some time in the past year. </p>
|
381,313 | <p>Solve the limits of below:</p>
<p>(1) $\lim\limits_{n \to \infty} \int_0^n (1+\frac{x}{n})^n e^{-2x}dx$.</p>
<p>(2) $\lim\limits_{n \to \infty} \int_0^n (1-\frac{x}{n})^n e^{\frac{x}{2}}dx$. </p>
<p>(3) $\int_0^{\infty} \frac{e^{-x}\sin^2 x}{x}dx$</p>
<p>(Hint: show $f(x,y)=e^{-x}\sin2xy$ is integrable on $[0,... | Henry T. Horton | 24,934 | <p>Definitely not.</p>
<p>Here's a simple example. Depending on the choice of transition functions, a bundle of the form
$$[-1,1] \hookrightarrow E \to S^1$$
and structure group $\Bbb Z/2 = \{-1,1\}$ (acting on the fiber $[-1,1]$ via multiplication) can have total space a cylinder or a Möbius band.</p>
<p><strong>Edi... |
2,348,641 | <p><span class="math-container">$f$</span> is a periodic continuous function of period <span class="math-container">$T > 0 $</span>.</p>
<ol>
<li><p>Let <span class="math-container">$ a > 0$</span>, prove that: <span class="math-container">$ \int_{0}^{+ \infty} f(t)e^{-at}dt $</span> is absolutely convergent.</p>... | Jack D'Aurizio | 44,121 | <p>$4$. Since $f$ is a bounded function the given limit equals $0$ for any $t>0$;</p>
<p>$3$. By the Cauchy-Schwarz inequality</p>
<p>$$ \left|\int_{0}^{T}f(t)\,e^{-at}\,dt\right|\leq \sqrt{\int_{0}^{T}e^{-at}\,dt\int_{0}^{T}f(t)^2 e^{-at}\,dt} $$
hence we may take $\mu$ as $\sqrt{\frac{1-e^{-aT}}{a}}$;</p>
<p>$2... |
1,540,082 | <p>I need to prove that the following sequence converges as $n\to\infty$:
$\sum\limits_{k=n+1}^{2n} 1/k$</p>
<p>The problem is that I've only ever seen sums from i to n for example. I'm confused because not only is 2n alien to me, but also k and n are related.</p>
<p>So far I've tried substituting k for n+1 to simpli... | Eric Wofsey | 86,856 | <p>Let us write $a_n=\sum_{k=n+1}^{2n} 1/k$, and let's compare $a_n$ to $a_{n+1}$. They have almost the same terms, except $a_n$ has one more term at the beginning and $a_{n+1}$ has two more terms at the end. We get that $$a_{n+1}-a_n=\frac{1}{2n+1}+\frac{1}{2n+2}-\frac{1}{n+1}>\frac{1}{2n+2}+\frac{1}{2n+2}-\frac{... |
3,279,965 | <p>Let <span class="math-container">$A$</span> be a non-symmetric <span class="math-container">$n\times n$</span> real matrix. Assume that all eigenvalues of <span class="math-container">$A$</span> have positive real parts. Could you please show some simple conditions such that <span class="math-container">$A+A^T$</spa... | Community | -1 | <p>As the OP writes, that follows is a class of <span class="math-container">$A$</span> s.t. <span class="math-container">$A+A^T>0$</span>.</p>
<p>Choose <span class="math-container">$A=[a_{i,j}]$</span> s.t. for every <span class="math-container">$i$</span>, </p>
<p><span class="math-container">$2a_{i,i}>\sum_... |
3,779,674 | <p>I've been working on this question for some time now, and have not had any significant progress towards a solution:</p>
<blockquote>
<p>Let <span class="math-container">$\Omega = \{z = x +iy \in \mathbb{C} : |y| \leq 1\}.$</span> If <span class="math-container">$f(z) = z^2 +2$</span>, then draw a sketch of
<span cla... | Ben Grossmann | 81,360 | <p>I am not sure how fruitful your approach will be. One method that will work is as follows: with polynomial long-division, we find that
<span class="math-container">$$
(t + 1)^{10}(t - 1)^2 + t = q(t)(t^3 - t^2 - 2t + 1) + 4631 t^2 + 3721 t - 2560.
$$</span>
Thus, we have
<span class="math-container">$$
(A + I)^{10... |
2,232,677 | <p>A parallellogram $ABCD$ is given, where $E$ is the midpoint of $BC$ and $F$ is a point on $AD$ so that $|FD| = 3|AF|$. $G$ is the point where $BF$ and $AE$ intersects.</p>
<p>Express the vector $AG$ in terms of vectors $AB$ and $AD$.</p>
<p>My solution to the problem is the following: Impose an <em>affine transfor... | Ted Shifrin | 71,348 | <p>Sure: My standard way of teaching this material is to set $\overrightarrow{AB}=\vec x$ and $\overrightarrow{AD} = \vec y$ and write everything appropriate in terms of those vectors. In particular, $\overrightarrow{AE} = \vec x + \frac12\vec y$ and $\overrightarrow{AF} = \frac14\vec y$. We want to find $t$ so that $\... |
1,575,920 | <p>The global error of $\int f(x) \mathrm{d}x$, between two
$x$-values by the trapezoidal rule is
$-(1/12)h^3f''(ξ)$</p>
<p>a) $f(x) = x^3, x := [0.2,0.5]$</p>
<p>Find the value for $ξ$. Not really sure where to start with this problem. We never did any examples in class.</p>
| marty cohen | 13,079 | <p>Do this:</p>
<p>Find the actual value of the integral.</p>
<p>Find the trapezoidal rule estimate.</p>
<p>Get the difference between the two.
That is the error.</p>
<p>Since the error is of the form
$-(1/12)h^3f''(ξ)$,
and you know $h$ and $f$
(and therefore $f''$),
find $ξ$
from the error,
$h$,
and $f''$.</p>
|
2,911,849 | <p>Let $S_i$ be a countable family of sets and $f$ a continuous map (say everything is in the complex plane).
Is it true that
$$
f( \overline{ \cup_i S_i } ) =
\overline{ \cup_i f( S_i ) }
$$
or perhaps the inclusion $\subseteq$?</p>
| Community | -1 | <p>The identity is not even true for <em>one</em> set, because some continuous functions simply do not send closed sets to closed sets. Take, for instance, $f(z)=i\Im z+\arctan\Re z$, which sends $\Bbb C$ to $\left\{z\in\Bbb C\,:\, \lvert\Re z\rvert<\frac\pi2\right\}$. The $\subseteq $ is true, because $\overline{\b... |
77,980 | <p>How do I check whether a give function is 'algebraic' or not? I have a function $m(z) = 2\pi i z^n$ where $z \in \mathbb{C} \backslash \mathbb{R}$ and $n \in \mathbb{Z}$. I can write this as $w - 2\pi i z^n = 0$ where $w = m(z)$ and it is a polynomial. I think this is an algebraic function? Is my understanding corre... | Cyril Banderier | 85,924 | <p>If $F(z)=z^\gamma$ is an algebraic function (over $\mathbb Q$),
then $F'(z)$ is also an algebraic function (over $\mathbb Q$),
and thus $F'(1)$ is an algebraic number (over $\mathbb Q$).
Therefore, $z^\gamma$ algebraic implies $\gamma$ algebraic.
Equivalently, if $\gamma$ is not algebraic, then $z^\gamma$ is not al... |
145,868 | <p>I have a data set like below:</p>
<pre><code>368 150 0.31895877
369 150 0.32141133
370 150 0.3190167
371 150 0.33890757
372 150 0.3649395
373 150 0.37585051
374 150 0.37916244
375 150 0.37619812
376 150 0.37287654
377 150 0.37249031
378 150 0.37102263
379 150 0.34839918
380 150 No Data
381 150 No Data
382 150 No Da... | halirutan | 187 | <p>As you said in the comments, the data for your histograms did contain negatives and you squared all of them. I suggest to consider looking at your values before you square them</p>
<p><img src="https://i.stack.imgur.com/0HVJp.png" alt="Mathematica graphics"></p>
<p>Although I have no insight into the underlying pr... |
1,977,345 | <p>Queston;</p>
<p>Given that 2 is a generator of cyclic group U(25), find all generators.</p>
<p>I am only conversant with the finding the mod which is very long with this question. Pls can someone enlighten me on how to get it done faster.</p>
<p>I am new to the cyclic group.</p>
<p>solution
U(25) = {1,2,3,4,6,7,... | Andreas Caranti | 58,401 | <p>The formula you need is the one that tells you that if $g$ is a generator of the cyclic group $G$ of order $n$, then $g^{k}$ is a generator if and only if $\gcd(n, k) = 1$.</p>
|
18,564 | <p>I want to get the eight points of intersection from the equations <code>2 Abs[x] + Abs[y] == 1</code> and <code>Abs[x] + 2 Abs[y] == 1</code>. To solve these equations, I tried</p>
<pre><code>Solve[{2 Abs[x] + Abs[y] == 1, Abs[x] + 2 Abs[y] == 1}, {x, y}]
</code></pre>
<p>but could only get the four points. I then... | kglr | 125 | <p>Using <code>MeshFunctions</code>:</p>
<pre><code>ContourPlot[{2 Abs[x] + Abs[y] == 1, Abs[x] + 2 Abs[y] == 1}, {x, -1, 1}, {y, -1, 1},
MeshFunctions -> {ConditionalExpression[#2, Abs[#] < 1] &,
ConditionalExpression[#, Abs[#2] < 1] &, Abs[#2] - Abs[#] &},
Mesh -> {{0}, {0}, {0}}, Mes... |
918,693 | <p>I've been at this for a few hours now, and it's frighteningly similar to the problem stated here: </p>
<p><a href="https://math.stackexchange.com/questions/388356/how-to-prove-int-0-infty-e-x2cos2bx-dx-frac-sqrt-pi2-e-b2">How to prove $\int_0^\infty e^{-x^2}cos(2bx) dx = \frac{\sqrt{\pi}}{2} e^{-b^2}$</a></p>
<p>b... | voldemort | 118,052 | <p>Just substitute $u^2=ax^2$, and use the result of the previous integration.</p>
<p>So, we must have $a >0$ (convergence issues), and thus, $u=\sqrt a x$, and $du=\sqrt a dx$.</p>
<p>After this substution, your integrand becomes $\frac{1}{\sqrt a} (e^{-u^2}\cos(\frac{b}{\sqrt a}u) du)$, and this is just the inte... |
962,301 | <p>$$(\sqrt 2-\sqrt 1)+(\sqrt 3-\sqrt 2)+(\sqrt 4-\sqrt 3)+(\sqrt 5-\sqrt 4)…$$</p>
<p>I have found partial sums equal to natural numbers. The first 3 addends sum to 1. The first 8 sum to 2. The first 15 sum to 3. When the minuend in an addend is the square root of a perfect square, the partial sum is a natural number... | please delete me | 168,166 | <p>By telescoping, we see that the $n$th partial sum is $\sqrt {n+1}-1$, so this series diverges.</p>
|
2,679,525 | <p>Find the $\gcd(x^3-6x^2+14x-15, x^3-8x^2+21x-18)$ over $\mathbb{Q}[x]$. Then find two polynomials $a(x),b(x) \in \mathbb{Q}[x]$ such that, $$a(x)(x^3-6x^2+14x-15) + b(x)(x^3-8x^2+21x-18)=\gcd(x^3-6x^2+14x-15, x^3-8x^2+21x-18)$$ </p>
<p>I have managed to find,
$$x^3-6x^2+14x-15=(x-3)(x^2-3x+5)$$
$$x^3-8x^2+21x-18=... | Community | -1 | <p>If I interpreted your question correctly and you want to do something like this, where in:</p>
<p>$a(x)(p)+b(x)(q)=gcd(p,q)$, $a=1$, and $b=1$</p>
<p>You can restate your expression as:</p>
<p>$$a(x)(x-3)(x^2-3x+5)+b(x)(x-3)(x-3)(x-2)$$
$$(x-3)\left(a(x)(x^2-3x+5)+b(x)(x-3)(x-2)\right)$$</p>
<p>Since this has to... |
13,478 | <p>This may be a silly question - but are there interesting results about the invariant: the minimal size of an open affine cover? For example, can it be expressed in a nice way? Maybe under some additional hypotheses?</p>
| Dan Petersen | 1,310 | <p>The minimal number of open affine sets needed to cover (the coarse moduli space of) $M_{g,n}$ is a famous open problem. The conjecture, which is due to Looijenga if I recall correctly, is that the minimum is given by $g - \delta_{n,0}$ if $g > 1$. Proving this would in one swoop give a proof of Diaz's theorem and... |
1,343,554 | <p>This is exercise 6.10 in Resnick's book "A Probability Path".</p>
<p>We're given a sequence of random variables $(X_n)$ and an increasing function
$f: [0, \infty) \rightarrow [0, \infty)$ such that </p>
<p>$$
\frac{f(x)}{x} \rightarrow \infty, \qquad x \rightarrow \infty,
$$</p>
<p>and</p>
<p>$$
\sup_{n \geq 1} ... | David C. Ullrich | 248,223 | <p>Hmm. Writing integrals as integrals: Say $\epsilon>0$. You need to show there exists $\delta>0$ so $P(A)<\delta$ implies $\int_A|X_n|\,dP<\epsilon$.
You know that $\int f(|X_n|)\,dP\le K$ for some fixed $K$. Hmm.</p>
<p>Ok. Say $L>0$ is a large number. Choose $M$ so $$\frac{f(t)}t>L\quad(t>M).$... |
1,463,887 | <p>$f(x,y)=x^2-y^2$ constraint to $x^2+y^2=1$</p>
<p>$f_x=2x$ and $f_y=-2y$ $\implies$ the critical point is at $(0,0)$</p>
<p>However, $(0,0)$ does not occur in the constraint. does that mean i don't have to consider it?</p>
<p>So if we work of the boundary curves $y=\pm\sqrt{1-x^2}$</p>
<p>we get a min @ $(0,1)$ ... | juantheron | 14,311 | <p>Given $x^2+y^2 = 1\;,$ Let $x=\cos \phi$ and $y=\sin \phi\;,$ Then $x^2-y^2 = \cos^2 \phi-\sin^2 \phi = \cos 2\phi$</p>
<p>Now Using $\bullet\; -1\leq \cos 2\phi \leq 1$</p>
<p>So we get $x^2-y^2\in \left[-1,1\right]$</p>
|
1,836,998 | <p>$$
\lim_{x\to 0^{+}}\left[\left(1+\frac{1}{x}\right)^x+\left(\frac{1}{x}\right)^x+\left(\tan(x)\right)^{\frac{1}{x}}\right]$$</p>
<p>Attempt: I used $\tan(x)\approx x$ also $(1+n)^{1/n}=e$ so I let $x=0+h$ and lim changes to lim h tending to $0$. But that gives me $e+\infty+h^{1/h}$. While the answer is an integer ... | egreg | 62,967 | <p>Since you seem to be losing yourself among infinitesimals, I can suggest a safer way; when exponentials are involved, passing to the logarithm is often simpler.</p>
<p>Besides, the substitution $x=0+h$ is of no consequence.</p>
<p>Consider
$$
\lim_{x\to0^+}\log\left(\left(1+\frac{1}{x}\right)^{x}\right)=
\lim_{x\t... |
1,911,601 | <p>Alright so I had to find the area of the common region determined by $y>=x^{0.5}$ and $x^2+y^2<2$. And I proceeded like this --
<a href="https://i.stack.imgur.com/R19kg.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/R19kg.jpg" alt="enter image description here"></a></p>
<p>But I'm not get... | heropup | 118,193 | <p>Your calculation is not correct, because the upper boundary of the integral $$\int_{x=0}^{\sqrt{2}} \sqrt{x} \, dx$$ is a vertical line at $x = \sqrt{2}$, not the arc of the circle centered at the origin with radius $\sqrt{2}$. Therefore, you cannot use this method to perform the area calculation; you have subtract... |
204,780 | <p>Let $X$ and $Y$ be independent discrete random variables, each taking values $1$ and $2$ with probability $1/2$ each. How do I calculate the covariance between $max(X,Y)$ and $min(X,Y)$?</p>
| robjohn | 13,854 | <p>First,
$$
\begin{array}{}
\mathrm{P}(\max(X,Y)=2)=\frac34\qquad\text{and}\qquad\mathrm{P}(\max(X,Y)=1)=\frac14\\
\mathrm{P}(\min(X,Y)=2)=\frac14\qquad\text{and}\qquad\mathrm{P}(\min(X,Y)=1)=\frac34\\
\end{array}
$$
Therefore,
$$
\mathrm{E}(\max(X,Y))=\frac74\qquad\text{and}\qquad\mathrm{E}(\min(X,Y))=\frac54
$$
Furt... |
1,798,710 | <blockquote>
<p>Prove that any continuous bijection $f:X \rightarrow Y$ from a compact space $X$ to a Hausdorff space $Y$ is a homeomorphism</p>
</blockquote>
<p>Requirements for a homeomorphism $f:X \rightarrow Y$:</p>
<ol>
<li>$f$ is continuous</li>
<li>$f$ is bijective</li>
<li>$f^{-1}$ is continuous</li>
</ol>... | Community | -1 | <p>Since the $f$ is continuous and bijective, it is a homeomorphism if and only if it is closed. But closed subset of a compact space are compact, image of compact is compact, and compact subsets of a Hausdorff space are closed.</p>
|
732,548 | <p>The proof for this using Dependent choice, is easy. I forgot how to prove this with only AC$_\omega$. Could someone help?</p>
<p>I feel really sorry that i keep asking these basic questions.. This would be my last one. Thank you in advance</p>
| Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcomma... |
3,629,790 | <p>So I am a bit stuck on applying the limit definition for this problem. This is what I have so far: </p>
<p><span class="math-container">$\lvert x - c \rvert < \delta $</span> then <span class="math-container">$\lvert f(x) - l \rvert < \epsilon$$ $$\lvert x + 2 \rvert < \delta$</span> <span class="math-cont... | John Titor | 770,201 | <p>If you choose ε so that <span class="math-container">$0<ε<2$</span> then from <span class="math-container">$|f(x)-l|<ε$</span> <span class="math-container">$\Rightarrow$</span> <span class="math-container">$|x^2+x-5+3|<ε$</span> <span class="math-container">$\Rightarrow$</span> <span class="math-containe... |
298,121 | <p>I was watching a video on functional programming and the speaker was introducing functional programming notation for functions:</p>
<p>". . . if <strong>g(a)</strong> is a function <strong>g</strong> with variable <strong>a</strong>, we can write it also as <strong>g a</strong> . . . now, this looks like <strong>g<... | Albert | 19,331 | <p>Recall that a function $f$ is called linear if it verifies $f(x+y)=f(x)+f(y)$ and $f(\lambda x)=\lambda f(x)$ for all $x,y$ in its domain of definition and $\lambda \in \mathbb{R}$.</p>
<p>It turns out that the only linear functions on $\mathbb{R}$ are the $f(x)=ax$, where $a$ is a constant : in other words, in one... |
84,255 | <p>It is just like the linear algebra over commutative ring (maybe advanced linear algebra), that is a nature extension and can make the structure of Lie algebra more algebraic, but I find little book discussing this topic.</p>
<p>I just know Weibel’s book “Homological Algebra” discuss something, but that is just a fo... | mathphysicist | 2,149 | <p>Actually, it is possible to go even further and define Lie algebras over <em>noncommutative</em> rings, see the paper <a href="http://math.uoregon.edu/~arkadiy/noncomloopalg.pdf">Lie algebras and Lie groups over noncommutative rings</a> by Berenstein and Retakh.</p>
|
84,255 | <p>It is just like the linear algebra over commutative ring (maybe advanced linear algebra), that is a nature extension and can make the structure of Lie algebra more algebraic, but I find little book discussing this topic.</p>
<p>I just know Weibel’s book “Homological Algebra” discuss something, but that is just a fo... | MTS | 703 | <p>One interesting fact about Lie algebras over commutative rings is that the PBW theorem can fail. I highly recommend the paper <a href="http://www.ams.org/mathscinet-getitem?mr=148717" rel="nofollow">A remark on the Birkhoff-Witt theorem</a> by P.M. Cohn. The fact that PBW can fail is probably addressed already in ... |
352,501 | <p>There are several things that confuse me about this proof, so I was wondering if anybody could clarify them for me.</p>
<blockquote>
<p><strong>Lemma</strong> Let <span class="math-container">$G$</span> be a group of order <span class="math-container">$30$</span>. Then the <span class="math-container">$5$</span>-Syl... | DonAntonio | 31,254 | <p>Short Answers:</p>
<p>1.- Because $\,P_5\le N_G(P_5)\,$ and $\,|N_G(P_5)|=|P_5|\,$</p>
<p>2.- The "2nd Noether Theorem" seems to be what others (like me) call one (the second or the third, usually) of the isomorphism theorems, and the formula you want to use is precisely the order of the group </p>
<p>$$P_5P_3/P_... |
2,007,123 | <p>Suppose $R=\mathbb{Q}[x]$ is a ring and define $T$ to be $T=\{f(x) \in R : f(0) \neq 0\}$ </p>
<p>Prove that $\frac{x}{1}$ is the only irreducible element in $T^{-1}R$ (disregarding associates).</p>
<hr>
<p>My approach: assume $\frac{x}{1}$ is reducible. So there exist non-unit elements $\frac{a}{b}, \frac{c}{d}$... | Han de Bruijn | 96,057 | <blockquote>
<p>Why a bilinear interpolation with a quadrilateral?</p>
</blockquote>
<p>As pointed out in the EDIT of the question, this issue is a bit more subtle.
<BR>
At first the comment by Rahul is applied here as a heuristics.
Rotate our $[-1,+1]×[-1,+1]$ square over $45^o$, with $xy \to \frac{1}{2}(y^2-x^2)$ ... |
2,943,637 | <p>I want to prove that if <span class="math-container">$|z|=1 $</span> then <span class="math-container">$z^8-3z^2+1 \neq 0$</span>. I tried to prove the reciprocal by taking norms in <span class="math-container">$z^8-3z^2+1= 0$</span> and then solving for <span class="math-container">$ |z|$</span> but it does not wo... | Umberto P. | 67,536 | <p>Try the reverse triange inequality:
<span class="math-container">$$|z^8 - 3z^2 + 1| \ge |3z^2| - |z^8| - 1.$$</span></p>
|
2,943,637 | <p>I want to prove that if <span class="math-container">$|z|=1 $</span> then <span class="math-container">$z^8-3z^2+1 \neq 0$</span>. I tried to prove the reciprocal by taking norms in <span class="math-container">$z^8-3z^2+1= 0$</span> and then solving for <span class="math-container">$ |z|$</span> but it does not wo... | Mohammad Riazi-Kermani | 514,496 | <p><span class="math-container">$$|z|=1 \implies |z^8|=1 $$</span></p>
<p>Therefore, <span class="math-container">$|3z^2-1|=|z^8|=1$</span></p>
<p>Note that <span class="math-container">$|3z^2-1| \ge ||3z^2|-1| =2$</span></p>
<p>Which is not consistent with <span class="math-container">$|3z^2-1|=1$</span></p>
|
3,401 | <p>Consider an idealized classical particle confined to a two-dimensional surface that is frictionless. The particle's initial position on the surface is randomly selected, a nonzero velocity vector is randomly assigned to it, and the direction of the particle's movement changes only at the surface boundaries where pe... | Andy Putman | 317 | <p>I'll interpret you question to be asking about whether the particle paths are "equidistributed" in the sense of dynamical systems. There is a large literature on this sort of thing, though usually instead of "particles" the authors talk about "billiards". While I don't know the answer to your question as stated, I... |
1,535,137 | <p>I recently used the method of least norms to solve an underdetermined system of linear equations for a problem at work. This got me thinking, if I were to think about this more generally, does such a solution exits?</p>
<p>That is, if $T$ is an operator on an inner-product space $V$, and if $x,y\in V$, is it possi... | user219967 | 219,963 | <p>When $n>m$, we have $rank(AB)\le min\{rank(A),rank(B)\}\Rightarrow rank(AB)\le m\Rightarrow Det(AB)=0$</p>
|
1,535,137 | <p>I recently used the method of least norms to solve an underdetermined system of linear equations for a problem at work. This got me thinking, if I were to think about this more generally, does such a solution exits?</p>
<p>That is, if $T$ is an operator on an inner-product space $V$, and if $x,y\in V$, is it possi... | Archis Welankar | 275,884 | <p>Let A be an m×n matrix and B an n×m matrix. Write [n] for the set { 1, ..., n }, and $\tbinom{[n]}m$ for the set of m-combinations of [n] (i.e., subsets of size m; there are $\tbinom nm of them)$. The Cauchy–Binet formula then states</p>
<p>$\det(AB) = \sum_{S\in\tbinom{[n]}m} \det(A_{[m],S})\det(B_{S,[m]})$ .a go... |
2,828,023 | <p><strong>Four cards are dealt off the top of a well-shuffled deck.
Find the chance that:</strong></p>
<p><strong>You get a queen or a king.</strong> </p>
<p>The solution shows $ 1- \frac { \binom {44}{4}}{ \binom {52}{4}}$ </p>
<p>I get that in this case it's easier to use compliment rule, however I'm trying anot... | SlipEternal | 156,808 | <p>$\dfrac{\dbinom{8}{1}\dbinom{44}{3}+\dbinom{8}{2}\dbinom{44}{2}+\dbinom{8}{3}\dbinom{44}{1}+\dbinom{8}{4}\dbinom{44}{0}}{\dbinom{52}{4}} = 1-\dfrac{\dbinom{8}{0}\dbinom{44}{4}}{\dbinom{52}{4}}$</p>
|
2,493,802 | <p>If $A \subset \mathbb R$, we have the distance between a point in $\mathbb R$ and $A$ as</p>
<p>$$d(x,A)=\inf\{|x-a| \mid a \in A\}.$$</p>
<p><strong>For $\varepsilon>0$, if we define $A(\varepsilon) = \{x \in \mathbb R \mid d(x,A) < \varepsilon \}$, how do we show this set is open?</strong></p>
<hr>
<p>I ... | Arun | 399,173 | <p>Geometrically one can guess that for any $b\in A(\varepsilon)$, the ball of radius <strong>$\frac{\varepsilon -d(b,A)}{2}$</strong> and centered at $b$ will lie inside $A(\varepsilon)$. Of course you need to prove it mathematically.</p>
|
3,930,373 | <p>Hi so im working on a question about finding the <span class="math-container">$rank(A)$</span> and the <span class="math-container">$dim(Ker(A))$</span> of a 7x5 Matrix. Without being given an actual matrix to work from.</p>
<p>I have been told that that the homogeneous equation <span class="math-container">$A\vec x... | Vercassivelaunos | 803,179 | <p>We can say so with perfectly good conscience. Ignoring all the notational baggage, the function in question is
<span class="math-container">$$f:\{0\}\longrightarrow\mathbb R,~0\mapsto0.$$</span>
Here, <span class="math-container">$\{0\}$</span> is a metric space with the only possible metric on a singleton set: <spa... |
2,972,611 | <p><strong>Q</strong>: Prove the given inequalities for positive a,b,c:<br><span class="math-container">$(i) \left[\frac{bc+ca+ab}{a+b+c}\right]^{a+b+c}>\sqrt{(bc)^a.(ca)^b.(ab)^c}$</span><br><span class="math-container">$(ii) \left(\frac{a+b+c}{3} \right)^{a+b+c}<a^ab^bc^c<\left(\frac{a^2+b^2+c^2}{a+b+c}\righ... | DeepSea | 101,504 | <p><span class="math-container">$a)$</span>We have by the weighted AM-GM inequality: <span class="math-container">$\sqrt{(ab)^{\frac{c}{a+b+c}}\cdot (bc)^{\frac{a}{a+b+c}}\cdot (ca)^{\frac{b}{a+b+c}}}\le \sqrt{\dfrac{3abc}{a+b+c}}\le \dfrac{ab+bc+ca}{a+b+c}\iff(ab+bc+ca)^2 \ge 3abc(a+b+c)\iff (ab)^2+(bc)^2+(ca)^2 \ge a... |
4,494,489 | <p>These are questions I’m stuck on:</p>
<p>(i) Prove for <span class="math-container">$k\in \mathbb{R}^+:\frac{2}{\sqrt{k+1}+\sqrt{k}}\lt\frac{1}{\sqrt{k}}$</span></p>
<p>(ii) Prove <span class="math-container">$16\lt\sum_{k=1}^{80}\frac{1}{\sqrt{k}}\lt17$</span></p>
<p>I did the first one but I just equated them unti... | true blue anil | 22,388 | <p>Why do you want to raise a hornet's nest when a simple solution exists ?</p>
<p>Anyway, rethink your formula considering that if event <span class="math-container">$E$</span> has occurred when <span class="math-container">$i=0$</span> (round <span class="math-container">$1$</span>), it can occur in the next round on... |
1,124,417 | <p>I was working my way through some Propositional Logic and had the following doubt :</p>
<blockquote>
<p><strong>Why is this true :</strong></p>
<p>((p <span class="math-container">$\Rightarrow$</span> r) <span class="math-container">$\land$</span> (q <span class="math-container">$\Rightarrow$</span> r)) <span class=... | Community | -1 | <p>One can get some information with basic geometry, using the fact that this equation corresponds to a <em>conic</em> plane curve. There are only two possible answers, and they are exact (no error terms); deciding which answer is correct seems harder. </p>
<p>Let me write my thoughts in case they are helpful to you.<... |
1,124,417 | <p>I was working my way through some Propositional Logic and had the following doubt :</p>
<blockquote>
<p><strong>Why is this true :</strong></p>
<p>((p <span class="math-container">$\Rightarrow$</span> r) <span class="math-container">$\land$</span> (q <span class="math-container">$\Rightarrow$</span> r)) <span class=... | Sergio Da Silva | 130,769 | <p>I want to add a tool into the mix which might simplify the situation for future computations. This comes from the theory of Frobenius splittings. Define a near splitting of $\mathbb{F}_q[x_1,..,x_n]$ by the map called $Tr(*)$ (the trace map) defined on monomials $m$ (and extended linearly) by: </p>
<p>$$Tr(m)=\begi... |
1,203,116 | <p>I'm trying to prove that in such triangle, AB>AC, but I can't suceed. I tried to use some properties of isósceles triangle since the sides are equal, but couldn't find something nice. Also, whats the criteria needed to prove that a side in a triangle is greater than the other? Thank you so much!</p>
| user26486 | 107,671 | <p><strong>Hint:</strong></p>
<p>(the diagram is not meant to be to scale)</p>
<p><span class="math-container">$$\angle BAD\stackrel{\text{isosc}}=\angle B:=a$$</span>
<span class="math-container">$$\angle ADC\stackrel{\text{isosc}}=\angle C\stackrel{(1)}=2a$$</span><br />
<span class="math-container">$\angle C>\ang... |
3,879,009 | <p>How to prove this lemma?</p>
<blockquote>
<p><span class="math-container">$\min(x,y,z) \leq ax+by+cz \leq \max(x,y,z)$</span>, with <span class="math-container">$a+b+c = 1$</span> for any real numbers <span class="math-container">$x,y,z$</span> and <span class="math-container">$a,b,c$</span> positive.</p>
</blockqu... | Jean Marie | 305,862 | <p>Proof with geometric flavour:</p>
<p>Consider <span class="math-container">$ax+by+cz$</span> as a <strong>barycentric combination</strong> of points <span class="math-container">$x,y,z$</span> with positive coefficients <span class="math-container">$a,b,c$</span> whose sum is <span class="math-container">$1$</span>.... |
2,996,476 | <p>So I was asked to solve the following sum using Fourier Series:</p>
<p><span class="math-container">$$\sum_{n\in\Bbb N}\frac{(-1)^n}{4n^2-1}^{(1)}$$</span></p>
<p>I decided to try splitting it up into two sums (for even and odd <span class="math-container">$n$</span>) in the hopes that it would telescope and be easi... | Batominovski | 72,152 | <p>Here is my attempt to follow the instruction to use Fourier series. First, as you did, I rewrite the sum as
<span class="math-container">$$\sum_{n=1}^\infty\,\frac{(-1)^n}{4n^2-1}=\sum_{n=1}^\infty\,\frac{(-1)^n}{2}\left(\frac{1}{2n-1}-\frac{1}{2n+1}\right)=\frac{1}{2}+\sum_{n=1}^\infty\,\frac{(-1)^n}{2n-1}\,.\tag{... |
2,996,476 | <p>So I was asked to solve the following sum using Fourier Series:</p>
<p><span class="math-container">$$\sum_{n\in\Bbb N}\frac{(-1)^n}{4n^2-1}^{(1)}$$</span></p>
<p>I decided to try splitting it up into two sums (for even and odd <span class="math-container">$n$</span>) in the hopes that it would telescope and be easi... | Community | -1 | <p>Consider a <span class="math-container">$2\pi$</span>-periodic function <span class="math-container">$f:\Bbb{R}\to\Bbb{C}$</span> whose values on <span class="math-container">$(-\pi,\pi]$</span> are given by <span class="math-container">$f(x)=\operatorname{sign}x\sin\alpha x$</span>. Then, <span class="math-contain... |
4,541,392 | <p>I have an integral that looks like the following:
<span class="math-container">$$
\int_a^b \frac{1}{\left(1 + cx^2\right)^{3/2}} \mathrm{d}x
$$</span>
I have seen a method of solving it being to substitute <span class="math-container">$x = \frac{\mathrm{tan}(u)}{\sqrt{c}}$</span>; however, this seems somewhat sloppy... | Accelerator | 1,065,595 | <p>Here's one idea that requires <span class="math-container">$c > 0$</span> and <span class="math-container">$a,b \in \mathbb{R}$</span>. Let <span class="math-container">$I$</span> be the given integral. If <span class="math-container">$x = \frac{\sinh{(u)}}{\sqrt{c}}$</span>, then the integral becomes</p>
<p><spa... |
2,058,560 | <p>Can anyone please simplify this boolean expression ? My answer always reduces to a single variable i.e $x$ but my instructor reduced to three literals.
$$x'y'z+x'yz'+xy'z'+xyz$$ </p>
| gt6989b | 16,192 | <p>We have
$$
\begin{split}
x'y'z+x'yz'+xy'z'+xyz
&= x'(y'z+yz') + x(y'z' + yz) \\
&= x'(\overline{yz'} + yz') + x(\overline{yz} + yz)
\end{split}
$$
and now note that if $y=0=z$ then the expression comes out $x$ and in all other cases, it will depend of $x'$ - you cannot really simplify any further</p>
|
100,957 | <p>I have this picture of small particles in a polymer film. I want to count how many particles in the figure, so that I can have a rough estimation of the particle density. But the image quality is poor, I had a hard time to do it.</p>
<p><a href="https://i.stack.imgur.com/ez50R.jpg" rel="noreferrer"><img src="https:... | M.R. | 403 | <p>In the post referenced in the comment, the key ideas is </p>
<blockquote>
<p>In order to separate the overlapping cells, I extract a set of markers
by finding local maximas of the DistanceTransform and use these as
markers in WatershedComponents</p>
</blockquote>
<p>However exactly following that example ver... |
990,467 | <p>I am having some trouble understanding these two questions. Any help is appreciated. Scanned questions are included at the end.</p>
<p>6) We are given the function $ f(x) =\frac{1 - 2x} {2x^2 - 3x - 2} $</p>
<p>6 a) Find the equation of the vertical asymptotes. Explain how.</p>
<p>For the above question how ... | robjohn | 13,854 | <p>Letting $u=\cos(x)$, then we need to have
$$
4^u=1-u^2
$$
which implies
$$
u^2+4^u=1
$$
that is, $(u,2^u)$ crosses the unit circle.</p>
<p>$\hspace{4.5cm}$<img src="https://i.stack.imgur.com/IoJWg.png" alt="enter image description here"></p>
<p>Now that the equation is $2^{\large \cos(x)}=|\!\sin(x)|$, we also nee... |
1,235,973 | <p>Im reaching a point in programing where I need to create basic shapes which I simply cant since my math skills are very bad. After finding out that the skills required are trigonometry I read a few books. The result was that I couldnt understand any word at all.</p>
<p>What is required in order to understand basic ... | Community | -1 | <p>I am useing Khan academy to catch up on things I have hut the algebra 1 and 2 sections so I am going to learn liner and diffrential algebra before I go back to the algebra 1 and 2 sections, but when I am doing liner algebra I have noticed like the Pythagorean theorem is in the geometry section and some of the algebr... |
2,248,303 | <p>Evaluate the integral $ \int_{0}^{\infty} {\frac{\sin ^3 (x)}{x} }\:dx$ </p>
| Jack D'Aurizio | 44,121 | <p>We have
$$ \int_{0}^{+\infty}\frac{\sin(mx)}{x}\,dx=\frac{\pi}{2} $$
for any $m\in\mathbb{N}^+$, hence it is enough to expand $\sin^3(x)$ as a Fourier sine series. Actually
$$ \sin^3(x) = \frac{3}{4}\sin(x)-\frac{1}{4}\sin(3x) $$
hence
$$ \int_{0}^{+\infty}\frac{\sin^3(x)}{x}\,dx = \left(\frac{3}{4}-\frac{1}{4}\righ... |
2,692,124 | <blockquote>
<p>A group <span class="math-container">$G$</span> is finite iff all its elements have the same finite order?</p>
</blockquote>
<p>One direction (when <span class="math-container">$G$</span> is finite) is trivial following from Lagrange's theorem. But if a each element of <span class="math-container">$G$</... | Exodd | 161,426 | <p>Try
$$\mathbb Z /2\mathbb Z \times \mathbb Z /2\mathbb Z \times\mathbb Z /2\mathbb Z \times\dots.$$
Every element has order 2.</p>
|
1,808,183 | <p>Consider the Cauchy problem:
$$\left\{\begin{array}{lll}
x^2\partial_x u+y^2\partial_yu=u^2\\
u(x,2x)=1
\end{array}\right.$$
It is easy to show that the characteristic equations are given by:
$$\frac{dx}{x^2}=\frac{dy}{y^2}=\frac{dz}{px^2+qy^2}=\frac{dp}{2pu-2xp}=\frac{dq}{2qu-2yq}=dt$$
By the Lagrange-Charpit's met... | izmirlig | 149,896 | <p>The explanation contains some errors so I will start at the beginning. <span class="math-container">$[X]$</span> is the distribution of the random variable, <span class="math-container">$X$</span>. <span class="math-container">$[X | Y]$</span> is the conditional distribution of <span class="math-container">$X$</span... |
2,070,353 | <p>Let there be two arbitrary functions $f$ and $g$, and let $g \circ f$ be their composition.</p>
<ol>
<li><p>Suppose $g \circ f$ is one-one. Then prove that this implies $f$ is also one-one.</p></li>
<li><p>Suppose $g \circ f$ is onto. Then prove that this implies $g$ is also onto.</p></li>
</ol>
<p>I tried to do b... | Fawad | 369,983 | <p>$$a_n= 2n+1$$
$$a_1 = 3$$
$$a_n=7$$ $$2n+1=7$$ $$n=3$$ $$a_3=7$$</p>
<p>So series will become 3,5,7,9,…</p>
<p>So before 7 there are 2 terms which are not included in your series.
So there are total $n-2$ terms</p>
|
2,657,112 | <p>I am currently reading the paper "PRIMES is in p" and have come across some notation that I don't quite understand in this following sentence</p>
<blockquote>
<p>Consider a prime $q$ that is a factor of $n$ and let $q^k || n$. Then...</p>
</blockquote>
<p>What does the notation $q^k || n$ mean here?</p>
<p>The ... | Community | -1 | <p>It means that $k$ is the higher power that divides $n$. In other words:
$$q^k\|n\iff q^k|n\wedge q^{k+1}\nmid n$$</p>
|
1,755,107 | <p>Let $(X,d)$ be a metric space and let $A \subseteq X$. We define the distance from a point $x \in X$ to $A$ by $d(x,A)= \inf \{ d(x,a) : a \in A \} $.</p>
<p>What will be the value of $d(x, \emptyset )$? I am confused between $+ \infty $ and $- \infty$. Also, is it possible to find $ \min \{ d(x,a) : a \in \emptyse... | bof | 111,012 | <p>$d(x,\emptyset)=\inf\emptyset=\infty.$ That makes sense because in general, if $A\subseteq B,$ then $d(x,A)\ge d(x,B);$ so $d(x,\emptyset)\ge d(x,B)$ for all $B.$</p>
<p>If the infimum of a set $S$ is <em>attained</em>, that is, if $\inf S\in S,$ then $\min S=\inf S;$ otherwise, $\min S$ does not exist. Thus $\min\... |
1,599,502 | <p>Solve for $x$ in exact value:</p>
<p>$\\3^{2x}-3^{x+2}+8=0$</p>
<p>I have tried substituting $3^x$ $=a$ but I didn't get anywhere.</p>
<p>$\\a^2-a^{1+\frac{2}{x}}+8=0$</p>
| Nate | 91,364 | <p>You need to be a bit more specific about what you mean by an algebra.</p>
<p>If you are in the setting of finite dimensional associative algebras then the Artin-Wedderburn theorem tells us that any such simple algebra is a matrix ring $M_n(D)$ of a finite dimensional division algebra $D$ over the ground field $k$. ... |
1,262,444 | <p>I study hard throughout the year and I am able to solve most problems in the text assigned to us and I am frequently the only one who can solve the hardest problems in the assignments or the problem sets. However, I can rarely do exceptionally well at math tests in my University and I frequently run out of time whil... | alkabary | 96,332 | <p>You know, Exams are like tournament or competitions. Some people are great at practice but when they enter a championship, and feel under pressure, they under perform. I am telling you this because I am student athlete , and I did track and field for 4 years while doing my double major in computer science and mathem... |
851,112 | <p>My goal is to obtain a reasonable approximation of the Gini index of a company (UBS).
I need to obtain an estimate of the salaries distribution from publicly available data:</p>
<ul>
<li>Nuber of employees=60205 </li>
<li>total compensation paid=15.182E9 CHF</li>
<li>min salary=50000 CHF</li>
<li>median salary = 10... | Matt L. | 70,664 | <p>Another nice way of seeing this equality is noting that</p>
<p>$$\sin\theta=\frac{1}{2i}(p-p^*)\quad\text{and}\quad\sin\phi=\frac{1}{2i}(q-q^*)$$</p>
<p>So the right-hand side is</p>
<p>$$\frac{p-p^*+q-q^*}{p-p^*-q+q^*}=\frac{(p+q)-(p+q)^*}{(p-q)-(p-q)^*}$$</p>
<p>Now note that $p^*=1/p$ and $q^*=1/q$, so we get... |
3,525,356 | <p><a href="https://i.stack.imgur.com/8OitL.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/8OitL.png" alt="enter image description here"></a></p>
<p>Can someone help me on this? Any help will be greatly appreciated. </p>
| ancient mathematician | 414,424 | <p>If <span class="math-container">$M$</span> has order <span class="math-container">$p$</span> then its minimal polynomial divides <span class="math-container">$X^p-1=(X-1)^p$</span>; and so is <span class="math-container">$(X-1)^2$</span>. So by the Jordan Canonical Form Theorem all elements of order <span class="mat... |
3,525,356 | <p><a href="https://i.stack.imgur.com/8OitL.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/8OitL.png" alt="enter image description here"></a></p>
<p>Can someone help me on this? Any help will be greatly appreciated. </p>
| Nourr Mga | 707,002 | <p>Since <span class="math-container">$GL_{2}(\Bbb Z_{p})$</span> contains <span class="math-container">$p+1$</span> Sylow p-subgroups and each of the them contains <span class="math-container">$p-1$</span> elements of order <span class="math-container">$p$</span>, then the total number of elements of order <span class... |
2,137,269 | <p>I just sat in on a lecture on exponential generating functions in combinatorics (I have no formal education in combinatorics myself). It was quite interesting, but I'm afraid I don't actually understand what the generating function is/does. I've tried doing some minimal research online, but everything I've seen seem... | Asinomás | 33,907 | <p>The generating function is a power series that is assigned to a sequence. If you have a sequence $\{a_n|n\in \mathbb N\}$ you can assign to this sequence a power series: $\sum\limits_{n=0}^\infty a_nx^n$.</p>
<p>Manipulating the power series using algebra (power series form a ring) can help to find a closed form fo... |
2,137,269 | <p>I just sat in on a lecture on exponential generating functions in combinatorics (I have no formal education in combinatorics myself). It was quite interesting, but I'm afraid I don't actually understand what the generating function is/does. I've tried doing some minimal research online, but everything I've seen seem... | Karl | 203,893 | <p>Suppose you have the sequence $2, 5, 8, 11,...$ </p>
<p>One idea might be to have the terms in the sequence as the coefficients of some polynomial. This polynomial is called a generating function of the sequence.</p>
<p>For example:</p>
<p>$$ g(x)=2x^1+5x^2+8x^3+11x^4+...$$</p>
<p>All seems fine but how do you ... |
785,603 | <p>What do Sylow 2-subgroups of finite simple groups look like?</p>
<p>It'd be nice to have explanations of the Sylow 2-subgroups of finite simple groups. There are many aspects to the question, so I envision many answers. I tried to give an answer of "broad strokes with references." It would be reasonable to answer j... | Jack Schmidt | 583 | <p>Nonabelian finite simple groups come in a few types:</p>
<ul>
<li>Alternating groups</li>
<li>Classical groups in odd characteristic</li>
<li>Exceptional groups in odd characteristic</li>
<li>Groups in even characteristic (classical or exceptional)</li>
<li>Sporadic groups</li>
</ul>
<p>In cases 1,2,3 the Sylow 2-... |
785,603 | <p>What do Sylow 2-subgroups of finite simple groups look like?</p>
<p>It'd be nice to have explanations of the Sylow 2-subgroups of finite simple groups. There are many aspects to the question, so I envision many answers. I tried to give an answer of "broad strokes with references." It would be reasonable to answer j... | Jack Schmidt | 583 | <h2>Normalizers and top fusion</h2>
<p>Kondratʹev (2005) contains a very clear description of the normalizers of Sylow 2-subgroups $S$ of finite simple groups. Typically there is not much going on: $N_G(S) = S$ unless:</p>
<ul>
<li>$G$ is a group of Lie type in even characteristic (then $N_G(S)$ is a Borel subgroup a... |
494,141 | <p>I'm slightly confused about representations of Simple functions. I take it that simple functions are of the form $\phi(x) = \sum_{k=1}^N a_k \chi_{A_k}(x)$ where $\chi$ is the indicator function and that $A_k$'s are measurable sets.</p>
<p>Now on building the theory of integration using simple functions, it appears... | QED | 91,884 | <p>Assume $f:(\Omega,\mathcal{A})\rightarrow\mathbb{R}$ is a simple function. Then $f$ is a measurable function that takes only finitely many real values. Let $\{c_1,\cdots,c_n\}$ be the set of possible values of $f$. Define $A_i=\{\omega\in\Omega:f(\omega)=c_i\}=f^{-1}\left(\{c_i\}\right)$. Then $A_i$'s are measurable... |
3,560,378 | <p>Let <span class="math-container">$f: \mathbb{R} \to \mathbb{R}$</span> be a convex function. Let <span class="math-container">$x,y,z \in \mathbb{R}$</span> and <span class="math-container">$a,b,c \geq 0$</span> such that <span class="math-container">$a+b+c=1$</span>. We want to show that <span class="math-container"... | Donald Splutterwit | 404,247 | <p><span class="math-container">\begin{eqnarray*}
f(ax+by+cz) &=& f \left( ax +(b+c) \frac{by+cz}{b+c} \right) \\
& \leq & a f(x) +(b+c) f \left( \frac{by+cz}{b+c} \right) \\
\end{eqnarray*}</span>
Now
<span class="math-container">\begin{eqnarray*}
f \left( \frac{by+cz}{b+c} \right) & \leq & \... |
3,203,460 | <p>The whole question looks like-<br></p>
<blockquote>
<p><span class="math-container">$f$</span> is a holomorphic function on the interior of the open unit disc
(say <span class="math-container">$D$</span>) around <span class="math-container">$0$</span> and <span class="math-container">$f(z)\ne0\ \forall z\in D$<... | Martin R | 42,969 | <p>You have to show that the given conditions imply that <span class="math-container">$|f(z)| \le 1$</span> in the unit disk <span class="math-container">$\Bbb D$</span>. Then the same reasoning can be applied to <span class="math-container">$1/f$</span> (since <span class="math-container">$f$</span> has no zeros). It ... |
3,203,460 | <p>The whole question looks like-<br></p>
<blockquote>
<p><span class="math-container">$f$</span> is a holomorphic function on the interior of the open unit disc
(say <span class="math-container">$D$</span>) around <span class="math-container">$0$</span> and <span class="math-container">$f(z)\ne0\ \forall z\in D$<... | Subham Jaiswal | 238,932 | <p>INCORRECT (1 does not imply 2, since in Cauchy's inequality <span class="math-container">$R$</span> gets smaller and smaller as we go close to the boundary. We do get uniform continuity on any sub disc of radius <span class="math-container">$r < 1$</span> for any <span class="math-container">$r$</span>).
I unable... |
406,252 | <p>How to define the amount of additions. E.g. $1+2+3+4+5+6+7+8+9$</p>
<p>Are there $9$ additions, because of the nine numbers that are added together.
Or can you also say that there are $8$ additions, because there are only $8$ '$+$' signs.</p>
| iostream007 | 76,954 | <p>Addition is a binary operator that uses two operands to execute.so if there are $n$ number of elements then number of addition will be $$\;n-1$$.In other answers people uses "addition of $0$",because $0$ is addition identity,it will give $n$ addition but it is only in case of certain operator like $+,-$ but there a... |
432,291 | <p>This question is related to <a href="https://mathoverflow.net/questions/421450/homotopy-type-theory-how-to-disprove-that-0-operatornamesucc0-in-the-ty">Homotopy type theory : how to disprove that <span class="math-container">$0=\mathrm{succ}(0)$</span>
in the type <span class="math-container">$\mathbb N$</span></a>.... | Mike Shulman | 49 | <p>Daniel's answer is correct that the judgmental distinctness of <span class="math-container">$0$</span> and <span class="math-container">$\mathsf{succ}(m)$</span> is not what justifies a definition by pattern-matching. However, it is still a meaningful question of how to prove that <span class="math-container">$0$</... |
2,179,012 | <p>Let $V_1$ be a Banach space and let $A_1 \subset V_1$ be a compact subset of $V_1$</p>
<p>Let $V_2$ be a Banach space and let $A_2 \subset V_2$ be a compact subset of $V_2$</p>
<p>Let $f: A_1 \to A_2 $ be a homeomorphism</p>
<p>Let $x_1,y_1 \in A_1$ such that $$\left\| x_1-y_1 \right\|=\sup_{v,u \in A_1} \left\| ... | Mahmoud Hassan | 282,900 | <p>Here are the three books I study from:</p>
<ul>
<li>S. Ponnusamy, Herb Silverman - Complex Variables with Applications.</li>
<li>Ravi P. Agarwal - An Introduction to Complex Analysis.</li>
<li>Jerrold E. Marsden, Michael J. Hoffman - Basic Complex Analysis</li>
</ul>
|
567,428 | <p>Let $A \ne V$ be a subspace of $V$ and $B$ a linearly independent subset of $V$. Prove that $B$ can be completed to a basis of $V$ with vectors from $V \setminus A$.</p>
<p>OK, I started with: </p>
<p>$K=\operatorname{lin}(v_1,v_2,v_3,\dotsc)$</p>
<p>and if any of $v_1,v_2,\dotsc$ belongs to $A$ we can replace th... | rnrstopstraffic | 218,011 | <p>Assume $$Dim(V)=m,$$ $$Dim(A)=n<m,$$ $$\text{and } B=\{v_1,...,v_n\}.$$ </p>
<p>Then $$Span(B)=A\neq V,$$</p>
<p>so $\exists v_{n+1}\in V\backslash A$, which means that there is no $n$-tuple $(c_1,...,c_n)$ with $v_{n+1}=c_1v_1+...+c_nv_n$ , i.e. $B\cup \{v_{n+1}\}$ is linearly independent. If $n+1=m$, we are d... |
6,056 | <p>Are there any good places for me to sell off my mathematics books online especially Springer and Dover books?</p>
<p>(I thought perhaps this was off topic, but then I thought everybody in math probably has the problem of having bought pricey books that they would prefer to get some money back for, at one point or a... | Jeremy Batterson | 2,302 | <p>Amazon will buy back a lot of textbooks at pretty fair rates. Check <a href="http://www.amazon.com/Textbooks-Trade-In/b/ref=bhp_9p2_A_txbq410_05?ie=UTF8&node=2205237011&pf_rd_m=ATVPDKIKX0DER&pf_rd_s=center-6&pf_rd_r=0RH8PF985VSEMWFKG1SV&pf_rd_t=101&pf_rd_p=1275995182&pf_rd_i=283155" rel="... |
310,238 | <p>The <a href="https://en.wikipedia.org/wiki/Weyl_character_formula" rel="nofollow noreferrer">Weyl character formula</a> and <a href="https://en.wikipedia.org/wiki/Weyl_character_formula#Weyl_denominator_formula" rel="nofollow noreferrer">the denominator identity</a> play important roles in the representation theory ... | Konstantinos Kanakoglou | 85,967 | <p>I agree with the suggestion in the comments for searching the front of the math arXiv (as an entry point), because this is a quite broad and active topic (and i am not sure it can be fully covered within a couple of references). </p>
<p>However, i think it might be particularly useful to mention: </p>
<ul>
<li>Th... |
1,564,750 | <p>Imagine that your country's postal system only issues 2 cent and 5 cent stamps. Prove that it possible to pay for postage using only these stamps for any amount n cents, where n is at least 4.</p>
<hr>
<p>My attempt (using Strong induction, I know we can use induction but since they say you can can apply strong in... | Mario Carneiro | 50,776 | <p>Here's another induction-free proof, using the fact that integers are even or odd:</p>
<p>If $n$ is even, then $n=2k$ and we are done ($k$ two-cent stamps).</p>
<p>If $n$ is odd, so $n=2k+1$, then since $n\ge4$ we have $k\ge\frac32$ whence $k\ge2$ because $k$ is an integer. Therefore $m=k-2\ge0$, and $n=2m+5$, so ... |
2,925,160 | <p>We know that <span class="math-container">$\mathbb{R}$</span> is a linear space over <span class="math-container">$\mathbb{Q}$</span>, denoted by <span class="math-container">$\mathbb{R_{\mathbb{Q}}}$</span>,where the vector addition is the real numbers and scalar multiplication is the number in <span class="math-co... | Lubin | 17,760 | <p>Here’s a way of looking at your problem that makes use of ramification theory. It may be too advanced for your taste, but at least it does provide an answer to your question.</p>
<p>You’re looking for many real quantities, preferably algebraic, that will be linearly independent over the real field <span class="math... |
217,580 | <p>Consider a compact manifold $M$ with boundary and corner. As an example, we could have the cube $\{(x_1, x_2,..x_n) \in \mathbb{R}^n : x_i \in [0,1]\}$. We could very well define the Laplacian $\Delta$ on such an $M$ with Dirichlet or Neumann boundary conditions. In such situations, are the eigenfunctions of the Lap... | Michael Renardy | 12,120 | <p>There is an extensive literature on elliptic problems on domains with corners. Grisvard's book (Elliptic Problems in Nonsmooth Domains) is a good place to start. </p>
|
3,507,927 | <p>Is there any simplification by the way of standard properties of Laplace Transform for this? <span class="math-container">$$L_{s} [t f(t) f'(t)]$$</span></p>
<p>where <span class="math-container">$f'(t)$</span> is <span class="math-container">$\frac{d}{dt} f(t)$</span></p>
<p>Alternately, is there a simplification... | user577215664 | 475,762 | <p>Since:
<span class="math-container">$$\mathcal {L}\{tg(t)\}=-\frac {d}{ds}\mathcal {L}\{g(t)\}$$</span>
We have:
<span class="math-container">$$\mathcal {L}[tf(t)f′(t)]=-\frac {d}{ds}\mathcal {L}\{f(t)f′(t)\}$$</span>
<span class="math-container">$$\mathcal {L}[tf(t)f′(t)]=-\frac 1 2\frac {d}{ds}\mathcal {L}\{(f^2(t... |
3,490,024 | <p>This is part of a physics problem I was doing yesterday. I am supposed to find the maximum value of
<span class="math-container">$$\mu^{-1}\cos\theta+\sin\theta$$</span>
This is is supposed to produce the result of <span class="math-container">$\sqrt{1+\mu^{-2}}$</span>. However I am not getting that. By taking the... | Abhishek Vangipuram | 630,593 | <p>It is known that for <span class="math-container">$A\cos(\theta) + B\sin(\theta)$</span>, we can represent it as 1 trigonometric function: <span class="math-container">$\sqrt{A^2+B^2}\cos(\theta + \phi)$</span> or <span class="math-container">$\sqrt{A^2+B^2}\sin(\theta+\omega)$</span>. Clearly it can be <span class=... |
1,836,236 | <p>For which $a$ and $b$ is this matrix diagonalizable?</p>
<p>$$A=\begin{pmatrix} a & 0 & b \\ 0 & b & 0 \\ b & 0 & a \end{pmatrix}$$</p>
<p>How to get those $a$ and $b$? I calculated eigenvalues and eigenvectors, but don't know what to do next?</p>
| DonAntonio | 31,254 | <p>That is a symmetric matrix: it is <strong>always</strong> diagonalizable over a field with characteristic$\;\neq2\;$. Not only that: it is <em>orthogonally</em> diagonalizable.</p>
|
3,676,003 | <p>I would like to find the parametric equation for a curve starting at radius of curvature 10 at angle 0 degrees and ending at radius of curvature 100 at 90 degrees. The equation for change in radius of curvature along the path will be specified. </p>
<p>One can imagine this as a 90 degree arc of a circle, except the... | Narasimham | 95,860 | <p>Natural/intrinsic differential equations need to be used. The Whewell's equation binding tangent rotation <span class="math-container">$\phi$</span> and arc length <span class="math-container">$s$</span> is more suitable compared to Cesàro equation <span class="math-container">$(\kappa, s)$</span> and it can be us... |
286,312 | <p>I'm starting to understand how induction works (with the whole $k \to k+1$ thing), but I'm not exactly sure how summations play a role. I'm a bit confused by this question specifically:</p>
<p>$$
\sum_{i=1}^n 3i-2 = \frac{n(3n-1)}{2}
$$</p>
<p>Any hints would be greatly appreciate</p>
| Sigur | 31,682 | <p>For $n=1$ you have $\sum_{i=1}^1(3i-2)=3-2=1=\frac{1(3\cdot 1-1)}{2}$.</p>
<p>Suppose that $$\sum_{i=1}^n(3i-2)=\frac{n(3n-1)}{2}$$ and prove that $$\sum_{i=1}^{n+1}(3i-2)=\frac{(n+1)(3(n+1)-1)}{2}.$$</p>
|
971,997 | <p>How do we find
$$\Re\left[\int_0^{\large\frac{\pi}{2}} e^{\Large e^{i\theta}}d\theta\right]$$</p>
<p>In the shortest and easiest possible manner?</p>
<p>I cannot think of anything good.</p>
| Leucippus | 148,155 | <p>Using
\begin{align}
e^{x} = \sum_{r=0}^{\infty} \frac{x^{r}}{r!}
\end{align}
then
\begin{align}
I &= \int_{0}^{\pi/2} e^{e^{i \theta}} \, d\theta = \sum_{r=0}^{\infty} \frac{1}{r!} \, \int_{0}^{\pi/2} e^{i \theta} \, d\theta \\
&= - i \,\sum_{r=0}^{\infty} \frac{e^{\pi i r/2} -1}{r \cdot r!} \\
&= -i \... |
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