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 |
|---|---|---|---|---|
2,075,122 | <p>While going through many proofs in graph theory, I noticed that for proof of statement like</p>
<blockquote>
<p>a if and only if b </p>
</blockquote>
<p>we need to prove </p>
<p>$a \rightarrow b$ and $b \rightarrow a$.</p>
<p>We know </p>
<p>$$ a\rightarrow b ={a}'+b$$</p>
<p>and</p>
<p>$$b\rightarrow ... | MPW | 113,214 | <p>You are correct. Proving the contrapositive is equivalent.</p>
|
2,075,122 | <p>While going through many proofs in graph theory, I noticed that for proof of statement like</p>
<blockquote>
<p>a if and only if b </p>
</blockquote>
<p>we need to prove </p>
<p>$a \rightarrow b$ and $b \rightarrow a$.</p>
<p>We know </p>
<p>$$ a\rightarrow b ={a}'+b$$</p>
<p>and</p>
<p>$$b\rightarrow ... | Joffan | 206,402 | <p>Another way to see it:</p>
<p>If you prove the following two statements:<br>
$\begin{align}
a &\implies b \\
\text{not }a &\implies \text{not }b
\end{align}$ </p>
<p>... you have shown that $a$ and $b$ are either both true or both false; the two statements are in lockstep. $a \mathord\iff b$.</p>
|
2,987,877 | <p>Find the inverse Laplace of <span class="math-container">$ \ \frac{2}{(s-1)^3(s-2)^2}$</span>. </p>
<p><strong>Answer:</strong></p>
<p>To do this we have to make partial fractions as follows:</p>
<p><span class="math-container">$ \frac{2}{(s-1)^3(s-2)^2}=\frac{A}{S-1}+\frac{B}{(s-1)^2}+\frac{C}{(s-1)}+\frac{J}{(... | Doug M | 317,162 | <p>You have one too many <span class="math-container">$\frac {1}{(s-1)}$</span> terms.</p>
<p><span class="math-container">$\frac {2}{(s-1)^3(s-2)^2} = \frac {A}{(s-1)} + \frac {B}{(s-1)^2} + \frac {C}{(s-1)^3} + \frac {D}{(s-2)} + \frac {E}{(s-2)^2}$</span></p>
<p>Traditionally, you would then multiply through to cl... |
1,882,401 | <p>Let $T$ be a linear operator on a finite dimensional vector space $V$ with dimension $n$. Then, is it necessary that $T$ has $n$ eigen values or could it be less than $n$?</p>
| michalOut | 356,317 | <p>If $T:\, V \rightarrow V$ then it admits the matrix representation with respect to some base of $V$. The eigenvalues are then computable (and defined) as the roots of the monic polynomial $p(\lambda )$
$$p(\lambda)=\text{det}(A-\lambda I),$$
where $A$ is the matrix representation of $T$ in the given base. As the <a ... |
1,882,401 | <p>Let $T$ be a linear operator on a finite dimensional vector space $V$ with dimension $n$. Then, is it necessary that $T$ has $n$ eigen values or could it be less than $n$?</p>
| Doug M | 317,162 | <p>It could be less than $n.$</p>
<p>Consider $\begin{bmatrix}1&1\\0&1\end{bmatrix}$</p>
<p>1 is an eigenvalue, and $\begin{bmatrix} 1\\0 \end{bmatrix}$ is an eigenvector, but there is not a second independent eigenvector. </p>
|
628,409 | <p>$$f(x)=\max\{x,0\}$$</p>
<p>I want to check whether this function is continuous in its domain $\mathbb{R}$ or not, but unfortunately I have no idea how to start.</p>
| Brian | 114,928 | <p>An alternative strategy to Edward's answer is to use a closed form expression for the $\max$ of two real numbers, $\max\left\{a,b\right\}=\frac{a+b+|a-b|}{2}$, show that $|\cdot|$ is continuous, and use the basic theorems of continuous functions. In fact, this can be generalized to show that $f=\max\left\{g,h\right\... |
312,249 | <p>I am attempting to solve an equation ${{n-2} \choose {2}} + {{n-3} \choose {2}} + {{n-4} \choose {2}} = 136$. With the formula for a combination being $\frac{n!}{r!(n - r)!}$, I simplified the given equation to:</p>
<p>$(n-2)! + (n-3)! + (n-4)! = 272n - 1088$ </p>
<p>However, I am not sure how I would solve for $\... | vonbrand | 43,946 | <p>Bad simplification, that one.
$$
\begin{align*}
\binom{n - 2}{2} + \binom{n - 3}{2} + \binom{n - 4}{2}
&= \frac{(n - 2)(n -3)}{2} + \frac{(n - 3) (n - 4)}{2} + \frac{(n - 4)(n - 5)}{2} \\
&= \frac{3 n^2 - 21 n + 38}{2}
\end{align*}
$$
The binomial coefficient $\binom{n}{k} = \binom{n}{n - k} = \frac{n (n... |
3,854,624 | <p>I'm trying to find <span class="math-container">$\int 1/\sin(x)\ dx$</span>, but I can't figure out how to do it. Also, what would be the value of
<span class="math-container">$$\int\limits_0^{2\pi} \frac{1}{\sin x} dx\quad ?$$</span>
Based on symmetry, I would try to say it is zero, but I've been told to be carefu... | Ken Hung | 626,360 | <p>By just checking the left end point only we already see the integral diverges. This is because on the interval <span class="math-container">$[0,2\pi]$</span>,
<span class="math-container">$$ |\! \sin x| \leq x \implies \left|\frac{1}{\sin x} \right| \geq \frac1x ,$$</span>
and the above leads us to
<span class="math... |
3,854,624 | <p>I'm trying to find <span class="math-container">$\int 1/\sin(x)\ dx$</span>, but I can't figure out how to do it. Also, what would be the value of
<span class="math-container">$$\int\limits_0^{2\pi} \frac{1}{\sin x} dx\quad ?$$</span>
Based on symmetry, I would try to say it is zero, but I've been told to be carefu... | Mr Pie | 477,343 | <p>Let <span class="math-container">$\;t=\tan \cfrac x2\;$</span> for <span class="math-container">$\;x\in(-\pi,\pi)\;$</span> then <span class="math-container">$\;\sin x = \cfrac{2t}{1+t^2}\;$</span> and <span class="math-container">$\;\mathrm dx=\cfrac 2{1+t^2}\,\mathrm dt$</span> <span class="math-container">\begin{... |
307,144 | <p>Are there irrational numbers for which we know that computing its nth digit would take (at least) linear/polynomial/exponential/superexponential time (wrt to length of n and with "big enough" n)?</p>
| Community | -1 | <blockquote>
<p>It seems to me that I could get from the first equation to the second equation by simply multiplying both sides by $A^T$, so in my conception this wouldn't change the solutions.</p>
</blockquote>
<p>It's not what you're missing, but what you're adding. You can indeed get from the first equation to th... |
1,534,693 | <p>I am a little stuck on coming up with geometrical explanation for why the following equalities are true. I tried arguing the $\cos(\theta)$ is the projection to the x-axis of a vector $r$ inside a unit circle, so as it goes around by $2 \pi$, the projections on both the positive and negative part of the x-axis cance... | Archis Welankar | 275,884 | <p>Hint you can use the formula $cosA+cosB+cosC=cosAcosBcosC[1-tanAtanB-tanBtanC-tanAtanC]$ and $sinA+sinB+sinC=cosAcosBcosC[tanA+tanB+tanC-tanAtanBtanC$ Note I have given it as a trigonometric proof for your formula. Its tidious work but you will surely get a trigonometric proof for it.Hope it helps you.</p>
|
3,282,719 | <p>In Javascript, the largest integer that can be represented exactly is <code>Number.MAX_SAFE_INTEGER</code>, with a value of <span class="math-container">$ 2^{53} - 1$</span>. What is the largest prime value that fits under this value threshold? I cannot find a suitable reference for this value on the internet.</p>
| Eric Towers | 123,905 | <p><span class="math-container">$9\,007\,199\,254\,740\,881$</span> is the largest prime less than or equal to <span class="math-container">$2^{53}$</span>.</p>
<p>This number is mentioned in this context <a href="https://www.nayuki.io/page/calculate-prime-factorization-javascript" rel="nofollow noreferrer">here</a> a... |
3,282,719 | <p>In Javascript, the largest integer that can be represented exactly is <code>Number.MAX_SAFE_INTEGER</code>, with a value of <span class="math-container">$ 2^{53} - 1$</span>. What is the largest prime value that fits under this value threshold? I cannot find a suitable reference for this value on the internet.</p>
| Henry | 6,460 | <p><span class="math-container">$2^{53}-111 = 9007199254740881$</span> is the largest prime below <span class="math-container">$2^{53}$</span></p>
<p>The next prime is <span class="math-container">$2^{53}+5 =9007199254740997$</span> but JavaScript will not represent this correctly</p>
|
4,768 | <p>How we can show every permutation is either even or odd,but not both......I can't arrive at a proof for this ..... Can anybody give me the proof...</p>
<p>Thanks in advance...</p>
| Gadi A | 1,818 | <p>One way is to define the sign of a permutation $\sigma$ using the polynomial $\Delta = \Pi (x_i-x_j)$ with $1\le i < j \le n$. </p>
<p>It is easy to see that $\sigma(\Delta) = \Pi (x_{\sigma(i)}-x_{\sigma(j)})$ satisfies $\sigma(\Delta)=\pm\Delta$. Now define the sign by $sign(\sigma)=\frac{\Delta}{\sigma(\Delta... |
4,768 | <p>How we can show every permutation is either even or odd,but not both......I can't arrive at a proof for this ..... Can anybody give me the proof...</p>
<p>Thanks in advance...</p>
| Douglas S. Stones | 139 | <p>There is a proof given here: <a href="http://www.jstor.org/stable/2316272" rel="noreferrer">An Historical Note on the Parity of Permutations</a>, T. L. Bartlow, American Mathematical Monthly Vol. 79, No. 7 (Aug. - Sep., 1972), pp. 766-769.</p>
<p>Here's an outline of Bartlow's proof (it matches the proof given in ... |
4,768 | <p>How we can show every permutation is either even or odd,but not both......I can't arrive at a proof for this ..... Can anybody give me the proof...</p>
<p>Thanks in advance...</p>
| ibnAbu | 334,224 | <p>To define the sign of a permutation <span class="math-container">$\sigma$</span> using the polynomial :
<span class="math-container">$$\begin{gathered} A = \left( {{x_1} β {x_2}} \right)\left( {{x_1} β {x_3}} \right)\left( {{x_1} β {x_4}} \right) \cdots \left( {{x_1} β {x_n}} \right) \\ \,\,\,\,\,\,\,\,\,\,\left( {... |
803,792 | <p>I have just taken calculus quiz but I could not find $\displaystyle \int_2^\infty\frac{\log^3(x-1)}{x^2}dx$? Any help would be appreciated. Thanks in advance.</p>
<p>EDIT:</p>
<p>Forgot to mention, my tutor gave us hints about this question.</p>
<ol>
<li>Use Taylor series</li>
<li>$\displaystyle \zeta(3)=\sum_{n... | gar | 138,850 | <p>By substitutions, the following integrals are equivalent:
\begin{align*}
\int_{2}^{\infty} \, \frac{\log^3(x-1)}{x^2}\, dx &= \int_{1}^{\infty} \, \frac{\log^3(x)}{(1+x)^2} \, dx\\
&= -\int_{0}^{1} \, \frac{\log^3(x)}{(1+x)^2}\, dx \tag 1
\end{align*}</p>
<p>$(1)$ can be written as a sum, consider:</p>
<... |
43,688 | <p>The nuclear norm of a matrix is defined as the sum of its singular values, as given by the singular value decomposition (SVD) of the matrix itself. It is of central importance in Signal Processing and Statistics, where it is used for matrix completion and dimensionality reduction. </p>
<p>A question I have is wheth... | mac | 9,390 | <p>A tentative answer: the nuclear norm of $A$ is the trace of $\sqrt{A^*A}$ where $A^*$ is the conjugate-transpose of $A$, and $\sqrt{\cdot}$ is the matrix square root. So provided you can calculate matrix square roots faster than singular value decompositions, this might be useful.</p>
|
7,940 | <p>I have multiple plots of permittivity against frequency, but they're hard to read because the frequency labels go from 4x10^14 to 1x10^15 in Hz. I want to label the frequency with THz instead of Hz, so each value would just get divided by 10^12. How do I do this?</p>
| Andreas Lauschke | 1,598 | <p>In <em>Mathematica</em> everything is an expression. That's the overarching concept that is fully pervasive throughout the entire Mathematica system. If you look at your plot expression with <code>FullForm</code>, you should be able to find your axes labels and tick marks and data. Then extract those expressions wit... |
7,940 | <p>I have multiple plots of permittivity against frequency, but they're hard to read because the frequency labels go from 4x10^14 to 1x10^15 in Hz. I want to label the frequency with THz instead of Hz, so each value would just get divided by 10^12. How do I do this?</p>
| kglr | 125 | <p>Get the tick specs using <code>AbsoluteOptions</code> and modify them as you like as follows:</p>
<pre><code>lp = ListPlot[RandomReal[{0, 1}, {10}], Joined -> True,
DataRange -> {10^10, 10^14}, Frame -> True];
fts = FrameTicks /. AbsoluteOptions[lp, FrameTicks];
fts[[1]] = ReplaceAll[#, {tick_, lbl_, {po... |
7,940 | <p>I have multiple plots of permittivity against frequency, but they're hard to read because the frequency labels go from 4x10^14 to 1x10^15 in Hz. I want to label the frequency with THz instead of Hz, so each value would just get divided by 10^12. How do I do this?</p>
| Michael E2 | 4,999 | <p>In V10, we can use the internal function <a href="https://mathematica.stackexchange.com/search?q=ScaledTicks"><code>Charting`ScaledTicks</code></a>.
Its use has the form</p>
<pre><code>Charting`ScaledTicks[{function, inversefunction}, options]
</code></pre>
<p>(The only options are <code>{Method -> Automatic, "... |
7,940 | <p>I have multiple plots of permittivity against frequency, but they're hard to read because the frequency labels go from 4x10^14 to 1x10^15 in Hz. I want to label the frequency with THz instead of Hz, so each value would just get divided by 10^12. How do I do this?</p>
| Edmund | 19,542 | <p>Starting in version 9 there is support for Units (<a href="http://reference.wolfram.com/language/guide/Units.html" rel="nofollow noreferrer">Units guide</a>). If your data is a <code>Quantity</code> then you can use <code>TargetUnits</code> to convert it to another <code>Quantity</code> in the plot.</p>
<pre><code... |
3,393,655 | <p>I want to show that:
<span class="math-container">$$
\lim_{n \rightarrow \infty} \int_0^{2\pi} \sin(x)^n \, dx=0
$$</span></p>
<p>and my idea was to use DCT (dominated convergenece theorem).</p>
<p>However, my textbook has the requirement that <span class="math-container">$u(x)=\lim_{n \rightarrow \infty} u_n(x)$<... | Sangchul Lee | 9,340 | <p>DCT only requires almost-everywhere convergence, which is the case in your limit. If the version in your textbook requires everywhere convergence, then here is a simple trick: <span class="math-container">$$\int_{0}^{2\pi}\sin^n(x)\,\mathrm{d}x=\int_{0}^{2\pi}u_n(x)\,\mathrm{d}x$$</span> where <span class="math-cont... |
1,112,926 | <p>Problem: For the sequence $r$ defined by </p>
<p>$$r_n = 3 \cdot 2^n - 4 \cdot 5^n, \ \ \ n \geq 0$$ <br></p>
<p>Prove that {$r_n$} satisfies <br></p>
<p>$$r_n = 7r_{n-1} - 10r_{n-2}, \ \ \ n \geq 2$$</p>
<p>Can this problem be explained and broken down and show the process? I'd like to follow your steps on my o... | Bumblebee | 156,886 | <p>$$7r_{n-1}-10r_{n-2}=7(3.2^{n-1}-4.5^{n-1})-10(3.2^{n-2}-4.5^{n-2})$$
$$7r_{n-1}-10r_{n-2}=(42-30).2^{n-2}+(-140+2=40)5^{n-2}$$
$$7r_{n-1}-10r_{n-2}=3.2^n-4.5^n$$
$$7r_{n-1}-10r_{n-2}=r_n$$</p>
|
5,238 | <p>For my homework, I have been asked to rationalise and simplify this surd;</p>
<p><span class="math-container">$$\frac{11}{3\sqrt{3}+7}$$</span></p>
<p>Each time I do this I get the wrong answer. The method I am using is;</p>
<p><span class="math-container">$$ \frac{11}{3\sqrt3+7} \times \frac{3\sqrt3-7}{3\sqrt3-7... | AgustΓ Roig | 664 | <p>Do you mean rationalise</p>
<p>$$
\frac{11}{3\sqrt{3}-7} \qquad \text{?}
$$</p>
<p>And are you sure you're trying</p>
<p>$$
\frac{11}{3\sqrt{3}-7} \cdot \frac{3\sqrt{3} + 7}{3\sqrt{3} + 7}\qquad \text{?}
$$</p>
<p>In general, <a href="http://en.wikipedia.org/wiki/Rationalisation_%28mathematics%29" rel="nofollow"... |
5,238 | <p>For my homework, I have been asked to rationalise and simplify this surd;</p>
<p><span class="math-container">$$\frac{11}{3\sqrt{3}+7}$$</span></p>
<p>Each time I do this I get the wrong answer. The method I am using is;</p>
<p><span class="math-container">$$ \frac{11}{3\sqrt3+7} \times \frac{3\sqrt3-7}{3\sqrt3-7... | Marvin | 544,299 | <p>Apparently you made a minor operational mistake when interpreting <span class="math-container">$ 3\sqrt{3} + 7 $</span> as its conjugate, <span class="math-container">$ 3\sqrt{3} - 7 $</span>. The proper way to rationalize the denominator containing a radical is to remove that radical by multiplying the numerator an... |
288,234 | <p>The inclusion $I\colon \mathbf{Grpd}\hookrightarrow\mathbf{Cat}$ of groupoids into categories has both a left and a right adjoint $L,R\colon \mathbf{Cat}\to \mathbf{Grpd}$, with $R(C)$ being largest groupoid contained in $C$ and $L(C) = C[C^{-1}]$ being $C$ with all morphisms brutally inverted. Going into $\infty$-c... | Martin Bidlingmaier | 84,063 | <p>As Valery Isaev has pointed out, $\mathbf{Kan} \hookrightarrow \mathbf{WKan}$ does not have a left adjoint. However, instead of considering $\mathbf{Kan} \hookrightarrow \mathbf{WKan}$, you can consider the forgetful functor $\mathbf{sKan} \hookrightarrow \mathbf{sWKan}$ of (weak) Kan complexes with <em>chosen</em> ... |
456,961 | <p>What is the maximum of ${\frac{(1-\cos x)}{x}}$ in the interval $[0, \pi]$?</p>
<p>I can show that the maximum is less than 1, but I want an exact value.</p>
| levitopher | 12,386 | <p>I do not know the answer analytically, but I suggest a numerical solution. It's pretty easy to find such things online; in the interest of homework rules, report back if you try and still can't find the answer.</p>
|
300,181 | <p>If we have $n$ different numbers from the set $\mathbb N$ what is the maximum possible number of numbers that we can contruct from these numbers by performing $m$ successive operations, where operation is addition or multiplication? To be more precise about the problem I will clarify it further with some examples, ... | Ivan Loh | 61,044 | <p><em>Note: The OP has clarified that brackets are not allowed. In other words, we have $m+1$ terms and $m$ successive operations in between, so terms like $(x_1+x_1)*x_1$ are not counted as a possibility. (We take it as $x_1+x_1*x_1=x_1+x_1^2$ instead)</em></p>
<p>In general, if you fix $m$, then $F(n,m)$ is a polyn... |
4,645,763 | <p>Recently I played a little bit around with GeoGebra and I constructed the in- and circumcircle of a <span class="math-container">$\triangle ABC$</span> with <span class="math-container">$A=(0,0)$</span> and <span class="math-container">$B=(1,0)$</span> and I asked myself if it is possible to construct the area where... | Jean Marie | 305,862 | <p>Here is a solution based on ratio <span class="math-container">$\frac{r}{R}$</span>, where <span class="math-container">$r$</span> is the inradius and <span class="math-container">$R$</span> the circumradius.</p>
<p>See Geogebra animation <a href="https://www.geogebra.org/calculator/zkfcq55v" rel="nofollow noreferre... |
2,274,514 | <p>I want to know how to evaluate $\int \frac{\log x}{x^2}$.
Using by parts, and after moving terms, we get something like
$$2 \int\frac{\log x}{x^2} = \frac{(\log x)^2}{x} + \int \frac{(\log x)^2}{x^2}$$
Using by parts again gives
$$\int \frac{(\log x)^2}{x^2} = \frac{2}{3}(\log x)^3 - \int \frac{(\log x)^3}{x^2}$$
At... | The Dead Legend | 433,379 | <p>$$\int\frac{\ln x}{x^2}dx$$
$$\int(\ln x) \frac{1}{x^2}dx$$
$$\ln x. (\frac{-1}{x})+\int\frac{1}{x^2}dx$$
$$\frac{-\ln x}{x^2}-\frac{1}{x}+C$$</p>
|
2,339,408 | <p>I see a question in Chinese senior high schools books:</p>
<blockquote>
<p>Throwing a fair coin until either there is one Head or four Tails.
Find the expectation of times of throwing.
(You start throwing a coin, if you see Head, then the game suddenly over; and if you see four Tail, the game is over too. Onl... | drhab | 75,923 | <p>You could take $\Omega=\{T,H\}^4$ as sample space where all outcomes are equiprobable, and prescribe random variable $X$ as the function $\Omega\to\mathbb R$ determined by:</p>
<ul>
<li>$X(\omega)=1$ if $\omega_1=H$</li>
<li>$X(\omega)=2$ if $\omega_1=T$ and $\omega_2=H$</li>
<li>$X(\omega)=3$ if $\omega_1=\omega_2... |
2,959,862 | <p>I'm having trouble finding the Fourier transform of <span class="math-container">$g(t) = \cos^2{a x}$</span>. </p>
<p>I know the answer has to be a summation of <span class="math-container">$3$</span> dirac delta functions, but I'm having trouble showing this. I'll show you where my work ran into a problem.</p>
<p... | user1098340 | 1,098,340 | <p>The simple way of how engineers do it:</p>
<p><span class="math-container">$$
\cos^2(t) = \left[ \frac{e^{jt} + e^{-jt}}{2} \right]^2 = \frac{1}{4}
e^{j2t} + \frac{1}{2} + \frac{1}{4} e^{-j2t}
$$</span>
We also know that:
<span class="math-container">$$
e^{2jt} \Longleftrightarrow 2 \pi \delta(\omega -2) \\
1 \Longl... |
1,729,434 | <p>I was able to prove the base case statement, where if you plug in $3$ for $n$ you get:</p>
<p>$19 β₯ 15$.</p>
<p>Next I supposed an arbitrary value $k$ where $k β₯ 3$ and $2k^2+1 β₯ 5k$. I know that next I need to prove that:</p>
<p>$2(k+1)^2+1 β₯ 5(k+1)$</p>
<p>But this is where I got stuck. </p>
| lulu | 252,071 | <p>Method I: Solving the quadratic $f(x)=2x^2-5x+1=0$ yields two solutions, approximately $.22\;\&\;2.28$ Thus for $x>2.28$ we have $f(x)>0$.</p>
<p>Method II: (induction). We let $f(n)=2n^2-5n+1$ and observe that $f(3)>0$. Now suppose that $f(n)>0$, we wish to verify that $f(n+1)>0$ but $$f(n+1... |
2,016,588 | <p>I'd really appreciate a push in the right direction for solving this. I just can't get it. Thanks</p>
<p>Prove $ ({x+1})^{1/3} < 1 + {\frac13}x $ for x > 0 </p>
| Bhaskara-III | 246,676 | <p>see my nice answer:</p>
<p>$$\int \frac{2\ dx}{(\cos x-\sin x)^2}=\int \frac{2\ dx}{\cos^2 x\left(1-\frac{\sin x}{\cos x}\right)^2}$$ $$=2\int \frac{\sec^2 x\ dx}{\left(1-\tan x\right)^2}$$
$$=-2\int \frac{d(1-\tan x)}{\left(1-\tan x\right)^2}$$
$$=-2 \frac{-1}{\left(1-\tan x\right)}+C$$$$=\frac{2\cos x}{\cos x-\s... |
2,182 | <p>I wondered if it is appropriate to ask on mathematica stack exchange a question about what they think about the ergonomy of mathematica in comparison to other softwars (matlab etc).</p>
<p>Because I find mathematica very unfriendly in comparison of everything I learnt but I would have to have other point of view to... | rcollyer | 52 | <p><strong>Completely separate option</strong>: </p>
<ul>
<li>Leave <a href="https://mathematica.stackexchange.com/questions/tagged/geographics" class="post-tag" title="show questions tagged 'geographics'" rel="tag">geographics</a> separate as while the creation of a <code>GeoGraphics</code> object, by hand, i... |
2,855,335 | <p><a href="https://i.stack.imgur.com/fAAXj.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fAAXj.png" alt="enter image description here" /></a></p>
<p>One should note that the family here may not countable. If it is countable, the it is a consequence of the following results;</p>
<p>Lemma 1. the pro... | Paul Frost | 349,785 | <p>The proof depends on which theorems you allow to use. You certainly can give a direct proof, but in my opinion the most elegant way is this:</p>
<p>Theorems used:</p>
<p>(1) Each uniform space $X$ has a <em>completion</em> $X'$ (i.e. a complete uniform space containing $X$ as a dense subset).</p>
<p>(2) If $X$ is... |
982,386 | <p>I am quite a beginner in linear algebra and matrix calculus. I was wondering what is the derivative of the matrix inverse when the matrix is symmetric. More precisely, I'm looking for $\frac{\partial}{\partial \mathbf{X}} \mathbf{X}^{-1}$ when $\mathbf{X}$ is a symmetric matrix.</p>
<p>I am asking this because I ha... | Avitus | 80,800 | <p>Hint: use $(X^{-1})_{ij}=\frac{C_{ji}}{\operatorname{det}(X)}$, with
$C_{ji}=(-1)^{i+j}X_{ji}=(-1)^{i+j}X_{ij}$, as $X$ is symmetric.</p>
<p>Then</p>
<p>$$\frac{\partial (X^{-1})_{ij}}{\partial X_{kl}}= \frac{\partial }{\partial X_{kl}}\left( \frac{(-1)^{i+j}X_{ij}}{\operatorname{det}(X)}\right).$$</p>
<p>Using ... |
1,249,248 | <p>Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function that is differentiable at the point $x_0$. Prove that $S_n(f(x_0))$ converges to $f(x_0)$, where $S_n$ denotes the partial sums of the Fourier series. </p>
<p>Note: We only covered the Riemann Lebesgue Lemma for the hypotheses that $f$ is continuous.... | Community | -1 | <p>The difference between the partial sum and $f(x)$ (I drop the subscript here) is given as you have given by <p>
${t_n}(x) - f(x) = = \frac{1}{\pi }\int_0^\pi {\left( {f(x + u) + f(x - u) - 2f(x)} \right){D_n}(u)du} $ <p></p>
<p>$ = \frac{1}{\pi }\int_0^\pi {\left( {f(x + u) + f(x - u) - 2f(x)} \right)\frac{{\sin... |
1,025,548 | <p>Can one help finding this limit</p>
<p>$$\lim_{x \to 0}\frac{x^{3}-\sin^{2}x\tan x}{\tan(\sin x) - \sin (\tan x)}$$</p>
<p>L'Hospital's rule is permited.</p>
<p>(<a href="https://math.stackexchange.com/questions/516483/find-lim-lim-x-to0-frac-tan-tan-x-sin-sin-x-tan-x-sin-x">Find lim:$\lim_{x\to0} \frac{\tan(\tan... | Olivier Oloa | 118,798 | <p><strong>Hint.</strong> Recall that, for $x$ near $0$, you have
$$\tan x = x+\dfrac{x^3}{3}+\dfrac{2x^5}{15}+\dfrac{17x^7}{315}+\mathcal{O}(x^9)$$
$$ \sin x = x-\dfrac{x^3}{6}+\dfrac{x^5}{120}-\dfrac{17x^7}{720}+\mathcal{O}(x^9)$$ to obtain (using only relevant terms) $$\tan(\sin x) = x+\frac{x^3}{6}-\frac{x^5}{40}-\... |
2,141,643 | <p>I am trying to solve the following differential equation:</p>
<p>$$yy''= 3(y')^{2} \\$$</p>
<p>I feel like there must be some substitution to turn this equation into an easier one, but I can not seem to find a substitution that works. Also, I have tried separating the equation into terms of a chain rule, but I can... | Lutz Lehmann | 115,115 | <p>If you are not at an equilibrium point, you can transform to
$$
\frac{y''}{y'}=3\frac{y'}{y}\implies \ln|y'|=3\ln|y|+c\implies y'=CΒ·y^3
$$
which is separable and can be solved as
$$
\frac1{y^2}=-2Cx+D.
$$</p>
|
1,046,229 | <p>For this problem in proving that the cardinality of <span class="math-container">$(0,1)$</span> is equal to that of the set of real numbers, would I just prove that <span class="math-container">$(0,1)$</span> is uncountable, and then use the theorem that the subset of an uncountable set is uncountable, by saying <sp... | MartΓn-Blas PΓ©rez Pinilla | 98,199 | <p>Bad argument. Works only if Continuum Hypothesis is true.</p>
|
3,692,997 | <p>A three-digit integer is chosen at random. What is the probability that it is possible to add a digit to its right end such that the resulting four-digit number is a multiple of <span class="math-container">$45$</span>?</p>
<p>Edit: I got <span class="math-container">$\frac{1}{15}$</span> as my answer can anyone co... | Hussain-Alqatari | 609,371 | <p>Simply,</p>
<p>We have the resulting numbers: <span class="math-container">$1000,1001,1002,1003,\dots,9999$</span></p>
<p>This is the sample space (All possible outcomes).</p>
<p>Next, we find out the series of integers that are divisible by <span class="math-container">$45$</span>: these integers are:</p>
<p><s... |
1,864,321 | <p>I tried to prove the following inequality which gives a lower bound to the Mathieu sum:
$$S=\sum_{k=1}^\infty\dfrac{2k}{(k^2+c^2)^2}$$
where $c\neq0$.
The Mathieu inequality states: $S\lt\dfrac{1}{c^2}$
The following inequality holds:
$$S\gt\dfrac{1}{c^2+\dfrac{1}{2}}$$
I tried to expand $S$ and I found an expressi... | Jack D'Aurizio | 44,121 | <p>It is a bit an overkill, but since for any $a,b>0$ we have:
$$ \int_{0}^{+\infty}x\sin(ax)e^{-bx}\,dx = \frac{2ab}{(a^2+b^2)^2} \tag{1}$$
it happens that:
$$ \sum_{k\geq 1}\frac{2k}{(k^2+c^2)^2}=\int_{0}^{+\infty}\sum_{k=1}^{+\infty}\frac{x\sin(cx)}{c}e^{-kx}\,dx = \frac{1}{c}\int_{0}^{+\infty}\frac{x\sin(cx)}{e^... |
1,923,038 | <p>I need to prove that $n^2 = {n \choose 2} + {n+1 \choose2}$. I have already proved this using algebra, but I am required to use both algebra and a formal combinatorial proof which demonstrates a bijection between the right and left hand sides. </p>
<p>If someone could show me how to get started with this proof and ... | Christian Blatter | 1,303 | <p>Define a map
$$f:\quad[n]\times[n]\to{[n]\choose 2}\sqcup{[n+1]\choose 2}$$
as follows: </p>
<p>$$f(x,y):=\left\{\eqalign{\{x,y\}\in{[n]\choose 2}\qquad&(x<y)\cr
\{x,n+1\}\in{[n+1]\choose 2}\qquad&(x=y)\cr
\{x,y\}\in{[n+1]\choose 2}\qquad&(x>y)\ ,\cr}\right.$$
and convince yourself that $f$ is bij... |
964,543 | <p>My question is what are the possible order of $A_6$? And how would I show I get $\frac{6!}{2}=360$. Any tips? I know that $A_6$ is the group of even permutations on six elements. I also know that $360=2^3*3^2*5^1$. How do I got about doing this?</p>
| rogerl | 27,542 | <p>Here's an alternative method (which is equivalent, but avoids the language of homomorphisms and kernels). Let $O\subset S_n$ be the set of odd permutations and $E\subset S_n$ be the set of even permutations. Choose any odd permutation $\pi$. Then since the product of odd permutations is even, and the product of an o... |
379,554 | <p>How can you fit a equilateral triangle on three arbitrary parallel lines with an edge and compass?</p>
<p><img src="https://i.stack.imgur.com/x8s9a.png" alt="enter image description here"></p>
| bob.sacamento | 61,250 | <p>Well, I'm late to this, but I've been obsessing over this far too much to back out now. So: Label your parallel lines $a$, $b$, and $c$ from bottom to top. Construct a line $d$ with a "positive slope" that crosses $b$ such that the top right angle of their intersection is $60^\circ$. Extend the line so that is cr... |
379,554 | <p>How can you fit a equilateral triangle on three arbitrary parallel lines with an edge and compass?</p>
<p><img src="https://i.stack.imgur.com/x8s9a.png" alt="enter image description here"></p>
| Roman Y. Andronov | 102,008 | <p>I've found a different invariant that solves this problem: two circles bisecting a perpendicular between any two parallels cut each other at points that are exact middles of the side opposite to the vertex on the third parallel.</p>
<p>If you want more explanations and sketches please go to:</p>
<p><a href="http:/... |
2,746,388 | <p>A common tangent to two curves is a line that is tangent to the two curves, but not necessarily at the same point.</p>
<p>Find, in terms of $a$ and $b$, the explicit equation of the common tangent to the two curves $y = x^2 + ax + b$ and $y = x^2 + bx + a$, where $a$ is not equal to $b$.</p>
<p>Also ο¬nd, in terms ... | Cesareo | 397,348 | <p>Calling</p>
<p>$$
\left\{
\begin{array}{rcl}
f_1(x,y) & = & x^2 + a x + b - y\\
f_2(x,y) & = & x^2 + b x + a - y\\
L& \rightarrow & p = p_1+ \lambda (p_2-p_1)
\end{array}
\right.
$$</p>
<p>with $p = (x,y), \ p_1 = (x_1,y_1) \ p_2 = (x_2,y_2)$ such that $f_1(p_1) = f_2(p_2) = 0$ the tangency... |
11,154 | <p>I have an image of gold electrodes on a flat substrate obtained with a scanning electron microscope.
I would like to colour in all the gold electrodes and particles that appear in the center of the image and make the background a different colour.</p>
<p>I'm trying to use some straightforward processing; applying <... | halirutan | 187 | <p>I think there is something wrong in your code. Why did you write <code># > -300 &</code> as criterion for the area.
Lets try another approach with only a simple gaussian filter:</p>
<pre><code>img = ImagePad[
ColorConvert[Import["http://i.stack.imgur.com/GdtlA.jpg"],
"Grayscale"], {{0, 0}, {-100, 0}... |
1,224,085 | <p>A series is an expression of the form
$$
\sum_{n=k}^{\infty} a_n
$$
where the $a_n$ are real numbers and they depend on $n$. If $a_n = b_n$ for all $n\geq k$, then I would assume that one would <em>say that</em> the two series
$$
\sum_{n=k}^{\infty} a_n\quad\text{and}\quad \sum_{n=k}^{\infty} b_n
$$
are the <em>same... | GEdgar | 442 | <p>Are the series
$$
0 + 1 + 2 + 3 + \dots \quad\text{and}\quad 1 + 2 + 3 + 4 + \dots
$$
<strong>the same</strong>? What if a summation method (like "zeta-regularization") assigns different "sums" to them? Then certainly we would require that the two series be considered "different".</p>
<p><em>further explanation</... |
2,675,565 | <p>I have a specific problem and I am kind of stuck. Don't know exactly where to begin defining what it is. Is someone could just give me a nodge in the right direction or even better, tell me what kind of problem it is, that would be nice. I can of course do the hard work myself but right now I need some help figuring... | quasi | 400,434 | <p>Enumerate the colors as:
$$
1{:}\;
\text{red},
\;\;\;
2{:}\;
\text{green},
\;\;\;
3{:}\;
\text{blue},
\;\;\;
4{:}\;
\text{yellow},
\;\;\;
5{:}\;
\text{purple},
\;\;\;
6{:}\;
\text{brown}
$$
and let $a_{i,j}$ be the number of balls of color $i$ in set $j$.
<p>
Then a solution to the problem is a $5$-tuple $x=(x_1,...... |
446,386 | <blockquote>
<p>I we are given a vector space $(V,+,\cdot)$ over a field $\mathbb K$ (where $\mathbb K=\mathbb R$ or $\mathbb K=\mathbb C$), is there the finest topology $\mathcal T$, such that $(V,\mathcal T)$ is a <a href="http://en.wikipedia.org/wiki/Topological_vector_space" rel="nofollow noreferrer">topological ... | Etienne | 80,469 | <p>I would say that there is a finest vector space topology. However, my argument may be completely wrong...</p>
<p>Let $(\tau_i)_{i\in I}$ be the family of all vector space topologies on $V$. Call a set $\mathcal E\subset V$ $elementary$ if it can be written as $\mathcal E=\mathcal U_{i_1}\cap \cdots \cap \mathcal U_... |
2,871,589 | <p>This may be a stupid question, but I was learning some set and group theory and it just made me think. Clearly the continuum is an infinite quantity $\mathfrak{c}$, but the set of all reals is also infinitely long. Or is it that $\sup(\mathbb{R})=\mathfrak{c}$ and $\mathfrak{c}\notin\mathbb{R}$. Regardless this is a... | angryavian | 43,949 | <p>Sketch:</p>
<p>Let $n = xy$. I assume the mines' locations as well as the choice of tiles to uncover are chosen without replacement (i.e. you cannot place a mine on a tile that already has a mine, and you cannot uncover a tile that you have already uncovered).</p>
<p>Instead of placing mines randomly on tiles, we ... |
3,267,311 | <p>The author in this example is trying to show that the norm of Hilbert matrix is less than or equal to <span class="math-container">$\pi$</span>. Hilbert matrix has entries
<span class="math-container">$$a_{ij}=\frac{1}{i+j+1};1\leq i,j \leq \infty$$</span> He used the fact that if <span class="math-container">$\exis... | Kavi Rama Murthy | 142,385 | <p>I think the sums starting from <span class="math-container">$i=0$</span> is a mistake. They should start from <span class="math-container">$i=1$</span>. Now <span class="math-container">$\int_{i-1+\frac 1 2}^{i+\frac 1 2} \frac 1 {(x+j+\frac 1 2)\sqrt x } dx > \frac 1 {(i+\frac 1 2 +j+\frac 1 2) \sqrt {i+\frac 1... |
22,892 | <p>Imagine the following scenario:
user A is developing a project with documentation in workbench. She wants to give her (unfinished) work to user B , so that user B can continue work on the project and eventually deploy it on her PC. The 2 users might have different operating systems.
What is the best way to accompli... | Leonid Shifrin | 81 | <p>I have done this many times, and to me this seems quite straight-forward.</p>
<h3>Export</h3>
<p>Select your project folder in the Navigator tab (usually on the left).</p>
<p>Then go to the <code>File</code> menu, then</p>
<pre><code>File -> Export -> General -> Archive file -> Browse
</code></pre>
<p>t... |
2,184,864 | <p>I am asked to prove the following, given that <span class="math-container">$a<b$</span>, and <span class="math-container">$c<d$</span>, such that <span class="math-container">$a,b,c,d>0$</span>:</p>
<blockquote>
<p>(1) <span class="math-container">$a+c<b+d$</span></p>
<p>(2) <span class="math-container">... | quanticbolt | 300,881 | <p>We want to show the following two things, (1) $a+c<b+d$, and (2) $ac<bd$.</p>
<p>The first is quite straightforward. First, we add $c$ to $a<b\to a+c<b+c$. Then, add, $b$ to both sides of $c<d\to b+c<b+d$. Hence, we have $$a+c<b+c<b+d\to a+c<b+d,$$ which is what we wanted to show.</p>
<p... |
2,359,797 | <p>I have a set defined by a lot of linear constraints. Namely, $\mathcal{X}=\{x:Ax\leq b\}$, where $x$ is a $j$- dimensional vector (each component can be either positive or negative). I want to know for each component $j$, $\bar x_j=\max_{x\in \mathcal{X}} x_j$ and $\underline x_j=\min_{x\in \mathcal{X}} x_j$. I know... | Johan LΓΆfberg | 37,404 | <p>Enumeration can be extremely expensive, so unless there is some particular structure in your case, the most efficient way is in general to solve $2j$ linear programs where you maximize and minimize along all directions.</p>
|
2,684,433 | <p>If I have a set $\{(1, H), (2, C), (3, F), (4, Z), (5, S), (6, L) \}$ is there any way to express this with set builder notation? If not, is there any other way to express this mathematically?</p>
| Nairit | 423,756 | <p>First, the base case $n=1$ is trivial but necessary.</p>
<p>This part is slightly tedious, but since you asked for induction, here.</p>
<p>Basically, we wish to prove that if we replace all occurrences of $n$ with $n+1$, we still have a multiple of 24. </p>
<p><a href="https://www.wolframalpha.com/input/?i=(n%2B1... |
317,904 | <p>Given, in 3D space: a point $P$ and a direction $v$, a point $Q$ and a direction $w$. So, two lines, $L_1 = P + tv$, $L_2 = Q + tw$.</p>
<p>I am looking for two points, one on each line, say P' and Q'. My requirement is that the distance from $P$ to $P'$ plus $Q$ to $Q'$ equals the distance from $P'$ to $Q'$:</p>
... | Hanul Jeon | 53,976 | <p><strong>Hint:</strong> $$\ln(1+x)=x+O(x^2)$$ as $x\to 0$. (So this limit is equal to 1.)</p>
<p>or you can use inequality
$$\ln(1+x)\le x.$$</p>
|
317,904 | <p>Given, in 3D space: a point $P$ and a direction $v$, a point $Q$ and a direction $w$. So, two lines, $L_1 = P + tv$, $L_2 = Q + tw$.</p>
<p>I am looking for two points, one on each line, say P' and Q'. My requirement is that the distance from $P$ to $P'$ plus $Q$ to $Q'$ equals the distance from $P'$ to $Q'$:</p>
... | lab bhattacharjee | 33,337 | <p>$$\lim_{n\to \infty} \left(1+\frac{1}{n!}\right)^{2n}=\lim_{n\to \infty}\left( \left(1+\frac{1}{n!}\right)^{n!}\right)^{\frac2{(n-1)!}}=e^0=1$$</p>
<hr>
<p>Alternatively,
Let $$A=\left(1+\frac{1}{n!}\right)^{2n}$$</p>
<p>So, $$\log A=2n\log \left(1+\frac{1}{n!}\right)\text { using }\log(1+x)=x-\frac{x^2}2+\frac{x... |
2,206,415 | <p>Let $n \in \mathbb{Z_+},$ use integration by parts to prove </p>
<p>$$\int_0^{\infty}x^ne^{-x}dx = n!$$</p>
<p>I know that if you repeatedly differentiate $x^n$ you get $n!$, but I don't know how to prove this?</p>
<p>A step by step answer would be helpful. Thanks!</p>
| mrnovice | 416,020 | <p>Let $I_{n} =\int_{0}^{\infty}x^ne^{-x}\, dx$</p>
<p>Let $u = x^n \implies \frac{du}{dx} = nx^{n-1}$</p>
<p>Let $v=\int e^{-x}\, dx = -e^{-x}$</p>
<p>Then $I_{n} = uv - \int_{0}^{\infty} v\frac{du}{dx}dx = [-x^ne^{-x}]_{0}^{\infty} + n\int_{0}^{\infty}x^{n-1}e^{-x}\, dx$</p>
<p>Note that $\lim\limits_{x \to \inft... |
2,553,073 | <p>For the question:</p>
<blockquote>
<p>For what values of $c$ does $8x+5y = c$ have exactly one strictly positive solution? </p>
</blockquote>
<p><a href="https://i.stack.imgur.com/LmheC.png" rel="nofollow noreferrer">The solution is this</a></p>
<p>So I have 3 questions.</p>
<p>I understand everything up unti... | JohnHalleck | 167,763 | <p>I make the assumption, since you didn't say what logic you are working in, that you are working in a classical two valued logic, which I'll just call PC.</p>
<p>There is a related (so called) paradox of material implication in PC called <a href="https://proofwiki.org/wiki/Disjunction_of_Conditional_and_Converse" re... |
3,798,735 | <p>Let <span class="math-container">$G(x,y): \mathbb{R}_0^+ \times \mathbb{R}_0^+ \to \{0,1,\infty\}$</span> be a (Borel) measurable function. If the set of values <span class="math-container">$(x,y) \in \mathbb{R}_0^+ \times \mathbb{R}_0^+ $</span> such that <span class="math-container">$G(x,y) = \infty$</span> has po... | Doug Spoonwood | 11,300 | <p>Yes, definitions can be statements.</p>
<p>For example, in the study of propositional calculi, a definition like</p>
<p>C <span class="math-container">$\delta$</span> Np <span class="math-container">$\delta$</span> Cp0 could get used.</p>
<p>In the above "C" gets used for material implication, "N"... |
193,481 | <p>$\#2^{\Omega} = 2^{\#\Omega}$</p>
<p>So far I know that when the size of $\Omega$ is 0, we have the $\emptyset$ and the size of the power set for $\Omega$ will be 1, or $2^{0} = 1 $ </p>
<p>How do I start by proving this for any size $n$ of $(\Omega)$. </p>
<p>Also does $\Omega$ have to be finite for this to be t... | William | 13,579 | <p>Suppose $\Omega$ is a finite set of $n$ elements. So we may as well call $\Omega = \{1, 2, ..., n\}$. Every subset $X \subset \Omega$ corresponds to a binary string $\sigma_X$ of length $n$ where the $k^\text{th}$ bit of $\sigma_X$ is $0$ if $k \notin X$ and $1$ if $k \in X$. Now by a counting argument, there are $2... |
1,698,257 | <p>How to find a function whose curl is $(7e^y,8x^7 e^{x^8},0)$?</p>
<p>I've tried several integration but can't find a trivial form. </p>
| WW1 | 88,679 | <p>Since $\vec\nabla \times \vec F(x,y,z) $ <br> has an x component that depends only on y <br> and a y component that depends only on x <br>and no z component </p>
<p>A good guess is that $\vec F(x,y,z) $ takes the form </p>
<p>$\vec F(x,y,z) = (0,0,g(x)+h(y)) $</p>
<p>$g(x) $ and $h(y) $ can be found easily by in... |
4,469,302 | <p>It is often written that all monads are functors, but it is quite hard to find an actual mathematical proof of it.</p>
<p>A functor is defined as a higher level type defining the <code>fmap</code> function:</p>
<pre><code>class Functor f where
fmap :: (a -> b) -> f a -> f b
</code></pre>
<p>It must als... | S.C. | 913,783 | <p>An alternate to the usual definition of a monad (as a functor plus some structure) that's similar to your Haskell definition has three pieces of structure and three equations.</p>
<p>Let <span class="math-container">$\mathcal C$</span> be a category. A monad on <span class="math-container">$\mathcal C$</span> consis... |
275,473 | <p>Let $f$ and $g$ be two functions with derivatives in some interval containing $0$, where $g$ is positive. Also</p>
<p>$$f(x)=o(g(x))~as~x \rightarrow0$$</p>
<p>Prove or dissprove:</p>
<p>1) $$\int_0^xf(t)dt=o\left(\int_0^xg(t)dt\right)$$
2) $$f'(x)=o(g'(x))$$</p>
<p>Now considering the first, my reasoning is as ... | ΛjuΛ.zΙ79365 | 79,365 | <p>1) Begin by writing down the meaning of $o$: for any $\epsilon>0$ there is $\delta>0$ such tha t $$|f(x)|\le \epsilon \, g(x) \quad \text{for all }\ x\in (-\delta,\delta) \tag1$$
Then use the integral triangle inequality:
$$\left|\int_0^x f(t)\,dt\right|\le \int_0^x |f(t)|\,dt \le \epsilon\,\int_0^x g(t)\,dt... |
541,761 | <p>Can we show that the ring of Gaussian integers $$\mathbb{Z}[\sqrt{17}]:=\{a+b\sqrt{17}:a,b\in\mathbb{Z}\}$$
$$\mathbb{Z}[\sqrt{11}]:=\{a+b\sqrt{11}:a,b\in\mathbb{Z}\}$$</p>
<p>equipped with standard addition and multiplication are not isomorphic?</p>
| Madrit Zhaku | 34,867 | <p>$$y'=-\frac{48x}{(x^2+12)^2}\Rightarrow y''=\left(-\frac{48x}{(x^2+12)^2}\right)'=-48\left(\frac{x}{(x^2+12)^2}\right)'$$$$=-48\frac{x'\cdot(x^2+12)^2-x((x^2+12)^2)'}{(x^2+12)^4)}=-48\frac{(x^2+12)^2-4x^2(x^2+12)}{(x^2+12)^4}$$
$$=-48\frac{(x^2+12)(x^2+12-4x^2)}{(x^2+12)^4}=-48\frac{12-3x^2}{(x^2+12)^3}=-48\cdot 3\f... |
2,835,802 | <p>I'm struggling to come up with the method to find the number of ways to take $k$ objects from $n$ groups which at least one object from each group is taken and order matters.</p>
<p>More specifically, I'm trying to order $8$ digits from the digit pool of $5-10$ ($6$ digits) and each digit must appear at least once ... | user | 505,767 | <p>We need to consider the following different sets of digits</p>
<ul>
<li>$ \{5,6,7,8,9,10,\color{red}{5,5}\},\{5,6,7,8,9,10,\color{red}{5,6}\},...,\{5,6,7,8,9,10,\color{red}{10,10}\}$</li>
</ul>
<p>which are $21$ ($6$ with one triple of repeated digits and $15$ with $2$ pairs of repeated digits) and we can permutat... |
626,700 | <p>Evaluate $\int\int (x+y+z)dS$, where $S$ is the boundary of the unit ball $B$; i.e $S$ is the set of $(x,y,z)$ with $x^2+y^2+z^2=1$.</p>
<p>I parametrized usually using $x=sin\theta cos\phi,y=sin\theta sin\phi,z=cos\theta$, where $0\le \theta \le \pi$, and $0\le \phi \le 2\pi$.</p>
<p>But in this way the integral ... | Disintegrating By Parts | 112,478 | <p>Why can't the integral be 0? For example, by symmetry across the x=0 plane,
$$
\int\int x dS = 0.
$$
The same argument applies to each of the three integrals.</p>
|
4,230,454 | <p>I have graphed two data sets, each with the same y-values but the two data sets have different x-values. I have graphed them and one had a positive correlation, while the other had no correlation.
in the x-axis I was given to label MPQ and in the y I was given to label T450. The question was whether increasing level... | Alphie | 522,332 | <p>A reasonable conclusion would be to say:</p>
<p>There was a positive linear association between MPQ and T450 in the first dataset, while no linear association was found between these variables in the second dataset.</p>
|
1,503,957 | <p>Okay, maybe I am just really bad with exponents or forgot how exponents work but how do you do these 2 problems, here's what I got so far. I need to state whether thee sequence is increasing, decreasing, and use the ratio rule and difference rule to figure it out. </p>
<blockquote>
<p><strong>Ratio rule</strong>:... | A.Ξ. | 253,273 | <p>Yet another approach for those who would like to avoid "messy" job with $n\times n$ Hessians. First we make a remark</p>
<p><strong>Remark:</strong> It is easy to see that the function $g(t)=\ln(e^t+1)$ is convex and increasing ($g'(t)\ge 0$ and $g''(t)\ge 0$). Thus, the function $g\circ F$ <a href="https://en.wiki... |
1,631,535 | <p>In the lecture notes for a course I'm taking, the definition of a convex function is given as follows:</p>
<p>"a function $f$ is convex if, for any $x_1$ and $x_2$, and for any $\alpha$ $\in$ [0,1], $\alpha f(x_1) + (1-\alpha)f(x_2) \ge f(\alpha x_1 + (1-\alpha ) x_2)$" </p>
<p>That is, if you draw a line segment ... | DanielWainfleet | 254,665 | <p>Without the AGM nor the weighted AGM inequality. It suffices to consider the case <span class="math-container">$x> y$</span> and <span class="math-container">$a=\alpha \in (0,1).$</span> Take a fixed <span class="math-container">$y>0$</span> and a fixed <span class="math-container">$a\in (0,1)$</span> and for ... |
4,189,140 | <p>If <span class="math-container">$G$</span> be a locally compact topological group. Show that <span class="math-container">$G$</span> is paracompact.</p>
<p>Note: If we restrict <span class="math-container">$G$</span> to be locally compact, connected topological group, this problem becomes easier by constructing a se... | Benjamin BenΔina | 946,790 | <p>Firstly, most of what you are asking can be found in Hewitt's and Ross' Abstract Harmonic Analysis: Volume I, Chapter 2, (8.13).</p>
<p>Secondly, I think @HennoBrandsma is correct, since for a topological group <span class="math-container">$T_0$</span> is not only equivalent to <span class="math-container">$T_2$</sp... |
4,189,140 | <p>If <span class="math-container">$G$</span> be a locally compact topological group. Show that <span class="math-container">$G$</span> is paracompact.</p>
<p>Note: If we restrict <span class="math-container">$G$</span> to be locally compact, connected topological group, this problem becomes easier by constructing a se... | Huy Nguyen | 922,437 | <p>You are almost there, Let <span class="math-container">$H= \cup_n \overline U_n$</span>, then <span class="math-container">$H$</span> is open and paracompact since it is a countable union of compact sets. <span class="math-container">$G$</span> is the union of all the cosets of <span class="math-container">$H$</span... |
4,608,240 | <p>I'm trying to show that <span class="math-container">$\sum_{n=1}^{\infty}\frac{n^{2}}{\left(n+1\right)!}=e-1$</span> using Taylor series, I know that <span class="math-container">$e=\sum_{n=0}^{\infty}\frac{1}{n!}$</span> and also that <span class="math-container">$e-1=\sum_{n=1}^{\infty}\frac{1}{n!}$</span> but i d... | geetha290krm | 1,064,504 | <p><span class="math-container">$e^{x}-1= \sum\limits_{n=1}^{\infty} \frac {x^{n}} {n!}=\sum\limits_{n=0}^{\infty} \frac {x^{n+1}} {(n+1)!}$</span>. Differentiate twice to get <span class="math-container">$e^{x}=\sum\limits_{n=0}^{\infty} \frac {(n+1)nx^{n}} {(n+1)!}=\sum\limits_{n=0}^{\infty} \frac {n^{2}x^{n}} {(n+1)... |
4,608,240 | <p>I'm trying to show that <span class="math-container">$\sum_{n=1}^{\infty}\frac{n^{2}}{\left(n+1\right)!}=e-1$</span> using Taylor series, I know that <span class="math-container">$e=\sum_{n=0}^{\infty}\frac{1}{n!}$</span> and also that <span class="math-container">$e-1=\sum_{n=1}^{\infty}\frac{1}{n!}$</span> but i d... | Z Ahmed | 671,540 | <p><span class="math-container">$$e=\sum_{k=0}^\infty \frac{1}{k!}$$</span>
Then <span class="math-container">$\sum_{k=0}^\infty \frac{1}{(k+1)!}=e-1$</span> and <span class="math-container">$\sum_{k=0}^{\infty} \frac{1}{(k-1)!}$</span> as <span class="math-container">$(-1)!\to \infty$</span></p>
<p>Now note that <spa... |
2,891,334 | <p>Let $T$ be the smallest positive integer which, when divided by $11,13,15$ leaves remainders in the sets $\{7,8,9\},\{1,2,3\},\{4,5,6\}$ respectively. What is the sum of the squares of the digits of $T$?</p>
<p>My working </p>
<p>After applying CRT, I got </p>
<p>$T\equiv 469+1365a+495b+286c\,\, \left(\mod 2145\r... | dan_fulea | 550,003 | <p>The simplest solution is to use computer support to verify. (There is nothing structural in the given setting, and it is a finite quick search.) For instance, using <a href="http://www.sagemath.org" rel="nofollow noreferrer">sage</a>:</p>
<pre><code>sage: for k in [1..10000]:
....: if k%11 in (7,8,9) and k%13 i... |
338,090 | <p>A is a $100 \times 100$ matrix.</p>
<p>The element in the $i^{th}$ row and $j^{th}$ column is given by $i^2 + j^2$</p>
<p>Find the rank</p>
| copper.hat | 27,978 | <p>Hint:</p>
<p>Let $[B]_{ij} = i^2$, $[C]_{ij} = j^2$. Notice that $A=B+C$ (indeed, $C=B^T$).</p>
<p>What can you say about the rank of $B,C$?</p>
<p>Also, notice that $A$ contains the submatrix $\begin{bmatrix} 2 & 5 \\ 5 & 8\end{bmatrix}$, which is invertible.</p>
<p>If we must use Matlab (or Octave, in ... |
2,714,418 | <p>Let $(f,g):\mathbb R\to \mathbb R$, $f(x) = x^2 - \frac{\cos x}{2}$ and $g(x)= \frac{x\sin x}{2}$</p>
<p>These are the given options :-</p>
<p>(A) $f(x) = g(x)$ for more than two values of $x$</p>
<p>(B) $f(x) \neq g(x)$, for all $x \in \mathbb R$</p>
<p>(C) $f(x) = g(x)$ for exactly one value of $x$</p>
<p>(D)... | JosΓ© Carlos Santos | 446,262 | <p>Note that $f(0)=-\frac12<0=g(0)$.</p>
<p>On the other hand, $f'(x)-g'(x)=\frac12x\bigl((4-\cos(x)\bigr)$ and therefore $f-g$ is strictly increasing in $[0,+\infty)$. This, together with fact that $f(\pi)>g(\pi)$, shows that the equality $f(x)=g(x)$ takes place exatly once in $[0,+\infty)$. And, since both fu... |
2,809,519 | <p>Question: Prove that $\pi$ is irrational, assuming the following result: if $x$ is rational, $tan(x)$ is not. </p>
<p>Proof: Let $x$ $\in$ $\mathbb Q$ </p>
<p>I have seen Lambert's proof, however I am severely confused on where to begin. Do I now suppose $tan(x)$ is rational, then proceed by contradiction? Hints ... | B. Mehta | 418,148 | <p>Hint: Suppose $\pi/4$ is rational. </p>
|
1,838,409 | <blockquote>
<p>In how many ways can you select two distinct integers from the set {1,
2, 3, . . . , 100} so that their sum is: (a) even? (b) odd?</p>
</blockquote>
<p>I'm studying for a discrete midterm this coming Monday and saw the following problem on a practice midterm my Professor posted. I know the amount h... | joriki | 6,622 | <p>As noted in a comment, the sum of an even number and an even number is an odd number. Thus you've already solved the problem; just add up the two contributions you found for even sums and subtract them from the total number $\binom{100}2$ of sums. Alternatively, independently pick an even and an odd number, each in ... |
2,439,140 | <blockquote>
<p>Let $\{a,b,c\}\subset\mathbb R$ such that $a+b+c=3$ and $abc\ge -4$. Prove that: $$3(abc+4)\ge 5(ab+bc+ca).$$</p>
</blockquote>
<hr>
<p>*) $ab+bc+ca<0$ This ineq is right</p>
<p>*) $ab+bc+ca\ge 0$ then in $ab,bc, ca$ at least a non-negative number exists assume is $ab$</p>
<p>$\Rightarrow \dis... | Michael Rozenberg | 190,319 | <p>I think your solution is true and very nice.</p>
<p>My proof.</p>
<p>We can assume that $ab+ac+bc\geq0$, otherwise the inequality is obvious.</p>
<p>Let $a+b+c=3u$, $ab+ac+bc=3v^2$ and $abc=w^3$.</p>
<p>Hence, we need to prove a linear inequality of $w^3$, </p>
<p>which says it's enough to prove our inequality ... |
3,020,674 | <p>I need to use Fermat's Theorem to prove that 10001 is not prime. I understand that I just need to find a counterexample where <span class="math-container">$a^{10000}$</span> mod 10001 = 1 mod 10001 does not hold true, but this seems kind of difficult with such large numbers. Any ideas?</p>
| Peter | 82,961 | <p>Another method, also based on Fermat's little theorem, is the following.</p>
<p>First notice <span class="math-container">$$10001=10^4+1=\frac{10^8-1}{10^4-1}$$</span></p>
<p>So finding a prime factor of <span class="math-container">$10^8-1$</span> that is not also a factor of <span class="math-container">$10^4-1$... |
560,234 | <p>Given the series</p>
<blockquote>
<p>$$\sum_{n=1}^{\infty}(-1)^n\sin\left(\frac{n}{\pi}\right)$$</p>
</blockquote>
<p>I need to test for convergence/divergence. I think the divergent test might work here. I could see that the $\lim_{n\rightarrow\infty}(-1)^n\sin(\frac{n}{\pi})$ might not exist, so the series is ... | JMag | 70,465 | <p>First, you want to think about $$\sin{\left(\frac{n}{\pi}\right)}.$$ This is similar to $\sin{(n)}$ except the period (of ocsillation) is different. We know that the limit of $\sin{(n)}$, as $n\rightarrow \infty$, does not exist since it oscillates indefinitely. Can you apply a similar argument to $\sin{(n/\pi)}$?</... |
2,217,890 | <p>The singular integrands I will ask about in this post about appear a lot in numerical integration so I would really like to understand theoretically what makes some of these integrands 'more integrable' (whatever that means) than others.</p>
<p><strong>Question 1:</strong></p>
<p>I experimented in Wolfram Alpha by... | Vidyanshu Mishra | 363,566 | <p>The tricks you are referring to are basically the results of special cases of standerd theorems and principles.</p>
<p>For instance, <a href="https://en.m.wikipedia.org/wiki/Stewart%27s_theorem" rel="nofollow noreferrer">Stewart theorem</a> gives is the length of any cevian in a triangle in terms of side of that tr... |
3,692,850 | <p>I want to know the convergence radius of <span class="math-container">$\displaystyle{\sum_{n=0}^{\infty}}(\sqrt{ 4^n +3^n}οΌ(-1)^n\sqrt{ 4^n-3^n})x^n$</span>.</p>
<p>Firstly, I tired to calculate <span class="math-container">$\lim_{k\to\infty}\left|\frac{a_{k}}{a_{k+1}}\right|$</span>,but I noticed this series does ... | Ian | 83,396 | <p>One way is to use the root test, which says the radius of convergence is always <span class="math-container">$\frac{1}{\limsup_{n \to \infty} |a_n|^{1/n}}$</span> where <span class="math-container">$a_n$</span> is the coefficient of <span class="math-container">$x^n$</span>. This works even if this root doesn't actu... |
195,898 | <p>I can't understand what is wrong with this paradox. How we should strictly mathematically explain it? </p>
<p>Mathematical induction:</p>
<p><strong>1. The basis:</strong> </p>
<p>$n=1,n=2$. Through any two (one) points on a plane we can draw a straight line.</p>
<p><strong>2. The inductive step:</strong></p>
... | Michael Joyce | 17,673 | <p>Hint: We know that through any 2 points in the plane we can draw a straight line, but that is not true for any 3 points. Thus, you should focus your attention on the argument that the claim is true for $n=2$ implies that it is true for $n=3$. Does the argument in 3. (Fake Pair Of Ducks) work for $k=2$ (3 points)?... |
3,207,628 | <p>There is this following statement which I need to evaluate to be true or false: </p>
<blockquote>
<p>Let <span class="math-container">$f(x) $</span>, <span class="math-container">$f:\mathbb{R} \rightarrow \mathbb{R_+} $</span> be a continuous probability density function. Then <span class="math-container">$\lim_{... | leonbloy | 312 | <p>Your proof is wrong, because, in general, <span class="math-container">$\lim_{x \to \infty}h(x) = 1$</span> does <em>not</em> imply <span class="math-container">$\lim_{x \to \infty}h'(x) = 0$</span> . See <a href="https://math.stackexchange.com/questions/162078/if-a-function-has-a-finite-limit-at-infinity-does-that... |
2,560,602 | <p>Part (a)
Suppose $f'(z)$ is a complex derivation of $f(z)$. Since $f'(z)$ takes complex values, $f'(z)$ is a $2 \times 1$ column vector in $\mathbb{R}^2$. </p>
<p>Part (b)
If I interpret $f$ as function between $\mathbb{R}^2 \rightarrow \mathbb{R}^2$, then the derivation $f'(z)$ becomes a $2 \times 2$ linear tranf... | Arthur | 15,500 | <p>The real derivative of a holomorphic function (which is required for the complex derivative to exist) at a point is the product of some multiple of the identity matrix with a rotation matrix (this is the reason for the name "holomorphic": it preserves angles, unless the derivative is $0$). Thus only a $2$-dimensiona... |
410,763 | <p>I am having hard time recalling some of the theorems of vector calculus. I want to calculate the volume integral of the curl of a vector field, which would give a vector as the answer. Is there any formula? As far as I can recall, maybe I can write</p>
<p><span class="math-container">$$\int \nabla \times \vec{A} \ \... | Alexys_from_Brazil | 101,836 | <p>You are right. The volume integral of the curl follows from the standard Gauss Theorem. This may be clearly seen if one analyzes each Cartesian component separately. To be concrete, let us consider the $z$ component: $$(\vec\nabla\times\vec F)_z= \left(\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\... |
3,347,170 | <blockquote>
<p><span class="math-container">$$|x+1|+|x|>3$$</span></p>
</blockquote>
<p>I have one simple problem. When I break the modulus function, ie. when I get the following expressions</p>
<p><span class="math-container">$$-x-1-x$$</span> <span class="math-container">$$x+1-x$$</span> and <span class="math... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: Consider the cases <span class="math-container">$x\geq 0$</span>, <span class="math-container">$-1\le x<0$</span> and <span class="math-container">$x<-1$</span>.
By the hint you will get
<span class="math-container">$$x+1+x>3$$</span>
<span class="math-container">$$x+1-x>3$$</span>
<span class="mat... |
209,757 | <p>Find the value of $c$ which makes it possible to solve:</p>
<p>$$u+v+2w=2,$$
$$2u+3v-w=5,$$
$$3u+4v+w=c$$</p>
| Souvik Dey | 42,041 | <p>If you add the second and third equation you get 5u+7v=5+c ; if you multiply the second equation by 2 and then add it to the first then you get 4u+u+6v+v=12=5u+7v , so c=7</p>
|
1,769,494 | <p>Say I have a function </p>
<p>$$ f(x) = \dfrac 1x$$</p>
<p>and I'm looking at its $n^{th}$ derivative and trying to come up with a formula.</p>
<p>I can easily get it because if forms a very consistent pattern and it somewhat reminds me of harmonic series.</p>
<p>The formula is </p>
<p>$$ f'^n (x) = (-1)^n {n!... | Vincent | 332,815 | <p>As you wrote it, $f''''=-4f$</p>
<p>It should be easy from here</p>
|
1,769,494 | <p>Say I have a function </p>
<p>$$ f(x) = \dfrac 1x$$</p>
<p>and I'm looking at its $n^{th}$ derivative and trying to come up with a formula.</p>
<p>I can easily get it because if forms a very consistent pattern and it somewhat reminds me of harmonic series.</p>
<p>The formula is </p>
<p>$$ f'^n (x) = (-1)^n {n!... | Element118 | 274,478 | <p>Notice a pattern?</p>
<p>$f(x) = e^x\cos x$</p>
<p>$f'(x) = e^x(\cos x-\sin x) = \sqrt{2}e^x\cos\left(x-\frac{\pi}{4}\right)$</p>
<p>$f''(x) = -2e^x\sin x = 2e^x\cos\left(x-\frac{\pi}{2}\right)$</p>
<p>$f'''(x) = -2e^x(\cos x+\sin x) = 2\sqrt{2}e^x\cos\left(x-\frac{3\pi}{4}\right)$</p>
<p>$f^{(4)}(x) = -4e^x\si... |
2,428,160 | <p>In the video game Player Unknown's Battlegrounds, you start with a circular safezone of diameter $n$. After some time, the safezone instantly shrinks to a circle with a diameter $\frac{n}{2}$. </p>
<p>The catch is that the smaller safezone will be entirely contained within the bounds of the previous, larger circl... | Robert Z | 299,698 | <p>Note that
\begin{align*}
f(x + h) - f (x)&=\left(\sqrt[4]{x+h}-\sqrt[4]{x}\right)\cdot
\frac{\sqrt[4]{x+h}+\sqrt[4]{x}}{\sqrt[4]{x+h}+\sqrt[4]{x}}\cdot
\frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}\\
&=\frac{\sqrt{x+h}-\sqrt{x}}{\sqrt[4]{x+h}+\sqrt[4]{x}}\cdot
\frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}... |
2,445,809 | <p>I want to calculate the derivative of an integral of a two-variable function, so</p>
<p>$\frac{d}{dy}\int_{0}^1f(x,y)\,dx$.</p>
<p>I am sorry if this is a basic question but a google search yields unusable results. I am 90% sure that the derivative can simply go under the integral, but I would like to be sure. Tha... | caverac | 384,830 | <p>You are right,</p>
<p>$$
\frac{{\rm d}}{{\rm d}y}\int_0^1{\rm d}x~f(x,y) = \int_0^1{\rm d}x~\frac{\partial f(x,y)}{\partial y}
$$</p>
|
2,763,466 | <p>I know there is no non-vanishing vector field on $S^2$, so I cannot comb the hair on a ball.
(I am treating $S^2$ as a manifold without the ambient space $\mathbb R^3$, which amounts to demanding that the vector field is tangential to $S^2$ at every point if you prefer the Euclidean point of view.)
Is this still imp... | Thomas Rot | 5,882 | <p>No. Line bundles on <span class="math-container">$S^2$</span> are classified by <span class="math-container">$H^1(S^2;\mathbb{Z}/2)=0$</span> which means that there is only the trivial line bundle <span class="math-container">$L$</span> on <span class="math-container">$S^2$</span>. But then <span class="math-contain... |
333,645 | <p>$$f(z)=\frac 1{\cos(z^4)-1}$$</p>
<p>$z=0$ is a pole of $f$, and I believe that the Laurent series centred at $0$ is $-\frac 2{z^8}-\frac 16+...$, which looks like the pole is of order $8$, but why does Wolfram Alpha claim that the pole is of order $2$?</p>
| Caran-d'Ache | 66,418 | <p>The order of the pole is the number of elements in the finite (!) principal part of the Laurent series of your function (or the highest negative power).
You have 4 poles (solve the equation $(\cos \left(z^4\right)-1)$: $\left\{\left\{x\to -\sqrt[4]{2 \pi } \left(\sqrt[4]{n}\right)\right\},\left\{x\to -i \sqrt[4]{2 \... |
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