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 |
|---|---|---|---|---|
2,048,282 | <p>I have been doing derivatives but I can't wrap my head around this question for whatever reason. Would appreciate anyone help.
$$g(x) = \tan(x)/e^x$$</p>
| K Split X | 381,431 | <p>You use the quotient rule.</p>
<p>Let <span class="math-container">$f\left(x\right) = \tan \left(x\right)$</span></p>
<p>Let <span class="math-container">$g\left(x\right) = e^x$</span></p>
<p>The quotient rule says that:</p>
<p><span class="math-container">$$\left(\frac{f\left(x\right)}{g\left(x\right)}\right)' ... |
2,048,282 | <p>I have been doing derivatives but I can't wrap my head around this question for whatever reason. Would appreciate anyone help.
$$g(x) = \tan(x)/e^x$$</p>
| Doug M | 317,162 | <p>You are going to hae to apply the product rule / quotient rule at least once. A question is wheter you want to apply it twice</p>
<p>$y = e^{-x} \tan x\\
\frac{dy}{dx} = (\frac {d}{dx} e^{-x}) \tan x + e^{-x}(\frac {d}{dx} \tan x)$</p>
<p>and $\tan x = \frac {\sin x}{\cos x}$</p>
<p>However this is an intersting... |
270,624 | <p>For a polynomial $f(x) = \sum_{i=0}^dc_ix^i \in \mathbb Z[x]$ of degree $d$, let</p>
<p>$$
H(f):=\max\limits_{i=0,1,\ldots, d}\{|c_i|\}
$$</p>
<p>denote the naive height. Further, define</p>
<p>$$
R(M, r, d) := \#\{f(x) \colon \text{$H(f) \leq M$, $\deg f = d$ and $f(x)$ has extactly $r$ real roots}\}.
$$</p>
<p... | Igor Rivin | 11,142 | <p>Liviu's answer is very informative, though it answers a question orthogonal to that of the OP. I believe (Liviu can correct me if I am wrong), the results don't actually depend on the coefficients being discrete, and qualitatively, the results are not much different when the coefficients are uniform centered real ra... |
1,301,509 | <p>I've the following integral, which should result in 1, as shown by the scetch, but in my calculation I get the result 0. What's my mistake?</p>
<p>Sorry the comments are in German and please note that a German 1 often looks like an English 7. Anything in the picture which looks like a 7 to you is in fact a 1.</p>
... | Cameron Williams | 22,551 | <p>The issue is you didn't change your differential. $dz = -dx$ fixes it. Your function is even so you could have simply worked with the integral from $0$ to $1$ instead.</p>
|
3,022,921 | <p>If 6 divides x and 8 divides x how do you deduce 24 divides x</p>
| Peter Szilas | 408,605 | <p>Given:</p>
<p>1)<span class="math-container">$x=6m$</span>, and </p>
<p>2)<span class="math-container">$x=8n$</span>;</p>
<p><span class="math-container">$x=6m=3(2m)$</span>; i .e. <span class="math-container">$3|x.$</span></p>
<p>Using 2): <span class="math-container">$3$</span> divides <span class="math-contai... |
33,005 | <p>Let n and p be any positive integer, make $p$ the subject of the equation: $(3n + p)\bmod4 = 0$. How is it done?</p>
<p>I've worked out that the only values for p are 1, 2, 3 and 0.</p>
<p>This formula is for calculating the amount of padding required in a bitmap's pixel array:</p>
<blockquote>
<p>Padding bytes... | picakhu | 4,728 | <p>It depends on the value of $n$. </p>
<p>If $n \equiv 1$ (mod 4), then, you have $3+p\equiv0$ (mod 4) which means that $p\equiv1$ (mod 4). This procedure can be repeated for other values of $n$, namely $n\equiv0,1,2,3$ (mod 4)</p>
<p>In addition to this, note that if $p\equiv n$ (mod 4), then you have that $3n+p=4n... |
1,757,092 | <p>I want to find an explicit formula for $\sum_{n=0}^\infty n^3x^n$ for $|x|\le1$.Is the idea that first to show that this series is convergent and then we can find the number that it converges to? I tried to use ratio test, but it didn't work. Any suggestion? Thanks!</p>
| DeepSea | 101,504 | <p>For $x = \pm 1$, you can find the value of the series separately and its not that hard. For $|x| < 1$, consider $f(x) = \displaystyle \sum_{n=0}^\infty x^n= \dfrac{1}{1-x}$, then find $xf'(x) = \displaystyle \sum_{n=0}^\infty nx^n= x(1-x)^{-2}$,and repeat this until you get to the desire series.</p>
|
444,448 | <p>Let $M$ be a set of prime numbers of $\mathbb{Q}$ . The limit
$$d(M)= \lim_{s\rightarrow 1^+} \frac{ \sum_{p \in M} p^{-s} }{ - \log(s-1)}$$
Where $p$ is a prime of $\mathbb{Q}$ is called <strong>Dirichlet Density</strong> of $M$. Also, the <strong>Natural density</strong> of $M$ is the limit
$$ \delta(M)= \lim_{x\... | Daniel Fischer | 83,702 | <p>Using $M(x)= \# \{ p \in M : p \leq x \}$, we observe that due to the definition of $M$, we have</p>
<p>$$M(10^k) \leqslant \pi(2\cdot 10^{k-1}), \text{ and } M(2\cdot 10^k) \geqslant \pi(2\cdot 10^k) - \pi(10^k).$$</p>
<p>Thus, supposing $k$ not too small, and using the prime number theorem, we obtain</p>
<p>$$... |
3,265,243 | <p>The number of ways of placing <span class="math-container">$n$</span> objects not in position is given by the inclusion-exclusion number <span class="math-container">$D_n$</span>: </p>
<p><span class="math-container">$n! \left( 1-\dfrac{1}{1!}+\dfrac{1}{2!}+....+(-1)^n\dfrac{1}{n!} \right)$</span></p>
<p>which can... | Martin R | 42,969 | <p>For <span class="math-container">$n=1$</span> both expressions are zero. For <span class="math-container">$n\ge 2$</span> we start with the second expression, shift the index by one, and increase the upper limit (which adds nothing to the sum):
<span class="math-container">$$
(n-1)!\sum_{i=1}^{n-1} (-1)^{i+1}\frac{n... |
4,022,415 | <p>My attempt:</p>
<p><span class="math-container">$\lbrace6,19,30\rbrace$</span> is sufficient to show that two sets are impossible.</p>
<p>Using a computer program with a brute force method I found that separating the numbers <span class="math-container">$1$</span> through <span class="math-container">$85$</span> int... | mathworker21 | 366,088 | <p>Not an answer.</p>
<p>If you want no solutions to <span class="math-container">$a+b=c^2$</span> with <span class="math-container">$a,b,c$</span> all in the same set, then <span class="math-container">$16$</span> sets suffice.</p>
<p>The sets are <span class="math-container">$$A_i := \{n \in \mathbb{N} : n \equiv i \... |
2,113,596 | <p>Questions with likely obvious answers, but I don't have the required intuition to go with the flow.</p>
<p>Consider $a+be^x + ce^{-x} = 0$. To solve it for the constants, we can try out different values of $x$ to get a system of $3$ equations and simply use a calculator (given technique). Why are we allowed to stic... | dxiv | 291,201 | <p>Considering the conversion to <code>C</code> as a function in <code>F</code> $\,\;C(F) = \frac{5}{9}(F-32)\,$, it is easy to see that it's a linear function, and its derivative is a constant $\frac{d\,C}{d\,F} = C' = \frac{5}{9}\,$.</p>
<p>Then, as for any linear function, $\Delta C = C' \,\Delta F\,$, so a change ... |
2,113,596 | <p>Questions with likely obvious answers, but I don't have the required intuition to go with the flow.</p>
<p>Consider $a+be^x + ce^{-x} = 0$. To solve it for the constants, we can try out different values of $x$ to get a system of $3$ equations and simply use a calculator (given technique). Why are we allowed to stic... | John Joy | 140,156 | <p>Hint: Solve for $\Delta C$.</p>
<p>$$C = \frac{5}{9}(F-32)$$
$$C + \Delta C = \frac{5}{9}((F+1)-32)$$</p>
<p>Can you generalize this for a temperature increase of any magnitude?</p>
|
2,801,406 | <blockquote>
<p>Find the coordinates of the points where the line tangent to the curve $$x^2-2xy+2y^2=4$$ is parallel to the $x$-axis, given that $$\frac{dy}{dx}=\frac{y-x}{2y-x}$$</p>
</blockquote>
<p>By letting $dy/dx = 0$ I get $y=x$ which is no help... what do I do?</p>
<p>Thanks</p>
| Christopher Marley | 510,133 | <p>You are definitely on the right track!
Yes, you should get $y=x$. Just plug it into the original equation and solve (because now you have two conditions that the point should satisfy: the equation and its derivative):</p>
<p>$x^2-2x^2+2x^2=4$<br>
$x=\pm 2$ --> $y=\pm2$</p>
<p>Final points: $(-2,-2), (2,2)$</p>
|
1,829,975 | <p>Find three different systems of linear equation whose solutions are $x_1 = 3, x_2 = 0, x_3 = -1$</p>
<p>I'm confused, how exactly can I do this?</p>
| Brian Fitzpatrick | 56,960 | <p>Note that your system is described by the matrix
$$
\left[\begin{array}{rrr|r}
1 & 0 & 0 & 3 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & -1
\end{array}\right]
$$
Performing any row operation on this matrix yields a system with the same solutions. For example, you could add $\DeclareMathOperat... |
2,694,525 | <p>I came across this exercise</p>
<p>$f(x,y)= \lim_{y\to\infty}{{1-y\sin{\pi x\over y}}\over \arctan x}$</p>
<p>The result I get is ${1-\pi x \over \arctan x}$, which depends on the value of $x$.</p>
<p>However, the question I have is that whatever $x$ is, since it's in the $\sin()$, which is a bounded function, sh... | Mohammad Riazi-Kermani | 514,496 | <p>You are finding the limit as y approaches infinity at a fixed point x. Thus $$f(x,y)= \lim_{y\to\infty}{{1-y\sin{\pi x\over y}}\over arctanx }={1-\pi x \over acrtanx}$$ is expected. </p>
<p>Of course the limit should depend on $x$ and it varies with x, but that does not mean that there is something wrong with the... |
4,030,296 | <p>I'm reading Carothers' Real Analysis, and I'm currently looking at <strong>homeomorphisms</strong>. The author says "two intervals that look different, are different" - i.e. they are not homeomorphic. The proof is done for the case <span class="math-container">$(0,1]$</span> and <span class="math-container... | TheSilverDoe | 594,484 | <ol start="0">
<li><p>Continuity still holds when you remove a point, because the restriction of a continuous map is still continuous.</p>
</li>
<li><p>A homeomorphism between <span class="math-container">$(0,1)$</span> and <span class="math-container">$\mathbb{R}$</span> is given by
<span class="math-container">$$x \m... |
238,659 | <p>Let $f : [a, b]\to R$ be a continuous function such that $[a,b] \subset [f(a), f(b)]$. Prove that there exists $x\in [a,b]$ such that $f(x) = x$.</p>
<p>My attempt:
I said let there be a $\delta > 0 $and defined $c$ and $d$ to be $x + \delta$ and $x-\delta$ respectively. From here since $f$ is continuous $[f(c)... | PtF | 49,572 | <p>In $\mathbb R^2$ draw the straight line $y=x$ and consider any interval at the positive part of $x$-axis, you'll get some intuition. Your problem is saying that under some conditions the graph of your function is going to intersept the straight line $y=x$ in some point. Think about the conditions for it to happen..<... |
2,549,690 | <p>Is a direct sum of cyclic groups cyclic? I know every abelian group is a direct sum of cyclic groups of prime power orders, but I can't make use of this.</p>
| user284331 | 284,331 | <p>The map $x\rightarrow x/r$ is bicontinuous and bijective.</p>
|
332,603 | <p>I've passed by this article:
<a href="http://gauravtiwari.org/2011/12/11/claim-for-a-prime-number-formula/" rel="noreferrer">http://gauravtiwari.org/2011/12/11/claim-for-a-prime-number-formula/</a></p>
<p>and this paper:
<a href="http://www.m-hikari.com/ams/ams-2012/ams-73-76-2012/kaddouraAMS73-76-2012.pdf" rel="no... | Michiel | 53,881 | <p>As you already mention yourself: it doesn't make sense to keep on looking for prime numbers with computer algorithms if there is a prime number equation.</p>
<p>Looking at the formulas on the site you provided, it seems to me that the formulas are really just an algorithm which allows you to determine whether some ... |
2,877,578 | <p>Yesterday, I asked the question: <a href="https://math.stackexchange.com/questions/2876740/prove-that-if-a-b-are-closed-then-exists-u-v-open-sets-such-that-u-cap?noredirect=1#comment5938458_2876740">Prove that if $A,B$ are closed then, $ \exists\;U,V$ open sets such that $U\cap V= \emptyset$</a>. </p>
<p>Here is th... | SFeesh | 346,530 | <p>Since $A$ and $B$ are disjoint, we can place an open ball $\mathcal B_a$ at each $a \in A$ such that $B \cap \overline{\mathcal B_a} = \varnothing$ (the bar denotes the closure). Then $\bigcup \mathcal B_a$ is an open neighbourhood of $A$. Because metric spaces are paracompact, we can choose a locally finite subcov... |
22,392 | <p>Consider the following polynomial series:</p>
<p>$S(x) = \sum_{i=1}^{\infty}(-1)^{i+1}x^{i^{2}}$</p>
<p>Between 0 and 1, this looks like a well-behaved function - is there any way to write this function in this interval without using a series?</p>
<p>Given $0 < S(x) < 1$, I need to solve the equation for $x... | S. Carnahan | 121 | <p>I don't know what you mean by "polynomial series" since your function doesn't seem to have much to do with polynomials (perhaps you could elaborate?).</p>
<p>$S(x) = -\frac12 \theta(\frac12, \frac{\log x}{\pi i }) - 1$, where $\theta$ is <a href="http://en.wikipedia.org/wiki/Theta_function" rel="nofollow">Jacobi's ... |
22,392 | <p>Consider the following polynomial series:</p>
<p>$S(x) = \sum_{i=1}^{\infty}(-1)^{i+1}x^{i^{2}}$</p>
<p>Between 0 and 1, this looks like a well-behaved function - is there any way to write this function in this interval without using a series?</p>
<p>Given $0 < S(x) < 1$, I need to solve the equation for $x... | Robin Chapman | 4,213 | <p>As other correspondents have pointed out, this is essentially a theta function.
You ask if you can write it in any other way. You can replace
the infinite series by an infinite product :-) One gets
$$1-2S(x)=\prod_{n=1}^\infty(1-x^{2m-1})^2(1-x^{2m}).$$
The equivalence is a special case of the Jacobi triple product:... |
238,052 | <p>If I have 4 different types of data such (that I get from an Excel file) as:</p>
<pre><code>https://pastebin.com/j3Bgfxqm
</code></pre>
<p>I am trying to implement a <code>Do</code> loop that extracts the data from the Excel file, superimposes the data in two different regions (as done here: <a href="https://mathema... | Daniel Huber | 46,318 | <p>After importing your data, I needed some fixes to convert it to numerical data:</p>
<pre><code>dat = Import["https://pastebin.com/j3Bgfxqm", "Data"][[1]];
dat = ToExpression[ StringCases[#,
"{" ~~ NumberString ~~ "," ~~ NumberString ~~ "}"]] & /@ dat;
</co... |
238,052 | <p>If I have 4 different types of data such (that I get from an Excel file) as:</p>
<pre><code>https://pastebin.com/j3Bgfxqm
</code></pre>
<p>I am trying to implement a <code>Do</code> loop that extracts the data from the Excel file, superimposes the data in two different regions (as done here: <a href="https://mathema... | Hugh | 12,558 | <p>I am sure that someone more expert in coding can simplify this but what I am going to do is to add the equation of a straight line to each curve and then choose values for the slope and intercept of the straight lines to minimise the differences between the curves. This will translate and rotate them to bring them t... |
886,626 | <p>I want to solve the following system of congruences:</p>
<p>$ x \equiv 1 \mod 2 $</p>
<p>$ x \equiv 2 \mod 3 $</p>
<p>$ x \equiv 3 \mod 4 $</p>
<p>$ x \equiv 4 \mod 5 $</p>
<p>$ x \equiv 5 \mod 6 $</p>
<p>$ x \equiv 0 \mod 7 $</p>
<p>I know, but do not understand why, that the first two congruences are redund... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\ x\equiv -1\ $ mod $\,2,3,4,5,6\iff x\equiv -1 \pmod m\ $ for $\, m = {\rm lcm}(2,3,4,5,6) = {\rm lcm}(4,5,6)$</p>
<p>because $\ 2,3,4,5,6\mid x\!+\!1\iff 4,5,6\mid x\!+\!1,\ $ since $\,4,6\mid x\!+\!1\,\Rightarrow\,2,3\mid x\!+\!1$</p>
|
779,042 | <p>Any pointers on how should I start?</p>
<p>$$I:=\int_ {0}^{\infty} \frac {\cos(ax)} {(x^2 + b^2)^n} \ \mathrm{d}x$$</p>
| Cody | 13,295 | <p>One may also do this one using residues, though the resulting series can be rather tedious.</p>
<p>But, it follows a binomial pattern. </p>
<p>I am going to allow the power on the denominator to be $n+1$ instead of $n$ as to find the nth derivative. </p>
<p>Consider $$\int_{C}\frac{e^{iaz}}{(z^{2}+b^{2})^{n+1}}$$... |
1,781,269 | <p>What's the general method to find the slope of a curve at the origin if the derivative at the origin becomes indeterminate. For Eg--</p>
<p>What is the slope of the curve <span class="math-container">$x^3 + y^3= 3axy$</span> at origin and how to find it because after following the process of implicit differentiatio... | Emilio Novati | 187,568 | <p>The curve of equation $x^3+y^3=3axy$ is a <a href="http://mathworld.wolfram.com/FoliumofDescartes.html" rel="nofollow">folium of Descartes</a>, and has a double point at $(x,y)=(0,0)$. So at this point the curve does not have a well defined tangent and, as noted in the answer of Hagen, it has no derivative nor slope... |
2,543,123 | <p>Let $\gcd(a, 11) = 1$. If $3a^7 \equiv 5 \pmod{11}$, show that $a \equiv 3 \pmod{11}$.</p>
<p>My first approach was to use Euler's theorem:</p>
<p>$a^{10} \equiv 1 \pmod{11}$</p>
<p>$3a^7 \equiv 5 \pmod{11}$ implies that $a^{-3} \equiv 9 \pmod{11}$ </p>
<p>I feel i'm not on the right track, hints are appreciate... | Michael Hardy | 11,667 | <p>The possible numbers are $n$ through $6n,$ not $1$ through $6n.$</p>
<p>Does "random" mean uniformly distributed? Not in the usage of probabilists, but many people, including many mathematicians, sometimes use the term that way.</p>
<p>Notice that if $n=2$ and $X$ is the sum of the two resulting numbers, then $\Pr... |
25,239 | <p>I am very new to <em>Mathematica</em> (first time!) and I'm already having troubles with arrays. I basically have a 4D tensor, that is a matrix (call it <code>M</code>) which has for each entry another matrix. I want to select the elements of the nested matrices that are bigger than zero, but <code>Select[M, # > ... | BoLe | 6,555 | <pre><code>m = RandomInteger[{1, 42}, {3, 3, 3, 3}];
</code></pre>
<p><code>Cases</code> will extract every element equal to 42 here. (There won't always be any for the random nature.)</p>
<pre><code>Cases[m, i_Integer /; i == 42, Infinity]
</code></pre>
<p>With <code>Position</code> you can identify where these ele... |
25,239 | <p>I am very new to <em>Mathematica</em> (first time!) and I'm already having troubles with arrays. I basically have a 4D tensor, that is a matrix (call it <code>M</code>) which has for each entry another matrix. I want to select the elements of the nested matrices that are bigger than zero, but <code>Select[M, # > ... | george2079 | 2,079 | <p>If you goal is a matrix of the origial dimensions retaining only the selectred values you could do something like this:</p>
<pre><code> Map[ If[# > 0, #, Null] & , m , {-1}]
</code></pre>
|
69,655 | <p>I'm facing a strange behavior of <code>HoldForm</code>.</p>
<p>I need to display <code>1/2*3/4</code> in LaTeX like this :
$$ \frac{1}{2} \times \frac{3}{4} $$</p>
<p>So I use Mathematica : <code>1/2* 3/4 // HoldForm // TeXForm</code>
BUT I get
$$ \frac{3}{2\ 4} $$</p>
<p>First the writing <code>2 space 4</code>... | Fred Simons | 20,253 | <p>This unexpected behaviour of HoldForm and Hold seems to be due to the function MakeExpression and not a bug in HoldForm or Hold.</p>
<p>Having entered</p>
<pre><code>Hold[1/2 3/4]
</code></pre>
<p>the frontend sends the command</p>
<pre><code>MakeExpression[BoxData[RowBox[{"Hold","[",RowBox[{RowBox[{"1","/","2"}... |
69,655 | <p>I'm facing a strange behavior of <code>HoldForm</code>.</p>
<p>I need to display <code>1/2*3/4</code> in LaTeX like this :
$$ \frac{1}{2} \times \frac{3}{4} $$</p>
<p>So I use Mathematica : <code>1/2* 3/4 // HoldForm // TeXForm</code>
BUT I get
$$ \frac{3}{2\ 4} $$</p>
<p>First the writing <code>2 space 4</code>... | Mr.Wizard | 121 | <p>I am posting a second answer because I am now taking a very different interpretation of your problem. In a comment below my first answer you state:</p>
<blockquote>
<p>Your function seems to correct one type problem with fraction. But I am more looking for something able to display TeX in the exact form I write ... |
734,248 | <p>Example of two open balls such that the one with the smaller radius contains the one with the larger radius.</p>
<p>I cannot find a metric space in which this is true. Looking for hints in the right direction. </p>
| George Kapoulas | 678,810 | <p>Try <span class="math-container">$p$</span>-adic numbers. The range of the metric are powers of <span class="math-container">$p$</span>.</p>
|
911,333 | <p>Show that the series</p>
<p>$$\sum_n \tan\left(\frac{1}{n}\right)$$</p>
<p>diverges.</p>
<p>I dont have any attempt to do, since I am having some troubles with series including geometric functions. I would be glad if I could get a detailed answer, if it is possible.</p>
<p>Thanks!</p>
| JimmyK4542 | 155,509 | <p><strong>Hint</strong>: For $x \in (0,\tfrac{\pi}{2})$ we have $\tan x > x$. Thus, $\tan \dfrac{1}{n} > \dfrac{1}{n}$. Now use the comparison test. </p>
|
911,333 | <p>Show that the series</p>
<p>$$\sum_n \tan\left(\frac{1}{n}\right)$$</p>
<p>diverges.</p>
<p>I dont have any attempt to do, since I am having some troubles with series including geometric functions. I would be glad if I could get a detailed answer, if it is possible.</p>
<p>Thanks!</p>
| davidlowryduda | 9,754 | <p>As $x \to 0$, we have that $\frac{\tan x}{x} \to 1$. So $\tan \frac{1}{n}$ behaves just like $\frac 1n$ as $n$ gets large.</p>
|
3,170,871 | <p>Could anyone please give me a hint on how to compute the following integral?</p>
<p><span class="math-container">$$\int \sqrt{\frac{x-2}{x^7}} \, \mathrm d x$$</span></p>
<p>I'm not required to use hyperbolic/ inverse trigonometric functions.</p>
| peter | 658,932 | <p>Write <span class="math-container">$y(x):=\sqrt{\frac {x-2} {x^7}}$</span>. </p>
<p>Note that <span class="math-container">$$y'(x)= \frac {7-3x}{x^8} \frac 1 y $$</span></p>
<p>Hence <span class="math-container">$$\frac d {dx} x^n y=n x^{n-1} y + x^{n-8} \frac {7-3x} y.$$</span></p>
<p>Do the ansatz <span class=... |
3,977,081 | <p>I’m just a high school student, so I may be somewhat logically flawed in understanding this.</p>
<p>According to wikipedia, the definition of function requires an input <span class="math-container">$x$</span> with its domain <span class="math-container">$X$</span> and an output <span class="math-container">$y$</span... | Timothy Chang | 871,381 | <p>I'm the OP, and I now kind of know what was causing me confusion back then.</p>
<p>I was actually thinking of function in a more computer science way. When we use the notation <span class="math-container">$y = f(x)$</span> in programming languages, it actually functions in the same way as I described in the question... |
4,064,209 | <p><a href="https://i.stack.imgur.com/Ux3cH.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Ux3cH.png" alt="enter image description here" /></a></p>
<p>Above is the exercise. Showing that <span class="math-container">$S$</span> is bounded is straightforward by <span class="math-container">$A$</span> ... | Igor Rivin | 109,865 | <p>Let <span class="math-container">$P(x) = \sum_{i=0}^\infty a_i x^i,$</span> and let <span class="math-container">$Q(x) = \sum_{j=0}^\infty b_j x^j.$</span>
Then, your limit just says that</p>
<p><span class="math-container">$P(1) Q(1) = (PQ)(1).$</span> Notice that for this to make sense, both series have to have ra... |
3,898,411 | <p>How can I prove that <span class="math-container">$ (a_n) = \frac{n^3 -1}{2n^3-n} $</span> converges?</p>
<p>I've calculated the limit and got a result of a 1/2.</p>
<p>Now I need to prove that this limit exists. So, I tried to use the definition and find an <span class="math-container">$M$</span> that <span class="... | player3236 | 435,724 | <p>This is a partial solution concerning <span class="math-container">$m\ne 1$</span>.</p>
<p>First we consider the cases <span class="math-container">$m\ge 2$</span>.</p>
<p>For case 1 (<span class="math-container">$2^m \ge 3^n$</span>), write <span class="math-container">$2^m = 3^n + k^2$</span>.</p>
<p>Taking modulo... |
504,431 | <p>I'm the teaching assistant for a first semester calculus course, and the professor has given the students the following problem:</p>
<blockquote>
<p>Find the points on the curve $xy=\sin(x+y)$ that have a vertical tangent line.</p>
</blockquote>
<p>Here is a picture of the curve:</p>
<p><img src="https://i.stac... | Calvin Lin | 54,563 | <p>(not a complete solution, but I wanted to add a graph)<br>
I suspect that it's a much more difficult question than intended.</p>
<hr>
<p>By implicit differentiation, $x \frac{dy}{dx} + y = \cos( x+y) ( 1 + \frac{dy}{dx} ) $.<br>
This gives $\frac{dy}{dx} = \frac{-y + \cos(x+y) } { x - \cos (x+y) }$.<br>
We want a ... |
2,894,376 | <blockquote>
<p>$2$ different History books, $3$ different Geography books and $2$ different Science books are placed on a book shelf. How many different ways can they be arranged? How many ways can they be arranged if books of the same subject must be placed together?</p>
</blockquote>
<p>For the first part of the ... | David G. Stork | 210,401 | <p>If the books of the same subject must be placed together, there are in essence three "packs," and these can be ordered in just $3! = 6$ ways, where I assume that the order <em>within</em> a pack is irrelevant. If that order is not irrelevant, you then have $3!=6$ ways to arrange the packs, then within the associate... |
275,820 | <p>I have a question concerning the definition of the square root of bounded linear operators. To introduce some notation: tr denotes the trace of linear operators and $\mathcal{L}(H)$ denotes the set of bounded linear operators, from H to H, where H symbolizes a Hilbert space. L' stands for the adjoint operator of L.
... | tomasz | 30,222 | <p>Notice that $LL'$ is positive definite. Square root is well-defined for bounded, positive definite operators (by functional calculus theorems).</p>
|
149,161 | <p>A finite simplicial set is a simplicial set having only a finite number of non degenerate simplicies. It is not hard to show that every finite simplicial set has only a finite number of simplicies in each degree. My question is: does the converse hold? that is, is every simplicial set, having a finite number of simp... | Benjamin Steinberg | 15,934 | <p>Let G be a finite group viewed as a one object category. Then the nerve BG is a simplicial set with finitely many simplices in each dimension but it is not finite.</p>
|
4,011,581 | <p>I know that a function is continuous at a point if the limit from left and right side exists and are equal and for a function to be continuous, the function should be continuous at all points. My question is that if I want to check continuity of a function, I cannot practically check continuity at each and every poi... | Tito Eliatron | 84,972 | <p>Well, that is not the rigurous definition of continuity but it works for most UNDERGRADUATE functions.</p>
<p>Some tricks to check continuity:</p>
<ul>
<li>You may know that elementary fucntions (<span class="math-container">$\sin$</span>, <span class="math-container">$\cos$</span>, <span class="math-container">$\ex... |
4,011,581 | <p>I know that a function is continuous at a point if the limit from left and right side exists and are equal and for a function to be continuous, the function should be continuous at all points. My question is that if I want to check continuity of a function, I cannot practically check continuity at each and every poi... | Jonas Linssen | 598,157 | <p>There are several „methods“ to check continuity of a function <span class="math-container">$f:\Bbb R \longrightarrow \mathbb{R}$</span>:</p>
<ul>
<li>show that given an arbitrary point <span class="math-container">$x$</span> and any sequence <span class="math-container">$x_n \rightarrow x$</span> converging to <span... |
198,132 | <p>Putting the equation $x^2 - x \sin(x) - \cos (x)$ into Wolfram Alpha, I am surprised that it has a nice <a href="http://www.wolframalpha.com/input/?i=roots%20of%20x%5E2%20-%20x%20sinx%20-%20cos%20x" rel="nofollow">parabolic shape</a>. Also, it has two complex roots.</p>
<p><strong>Question</strong></p>
<p>Is it po... | N. S. | 9,176 | <p>$f(0)=-1$ and $f(2)=4-2\sin(2)-\cos(2)\geq 4-2-1=1 \,.$</p>
<p>Then, by IVT it has a root between $0$ and $2$. Similarly, it has a second root between $-2$ and $0$. </p>
|
1,276,264 | <p>So, I was wondering if it is possible to solve for $n$ in $2^n=8$ (or any other question where $n$ is a power) using $9^{th}$ grade math. Please excuse my naïveté if this is extremely stupid/simple. </p>
<p>Thanks so much in advance! –– come to think of it: Is it possible at all?</p>
| Mathmo123 | 154,802 | <p>The best way to solve problems like this is using <em>logarithms</em>. If you have an equation
$$10 ^x = y$$
where $y$ is any positive number, and you wish to find $x$, then the value of $x$ will be (by definition) $\log y$ (this is what the $\log$ button on your calculator is for). </p>
<p>For example, to solve
... |
529,053 | <p>I have already proved that if ${X_k}$ converges to a limit $L$, then any subsequence of it also converges to $L$. And now the question asks to show that if ${X_k}$ has two subsequence which converge to two different limits, then ${X_k}$ can not be convergent.</p>
| Hagen von Eitzen | 39,174 | <p><em>Hint:</em> Assume the sequnce converges to some limit $L$. What can be said about the so-called different limits of the subsequences.</p>
|
529,053 | <p>I have already proved that if ${X_k}$ converges to a limit $L$, then any subsequence of it also converges to $L$. And now the question asks to show that if ${X_k}$ has two subsequence which converge to two different limits, then ${X_k}$ can not be convergent.</p>
| tylerc0816 | 53,243 | <p>A sequence $\{ x_n \}$ converges to $L$ if and only if <em>every</em> subsequence of $\{ x_n \}$ converges to $L$. Therefore, if there exists two subsequences $\{ x_{n_k} \}$ and $\{ x_{n_l} \}$ converging to two different limits $L'$ and $L''$, then $\{ x_n \}$ cannot be convergent.</p>
|
135,012 | <p>How to prove (or to disprove) that all the roots of the polynomial of degree $n$ $$\sum_{k=0}^{k=n}(2k+1)x^k$$ belong to the disk $\{z:|z|<1\}?$ Numerical calculations confirm that, but I don't see any approach to a proof of so simply formulated statement. It would be useful in connection with an irreducibility p... | Chris Godsil | 1,266 | <p>The standard approach to this type of question is to use the Schur-Cohn procedure.</p>
|
83,607 | <p>While solving the heat equation in one spatial variable $u_t = u_{xx} $ (x goes from 0 to L) with the initial temperature distribution $T_0 \frac{x(L-x)}{L^2}$ , and with neumann boundary conditions $u_x(0,t) = u_x(L,t) = 0$, I got some really weird behaviour from NDSolve.</p>
<p>My code looks like this:</p>
<pre... | Michael E2 | 4,999 | <p>You can use the finite element method with the method of lines as <a href="https://mathematica.stackexchange.com/users/120/toadatrix">@toadatrix</a> suggested, but for the FEM method to work, you need to do a little more. The Neumann boundary conditions need to be specified using <a href="http://reference.wolfram.c... |
878,517 | <p>Is there any more solutions to this functional equation $f(f(x))=x$?</p>
<p>I have found: $f(x)=C-x$ and $f(x)=\frac{C}{x}$.</p>
| Micah | 30,836 | <p>If you don't make any niceness assumptions about $f$, there are lots. Partition $\Bbb{R}$ (or whatever you want $f$'s domain to be) into $1$- and $2$-element subsets, in any way you like. Then define $f(x)=y$, if $\{x, y\}$ is in your partition, or $f(x)=x$, if $\{x\}$ is in your partition.</p>
<p>Moreover, any suc... |
878,517 | <p>Is there any more solutions to this functional equation $f(f(x))=x$?</p>
<p>I have found: $f(x)=C-x$ and $f(x)=\frac{C}{x}$.</p>
| Michael Albanese | 39,599 | <p>This answer will only deal with continuous maps.</p>
<hr>
<p>A map $f : X \to X$ such that $f \circ f = \operatorname{id}_X$ is called an <em>involution</em> of $X$. Two involutions of $X$, $f$ and $g$, are said to be <em>equivalent</em> if there is a self-homeomorphism $h$ such that $f\circ h = h\circ g$ (or writ... |
123,202 | <blockquote>
<p>Let $X,Y$ be vectors in $\mathbb{C}^n$, and assume that $X\ne0$. Prove
that there is a symmetric matrix $B$ such that $BX=Y$.</p>
</blockquote>
<p>This is an exercise from a chapter about bilinear forms. So the intended solution should be somehow related to it.</p>
<p>Pre-multiplying both sides by... | bfhaha | 128,942 | <p>My solution don't use bilinear form.</p>
<p>Suppose that
$$X=\vec{A}+\vec{B}i=
\left(
\begin{array}{c}
a_1+b_1 i \\
a_2+b_2 i \\
\vdots \\
a_n+b_n i \\
\end{array}
\right),
Y=\vec{C}+\vec{D}i=
\left(
\begin{array}{c}
c_1+d_1 i \\
c_2+d_2 i \\
\vdots \\
c_n+d_n i \\
\end{array... |
2,036,943 | <p>any riemann integrable function is a smooth piecewise function?</p>
<p>Is true for fundamental theorem calculus?</p>
| Marko Karbevski | 45,470 | <p>No, and here are a few examples:</p>
<ul>
<li><p>The indicator function of the Cantor set</p></li>
<li><p>Any nowhere differentiable continuous function. As they are locally bounded and continuous, they are integrable on any compact interval</p></li>
<li><p>As @user284331 and @ Henry W. pointed out, you can look up... |
1,279,564 | <p>I try to be rational and keep my questions as impersonal as I can in order to comply to the community guidelines. But this one is making me <strong>mad</strong>. Here it goes.
Consider the uniform distribution on $[0, \theta]$. The likelihood function, using a random sample of size $n$ is
$\frac{1}{\theta^{n}}$.<b... | Seyhmus Güngören | 29,940 | <p>What you are doing is wrong. You must find the likelihood function. What you found is $1/\theta^n$? so where is it defined? It is true that $X_n$ is the maximum likelihood estimator because it maximizes the true likelihood function. How do you find it?</p>
<p><strong>Added</strong>: Your answer is actually on the r... |
1,085,702 | <p>It's said that a computer program "prints" a set <span class="math-container">$A$</span> (<span class="math-container">$A \subseteq \mathbb N$</span>, positive integers.) if it prints every element of <span class="math-container">$A$</span> in ascending order (even if <span class="math-container">$A$</span... | Xoff | 36,246 | <p>An explicit solution in your system could be :</p>
<ol>
<li>Consider an enumeration of your programs ($P_0$ is the first program, $P_1$ the second, and so on)</li>
<li>Let denote by $P_{\! n}(m)$ the m$^\mbox{th}$ number printed by program $P_{\!n}$ (if it exists)</li>
<li>Consider the integers $i_n$ defined by
$$... |
1,531,646 | <p>Find the following limit</p>
<p>$$
\lim_{x\to0}\left(\frac{1+x2^x}{1+x3^x}\right)^\frac1{x^2}
$$</p>
<p>I have used natural logarithm to get</p>
<p>$$
\exp\lim_{x\to0}\frac1{x^2}\ln\left(\frac{1+x2^x}{1+x3^x}\right)
$$</p>
<p>After this, I have tried l'opital's rule but I was unable to get it to a simplified for... | Michael Medvinsky | 269,041 | <p>Note that $\lim\limits_{x\to0} xn^x=0$.
Next, use the L'Hospital's rule twice to get
$$\lim_{x\to0}\frac{\ln \left({1+xn^x}\right)}{x^2}=
\lim_{x\to0}\frac{\frac{x n^x \ln ^2n+2 n^x \ln n}{x n^x+1}-\frac{\left(n^x+x n^x \ln n\right)^2}{\left(x n^x+1\right)^2}}{2}
=\frac{2 \ln n-1}{2}
$$
Now
$$\lim_{x\to0}\frac{\ln\... |
3,567,662 | <p>I have just learned how to convert a plane in R3 from Cartesian to parametric form, by setting 2 variables to 0 and solving for the 3rd one in order to obtain 3 points on the plane, and solve from there. However, this does not work when 1 or 2 of the variables are 0, as it is not possible to find 3 points on the pla... | Oliver Kayende | 704,766 | <p>Another way without the corollary but intuitive. By the fundamental theorem of arithmetic <span class="math-container">$(x,y)\mapsto 2^{x'}3^{y'}$</span> defines an injection from <span class="math-container">$X\times Y$</span> into <span class="math-container">$\Bbb N$</span> given bijections <span class="math-cont... |
1,761,857 | <p>I need to show that, for $f:X\to \mathbb{R}$ bounded, we have:</p>
<p>$$\sup\{|f(x)-f(y)|, x,y\in X\}= \sup f - \inf f$$</p>
<p>Well, I know that </p>
<p>$$\sup\{|f(x)-f(y)|, x,y\in X\}\ge |f(x)-f(y)|$$ but in what this helps? I really have no idea in how to prove this one</p>
| triple_sec | 87,778 | <p>Fix any $x_0\in X$ and $y_0\in X$. Then, one has that $$f(x_0)-f(y_0)\leq|f(x_0)-f(y_0)|\leq\sup_{x,y\in X}\{|f(x)-f(y)|\},$$ since every real number is less than or equal to its absolute value. Rearranging yields: $$f(x_0)\leq\sup_{x,y\in X}\{|f(x)-f(y)|\}+f(y_0).$$ This is true for any $x_0\in X$, so taking suprem... |
4,438,512 | <p>Recently I learned about dividing and forking of formula / partial types:</p>
<p>We say that <span class="math-container">$φ(x,b)$</span> divides over <span class="math-container">$C$</span> (where <span class="math-container">$b$</span> in the monster and <span class="math-container">$C$</span> is small) if there e... | Primo Petri | 137,248 | <p>Non-forking and invariance (a notion easier to grasp) coincide in a large class of theories.</p>
<p>Below I elaborate with a model <span class="math-container">$M$</span> instead of a set <span class="math-container">$C$</span> to avoid some technical issues (b.t.w., these technical issues are quite interesting, jus... |
232,276 | <p>I can prove with the triangle inequality that the unit sphere in $R^n$ is convex, but how to show that it is strictly convex?</p>
| agt | 6,752 | <p>I will show that if $(E,\langle\dot,\dot\rangle)$ is an inner product space, then the normed space $(E,\|\dot\|)$ is strictly convex, where $\|x\|:=\sqrt{\langle x,x\rangle}.$</p>
<p>Take any two point $x,y\in E,$ with $\|x\|=\|y\|=1$ and $x\neq y.$
Then for any $0<\alpha<1,$ we have $\|\alpha x+(1-\alpha) y\... |
2,135,228 | <p>Find</p>
<p>(a) $P\{A \cup B\}$</p>
<p>(b) $P\{A^c\}$</p>
<p>(c) $P\{A^c \cap B\}$</p>
<p>This is what I have right now:</p>
<p>(a) $P\{A \cup B\}=0.4+0.5=0.90$</p>
<p>(b) $P\{A^c\}= 1-0.4=0.60$</p>
<p>(c) $P\{A^c \cap B\}= (0.6)\cdot(0.5)=0.30$</p>
<p>Am I doing it correctly?</p>
| MPW | 113,214 | <p><strong>Hint:</strong> The first one is correct. Now what happens when you change the association by moving the grouping symbols to the pair in the other $\oplus$? For associativity to hold, the results must be equal for all choices of $u,v,w$. Is that the case?</p>
|
600,097 | <p>I am stuck on the following problem from an exercise in my analysis book: </p>
<blockquote>
<p>Show that $$\int_0^4 x \mathrm d(x-[x])=-2$$ where $[x]$ is the greatest integer not exceeding $x$. </p>
</blockquote>
<p>I think I have to partition the interval $[0,4]$ into some suitable subintervals and here I see ... | GEdgar | 442 | <p>What happens if you try partitions like
$$
0,1-\delta,1,2-\delta,2,3-\delta,3,4-\delta,4
$$
where $\delta>0$ is very small?</p>
|
600,097 | <p>I am stuck on the following problem from an exercise in my analysis book: </p>
<blockquote>
<p>Show that $$\int_0^4 x \mathrm d(x-[x])=-2$$ where $[x]$ is the greatest integer not exceeding $x$. </p>
</blockquote>
<p>I think I have to partition the interval $[0,4]$ into some suitable subintervals and here I see ... | DonAntonio | 31,254 | <p>Using <a href="http://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral" rel="nofollow">integration by parts for the Riemann-Stieltjes integral</a>, we get:</p>
<p>$$\int\limits_0^4xd(x-\lfloor x\rfloor)=\left.x(x-\lfloor x\rfloor)\right|_{x=4}-\left.x(x-\lfloor x\rfloor)\right|_{x=0}-\int\limits_0^4(x-\lflo... |
4,007,987 | <p>So define a polynomial <span class="math-container">$P(x) = 4x^3 + 4x - 5 = 0$</span>, whose roots are <span class="math-container">$a, b $</span> and <span class="math-container">$c$</span>. Evaluate the value of <span class="math-container">$(b+c-3a)(a+b-3c)(c+a-3b)$</span></p>
<p>Now tried this in two ways (both... | Mark Bennet | 2,906 | <p>Hint: <span class="math-container">$b+c-3a = (a+b+c)-4a = -4a$</span> and everything simplifies easily.</p>
|
738,455 | <p>Sources: <a href="https://rads.stackoverflow.com/amzn/click/0495011665" rel="nofollow noreferrer"><em>Calculus: Early Transcendentals</em> (6 edn 2007)</a>. p. 890, Section 14.3. Exercise 50b, c.. </p>
<blockquote>
<p><img src="https://i.stack.imgur.com/8ggG2.png" alt="enter image description here"></p>
</blo... | Hagen von Eitzen | 39,174 | <p>The function $f$ depends on only one variable - there's no comma between the parentheses. So for the given function $t\mapsto f(t)$, there is a single derivative $t\mapsto f'(t)=\lim_{h\to0}\frac{f(t+h)-f(t)}{h}$. Then $f'(xy)$ is just obtained by pugging $t=xy$ into $f'(t)$. This should not be mixed with the partia... |
2,965,193 | <p>Basically the question is asking us to prove that given any integers <span class="math-container">$$x_1,x_2,x_3,x_4,x_5$$</span> Prove that 3 of the integers from the set above, suppose <span class="math-container">$$x_a,x_b,x_c$$</span> satisfy this equation: <span class="math-container">$$x_a^2 + x_b^2 + x_c^2 = 3... | SQB | 106,234 | <p>Any integer is of one of the following forms:</p>
<ul>
<li><span class="math-container">$3k + 0$</span> <em>(these are the multiples of 3)</em></li>
<li><span class="math-container">$3k + 1$</span></li>
<li><span class="math-container">$3k + 2$</span></li>
</ul>
<p>where <span class="math-container">$k$</span> is ... |
3,699,645 | <p>In the evaluation of <span class="math-container">$$\sum_{k=1}^\infty \sum_{\ell=1}^{k-1}\sum_{m=1}^{\ell-1}\frac{\delta_{k, 2\ell-2m}}{m\left(\ell-m\right)\left(k-\ell\right)}.$$</span>
Here <span class="math-container">$\delta_{k, 2\ell-2m}$</span> denotes the "Kronecker delta" (see <a href="https://en.wikipedia.o... | Varun Vejalla | 595,055 | <p>You did not reduce the sum properly. For a given even <span class="math-container">$k$</span>, the set of <span class="math-container">$m$</span> that satisfies <span class="math-container">$k = 2\ell - 2m, 1 \le \ell \le k-1, 1 \le m \le \ell-1$</span> is the positive natural numbers less than or equal to <span cla... |
3,699,645 | <p>In the evaluation of <span class="math-container">$$\sum_{k=1}^\infty \sum_{\ell=1}^{k-1}\sum_{m=1}^{\ell-1}\frac{\delta_{k, 2\ell-2m}}{m\left(\ell-m\right)\left(k-\ell\right)}.$$</span>
Here <span class="math-container">$\delta_{k, 2\ell-2m}$</span> denotes the "Kronecker delta" (see <a href="https://en.wikipedia.o... | xpaul | 66,420 | <p>It is not a complete answer. Let
<span class="math-container">\begin{eqnarray}
f(x)&=&\sum_{m=1}^\infty \frac{H_{m-\frac32}}{m^2}x^m
\end{eqnarray}</span>
and then
<span class="math-container">\begin{eqnarray}
f'(x)&=&\sum_{m=1}^\infty \frac{H_{m-\frac32}}{m}x^{m-1}\\
(xf'(x))'&=&\sum_{m=1}^\... |
15,033 | <p>I have one incident edges and multiple outgoing Edges, for which I want to pick an outgoing edge such that the angles between the outgoing edge and the incoming edge is the smallest of all. We know the coordinates for the vertex $V$ .</p>
<p>The angle must start from the incoming edge ($e_1$) and ends at another ed... | Eric Beaudoin | 5,292 | <p>If you are trying to put this into code you may want to consider the function atan2(y,x), it is available in most languages. Note that it takes the y coordinate as the first parameter. Since atan2 takes two parameter it can figure out in which quadrant your vector lies, and it can output the angle in the range $-\pi... |
4,544,450 | <p>im trying to solve this logical equation</p>
<p>p≡((p∧∼q)→q)→p
i know i have to solve for the right side and im pretty certain the final step must be the absorption law 10). But the ∼q is bugging me</p>
| Hongleng Fu | 557,275 | <p>One key technique here is to use other sentence to rephrase <span class="math-container">$a \to b$</span>.</p>
<p>It is known that <span class="math-container">$a \to b$</span> is false only if <span class="math-container">$a$</span> is true and <span class="math-container">$b$</span> is false.</p>
<p>Using the tabl... |
242,203 | <p>What's the derivative of the integral $$\int_1^x\sin(t) dt$$</p>
<p>Any ideas? I'm getting a little confused.</p>
| Ormi | 49,301 | <p>$ \frac{d}{dt}\int_1^x\sin(t)dt = \frac{d}{dt} [-\cos t]_1^x = \frac{d}{dt}[-\cos x+\cos(1)] = \sin x $</p>
|
2,164,994 | <p>Is the ratio test for convergence applicable to the below series:</p>
<p>$$\sum_{n=1}^\infty \frac{n^3+1}{\sqrt[3]{n^{10} + n}}$$</p>
<p>I already know that the series diverge. I want to confirm if the ratio test is applicable or not?</p>
| DeepSea | 101,504 | <p><strong>hint</strong>: Compare your series with $\displaystyle \sum_{n = 1}^{\infty} \dfrac{1}{n^{\frac{1}{3}}}$, which diverges.</p>
|
2,164,994 | <p>Is the ratio test for convergence applicable to the below series:</p>
<p>$$\sum_{n=1}^\infty \frac{n^3+1}{\sqrt[3]{n^{10} + n}}$$</p>
<p>I already know that the series diverge. I want to confirm if the ratio test is applicable or not?</p>
| marwalix | 441 | <p>Let's compute the ratio</p>
<p>$${a_{n+1}\over a_n}={(n+1)^3+1\over n^3+1}\cdot {\sqrt[3]{(n+1)^{10}+n+1}\over \sqrt[3]{n^{10}+n}}\sim{n^{1\over 3}\over(n+1)^{1\over3}}\to 1$$</p>
<p>We cannot conclude with the ratio test</p>
|
664,152 | <p>I have a homework problem that I'm very stuck on. The problem statement is as follows:</p>
<p>"Suppose that $X$ is a metric space, and that for any sets $E,F \subseteq X$, if dist$(E,F) > 0$ then $\mu^*(E \cup F) = \mu^*(E) + \mu^*(F)$. Prove that every open set is a splitting set. (Recall that the distance b... | phaiakia | 81,511 | <p>Felt like I should come back and add the correct answer. Let $U$ be an open set in $X$ and consider $U^c$. For each $n$, let $U_n = \{ x \in U : d(x,y) > 1/n \; \forall y \in U^c \}$. Then $\mu^*(E \cap (U_n \cup U^c)) = \mu^*(E \cap U_n) + \mu^*(E \cap U^c)$. Then taking the limit of both sides, continuity f... |
788,245 | <p>$$\sum_{n=1}^{\infty}\frac{(n+2)!}{(3n-1)}$$ I know this series does not converge. Can someone show me how to prove that? Should i use criteria of Dalamber or any other criteria?</p>
| Santosh Linkha | 2,199 | <p>The part of the series itself diverges. Use integral test to check it out. Forget about the whole series.
$$\sum_{n=1}^{\infty}\frac{1}{(3n-1)} \le \sum_{n=1}^{\infty}\frac{(n+2)!}{(3n-1)}$$
also note that the first <strong>necessity</strong> for convergence of series is that it's limit $n\to\infty$ must be zero. i.... |
3,773,933 | <p>Question goes like this:</p>
<p>In a box containing <span class="math-container">$36$</span> strawberries, <span class="math-container">$2$</span> of them are rotten. Kyle randomly picked <span class="math-container">$5$</span> of these strawberries.<br>
a. What is the probability of having at least 1 rotten strawbe... | N. F. Taussig | 173,070 | <blockquote>
<p>In a box containing <span class="math-container">$36$</span> strawberries, <span class="math-container">$2$</span> of them are rotten. Kyle randomly picked <span class="math-container">$5$</span> of these strawberries. What is the probability of at least one rotten strawberry among the five?</p>
</blo... |
880,928 | <p>$A,B,C,D,E,F,G$</p>
<p>A list consists of all possible three-letter arrangements formed by using the letters above such that the first letter is $D$ and one of the remaining letters is $A$. If no letter is used more than once in an arrangement in the list and one three-letter arrangement is randomly selected from t... | Asimov | 137,446 | <p>Well, if the first letter is D, we don't have to worry about the first choice.</p>
<p>The second letter has 6 possibilities</p>
<p>If it is A, then the last letter has 5 possibilities</p>
<p>If it is not A (which there are 5 letters that aren't A), then the last letter must be A, so for each of them, there is onl... |
371,999 | <p>We are trying to estimate the cardinality $K(n,p)$ of so-called Kuratowski monoid with $p$ positive and $n$ negative linearly ordered idempotent generators. In particular, we are interesting in the case $n=p$.</p>
<p>This notion arose in an intersection of general topology and abstract algebra. We calculated the ex... | Qiaochu Yuan | 232 | <p>The argument you've written down shows that </p>
<p>$$K(n, n) \le \sum_{k=0}^{2n} {2k \choose k}.$$</p>
<p>Stirling's formula gives the asymptotic</p>
<p>$${2k \choose k} \sim \frac{4^k}{\sqrt{\pi k}}$$</p>
<p>so this sum behaves approximately like a geometric series, and we expect that</p>
<p>$$\sum_{k=0}^{2n}... |
371,999 | <p>We are trying to estimate the cardinality $K(n,p)$ of so-called Kuratowski monoid with $p$ positive and $n$ negative linearly ordered idempotent generators. In particular, we are interesting in the case $n=p$.</p>
<p>This notion arose in an intersection of general topology and abstract algebra. We calculated the ex... | Alex Ravsky | 71,850 | <p><strong>Theorem</strong>. $\lim_{n\to\infty} K(n,n)/\binom {2n}n^2=16/9$.</p>
<p>This theorem is a corollary of the following results.</p>
<hr>
<p>For every integers $0\le a,b\le n$ put $c_{a,b}(n)=\binom {2n-a-b}{n-a}/\binom {2n}n$.</p>
<p><strong>Lemma 1</strong>. For every integers $a,b\ge 0$ there exists a l... |
2,332,741 | <p>I have the following problem:</p>
<blockquote>
<p>Let $\Omega \subset \mathbb{R}^3$ be an open bounded set with a smooth boundary $\partial \Omega$ and the unit normal $v$. Calculate for the vector field $a(x,y,z)=(0,0,-pz)$ with $p>0$ the value of -$\int_{\partial\Omega}\langle a,v\rangle d\mu_{\partial\Omega... | Gabriel Soranzo | 51,400 | <p>As the surface <span class="math-container">$X$</span> is supposed to be projective there exist an ample divisor <span class="math-container">$H$</span> on <span class="math-container">$X$</span>. Following the same reasoning as the begining of the proof of V.1.1 the exist an integer <span class="math-container">$n&... |
1,028,695 | <p>While reading though some engineering literature, I came across some logic that I found a bit strange. Mathematically, the statement might look something like this: </p>
<p>I have a linear operator $A:L^2(\Bbb{R}^3)\rightarrow L^2(\Bbb{R}^4)$, that is a mapping which takes functions of three variables to functions... | fodon | 64,111 | <p>I think it always has to be true. It is like mapping points in a flat plane to a 3D space. You get a volume 0 manifold in the 3D space if there is a 1 to 1 mapping.</p>
|
4,241,919 | <p>Let <span class="math-container">$x, y, z$</span> are three <span class="math-container">$n\times 1$</span> vectors. For each vector, every element is between 0 and 1, and the sum of all elements in each vector is 1. Now I am wonder why the following inquality holds:</p>
<p><span class="math-container">$x^Ty+y^Tz-x^... | blundered_bishop | 508,406 | <p>Note that your conditions imply <span class="math-container">$x \cdot x \le1$</span> for every <span class="math-container">$x$</span>.
From <span class="math-container">$(x-y) \cdot (x-y) \ge0$</span> expand to get</p>
<p><span class="math-container">$$2x\cdot y \le x\cdot x + y\cdot y\le2 \Rightarrow x\cdot y \le ... |
4,241,919 | <p>Let <span class="math-container">$x, y, z$</span> are three <span class="math-container">$n\times 1$</span> vectors. For each vector, every element is between 0 and 1, and the sum of all elements in each vector is 1. Now I am wonder why the following inquality holds:</p>
<p><span class="math-container">$x^Ty+y^Tz-x^... | Calvin Lin | 54,563 | <p><strong>Hint:</strong> Show that for real numbers <span class="math-container">$ a, b, c \in [0, 1 ]$</span>, we have</p>
<p><span class="math-container">$$ b ( a + c ) - ac \leq b. $$</span></p>
<blockquote class="spoiler">
<p> The expression is linear in each variable, so we just have to check the 8 end points.<... |
2,572,032 | <p>I'm looking for help with <strong>(b)</strong> and <strong>(c)</strong> specifically. I'm posting <strong>(a)</strong> for completeness.</p>
<p><strong>(a)</strong> Show convergence for $a_n=\sqrt{n+1}-\sqrt{n}$ towards $0$ and test $\sqrt{n}a_n$ for convergence.</p>
<p><strong>(b)</strong> Show $b_n=\sqrt[k]{n+1}... | José Carlos Santos | 446,262 | <p>In order to solve (b), let $a=\sqrt[k]{n+1}$ and $b=\sqrt[k]n$. Then\begin{align}\sqrt[k]{n+1}-\sqrt[k]n&=a-b\\&=\frac{(a-b)(a^{k-1}+a^{k-2}b+\cdots+ab^{k-2}+b^{k-1})}{a^{k-1}+a^{k-2}b+\cdots+ab^{k-2}+b^{k-1}}\\&=\frac{a^k-b^k}{a^{k-1}+a^{k-2}b+\cdots+ab^{k-2}+b^{k-1}}\\&=\frac1{\sqrt[k]{n+1}^{k-1}+\... |
2,572,032 | <p>I'm looking for help with <strong>(b)</strong> and <strong>(c)</strong> specifically. I'm posting <strong>(a)</strong> for completeness.</p>
<p><strong>(a)</strong> Show convergence for $a_n=\sqrt{n+1}-\sqrt{n}$ towards $0$ and test $\sqrt{n}a_n$ for convergence.</p>
<p><strong>(b)</strong> Show $b_n=\sqrt[k]{n+1}... | rtybase | 22,583 | <p><strong>Hint:</strong> Use the fact that $a^k-b^k=(a-b)(a^{k-1}+a^{k-2}b+...+ab^{k-2}+b^{k-1})$ where $a=\sqrt[k]{n+1}$ and $b=\sqrt[k]{n}$
or
$$0<a-b=\frac{a^k-b^k}{(a^{k-1}+a^{k-2}b+...+ab^{k-2}+b^{k-1})}<\frac{1}{kb^{k-1}}=\frac{1}{k\sqrt[k]{n^{k-1}}}$$</p>
|
2,741,229 | <p>I have searched a lot, but i haven't found any proof about that statement. I have checked the proof of</p>
<blockquote>
<p>If <span class="math-container">$f$</span> is differentiable, then <span class="math-container">$f$</span> is continuous</p>
</blockquote>
<p>but it's not the same argument I think. Also, I ... | Steven Alexis Gregory | 75,410 | <p>$f'$ need not be continuous.</p>
<p>Suppose that $f'(x)$ exists in the interval $(a,b)$. If $\xi \in (a,b)$, then $f'(\xi)$ exists. Hence $f$ is continuous at $\xi$. Since this is true for all $\xi$ in $(a,b)$, then $f$ is continuous on $(a,b)$.</p>
|
129,132 | <p>Both the ratio test and the root test define a number (via a limit).</p>
<p>If both limits exist (and shows that the series is convergent), what (if any) is the relation between the 2 numbers ? are they equal ?
What is the relation (if any) between them and the original series (other than the fact that they say th... | fny | 28,533 | <p>Both tests will yield the same property about the series:</p>
<ol>
<li>if $L<1$ the series is absolutely convergent</li>
<li>if $L>1$ the series is divergent.</li>
<li>if $L=1$ the series may be divergent, conditionally convergent, absolutely convergent</li>
</ol>
<p>Edit: Note that if L=1 in the ratio test,... |
3,293,955 | <p>Here is the curve i wish to plot with a function:</p>
<p><a href="https://i.stack.imgur.com/4Smib.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/4Smib.png" alt="eliptical curve"></a></p>
<p>I expect the curve to be 1/4 of an elipse but I only have the coordinates to work with (minx,miny and max... | David C. Ullrich | 248,223 | <p>This immediate from two facts:</p>
<p>(i) He might have been more explicit about the definition; in fact if <span class="math-container">$f\in L^1$</span> the maximal function is <span class="math-container">$$Mf(x)=\sup_{r>0}\frac 1{m(B(x,r))}\int_{B(x,r)}|f|.$$</span></p>
<p>(ii) If <span class="math-containe... |
2,794,962 | <p>Give is: $C$ which is a closed curve which forms the surface $\Sigma$., $\vec{v} $ which is a constant vector. </p>
<p>I should prove the following expression without using Stokes' Theorem: </p>
<p>$$\oint_C \vec{v} \cdot d\vec{l} = 0$$</p>
<p>How do I go about doing it for an arbitrarily closed (even overlappin... | Arnaud Mortier | 480,423 | <p>Call $X$ the number of times your event occurs. Assume first that the number of occurrences is theoretically unlimited. Then $X+1$ follows a <a href="https://en.wikipedia.org/wiki/Geometric_distribution" rel="nofollow noreferrer">geometric distribution</a> of parameter $1-p$: $$\Bbb P(X+1=n)=p^{n-1}(1-p)\qquad \text... |
1,652,297 | <p><strong>Thm</strong>
Let $V$ and $W$ be Vector spaces and let $T:V \to W$ be linear </p>
<p>If $\beta = \{ v_1,\dots ,v_n \}$ is a basis for $V$
then $$ R(T)=\text{span}(T(\beta))=\text{span}(\{ T(v_1),\dots,T(v_n) \} ) $$</p>
<hr>
<p><strong>Dimension Theorem</strong> </p>
<p>Let $V$ and $W$ be Vector spac... | Alex Wertheim | 73,817 | <p>Let's look at your specific claim: that if $\dim V$ is $2$, then are $2$ vectors in the basis of $V$ (call them $v_{1}, v_{2}$) and the images $T(v_{1}), T(v_{2})$ form a basis for the image of $T$. This is a perfectly reasonable sounding thing to suggest - unfortunately, it's also totally false! </p>
<p>Here's an ... |
2,829,990 | <p>I want to calcurate</p>
<p><span class="math-container">$$ \lim_{n \to \infty} \int_{(0,1)^n} \frac{n}{x_1 + \cdots + x_n} \, dx_1 \cdots dx_n $$</span></p>
<p>I met this in studying Lebesgue integral. But, I don't know how to do at all. I would really appreciate if you could help me!</p>
<p>[Add]</p>
<p>Thanks to e... | achille hui | 59,379 | <p>The limit is $2$. </p>
<p>This might be an overkill but it allow you to compute the asymptotic expansion instead of just the limit.</p>
<hr>
<p>Let $d^n x$ be a short hand for $dx_1\cdots dx_n$</p>
<p>Notice for $x_1,\ldots,x_n > 0$, we have
$\displaystyle\;\frac{1}{\sum_{k=1}^n x_k} = \int_0^\infty e^{-t\sum... |
172,617 | <p>I need to plot two datasets on the same plot. The datasets have the same x-range. However, I want to show only parts of the plot. </p>
<p>A minimal example would be</p>
<pre><code> h = π/100.;
i1 = ListLinePlot[Table[{i*h, Sin[i*h]}, {i, 0, 100}], PlotStyle -> Red];
i2 = ListLinePlot[Table[{i*h, Cos[... | Nasser | 70 | <p>This is a hack may be as I do not see now a direct option to do this.</p>
<p>Combine the 2 list plots into one, using <code>Epilog</code> then use <code>Show</code></p>
<pre><code>red = Table[{i*h,Sin[i*h]},{i,0,100}];
blue = Table[{i*h,Cos[i*h]},{i,0,100}];
vLine = Graphics[{Black, Dashed, Line[{{Pi/2, -1}, {P... |
954,376 | <p>Beltrami made (out of thin paper or stiff or starched cloth not mentioned) a model of a surface of constant negative Gauss curvature <span class="math-container">$ K=-1/c^2$</span>. The original might have resembled a <em>large saddle shaped</em> Pringles chip, and frills might have developed by sagging with time, i... | Alan | 92,834 | <p><img src="https://i.stack.imgur.com/KuuGK.jpg" alt="enter image description here" /></p>
<p>Here is a parametrization of the Pseudosphere:</p>
<p><span class="math-container">$$ x (u,v) = a \frac {cos(v)}{cosh(u)} $$</span></p>
<p><span class="math-container">$$ y (u,v) = a \frac {sin(v)}{cosh(u)} $$</span></p>
<p><... |
2,533,834 | <p>For a complex number $z$, I came across a statement that $\ln(e^{z})$ is not always equal to $z$. Why is this true?</p>
<p>Thanks for the help.</p>
| Oscar Lanzi | 248,217 | <p>Render</p>
<p>$\ln(n)=\int_1^n\frac{dx}{x}$</p>
<p>Show, by putting in $u=x/n$, that</p>
<p>$\ln(2n)-\ln(n)=\int_1^2\frac{du}{u}=ln(2)$</p>
<p>Then for instance</p>
<p>$\ln(5)>\ln(4)=\ln(2)+\ln(2)=2\ln(2),$</p>
<p>$\ln(12)>\ln(8)=\ln(2)+\ln(4)=3\ln(2),$</p>
<p>and as $n$ increases without bound the coef... |
195,790 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/19796/name-of-this-identity-int-e-alpha-x-cos-beta-x-space-dx-frace-al">Name of this identity? $\int e^{\alpha x}\cos(\beta x) \space dx = \frac{e^{\alpha x} (\alpha \cos(\beta x)+\beta \sin(\beta x))}{\alpha... | DonAntonio | 31,254 | <p>$$\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to 0}\frac{\frac{2}{(x+h)^2}-\frac{2}{x^2}}{h}=\lim_{h\to 0}\frac{-4xh-2h^2}{x^2(x+h)^2h}=$$</p>
<p>$$=\lim_{h\to 0}\frac{-4x-2h}{x^2(x+h)^2}=-\frac{4x}{x^4}=-\frac{4}{x^3}$$</p>
|
2,508,499 | <p>How is this hold that $\mathbb R \subseteq B(0,2)$ where
$\big<\mathbb R,d\big>$ and d is a discrete metric?</p>
<p>By doing so we showed that $\mathbb R $ is bounded</p>
| Ross Millikan | 1,827 | <p>Look up the definition of a ball. $B(a,b)$ is all the points within a distance $b$ of $a$ with the given metric, that is $\{c|d(a,c) \lt b\}$. What is the distance between $0$ and $1000$ in the discrete metric? Hint: it is not $1000$.</p>
|
276,200 | <p>could any one tel me how to show the following? I am not getting any idea, thank you for help.</p>
<p>$1)$ $C[0,1]$ is not locally compact.</p>
<p>$2$) $\mathbb{R}^{\infty}$ is not locally compact where $\mathbb{R}^{\infty}=\{x=\{x_n\}:\sum_{n=1}^{\infty} |x_n|^2<\infty\}$ and $||x||=(\sum_{n=1}^{\infty} |x_n|^... | Marc van Leeuwen | 18,880 | <p>I supposing you mean the ring of polynomials over power series $\mathbf C[[x]][y]$. A nontrivial decomposition of those polynomials, made monic in $y$, would require finding in $\mathbf C[[x]]$ a square root of the negated constnat term (coefficient of $y^0$) in both cases. Now $x^3$ cannot have a square root becaus... |
276,200 | <p>could any one tel me how to show the following? I am not getting any idea, thank you for help.</p>
<p>$1)$ $C[0,1]$ is not locally compact.</p>
<p>$2$) $\mathbb{R}^{\infty}$ is not locally compact where $\mathbb{R}^{\infty}=\{x=\{x_n\}:\sum_{n=1}^{\infty} |x_n|^2<\infty\}$ and $||x||=(\sum_{n=1}^{\infty} |x_n|^... | Ted | 15,012 | <p>Assuming C{x} means the power series ring in $x$ over the complex numbers,</p>
<p>$$f(x,y) = y^2 + x^3 - x^2 = y^2 - (x^2 - x^3) = y^2 - x^2(1-x) = (y - x \sqrt{1-x})(y + x \sqrt{1-x})$$ and $\sqrt{1-x}$ can be expanded as a power series is $x$. Therefore $f$ is reducible.</p>
|
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