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 |
|---|---|---|---|---|
2,952,803 | <blockquote>
<p>Let <span class="math-container">$a<b$</span> and <span class="math-container">$a,b\in\Bbb R$</span>. Then there is <span class="math-container">$c\in\Bbb R\setminus\Bbb Q$</span> such that <span class="math-container">$a<c<b$</span>.</p>
</blockquote>
<hr>
<p><strong>My attempt:</strong></... | Cesareo | 397,348 | <p><span class="math-container">$$
-\frac{d}{dt}(M-P)=k(M-P)
$$</span>
so
<span class="math-container">$$
\frac{d(M-P)}{M-P} = -k dt
$$</span></p>
<p>and integrating</p>
<p><span class="math-container">$$
\ln|M-P| = -k t - C_0\Rightarrow P = M-C_1e^{-kt}
$$</span></p>
<p>now if <span class="math-container">$P(0) = 0... |
2,866,423 | <p>When I taught my student the logarithm, he asked me about the historical definition of $\ln(x)$. </p>
<ol>
<li>The first definition I found is that $$\ln(x)=\int_{1}^{x}{ \frac{dt}{t} } $$</li>
<li>Defined as the logarithm to base $e$ or the inverse function of the exponentiation to base
$e$: $$\ln(x)=y \Longleft... | Theo Bendit | 248,286 | <p>I'm no expert on maths history, but logarithms are old enough not to have a "historical definition" that meets our standards of what a definition should be. I think the integral definition of the logarithm is the better one to teach, for a few reasons:</p>
<ol>
<li><p>What is exponentiation? Even if you define $e$,... |
2,866,423 | <p>When I taught my student the logarithm, he asked me about the historical definition of $\ln(x)$. </p>
<ol>
<li>The first definition I found is that $$\ln(x)=\int_{1}^{x}{ \frac{dt}{t} } $$</li>
<li>Defined as the logarithm to base $e$ or the inverse function of the exponentiation to base
$e$: $$\ln(x)=y \Longleft... | fleablood | 280,126 | <p>The fact that they call it a "logarithm" implies the must have had a concept that it is the logarithm of <em>some</em> base. So when the <em>defined</em> they must have been using the concept $\ln x = y \iff e^y = x$. And I even imagine they would be aware that $\frac {db^x}{dx} = C_b*b^x$ (for rational values of ... |
504,524 | <p>I'm trying to learn probability and statistics but I can't really get my head around this one. I realize after drawing the first card there will only be 51 cards in the deck but I'm having trouble calculating the chance that the second one is an Ace if I don't know what the first card is?</p>
<p>Assuming that the i... | Trevor Wilson | 39,378 | <p>For any given position $n$ in the deck (here $n=2$, meaning the second card,) if the deck is shuffled then the probability that the card in position $n$ is an ace is $4/52 = 1/13$, because $4$ out of the $52$ cards in the deck are aces.</p>
|
2,898,390 | <p>Is there any algorithm or a technique to calculate how many prime numbers lie in a given closed interval [a1, an], knowing the values of a1 and an, with a1,an ∈ ℕ?</p>
<p>Example: </p>
<p>[2, 10] --> 4 prime numbers {2, 3, 5, 7}</p>
<p>[4, 12] --> 3 prime numbers {5, 7, 11}</p>
| Adrian Keister | 30,813 | <p>I'll do my best to type up the K-maps in MathJax. For DNF, you just go with $G$ as follows:
$$
\begin{array}{c|c|c|c|c|}
AB &00 &01 &11 &10 \\ \hline
CD \\ \hline
00 &1 &1 &0 &0 \\ \hline
01 &1 &1 &0 &0 \\ \hline
11 &1 &0 &0 &1 \\ \hline
10 &1 &... |
337,930 | <p>Given two polynomials</p>
<p>$$
p(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_{n-1}x^{n-1} \\
q(x) = b_0 + b_1 x + b_2 x^2 + \ldots + b_{n}x^{n}
$$</p>
<p>And the series expansion from their rational polynomial</p>
<p>$$
\frac{p(x)}{q(x)} = c_0 + c_1 x + c_2 x^2 + \ldots
$$</p>
<p>is it possible to recover the the o... | Did | 6,179 | <p><em>(Reproduced from <a href="https://math.stackexchange.com/a/47872/6179">there</a>.)</em></p>
<p>Since ${R\choose k}$ is the coefficient of $x^k$ in the polynomial $(1+x)^R$ and ${M\choose n-k}$ is the coefficient of $x^{n-k}$ in the polynomial $(1+x)^M$, the sum $S(R,M,n)$ of their products collects all the con... |
2,523,342 | <p>Assume $f\in L^p(\Bbb R^d) $ and $g\in L^q(\Bbb R^d) $
Where, $1<p<\infty$ and $1<q<\infty$ are dual ecxponents namely, $$\frac1p+\frac1q =1$$
Then for every $s\in\Bbb R$ such that, $sp\le d$ show that,
$$\lim_{j\to\infty} \int_{\Bbb R^d}f_j(x)g(x)dx = 0$$</p>
<p>Where, $$f_j(x) = j^sf(jx)~~~$$</p>
... | Guy Fsone | 385,707 | <p>This is just a reminiscent of the this:
<a href="https://math.stackexchange.com/questions/2521555/how-to-prove-that-lim-limits-k-rightarrow-infty-int-bbbr-f-kx-gx/2521573#2521573">How to prove that, $\lim\limits_{k \rightarrow \infty} \int_{\Bbb{R}} f_k(x) g(x) dm(x) = 0$</a>.</p>
<p>If $sp =d$ then, Proceed as fol... |
442,950 | <p>I would like to show <span class="math-container">$\lim\limits_{r\to\infty}\int_{0}^{\pi/2}e^{-r\sin \theta}\text d\theta=0$</span>.</p>
<p>Now, of course, the integrand does not converge uniformly to <span class="math-container">$0$</span> on <span class="math-container">$\theta\in [0, \pi/2]$</span>, since it has... | Mhenni Benghorbal | 35,472 | <p>Note that, the integral converges uniformly, since</p>
<p>$$ e^{-r\sin(\theta)} \leq e^{-\sin(\theta)}, \quad r\geq 1, $$</p>
<p>which justifies changing the limit with the integral and the answer is $0$. </p>
|
2,970,836 | <p>Imagine that <span class="math-container">$f(z)$</span> is holomorphic in some subsect of the complex plane. Is it true that
<span class="math-container">$$
\bar{f}(z)=f(\bar{z})
$$</span></p>
| ajotatxe | 132,456 | <p>It is false. Take for example <span class="math-container">$f(z)=iz$</span>.</p>
|
2,970,836 | <p>Imagine that <span class="math-container">$f(z)$</span> is holomorphic in some subsect of the complex plane. Is it true that
<span class="math-container">$$
\bar{f}(z)=f(\bar{z})
$$</span></p>
| Mohammad Riazi-Kermani | 514,496 | <p>The answer is no.</p>
<p>Consider the function <span class="math-container">$f(z)=e^{iz}$</span> which is holomorphic.</p>
<p><span class="math-container">$$f(z)=e^{iz}=e^{-y+ix}=e^{-y}(\cos x + i \sin x)$$</span></p>
<p><span class="math-container">$$\bar {f(z)}= e^{-y}(\cos x - i \sin x)$$</span></p>
<p><span ... |
3,889,175 | <p>Is there a way to calculate:<span class="math-container">$\sum_{i=0}^{k} {2k+1\choose k-i}$</span> using only:</p>
<ul>
<li>symmetry;</li>
<li>pascal's triangle;</li>
<li>one of these sums: <span class="math-container">$$\sum_{i=0}^{k} {n+i\choose i}={n+k+1\choose k}$$</span> and <span class="math-container">$${p\ch... | Z Ahmed | 671,540 | <p>Declare <span class="math-container">$x\ne 1$</span> and <span class="math-container">$x\ne 2$</span> to avoid the division by zero. As you do ordinary algebraic manipulation, you get <span class="math-container">$3(x-1)-(x-2)-7 =0 \implies x=4$</span> which does not contradict the declarations made, this solution i... |
422,225 | <p>The proof uses this lemma which I understand: </p>
<p>$\mathbf {Lemma}$: Suppose $x$ and $y$ are positive real numbers such that $x>y$. If we decrease $x$ and increase $y$ by some positive quantity $E$ such that $x-E \ge y+E$, then $(x-E)(y+E) \gt xy$ . $\;$Hence, by subtracting $E$ from $x$ and adding it to $y$... | vadim123 | 73,324 | <p>1) You are correct, there needs to be some tweaking.</p>
<p>2) It makes one more number equal to A, so by induction eventually they all will be.</p>
<p>3) You don't make both equal to A; you make at least one equal to A.</p>
<p>Perhaps a simpler proof of the middle part, avoiding the first issue, is this: Let $... |
4,112,308 | <p>I was just exploring a little bit on Desmos, and was trying to figure out something somewhat interesting. I'm familiar that this is an elliptic curve, but ALL I know about them is that they are of the form <span class="math-container">$y^2=x^3+ax+b$</span>. Nothing else, really....</p>
<p>So, here's what I'm thinkin... | Ayoub | 536,671 | <p>Let <span class="math-container">$f(x)=x^3-x+b$</span> and consider the curve <span class="math-container">$y^2=f(x)$</span>.</p>
<p>To say that the curve "self-intersects" at <span class="math-container">$(x_0,0)$</span> means <span class="math-container">$f$</span> has a double zero at <span class="math... |
3,515,649 | <blockquote>
<p><a href="https://i.stack.imgur.com/qqIay.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/qqIay.jpg" alt="enter image description here" /></a></p>
</blockquote>
<p><strong>My try</strong> :</p>
<p>we can deal three different cases of removing two biased coin and 1 unbiased coin.</p>
<p... | Henry | 6,460 | <p>There are four possible combinations:</p>
<ul>
<li>Double-headed and shows heads with probability <span class="math-container">$\frac{4}{50}\times 1= 0.08$</span></li>
<li>Double-tailed and shows tails with probability <span class="math-container">$\frac{1}{50}\times 1 =0.02 $</span></li>
<li>Fair coin and shows he... |
1,499,423 | <p>Are there a group of numbers whose squares are made up of squares? For example, $7$ would be one because $7^2$ is $49$ which has $2^2$ and $3^2$. $20$ would be another example.</p>
<p>What are these numbers called?</p>
<p>Please help me find good <strong>tags</strong> for this question.</p>
| Peter Taylor | 5,676 | <p>OEIS calls them <a href="http://oeis.org/A128783" rel="nofollow">Numbers whose square is a nontrivial concatenation of other squares</a>, and calls the sequence of their squares <a href="http://oeis.org/A019547" rel="nofollow">Squares which are a decimal concatenation of two or more squares</a>. One of the reference... |
2,385,369 | <p>Explain why $K_{2,3}$ cannot have a Hamilton cycle.</p>
<p>I can visibly see and show why this is the case, but is there a mathematical proof or specific way of explaining how this Hamilton cycle cannot exist? Thanks a ton for all the help!</p>
| Community | -1 | <p>The graph $K_{2,3}$ is bipartite, and a bipartite graph cannot contain an odd cycle. Since $|V(K_{2,3})| = 5$, this graph can not contain a hamiltonian cycle.</p>
|
2,064,643 | <p>Let $a_n,b_n > 0, \sum_{n=1}^{\infty}a_n < \infty, \sum_{n=1}^{\infty}b_n = \infty$. Is it possible for the Cauchy product of the two series to converge?</p>
| Daniel Fischer | 83,702 | <p>No, under the given assumptions, the Cauchy product always diverges. We have</p>
<p>$$c_n = \sum_{k = 1}^{n-1} a_k b_{n-k} \geqslant a_1\cdot b_{n-1}$$</p>
<p>for all $n\geqslant 2$, so</p>
<p>$$\sum_{n = 2}^N c_n \geqslant a_1\sum_{m = 1}^{N-1} b_m \to +\infty.$$</p>
|
279,043 | <p>I would like to plot a complex graph of the Riemann zeta function on the Argand diagram <span class="math-container">$ς(s)$</span>, where <span class="math-container">$s = \frac{1}{2} + i t $</span>, and the value of <span class="math-container">$t$</span> is varied to get a graph in the polar form.</p>
<p>Can anyon... | Artes | 184 | <p>The simplest way to get an expected plot exploits <code>ParametricPlot</code> and for the sake of clearer visualization we can take advantage of <code>ListAnimate</code> e.g.</p>
<pre><code>anim = Table[
ParametricPlot[ReIm@Zeta[1/2 + I t], {t, 0, k},
PlotRange -> {{-2, 4}, {-2.3, 2.3}}, Plo... |
1,341,486 | <p>Problem:
Find the sum to $n$ terms of
\begin{eqnarray*}
\frac{1}{1\cdot 2\cdot 3} + \frac{3}{2\cdot 3\cdot 4} + \frac{5}{3\cdot 4\cdot 5} +
\frac{7}{4\cdot 5\cdot 6}+\cdots \\
\end{eqnarray*}
Answer:
The way I see it, the problem is asking me to find this series:
\begin{eqnarray*}
S_n &=& \sum_{i=1}^... | mathlove | 78,967 | <p>Setting </p>
<p>$$\frac{2n-1}{n(n+1)(n+2)}=\frac{An+B}{n(n+1)}-\frac{A(n+1)+B}{(n+1)(n+2)}$$
gives us $A=2,B=-\frac 12$, i.e.
$$\frac{2n-1}{n(n+1)(n+2)}=\frac{2n-\frac 12}{n(n+1)}-\frac{2(n+1)-\frac 12}{(n+1)(n+2)}.$$</p>
<p>Hence, we have
$$\begin{align}\sum_{i=1}^{n}\frac{2i-1}{i(i+1)(i+2)}&=\sum_{i=1}^{n}\l... |
1,170,627 | <p>Is it true that every non-empty open set has Lebesgue measure greater than zero? </p>
<p>I could think of a proof along the following lines but not sure if that would be right:</p>
<p>Since every non-empty open set is a finite or countable union of open intervals where at least one open interval is nonempty, and s... | Ivo Terek | 118,056 | <p>Yes, take a point $x \in A$, with $A$ open. Then exists $\epsilon > 0$ such that $(x-\epsilon, x+\epsilon) \subset A$. So: $${\frak m}(x-\epsilon,x+\epsilon) \leq {\frak m}A \implies 0 < 2\epsilon \leq {\frak m}A \implies {\frak m}A > 0.$$</p>
|
1,170,627 | <p>Is it true that every non-empty open set has Lebesgue measure greater than zero? </p>
<p>I could think of a proof along the following lines but not sure if that would be right:</p>
<p>Since every non-empty open set is a finite or countable union of open intervals where at least one open interval is nonempty, and s... | 5xum | 112,884 | <p>Every non-empty open set contains at least one open interval, and open intervals have a positive measure.</p>
|
1,876,287 | <p><strong>Question:</strong></p>
<p>Let P be a point where the normal (in the point where the x-coordinate is h) to the curve</p>
<p>$$y = e^{2x} - 2x$$</p>
<p>cuts the y-axis. Determine the y-coordinates of P when h goes to 0.</p>
<p><strong>Attempted solution:</strong></p>
<p>I first decided to draw the followi... | Community | -1 | <p>The LHS in the equation of the normal should be $y - y(h)$, i.e. $y- e^{2h} - 2h$. Next, the point $P$ corresponds to $x =0$, so it has:</p>
<p>$$y_P = e^{2h} - 2h + \frac{h}{2e^{2h} - 2}$$</p>
<p>Using L'Hôpital's rule, we find: $y_P \to 1 + 1/4 = 5/4$</p>
|
1,876,287 | <p><strong>Question:</strong></p>
<p>Let P be a point where the normal (in the point where the x-coordinate is h) to the curve</p>
<p>$$y = e^{2x} - 2x$$</p>
<p>cuts the y-axis. Determine the y-coordinates of P when h goes to 0.</p>
<p><strong>Attempted solution:</strong></p>
<p>I first decided to draw the followi... | MathInferno | 203,291 | <p>Here is how one would do the limit without L'Hôpital's rule for completeness:</p>
<p>$$\lim_{h \rightarrow 0} \left(e^{2h} -2h + \frac{h}{2e^{2h} - 2}\right) = 1 - 0 + \lim_{h \rightarrow 0} \frac{h}{2e^{2h} - 2} = 1 + \frac{1}{2} \lim_{h \rightarrow 0} \frac{h}{e^{2h} - 1}$$</p>
<p>$$\lim_{h \rightarrow 0} \frac{... |
858,576 | <p>Prove that the union of three subspaces of V is a subspace iff one of the subspaces contains the other two.</p>
<p>I can do this problem when I am working in only two subspaces of $V$ but I don't know how to do it with three. </p>
<p>What I tried is:
If one of the subspaces contains the other two, Then their union... | JeffW89 | 200,928 | <p>Gina's answer is great, but I think we can clean it up a bit.</p>
<p>Let <span class="math-container">$U_1,U_2,U_3$</span> be subspaces of <span class="math-container">$V$</span> over a field <span class="math-container">$k\neq \mathbb{F}_2$</span>.</p>
<p><span class="math-container">$(\Leftarrow)$</span> Suppose t... |
2,456,976 | <p>$ f(x,y) = \begin{cases} \dfrac{x^3+y^3}{x^2+y^2} &\quad\text{if} [x,y] \neq [0,0]\\[2ex] 0 &\quad\text{if}[x,y] = [0,0]\\ \end{cases} $</p>
<p>The only point it could be discontinuous in is <code>[0,0]</code>. How do I find the limit of the function for $(x,y) \rightarrow (0,0)$? $ \lim_{(x,y) \rightarrow... | SlowerPhoton | 440,566 | <p>$x = r\cos \theta$, $y = r\sin \theta$</p>
<p>instead of $(x,y) \rightarrow (0,0)$ I can now use $r\rightarrow0$</p>
<p>$$\begin{align}
\lim_{r\to0} \frac{r^3\cos^3\theta + r^3\sin^3\theta}{r^2\cos^2\theta + r^2\sin^2\theta}
&=\, \lim_{r\to0} \frac{r (\cos^3\theta + \sin^3\theta)}{\cos^2\theta + \sin^2\theta} ... |
1,105,454 | <p>We have $f: \Bbb{R} \rightarrow \Bbb{R}$ defined as follows:</p>
<p>$$f(x) = \begin{cases} a, & \mbox{if } x=0 \\ \sin\frac{b}{|x|}, & \mbox{if } x\neq 0 \end{cases}$$</p>
<p>The problem asks us to tell for which $a,b \in \Bbb{R}$, $f$ is continuous.</p>
<p>Intuitively I should find $\lim_{x\rightarrow 0}... | Senex Ægypti Parvi | 89,020 | <p>You show "sum" as the result of the function "sum."<br>
If you do not insist on the sought-after word beginning with "oper-," how about -- "result?"</p>
|
2,661,468 | <p>the number of not identically zero functions $f:\mathbb{R} \to \mathbb{R}$ satisfying the equation $f(xy)=f(x)f(y)$ and $f(x+z)=f(x)+f(z)$ for some $z$ not equal to zero</p>
<ol>
<li>one</li>
<li>finite</li>
<li>countable</li>
<li>uncountable</li>
</ol>
<p>it seems like the question asks about the number of homomo... | 5xum | 112,884 | <p>The mistake in your thinking is, primarily, that guessing the answer and not doing any actual mathematical work will help you in finding the solution.</p>
<p>If you think there are uncountable many functions, then try to <strong>show</strong> that. Construct at least a couple of them, and then possibly "construct" ... |
891,575 | <p>The circumference of a circle has length 90 centimeters, Three points on the circle divide the circle into three equal lengths. Three ants A, B, and C start to crawl clockwise on the circle, with starting from one of the three points. Initially A is ahead of B and B is ahead of C. Ant A crawls 3 centimeters per seco... | Seyed Mohsen Ayyoubzadeh | 165,227 | <p>I suggest you use the angle as the ant's position indicator. Note that $$\theta = \frac{s}{r} = \frac{s}{{\frac{{90}}{{2\pi }}}} = \frac{{2\pi s}}{{90}}$$Now:$$\begin{array}{l}{\theta _{\rm{A}}}(t) = {\omega _{\rm{A}}}t + {\theta _{{{\rm{A}}_0}}}\\{\theta _{\rm{B}}}(t) = {\omega _{\rm{B}}}t + {\theta _{{{\rm{B}}_0}... |
891,575 | <p>The circumference of a circle has length 90 centimeters, Three points on the circle divide the circle into three equal lengths. Three ants A, B, and C start to crawl clockwise on the circle, with starting from one of the three points. Initially A is ahead of B and B is ahead of C. Ant A crawls 3 centimeters per seco... | shooting-squirrel | 68,659 | <p>Suppose, without affecting the result of the problem, that ant $A$ starts at the polar coordinates of angle $0$ on the unit circle, the other ants will be at position $2\frac{\pi}{3}$ and $4\frac{\pi }{3}$.
Let's try to find three functions, $f$, $g$, $h$, that takes $t$ (time) as argument, and yields the angle at w... |
630,966 | <p>Most universities have a 3rd year undergraduate analysis course in which metric spaces are studied in depth (compactness, completeness, connectedness, etc...). However, in practice it seems that most of these metric spaces are normed vector spaces. Why not just cover normed vector spaces instead of metric spaces? </... | Igor Rivin | 109,865 | <p>Metric spaces are more general than normed spaces, because they need not be vector spaces. They are easier than general topological spaces, but introduce all of the relevant concepts.</p>
|
630,966 | <p>Most universities have a 3rd year undergraduate analysis course in which metric spaces are studied in depth (compactness, completeness, connectedness, etc...). However, in practice it seems that most of these metric spaces are normed vector spaces. Why not just cover normed vector spaces instead of metric spaces? </... | Matt E | 221 | <p>You write in a comment that metric spaces "don't come up very much in practice".
This is not true (although may reflect the mathematics you have seen so far).</p>
<p>Metric spaces (and, more generally, topological spaces) occur all over the place. I am a working number theorist, and I use the concepts of topology ... |
630,966 | <p>Most universities have a 3rd year undergraduate analysis course in which metric spaces are studied in depth (compactness, completeness, connectedness, etc...). However, in practice it seems that most of these metric spaces are normed vector spaces. Why not just cover normed vector spaces instead of metric spaces? </... | Jacob Denson | 120,724 | <p>Another perspective might be useful for others looking at this question. Studying subsets of normed spaces is essentially the same as studying metric spaces. Given a metric space <span class="math-container">$(M,d)$</span>, we can define a vector space <span class="math-container">$L^\infty(M)$</span> of bounded fun... |
446,499 | <p>I have just learned the definition of connectedness and wikipedia gives an example of a disconnected set: <span class="math-container">$(0,1)\cup \left\{ 3 \right\}$</span> (<a href="https://en.wikipedia.org/wiki/Connected_space#Examples" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Connected_space#Exampl... | Neal | 20,569 | <p>In the subspace topology inherited from $\mathbb{R}$, the space $X = (0,1)\cup\{3\}$ can be covered by two open sets, $(0,1)$ and $\{3\}$, which are disjoint. Hence $X$ is disconnected.</p>
<p>It is also arcwise disconnected because no path can be found connecting any $x\in(0,1)$ with $3$.</p>
|
2,694,740 | <p>$$\frac{2.10^{-7} - 0,4.10^{-6}}{10^{-8}} = ? $$</p>
<p>These questions are making me confused because we're dealing with the terms like $10^x$. What are your professional tips? </p>
<p><strong>My attempt:</strong></p>
<p>$$\frac{2.10^{-7} - 4.10^{-7}}{10^{-8}} \tag{1} $$
$$\frac{ -8.10^{-7}}{10^{-8}} \tag{2} $$<... | user | 505,767 | <p>From here</p>
<p>$$\frac{2\cdot 10^{-7} - 4\cdot 10^{-7}}{10^{-8}}=\frac{-2\cdot 10^{-7}}{10^{-8}}=-2\cdot 10^{-7}\cdot 10^{8}=-20$$</p>
<p>or as an alterntive</p>
<p>$$\frac{2\cdot 10^{-7} - 0.4\cdot 10^{-6}}{10^{-8}}=2\cdot 10^{-7}\cdot 10^{8} - 0.4\cdot 10^{-6}\cdot 10^{8}=20-40=-20$$</p>
|
1,521,779 | <p>I have a homework question that I want to make sure I'm getting it right.</p>
<p>This is a joint probability table for the proportions of survey respondents who smoke and who have had heart attacks.</p>
<p><kbd> &n... | Will Jagy | 10,400 | <p>I think I will throw in an advertisement for quadratic forms. Solve $u^2 \equiv -1 \pmod p.$ This could be by hand for small primes or primes of very special forms, otherwise it is Cornacchia or Tonelli-Shanks. Next we have $(2u)^2 \equiv -4 \pmod {4p},$ or
$$ (2u)^2 = - 4 + 4 p t, $$
$$ (2u)^2 - 4 pt = -4. $$
Th... |
345,735 | <p>If <strong>two planes</strong> are <strong>intersected</strong> <em>by making a straight line, like <span class="math-container">$AB$</span></em> then</p>
<blockquote>
<p>Does the angle between two planes (see figure) <strong>always</strong> given by the
angle between normal vectors (<span class="math-container">$n_... | Cameron Buie | 28,900 | <p>Yes, it is. ${}{}{}{}{}{}{}{}$</p>
|
1,419,315 | <p>I have a particular scenario.</p>
<p>In this scenario, we have the standard cubic equation,</p>
<pre><code>ax^3 + bx^2 + cx + d = y
</code></pre>
<p>as well as 3 points that are graphed, <a href="https://i.imgur.com/VCZKuGW.png" rel="nofollow noreferrer">as can be seen in this graph</a>. (The line is irrelevant ... | mickep | 97,236 | <p>If you want to proceed with this solution, you should complete the square. It is important that you "complete one variable completely every time". We write
$$\begin{aligned}
x_1^2+x_2^2+x_3^2-x_1x_2-x_1x_3-x_2x_3&=\Bigl(x_1-\frac{1}{2}x_2-\frac{1}{2}x_3\Bigr)^2+\frac{3}{4}x_2^2+\frac{3}{4}x_3^2-\frac{3}{2}x_2x_3... |
3,748,739 | <p>Let <span class="math-container">$X\perp Y$</span> with <span class="math-container">$X,Y\sim N(0,1)$</span>. Let <span class="math-container">$U=\frac{(X+Y)}{\sqrt{2}}$</span> and <span class="math-container">$V=\frac{(X-Y)}{\sqrt{2}}$</span>.</p>
<ol>
<li>Find the law of <span class="math-container">$(U,V)$</span>... | user21820 | 21,820 | <p><strong>Hint</strong>: Show that if <span class="math-container">$f$</span> has nonzero gradient at any point, then it cannot be bounded.</p>
|
393,250 | <p>Let <span class="math-container">$G$</span> be a finitely generated residually finite group and let <span class="math-container">$M$</span> be a finitely generated <span class="math-container">$\mathbb{Z}[G]$</span>-module.</p>
<p><strong>Question</strong>: Must <span class="math-container">$M$</span> be residually ... | Benjamin Steinberg | 15,934 | <p>Here is another example with different groups. <a href="https://www.sciencedirect.com/science/article/pii/0021869373900112" rel="nofollow noreferrer">Formanek</a> showed that the group ring over any field of a free product of non-trivial groups (and not both order 2) is primitive, has a faithful simple module. Tha... |
3,537,654 | <p><span class="math-container">$$\lim_{x\to 0^{+}} (\tan x)^x$$</span></p>
<p><span class="math-container">$$\lim_{x\to 0^{+}} e^{\ln((\tan x)^x)}=\lim_{x\to 0^{+}} e^{x\ln(\tan x)}=\lim_{x\to 0^{+}} e^{x[\ln(\sin x)-\ln(\cos x)]}$$</span></p>
<p>We can continue to create an expression that may help us use L'Hospita... | lab bhattacharjee | 33,337 | <p><span class="math-container">$$\lim_{x\to0^+}(\tan x)^x=\left(\lim_{x\to0^+}\tan x^{\tan x}\right)^{\lim_{x\to^+}\dfrac x{\tan x}}$$</span></p>
<p>For the inner limit use <a href="https://math.stackexchange.com/questions/637401/limit-of-xx-as-x-tends-to-0">Limit of $x^x$ as $x$ tends to $0$</a></p>
|
2,792,651 | <p>I want to prove that $h_K=2$ if $K=\mathbb{Q}[\sqrt{-6}]$. Using Minkowski Theorem I have that $Cl_K=\{(1),(3,\sqrt{-6}),(2,\sqrt{-6})\}$, and I thought it was a good idea to use Lagrange Theorem (order of an element divides order of the gorup).</p>
<p>The main problem is that I can't reduce $(2,\sqrt{-6})^2$: </p>... | DonAntonio | 31,254 | <p>$$(2,\sqrt{-6})^2=(2,2\sqrt{-6},\,-6)=(2)\ldots$$</p>
|
2,792,651 | <p>I want to prove that $h_K=2$ if $K=\mathbb{Q}[\sqrt{-6}]$. Using Minkowski Theorem I have that $Cl_K=\{(1),(3,\sqrt{-6}),(2,\sqrt{-6})\}$, and I thought it was a good idea to use Lagrange Theorem (order of an element divides order of the gorup).</p>
<p>The main problem is that I can't reduce $(2,\sqrt{-6})^2$: </p>... | Bill Wallis | 350,028 | <p>It's also worth noting that the ideals
$$
(3, \sqrt{-6}) \quad\text{and}\quad (2, \sqrt{-6})
$$
belong to the same ideal class in $\mathrm{Cl}(K)$ since
$$
(3, \sqrt{-6})(2, \sqrt{-6}) = (6, 2\sqrt{-6}, 3\sqrt{-6}, 6) = (\sqrt{-6})
$$
and both ideals are equal to their inverse in the class group, since they're of or... |
126,549 | <p>For a quadratic form $q(\mathbf{v})$, when you change the basis do you <em>always</em> change the quadratic form? Can you have the same quadratic form with respect to different basis? Or is the quadratic form unique to the basis. </p>
<p>Also, if you're given a quadratic form say $q(\mathbf{v}) = 3x^2 + y^2 - 2z^2 ... | Marc van Leeuwen | 18,880 | <p>Your question is not too clear, but in any case a change of basis does not necessarily change the matrix describing the quadratic form, and so conversely changing the basis while keeping the same matrix does not always change the form. An extreme case is the null matrix which always defines the null form, no matter ... |
515,491 | <p>Let $σ(n)$ denote the sum of all the positive divisors of $n∈ \mathbb N$. I think that $6$ divides $σ(6n-1)$ for all $n∈ \mathbb N$ , but I am not able to prove it. So, a proof of the result (if it is true and I think it is) will be much appreciated.</p>
| wendy.krieger | 78,024 | <p>The sum of divisors is weakly multiplicative, which means that if $\operatorname{gcd}(a,b)=1$, then $\sigma(ab) = \sigma(a)\sigma(b)$. This allows to divide it into its prime-power divisors, eg $\sigma(2^a 3^b \dots) = \sigma(2^a)\sigma(3^b)\dots$. </p>
<p>A number of the form of $6n-1$ must have at least one pri... |
515,491 | <p>Let $σ(n)$ denote the sum of all the positive divisors of $n∈ \mathbb N$. I think that $6$ divides $σ(6n-1)$ for all $n∈ \mathbb N$ , but I am not able to prove it. So, a proof of the result (if it is true and I think it is) will be much appreciated.</p>
| Hanul Jeon | 53,976 | <p>If $n=6k-1$ then $n$ is not square. (You can easily check this fact.) So if $d\mid n$ then $d\neq n/d$. Furthermore, we get $(d,2)=(d,3)=1$. That is, $d\equiv\pm1 \pmod 6$ so we get
$$
d+\frac{n}{d}=d+\frac{6k-1}{d}\equiv d-\frac{1}{d} \equiv 0 \pmod 6.
$$</p>
<p>Therefore $6\mid d+n/d$ for all $d\mid n$. Since $n$... |
16,982 | <p>one can obtain solutions to the <a href="http://en.wikipedia.org/wiki/Laplace%27s_equation" rel="noreferrer">Laplace equation</a>
<span class="math-container">$$\Delta\psi(x) = 0$$</span></p>
<p>or even for the <a href="http://en.wikipedia.org/wiki/Poisson%27s_equation" rel="noreferrer">Poisson equation</a> <span cl... | George Lowther | 1,321 | <p>The general form for the infinitesimal generator of a continuous diffusion in $\mathbb{R}^n$ is
$$
Af(x) = \frac12\sum_{ij}a_{ij}\frac{\partial^2 f(x)}{\partial x_i\partial x_j}+\sum_ib_i\frac{\partial f(x)}{\partial x_i}-cf(x).\qquad{\rm(1)}
$$
Here, $a_{ij}$ is a positive-definite and symmetric nxn matrix, $b_i$... |
2,204,944 | <p>A line is a collection of infinitely many points. By definition, a point has no dimensions. But, how can infinitely many dimensionless points give rise to a line with a dimension. This is the same case with planes, solids and higher dimensions too...</p>
<p>Thanks in advance for any help..!!</p>
| Ryan | 258,987 | <p>$$Pr(x_2 = 1) = Pr(x_1 = 1)*0 + Pr(x_1 \neq1) * Pr(x_2 = 1| x_1 \neq 1) = 0 + \frac{1}{3} * \frac{3}{4} = \frac {1}{4}$$</p>
<p>Similarly, we have</p>
<p>$$Pr(x_2 = i) = \frac {1}{4}, \quad i = 1,2,3,4$$.</p>
<p>Proceeding forward (i.e., regarding $X_2$ as $X_1$), we can derive</p>
<p>$$Pr(x_3 = i) = \frac {1}{4... |
2,129,764 | <p>Hey guys I have a problem that I'm having trouble solving. Here is the question:</p>
<p><strong>Consider events $A, B, C$ such that $P(A\mid B) > P(A)$ and $P(B\mid C) > P(B)$. Does it follow that $P(A\mid C) > P(A)$? Either prove it to be so or provide a counterexample.</strong></p>
<p>And here is what I... | JMoravitz | 179,297 | <p>$A=\{1,2\}, B=\{2,3\}, C=\{3,4\}, \Omega=\{1,2,3,4,5\}$ with the uniform probability measure.</p>
<p>$\Pr(A)=\Pr(B)=\Pr(C)=\frac{2}{5}$</p>
<p>$\Pr(A\mid B)=\Pr(B\mid C)=\frac{1}{2}>\frac{2}{5}$</p>
<p>$\Pr(A\mid C)=0\not > \frac{2}{5}$</p>
|
250,454 | <p>Is there a <code>ReplaceOnce</code> which does only one replacement if possible by trying the rules sequentially in order. Consider the following as an example:</p>
<pre><code>ReplaceOnce[{"May","5","May","5"},{"May"->1,"5"->2}]
</code></pre>
<p>shoul... | andre314 | 5,467 | <p>One approach (maybe not suited to your problem) is to use as pattern the whole expression instead of the elements of the expression. For example :</p>
<pre><code>Replace[
{"May","5","May","5"}
,{{a___,"May",b___} :> {a,1,b}}
,{0}]
</code></pre>
<p>returns :</p>
... |
78,641 | <p>I am interested in the relation between the property of countable chain condition (ccc) and the property of separable. Could someone recommend some papers or books about this to me? thanks in advance.</p>
| Andreas Blass | 6,794 | <p>The following is a special case of the last two sentences of Stefan Geschke's answer, but it may be more accessible than the general case to non-set-theorists. The quotient of the Boolean algebra of Borel sets of reals modulo the ideal of sets of measure 0 is an example of a ccc but not $\sigma$-centered Boolean al... |
3,212,812 | <p>Does a REAL, everywhere continuous function exist which has an infinite number of turning points but does NOT use trig functions ? I think not but having trouble formalising my reasons. (The function need not be periodic but its domain and range must be all REAL x)</p>
| tia | 578,988 | <p>Of course such functions exist. The <a href="https://en.wikipedia.org/wiki/Airy_function" rel="nofollow noreferrer">Airy functions</a> come to mind.</p>
|
3,212,812 | <p>Does a REAL, everywhere continuous function exist which has an infinite number of turning points but does NOT use trig functions ? I think not but having trouble formalising my reasons. (The function need not be periodic but its domain and range must be all REAL x)</p>
| Gerry Myerson | 8,269 | <p>How about the piecewise linear function whose graph goes through <span class="math-container">$$\dots,(-4,0),(-3,1),(-2,0),(-1,1),(0,0),(1,1),(2,0),(3,1),(4,0),\dots$$</span> </p>
<p>You could smooth it out by making it piecewise quadratic instead. </p>
<p>(EDIT:) Explicitly: write <span class="math-container">$\o... |
781,776 | <blockquote>
<p>A red die, a blue die, and a yellow die (all six sided) are rolled. Given that no two of the dice land on the same number, what is the conditional probability that blue is less than yellow which is less than red?</p>
</blockquote>
<p>The Answer is a sixth. I have absolutely no idea how to do this tho... | David | 119,775 | <p>First ignore the restriction concerning the first office. The number of ways to distribute the $15$ identical computers is $C(17,2)$.</p>
<p>Now any distribution in which the first office gets $5$ computers or more should not have been included. To count these distributions, put $5$ computers into the first offic... |
2,624,498 | <p>Evaluate $$\lim_{n \rightarrow\infty} \sqrt[n]{3^{n} +5^{n}}$$</p>
<p>Attempt:</p>
<p>The only sort of manipulation that has come to mind is: $$e^{\frac{1}{n}ln(e^{n\ln(3)} + e^{n\ln(5)})}$$</p>
<p>So what is the trick to successfully evaluate this?</p>
| DeepSea | 101,504 | <p><strong>hint:</strong> There is a standard trick....: $5^n < 3^n+5^n < 2\cdot 5^n$, and in general if $0 < a < b$, then $ \displaystyle \lim_{n \to \infty} \sqrt[n]{a^n+b^n} = b$</p>
|
2,746,637 | <p>We have $*$ the Hodge operator, and $d $ the exterior derivative. We define $\delta=\pm *d*$ and $\triangle=d\delta+\delta d $. <a href="https://rads.stackoverflow.com/amzn/click/0387908943" rel="nofollow noreferrer">Warner</a> (pp. 223) says that we have
$$
\triangle (E^p (M))=d\delta (E^p (M))\oplus \delta d (E^p... | Amitai Yuval | 166,201 | <p>Warner actually says more than that, and the part you omitted is the whole point.</p>
<p>If I remember correctly, this is what Warner calls the Hodge decomposition theorem:$$\begin{align}E^p(M)&=\Delta(E^p(M))\oplus\mathcal{H}^p(M)\\&=d\delta(E^p(M))\oplus\delta d(E^p(M))\oplus\mathcal{H}^p(M)\\&=d(E^{... |
91,302 | <p>So, we represent numbers usually in a form of a sequence of digits where each one of them multiplies the power of a base:</p>
<p>$13.2 = 1 * 10^1 + 3 * 10^0 + 2 * 10^{-1}$</p>
<p>So that much is clear, perfectly. But what interests me is the "symmetry" between the left and right of the radix point which separates ... | hmakholm left over Monica | 14,366 | <p>Except for typographic conventions, the rule is the same on both sides of the point: A zero matters <em>if and only if</em> it comes <em>between</em> a nonzero digit and the decimal point.</p>
|
1,028,371 | <p>I have been trying to prove this, but I am having trouble understanding how to prove the following mapping I found is injective and surjective. Just as a side note, I am trying to show the complex ring is isomorphic to special $2\times2$ matrices in regard to matrix multiplication and addition. Showing these hold is... | Rudy the Reindeer | 5,798 | <p>For injectivity assume that for some $(a,b), (a',b') \in \mathbb C$:</p>
<p>$$ \phi (a,b)
= \left (\begin{array}{cc} a & -b \\ b & a \end{array}\right ) =
\left (\begin{array}{cc} a' & -b' \\ b' & a' \end{array}\right ) = \phi(a',b')$$</p>
<p>Then since two matrices are equal if and only if each e... |
1,028,371 | <p>I have been trying to prove this, but I am having trouble understanding how to prove the following mapping I found is injective and surjective. Just as a side note, I am trying to show the complex ring is isomorphic to special $2\times2$ matrices in regard to matrix multiplication and addition. Showing these hold is... | cansomeonehelpmeout | 413,677 | <ul>
<li><p>For <strong>injectivity</strong> you need to show that if $\phi(z_1)=\phi(z_2)$ then $z_1=z_2$. So assume that $$\phi(a+bi)=\begin{pmatrix}a&-b\\b&a\end{pmatrix}=\begin{pmatrix}a'&-b'\\b'&a'\end{pmatrix}=\phi(a'+b'i)$$ then $$\begin{pmatrix}a-a'&-b+b'\\b-b'&a-a'\end{pmatrix}=\begin{p... |
2,990,580 | <p>I am doing my maths A-level*. Often when I am at home I get questions about why we solve certain problem types in a certain way. One example is "why does completing the square work?"</p>
<p>Is there a website which collects explanations like these together for me to read? <strong><em>Preferably one that is aimed at... | Martín Vacas Vignolo | 297,060 | <p>The implication <span class="math-container">$x\in A \to x \in A\cap C$</span> is false. Why <span class="math-container">$x\in C$</span>?</p>
|
13,951 | <p>let $f$ and $g $ be two real valued function , I have asked many students what is the derivative of $(fg)'$ they answered me :it is $f' \cdot g'$, then I seek why most people (students) guess that ?</p>
| guest | 9,519 | <p>A. Maybe because the dash is in the exponent place.</p>
<p>B. It's not a bad guess, if you had little info and had to make a quick guess.</p>
<p>C. Humans are not computers and reason by analogy, first.</p>
<hr>
<p>On the practical side, who cares? Will knowing why people guess that make the problem go away t... |
3,403,855 | <p>Construct an example of a set of real numbers E that has no points of accumulation and yet has the property
that for every ε > 0 there exist points x, y ∈ E so that 0 < |x − y| < ε.</p>
<p>so i know we need a convergent sequence to show that the difference between two elements can be as small as we like, also... | Olivier Roche | 649,615 | <p>Consider <span class="math-container">$E := \mathbb{N}\setminus\{0\} \cup \{n + \frac{1}{n} \big| \ n \in \mathbb{N}\setminus\{0\} \}$</span></p>
<p><span class="math-container">$E$</span> satisfies <span class="math-container">$\forall \epsilon > 0 \ \exists x,y\in E, \, |x-y|<\epsilon$</span>.</p>
<p>There... |
85,309 | <p>Let A and B be positive semidefinite matrices. It is not hard to see that $(A-B)^2 \leq 2A^2 + 2B^2$. In fact, $2A^2 + 2B^2 - (A-B)^2 = (A+B)^2$ is positive semidefinite. </p>
<p>My question is: Is there a constant c (independent of A and B and the dimension) such that</p>
<p>$$(A-B)^2 \leq c (A+B)^2?$$</p>
<p>Th... | M. Lin | 54,458 | <p>A weaker version is also not true, that is, there is no constant $c$ such that
$$\lambda_j(A-B)^2 \leq c \lambda_j(A+B)^2, ~~ j=1, 2,\ldots$$
where $\lambda_j(A)$ means the $j$-th largest eigenvalue of $A$.</p>
<p>Nevertheless, it is trivial to see that<br>
$$trace(A-B)^2 \leq trace(A+B)^2.$$</p>
|
356,306 | <p>If $f:X_1 \rightarrow X_2$ and $g:X_2 \rightarrow X_3$ are homomorphisms.
If $g \circ f =0$ does it imply that $Im f \subseteq ker g$? and how to show that? do you have an example?
thanks :)</p>
| muzzlator | 60,855 | <p>If $f(x) \in \operatorname{im}f$ was not in $\ker g$, then $g(f(x)) \neq 0$</p>
|
89,621 | <p>All geometry in computer graphics are transformed by position * transform matrix; The issue is the fact that position is a vector with 3 components (x,y,z); And transform matrix is a 4 by 4 with one column that can be dumped(at least in my case). So my transform matrix is now a 3 by 4 matrix:<br>
axis x { x... | Palax | 20,875 | <p>Thanks for the answer hardmath; The solution is at <a href="http://www.euclideanspace.com/maths/geometry/affine/matrix4x4/index.htm" rel="nofollow">here</a>.
More precisely to move and rotate a point (vector x y z) with a transform matrix (4 by 4) you must add to the point a new component. This will make the point a... |
632,891 | <p>I'm trying to solve this limit, for which I already know the solution thanks to Wolfram|Alpha to be $\sqrt[3]{abc}$:</p>
<p>$$\lim_{n\rightarrow\infty}\left(\frac{a^\frac{1}{n}+b^\frac{1}{n}+c^\frac{1}{n}}{3}\right)^n:\forall a,b,c\in\mathbb{R}^+$$</p>
<p>As this limit is an indeterminate form of the type $1^\inft... | Ragnar | 91,741 | <p>This is not an answer to the question itself I think, but still a useful observation.</p>
<p>This can be found using the <a href="http://en.wikipedia.org/wiki/Generalized_mean" rel="nofollow">Generalized means</a>. Say that
$$
M_p(x_1,\dots,x_n)=\left(\frac 1n\sum x_i^p\right)^\frac 1p
$$
In general, we have $M_m(x... |
2,436,268 | <p>My problem is evaluating the following limit:
$$\lim_{(x,y)\to(0,0)}\frac{x^5+y^2}{x^4+|y|}$$
The answer should be 0. I tried to convert the limit into polar form, but it didn't help because I couldn't isolate the $r$ and $\theta$-variables of the expression. My "toolbox" for solving problems like these is very limi... | zwim | 399,263 | <p>A common tool is to homogenise the denominator by setting $\displaystyle u=\frac{y}{x^4}$, then discuss the limit in function of $u$.</p>
<p>$\displaystyle f(x,y)=\frac{x^5+y^2}{x^4+|y|}=\frac{x^5+u^2x^8}{x^4+|u|x^4}=\frac{x+u^2x^4}{1+|u|}=\frac{x+uy}{1+|u|}=x\left(\frac 1{1+|u|}\right)+y\left(\frac u{1+|u|}\right)... |
3,640,298 | <p><span class="math-container">$$ f(x,y,z)= \int_{-\infty}^{+\infty} e^ {-t(t-x)(t-y)(t-z)}\;dt$$</span> </p>
<p><span class="math-container">$$t=p+x$$</span>
<span class="math-container">$$ f(x,y,z)= \int_{-\infty}^{+\infty} e^ {-p(p+x)(p-(y-x))(p-(z-x))}\;dp$$</span> </p>
<p><span class="math-container">$$ ... | Bill Dubuque | 242 | <p><strong>Hint</strong> <span class="math-container">$ $</span> Let <span class="math-container">$\, c_a := d p_1^{r_1}\cdots p_t^{r_t}$</span> be the product of all prime factors of <span class="math-container">$\,a\,$</span> which also divide <span class="math-container">$\,b,\,$</span> and similarly for <span class... |
3,640,298 | <p><span class="math-container">$$ f(x,y,z)= \int_{-\infty}^{+\infty} e^ {-t(t-x)(t-y)(t-z)}\;dt$$</span> </p>
<p><span class="math-container">$$t=p+x$$</span>
<span class="math-container">$$ f(x,y,z)= \int_{-\infty}^{+\infty} e^ {-p(p+x)(p-(y-x))(p-(z-x))}\;dp$$</span> </p>
<p><span class="math-container">$$ ... | BelowAverageIntelligence | 441,199 | <p>As lulu noted, the order of any prime dividing <span class="math-container">$ab$</span> must be divisible by 3, because <span class="math-container">$ab=m^3$</span>. Now the product above just results from the fact that we can break up the primes dividing <span class="math-container">$m$</span> into those which divi... |
3,549,072 | <p>The following are given</p>
<p><span class="math-container">$$ \lim_{x \to \infty}{\log(x)} = \infty$$</span></p>
<p><span class="math-container">$$ \lim_{x \to \infty}{\cosh(x)} = \infty$$</span></p>
<p><span class="math-container">$$ \lim_{x \to \infty}{\sinh(x)} = \infty$$</span></p>
<p><span class="math-cont... | gfppoy | 390,474 | <p><span class="math-container">$\tanh(x)-1=\frac{e^x-e^{-x}}{e^x+e^{-x}}-1=\frac{e^x-e^{-x}}{e^x+e^{-x}}-\frac{e^x+e^{-x}}{e^x+e^{-x}}=\frac{-2e^{-x}}{e^x+e^{-x}}$</span></p>
<p><span class="math-container">$\implies \frac{\tanh(x)-1}{e^{-2x}}=\frac{-2e^{x}}{e^x+e^{-x}} \to -2$</span> as <span class="math-container">... |
3,082,080 | <p><a href="https://i.stack.imgur.com/Dcf40.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Dcf40.png" alt="enter image description here"></a></p>
<blockquote>
<p>The sides <span class="math-container">$AB, BC, CD$</span> of trapezoid <span class="math-container">$ABCD$</span> touches the circle w... | PrincessEev | 597,568 | <p><strong><em>Warning:</strong> might be a little convoluted, uses a bit of assumed knowledge, and is probably not the most elegant/efficient solution. I've triple-checked the method and arithmetic though so unless there's a fundamental flaw with my solution, I think this should be correct.</em></p>
<hr>
<p>We make ... |
716,498 | <p>My Algebraic Topology book says </p>
<blockquote>
<p>Let $\Bbb{R}^n$ denote Euclidean n-space. Then $\pi_1(\Bbb{R}^n,x_0)$ is the trivial subgroup (the group consisting of the identity alone).</p>
</blockquote>
<p>I wonder why that is. I can imagine infinite continuous "loops" in $\Bbb{R}^3$ that start and end a... | Community | -1 | <p>$\mathbb{R}^n$ is contractible (homotopy equivalent to a point). Hence its fundamental group is the same as the fundamental group of a singleton, i.e. trivial.</p>
|
716,498 | <p>My Algebraic Topology book says </p>
<blockquote>
<p>Let $\Bbb{R}^n$ denote Euclidean n-space. Then $\pi_1(\Bbb{R}^n,x_0)$ is the trivial subgroup (the group consisting of the identity alone).</p>
</blockquote>
<p>I wonder why that is. I can imagine infinite continuous "loops" in $\Bbb{R}^3$ that start and end a... | Christoph | 86,801 | <p>Let $\gamma\colon[0,1]\to\mathbb R^n$ be a loop in $x_0$, then
\begin{align}
H\colon [0,1]\times [0,1] &\longrightarrow \mathbb R^n\\
(s, t) &\longrightarrow (1-t)\cdot\gamma(s)+t\cdot x_0
\end{align}
is a continous map such that</p>
<ol>
<li>$H(0,t) = H(1,t) = x_0$ for all $t\in[0,1]$,</li>
<li>$H(s,0) = \... |
323,109 | <p>Could someone help with the following integration:
$$\int^0_\pi \frac{x \sin x}{1+\cos^2 x}$$</p>
<p>So far I have done the following, but I am stuck:</p>
<p>I denoted $ y=-\cos x $ then:
$$\begin{align*}&\int^{1}_{-1} \frac{\arccos(-y) \sin x}{1+y^2}\frac{\mathrm dy}{\sin x}\\&= \arccos(-1) \arctan 1+\a... | robjohn | 13,854 | <p>Another approach, using integration by parts and symmetry about <span class="math-container">$\pi/2$</span>:
<span class="math-container">$$
\begin{align}
\int_0^\pi\frac{x\sin(x)}{1+\cos^2(x)}\,\mathrm{d}x
&=-\int_0^\pi x\,\mathrm{d}\arctan(\cos(x))\tag{1a}\\
&=\left.-x\arctan(\cos(x))\,\vphantom{\int}\righ... |
795 | <p>Please observe the following thread <a href="https://math.stackexchange.com/questions/4489/proving-that-the-given-diophantine-equation-has-a-solution">Proving that the given Diophantine equation has a solution</a>.</p>
<p>There is a long boring argument/discussion about whether it should be posted, who should post ... | Jonas Meyer | 1,424 | <p>I may as well respond as one of the posters of comments there. The discussion was started there partly as a reaction to the OP's past practices. I have gathered from browsing around here that posting on meta has had little to no effect on these practices. I don't know whether there is a serious problem here, but ... |
3,845,570 | <p>Premises: <span class="math-container">$\neg(A \to B)\ ,\ \neg B \to C$</span> .</p>
<p>Conclusion: <span class="math-container">$C$</span></p>
<p>My intuition is that I should do a sub-derivation where I prove <span class="math-container">$\neg C$</span> is an absurdity. However, I soon run into issues. If I could... | Mauro curto | 781,761 | <p>There is no need to assume <span class="math-container">$¬C$</span>, here is an intuitionistic derivation:</p>
<p><span class="math-container">$1). ¬(A → B) - premise$</span></p>
<p><span class="math-container">$2). (¬B → C) - premise$</span></p>
<p><span class="math-container">$3). B - assumption$</span></p>
<p><sp... |
3,845,570 | <p>Premises: <span class="math-container">$\neg(A \to B)\ ,\ \neg B \to C$</span> .</p>
<p>Conclusion: <span class="math-container">$C$</span></p>
<p>My intuition is that I should do a sub-derivation where I prove <span class="math-container">$\neg C$</span> is an absurdity. However, I soon run into issues. If I could... | user2661923 | 464,411 | <p><span class="math-container">$A \Rightarrow B$</span> is logically equivalent to <span class="math-container">$(\neg A) \vee B$</span>.</p>
<p><strong>Start of Edit Insert</strong><br>
See lemontree's comments/reactions to my answer which seem to indicate that the above statement can not be assumed, but rather must ... |
2,352,721 | <h2>Question</h2>
<blockquote>
<p>Four fair six-sided dice are rolled. The probability that the sum of the results being <span class="math-container">$22$</span> is <span class="math-container">$$\frac{X}{1296}.$$</span> What is the value of <span class="math-container">$X$</span>?</p>
</blockquote>
<h2>My Approach</h2... | adhg | 17,222 | <p>On an intuitive level, simply count how many ways are there to get 22. There are 2 ways to do so: </p>
<p><code>(A) 6,6,6,4
(B) 6,6,5,5</code></p>
<p>Each way can have different permutation (example 6,6,6,4, 6,6,4,6 etc) so for the first way (A), you can have 4 ways to arrange the result because: <code>4! / 3! =... |
169,097 | <p>In the beginning of chapter two in The HoTT Book there is a discussion about synthetic vs. analytic geometry:</p>
<blockquote>
<p>An important difference between homotopy type theory and classical homotopy theory is that homotopy type theory provides a <em>synthetic</em> description of spaces, in the following se... | François G. Dorais | 2,000 | <p>The (homotopy-theoretic) circle is discussed in Chapter 6 of the book and it does have points. In fact, it is defined as a higher inductive type with an explicit point $\mathsf{base}:S^1$ and one nontrivial identification $\mathsf{loop}:\mathsf{base} =_{S^1} \mathsf{base}$.</p>
<p>This paragraph is about something ... |
60,326 | <p>If f is a weight 2 cuspidal newform, then it is common for L(f,1) to vanish. Indeed, the sign of the functional equation of f can force such vanishing. However, if f has weight k>2, then there is no a priori reason why L(f,1) will vanish. </p>
<p>My question: are there known examples where L(f,1)=0 for a newform... | Rob Harron | 1,021 | <p>Based on your normalization, $L(s,f)$ is defined as an Euler product for $\Re(s)>\frac{k+1}{2}$, so $L(s,f)$ is non-zero in that right-half plane. Now Jacquet–Shalika <a href="http://www.ams.org/mathscinet-getitem?mr=432596" rel="noreferrer">MR0432596</a> showed that that non-zero region extends to the line $\Re(... |
6,741 | <p>I would like to have a plot that is filled to the axis with <code>Green</code> if the <code>y</code> value is greater than 10, and <code>Blue</code> if it is less than 10.</p>
<p>I attempted to do the following:</p>
<p><code>Plot[x, {x, 0, 20}, Filling -> Axis,
ColorFunction ->
Function[{x, y}, Piecewis... | Heike | 46 | <p>The problem is not that the test is only evaluated once but that by default <code>ColorFunctionScaling</code> is set to <code>True</code> which means that the coordinates are rescaled to lie in the interval $[0,1]$ before being fed to <code>ColorFunction</code>. Try this instead</p>
<pre><code>Plot[x, {x, 0, 20}, F... |
946,973 | <p>After completing the square, what are the solutions to the quadratic equation below?
<span class="math-container">$$x^2 + 2x = 25$$</span></p>
<p><img src="https://i.stack.imgur.com/AoFhV.png" alt="enter image description here" /></p>
<p>Honstely I think it's B. But I'm not sure.</p>
| Mike | 17,976 | <p>For the quadratic equation $x^2+bx+c=0$, the sum of the roots is $-b$ and the product is $c$. So for $x^2+2x-25=0$, the $2$ roots sum to $-2$, which eliminates A and B. The product is $-25$. The product of the roots for D is clearly $-24$, which eliminates this answer, leaving C. And it can be quickly verified</... |
1,111,952 | <p><strong>My Try:</strong> </p>
<p>We substitute $y = x^{2/3}$. Therefore, $x = y^{3/2}$ and $\frac{dx}{dy} = \frac{2}{3}\frac{dy}{y^{1/3}}$</p>
<p>Hence, the integral after substitution is: </p>
<p>$$ \frac{3}{2} \int_0^\infty \sin(y)\sqrt{y} dy$$</p>
<p>Let's look at:</p>
<p>$$\int_0^\infty \left|\sin(y)\sqrt{... | GEdgar | 442 | <p>Maple writes the indefinite integral as
$$
\int \sin(x^{2/3})\,dx = \frac{-3x^{1/3}\cos(x^{2/3})}{2}+
\frac{3\sqrt{\pi}\;C\left(\displaystyle \frac{\sqrt{2} x^{1/3}}{\sqrt{\pi}}\right)}{2\sqrt{2}}
$$
where $C$ is the Fresnel C function. The term with the Fresnel function does converge, but the first term oscillates... |
4,547,918 | <p>Given the torus and given the point p <span class="math-container">$\in$</span> M corresponding to the parameters <span class="math-container">$s=\frac{\pi }{4}$</span> and <span class="math-container">$t=\frac{\pi }{3}$</span>.
Determine the cartesian equation of the tangent plane to M in p.</p>
<p><span class="mat... | Suzu Hirose | 190,784 | <p>Did Wolfram Alpha really produce that?</p>
<p><span class="math-container">$\cos^2 x=1-\sin^2x=(1+\sin x)(1-\sin x)$</span>.</p>
<p>You can do the rest yourself.</p>
|
3,444,214 | <p>Let <span class="math-container">$\zeta = e^{2\pi i / 7}$</span>. I know the minimal polynomial of <span class="math-container">$\zeta$</span> over <span class="math-container">$\mathbb{Q}$</span> is <span class="math-container">$\sum_{i=0}^{6} x^{i}$</span>. But what is <span class="math-container">$[ \mathbb{Q}(\z... | nguyen quang do | 300,700 | <p>For an odd prime, the cyclotomic field <span class="math-container">$K=\mathbf Q(\zeta_p)$</span> has cyclic Galois group of order <span class="math-container">$\frac {p-1}2$</span>, hence admits a unique subextension <span class="math-container">$L$</span> s.t. <span class="math-container">$[K:L]=2$</span>, which i... |
1,817,035 | <p>Gradient descent reduces the value of the objective function in each iteration. This is repeated until convergence happens.</p>
<p>The question is if the norm of gradient has to decrease as well in every iteration of gradient descent?</p>
<hr>
<p><strong>Edit:</strong> How about when the objective is a convex fun... | Luca Citi | 197,925 | <p>No. Take for example $f(x)=\sqrt{|x|}$. While $f(x)$ decreases monotonically at each step under the assumption that steps are small enough (or if you apply backstepping to enforce monotonicity), the gradient $(2 \, \mathrm{sign}(x) \, f(x))^{-1}$ gets larger in modulus as the current iterate $x$ approaches the solut... |
492,407 | <p>I was searching for methods on how to calculate the area of a polygon and stubled across this: <a href="http://www.mathopenref.com/coordpolygonarea.html" rel="nofollow noreferrer">http://www.mathopenref.com/coordpolygonarea.html</a>.
$$
\mathop{area} =
\left\lvert\frac{(x_1y_2 − y_1x_2) + (x_2y_3 − y_2x_3) + \cdots... | Christian Blatter | 1,303 | <p>The formula in question can be explained by means of Green's formula in the plane.</p>
<p>Let ${\bf F}=(P,Q)$ be a force field in the plane, and assume that $\Omega$ is a finite domain with boundary cycle $\partial\Omega$. Then Green's formula says that
$$\int\nolimits_{\partial B} (P\>dx+Q\>dy)=\int\nolimits... |
289,864 | <p>Let $C_{1}$ be a circle of unit radius. Let A and B be two points inside $C_{1}$. Now I want to construct another circle $C_{2}$ such that A and B lie on $C_{2}$ and $C_{2}$ is orthogonal to $C_{1}$ at their point of intersection(I want $C_{2}$ in such a way that it intersects $C_{1}$). I tried and failed to find a ... | Sigur | 31,682 | <p>Take the inverse of $A$ with respect to the circle $C_1$ (<a href="https://math.stackexchange.com/q/146264/31682">see here</a>) to obtain the third point $C$. Now construct the circle containing $A,B,C$.</p>
<p><img src="https://i.stack.imgur.com/UNdqV.png" alt="enter image description here"></p>
|
514,912 | <p>I have what may seem a very trivial question, but how it is answered may affect how a proof of mine is structured. It pertains to formatting and convention. When 'recursively' defining a function does it make sense to use quantifiers? </p>
<p>For example would:</p>
<p>$ 5 \in R $</p>
<p>If $ r \in R $, then $ \fo... | Amateur | 80,208 | <p>Here's for
$$
\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{\cdots}}}}.
$$
First define a sequence recursively by $a_1 := \sqrt{6}$ and $a_n := \sqrt{6+a_{n-1}}$ ($n \geq 2)$. Then we wish to compute $\lim_{n \to \infty} a_n$. In order to find this limit, we need, of course, to know that it exists. But instead of trying to show rig... |
3,521,525 | <p>How many three-digit numbers are there whose digits in the hundreds place and ones place are the same? (Assume that a nonzero digit is in the hundreds place.) </p>
<p>Please try to simplify the solution so that a child under 14 may understand this. Also, it would help if you included a formula that <em>may</em> be... | David G. Stork | 210,401 | <p>All the numbers are of the form <span class="math-container">$XYX$</span>. Note that <span class="math-container">$X$</span> can be <span class="math-container">$1 \to 9$</span>. For each of these <span class="math-container">$9$</span> cases the middle digit <span class="math-container">$Y$</span> can be any of t... |
4,047,601 | <p>I did a question <span class="math-container">$\int_{0}^{1}\frac{1}{x^{\frac{1}{2}}}\,dx$</span>, and evaluating this is divergent integral yes? Then as a general form <span class="math-container">$\int_{0}^{1} \frac{1}{x^p}\,dx$</span>, <span class="math-container">$p \in \mathbb{R}$</span>, what values of <span cl... | pawel | 879,543 | <p>First of all <span class="math-container">$$\int_{0}^{1}\frac{1}{x^\alpha}$$</span> is divergent if <span class="math-container">$\alpha>1$</span>, so the first case cited by you (for <span class="math-container">$\alpha=\frac{1}{2}$</span>) corresponds to convergence.</p>
<p>Now let's observe that:
<span class="... |
2,498,359 | <p>This is a basic probability question. </p>
<p>Persons A and B decide to arrive and meet sometime between 7 and 8 pm. Whoever arrives first will wait for ten minutes for the other person. If the other person doesn't turn up inside ten minutes then the person waiting will leave. What is the probability that they will... | Abhiram Natarajan | 481,835 | <p>My favourite way of solving uniform distribution related problems is to try and think of them in terms of relative volume/area/length. For instance, take a continuous uniform distribution on $[0, 1]$. The probability that $X \in (a, b)$, where $0 \le a \le b \le 1$, and $X$ is a random variable drawn from that distr... |
3,908,955 | <p>Is the given series convergent or divergent? Give a reason. Show details.</p>
<p><span class="math-container">$$\sum_{n=2}^{\infty} \frac{(-i)^n}{ln \ n}$$</span></p>
<p>So maybe I'll try using the ratio test?</p>
<p>So the series converges if <span class="math-container">$$\left| \frac{z_{n+1}}{z_n} \right| < 1$... | user2661923 | 464,411 | <p>Normally, before or while I present an answer, I am supposed to respond to the OP's question(s) and point out any errors or omissions. I simply can't do that here. I have to agree with Ted Shifrin's comment.</p>
<p>Let <span class="math-container">$f(x) = x^2 + x + 1.$</span></p>
<p>To Prove:</p>
<p><span class="m... |
4,069,499 | <p>If we let <span class="math-container">$x = 0$</span>.</p>
<p><span class="math-container">\begin{align*}
3(0+7)-y(2(0)+9) \\
21-9y \\
\end{align*}</span></p>
<p>Then <span class="math-container">$9y$</span> should always equal <span class="math-container">$21$</span>?
Solving for <span class="math-container">$y$</... | Deepak | 151,732 | <p>The easiest way to think about this is to realise you need the expression to be independent of the value of <span class="math-container">$x$</span>. Which means the coefficient of the <span class="math-container">$x$</span> term has to vanish, leaving only a constant term.</p>
<p>That means <span class="math-contain... |
3,880,743 | <p>If <span class="math-container">$T:\mathbb{R}^2 \rightarrow \mathbb{R}$</span> is a function such that <span class="math-container">$T(\alpha v)=\alpha T(v)$</span> <span class="math-container">$\forall \alpha \in \mathbb{R}$</span> and <span class="math-container">$v \in \mathbb{R}^2$</span>, is T necessarily a lin... | Kenta S | 404,616 | <p>Let <span class="math-container">$T(x,y)=\sqrt[3]{x^3+y^3}$</span>. Clearly <span class="math-container">$T(\alpha x,\alpha y)=\alpha T(x,y)$</span> for all <span class="math-container">$x,y,\alpha\in\mathbb R$</span>, but <span class="math-container">$T(1,1)=\sqrt[3]2\ne T(0,1)+T(1,0)=2$</span>.</p>
|
1,783,601 | <p>Find the residue at $\pi$ for the function defined by $$\dfrac{z^2+\sin\left(z\right)}{\left(z-\pi\right)^4}$$</p>
<p>I thought I could do this using the 'gh rule' however this gives $$\dfrac{\pi^2+\sin\left(\pi\right)}{4\left(\pi-\pi\right)^3}$$ which is undefined.
Is there an alternative way to calculate this? </... | bambihunter | 339,317 | <p>multiply the integrand by </p>
<p>$$
1=\frac{\left(-\sqrt{1-x}-\sqrt{x+1}+2\right) \left(\sqrt{1-x^2}+1\right)}{\left(-\sqrt{1-x}-\sqrt{x+1}+2\right) \left(\sqrt{1-x^2}+1\right)}
$$</p>
<p>doing the alegbra correctly this indeed eliminates the roots in the denominator and we end up with
$$
I=\frac{1}{2}\int_{0}^{... |
1,783,601 | <p>Find the residue at $\pi$ for the function defined by $$\dfrac{z^2+\sin\left(z\right)}{\left(z-\pi\right)^4}$$</p>
<p>I thought I could do this using the 'gh rule' however this gives $$\dfrac{\pi^2+\sin\left(\pi\right)}{4\left(\pi-\pi\right)^3}$$ which is undefined.
Is there an alternative way to calculate this? </... | Noam Shalev - nospoon | 219,995 | <p>I'm not sure this is the most straight forward way, but here it is anyway.</p>
<p>Start with the substitution $\sqrt{1-x}\mapsto x $. The integral transforms into</p>
<p>$$I=\int_0^1 \frac{2 x}{2+x+\sqrt{2-x^2}}dx.$$
Next, rationalize the denominator by multiplying the numerator and the denominator by $2+x-\sqrt{2... |
2,825,652 | <p>I have a magnetometer sensor on each vertices of an isosceles triangle. I also have a magnet that can be anywhere on the triangle (inside, on edges, etc). I have the magnitude reading from each sensor (essentially giving me the distance the magnet is from each vertices of the triangle). I'd like to calculate the x,y... | coffeemath | 483,139 | <p>I'll assume the magnet inside the closure of the triangle. It can be done using only two of the distances, if things are set up right. After a translation and a rotation, we may assume $A=(0,0),B=(0,c),$ and $C=(c_1,c_2)$ where $c_2>0.$ Here $c$ is the length of the side opposite $C.$</p>
<p>Now let $P=(x,y)$ be... |
1,328,909 | <p>I know how to find for which $n$ $\phi(n)=n/2$ or $\phi(n)=n/3$, my method for finding those was simply to find primes $p$ that satisfy $\Pi_p$$_|$$_n$$1-1/p$ $ = 1/2$ or $1/3$.</p>
<p>However, I don't know how to find $\Pi_p$$_|$$_n$$1-1/p = n/6$. Intuitively it seems that if I combine results for both $\phi(n) =... | Adelafif | 229,367 | <p>If we take the Cantor's set C and and take its complement E and take the characteristic function f of E then f is continuous on E(being a countable disjoint union of intervals) and it is discontinuous on C since it is perfect and nowhere dense in R and since f is 0 on C. </p>
|
2,022,423 | <p>You are asked to <strong>permute the neighboring sub-sequence</strong> of the sequence $n,n-1,n-2,\cdots,1$ until the sequence is brought to the increasing order. </p>
<p>By <em>permute the neighboring sub-sequence</em> I mean for example:
$5,4,3,2,1 \to 5,3,4,2,1 $ or $5,4,3,2,1\to 5,2,4,3,1$ or $5,4,3,2,1\to5,2,... | Alex Ravsky | 71,850 | <blockquote>
<p>A detailed canonical answer is required</p>
</blockquote>
<p>As far as I understood, you are asking about a minimal number <span class="math-container">$d(\iota)$</span> of transpositions (swaps of adjacent blocks) required to sort (that is, transform to identity) the completely reverted permutation ... |
1,942,854 | <blockquote>
<p>Does there exist an $n \not \equiv 3 \pmod{4}$ where $n \in \mathbb{N}$ and is greater than $1$ such that there exists a prime $p >5$ such that \begin{cases}3^{n^2-1} &\equiv 1 \pmod{p}\\2^{n^2-1} &\equiv 1 \pmod{p}?\end{cases}</p>
</blockquote>
<p>I tried finding such a prime but I could... | fleablood | 280,126 | <p>$f (x) =-\frac 23 x^{-\frac 43}=-\frac 23 \frac 1 {x^{4/3}}=-\frac 23 \sqrt [3]{1/x^4} $. </p>
<p>As $1/x^4 $ is always positive. $f $ is always negative. Not just for $x = -5$ but <strong>all</strong> $x $. (Except $x= 0$ where the function isn't defined.)</p>
|
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