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 |
|---|---|---|---|---|
3,746,597 | <p>My assumption would be</p>
<p><span class="math-container">$$\int_{-a}^a x\ dx=0$$</span></p>
<p>Am I on the right track here? Also, for indefinite integrals</p>
<p><span class="math-container">$$\int (f)x\ dx$$</span></p>
<p>would this be correct as well?</p>
<p><strong>Background</strong></p>
<p>My professor raise... | user0102 | 322,814 | <p>Here it is a more general result which may help you.</p>
<p>Consider that the function <span class="math-container">$f:\textbf{R}\to\textbf{R}$</span> is odd. This means that <span class="math-container">$f(-x) = -f(x)$</span>. Thus one has that
<span class="math-container">\begin{align*}
\int_{-a}^{a}f(x)\mathrm{d}... |
4,566,254 | <p>Let <span class="math-container">$F$</span> be a functor <span class="math-container">$\mathscr{C}^\text{op}\times\mathscr{C}\to\mathbf{Set}$</span>, and let <span class="math-container">$S$</span> be an arbitrary set. Can we write the following?
<span class="math-container">$$
\int^{C:\mathscr{C}} S\times F(C,C) \c... | Dabouliplop | 426,049 | <p>Yes, this is true. It can be proved in several way. Let's see that in a concrete way. A point of <span class="math-container">$∫^c S×F(c,c)$</span> is given by some triple <span class="math-container">$(c,s,x)$</span> with <span class="math-container">$\newcommand{\C}{\mathscr{C}}c∈\C$</span>, <span class="math-cont... |
1,302,708 | <p>I have 2 sets of elements, say $A=\{a\}$ (only 1 element) and $B = \{b_1, b_2,..., b_n\}$.</p>
<p>The probability of picking $A$ is $0.3$ and the probability of picking $B$ is $0.7$, and all elements in $B$ have equal chances.</p>
<p>Now combining everything together:</p>
<p>What would be the probability of picki... | mjqxxxx | 5,546 | <p>If either of the exponents is strictly positive $(\alpha > 0)$, then $(0,1)$ or $(1,0)$ is a solution, because $0^{\alpha} + 1^{\beta} = 1$ for any $\beta$. This occurs for all $\theta \in (-\pi/2, \pi)$, and leaves two cases:</p>
<ul>
<li>One exponent is zero and the other is strictly negative: $\theta\in\{-\p... |
1,302,708 | <p>I have 2 sets of elements, say $A=\{a\}$ (only 1 element) and $B = \{b_1, b_2,..., b_n\}$.</p>
<p>The probability of picking $A$ is $0.3$ and the probability of picking $B$ is $0.7$, and all elements in $B$ have equal chances.</p>
<p>Now combining everything together:</p>
<p>What would be the probability of picki... | Piquito | 219,998 | <p>For all couple of natural numbers (n,m) we have $\frac{1}{n} + \frac{1}{m}$$\leq$ $n^{cos\theta}$+ $m^{sin\theta}$$\leq n+m$ because<br>
$\frac{1}{n}$$\leq$ $n^{cos\theta}$$\leq{n}$ and the same for the sinus; therefore, when $\frac{1}{n} + \frac{1}{m}$<1 one has by continuity, a value of $\theta$ (actually in... |
4,126,470 | <p><span class="math-container">$$\sum_{n=2}^{∞} \frac{1}{n\left(\left(\ln\left(n\right)\right)^3+\ln\left(n\right)\right)}$$</span></p>
<p>I know that there are several methods of finding the convergence of a series. The ratio test, the comparison test, the limit comparison test. There is also this theorem: If a serie... | Gary | 83,800 | <p>You can use comparison and integral test. The terms of the series are all positive and we have
<span class="math-container">\begin{align*}
\sum\limits_{n = 2}^\infty {\frac{1}{{n(\log ^3 n + \log n)}}} & \le \sum\limits_{n = 2}^\infty {\frac{1}{{n\log ^2 n}}} < \frac{1}{{2\log ^2 2}} + \sum\limits_{n = 3}^... |
2,168,906 | <blockquote>
<p>The task is to find necessary and sufficient condition on <span class="math-container">$b$</span> and <span class="math-container">$c$</span> for the equation <span class="math-container">$x^3-3b^2x+c=0$</span> to have three distinct real roots.</p>
</blockquote>
<p>Are there any formulas (such as <spa... | Mark Viola | 218,419 | <p>If $y=e^{x^x}$, then $\log(y)=x^x\ne x\log(e^x)=\log(e^{x^2})$</p>
<p>Note that $e^{x^x}\ne (e^x)^x=e^{x^2}$.</p>
<p>So, to differentiate $y$, we use $\log(\log(y))=x\log(x)$. Then, </p>
<p>$$\frac{d\log(\log(y))}{dx}=\log(x)+1=\frac{1}{y\log(y)}\frac{dy}{dx}$$</p>
<p>whence solving for $\frac{dy}{dx}$ and usin... |
1,101,371 | <p>Any book that I find on abstract algebra is somehow advanced and not OK for self-learning. I am high-school student with high-school math knowledge. Please someone tell me a book can be fine on abstract algebra? Thanks a lot. </p>
| Theodore Sternberg | 207,475 | <p>Try Pinter's <em>A Book of Abstract Algebra</em>. Over half the book is extended problems that you're gently led through as you sort of discover algebra on your own. There are nice applications to computer science, genetics and kinship networks too. You'll have a great time!</p>
<p>Stahl's <em>Introductory Moder... |
184,601 | <p>A user on the chat asked how could he make something that would cap when it gets a specific value like 20. Then the behavior would be as follows:</p>
<p>$f(...)=...$</p>
<p>$f(18)=18$</p>
<p>$f(19)=19$</p>
<p>$f(20)=20$</p>
<p>$f(21)=20$</p>
<p>$f(22)=20$</p>
<p>$f(...)=20$</p>
<p>He said he would like to pe... | celtschk | 34,930 | <p>While we are at fancy expressions, what about
$$20-\lim_{n\to\infty}\frac1n\ln\left(1+\mathrm e^{n(20-x)}\right)$$</p>
|
245,464 | <p>I only have one region plot and still want to get the legend (both marker and label). I tried the following, but why the legend market does not show up?</p>
<p><code>RegionPlot[x^2 < y^3 + 1 && y^2 < x^3 + 1, {x, -2, 5}, {y, -2, 5}, PlotLegends -> Placed["MyLegend", {0.15, 0.08}]]</code>... | kglr | 125 | <p>If the only reason for using <code>Mesh</code> is to get a hatch filling for the region, you can use <a href="https://reference.wolfram.com/language/ref/HatchFilling.html" rel="nofollow noreferrer"><code>HatchFilling</code></a>:</p>
<pre><code>RegionPlot[x^2 < y^3 + 1 && y^2 < x^3 + 1, {x, -2, 5}, {y, ... |
2,666,772 | <blockquote>
<p>$W$ = $\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 \end{bmatrix}$. Use W to build an 8x8 matrix encoding an orthonormal basis in $R^8$ by scaling A = $\begin{bmatrix} W & W \\ W & -W \end{bmatrix}$ in t... | user | 505,767 | <p>Note that $$W^T\cdot W=4I$$ then</p>
<p>$$\begin{bmatrix}W&0\\0&W\end{bmatrix}^T\begin{bmatrix}W&0\\0&W\end{bmatrix}=4\begin{bmatrix}I&0\\0&I\end{bmatrix}$$</p>
<p>then assume</p>
<p>$$U=\begin{bmatrix} \frac1{\sqrt4} & \frac1{\sqrt4} & \frac1{\sqrt4} & \frac1{\sqrt4} \\ \frac1... |
3,773,695 | <p>I have been trying to get some upper bound on the coefficient of <span class="math-container">$x^k$</span> in the polynomial
<span class="math-container">$$(1-x^2)^n (1-x)^{-m}, \text{ $m \le n$}.$$</span></p>
<p>A straightforward calculation shows that for even <span class="math-container">$k$</span>, the coefficie... | Brian Moehring | 694,754 | <p><strong>A short disclaimer/warning:</strong> I haven't studied these types of problems formally, so there may be a standard, simpler way to deal with this. This is just the method that occurred to me.</p>
<p>Also, in the general proof, I abuse the symbol <span class="math-container">$y'$</span> a fair bit to mean t... |
3,277,555 | <p>For a math class I was given the assignment to make a game of chance, for my game the person must roll 4 dice and get a 6, a 5, and a 4 in a row in 3 rolls or less to qualify. the remaining dice must be over 3 for you to win. my question though is how can I find out the probability of rolling the 6,5, and 4 in a sin... | drhab | 75,923 | <p>If I understand well then the question is: "if <span class="math-container">$4$</span> dice are thrown then what is the probability that <span class="math-container">$4$</span>, <span class="math-container">$5$</span> and <span class="math-container">$6$</span> are among the results?" (please correct me if I am wron... |
1,110,543 | <p>I am dealing with galois theory at the moment and I came across with an example in the lecture and I got a question:</p>
<p>Let $K=\mathbb Q$ and $L=\mathbb Q(\sqrt{2},\sqrt{3})\subset \mathbb C$. Lets consider the field-extension $L/K$</p>
<p>Because of $[L:K]=[L:K(\sqrt{2})]\cdot [K(\sqrt{2}):K]=4$</p>
<p><stro... | Andrea Mori | 688 | <p>(1) Suppose that $\sqrt{3}=a+b\sqrt{2}$ with $a,b\in\Bbb Q$. Taking squares
$$
(a^2+2b^2)+2ab\sqrt{2}=3.
$$
Can you derive a contradiction?</p>
<p>(2) The cardinality of the Galois group of a Galois extension always coincides with the degree. What are the generators in this case, and what is the effect of the autom... |
3,504,422 | <blockquote>
<p>Find: <span class="math-container">$$\displaystyle\lim_{x\to
\infty}\left(\frac{\ln(x^2+3x+4)}{\ln(x^2+2x+3)}\right)^{x\ln x}$$</span></p>
</blockquote>
<p>My attempt:</p>
<p><span class="math-container">$\displaystyle\lim_{x\to\infty}\left(\frac{\ln(x^2+3x+4)}{\ln(x^2+2x+3)}\right)^{x\ln x}=\lim_{... | Michael Rozenberg | 190,319 | <p>By your work and since <span class="math-container">$e^x$</span> and <span class="math-container">$\ln$</span> are continuous function, we obtain: <span class="math-container">$$\lim_{x\rightarrow+\infty}\left(\frac{\ln(x^2+3x+4)}{\ln(x^2+2x+3)}\right)^{x\ln{x}}=\lim_{x\rightarrow+\infty}\left(1+\frac{\ln\frac{x^2+3... |
441,962 | <p>I looking for a proof for the theorem but I have not find yet.</p>
<p>A link or even sketch for How it goes will be very appreciate.</p>
<p>A linear map is self adjoint </p>
<p>iff </p>
<p>the matrix representation according to orthonormal basis is self adjoint.</p>
<p>by the way is not that true for all self a... | Hagen von Eitzen | 39,174 | <p>The adjoint $A^*$ of a linear map $A$ on a space with scalar product is determined by the property $\langle Ax,y\rangle = \langle x, A^*y\rangle$ for all $x,y$. If we let $x,y$ run through the base vectors of an ON basis, then $\langle Ae_i,e_j\rangle$ is the $i,j$ entry of the matrix for $A$ and $\langle e_i,A^*e_j... |
979,144 | <p>I am searching for a formula of sum of binomial coefficients $^{n}C_{k}$ where $k$ is fixed but $n$ varies in a given range? Does any such formula exist?</p>
| robjohn | 13,854 | <p>As shown in <a href="https://math.stackexchange.com/a/399213">this answer</a>, we have the formula
$$
\sum_{j=k}^{n-m}\binom{j}{k}\binom{n-j}{m}=\binom{n+1}{k+m+1}
$$
If we set $m=0$, we get
$$
\sum_{j=k}^n\binom{j}{k}=\binom{n+1}{k+1}
$$</p>
|
3,702,649 | <p>It's obvious that it's symmetric because <span class="math-container">$a_{\left(i+1\right)j}=\left(m+1\right)\left(i+1+j\right) = a_{i\left(j+1\right)}=\left(m+1\right)\left(i+j+1\right)$</span>, but how can I prove that it's a Latin square and that it's diagonal consists of different elements?</p>
<p>I thought abo... | Rebecca J. Stones | 91,818 | <p>It's true if and only if <span class="math-container">$m+1$</span> and <span class="math-container">$n$</span> are coprime. In particular, the cells <span class="math-container">$(i,j)$</span> and <span class="math-container">$(i,j+n/\gcd(m+1,n))$</span> are in the same row and contain the same symbol. However, th... |
148,032 | <p>What is the larger of the two numbers?</p>
<p>$$\sqrt{2}^{\sqrt{3}} \mbox{ or } \sqrt{3}^{\sqrt{2}}\, \, \; ?$$
I solved this, and I think that is an interesting elementary problem. I want different points of view and solutions. Thanks!</p>
| Robert Mastragostino | 28,869 | <p>$$\sqrt2^{\sqrt 3}<^?\sqrt3^{\sqrt 2}$$
Raise both sides to the power $2\sqrt 2$, and get an equivalent problem:
$$2^{\sqrt 6}<^?9$$
Since $\sqrt 6<3$, we have:
$$2^{\sqrt 6}< 2^3 = 8 <9$$
So ${\sqrt 2}^{\sqrt 3}$ is smaller than $\sqrt3^{\sqrt 2}$.</p>
|
1,243,661 | <p>Let $\Theta$ be an unknown random variable with mean $1$ and variance $2$. Let $W$ be another unknown random variable with mean $3$ and variance $5$. $\Theta$ and $W$ are independent.</p>
<p>Let: $X_1=\Theta+W$ and $X_2=2\Theta+3W$. We pick measurement $X$ at random, each having probability $\frac{1}{2}$ of being c... | BruceET | 221,800 | <p>First, $Var(X_2) = Var(2\Theta + 3W) = 4Var(\Theta) + 9Var(W)$ and
similarly for $Var(X_1)$. However, you cannot average the two variances to get $Var(X)$. This does not take account of the
randomness of the coin toss to decide whether to pick $X_1$ or $X_2.$
[And even if you deterministically chose each $X_i$ alter... |
596,671 | <blockquote>
<p>$$\sum^{\infty}_{n=1}\frac{(-1)^n}{n^a\ln n}$$
$$a>0$$</p>
<p>Does the series converge/converge absolutely/diverge ?</p>
</blockquote>
<p>I tried to divide to cases and factor the series:</p>
<p>$\sum^{\infty}_{n=1}\frac{(-1)^n}{n^a\ln n}=\sum\frac{(-1)^n}{n^a}\sum\frac{1}{\ln n}$</p>
<p>... | Community | -1 | <p>Firstly note that your series should be $$\sum_{n=2}^{\infty}\frac{(-1)^n}{n^a \ln n}$$ since the logarithm function becomes zero at $n=1$. Let us consider the absolute valued series first,</p>
<p>$$\sum_{n=2}^{\infty}\frac{1}{n^a \ln n}$$</p>
<p>Note that, </p>
<p>$$\frac{1}{n^a \ln n}<\frac{1}{n^a}\mbox{ for... |
596,671 | <blockquote>
<p>$$\sum^{\infty}_{n=1}\frac{(-1)^n}{n^a\ln n}$$
$$a>0$$</p>
<p>Does the series converge/converge absolutely/diverge ?</p>
</blockquote>
<p>I tried to divide to cases and factor the series:</p>
<p>$\sum^{\infty}_{n=1}\frac{(-1)^n}{n^a\ln n}=\sum\frac{(-1)^n}{n^a}\sum\frac{1}{\ln n}$</p>
<p>... | robjohn | 13,854 | <p>The <a href="http://en.wikipedia.org/wiki/Alternating_series_test" rel="nofollow">alternating series test</a> says that
$$
\sum_{n=2}^\infty\frac{(-1)^n}{n^a\log(n)}
$$
converges conditionally since $\frac1{n^a\log(n)}$ monotonically converges to $0$.</p>
<p>For absolute convergence, by comparison to
$$
\sum_{n=3}^... |
1,567,229 | <p>Let Y1 and Y2 have the joint probability density function given by:</p>
<p>$ f (y_1, y_2) = 6(1−y_2), \text{for } 0≤y_1 ≤y_2 ≤1$</p>
<p>Find $P(Y_1≤3/4,Y_2≥1/2).$</p>
<p>Answer:</p>
<p>$$\int_{1/2}^{3/4}\int_{y_1}^{1}6(1− y_2 )dy_2dy_1 + \int_{1/2}^{1}\int_{1/2}^{1}6(1− y_2 )dy_1dy_2 = 7/64 + 24/64 = 31/64 $$... | Alex | 38,873 | <p>8 digit number cant start with 0. Hence: choose 3 out of 7 for 0s, then 3 out of 5 for 2s, multiply it by 2!</p>
|
1,567,229 | <p>Let Y1 and Y2 have the joint probability density function given by:</p>
<p>$ f (y_1, y_2) = 6(1−y_2), \text{for } 0≤y_1 ≤y_2 ≤1$</p>
<p>Find $P(Y_1≤3/4,Y_2≥1/2).$</p>
<p>Answer:</p>
<p>$$\int_{1/2}^{3/4}\int_{y_1}^{1}6(1− y_2 )dy_2dy_1 + \int_{1/2}^{1}\int_{1/2}^{1}6(1− y_2 )dy_1dy_2 = 7/64 + 24/64 = 31/64 $$... | true blue anil | 22,388 | <p><em>A simple way</em></p>
<p>You computed the ans as $1120$.<br>
Just multiply it by $\dfrac58$, the probability that it starts with a non-zero</p>
<p>The ans it yields (which others have also got) is $700$,</p>
<p>the options are definitely wrong.</p>
|
362,854 | <blockquote>
<p>Show that every subgroup of $Q_8$ is normal.</p>
</blockquote>
<p>Is there any sophisticated way to do this ? I mean without needing to calculate everything out.</p>
| Geoff Robinson | 13,147 | <p>It depends what you call sophisticated. There is only one subgroup of order $8,$ one subgroup of order $2$ and one subgroup of order $1$ in $Q_{8},$ so each of those is normal. In any finite $p$-group, every maximal subgroup is normal, so each subgroup of order $4$ of $Q_{8}$ is normal. This is not really substantia... |
221,729 | <p>Till now, I have proved followings;</p>
<p>Suppose $X,Y$ are metric spaces and $E$ is dense in $X$ and $f:E\rightarrow Y$ is uniformly continuous. Then,</p>
<ol>
<li><p>$Y=\mathbb{R}^k \Rightarrow \exists$ a continuous extension.</p></li>
<li><p>$Y$ is compact $\Rightarrow \exists$ a continuous extension.</p></li>... | Tsz Chung Ho | 136,092 | <p>The definition of the derivative can be expressed using asymptotic notation.</p>
<p>We say f has a derivative at x if there exists M such that:</p>
<p><span class="math-container">$$f(x+\epsilon) = f(x) + M\epsilon + o(\epsilon)$$</span></p>
<p>We denote this M as f '(x)</p>
<p>(edited as per Antonio's correctio... |
546,701 | <p>Find the number of positive integers $$n <9,999,999 $$ for which the sum of the digits in n equals 42.</p>
<p>Can anyone give me any hints on how to solve this?</p>
| Brian M. Scott | 12,042 | <p>Let the digits be $d_1,d_2,d_3,d_4,d_5,d_6$, and $d_7$, where we allow leading zeroes so as to make each number in the specified interval a seven-digit integer. You’re looking for all solutions in non-negative integers to </p>
<p>$$d_1+d_2+d_3+d_4+d_5+d_6+d_7=42\;,$$</p>
<p>with the restriction that $d_k\le 9$ for... |
1,033,208 | <p>What do square brackets mean next to sets? Like $\mathbb{Z}[\sqrt{-5}]$, for instance. I'm starting to assume it depends on context because google is of no use.</p>
| annimal | 157,326 | <p>it's the polynomial ring in the bracket with coefficients $\mathbf{Z}$, for example, $\mathbf{Z}[x]$ is the polynomial of x with coefficients $\mathbf{Z}$, like $x^3 + 2x^2 + 3$, and for $\mathbf{Z}[\sqrt{-5}]$, just replace x by $\sqrt-5$, the only difference is that $(\sqrt{-5}) ^2 = 5$, which is in $\mathbf{Z}$,... |
3,276,332 | <p>I'm studying for my exam in discrete mathematics and found the following problem on last years exam:</p>
<p>Find a closed formula without using induction for <span class="math-container">$\sum_{k=0}^n k^3$</span>.</p>
<p>I tried it by finding the Generating Function first:</p>
<p><span class="math-container">$F(x... | Acccumulation | 476,070 | <p>There are several methods. There's the linear algebra method of finding difference formulas for <span class="math-container">$k^m$</span>, treating a difference formula for a particular <span class="math-container">$m$</span> as a vector, then finding <span class="math-container">$\sum k^m$</span> in terms of those ... |
3,276,332 | <p>I'm studying for my exam in discrete mathematics and found the following problem on last years exam:</p>
<p>Find a closed formula without using induction for <span class="math-container">$\sum_{k=0}^n k^3$</span>.</p>
<p>I tried it by finding the Generating Function first:</p>
<p><span class="math-container">$F(x... | Community | -1 | <p><span class="math-container">$$\sum_{k=0}^0 k^3=0,
\\\sum_{k=0}^1 k^3=1,
\\\sum_{k=0}^2 k^3=9,
\\\sum_{k=0}^3 k^3=36,
\\\sum_{k=0}^4 k^3=100.
$$</span></p>
<p>The requested formula must be a quartic polynomial, because the difference <span class="math-container">$P(n)-P(n-1)=n^3$</span> is a cubic polynomial. This ... |
3,251,233 | <blockquote>
<p>Calculate <span class="math-container">$\int_3^4 \sqrt {x^2-3x+2}\, dx$</span> using Euler's substitution</p>
</blockquote>
<p><strong>My try:</strong>
<br><span class="math-container">$$\sqrt {x^2-3x+2}=x+t$$</span>
<span class="math-container">$$x=\frac{2-t^2}{2t+3}$$</span>
<span class="math-conta... | egreg | 62,967 | <p>You can first observe that
<span class="math-container">$$
\sqrt{x^2-3x+2}=\frac{1}{2}\sqrt{4x^2-12x+8}=\frac{1}{2}\sqrt{(2x-3)^2-1}
$$</span>
so with <span class="math-container">$2x-3=t$</span>, you get
<span class="math-container">$$
\frac{1}{4}\int_3^5\sqrt{t^2-1}\,dt
$$</span>
Now use the Euler substitution <sp... |
1,444,820 | <p>I want to solve the following funktion for $x$, is that possible? And how woult it look like?</p>
<p>$y = xp -qx^{2}$</p>
<p>Thanks for Help!</p>
| Titus | 156,008 | <p>We can reduce the problem to the case that $g(x) = x^k$ for $k \in \mathbb{N}$. We find that if $f(x) = a_0 + a_1 x + a_2x^2$, then
$$ \int_0^1 f(x)g(x) dx = g(1/2) ~~ \textrm{ implies } ~~ {a_0 \over k+1} + {a_1 \over k+2} + {a_2 \over k+3} = {1 \over 2^k} ~~ \forall k \in \mathbb{N}.$$
Writing this equation for ... |
358,786 | <p>Why does Egorov's theorem not hold in the case of infinite measure? It turns out that, for example, $f_n = \chi_{[n,n+1]}x$ does not converge nearly uniformly, that is, it does not converge on E such that for a set F m(E\F) < $\epsilon$. Is this simply true because it takes on the value 1 for each n but suddenl... | Alp Uzman | 169,085 | <p>Set $\forall n\geq1: f_n:[0,\infty[\to\{0,1\}, f_n:=\chi_{[n-1,n]}$. Then $f_n\to0$ pointwise on $\mathbb{R}$. Suppose $\exists F\subseteq\mathbb{R}: f_n\stackrel{u.}{\to}0$ on $F$, i.e. that</p>
<p>$$\forall \epsilon>0,\exists N,\forall n\geq N,\forall x\in F: |f_n(x)|<\epsilon.$$</p>
<p>For $\epsilon:=1, \... |
594,975 | <p>What will be the basis of vector space <span class="math-container">$\Bbb C$</span> over a field of rational numbers <span class="math-container">$\Bbb Q$</span>?</p>
<p>I think it will be an infinite basis! I think it will be <span class="math-container">$B=\{r_1+r_2i \mid r_1, r_2 \in \Bbb Q^{c}\}\cup\{1,i\}$</s... | Dylan Yott | 62,865 | <p>Any basis for $\Bbb C$ as a $\Bbb Q$ vector space must be infinite, since any finite dimensional $\Bbb Q$ vector space is countable, but $\Bbb C$ is not. Constructing such a basis requires the axiom of choice.</p>
|
594,975 | <p>What will be the basis of vector space <span class="math-container">$\Bbb C$</span> over a field of rational numbers <span class="math-container">$\Bbb Q$</span>?</p>
<p>I think it will be an infinite basis! I think it will be <span class="math-container">$B=\{r_1+r_2i \mid r_1, r_2 \in \Bbb Q^{c}\}\cup\{1,i\}$</s... | Ryan Reich | 3,547 | <p>The straightforward answer for "what is a basis for $\mathbb{C}/\mathbb{Q}$" is that we don't know. The sneaky answer is that we do know there is one, because any maximal linearly independent set is a basis, and exists by Zorn's Lemma. This shows that $\mathbb{C}$ is a free $\mathbb{Q}$-module, since that concept ... |
4,015,203 | <p>How would you prove the following property about covariances?</p>
<p><a href="https://i.stack.imgur.com/y18wd.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/y18wd.png" alt="enter image description here" /></a></p>
<p>I found it here:</p>
<p><a href="https://www.probabilitycourse.com/chapter5/5_3_... | Vons | 274,987 | <p>You can prove the result <span class="math-container">$Cov(aX+bY,cW+dZ)=abCov(X,Y)+acCov(X,W)+bcCov(Y,W)+bdCov(Y,Z)$</span> as follows.</p>
<p><span class="math-container">$$\begin{split}Cov(aX+bY,cW+dZ)&=E[(aX+bY-E(aX+bY))(cW+dZ-E(cW+dZ))]\\
&=E[(a(X-E(X)+b(Y-E(Y))(c(W-E(W)+d(Z-E(Z))]\\
&=E[ac(X-E(X))(W... |
3,484,293 | <p>In the <span class="math-container">$xy$</span> - plane, the point of intersection of two functions <span class="math-container">$f(x) = x^2$</span> and <span class="math-container">$g(x) = x + 2$</span> lies in which quadrant/s ?</p>
<p>I have no idea how to begin with this question.</p>
| Community | -1 | <p>Go forward like you would mathematically, That is equating the two functions.</p>
<p><span class="math-container">$f(x) = g(x)$</span></p>
<p><span class="math-container">$x^2 = x + 2$</span></p>
<p><span class="math-container">$x^2 - x - 2 = 0$</span></p>
<p><span class="math-container">$x = -1$</span> or <spa... |
2,368,179 | <p>Answer should be in radians
Like π/4 (45°) π(90°).
I used $\tan(A+B)$ formula and got $5/7$ as the answer, but that's obviously wrong.</p>
| Dr. Sonnhard Graubner | 175,066 | <p>use that $$\tan(A+B)=\frac{\tan(A)+\tan(B)}{1-\tan(A)\tan(B)}$$</p>
|
2,546,161 | <p>I came across this interesting question in an interview:</p>
<p>Given $X$ and $Y$, these two independent standard normal. We have the following probability of $P(X>0| X+Y>0) = 0.75$. One can get this easily by draw a 2d plane and find out the required area.</p>
<p>Now, if $X$ and $Y$ are joint normal with co... | Did | 6,179 | <p>For approaches "less cumbersome than the double gaussian integral", start from the fact that if the 2D standard normal distribution is invariant by the rotations, thus, for every $(X_0,Y_0)$ standard normal and every <strong>angular sector $S$ of angle $\vartheta$</strong>, $$P((X_0,Y_0)\in S)=\vartheta/(2\pi)$$
If ... |
2,773,515 | <p>Given $X_1 \sim \exp(\lambda_1)$ and $X_2 \sim \exp(\lambda_2)$, and that they are independent, how can I calculate the probability density function of $X_1+X_2$? </p>
<hr>
<p>I tried to define $Z=X_1+X_2$ and then: $f_Z(z)=\int_{-\infty}^\infty f_{Z,X_1}(z,x) \, dx = \int_0^\infty f_{Z,X_1}(z,x) \, dx$.<br>
An... | tarkovsky123 | 518,320 | <p>Just as has been pointed out by the other answers, you can simply calculate the pdf for $X_1 + X_2$ by using the principle of <em>convolution</em>. In fact, in general one can show that if $X_1,X_2,...X_n$ are i.i.d variables with exponential distribution with parameter $\lambda$ then $S = \sum_{k=1}^{n}X_k \sim \Ga... |
3,692,083 | <p>I wish to show that the closed unit ball in <span class="math-container">$l^1$</span> is not compact, for which I believe it would be easiest to show that it is not bounded. For this I want to consider the sequence {1, 1/2, 1/3, ... , 1/n, ...}, since the harmonic series is known to be divergent. But will this seque... | Kavi Rama Murthy | 142,385 | <p>The closed unit ball in any normed linear space is bounded. Your sequence does not belong to <span class="math-container">$\ell^{1}$</span>. </p>
<p>To show that that closed unit ball in <span class="math-container">$\ell^{1}$</span> is not compact consider the elements <span class="math-container">$e_n=(0,0,...,0,... |
2,420,255 | <p>If the price of an article is increased by percent $p$, then the decrease in percent of sales must not exceed $d$ in order to yield the same income. The value of $d$ is:
$\textbf{(A)}\ \frac{1}{1+p} \qquad \textbf{(B)}\ \frac{1}{1-p} \qquad \textbf{(C)}\ \frac{p}{1+p} \qquad \textbf{(D)}\ \frac{p}{p-1}\qquad \textbf... | R. J. Mathar | 478,393 | <p>The proof is easy with standard vector algebra: The line from (B,D) to (A,C) is parametrized by point coordinates (x,y) = (B,D)+t(A-B,C-D), 0<=t<=1, and the line from (C,D) to (A,B) is parametrized by point coordinates (x,y) = (C,D)+t'(A-C,B-D), 0<= t'<=1. The intersection between the two lines is found ... |
518,140 | <p>What is the relation between the definition of homotopy of two functions</p>
<blockquote>
<p>"A homotopy between two continuous functions $f$ and $g$ from a topological space $X$ to a topological space $Y$ is defined to be a continuous function $H : X × [0,1] → Y$ from the product of the space $X$ with the unit i... | Asaf Karagila | 622 | <p>When I took a course in mathematical history, the only real achievement of the medieval mathematics (according to the professor teaching the course) was the following:</p>
<blockquote>
<p>$$\sum_{n=1}^\infty\frac1n=\infty$$</p>
</blockquote>
<p>The proof is due to Oresme, who gave a very nice proof of the follow... |
518,140 | <p>What is the relation between the definition of homotopy of two functions</p>
<blockquote>
<p>"A homotopy between two continuous functions $f$ and $g$ from a topological space $X$ to a topological space $Y$ is defined to be a continuous function $H : X × [0,1] → Y$ from the product of the space $X$ with the unit i... | Per Erik Manne | 33,572 | <p>You can find a list of mathematicians from this period <a href="http://www-history.mcs.st-and.ac.uk/Indexes/500_999.html" rel="nofollow noreferrer">here</a> and <a href="http://www-history.mcs.st-and.ac.uk/Indexes/1000_1499.html" rel="nofollow noreferrer">here</a>, with links to biographies.</p>
<p>Alcuin of York wr... |
230,887 | <p>Let $(F^\bullet,d_F)$ and $(G^\bullet,d_G)$ be two complexes in an abelian category $\mathbf{A}$.</p>
<p>The complex cone $Cone(\varphi)^\bullet$ of a morphism of complexes $\varphi:F^\bullet \to G^\bullet$ is defined as</p>
<p>$$Cone(\varphi)^i=G^i\oplus F^{i+1},$$</p>
<p>and its differential is</p>
<p>$$d(g^i,... | Jason Starr | 13,265 | <p>You might want to take a look at Problem 1 on the following problem set from a course on homological algebra. <br> <a href="http://www.math.stonybrook.edu/~jstarr/M536f15/M536f15ps10.pdf" rel="nofollow">http://www.math.stonybrook.edu/~jstarr/M536f15/M536f15ps10.pdf</a> <br>
In particular, the mapping complex satisf... |
2,408,521 | <blockquote>
<p>The plane $x - y = 0$ </p>
</blockquote>
<p>This seems very easy, but I will do it in case I'm barking up the wrong tree. Also, if there is a more efficient way to do it please tell me.</p>
<p>$y$ is a free variable so let it equal $r$<br>
$x = y = r$
therefore,<br>
$$v = \begin{bmatrix}
r \\
r \\
... | Shaun | 104,041 | <p>It's already in conjunctive normal form.</p>
|
2,408,521 | <blockquote>
<p>The plane $x - y = 0$ </p>
</blockquote>
<p>This seems very easy, but I will do it in case I'm barking up the wrong tree. Also, if there is a more efficient way to do it please tell me.</p>
<p>$y$ is a free variable so let it equal $r$<br>
$x = y = r$
therefore,<br>
$$v = \begin{bmatrix}
r \\
r \\
... | skyking | 265,767 | <p>It's both in conjunctive and disjunctive normal form. However the terms/factors are different. To make it more clear in conjunctive normal form it's</p>
<p>$$(\neg a) \land (\neg b)$$</p>
<p>and in disjunctive normal form it's</p>
<p>$$(\neg a \land \neg b)$$</p>
<p>Where the factors/terms are inside the parenth... |
3,896,562 | <p>Suppose <span class="math-container">$f:\mathbb{S}^n\rightarrow Y$</span> is a continuous map null homotopic to a constant map <span class="math-container">$c$</span>. In other words: <span class="math-container">$F: f\simeq c$</span> , where <span class="math-container">$c(x)=y$</span></p>
<p>Now, we may extend <sp... | NHL | 841,510 | <p>let <span class="math-container">$h=g-f>0$</span> on <span class="math-container">$[a,b]$</span>. Since <span class="math-container">$h$</span> is continuous on a closed interval, it has a global minimum (extreme value theorem).</p>
<p>then let <span class="math-container">$\delta=\min_{x\in[a,b]}{h(x)}$</span></... |
194,096 | <p>Is it possible to find an expression for:
$$S(N)=\sum_{k=0}^{+\infty}\frac{1}{\sum_{n=0}^{N}k^n}?$$</p>
<p>For $N=1$ we have</p>
<p>$$S(1) = \displaystyle\sum_{k=0}^{+\infty}\frac{1}{1 + k} = \displaystyle\sum_{k=1}^{+\infty}\frac{1}{k}$$</p>
<p>which is the (divergent) harmonic series. Thus, $S (1) = \infty$.</p... | Marko Riedel | 44,883 | <p>Here is an approach using Mellin transforms to enrich the collection
of solutions. Write</p>
<p>$$S(N) = 1 + \frac{1}{N+1} + \sum_{k\ge 2} \frac{1}{\sum_{n=0}^N k^n}
= 1 + \frac{1}{N+1} + \sum_{k\ge 2} \frac{k-1}{k^{N+1}-1}
\\= 1 + \frac{1}{N+1} + \sum_{k\ge 2} \frac{k}{k^{N+1}-1}
- \sum_{k\ge 2} \frac{1}{k^{N+1}... |
1,883,459 | <p>I am trying to make a block diagonal matrix from a given matrix by multiplying the given matrix to some other matrices.
Say $A$ is an $N \times N$ matrix, I want to make an $A^\prime$ matrix with size $kN \times kN$ such that $A^\prime$ has $A$ as its diagonal element $k$ times. In fact $A^\prime$ is the direct sum... | Nick Alger | 3,060 | <p>As Med points out, it is generally not possible to do what you want with only one "matrix sandwich". However, it is possible with $k$ "matrix sandwiches" as follows:</p>
<p>$$A' = \begin{bmatrix}
I \\ 0 \\ \vdots \\ 0
\end{bmatrix}A
\begin{bmatrix}
I & 0 & \dots & 0
\end{bmatrix}
+
\begin{bmatrix}
0 \\... |
1,555,429 | <p>Hi I am trying to solve the sum of the series of this problem:</p>
<p>$$
11 + 2 + \frac 4 {11} + \frac 8 {121} + \cdots
$$</p>
<p>I know its a geometric series, but I cannot find the pattern around this. </p>
| Esperluet | 293,654 | <p>Your series is a geometric series with a common ration of $q =\dfrac{2}{11}$ and a first term $a_0$ equal to $11$.<br>
$-1 < \dfrac{2}{11}< 1$ so the series is convergent.<br>
Its sum is equal to
$$a_0 \cdot \dfrac{1}{1-q} = 11 \cdot \dfrac{1}{1-\dfrac{2}{11}} = \dfrac{121}{9}$$</p>
|
3,735,904 | <p><span class="math-container">$\mathbf{Question:}$</span> Prove that <span class="math-container">$(A\cap C)-B=(C-B)\cap A$</span></p>
<p><span class="math-container">$\mathbf{My\ attempt:}$</span></p>
<p>Looking at LHS, assuming <span class="math-container">$(A\cap C)-B \neq \emptyset$</span></p>
<p>Let <span class=... | user0102 | 322,814 | <p><span class="math-container">\begin{align*}
(A\cap C) - B = (A\cap C)\cap\overline{B} = (C\cap\overline{B})\cap A = (C - B)\cap A
\end{align*}</span></p>
|
4,551,674 | <p>The question is:</p>
<blockquote>
<p><span class="math-container">$f: A\to R$</span> is a continuous, real-valued function, where <span class="math-container">$A\subseteq\mathbb{R}^n$</span>.</p>
<p>If <span class="math-container">$f(x)\to\infty$</span> as <span class="math-container">$\|x\|\to\infty,$</span> show t... | mathcounterexamples.net | 187,663 | <p>The result is false without further hypothesis on <span class="math-container">$A$</span>. For example if <span class="math-container">$n=1$</span> and <span class="math-container">$A=\mathbb Q \subset \mathbb R$</span>, then the map <span class="math-container">$f(x)=(x-\sqrt 2)^2$</span> satisfies the hypothesis o... |
3,989,591 | <p>The following question I read in a book, but the book does not give proof. I doubt the correctness of the result</p>
<p>let <span class="math-container">$p>3$</span> be prime number. prove or disprove
<span class="math-container">$$(x+1)^{2p^2}\equiv x^{2p^2}+\binom{2p^2}{p^2}x^{p^2}+1\pmod {p^2}\tag{1}$$</span><... | Kenta S | 404,616 | <p><span class="math-container">\begin{align}
\int_0^2e^xdx&=\lim_{n\to\infty}\sum_{i=0}^{n-1}\frac2{n}e^{2i/n}\\
&=\lim_{n\to\infty}\frac2{n}\left(\frac{e^{2}-1}{e^{2/n}-1}\right)\\
&=e^2-1.
\end{align}</span></p>
|
629,347 | <p>I understand <strong>how</strong> to calculate the dot product of the vectors. But I don't actually understand <strong>what</strong> a dot product is, and <strong>why</strong> it's needed.</p>
<p>Could you answer these questions?</p>
| Michael Hoppe | 93,935 | <p>If the length of $B$ is $1$ then $\langle A,B\rangle$ is the coordinate of $A$ in direction $B$.</p>
<p>There is a nice interpretation of the scalar product where $B$ has arbitrary length. Let $B=(b_1,b_2)$, then define $J(B):=(-b_2,b_1)$; you'll get $J(B)$ by rotating $B$ counterclockwise by $\pi/2$. Observe tha... |
2,956,330 | <p>Hi all I think I found a new proof that <span class="math-container">$[0, 1]$</span> is compact but I am not 100% if it is correct, could you help me check? In the usual proof we just take the <code>sup{x in [0, 1] : [0, x] is covered by finitely many intervals}</code> etc.</p>
<p>My proof goes like this:</p>
<p>... | Calum Gilhooley | 213,690 | <p>It's not clear to me how the selection procedure in your third and fourth paragraphs would work.</p>
<p>Let <span class="math-container">$\alpha = 1/\sqrt{2}$</span>, and consider the cover of <span class="math-container">$[0, 1]$</span> by the uncountable collection consisting of (i) the open interval <span class=... |
2,956,330 | <p>Hi all I think I found a new proof that <span class="math-container">$[0, 1]$</span> is compact but I am not 100% if it is correct, could you help me check? In the usual proof we just take the <code>sup{x in [0, 1] : [0, x] is covered by finitely many intervals}</code> etc.</p>
<p>My proof goes like this:</p>
<p>... | DanielWainfleet | 254,665 | <p>I don't see how your 2nd step is guaranteed to produce a countable cover of all the irrationals in <span class="math-container">$[0,1].$</span></p>
<p>I suggest this: Let <span class="math-container">$C$</span> be a cover of <span class="math-container">$[0,1]$</span> by open subsets of <span class="math-containe... |
312,238 | <p>Reading my textbook, I came across exercises for nested quantifiers.</p>
<p>The question: Let $L(x, y)$ be the statement “$x$ loves $y$,” where the domain for both $x$ and $y$ consists of all people in the world.
Use quantifiers to express each of these statements.</p>
<p>i) Everyone loves himself or herself.</p>
... | Peter Smith | 35,151 | <p>Petr Pudlák's answer is of course exactly right about the equivalence, and he gives a pair of proofs which shows why it holds. But it is worth remarking as a footnote that this equivalence is (of course!) only available if you are already using the language of first-order logic <em>with identity</em>. </p>
<p>Now, ... |
1,119,010 | <p>Write down the assumptions in a form of clauses and give a resolution proof that the proposition
$$\Big((p \rightarrow q) \land ( q \rightarrow r) \land p \Big) \rightarrow r$$
is a tautology.</p>
| amWhy | 9,003 | <p>Well, you have $p$ in the antecedent, and you have $p\rightarrow q$, and together, by modus ponens, you get $q$. Now, $q$, with the implication $q\rightarrow r$ give you $r$, again, using modus ponens.</p>
<p>So the conjunction in the antecedent (i.e. the three conjuncts in the antecedent) imply $r$. </p>
<p>Can y... |
3,971,833 | <p>If there is an <span class="math-container">$n$</span> by <span class="math-container">$n$</span> matrix where each element is either 1 or -1, how many unique matrices are there such that each row and each column multiplies to 1?</p>
<p>I solved for the trivial case of <span class="math-container">$n = 2$</span>, wh... | Donald Splutterwit | 404,247 | <p>Each row can have <span class="math-container">$1$</span>'s & <span class="math-container">$-1$</span>'s in <span class="math-container">$2^{n-1}$</span> ways. Fill in the first <span class="math-container">$n-1$</span> rows & then complete the last row to give each column a positive parity. So there are <sp... |
3,622,508 | <p>I’m not sure exactly about the conditions needed for a subset <span class="math-container">$S$</span> to localise a ring <span class="math-container">$R$</span>. I know <span class="math-container">$S$</span> has to be multiplicative. But does <span class="math-container">$S$</span> also have to be a subset of the n... | Andrea Mori | 688 | <p>The only thing that is "forbidden" is that <span class="math-container">$0\in S$</span> or otherwise the localized ring is the <span class="math-container">$0$</span>-ring. Also you better have <span class="math-container">$1\in S$</span> (or to be precise in the saturation of <span class="math-container">$S$</span>... |
1,158,970 | <p>In the lectures notes <a href="http://users.jyu.fi/~pkoskela/quasifinal.pdf" rel="nofollow">http://users.jyu.fi/~pkoskela/quasifinal.pdf</a> (Prof. Koskela has made them freely available from his webpage, so I am guessing is OK that I paste the link here) Quasiconformality is defined by saying that $\displaystyle \l... | matthew | 233,974 | <p>Regarding point (3), you could not have "infinite distortion at some points" and still have an injective function. But, you can have, say, points with arbitrarily large distortion as you approach the boundary of your domain. A relatively simple example on the unit disk is $f(z) = \text{Re}\left( \frac{i}{2} \text{... |
3,628,159 | <p>I have <span class="math-container">$1,2,\ldots, n$</span> numbers and I want pick <span class="math-container">$k$</span> of them with replacement and such that order matters. </p>
<p>So for <span class="math-container">$n=10$</span> and <span class="math-container">$k=4$</span> I can get: <span class="math-contai... | drhab | 75,923 | <p>Assume that it is not true for every positive integer <span class="math-container">$n$</span>.</p>
<p>Then according to WOP a positive integer <span class="math-container">$m$</span> exists such that <span class="math-container">$3\nmid4^m+5$</span> and <span class="math-container">$3\mid4^n+5$</span> for every pos... |
390,145 | <p>Let p be a prime and a belong to Z. Find all solutions to the equation
$$(x-a)^2(x-a-1) + p \equiv 0 \bmod{p^3}$$</p>
<p>I'm having a hard time working with this as such few variables are given. We know p is prime and a is an integer, and we are solving for x. I tried letting another variable $y=x-a$, but that lea... | Ivan Loh | 61,044 | <p>Note that $p \|(x-a)^2(x-a-1)$, so $p \nmid x-a$ and $p \| x-a-1$. Write $x-a=rp+1$, where $p \nmid r$. We have $(rp+1)^2rp+p \equiv 0 \pmod{p^3}$, so $0 \equiv (rp+1)^2r+1 \equiv 2r^2p+(r+1) \pmod{p^2}$. Thus $p \mid r+1$. Write $r=sp-1$. We have $2(sp-1)^2p+sp \equiv 0\pmod{p^2}$, so $0 \equiv 2(sp-1)^2+s \equiv 2... |
390,145 | <p>Let p be a prime and a belong to Z. Find all solutions to the equation
$$(x-a)^2(x-a-1) + p \equiv 0 \bmod{p^3}$$</p>
<p>I'm having a hard time working with this as such few variables are given. We know p is prime and a is an integer, and we are solving for x. I tried letting another variable $y=x-a$, but that lea... | Robert Israel | 8,508 | <p>First of all, consider it mod $p$. Either $x \equiv a$ or $a+1 \mod p$. Then write $x = a + t p$ or $x = a+1 + tp$ ...</p>
|
1,031,632 | <p>I have problem with the sum:</p>
<p>$$
\sum_{k=0}^n \dbinom{n}{k}(\cos \alpha)^k(i\sin \alpha)^{n-k}\,\,
$$
Apparantly, I have an imaginary unit therefore I need to distinguish even and odd powers of $i$ to do so I need to introduce $2k$ as in:
$$
\sum_{k=0}^n f(k) = \sum_{k=0}^{n/2} g(2k)
$$
and eventually find $g... | epi163sqrt | 132,007 | <blockquote>
<p><em>Hint:</em> Using a slightly <em>extended version</em> of the binomial coefficient (see e.g. <a href="http://www.math.upenn.edu/~wilf/gfology2.pdf" rel="nofollow">Wilf</a> p.15) with </p>
<p>$$\binom{n}{k}=0\qquad k>n$$</p>
<p>the calculation can be written more compactly:</p>
</blockq... |
4,105,812 | <p>could someone help me check if my proof is valid?</p>
<p>Use direct proof to prove the following theorem: <span class="math-container">$$ A \lor (B \rightarrow A), B \vdash_R A $$</span></p>
<p>We aren't allowed to use proof by resolution, we can only use logic axioms and inference rules such as hypothetical and dis... | Lutz Lehmann | 115,115 | <p>Take the derivative of the equation and test if the coefficient of <span class="math-container">$y''$</span> can be divided out, modulo the original equation. If the equation can be transformed into a Clairaut equation, then this should work.
<span class="math-container">\begin{align}
x(y')^2+yy'&=\frac3{5x^2}\\... |
52,874 | <p>Consider a coprime pair of integers $a, b.$ As we all know ("Bezout's theorem") there is a pair of integers $c, d$ such that $ac + bd=1.$ Consider the smallest (in the sense of Euclidean norm) such pair $c_0, d_0$, and consider the ratio $\frac{\|(c_0, d_0)\|}{\|(a, b)\|}.$ The question is: what is the statistics of... | Gerry Myerson | 3,684 | <p>I did a little experiment. Fix $a=29$, let $b=1,2,\dots,28$. So, you get 28 data points. Well, these points are already extremely regularly distributed. Taking just the first half, $1\le b\le14$, and rearranging the ratios in increasing order, they are (to three decimals) $$.034,.069,.103,.138,.172,.207,.242,.275,.3... |
1,567,152 | <blockquote>
<p>Theorem: $X$ is a finite Hausdorff. Show that the topology is discrete.</p>
</blockquote>
<p>My attempt: $X$ is Hausdorff then $T_2 \implies T_1$ Thus for any $x \in X$ we have $\{x\}$ is closed. Thus $X \setminus \{x\}$ is open. Now for any $y\in X \setminus \{x\}$ and $x$ using Hausdorff property, ... | skyking | 265,767 | <p>You're a bit sloppy in assuming that $\{x\}$ is open.</p>
<p>The thing you have to prove is that any subset of $X$ is open. This is quite straight forward as every subset of $X$ is $X$ minus a finite number of points, if it's not $X$ itself (which is open anyway) it's minus a finite positive number of points. That ... |
1,567,152 | <blockquote>
<p>Theorem: $X$ is a finite Hausdorff. Show that the topology is discrete.</p>
</blockquote>
<p>My attempt: $X$ is Hausdorff then $T_2 \implies T_1$ Thus for any $x \in X$ we have $\{x\}$ is closed. Thus $X \setminus \{x\}$ is open. Now for any $y\in X \setminus \{x\}$ and $x$ using Hausdorff property, ... | Community | -1 | <p>Let <span class="math-container">$X$</span> be a finite Hausdorff space. Let <span class="math-container">$x\in X$</span>. For each <span class="math-container">$y\not =x\in X$</span> let <span class="math-container">$U_y$</span> and <span class="math-container">$V_y$</span> be disjoint open sets with <span class... |
3,235,300 | <p>I tried with , whenever <span class="math-container">$x > y$</span> implies <span class="math-container">$p(x) - p(y) =( 5/13)^x (1-(13/5)^{(x-y)}) + (12/13)^x (1- (13/12)^{(x-y)}) > 0 $</span>.
But here I don't understand why the answer is no.</p>
| auscrypt | 675,509 | <p>No, it is in fact strictly decreasing. Note that <span class="math-container">$\frac{5}{12}$</span> and <span class="math-container">$\frac{12}{13}$</span> are both less than <span class="math-container">$1$</span>, and so <span class="math-container">$\left(\frac{5}{12}\right)^x$</span> and <span class="math-contai... |
2,405,505 | <p>How to prove that the infinite product $\prod_{n=1}^{+\infty} \left(1-\frac{1}{2n^2}\right)$ is positive ?</p>
<p>Thanks</p>
| Zhihao.Lu | 474,641 | <p>$1-\dfrac{1}{n^2}<1-\dfrac{1}{2n^2}<\left(1-\dfrac{1}{n^2}\right)^2$,So you can see Its limit exist.</p>
<p>Sorry For I cannot use LaTex and My poor English.</p>
|
4,286,136 | <p>I'm trying to find the general solution to <span class="math-container">$xy' = y^2+y$</span>, although I'm unsure as to whether I'm approaching this correctly.</p>
<p>What I have tried:</p>
<p>dividing both sides by x and substituting <span class="math-container">$u = y/x$</span> I get:</p>
<p><span class="math-cont... | Botnakov N. | 452,350 | <p>You say that
<span class="math-container">$$y' = u^2x^2+u$$</span>
but <span class="math-container">$$y' = \frac{y^2+y}{x} = \bigg(\frac{y}{x}\bigg)^2 x + u = u^2 x +u.$$</span> So here's a mistake.</p>
<p>The right solution:</p>
<p><span class="math-container">$$xdy = (y^2 + y)dx$$</span>
<span class="math-containe... |
403,184 | <p>A (non-mathematical) friend recently asked me the following question:</p>
<blockquote>
<p>Does the golden ratio play any role in contemporary mathematics?</p>
</blockquote>
<p>I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for al... | Matt Zaremsky | 164,670 | <p>The "Cleary group" <span class="math-container">$F_\tau$</span> is a version of Thompson's group <span class="math-container">$F$</span>, introduced by Sean Cleary, that is defined using the golden ratio, and it's definitely of interest in the world of Thompson's groups. See <em><a href="https://arxiv.org/... |
403,184 | <p>A (non-mathematical) friend recently asked me the following question:</p>
<blockquote>
<p>Does the golden ratio play any role in contemporary mathematics?</p>
</blockquote>
<p>I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for al... | wlad | 75,761 | <p>Consider the function which is the limit of the sequence <span class="math-container">$x^{2^0}, \sqrt{1 + x^{2^1}}, \sqrt{1 + \sqrt{1 + x^{2^2}}}, \dotsc$</span> Call this function <span class="math-container">$U(x)$</span>, with domain <span class="math-container">$\{x \in \mathbb R \mid x \geq 0\}$</span>. Notice ... |
403,184 | <p>A (non-mathematical) friend recently asked me the following question:</p>
<blockquote>
<p>Does the golden ratio play any role in contemporary mathematics?</p>
</blockquote>
<p>I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for al... | Alexandre Eremenko | 25,510 | <p>Yes, it does:</p>
<p>Lyubich, Mikhail; Milnor, John The Fibonacci unimodal map. J. Amer. Math. Soc. 6 (1993), no. 2, 425–457.</p>
<p>and two more papers of the same authors studying what they call Fibonacci map.</p>
|
403,184 | <p>A (non-mathematical) friend recently asked me the following question:</p>
<blockquote>
<p>Does the golden ratio play any role in contemporary mathematics?</p>
</blockquote>
<p>I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for al... | Roland Bacher | 4,556 | <p>The golden ratio occurs in the asymptotic growth-rate for the number of numerical semigroups of given genus:</p>
<p>A numerical semigroup of genus <span class="math-container">$g$</span> is a subset <span class="math-container">$S$</span> of <span class="math-container">$\mathbb N=\{0,1,2,\ldots\}$</span> such that ... |
403,184 | <p>A (non-mathematical) friend recently asked me the following question:</p>
<blockquote>
<p>Does the golden ratio play any role in contemporary mathematics?</p>
</blockquote>
<p>I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for al... | David Richter | 11,264 | <p>See the article "<a href="https://doi.org/10.1007/BF01388660" rel="nofollow noreferrer">Generalized Dehn–Sommerville relations for polytopes,
spheres and Eulerian partially ordered sets</a>" by Margaret Bayer and Louis Billera. These authors aimed to extend the Dehn–Sommerville equations to homology spher... |
403,184 | <p>A (non-mathematical) friend recently asked me the following question:</p>
<blockquote>
<p>Does the golden ratio play any role in contemporary mathematics?</p>
</blockquote>
<p>I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for al... | Sylvain JULIEN | 13,625 | <p>I'm not sure whether it qualifies or not, but I found a link between the golden ratio and the Riemann Hypothesis some years ago: <a href="https://math.stackexchange.com/questions/2380532/is-there-a-hidden-connection-between-rh-and-the-golden-ratio">Is there a hidden connection between RH and the golden ratio?</a>.</... |
1,147,808 | <p>I try to prove that ${2^{n-1}}$ elements of the field $\mathbf{F}_{2^{n}}$ have a Trace with value 1, while the other ${2^{n-1}}$ elements have a Trace with value 0.</p>
<p>I started to show that Trace(1) = 1, and I tried to use the additivity of the Trace but I wasn't successful. Any advice ?</p>
| Timbuc | 118,527 | <p>This is just linear algebra: the trace map is a linear functional $\;\Bbb F_{2^n}\to\Bbb F_2\;$, and since the extension $\;\Bbb F_{2^n}/\Bbb F_2\;$ is separable it is <strong>not</strong> the zero functional (or just show there's some element with trace different from zero), from where it follows that it is onto (... |
426,306 | <p>If <span class="math-container">$K = \mathbb{Q}(\sqrt{d})$</span> is a real quadratic field, then any unit <span class="math-container">$u \in \mathcal{O}_K^\times$</span> with <span class="math-container">$u > 1$</span> must not be too small: indeed, such a <span class="math-container">$u = u_1 + u_2 \sqrt{d}$</... | KConrad | 3,272 | <p>Asking about a smallest unit bigger than <span class="math-container">$1$</span> in a unit group of rank greater than <span class="math-container">$1$</span> feels like the wrong question, sort of like asking for a smallest algebraic integer of absolute value greater than <span class="math-container">$1$</span> in a... |
390,640 | <p>Please help me to find a closed form for the following integral:
$$\int_0^1\log\left(\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\right)\,{\mathrm d}x.$$</p>
<p>I was told it could be calculated in a closed form.</p>
| Start wearing purple | 73,025 | <p>$$\boxed{\displaystyle\int_0^1\log\log\left(\frac1x+\sqrt{\frac1{x^2}-1}\right)\mathrm dx=-\gamma-2\ln\frac{2\Gamma(3/4)}{\Gamma(1/4)}}\tag{$\heartsuit$}$$</p>
<hr>
<p><strong>Derivation</strong>:</p>
<p>After the change of variables $x=\frac{1}{\cosh u}$ the integral becomes
$$\int_0^{\infty}\ln u \frac{\sinh u}... |
684,755 | <p>In section 10 of <em>Topology</em> by Munkres, the minimal uncountable well-ordered set $S_{\Omega}$ is introduced. Furthermore, it is remarked that,</p>
<blockquote>
<p>Note that $S_{\Omega}$ is an uncountable well-ordered set every section of which is countable. Its order type is in fact uniquely determined by ... | Unwisdom | 124,220 | <p>Suppose that $\alpha$ and $\beta$ are two uncountable ordinals. Suppose further that for every $\gamma<\alpha$, $\gamma$ is countable. Likewise, suppose that for very $\eta<\beta$, $\eta$ is countable. You want to show that $\alpha=\beta$. </p>
<p>Suppose that $\alpha<\beta$. Then $\eta=\alpha$ is unco... |
148,037 | <p>for example I have a data</p>
<pre><code>Clear[data];
data[n_] :=
Join[RandomInteger[{1, 10}, {n, 2}], RandomReal[1., {n, 1}], 2];
</code></pre>
<p>then <code>data[3]</code> gives</p>
<pre><code>{{4, 8, 0.264842}, {9, 5, 0.539251}, {3, 1, 0.884612}}
</code></pre>
<p>in each sublist, first two value is matrix ... | Edmund | 19,542 | <p>You may use <a href="http://reference.wolfram.com/language/ref/GroupBy.html" rel="nofollow noreferrer"><code>GroupBy</code></a>.</p>
<pre><code>Clear[toSparse]
toSparse[data_] := SparseArray@Normal@GroupBy[data, Most -> Last, Total]
</code></pre>
<p>Then</p>
<pre><code>toSparse[data[1000]]; // AbsoluteTiming
<... |
2,877,916 | <p>Can you some one please tell how to prove Holder Space is Normed Linear Space</p>
<p>The Holder Space $C^{k,\gamma}(\bar{U})$ consisting of the all $u \in C^k(\bar{U})$ for which the norm</p>
<p>$$\|u\|_{C^{k,\gamma}(\bar{U})}:= \sum_{|\alpha|\le k} \|D^\alpha u \|_{C(\bar{U})}+\sum_{|\alpha|=k} [D^\alpha u]_{C^{... | Community | -1 | <p>Pick any of the points remaining in the square. Draw loops starting at that point, encircling a single one of the removed points, without self intersections, and without intersecting the other two loops.</p>
<p>Since the interior of the loops have a removed point each, then they can be retracted to the points of th... |
1,037,736 | <p>$$\sum \limits_{v=1}^n v=\frac{n^2+n}{2}$$</p>
<p>please don't downvote if this proof is stupid, it is my first proof, and i am only in grade 5, so i haven't a teacher for any of this 'big sums'</p>
<p>proof:</p>
<p>if we look at $\sum \limits_{v=1}^3 v=1+2+3,\sum \limits_{v=1}^4 v=1+2+3+4,\sum \limits_{v=1}^5 v=... | Deepak | 151,732 | <p>I'm not familiar with "rainbow numbers", and I'm afraid I can't follow every step of your proof. But if you're just looking for a very elementary proof of this, here's the easiest one I can think of:</p>
<p>Write the sum forwards:</p>
<p>$S_n = 1 + 2 + 3 + ... + n$</p>
<p>and then backwards:</p>
<p>$S_n = n + (n... |
164,060 | <p>When I plot the data I have using <code>ListStepPlot</code> and <code>ListLinePlot[data,InterpolationOrder -> 0]</code> I am getting two different plot. I guess there is a bug in <code>ListStepPlot</code>. </p>
<pre><code>data={{{0, 1}, {0.0582215, 2}, {0.597255, 3}, {1.17158, 4}}, {{1.17158,
4}, {1.36478, 5... | m_goldberg | 3,066 | <p>There is no bug. <code>ListStepPlot</code> gives a different result from <code>ListLinePlot</code> because it is using a different plotting algorithm.</p>
<p><code>ListStepPlot</code> draws steps (horizontal lines) through the data points and gives the user a choice of three positions for where the data point stand... |
2,556,339 | <p>This is the function $f(x)$$=\frac{1}{\sqrt{3x-2}}$ .
I wrote that $$\lim_{h\to 0}\frac{\frac{\sqrt{3x+3h-2}}{3x+3h-2}-\frac{\sqrt{3x-2}}{3x-2}}{h}.$$
I am not able to continue further.</p>
| Mark Viola | 218,419 | <p><strong>HINT:</strong></p>
<p>Note that </p>
<p>$$\begin{align}
\frac{1}{\sqrt{3(x+h)-2}}-\frac{1}{\sqrt{3x-2}}&=\frac{\frac{1}{3(x+h)-2}-\frac{1}{3x-2}}{\frac{1}{\sqrt{3(x+h)-2}}+\frac{1}{\sqrt{3x-2}}}\\\\
&=\frac{\frac{-3h}{(3(x+h)-2)(3x-2)}}{\frac{1}{\sqrt{3(x+h)-2}}+\frac{1}{\sqrt{3x-2}}}
\end{align}$$... |
1,364,936 | <p>How can one prove that the real cubic equation $$P(X)=X^3+pX+q$$ is not solvable by <strong>real radicals</strong> when $$D=-4p^3 - 27q^2 >0?$$</p>
<p>Which means that there is no sequence of extension:
$$\mathbb R=L_0 \subset L_1 \subset ... \subset L_n=L$$
with $a\in L$ root of $P$ and for $0 \leqslant i \leqs... | Jack D'Aurizio | 44,121 | <p>$$\begin{eqnarray*}S_N=\!\!\!\sum_{\substack{-N\leq z_1,z_2\leq N\\ z_1\neq z_2}}\!\!(z_1^2-2z_1 z_2)&=&\sum_{-N\leq z_1,z_2\leq N}(z_1^2-2z_1 z_2)+\sum_{-N\leq z\leq N}z^2\\&=&(2N+2)\sum_{-N\leq z\leq N}z^2-2\,\left(\sum_{-N\leq z\leq N}z\right)^2\\&=&(4N+4)\sum_{n=1}^{N}n^2\\&=&\fra... |
1,292,490 | <blockquote>
<p>Let $(a_{ij})$ be a real $n \times n$ matrix satisfying,</p>
<ol>
<li>$a_{ii} > 0 \space (1 \leq i \leq n) ,$</li>
<li>$a_{ij} \leq 0 \space (i \ne j, 1 \leq i,j \leq n) ,$</li>
<li>$\sum_{i=1}^ {i=n} \space a_{ij} > 0 (1 \leq j \leq n).$ </li>
</ol>
<p>Then $\det (A) > 0... | AlexR | 86,940 | <p><strong>Hint</strong><br>
$A$ has positive eigenvalues because it is diagonally dominant (<em>why?</em>) and the diagonal entries are positive. This suffices to show that $\det A = \prod_{i=1}^n \lambda_i > 0$ where $\lambda_i$ are the eigenvalues of $A$.</p>
|
3,407,852 | <p>A space X is said to be h-homogeneous if
every non-empty clopen subset of <span class="math-container">$X$</span> is homeomorphic to <span class="math-container">$X.$</span></p>
<p>Is the space <span class="math-container">$L = 2^{\mathbb N} - \{p\}$</span> for <span class="math-container">$p \in 2^{\mathbb N}$</s... | David G. Stork | 210,401 | <p><span class="math-container">$$n = 10^{10} - \left( \lfloor \sqrt{10^{10}} \rfloor + \lfloor \sqrt[3]{10^{10}} \rfloor + \lfloor \sqrt[6]{10^{10}} \rfloor - \lfloor \sqrt[6]{10^{10}} \rfloor - \lfloor \sqrt[15]{10^{10}} \rfloor - \lfloor \sqrt[10]{10^{10}} \rfloor + \lfloor \sqrt[30]{10^{10}} \rfloor \right) = 9,999... |
3,401,044 | <p>I am solving Section38 Exercise 5 in Topology, Munkres.</p>
<p>I solved that there is continuous surjectice closed
<span class="math-container">$$f : \beta(S_\Omega) \rightarrow Y$$</span>
for any compactification <span class="math-container">$Y$</span> of <span class="math-container">$S_\Omega$</span></p>
<p>A... | Henno Brandsma | 4,280 | <p>In part a) of that exercise it is shown that any continuous function <span class="math-container">$f: S_\Omega \to \Bbb R$</span> is eventually constant, in the sense that there is some <span class="math-container">$\alpha_0 \in S_\Omega$</span> and some <span class="math-container">$p \in \Bbb R$</span> such that <... |
2,332,419 | <p>What's the angle between the two pointers of the clock when time is 15:15? The answer I heard was 7.5 and i really cannot understand it. Can someone help? Is it true, and why?</p>
| MPW | 113,214 | <p>If this is a 12-hour clock, then the minute hand is at 3 and the hour hand is 1/4 of the way between 3 and 4. Thus the angle between them is $\frac14(\frac{360^{\circ}}{12})=7.5^{\circ}$.</p>
<p>(Note that the angle between two successive numbers on the face, like 3 and 4, is 1/12 of the full circle; that's where t... |
569,012 | <p>Let $I$ be the incenter of $\triangle{ABC}$. Let $R$ be the radius of the circle that circumscribes $\triangle{IAB}$. Find a formula for $R$ in term of other elements $a, b, c, A, B, C, r, R$ of $\triangle{ABC}$. I need this formula in order to prove a geometric inequality.</p>
| chloe_shi | 45,070 | <p><img src="https://i.stack.imgur.com/pQXw1.gif" alt="enter image description here"></p>
<p>suppose that $AI$ cut the circumcircle of $\triangle ABC$ at $D$.<br>
$\angle DBI=\angle IBC+\angle DBC=\angle IBA+\angle BAD=\angle BID$<br>
thsu, $\triangle DBI$ is isosceles and likewise, $\triangle DCI$ is isosceles.<br>
t... |
2,886,973 | <p>The Wikipedia article gives an <a href="https://en.wikipedia.org/wiki/Gauss%E2%80%93Bonnet_theorem#Interpretation_and_significance" rel="nofollow noreferrer">interesting example</a> of the Gauss-Bonnet theorem:</p>
<blockquote>
<p>As an application, a torus has Euler characteristic 0, so its total curvature must ... | Angina Seng | 436,618 | <p>When you join up the edges of the flat square to make a projective plane, what happens to its corners? Each corner gets identified with the opposite corner.
So at two points in the projective plane you have two corner bits of squares
identified. They may be flat, but round the corner, you have only $\pi$
worth of an... |
1,252,167 | <p>I'm trying to understand what a vector of functions is, from trying to understand how to solve linear homogeneous differential equations. </p>
<p>It seems that functions can be manipulated as vectors as long as they are not interpreted as having real values.<br>
Suppose the solution space of a linear homogeneous di... | Brian M. Scott | 12,042 | <p>As you’ve already discovered, the answer is <em>no</em>. In fact, you can’t even guarantee an irreducible open refinement. For $n\in\Bbb N$ let $U_n=\{k\in\Bbb N:k<n\}$, and let $\tau=\{U_n:n\in\Bbb N\}\cup\{\Bbb N\}$; then $\tau$ is a $T_0$ topology on $\Bbb N$, and $\tau\setminus\{\Bbb N\}$ is an open cover of ... |
96,970 | <p>I would need to identify the types of regular polygons forming the surface of a convex hull of 3D points.
If I e.g. take the following example of a regular polyhedron</p>
<pre><code>ConvexHullMesh[N[PolyhedronData["Dodecahedron", "VertexCoordinates"]]]
</code></pre>
<p>The convex hull routine returns a triangulate... | Taiki | 5,906 | <p>The procedure groups triangles based on the same unit normal vector, then uses the vertices in each group to form a new polygon. The vertices are sorted in such a way that their polygon is not self-intersecting.</p>
<p>This method doesn't allow for coplanar tolerance. Triangles in the same group have the same unit ... |
96,970 | <p>I would need to identify the types of regular polygons forming the surface of a convex hull of 3D points.
If I e.g. take the following example of a regular polyhedron</p>
<pre><code>ConvexHullMesh[N[PolyhedronData["Dodecahedron", "VertexCoordinates"]]]
</code></pre>
<p>The convex hull routine returns a triangulate... | J. M.'s persistent exhaustion | 50 | <p>Here is a solution that uses undocumented functionality to generate an appropriate <code>MeshRegion[]</code> object:</p>
<pre><code>Graphics`Mesh`MeshInit[];
FirstCase[ConvexHull3D[N[PolyhedronData["Dodecahedron", "VertexCoordinates"]],
FlatFaces -> False],
GraphicsComplex[pts_,... |
167,262 | <p>I make a circle with radius as below</p>
<pre><code>Ctest = Table[{0.05*Cos[Theta*Degree], 0.05*Sin[Theta*Degree]}, {Theta, 1, 360}] // N;
</code></pre>
<p>And herewith is my list of data points</p>
<pre><code>pts = {{0., 0.}, {0.00493604, -0.00994539}, {0.00987001, -0.0198918}, {0.0148019, -0.0298392}, {0.019731... | OkkesDulgerci | 23,291 | <pre><code> eq = 2 x11 + 8 x12 + 6 x13 - 4 x14 + 7 x15 + 3 x16 - 5 x17 + 4 x18 +
2 x19 + 2 x20 + 10 x21 + 10 x22 + 4 x23 + 3 x24 + 2 x25 + 2 x26 -
5 x27 + 9 x28 + 6 x29 + 9 x30 + 7 x31 - x32 - 4 x33 - 3 x34 +
x35 + 2 x36 + x37 - 2 x38 - x39 + x40;
var = Variables@eq;
NSolve[eq == 0 && And @@ (0 &l... |
85,717 | <p>Nowadays we can associate to a topological space $X$ a category called the fundamental (or Poincare) $\infty$-groupoid given by taking $Sing(X)$.</p>
<p>There are many different categories that one can associate to a space $X$. For example, one could build the small category whose object set is the set of points wi... | Toby Bartels | 8,508 | <p>I would call this an <a href="http://ncatlab.org/nlab/show/internal+category">internal category</a> in the category of topological spaces and continuous maps.</p>
|
97,877 | <p>Does anyone know a reference for the 2-dimensional version of the Schoenflies theorem? To be precise, I'd like a reference for the fact that every continuous, 1-1 map $S^1\rightarrow \mathbb{R}^2$ extends to a homeomorphism $\mathbb{R}^2 \rightarrow \mathbb{R}^2$. The discussions of the Jordan Curve Theorem that I... | Ryan Budney | 1,465 | <p>In the smooth case the idea is to take a linear height function on the plane, which is generically Morse on the curve. Apply the Jordan curve theorem + basic Morse theory, this tells you the compact region bounded by the curve is a union of discs, glued together along common arcs, and the "gluing pattern" is that of... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.