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 |
|---|---|---|---|---|
3,082,873 | <p>Consider the set of all boolean square matrices of order <span class="math-container">$3 \times 3$</span> as shown below where a,b,c,d,e,f can be either 0 or 1.</p>
<p><span class="math-container">$\begin{bmatrix}
a&b&c\\
0&d&e\\
0&0&f
\end{bmatrix}$</span></p>
<p>Out of all p... | DonAntonio | 31,254 | <p>It looks basically correct but I'd word it (fully) as follows: </p>
<p>Any given square matrix (over a field) is singular if and only if its determinant is zero. The given matrix is a triangular one and thus its determinant is simply the product of the elements in the main diagonal. Then the matrix is singular iff ... |
2,103,624 | <p>I see that this is true, by doing some examples:</p>
<p>For instance, if $$u = x^2$$ we have,</p>
<p>$$uu_x = 2x^3$$
$$( \frac12 u^2 )_x = 2x^3$$</p>
<p>How can we manipulate one side to show it is equal to the other side?</p>
| Siong Thye Goh | 306,553 | <p>This is chain rule</p>
<p>$$\frac{d}{dx}u(x)^2=2u(x)u'(x)$$</p>
|
2,103,624 | <p>I see that this is true, by doing some examples:</p>
<p>For instance, if $$u = x^2$$ we have,</p>
<p>$$uu_x = 2x^3$$
$$( \frac12 u^2 )_x = 2x^3$$</p>
<p>How can we manipulate one side to show it is equal to the other side?</p>
| F. Conrad | 403,916 | <p>Its just an application of the chainrule. Taking $f(x)=\frac{1}{2}x^2$ and looking at $f(u(x))$ you get:
$$
(f(u))_x=f_x(u) \cdot u_x=\frac{1}{2}\, 2\, u\cdot u_x=uu_x
$$</p>
|
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>
| kingW3 | 130,953 | <p>Note that your $f$ has to satisfy the first $3$ equations you got hence it has to satisfy $a_0=-\frac{3}{2},a_1=15,a_2=-15$ but since you found a counter example that means that $f(x)=\frac{-3}{2}+15x-15x^2$ doesn't satisfy the condition that means that no polynomial satisfy the condition.By the way the polynomial I... |
2,897,340 | <p>my attempt for
(i)</p>
<p>$\left. \begin{array} { l } { \cot ( \theta ) = - \frac { 1 \cdot \sqrt { 3 } } { \sqrt { 3 } \sqrt { 3 } } } \\ { 1 \cdot \sqrt { 3 } = \sqrt { 3 } } \end{array} \right.$</p>
<p>$\cot ( \theta ) = - \frac { \sqrt { 3 } } { 3 }$</p>
<p>(ii)</p>
<p>$\left. \begin{array} { l } { \text { ... | lab bhattacharjee | 33,337 | <p>$$1=4\cos^2t=2(1+\cos2t)$$</p>
<p>$$\iff\cos2t=?$$</p>
<p>$$2t=2n\pi\pm\dfrac{2\pi}3$$</p>
|
1,780,253 | <p>If I have two points $p_1, p_2$ uniformly randomly selected in the unit ball, how can I calculate the probability that one of them is closer to the center of the ball than the distance between the two points?</p>
<p>I know how to calculate the distribution of the distance between two random points in the ball, same... | joriki | 6,622 | <p>$d(p_1,O)\lt d(p_1,p_2)$ or $d(p_2,O)\lt d(p_1,p_2)$ if and only if one of $d(p_1,O)$ and $d(p_2,O)$ is the least of the three distances. By symmetry, the probability for this is twice the probability that $d(p_2,O)$ is the least of the three distances.</p>
<p>Fix $p_1$ at $(0,r)$. Then $d(p_2,O)$ is the least of t... |
832,386 | <p>From the first page of chapter 1 of George Andrews "Theory of Partitions" (Rather ominous place to get stuck):</p>
<p><img src="https://i.stack.imgur.com/ijmg4.png" alt="enter image description here"></p>
<p>What do these last two sentences mean? I don't get "where exactly $f_l$ of the $\lambda_j$ are equal to $i... | John Machacek | 155,418 | <p>It means for example that $\lambda = (1^22^33^04^05^1)$ another notation for $\lambda = (1,1,2,2,2,5)$. That is the $f_l$ superscripts tells how many parts of a given size you have.</p>
|
832,386 | <p>From the first page of chapter 1 of George Andrews "Theory of Partitions" (Rather ominous place to get stuck):</p>
<p><img src="https://i.stack.imgur.com/ijmg4.png" alt="enter image description here"></p>
<p>What do these last two sentences mean? I don't get "where exactly $f_l$ of the $\lambda_j$ are equal to $i... | David | 119,775 | <p>It just means that the value $i$ is repeated $f_i$ times. For example the notation $(1^42^23^04^15^1)$ means the same as $(1,1,1,1,2,2,4,5)$. Since the parts have to add up to the integer $n$, the sum
$$\sum_{i\ge1}f_ii=n$$
in this example is just another (IMHO unnecessarily complicated) way of writing
$$1+1+1+1+2... |
3,484,293 | <p>In the <span class="math-container">$xy$</span> - plane, the point of intersection of two functions <span class="math-container">$f(x) = x^2$</span> and <span class="math-container">$g(x) = x + 2$</span> lies in which quadrant/s ?</p>
<p>I have no idea how to begin with this question.</p>
| QC_QAOA | 364,346 | <p>First, let us solve</p>
<p><span class="math-container">$$f(x)=g(x)$$</span></p>
<p><span class="math-container">$$x^2=x+2$$</span></p>
<p><span class="math-container">$$0=x^2-x-2$$</span></p>
<p>By the quadratic formula, we know</p>
<p><span class="math-container">$$x=-1\text{ or }x=2$$</span></p>
<p>Now, for... |
3,699,439 | <p>I am interested if there is geometric meaning (using graphs) of <span class="math-container">$(1 + \frac{1}{n})^n$</span> when <span class="math-container">$n \rightarrow \infty$</span>. Also, is there visual explanation of why is <span class="math-container">$e^x = (1 + \frac{x}{n})^n$</span> when <span class="math... | Community | -1 | <p>Exponentiation turns addition to product, <span class="math-container">$$a^{b+c}=a^ba^c$$</span> (in the naturals, this is immediate from the definition). This corresponds to a "translation" property: shifting the argument amounts to a multiplication by a constant, and conversely, multiplying by a constant... |
4,410,917 | <p>A student is looking for his teacher. There is a 4/5 chance that the teacher is in one of 8 rooms, and he has no specific room preferences. Student checked 7 of the rooms, but the teacher wasn't in any of them. What's the probability that he is in one of the 8 rooms?</p>
<p>I tried dividing the P(4/5) by 8 and getti... | A.M. | 454,779 | <p>You have found out:</p>
<ul>
<li><span class="math-container">$0.1$</span> is the probability that the teacher is in a particular room, for any of the <span class="math-container">$8$</span> particular rooms (assuming that we haven't started looking for him).</li>
<li><span class="math-container">$0.7$</span> is the... |
2,107,787 | <p>I am a a student and I am having difficulty with answering this question. I keep getting the answer wrong. Please may I have a step by step solution to this question so that I won't have difficulties with answering these type of questions in the future.</p>
<p><em>n</em> is a number. 100 is the LCM of 20 and <em>n<... | user35359 | 35,359 | <p>Yes, assuming that none of the $U_{\alpha}$ is empty. To show that there is a connection between $v_{\alpha}$ and $v_{\beta}$ in the graph, consider a path $p:[0,1]\to X$ that connects some arbitrary point $x_{\alpha}\in U_{\alpha}$ with some point $x_{\beta}\in U_{\beta}$. $[0,1]$ is covered by the preimages $p^{-1... |
3,361,833 | <p><a href="https://i.stack.imgur.com/HVt8W.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/HVt8W.png" alt="enter image description here"></a> </p>
<p>Please Refer to answer 2 of the material above. I can follow the text up to that point. I always seem to loose conceptualisation when there's no illu... | Lutz Lehmann | 115,115 | <p>If the stack has size <span class="math-container">$n$</span> at some time, then you will have performed <span class="math-container">$i$</span> size doublings so that after the last doubling the size <span class="math-container">$2^i$</span> is larger than <span class="math-container">$n$</span>, that is <span clas... |
3,361,833 | <p><a href="https://i.stack.imgur.com/HVt8W.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/HVt8W.png" alt="enter image description here"></a> </p>
<p>Please Refer to answer 2 of the material above. I can follow the text up to that point. I always seem to loose conceptualisation when there's no illu... | kingW3 | 130,953 | <p><span class="math-container">$n$</span> is the number of operations, if <span class="math-container">$n=1$</span> then we don't need to resize the array (each array has at least 1 element);if we look at <span class="math-container">$1, 2,4,8,\cdots$</span> then <span class="math-container">$n$</span> is between two ... |
1,388,434 | <p>I was brushing up on some calculus and I was thinking about the following function:</p>
<blockquote>
<p>Let$$f(x) =
\begin{cases}
\frac{yx^{6} +y^{3}+x^{3}y}{x^{6}+y^{2}} & \text{for $(x,y)$ $\neq (0,0)$,} \\
0 & \text{for $(x,y)$ $=$ $(0,0)$. } \\
\end{cases}$$The function $f$ is continuous over the en... | user135520 | 135,520 | <p>We have that differentiability implies continuity, therefore if $f$ is differentiable at the origin then it will be continuous at the origin. Which means we must have</p>
<p>$\lim_{(x,y) \to (0,0)} f(x,y)=(0,0)$</p>
<p>We consider approaching the origin along the path $(x,x^{3})$ where </p>
<p>$\lim_{(x,y) \to (0... |
2,378,577 | <p>How do I prove or disprove that for a rational number x and an irrational number y, $\ x^y\ $ is irrational?</p>
| Michael Hartley | 96,763 | <p>Did you have some specific $x$ and $y$ in mind? Because the general statement isn't true: let $x=2$, and let $x^y=3$, for example. </p>
|
32,088 | <h2>Motivation</h2>
<p>One of the methods for strictly extending a theory <span class="math-container">$T$</span> (which is axiomatizable and consistent, and includes enough arithmetic) is adding the sentence expressing the consistency of <span class="math-container">$T$</span> ( <span class="math-container">$Con(T)$</... | Timothy Chow | 3,106 | <p>The short answer is no. Con(T) is a very weak assumption and it is asking a lot for it to have interesting mathematical consequences. A slightly less ambitious question is whether "ZFC + the consistency of some large cardinal axiom" has any interesting mathematical consequences. Here the work of Harvey Friedman i... |
106,000 | <p>I have the following data </p>
<pre><code> hours={38.9, 39, 38.9, 39, 39.3, 39.7, 39.2, 38.8, 39.6, 39.8, 39.9, 40.3, \
40, 40.2, 40.8, 40.7, 40.8, 41.2, 40.6, 40.7, 40.7, 40.9, 40.6, 40.8, \
40.3, 40.4, 40.7, 40.5, 40.7, 41.2, 40.3, 39.7, 40.4, 40.1, 40.3, \
40.6, 40.1, 40.5, 40.8, 40.8, 40.9, 41.7}
</code></pr... | Basheer Algohi | 13,548 | <pre><code>ListPlot[MapThread[
Labeled[#1, Style[#2, Bold, 12]] &, {hours,
PadRight[Range[12], Length[hours], "Periodic"]}]]
</code></pre>
<p><a href="https://i.stack.imgur.com/0zO5B.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/0zO5B.jpg" alt="enter image description here"></a></p>
|
268,091 | <p><a href="https://i.stack.imgur.com/PAO6T.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PAO6T.jpg" alt="enter image description here" /></a></p>
<p>I have to solve ODE x'(t)=1/2(x(t))-t, x(0)
The existence of solutions of this IVP is equivalent to finding a fixed point of integral operator T:C[0,... | Michael E2 | 4,999 | <p>Using <code>FixedPoint</code> to implement the Picard iteration, both numerically and symbolically.</p>
<p>First, define the ODE:</p>
<pre><code>odeFN = Function[{t, x}, x[t]/2 - t];
</code></pre>
<p>We also need to pass <code>FixedPoint[]</code> a <code>SameTest</code> that will stop the iteration after finitely ma... |
1,454,500 | <p>I am self studying mathematics for Physics by reading book <strong>Mathematical methods in Physical Sciences</strong>. I am stuck at this problem for days:</p>
<pre><code>Prove the following by appropriate manipulations using Fact 1 to 4;
do not just evaluate the determinants.
| 1 a bc | | 1 a a^2 | ... | Alex G. | 130,309 | <p>No. In general, for $a,b,c,d\in \Bbb N$ with $\frac{a}{b} < \frac{c}{d}$, $$\frac{a}{b} <\frac{a+c}{b+d}<\frac{c}{d}.$$
<strong>Proof</strong>: The left inequality is equivalent to $$a(b+d) < b(a+c) \iff ad < bc \iff \frac{a}{b}<\frac{c}{d}.$$ Likewise for the right inequality.</p>
|
3,896,562 | <p>Suppose <span class="math-container">$f:\mathbb{S}^n\rightarrow Y$</span> is a continuous map null homotopic to a constant map <span class="math-container">$c$</span>. In other words: <span class="math-container">$F: f\simeq c$</span> , where <span class="math-container">$c(x)=y$</span></p>
<p>Now, we may extend <sp... | Didier | 788,724 | <p>The idea is that you want to prove that there exists some constant <span class="math-container">$\delta$</span> such that <span class="math-container">$g(x)-f(x) \geqslant \delta >0$</span> for all <span class="math-container">$x \in [a,b]$</span>. Your only hypothesis are the continuity of <span class="math-cont... |
194,096 | <p>Is it possible to find an expression for:
$$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}$$</p>
<p>which is the (divergent) harmonic series. Thus, $S (1) = \infty$.</p... | Did | 6,179 | <p>$$
S(N)=1+\frac1{N+1}+\sum_{k=1}^{+\infty}\left(\zeta((N+1)k-1)-\zeta((N+1)k)\right)
$$</p>
|
1,563,205 | <p>I have 3 points: $A(0;0;0), B(0;0;1), C(2;2;1) $. They exist on the plane.
I assumed that scalar product of the normal vector and a line which exists on the same plane will be equal to 0. Scalar product equals to $x*2+y*2+z*1=0$ where $x,y,z$ are coordinates of the normal vector. Finally i can get $x,y,z$ using a se... | Nathan Marianovsky | 257,054 | <p>Let us say that we have any three arbitrary points $A$, $B$, $C$ that are unique. Define the following vectors:</p>
<p>$$\overrightarrow{AB} = B - A$$</p>
<p>$$\overrightarrow{AC} = C - A$$</p>
<p>which both lie along the plane that contains the three points. Now we form the normal vector by using the cross produ... |
2,463,565 | <p>I want to use the fact that for a $(n \times n)$ nilpotent matrix $A$, we have that $A^n=0$, but we haven't yet introduced the minimal polynomials -if we had, I know how to prove this.</p>
<p>The definition for a nilpotent matrix is that there exists some $k\in \mathbb{N}$ such that $A^k=0$.</p>
<p>Any ideas?</p>
| M. Van | 337,283 | <p>Let $k$ be the smallest positive integer such that $A^k=0$. Then we are done if $k \leq n$, since then $A^n=A^kA^{n-k}=0$. Suppose $k>n$. We look at $A$ as a linear map $\mathbb{R}^n \rightarrow \mathbb{R}^n$. Notice that we have the following sequence of nested subspaces of $\mathbb{R}^n$:
$$\mathbb{R}^n \sups... |
14,515 | <p>My problem is:
What is the expression in $n$ that equals to $\sum_{i=1}^n \frac{1}{i^2}$?</p>
<p>Thank you very much~</p>
| Derek Jennings | 1,301 | <p>I not sure of the thrust of your question but maybe the generalised harmonic numbers are what you want
$$ H_{n,r} = \sum_{k=1}^n \frac{1}{k^r} , $$
</p>
<p>and in particular $H_{n,2}$
</p>
<p>You can find more information <a href="http://mathworld.wolfram.com/HarmonicNumber.html">here,</a> including a very nice id... |
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>
| Eric Auld | 76,333 | <p>That's a huge question. It's what's called an <em>inner product</em>. A good short answer is that it gives you a way to make sense of what an angle between two vectors is. $$\theta = \cos^{-1}\left( \frac{a\cdot b }{|a||b|} \right)$$</p>
|
3,933,851 | <p>Suppose <span class="math-container">$N$</span> is called a magic number if it is a positive integer and when you stick <span class="math-container">$N$</span> on the end of any positive integer, the resulting integer is divisible by <span class="math-container">$N.$</span> How many magic numbers are there less than... | user2661923 | 464,411 | <p>Hint</p>
<p><span class="math-container">$n$</span> can have at most <span class="math-container">$4$</span> digits.</p>
<p>Further, it is presumed that the leftmost digit of <span class="math-container">$n$</span> can not be <span class="math-container">$0$</span>.</p>
<p>If <span class="math-container">$n$</span> ... |
3,715,987 | <p>The domain is: <span class="math-container">$\forall x \in \mathbb{R}\smallsetminus\{-1\}$</span></p>
<p>The range is: first we find the inverse of <span class="math-container">$f$</span>:
<span class="math-container">$$x=\frac{y+2}{y^2+2y+1} $$</span>
<span class="math-container">$$x\cdot(y+1)^2-1=y+2$$</span>
<sp... | L F | 221,357 | <p>Why? Just write the function as follows:
<span class="math-container">$$f(x) = \frac{(x+1)+1}{(x+1)^2} = \frac{x+1}{(x+1)^2}+ \frac{1}{(x+1)^2}
$$</span>
later,
<span class="math-container">$$f(x)= \frac{1}{(x+1)}+\frac{1}{(x+1)^2}=u+u^2 = (u+0.5)^2-0.25 = \left(\frac{1}{x+1}+0.5\right)^2-0.25$$</span></p>
<p>Sin... |
3,715,987 | <p>The domain is: <span class="math-container">$\forall x \in \mathbb{R}\smallsetminus\{-1\}$</span></p>
<p>The range is: first we find the inverse of <span class="math-container">$f$</span>:
<span class="math-container">$$x=\frac{y+2}{y^2+2y+1} $$</span>
<span class="math-container">$$x\cdot(y+1)^2-1=y+2$$</span>
<sp... | hamam_Abdallah | 369,188 | <p><span class="math-container">$$f(x)=\frac{x+2}{(x+1)^2}$$</span></p>
<p><span class="math-container">$$f'(x)=\frac{(x+1)-2(x+2)}{(x+1)^3}$$</span>
<span class="math-container">$$=\frac{-x-3}{(x+1)^3}$$</span>
so</p>
<p><span class="math-container">$$f((-\infty,-1))=[f(-3),+\infty)$$</span>
and
<span class="math-co... |
3,629,186 | <p>Assume that <span class="math-container">$x=x(t)$</span> and <span class="math-container">$y=y(t)$</span>. Find <span class="math-container">$dx/dt$</span> given the other information.</p>
<p><span class="math-container">$x^2−2xy−y^2=7$</span>; <span class="math-container">$\frac{dy}{dt} = -1$</span> when <span cla... | user577215664 | 475,762 | <p><span class="math-container">$$2x\left(\frac{dy}{dt}\right)-2\left[x(\frac{dy}{dt})+y(\frac{dx}{dt})\right]-2y\left(\frac{dx}{dt}\right) = 0$$</span>
You are almost there. It should be:
<span class="math-container">$$2x\left(\color{red}{\frac{dx}{dt}}\right)-2\left[x(\frac{dy}{dt})+y(\frac{dx}{dt})\right]-2y\left(\c... |
4,069,120 | <p>I am confused with the definition of 'basis'. <br/>
A basis <span class="math-container">$\beta$</span> for a vector space <span class="math-container">$V$</span> is a linearly independent subset of <span class="math-container">$V$</span> that generates <span class="math-container">$V$</span>. And span(<span class="... | José Carlos Santos | 446,262 | <p>Since <span class="math-container">$V$</span> is the whole space, the possibility “there might exist <span class="math-container">$b\in\operatorname{span}(\beta)$</span> s.t. <span class="math-container">$b\notin V$</span>” isn't real. So, <span class="math-container">$V$</span> being generated by <span class="math... |
4,569,458 | <p>I wonder if there is any trick in this problem. The following graph is a regular hexagon with its center <span class="math-container">$C$</span> and one of the vertices <span class="math-container">$A$</span>. There are <span class="math-container">$6$</span> vertices and a center on the graph, and now assume we per... | lonza leggiera | 632,373 | <p>Label the vertices of the hexagon as in the diagram below. Let <span class="math-container">$\ P_i\ $</span> be the probability that the walk returns to <span class="math-container">$\ A\ $</span> from <span class="math-container">$\ B_i\ $</span> without visiting <span class="math-container">$\ C\ $</span>. Then,... |
2,936,269 | <p>How do you simplify: <span class="math-container">$$\sqrt{9-6\sqrt{2}}$$</span></p>
<p>A classmate of mine changed it to <span class="math-container">$$\sqrt{9-6\sqrt{2}}=\sqrt{a^2-2ab+b^2}$$</span> but I'm not sure how that helps or why it helps.</p>
<p>This questions probably too easy to be on the Math Stack Exc... | Noah Schweber | 28,111 | <p>Let's forget about <span class="math-container">$9-6\sqrt{2}$</span> for a second and just think about the expression your classmate thinks is useful:<span class="math-container">$$a^2-2ab+b^2.$$</span></p>
<p>And let's keep in mind our goal here. We're looking for something which is a perfect square (since we want... |
126,983 | <p>I am working on an integral on the following trigonometric functions</p>
<p>$$\int_{-\pi}^\pi \frac{\cos[(4m+2)x] \cos[(4m+1)x]}{\cos x}dx$$</p>
<p>where $m$ is positive integer. I am running the following code in mathematica </p>
<pre><code>Assuming[Element[m, Integers] && (m > 0),
Integrate[
Cos[(... | Dr. Wolfgang Hintze | 16,361 | <p>The problem is very similar to that of [1] <a href="https://mathematica.stackexchange.com/questions/126984/how-to-compute-the-integral-of-trigonometic-function-with-multiple-angle">How to compute the integral of trigonometic function with multiple angle</a></p>
<p>Hence we don't repeat what we have discussed there ... |
1,031,632 | <p>I have problem with the sum:</p>
<p>$$
\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)
$$
and eventually find $g... | mike | 75,218 | <p>Follow Christopher's suggestion.
Suppose that $n=2m+1$ then
$$A_{2m+1}=\sum_{k=0}^m \dbinom{2m+1}{2k}(\cos \alpha)^{2m+1-2k}(i\sin \alpha)^{2k}$$
$$+\sum_{k=0}^m \dbinom{2m+1}{2k+1}(\cos \alpha)^{2m+1-(2k+1)}(i\sin \alpha)^{2k+1}$$</p>
<p>$$Re(A_{2m+1})=\sum_{k=0}^m \dbinom{2m+1}{2k}(\cos \alpha)^{2m+1-2k}(-1)^k(\s... |
3,671,608 | <p>Find the number of ways to distribute <span class="math-container">$7$</span> red balls, <span class="math-container">$8$</span> blue ones and <span class="math-container">$9$</span> green ones to two people so that each person gets <span class="math-container">$12$</span> balls. The balls of one color are indisting... | JMoravitz | 179,297 | <p><strong>Hint:</strong></p>
<p><a href="https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)" rel="nofollow noreferrer">Stars and bars</a> can get you almost there.</p>
<p>Letting <span class="math-container">$R,B,G$</span> denote the number of red, blue, and green balls that the first person gets respectiv... |
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... | Aaron Meyerowitz | 8,008 | <p>Here is an attempt to give a somewhat finer grained view of the distribution. The set of ratios $\sqrt{\frac{s^2+t^2}{a^2+b^2}} \subset(0,\frac{1}{2})$ are essentially the values in the first half of the Farey sequence $\lbrace \frac{p}{q} | \gcd(p,q)=1,\ 2p \le q<N\ \rbrace$. This has already been pointed out ... |
1,898,803 | <p>So, I looked up this question from G.H. Hardy's <em>A Course of Pure Mathematics</em> and found one examination question from the Cambridge Mathematical Tripos and it has baffled me ever since. I am supposed to sketch</p>
<p>$\lim_{n\to\infty}\dfrac{x^{2n}\sin{(\pi x/2)}+x^2}{x^{2n}+1}$</p>
<p>I have found out tha... | Robert Z | 299,698 | <p>We have that
$$f_n(x):=\dfrac{x^{2n}\sin{(\pi x/2)}+x^2}{x^{2n}+1}=\sin(\pi x/2)+\dfrac{x^2-\sin{(\pi x/2)}}{x^{2n}+1}.$$
Moreover,
$$\lim_{n\to \infty}x^{2n}=\left\{\begin{array}{ll}
0 & \text{if $|x|<1$, } \\
1 & \text{if $|x|=1$, } \\
+\infty & \text{if $|x|>1$, }
... |
4,286,136 | <p>I'm trying to find the general solution to <span class="math-container">$xy' = y^2+y$</span>, although I'm unsure as to whether I'm approaching this correctly.</p>
<p>What I have tried:</p>
<p>dividing both sides by x and substituting <span class="math-container">$u = y/x$</span> I get:</p>
<p><span class="math-cont... | user577215664 | 475,762 | <p><span class="math-container">$$u'x+u = u^2x^2+u $$</span>
<span class="math-container">$$\implies u' = u^2x \implies \int\frac{du}{u^2}=\int x dx$$</span>
<span class="math-container">$$\color{red}{\frac{1}{u}}=\frac{x^2}{2}+c\implies y = \frac{2x}{x^2+c}$$</span>
You made a sign mistake:
<span class="math-container... |
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... | JoshuaZ | 127,690 | <p>I'm not sure how recent "contemporary" mathematics means so I'll mention a few things, some somewhat classical but all connected to the fact that <span class="math-container">$\phi=\frac{1+\sqrt{5}}{2}$</span> is closely connected to the Fibonacci sequence. I will endeavor to order these roughly in oldest ... |
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... | Joseph Van Name | 22,277 | <p>One way to make the Golden ratio more interesting to mathematicians is to generalize the Fibonacci sequence to a non-commutative and possibly non-associative context.</p>
<p>Define the Fibonacci terms <span class="math-container">$(t_{n})_{n\geq 1}$</span> with respect to a binary operation <span class="math-contain... |
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... | Sebastien Palcoux | 34,538 | <p>The golden ratio <span class="math-container">$\phi$</span> is present in category theory and quantum algebra.</p>
<p>It is the smallest possible non-integral Frobenius-Perron dimension of a fusion ring/category, the one with simple objects <span class="math-container">$\{1,X\}$</span> and <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... | polfosol | 93,602 | <p>Here is <a href="https://mathoverflow.net/q/34052/93602" title="Function satisfying f^{-1} = f'">an interesting question</a> that asks whether there is any function that satisfies <span class="math-container">$f^{-1}=f'\;$</span> i.e. its derivative is equal to the inverse function. In addition to the answers provid... |
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... | John Jiang | 4,923 | <p>Golden ratio appeared in the recent breakthrough <a href="https://arxiv.org/abs/2211.09055" rel="nofollow noreferrer">A constant lower bound for the union-closed sets conjecture</a> of Frankl's Union Closed Set conjecture by Justin Gilmer, as well as the subsequent optimization <a href="https://arxiv.org/abs/2211.11... |
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... | Alessandro Della Corte | 167,834 | <p>Let <span class="math-container">$\rho$</span> be the binary substitution defined by: <span class="math-container">$$\rho(00)=\text{empty word}\quad\rho(01)=1\quad\rho(10)=0\quad\rho(11)=01.$$</span>
Let <span class="math-container">$R$</span> be the self-map of <span class="math-container">$[0,1]$</span> associatin... |
38,206 | <p>Simple question (I seem have asked a few like this...)</p>
<p>What is $\mbox{Hom}(\mathbb{Z}/2,\mathbb{Z}/n)$? (for $n \ne 2$)</p>
| Zev Chonoles | 264 | <p>If $f:G\rightarrow H$ is a group homomorphism, then $\text{ord}(f(g))\mid\text{ord}(g)$ for all $g\in G$ because $g^n=e_G$ implies $f(g)^n=f(g^n)=f(e_G)=e_H$. </p>
<p>Thus, if $f:\mathbb{Z}/(2)\rightarrow H$ is a homomorphism, we know that $f(0+(2))=e_H$ because $f$ is a homomorphism, and $f(1+(2))$ must be an elem... |
38,206 | <p>Simple question (I seem have asked a few like this...)</p>
<p>What is $\mbox{Hom}(\mathbb{Z}/2,\mathbb{Z}/n)$? (for $n \ne 2$)</p>
| t.b. | 5,363 | <p>More generally $\operatorname{Hom}{(\mathbb{Z}/m,\mathbb{Z}/n)}$ is cyclic of order $\operatorname{gcd}(m,n)$. I strongly recommend you to try to prove this on your own. In case of emergency, google for <code>hom z nz z mz</code>.</p>
|
1,838,357 | <p>Let us consider hyperbolic disc. I use uniform tessellation {5,4}.Here 5 stands for pentagon, 4 for number of polygons sharing the same vertex. </p>
<p><a href="https://i.stack.imgur.com/GeTny.gif" rel="nofollow noreferrer">{hyperbolic disc}</a>
There exists formula which defines area of convex hyperbolic polygon:
... | Lee Mosher | 26,501 | <p>The computation looks fine. </p>
<p>And yes, there is a tesselation of type $\{5,n\}$ for every $n \ge 4$. </p>
<p>To see why, you need the fact that if $\alpha_0$ is the interior angle of a regular Euclidean pentagon then for any $\alpha < \alpha_0$ one can construct a regular hyperbolic pentagon $P$ having in... |
1,838,357 | <p>Let us consider hyperbolic disc. I use uniform tessellation {5,4}.Here 5 stands for pentagon, 4 for number of polygons sharing the same vertex. </p>
<p><a href="https://i.stack.imgur.com/GeTny.gif" rel="nofollow noreferrer">{hyperbolic disc}</a>
There exists formula which defines area of convex hyperbolic polygon:
... | Willemien | 88,985 | <p>First I have to point you at a misconception you have, and I fear it will be confusing.
(I only understand just about half of it myself)</p>
<p>The disk model you use is the Poincare disk model (see <a href="https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model" rel="nofollow">https://en.wikipedia.org/wiki/Poinca... |
1,114,767 | <p>I would like a reference for the following result (you can assume more regularity and replace $C^2(\bar\Omega)$ with $C^2(\mathbb R^n)$ if needed):</p>
<blockquote>
<p>Let $\Omega\subset\mathbb R^n$ be a bounded domain with a $C^2$ boundary. Let $f\in C^2(\bar\Omega)$ and $\gamma\in C^2(\bar\Omega)$ with $\gamma&... | Dirk | 3,148 | <p>I would be very surprised if this would not be in Gilbarg/Trudinger "Elliptic Partial Differential Equations of Second Order" - I would start looking in Chapter 6 "Classical Solutions" (in fact they distinguish between "classical solutions" that are $C^2$ and "strong solutions" that are $W^{2,p}$).</p>
<hr>
<p>An ... |
271,915 | <p>Not clear from <code>DayMatchQ</code> <a href="https://reference.wolfram.com/language/ref/MatchQ.html" rel="nofollow noreferrer">doc page</a> but doesn't seem to work for say alternatives the way <code>MatchQ</code> does, eg</p>
<p><code>DayRange[Today,DayPlus[Today,30]] // Select[DayMatchQ[#,Monday | Wednesday | Fr... | Ben Izd | 77,079 | <p>If we assume our data is store in <code>data</code> variable:</p>
<pre><code>data = DayRange[Today, DayPlus[Today, 30]];
</code></pre>
<ol>
<li>We have <a href="http://reference.wolfram.com/language/ref/DateSelect.html" rel="nofollow noreferrer"><code>DateSelect</code></a> for date operations:</li>
</ol>
<pre><code>... |
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>
| Jyrki Lahtonen | 11,619 | <p>Because $tr(x)$ is a polynomial of degree $2^{n-1}$ it can take the value zero at most $2^{n-1}$ times. Because it is linear it takes the value zero at least $2^{n-1}$ times (rank-nullity tells that the kernel has dimension $\ge n-1$). Because $tr(x)$ is either zero or $1$, it has to take both values equally often.<... |
3,803,360 | <p>Convergence of <span class="math-container">$\sum\sum_{k, n=1}^\infty\frac{1}{(n+3)^{2k}}$</span>.</p>
<p>What I tried:</p>
<p>For the iterated summation, <span class="math-container">$\sum_{n=1}^\infty(\sum_{k=1}^\infty\frac{1}{(n+3)^{2k}})=\sum_{n=1}^\infty\lim_{k\to\infty}\frac{1-(\frac{1}{n+3})^{2k}}{1-(\frac{1}... | Greg Martin | 16,078 | <p>The idea is completely right!—you just made a typo involving the first term of each geometric series:
<span class="math-container">$$
\sum_{n=1}^\infty \bigg( \sum_{k=1}^\infty\frac{1}{(n+3)^{2k}} \bigg) = \sum_{n=1}^\infty\lim_{K\to\infty}\frac{\frac{1}{(n+3)^2}-\frac{1}{(n+3)^{2K+2}}}{1-\frac{1}{(n+3)^2}}=\sum_{n=... |
4,329,043 | <blockquote>
<p>Let {<span class="math-container">$X_n, n\geq1$</span>} be a sequence of i.d.d random variables. Suppose that <span class="math-container">$X_1$</span> follows a uniform distribution on (-1, 1).<br />
Let <span class="math-container">$$ \begin{gather} Y_n = \frac{\sum_{i=1}^{n} X_i}{\sum_{i=1}^{n}X_i^2 ... | Community | -1 | <p>Note that
<span class="math-container">\begin{gather}
\sqrt{n}Y_n = \frac{\frac{1}{\sqrt{n}}\sum_{i=1}^{n} X_i}{\frac{1}{n}\sum_{i=1}^{n}X_i^2 + \frac{1}{n}\sum_{i=1}^{n}X_i^3}.
\end{gather}</span>
The numerator converges in distribution to <span class="math-container">$N(0,1/3)$</span>, and the denominator converge... |
4,329,043 | <blockquote>
<p>Let {<span class="math-container">$X_n, n\geq1$</span>} be a sequence of i.d.d random variables. Suppose that <span class="math-container">$X_1$</span> follows a uniform distribution on (-1, 1).<br />
Let <span class="math-container">$$ \begin{gather} Y_n = \frac{\sum_{i=1}^{n} X_i}{\sum_{i=1}^{n}X_i^2 ... | tommik | 791,458 | <blockquote>
<p>I think this problem has to do with the central limit theorem somehow.</p>
</blockquote>
<p>Yes, CLT along with Slutsky's Theorem. In fact, as already shown,</p>
<p><span class="math-container">$$\sqrt{n}Y_n=A_nB_n$$</span></p>
<p>where</p>
<p><span class="math-container">$$A_n\xrightarrow{\mathcal{P}}3... |
2,985,172 | <p>Let <span class="math-container">$f(x) = \dfrac{1}{3}x^3 - x^2 - 3x.$</span> Part of the graph <span class="math-container">$f$</span> is shown below. There is a maximum point at <span class="math-container">$A$</span> and a minimum point at <span class="math-container">$B(3,-9)$</span>. </p>
<p><a href="https://i.... | PrincessEev | 597,568 | <p>In regards to part b i, the reflection in the <span class="math-container">$y$</span>-axis would be basically multiplying the <span class="math-container">$x$</span>-coordinate of <span class="math-container">$B$</span> by <span class="math-container">$-1$</span>. So you did that correctly.</p>
<p>In regards to par... |
4,563,707 | <p>Sequence given : 6, 66, 666, 6666. Find <span class="math-container">$S_n$</span> in terms of n</p>
<p>The common ratio of a geometric progression can be solved is <span class="math-container">$\frac{T_n}{T_{n-1}} = r$</span>, where r is the common ratio and n is the</p>
<p>When plugging in 66 as <span class="math-c... | fleablood | 280,126 | <p>When I go way back to when I first learned and thought of this and the concepts that <em>clicked</em> back before I got so familiar and jaded to things I started explaining them as though they were totally obvious, I'd have thought of it like this:</p>
<p><span class="math-container">$S_n= 6 + 66 + 666 + 6666 + ....... |
7,080 | <p>What is the right definition of the symmetric algebra over a graded vector space V over a field k?</p>
<p>More generally: What is the right definition of the symmetric algebra over an object in a symmetric monoidal category (which is suitably (co-)complete)?</p>
<p>Two possible definitions come to my mind:</p>
<p... | Mark Hovey | 1,698 | <p>In simple terms, it seems to me to depend on your definition of "graded commutative k-algebra". I would take this mean that </p>
<p>$xy =(-1)^{|x||y|}yx$</p>
<p>where |x| denotes the degree of the homogenous element x. So I would divide by the ideal generated by </p>
<p>$xy -(-1)^{|x||y|}yx$</p>
<p>for homogen... |
7,080 | <p>What is the right definition of the symmetric algebra over a graded vector space V over a field k?</p>
<p>More generally: What is the right definition of the symmetric algebra over an object in a symmetric monoidal category (which is suitably (co-)complete)?</p>
<p>Two possible definitions come to my mind:</p>
<p... | Theo Johnson-Freyd | 78 | <p>This is not an answer, as I think Scott did a better job than I could have. Another algebra that generalizes the symmetric algebra in characteristic non-zero is the algebra of divided polynomials. Let <em>V</em> be a finite-dimensional vector space over <em>k</em> a field, and let <em>V*</em> be its dual space. F... |
7,080 | <p>What is the right definition of the symmetric algebra over a graded vector space V over a field k?</p>
<p>More generally: What is the right definition of the symmetric algebra over an object in a symmetric monoidal category (which is suitably (co-)complete)?</p>
<p>Two possible definitions come to my mind:</p>
<p... | Leonid Positselski | 2,106 | <p>The right definition is: take the free associative (tensor) algebra generated by $V$; divide out the ideal generated by the elements $xy-(-1)^{|x||y|}yx$ for all homogeneous $x$, $y\in V$ and $z^2=0$ for all odd $z\in V$. This should commute with the base change well (when $V$ is flat over your base).</p>
|
1,903,520 | <p>By generalizing the approach in <a href="https://math.stackexchange.com/questions/1903152/integral-involving-a-dilogarithm-versus-an-euler-sum">Integral involving a dilogarithm versus an Euler sum.</a> meaning by using the integral representation of the harmonic numbers and by computing a three dimensional integral... | Przemo | 99,778 | <p>Here we provide a closed form for another related sum. We have:
\begin{eqnarray}
&&\sum\limits_{n=1}^\infty \frac{H_n^3}{n} \cdot x^n =\\
&&3 \zeta(4)-3 \text{Li}_4(1-x)+3 \text{Li}_3(1-x) \log (1-x)+\log (x) \log ^3(1-x)+\\
&&\text{Li}_2(x){}^2-2 \text{Li}_4(x)-3 \text{Li}_4\left(\frac{x}{x-... |
3,958,133 | <p>How to simplify the following probability</p>
<p><span class="math-container">$\operatorname{P} ( { C |B,A} )P( { B |A} )P( A ) + P( { {\bar B} |A} )P( A )$</span></p>
<p><span class="math-container">$ = P\left( {A,B,C} \right) + P\left( {A,\bar B} \right)$</span></p>
<p>Can <span class="math-container">$P\left( {A,... | Wuestenfux | 417,848 | <p>Hint: <span class="math-container">$ P(C|B,A)P(B|A)P(A)=P(A,B,C)$</span>.</p>
|
2,225,150 | <p>I am seek for a rigorous proof for the following identity</p>
<p>$\sum_{i = 0}^{T} x_i \sum_{j = 0}^{i} y_j = \sum_{i = 0}^{T}y_i\sum_{j = i}^{T} x_j$. </p>
<p>By setting some small $T$ and expand the formulas, it is then clear to see the result. I am asking for help to give a formal proof of this identity, by reo... | Alex R. | 22,064 | <p>The usual strategy is to "draw a triangle." Let $a_{ij}:=x_iy_j$. Then:</p>
<p>$a_{11}$</p>
<p>$a_{21},a_{22}$</p>
<p>$a_{31},a_{32},a_{33}$</p>
<p>$\vdots$</p>
<p>$a_{T1},a_{T2},a_{T3},\ldots ,a_{TT}.$</p>
<p>Since both sums are finite, there's no issue with swapping them. </p>
<p>Then $\sum_{i=0}^T\sum_{j=0... |
4,304,724 | <p><strong>Here's the question</strong>: <em>Suppose there's a bag filled with balls numbered one through fifty. You reach in and grab three at random, put them to the side, and then replace the ones you took so that the bag is once again filled with fifty distinctly numbered balls. Do this five times, so you have 5... | user2661923 | 464,411 | <p>Addendum of clarifications added in response to the comment/questions of Jotak.</p>
<hr />
<p><span class="math-container">$\underline{\text{First Problem : General Considerations and Choice of Denominator}}$</span></p>
<p>In this section, I discuss the specific problem that the OP (i.e. original poster) tackled - w... |
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$$
with $a\in L$ root of $P$ and for $0 \leqslant i \leqs... | Batominovski | 72,152 | <p>Hint: $\sum_{z_1 \neq z_2} \,z_1z_2=\sum_{z_1,z_2}\,z_1z_2-\sum_{z_1}\,z_1^2=\left(\sum_{z_1}\,z_1\right)\left(\sum_{z_2}\,z_2\right)-\sum_{z_1}\,z_1^2$ and $\sum_{z_1 \neq z_2}\,z_1^2 = 2N\,\sum_{z_1}\,z_1^2$.</p>
<p>After some algebraic manipulation, the answer should be $$(2N+2)\cdot 2\left(\frac{N(N+1)(2N+1)}{6... |
2,601,601 | <p>Consider the complex matrix $$A=\begin{pmatrix}i+1&2\\2&1\end{pmatrix}$$ and the linear map $$f:M(2,\mathbb{C})\to M(2,\mathbb{C}),\qquad X\mapsto XA-AX.$$</p>
<p>I want to find a basis of $\ker f$.</p>
<p>I already know the canonical basis $\{E_{11},E_{12},E_{21},E_{22}\}$ and computed $$f(E_{11})=\begin{... | egreg | 62,967 | <p>Yes, it helps. The matrix of $f$ with respect to this basis is
$$
\begin{bmatrix}
0 & 2 & -2 & 0 \\
2 & 0 & 0 & -2 \\
-2 & 0 & 0 & 2 \\
0 & -2 & 2 & 0
\end{bmatrix}
\xrightarrow{\text{Gaussian elimination}}
\begin{bmatrix}
1 & 0 & 0 & -1 \\
0 & 1 & ... |
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... | DanielWainfleet | 254,665 | <p>Let <span class="math-container">$S$</span> be a Tychonoff space and let <span class="math-container">$id_S:S\to cS$</span> be a compactification of <span class="math-container">$S.$</span></p>
<p>(i). If <span class="math-container">$p\in cS \setminus S$</span> and <span class="math-container">$U$</span> is an ope... |
1,374,977 | <blockquote>
<p>Which one of the following is true.</p>
<p>$(a.)\ \log_{17} 298=\log_{19} 375 \quad \quad \quad \quad (b.)\ \log_{17} 298<\log_{19} 375\\
(c.)\ \log_{17} 298>\log_{19} 375 \quad \quad \quad \quad (d.)\ \text{cannot be determined} $</p>
</blockquote>
<p>$17^{2}=289 $ it has a differen... | Marconius | 232,988 | <p>Let $x=\log_{17}{298}, y=\log_{19}{375}$.</p>
<p>By definition of logarithms,</p>
<p>$17^x = 298$ and $19^y=375$</p>
<p>So</p>
<p>$17^{x-2} = \dfrac{298}{289} = 1 + \dfrac{9}{289} \tag{1}$ and $19^{y-2}=\dfrac{375}{361} = 1 + \dfrac{14}{361} \tag{2}$.</p>
<p>Now take natural logarithms</p>
<p>$(x-2)\ln{17} = \... |
1,374,977 | <blockquote>
<p>Which one of the following is true.</p>
<p>$(a.)\ \log_{17} 298=\log_{19} 375 \quad \quad \quad \quad (b.)\ \log_{17} 298<\log_{19} 375\\
(c.)\ \log_{17} 298>\log_{19} 375 \quad \quad \quad \quad (d.)\ \text{cannot be determined} $</p>
</blockquote>
<p>$17^{2}=289 $ it has a differen... | Tzara_T'hong | 259,298 | <p>The following method does not use approximate calculations.</p>
<p>First of all note that $\ln 19 <\frac{7}{6}\ln 17$.</p>
<p>$$\log_{17} 298 \vee \log_{19} 375$$
$$\frac{\ln 298}{\ln 17}\vee \frac{\ln 375}{\ln 19}$$
$$\frac{\ln 298}{\ln 17}-2\vee \frac{\ln 375}{\ln 19}-2$$
$$\frac{\ln \frac{298}{17^2}}{\ln 17}... |
3,745,551 | <p>I often see people say that if you have 2 IID gaussian RVs, say <span class="math-container">$X \sim \mathcal{N}(\mu_x, \sigma_x^2)$</span> and <span class="math-container">$Y \sim \mathcal{N}(\mu_y, \sigma_y^2)$</span>, then the distribution of their sum is <span class="math-container">$\mathcal{N}(\mu_x + \mu_y, \... | Alex Ortiz | 305,215 | <p>I think your confusion does not really have anything to do with random variables, but rather simply with interpreting units. If I tell you I have <span class="math-container">$10$</span> apples and <span class="math-container">$3$</span> oranges, then <span class="math-container">$10 + 3 = 13$</span>, no matter what... |
2,564,321 | <p>I'm trying to prove </p>
<p>$$e^x\leq e^a\frac{b-x}{b-a}+e^b\frac{x-a}{b-a}$$</p>
<p>for any $x\in[a,b]$. Since this looks reminiscent of the mean value theorem or linear approximations I jotted down some equations relating to those, but didn't see any way of making progress with them. I know that $e^x$ is an in... | Paramanand Singh | 72,031 | <p>We can consider the function $$g(x) =f(b) - f(x) - \frac{f(b) - (a)} {b-a} (b-x) $$ where $f(x) = e^{x} $. We have to prove that $g(x) \geq 0$ for all $x\in[a, b] $. We have via mean value theorem $$f'(c) =\frac{f(b) - f(a)} {b-a} $$ for some $c\in(a, b) $ and since $f'(c) =f(c) $ we get $$g(x) =f(b) - f(x) - (b-x) ... |
216,099 | <p>$$x=\int \sqrt{\frac{y}{2a-y}}dy$$</p>
<p>According to my textbook, it says that the substitution by $y=a(1-\cos\theta)$ will easily solve the intergral. Why does this work?</p>
| DonAntonio | 31,254 | <p>$$y=a(1-\cos\theta)\Longrightarrow dy=a\sin\theta\,d\theta\Longrightarrow$$</p>
<p>$$\int\sqrt{\frac{y}{2a-y}}\,dy=\int\sqrt{\frac{a(1-\cos\theta)}{a(1+\cos\theta)}}\,a\sin\theta\,d\theta$$</p>
<p>But I can't see an easy way to solve the above, which according to WA is a rather ugly expression which, assuming posi... |
216,099 | <p>$$x=\int \sqrt{\frac{y}{2a-y}}dy$$</p>
<p>According to my textbook, it says that the substitution by $y=a(1-\cos\theta)$ will easily solve the intergral. Why does this work?</p>
| Mike | 17,976 | <p>Let me try to answer this. First thing I'd do is try to rewrite the integral as</p>
<p>$$\int\sqrt{\frac{2a}{2a-y}-1}dy$$</p>
<p>From here, I'd attempt to eliminate the square root by letting</p>
<p>$$\frac{2a}{2a-y}=\sec^2\theta$$</p>
<p>$$2a=2a\sec^2\theta-y\sec^2\theta$$</p>
<p>$$y=\frac{2a(\sec^2\theta-1)}... |
1,824,966 | <p>Ok, I was asked this strange question that I can't seem to grasp the concept of..</p>
<blockquote>
<p>Let $T$ be a linear transformation such that:
$$T \langle1,-1\rangle = \langle 0,3\rangle \\
T \langle2, 3\rangle = \langle 5,1\rangle $$
Find $T$.</p>
</blockquote>
<p>Is there suppose to be a funct... | Jeff L. | 310,106 | <p>Technically the $T$ you are asked to find is a function and not a matrix. Though you can find the matrix representation of $T$ as others have noted in their answers. I think that the point of this question is to make sure you understand how bases and linear transformations (as functions) work. The question isn't wor... |
37,052 | <p>This is my first question with mathOverflow so I hope my etiquette is up to par here.</p>
<p>My question is regarding a <span class="math-container">$3\times3$</span> magic square constructed using the la Loubère method (see <a href="http://en.wikipedia.org/wiki/Magic_square#Method_for_constructing_a_magic_square_of... | David Bar Moshe | 1,059 | <p>The article <em><a href="https://doi.org/10.1090/S0002-9939-98-05028-X" rel="nofollow noreferrer">Wick products of the CAR algebra</a></em> by E. R. Negrin provides the required formula for the antisymmetric Fock space
in the corollary on page 3644.</p>
<p>I want to point out that the Wick products (for the antisymm... |
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... | Ulrich Neumann | 53,677 | <p>You could solve your problem in the way I do it examplary for simpler problem:</p>
<pre><code>NSolve[{x1 + 2 x2 - x3 == 0,
0 <= x1 <= 1, 0 <= x2 <= 1, 0 <= x3 <= 1}, {x1, x2, x3}, Integers]
(* {{x1 -> 0, x2 -> 0, x3 -> 0}, {x1 -> 1, x2 -> 0, x3 -> 1}} *)
</code></pre>
|
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... | gwr | 764 | <p><strong>Restricting Variables to Specific Values</strong></p>
<p>Restricting the variables to 0 or 1 might more easily be done using regions:</p>
<pre><code>poly = 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 x2... |
2,633,392 | <p>I was recently reading about sets and read that $B$ is a subset of $A$ when each member of $B$ is a member of $A$. However, I am not sure about whether this requires the members of $A$ to simply be members of $B$, or if they could be part of $A$ in some other way - i.e. embedded within a set inside $A$.</p>
<p>I tr... | Akababa | 87,988 | <p>In general you can't consider $x$ and $\{x\}$ as the same thing, so $Y\subseteq X$ means $$\forall y\in Y(y\in X)$$</p>
|
49,281 | <p>I have two ingredients here:</p>
<ul>
<li>a big dataset contained in a list, with ~ 20M values. </li>
<li>a function that takes as each element of the list as input and yields True or False</li>
</ul>
<p>I want to save somewhere the elements of the list that yielded True.
Usually I would do something like that:</p... | Ken | 15,848 | <p>If the data set is truly large, I'd suggest saving some results to the disk if it is an option. This way, if anything goes wrong at any point, you don't have to re-compute everything all over again.</p>
<p>Of course, the disk operations will be slow, but I sometimes find persisting the state of the computation hel... |
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... | David Roberts | 4,177 | <p>Topological categories were invented by Charles Ehresmann in the late 1950s, and can be seen in his 1959 paper I think called Catégories topologique et catégories differentiable. The usage 'topological category' for a Top-category is much newer.</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... | Scott Taylor | 23,938 | <p>Thomassen's paper on triangulating surfaces addresses this as well.
See: <a href="https://mathoverflow.net/questions/17578/triangulating-surfaces">Triangulating surfaces</a></p>
|
1,743,482 | <p>I was doing this question on convergence of improper integrals where in our book they have used the fact that $2+ \cos(t) \ge1$. Can somebody prove this?</p>
| DonAntonio | 31,254 | <p>Suppose </p>
<p>$$\;p=1\pmod4\implies p-1=0\pmod4\implies \;\text{the cyclic group}\;\;\Bbb F_p^*\;$$
of order $\;p-1\;$ has a unique subgroup of order $\;4\;$, say $\;T:=\langle x\rangle\;$ , and since</p>
<p>$$\text{ord}\,(x)=4\implies x^2=-1$$</p>
|
252,147 | <blockquote>
<p><strong>Problem:</strong> Let $G$ be an infinite abelian group. Show that if $G$ has a nontrivial subgroup $K$ such that $K$ is contained in all nontrivial subgroups of $G$, then $G$ is a $p$- group for some prime $p$. Moreover, $G$ is of type $p^\infty$ (quasicyclic) group.</p>
</blockquote>
<p>I h... | Mikasa | 8,581 | <p>Hint: If it is as @Alexander noted, show that the group is torsion. So you can write $G$ for some prime $p$ as $$G\cong G_p$$ where $G_p=\{x\in G\mid p^nx=0\}$ for some $n\geq 0$. This can be show that the group in a $p$ group. For the rest, I think we should focus on proving $G$ is dividable.</p>
|
174,573 | <p>I am generating a random matrix $F$ and then plotting norm of the matrices $E+Ft$, representing them as a table. But I also want to print the matrix $F$ near every plot. If I remove the semicolon from <code>F // MatrixForm;</code> in the code below I get some errors.</p>
<p>My code:</p>
<pre><code>Dimension := 3;
... | t-smart | 53,560 | <pre><code>a={{0, 0, 0}, {0, 0, 1}, {0, 1, 0}, {0, 1, 1}, {1, 0, 0}, {1, 0, 1}, {1, 1, 0}, {1, 1, 1}};
Cases[a, {b_, _, _} -> If[b == 0, 1, 0]]
Cases[a, {b_, c_, _} -> If[{b,c} =={1,0}, 1, 0]]
</code></pre>
<p>For changeable length, use ___ in place of _, like <code>{a_,b_,___}</code>
This would do just fine. Si... |
2,604,178 | <p>Let $(G=(a_1,...,a_n),*)$ be a finite Group. Define for a element $a_i \in G$ a permutation $\phi = \phi(a_i)$ by left multiplication:</p>
<p>$$
\begin{bmatrix}
a_1 & a_2 & ... & a_n \\
a_i*a_1 & a_i*a_2 & ... & a_i*a_n \\
\end{bmatrix}
$$
I am struggling to understand why this is the permu... | B. Goddard | 362,009 | <p>There's a typo in your displayed equation. The $x$ in the numerator should be $x_n$. Then one just writes</p>
<p>$$x_{n+1} = x_n -\frac{1+x_n^2}{2x_n} = x_n\frac{2x_n}{2x_n} - \frac{1+x_n^2}{2x_n} =\frac{2x_n^2-x_n^2-1}{2x_n}. $$</p>
|
2,604,178 | <p>Let $(G=(a_1,...,a_n),*)$ be a finite Group. Define for a element $a_i \in G$ a permutation $\phi = \phi(a_i)$ by left multiplication:</p>
<p>$$
\begin{bmatrix}
a_1 & a_2 & ... & a_n \\
a_i*a_1 & a_i*a_2 & ... & a_i*a_n \\
\end{bmatrix}
$$
I am struggling to understand why this is the permu... | Mohammad Riazi-Kermani | 514,496 | <p>Newton's Method works only if the sequence of iterates converges. This is not always the case. </p>
<p>For example if you choose $f(x)=\sqrt[3] x$ and try Newton's Method to find the root $x=0$ </p>
<p>We get $$ x_{n+1}=x_n - \frac{f(x_n)}{f'(x_n)} = x_n - \frac{x_n^{1/3}}{(1/3) x_n^{-2/3}}=-2x_n $$ The sequence o... |
604,824 | <p>So the puzzle is like this:</p>
<blockquote>
<p>An ant is out from its nest searching for food. It travels in a straight line from its nest. After this ant gets 40 ft away from the nest, suddenly a rain starts to pour and washes away all its scent trail. This ant has the strength of traveling 280 ft more then it ... | copper.hat | 27,978 | <p>Here is a slightly shorter solution:</p>
<p><img src="https://i.stack.imgur.com/8zKSr.png" alt="enter image description here"></p>
<p>$40(\sqrt{2}+1 + \pi +1) \approx 262.2$.</p>
|
604,824 | <p>So the puzzle is like this:</p>
<blockquote>
<p>An ant is out from its nest searching for food. It travels in a straight line from its nest. After this ant gets 40 ft away from the nest, suddenly a rain starts to pour and washes away all its scent trail. This ant has the strength of traveling 280 ft more then it ... | Gottfried Helms | 1,714 | <p>This is only a partial proof and it is intended only as an illustration of my comment to @copper.hat - a full proof seems already be given the other answer... </p>
<p>I'm not sure with the second part: The second part is, that we <strong><em>assume</em></strong> , the minimal path ends with a path walking on the ... |
423,718 | <p>I have a general question, that deals with the question how I am able to find out whether a particular curve(in $\mathbb{C}$) is positively oriented? Take e.g. $ y(t)=a+re^{it}$. Obviously this one is positively oriented, but is there a fast general method to proof this? </p>
| Jim Belk | 1,726 | <p>The question you are asking is how to prove that the <a href="http://en.wikipedia.org/wiki/Turning_number" rel="nofollow">turning number</a> of a simple closed curve is equal to one. This is the same as saying that the <a href="http://en.wikipedia.org/wiki/Winding_number" rel="nofollow">winding number</a> of the cu... |
1,443,335 | <p>The rank of a linear transformation from V into W is defined:</p>
<blockquote>
<p>If V is finite-dimensional, the <em>rank</em> of T is the dimension of the range of T and ...</p>
</blockquote>
<p>However, there is no guarantee the range of T is finite-dimensional, in which case the dimension of it cannot be def... | Brian M. Scott | 12,042 | <p>If you divide out the fraction, you get</p>
<p>$$\frac{17+n}{15-2n}=-\frac12+\frac{49/2}{15-2n}=-\frac12+\frac12\cdot\frac{49}{15-2n}=\frac12\left(\frac{49}{15-2n}-1\right)\;.$$</p>
<p>This is an integer if and only if $\dfrac{49}{15-2n}$ is an odd integer. Since all divisors of $49$ are odd, we need only look for... |
697,984 | <p>I want to check whether the position operator $A$, where $Af(x)=xf(x)$ , is self-adjoint. For this to be true it has to be Hermitian and also the domains of it and its adjoint must be equal. The Hilbert space I'm working with is of course $L^2(\mathbb{R}) $ with the natural inner product. The problem I'm having is w... | Tim Seguine | 15,382 | <p>Notice that: $$A\cup B= (A \cap B^C)\cup(A^C \cap B)\cup(A\cap B)$$ That is, $A \cup B$ is the disjoint union of three sets.</p>
<p>Now, remember that disjoint unions are nice, because we can distribute over them. (i.e. $A \cap B=\emptyset \implies P(A\cup B)=P(A)+P(B)$)</p>
<p>Combining that idea with the first l... |
1,531,291 | <p>I want to find the radius of convergence of </p>
<p><span class="math-container">$$\sum_{k = 0}^{\infty}\frac{ k^{2 k + 5} \ln^{10} k \ln \ln k}{\left(k!\right)^2} \,x^k$$</span></p>
<p>I know formulae
<span class="math-container">$$R=\dfrac{1}{\displaystyle\limsup_{k\to\infty} \sqrt[k]{\left\lvert a_k\right\rver... | Mark Viola | 218,419 | <p><strong>HINT:</strong></p>
<p><a href="https://en.wikipedia.org/wiki/Stirling%27s_approximation#Derivation" rel="nofollow">Stirling's Formula</a> states</p>
<p>$$k! =\sqrt{2\pi k}\left(\frac ke\right)^k \left(1+O\left(\frac1k\right)\right)$$</p>
<p>Then, </p>
<p>$$\left(\frac{k^{2k+5}\,(\log k)^{10}\,\log (\log ... |
1,531,291 | <p>I want to find the radius of convergence of </p>
<p><span class="math-container">$$\sum_{k = 0}^{\infty}\frac{ k^{2 k + 5} \ln^{10} k \ln \ln k}{\left(k!\right)^2} \,x^k$$</span></p>
<p>I know formulae
<span class="math-container">$$R=\dfrac{1}{\displaystyle\limsup_{k\to\infty} \sqrt[k]{\left\lvert a_k\right\rver... | Community | -1 | <p>We have $$\sqrt[k]{\frac{ k^{2 k + 5} \ln^{10} k \ln \ln k}{\left(k!\right)^2}} = \left(\frac {k^4 e^{2k} \log^{10} k \log \log k} {2 \pi} \left(1 + O \left(\frac {1} {k}\right)\right)\right)^{1/k} \to e^2,$$ so $$R = \frac {1} {e^2}.$$</p>
|
2,832,311 | <p>Suppose I draw 10 tickets at random with replacement from a box of tickets, each of which is labeled with a number. The average of the numbers on the tickets is 1, and the SD of the numbers on the tickets is 1. Suppose I repeat this over and over, drawing 10 tickets at a time. Each time, I calculate the sum of the n... | StackTD | 159,845 | <blockquote>
<p>$(a_{n}) \rightarrow a$ if for every $\epsilon>0 , \exists N \in \mathbb{N}$ such that whenever $n \geq N$ it follows that $|a_{n}-a|< \epsilon$</p>
</blockquote>
<ul>
<li><strong>first</strong> you pick a (positive) margin of error, $\epsilon>0$;</li>
<li>then you find an index $N$ such tha... |
844,700 | <p>I am looking for a calculator which can calculate functions like $f(x) = x+2$
at $x=a$ etc; but I am unable to do so. Can you recommend any online calculator?</p>
| lhf | 589 | <p>Try the <a href="http://www.desmos.com/calculator" rel="nofollow">Desmos Graphing Calculator</a>. The output looks really nice.</p>
<p>See an <a href="http://www.desmos.com/calculator/59qdbtnlzy" rel="nofollow">interactive example of drawing lines</a>.</p>
|
2,878,508 | <p>How can I determine $ f(x)$ if $f(1-f(x))=x$ for all real $x$?
I have already recognized one problem caused from this: it
follows that $ f(f(x))=1-x $, which is discontinuous. So how can I construct a function $f(x)$?</p>
<p>Best regards and thanks,
John</p>
| Batominovski | 72,152 | <p>This answer is heavily inspired by Adrian Keister's work. Define
<span class="math-container">$$g(x):=f\left(x+\frac{1}{2}\right)-\frac12\text{ for each }x\in\mathbb{R}\,.$$</span>
(Note that <span class="math-container">$f(x)=g\left(x-\dfrac12\right)+\dfrac12$</span> for all <span class="math-container">$x\in\math... |
834,949 | <p>I have this HW where I have to calculate the $74$th derivative of $f(x)=\ln(1+x)\arctan(x)$ at $x=0$.
And it made me think, maybe I can say (about $\arctan(x)$ at $x=0$) that there is no limit for the second derivative, therefore, there are no derivatives of degree grater then $2$.
Am I right?</p>
| amWhy | 9,003 | <p>You're incorrect. The second derivative of $f(x)$ exists, and furthermore, $f''(0) = 2$.</p>
<p>In particular, there is no problem calculating the derivative at any order of the function $g(x) = \arctan(x)$. For example, $g(x) = \arctan(x) \implies g'(x) = \frac 1{1 + x^2}$, and $g''(x) = -\frac{2x}{(1 + x^2)^2}$.... |
510,151 | <p>Prove by induction that $2k(k+1) + 1 < 2^{k+1} - 1$ for $ k > 4$.
Can some one pls help me with this?</p>
<p>I reformulated like this</p>
<p>$ 2k(k+1) + 1 < 2^{k+1} - 1 $</p>
<p>$ 2k^2+2k+2<2^{k+1}$</p>
<p>and I tried like this
Take $k=k+1$</p>
<p>$ 2^{k+2} -1 > 2(k+1)(k+2) + 1 $</p>
<p>$... | Umberto P. | 67,536 | <p>The sequence fails to be Cauchy because the distance between any two terms in the sequence is $1$.</p>
|
2,878,814 | <ol>
<li>If a function $f(x)$ is continuous and increasing at point $x=a,$ then there is a nbhd $(x-\delta,x+\delta),\delta>0$ where the function is also increasing.</li>
<li>if $f' (x_0)$ is positive, then for $x$ nearby but smaller than
$x_0$ the values $f(x)$ will be less than $f(x_0)$, but for $x$
nearby but la... | Doug M | 317,162 | <p>I am going to guess that $AB = AB'$</p>
<p>In which case $AC,AC', BC, BC'$ are proportional to $\cos\theta, \cos (\theta + \alpha),\sin\theta, \sin(\theta + \alpha)$</p>
<p>And, you are trying to show.</p>
<p>$\sin \theta = \frac {cos\theta\cos\alpha - \cos (\theta+\alpha)}{\sin\alpha}$</p>
<p>Which simplifies ... |
456,583 | <p>I was searching for a Latex symbol that indicates $A \Rightarrow B$ and $A \not\Leftarrow B$ ($B$ if not only if $A$, $B$ ifnf $A$). I thought of using $A \Leftrightarrow B$ with the left arrow tick <code><</code> crossed out. Since I did not find such a symbol:</p>
<p>Is there a Latex symbol for this?</p>
<p>H... | Greg Nisbet | 128,599 | <p>You need to be careful with this connective (in classical logic) <em>because</em> it doesn't let you play fast and loose with the scope of free variables the way that <span class="math-container">$\to$</span> and <span class="math-container">$\leftrightarrow$</span> do.</p>
<p>I think using words in your expression ... |
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