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 |
|---|---|---|---|---|
3,052,746 | <p>I want to solve <span class="math-container">$2x = \sqrt{x+3}$</span>, which I have tried as below:</p>
<p><span class="math-container">$$\begin{equation}
4x^2 - x -3 = 0 \\
x^2 - \frac14 x - \frac34 = 0 \\
x^2 - \frac14x = \frac34 \\
\left(x - \frac12 \right)^2 = 1 \\
x = \frac32 , -\frac12
\end{equation}$$</span>... | fleablood | 280,126 | <p><span class="math-container">$\require{cancel}$</span>Additional details in <span class="math-container">$\color{blue}{blue}$</span>. Important detail in <span class="math-container">$\color{green}{green}$</span>. Mistakes in <span class="math-container">$\color{red}{\cancel{\text{canceled red}}}$</span>. Correcti... |
2,343,027 | <p>I am having a problem in proving this map to be one-one. It is not said anything about the relationship about $K$ and $R$. Or is it not necessary that they be related somehow. Please help.</p>
| egreg | 62,967 | <p>Hint: a ring homomorphism $f$ is 1-1 if and only if $\ker f=\{0\}$; what are the ideals of $K$?</p>
|
2,343,027 | <p>I am having a problem in proving this map to be one-one. It is not said anything about the relationship about $K$ and $R$. Or is it not necessary that they be related somehow. Please help.</p>
| Pierre-Yves Gaillard | 660 | <p>The statement is false: consider the unique morphism from a field to the zero ring.</p>
<p><strong>Edit.</strong> The above lines answer the question </p>
<blockquote>
<p>Let $f: K \rightarrow R$ be a ring homomorphism, where $K$ is a field and $R$ is a commutative ring with unity. Prove that $f$ is one-to-one,<... |
1,416,275 | <p>I've just began the study of linear functionals and the dual base. And this book I'm reading says the dual space $V^{*}$ may be identified with the space of row vectors. This notion seems very important, but I'm having trouble understanding it. Here is the text:</p>
<blockquote>
<p>Let $\sigma$ be an element of t... | Matematleta | 138,929 | <p>The elements of $V^*$ operate on $V$ in the obvious way: $(\varphi ,v)\mapsto \varphi (v)$.</p>
<p>We ask how can we represent this using the bases for $V$ and $V^*$.</p>
<p>Given the basis $\mathcal B=\left \{ v_{1},\cdots ,v_{n} \right \}$ for $V$, there is the natural basis $\mathcal B^*=\left \{ v_{1}^*,\cdot... |
80,056 | <p>I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the possible pros and cons of doing this. </p>
<p>Obviously, slides make it possible to produce and show clear graphs/pictu... | Jaap Eldering | 3,928 | <p>I just finished teaching a course on linear algebra to non-math students. I used a combination of latex-beamer slides and blackboard. One advantage of the slides was being able to do examples of Gauss elimination and inversion of matrices quicker than on the blackboard and without making mistakes. On the other hand,... |
80,056 | <p>I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the possible pros and cons of doing this. </p>
<p>Obviously, slides make it possible to produce and show clear graphs/pictu... | Adrien Hardy | 15,517 | <p>It also depends on how do you think it is the best for your students to learn : By listening (hopefully carefully) to the course, and then reading notes you'll provide them, OR by letting them write themselves the content. </p>
<p>I don't like to much the first option, certainly because I've not been used too, and ... |
80,056 | <p>I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the possible pros and cons of doing this. </p>
<p>Obviously, slides make it possible to produce and show clear graphs/pictu... | GMark | 17,764 | <p>My solution is to use a tablet PC (the pen-enabled kind, not the modern entertainment tablets like the Ipad),hooked up to a data projector. </p>
<p>I have "lecture templates" which contain the copying intensive stuff (statements of theorems, definitions, graphs, complex diagrams) on the page, along with plenty of b... |
424,675 | <p>Just one simple question:</p>
<p>Let $\tau =(56789)(3456)(234)(12)$.</p>
<p>How many elements does the conjugacy class of $\tau$ contain? How do you solve this exersie?</p>
<p>First step is to write it in disjunct cyclces I guess. What's next? :)</p>
| Dr Anil Kumar | 149,989 | <p>finite difference method is the oldest method to find the limited region or close region and FEM is the structural method to solve the partial deferential equation. </p>
|
189,293 | <p><img src="https://i.stack.imgur.com/Ngfb2.jpg" alt="Vesica Pisces"></p>
<p>I have the radius and center $(x,y)$ on both circles, but how do I get the $(x,y)$ of the red circle, or in other words how do I get the $(x,y)$ position of where the circles intersect at the top or bottom?</p>
| Sasha | 11,069 | <p>Let $O_1$ and $O_2$ denote centers of each circle, and $r_1$ and $r_2$ denote their radii. Let $P$ denote the point of intersection you are interested in. We know length of each side of the triangle $\triangle O_1 O_2 P$, hence we can determine its height $h$, i.e. distance from $P$ to the line passing through $O_1$... |
466,576 | <p>Some conventional math notations seem arbitrary to English speakers but were mnemonic to non-English speakers who started them. To give a simple example, Z is the symbol for integers because of the German word <em>Zahl(en)</em> 'number(s)'. What are some more examples?</p>
| Mikhail Katz | 72,694 | <p>The reason the "Klein bottle" is called a bottle has its origin in something of a German pun on Fläche/Flasche; see <a href="http://en.wikipedia.org/wiki/Klein_bottle">here</a></p>
|
466,576 | <p>Some conventional math notations seem arbitrary to English speakers but were mnemonic to non-English speakers who started them. To give a simple example, Z is the symbol for integers because of the German word <em>Zahl(en)</em> 'number(s)'. What are some more examples?</p>
| Pierre-Yves Gaillard | 660 | <p>Gabriel introduced the notation $\text{Sex}(\mathcal A,\mathcal B)$ to denote the category of left exact functors from $\mathcal A$ to $\mathcal B$. This because the Latin word for <em>left</em> (which is <em>sinister</em>) starts with an S.</p>
|
466,576 | <p>Some conventional math notations seem arbitrary to English speakers but were mnemonic to non-English speakers who started them. To give a simple example, Z is the symbol for integers because of the German word <em>Zahl(en)</em> 'number(s)'. What are some more examples?</p>
| Jean-Sébastien | 31,493 | <p>QED comes from the latin <em>quod erat demonstrandum</em></p>
|
466,576 | <p>Some conventional math notations seem arbitrary to English speakers but were mnemonic to non-English speakers who started them. To give a simple example, Z is the symbol for integers because of the German word <em>Zahl(en)</em> 'number(s)'. What are some more examples?</p>
| MJD | 25,554 | <p>The identity element of a group is $e$ for <em>einheit</em>, the German word for "identity".</p>
|
1,354,745 | <p>Let polynomial $p(z)=z^2+az+b$ be such that $a$and $b$ are complex numbers and $|p(z)|=1$ whenever $|z|=1$. Prove that $a=0$ and $b=0$.</p>
<p>I could not make much progress.
I let $z=e^{i\theta}$ and $a=a_1+ib_1$ and $b=a_2+ib_2$ </p>
<p>Using these values in $P(z)$ i got
$|P (z)|^2=1=(\cos (2\theta)-a_2\sin (\th... | Dleep | 240,562 | <p>Hint: What happens when $ z = 1 $, $ z = i $, $z = -1$, $z = -i$ ?</p>
|
4,498,203 | <p>We know that each row (and each column) of composition table of a finite group, is a rearrangement (permutation) of the elements of the group.</p>
<p>How about the other way round? If we have a composition table where each row and each column is a permutation of the elements of a set, does this composition table nec... | Shaun | 104,041 | <p>No.</p>
<p>Consider</p>
<p><span class="math-container">$$\begin{array}{c|ccc}
\ast & e & a & b\\
\hline
e & e & a & b\\
a & b & e & a\\
b & a & b & e
\end{array}.$$</span></p>
<p>There is no identity, so it cannot be a group.</p>
|
2,906,865 | <p>I am trying to find general formula of the sequence $(x_n)$ defined by
$$x_1=1, \quad x_{n+1}=\dfrac{7x_n + 5}{x_n + 3}, \quad \forall n>1.$$
I tried
put $y_n = x_n + 3$, then $y_1=4$ and
$$\quad y_{n+1}=\dfrac{7(y_n-3) + 5}{y_n }=7 - \dfrac{16}{y_n}, \quad \forall n>1.$$
From here, I can't solve it. How can ... | Daniel Schepler | 337,888 | <p>Since the function being iterated is a projective-linear function, it follows that if you let
$$ \begin{bmatrix} a_n \\ b_n \end{bmatrix} := \begin{bmatrix} 7 & 5 \\ 1 & 3 \end{bmatrix} ^{n-1} \begin{bmatrix} 1 \\ 1\end{bmatrix}$$
then $x_n = \frac{a_n}{b_n}$. Now, to find the powers of the matrix $\begin{b... |
2,906,865 | <p>I am trying to find general formula of the sequence $(x_n)$ defined by
$$x_1=1, \quad x_{n+1}=\dfrac{7x_n + 5}{x_n + 3}, \quad \forall n>1.$$
I tried
put $y_n = x_n + 3$, then $y_1=4$ and
$$\quad y_{n+1}=\dfrac{7(y_n-3) + 5}{y_n }=7 - \dfrac{16}{y_n}, \quad \forall n>1.$$
From here, I can't solve it. How can ... | minhthien_2016 | 336,417 | <p>The chracteristic equation of the given sequence is
<span class="math-container">$$y=\dfrac{7y+5}{y+3} \Leftrightarrow y_1 = 5 \lor y_2 = -1.$$</span>
Let us consider the sequence
<span class="math-container">$$b_n = \dfrac{x_n-y_1}{x_n - y_2}=\dfrac{x_n-5}{x_n+1}.$$</span>
We note that
<span class="math-container">... |
3,002,668 | <p>I have to solve this inequality:</p>
<p><span class="math-container">$$5 ≤ 4|x − 1| + |2 − 3x|$$</span></p>
<p>and prove its solution with one (or 2 or 3) of this sentences:</p>
<p><span class="math-container">$$∀x∀y |xy| = |x||y|$$</span></p>
<p><span class="math-container">$$∀x∀y(y ≤ |x| ↔ y ≤ x ∨ y ≤ −x)$$</s... | Will Jagy | 10,400 | <p>Part (I) fill in this table
<span class="math-container">$$
\begin{array}{c|c|c|c|c|c|c|c}
x & x-1&|x-1|&4|x-1| &-3x&2-3x&|2-3x|& y = 4|x-1| +|2-3x| \\ \hline
\frac{-1}{6} & &&&&&& \\ \hline
0 & &&&&&& \\ \hline
\frac{1}{6} &... |
179,583 | <p>I have a fairly large array, a billion or so by 500,000 array. I need to calculate the singular value decomposition of this array. The problem is that my computer RAM will not be able to handle the whole matrix at once. I need an incremental approach of calculating the SVD. This would mean that I could take one or a... | Hans Engler | 9,787 | <p>You could compute the SVD of randomly chosen submatrices of your original matrix, as shown e.g. in <a href="http://www.cc.gatech.edu/fac/vempala/papers/dfkvv.pdf" rel="nofollow">the 2004 paper by Drineas, Frieze, Kannan, Vempala and Vinay</a>, and scale the result to obtain an approximate SVD of the original matrix.... |
1,396,067 | <p><strong>Question:</strong><br/>
The bacteria in a certain culture double every $7.3$ hours. The culture has $7,500$ bacteria at the start.
How many bacteria will the culture contain after $3$ hours?
<br />
<br />
<strong>Possible Answers:</strong><br/>
a. $9,449$ bacteria<br/>
b. $9,972$ bacteria<br/>
c. $40,510$ ba... | Community | -1 | <p>If $x\in n\mathbb{Z}\cap m\mathbb{Z}$, then $x$ is an integer multiple of both $m$ and $n$, and therefore an integer multiple of $\text{lcm}(n,m)$. That is, if both $n$ and $m$ divide $x$, then $\text{lcm}(n,m)$ divides $x$. Many proofs of this are given <a href="https://math.stackexchange.com/questions/727544/cant-... |
17,143 | <p>My next project I'd like to start working on is Domain Coloring. I am aware of the beautiful discussion at:</p>
<p><a href="https://mathematica.stackexchange.com/questions/7275/how-can-i-generate-this-domain-coloring-plot">How can I generate this "domain coloring" plot?</a></p>
<p>And I am studying it. H... | J. M.'s persistent exhaustion | 50 | <p>I might as well put in my own take. I tried a bit to match the coloring used by Hans in his pics, and here's the result:</p>
<pre><code>SetAttributes[colorize, Listable];
colorize[z_] :=
Darker[Blend[{{0, Black}, {1/2, Red}, {3/4, Orange}, {1, Yellow}},
Mod[Arg[z]/(2 π), 1]], If[z =... |
2,222,966 | <p>Given the three line segments below, of lengths a, b and 1, respectively:<a href="https://i.stack.imgur.com/HWoz8.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/HWoz8.jpg" alt="enter image description here"></a></p>
<p>construct the following length using a compass and ruler: $$\frac{1}{\sqrt{b+... | Will Jagy | 10,400 | <p>It is all similar right triangles, along with the theorem that, when a triangle has all three vertices on a circle and two of them on a diameter, then it is a right triangle. </p>
<p><a href="https://i.stack.imgur.com/GVY8h.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/GVY8h.jpg" alt="enter ima... |
590,219 | <p>Disclaimer: this is a homework question. I'm looking for direction, not an answer.</p>
<blockquote>
<p>Given a field <span class="math-container">$F$</span>, show that <span class="math-container">$F[x,x^{-1}]$</span> is a principal ideal domain.</p>
</blockquote>
<p>I'm unsure how to proceed. Would it be better... | user52045 | 52,045 | <p>Polynomial ring over a field $k$ is a PID. Notice that $S^{-1}k[x]=k[x,x^{-1}]$, where $S=\{x^i:i\in \mathbb N\}$. Now use the fact that <a href="https://math.stackexchange.com/questions/536624/is-the-localization-of-a-pid-a-pid">localization of a PID is a PID</a>.</p>
|
2,874,763 | <p>I know for 3-D $$\nabla^2 \left(\frac1r\right)=-4\pi\, \delta(\vec{r})\,.$$
I would like to know, what is $$\text{Div}\cdot\text{Grad}\left(\frac{1}{r^2}\right)$$ in 4-Dimensions ($r^2=x_1^2+x_2^2+x_3^2+x_4^2$)?</p>
| Maxim | 491,644 | <p>Since the Fourier transform of a radial function is also a radial function and the transform of $r^\lambda$ is a homogeneous function of degree $-\lambda - n$ for $(-\lambda - n)/2 \notin \mathbb N^0$,
$$\mathcal F[r^\lambda] =
(r^\lambda, e^{i \boldsymbol x \cdot \boldsymbol \xi}) =
C_{\lambda, n} \rho^{-\lambda - ... |
929,598 | <p>A rectangle is 4 times as long as it is wide. If the length is increased by 4 inches and the width is decreased by 1 inch, the area will be 60 square inches. What were the dimensions of the original rectangle? Explain your answer.</p>
| mathlove | 78,967 | <p>HINT :
$$(2k+1)!!=1\cdot 3\cdot 5\cdot \cdots (2k-1)\cdot (2k+1)$$
$$(2k+1)!=1\cdot 2\cdot 3\cdot \cdots (2k)\cdot (2k+1) $$
$$2^k\cdot k!=2^k\times \{1\cdot 2\cdot 3\cdot \cdots (k-1)\cdot k\}=2\cdot 4\cdot 6\cdot\cdots (2k-2)\cdot (2k).$$</p>
|
1,413,150 | <p>So for a periodic function <span class="math-container">$f$</span> (of period <span class="math-container">$1$</span>, say), I know the Riemann-Lebesgue Lemma which states that if <span class="math-container">$f$</span> is <span class="math-container">$L^1$</span> then the Fourier coefficients <span class="math-cont... | Robert Israel | 8,508 | <p>Let
$$ \eqalign{f(n) = \dfrac{1}{n} + \left( 1 + \dfrac{1}{n}\right)^n &= \dfrac{1}{n} + \exp\left( n \ln\left(1+\dfrac{1}{n}\right)\right) \cr &=
\dfrac{1}{n} + \exp\left(1 - \dfrac{1}{2n} + \dfrac{1}{3n^2} + O\left(\dfrac{1}{n^3}\right)\right) \cr &= e - \dfrac{e-2}{2n} + \dfrac{11e}{24 n^2} + O\left(\... |
1,074,341 | <p>Prove that a Covering map is proper if and only if it is finite-sheeted.</p>
<p>First suppose the covering map $q:E\to X$ is proper, i.e. the preimage of any compact subset of $X$ is again compact. Let $y\in X$ be any point, and let $V$ be an evenly covered nbhd of $y$. Then since $q$ is proper, and $\{y\}$ is comp... | Georges Elencwajg | 3,217 | <p>$\Rightarrow$ Each fiber is compact (by properness) and discrete (from definition of covering space) hence is finite.</p>
<p>$\Leftarrow$ You have to prove that for $K\subset X$ the inverse image $q^{-1}(K)$ is compact.<br>
Since $\operatorname {res} q:q^{-1}(K) \to K$ is a finite covering space in its own right a... |
4,147,126 | <p>Let <span class="math-container">$\ p_n\ $</span> be the <span class="math-container">$\ n$</span>-th prime number.</p>
<blockquote>
<p>Does the <a href="https://en.wikipedia.org/wiki/Prime_number_theorem" rel="nofollow noreferrer">prime number theorem</a> ,</p>
<p><span class="math-container">$\Large{\lim_{x\to\inf... | Aphelli | 556,825 | <p>Indeed, yes! It can be shown elementarily that the statement of the PNT that you gave is equivalent to <span class="math-container">$\frac{p_n}{n\ln{n}} \rightarrow 1$</span>. Since <span class="math-container">$\frac{n+1}{n} \rightarrow 1$</span>, <span class="math-container">$\frac{\ln(n+1)}{\ln{n}}\rightarrow 1$<... |
655,981 | <p>How to calculate this complex integral?
$$\int_0^{2\pi}\cot(t-ia)dt,a>0$$</p>
<p>I got that the integral is $2\pi i$ if $|a|<1$ and $0$ if $a>1$
yet, friends of mine got $2\pi i$ regardless the value of $a$.
looking for the correct way</p>
| Daniel Fischer | 83,702 | <blockquote>
<p>so, I'll use the residue theorem</p>
</blockquote>
<p>Be careful. You need a closed contour for the residue theorem, but the interval $[0,2\pi]$ isn't a closed contour.</p>
<p>There are of course several methods to evaluate the integral. One particularly nice way, since $\cos = \sin'$ and $\sin$ is ... |
935,506 | <p>I'm a bit puzzled by this one.</p>
<p>The domain $X = S(0,1)\cup S(3,1)$ (where $S(\alpha, \rho)$ is a circular area with it's center at $\alpha$ and radius $\rho$). So the domain is basically two circles with radius 1 and centers at 0 and 3.</p>
<p>I'm supposed to find analytic function $f$ defined on $X$ where t... | ncmathsadist | 4,154 | <p>Yes. You can try one of two things. Integrate by parts with
$$u = t$$ and
$$dv = {2t\,dt\over \sqrt{1 - t^2}}$$
or a trig sub $t = \sin(\theta)$. The or is inclusive or here.</p>
|
935,506 | <p>I'm a bit puzzled by this one.</p>
<p>The domain $X = S(0,1)\cup S(3,1)$ (where $S(\alpha, \rho)$ is a circular area with it's center at $\alpha$ and radius $\rho$). So the domain is basically two circles with radius 1 and centers at 0 and 3.</p>
<p>I'm supposed to find analytic function $f$ defined on $X$ where t... | Bouzari Abdelkader | 727,256 | <p><span class="math-container">$$\int4x\sqrt{1-x^{4}}dx\\
x^{2}=\sin y\Rightarrow 2xdx=\cos ydy\\
\int4x\sqrt{1-x^{4}}dx=2\int \cos^{2}ydy=\int(1+\cos 2y)dy=y+\frac{\sin 2y}{2}+C\\
\int4x\sqrt{1-x^{4}}dx=\arcsin x^{2}+x^{2}\sqrt{1-x^{4}}+C\\
$$</span></p>
|
1,904,903 | <p>Taken from Soo T. Tan's Calculus textbook Chapter 9.7 Exercise 27-</p>
<p>Define $$a_n=\frac{2\cdot 4\cdot 6\cdot\ldots\cdot 2n}{3\cdot 5\cdot7\cdot\ldots\cdot (2n+1)}$$
One needs to prove the convergence or divergence of the series $$\sum_{n=1}^{\infty} a_n$$</p>
<p>upon finding the radius of convergence for $\su... | Community | -1 | <p>Let</p>
<p>$$ a = \frac{2}{3} \cdot \frac{4}{5} \cdots \frac{2n}{2n+1} , \quad
b = \frac{1}{2} \cdot \frac{3}{4} \cdots \frac{2n-1}{2n} $$</p>
<p>Then $a > b$ and $ab = \dfrac{1}{2n+1}$, so actually $a > \dfrac{1}{\sqrt{2n+1}}$ - stronger than what you needed.</p>
<p>In other words: You can use this to p... |
246,606 | <p>I have matrix:</p>
<p>$$
A = \begin{bmatrix}
1 & 2 & 3 & 4 \\
2 & 3 & 3 & 3 \\
0 & 1 & 2 & 3 \\
0 & 0 & 1 & 2
\end{bmatrix}
$$</p>
<p>And I want to calculate $\det{A}$, so I have written:</p>
<p>$$
\begin{array}{|cccc|ccc}
1 & 2 & 3 & 4 & 1 & 2 ... | martini | 15,379 | <p><a href="http://en.wikipedia.org/wiki/Rule_of_Sarrus">Sarrus's rule</a> works only for $3\times 3$-determinants. So you have to find another way to compute $\det A$, for example you can apply elementary transformations not changing the determinant, that is e. g. adding the multiple of one row to another:
\begin{alig... |
2,195,287 | <blockquote>
<p>Knowing that $p$ is prime and $n$ is a natural number show that
$$n^{41}\equiv n\bmod 55$$
using Fermat's little theorem
$$n^p\equiv n\bmod p$$</p>
</blockquote>
<p>If the exercise was to show that
$$n^{41}\equiv n\bmod 11$$ I would just rewrite $n^{41}$ as a power of $11$ and would easily prov... | lhf | 589 | <p>We have</p>
<ul>
<li><p>mod $\ \ 5:\quad$ $n^{41} \equiv (n^8)^5n \equiv n^8 n \equiv n^9 \equiv n^5 n^4 \equiv n n ^4 \equiv n^5 \equiv n$</p></li>
<li><p>mod $11:\quad$ $n^{41} \equiv (n^3)^{11} n^8 \equiv n^3 n^8 \equiv n^{11} \equiv n$</p></li>
</ul>
<p>Now apply the Chinese reminder theorem.</p>
|
1,985,427 | <p>$$
A= \begin{bmatrix}
2 & 1 & -1 \\
-2 & -2 & 1 \\
0 & -2 & 1 \\
\end{bmatrix}
$$</p>
<p>Can someone show me the best way to approach this? Should I use pivoting? I tried using the formula, but I think that only works for 2 x 2 matrices. </p>
| Deepak Suwalka | 371,592 | <p>$\begin {bmatrix} 2&1&-1\\-2&-2&1\\0&-2&1 \end {bmatrix}$</p>
<p>We can also solve it using determinants-</p>
<p>Calculating determinant of the matrix </p>
<p>Let $A=2[-2+2]-1[-2]-1[4]$</p>
<p>$=2-4=-2$</p>
<p>Calculating matrix of minors</p>
<p>$\begin {bmatrix} -2+2&&-2-0&... |
408,601 | <p>I am asked to find the derivative of $\left(x^x\right)^x$. So I said let $$y=(x^x)^x \Rightarrow \ln y=x\ln x^x \Rightarrow \ln y = x^2 \ln x.$$Differentiating both sides, $$\frac{dy}{dx}=y(2x\ln x+x)=x^{x^2+1}(2\ln x+1).$$</p>
<p>Now I checked this answer with Wolfram Alpha and I get that this is only correct whe... | iostream007 | 76,954 | <p>in step
$$\dfrac{dy}{dx}=y(2x\ln x+x)$$
$$\dfrac{dy}{dx}=y(x\ln x+x\ln x+x)$$
in question $y=(x^x)^x$
$$\dfrac{dy}{dx}=(x^x)^x(x\ln x+\ln x^x+x)$$
just rearrange your second last step</p>
|
2,611,382 | <p>Solve the equation,</p>
<blockquote>
<p>$$
\sin^{-1}x+\sin^{-1}(1-x)=\cos^{-1}x
$$</p>
</blockquote>
<p><strong>My Attempt:</strong>
$$
\cos\Big[ \sin^{-1}x+\sin^{-1}(1-x) \Big]=x\\
\cos\big(\sin^{-1}x\big)\cos\big(\sin^{-1}(1-x)\big)-\sin\big(\sin^{-1}x\big)\sin\big(\sin^{-1}(1-x)\big)=x\\
\sqrt{1-x^2}.\sqrt{2x... | Barry Cipra | 86,747 | <p>Here's a way to avoid the extraneous solution. Note that $\arcsin u+\arccos u={\pi\over2}$ for all $u\in[-1,1]$. Thus we can rewrite $\arcsin x+\arcsin(1-x)=\arccos x$ as </p>
<p>$$\arcsin x+{\pi\over2}-\arccos(1-x)={\pi\over2}-\arcsin x$$</p>
<p>which simplifies to</p>
<p>$$2\arcsin x=\arccos(1-x)$$</p>
<p>App... |
1,424,124 | <p>If $a,b$ be two positive integers , where $b>2 $ , then is it possible that $2^b-1\mid2^a+1$ ? I have figured out that if $2^b-1\mid 2^a+1$, then $2^b-1\mid 2^{2a}-1$ , so $b\mid2a$ and also $a >b$ ; but nothing else. Please help. Thanks in advance</p>
| Community | -1 | <p>Assume that it is possible.Then obviously $a>b$ so write $a=bx+r$ with $r \leq b-1$ .Then :
$$2^b-1 \mid 2^{bx}-1$$</p>
<p>Multiply by $2^r$ to get :
$$2^b-1 \mid 2^{bx+r}-2^r=2^a-2^r$$ </p>
<p>But we known that $2^b-1 \mid 2^a+1$ so subtracting them we get :</p>
<p>$$2^b-1 \mid 2^r+1$$ </p>
<p>This means tha... |
3,030,753 | <p>Let <span class="math-container">$f:\mathbb R \rightarrow \mathbb R$</span> be a continuous function and <span class="math-container">$x_0 \in \mathbb R$</span> such that f is differentiable on both intervals <span class="math-container">$(-\infty, x_0]$</span> and <span class="math-container">$[x_0, +\infty)$</span... | Robert Z | 299,698 | <p>Hint (to be read after copper.hat hint). </p>
<p>Let us consider the following two differentiable extensions of <span class="math-container">$f$</span>:
<span class="math-container">$$F_+(x) = \begin{cases} f(x), & x \ge x_0, \\
f(x_0)+f'_+(x_0)(x-x_0), & x \le x_0, \end{cases}$$</span>
and
<span class="ma... |
410,013 | <p><strong>Short question:</strong> Is there a standard term for a set <span class="math-container">$F$</span> such that there does not exist a surjection <span class="math-container">$F \twoheadrightarrow \omega$</span> (in the context of ZF)?</p>
<p><strong>More detailed version:</strong> Consider the following four ... | Guozhen Shen | 101,817 | <p>I suggest the terminology "power Dedekind finite" by Andreas Blass in his paper <em>Power-Dedekind Finiteness</em>, and I use this terminology throughout all my papers. By Kuratowski's celebrated theorem, a set <span class="math-container">$F$</span> does not map onto <span class="math-container">$\omega$<... |
97,340 | <p>Fellow Puny Humans, </p>
<p>A <em>geometric net</em> is a system of points and lines that obeys three axioms:</p>
<ol>
<li>Each line is a set of points.</li>
<li>Distinct line has at most one point in common.</li>
<li>If $p$ is a point and $L$ is a line with $p \notin L$, then there is exactly one line $M$ such th... | Community | -1 | <p>It is exactly the number of lines you have in a class, simply because <em>equal or parallel</em> is an equivalence relation. Let me clarify:</p>
<p>Say $C_1, C_2,\dots, C_m$ are your equivalence classes. Say $L\in C_i$ is a line in $C_i$, for some $i\in\{1,\dots,m\}$. Say $1\leq j \leq m$, $j\neq i$, and $M\in C_j$... |
966,570 | <p>My attempt: Let the characteristic be $n$. </p>
<p>Then, $n \cdot (1_6, 1_{15}) = (0_6, 0_{15})$,</p>
<p>i.e. $n \cdot 1_6=0_6$
and $n \cdot 1_{15}=0_{15}$</p>
<p>The least $n$ for which both are true is $30$, so $30$ is the characteristic.</p>
<p>Is my method correct? If so, if my writing ok?</p>
| Harry Wilson | 67,863 | <p>So, you are asking for a metric that takes two matrices, $A$ and $B$, and outputs a real number $d(A,B)$ obeying the principles one expects from a distance function : symmetry, reflexivity, the triangle equality.
One way to view this is to view that space of $n \times n$ real matrices as the vector space $\mathbb{R}... |
966,570 | <p>My attempt: Let the characteristic be $n$. </p>
<p>Then, $n \cdot (1_6, 1_{15}) = (0_6, 0_{15})$,</p>
<p>i.e. $n \cdot 1_6=0_6$
and $n \cdot 1_{15}=0_{15}$</p>
<p>The least $n$ for which both are true is $30$, so $30$ is the characteristic.</p>
<p>Is my method correct? If so, if my writing ok?</p>
| Pieter21 | 170,149 | <p>Depends on what would you like the distance between for instance $I$ and $-I$ to be. Or even the distance between $I$ and null/zero matrix $0$.</p>
<p>But I expect that in most cases you should also take into account the vectors you want to multiply with.</p>
<p>In the general case, I don't expect such distance wo... |
2,788,015 | <p>I'm trying to solve an exercise that says</p>
<blockquote>
<p>Show that a locally compact space is $\sigma$-compact if and only if is separable.</p>
</blockquote>
<p>Here locally compact means that also is Hausdorff. I had shown that separability imply $\sigma$-compactness but I'm stuck in the other direction.</... | Mirko | 188,367 | <p>take $\omega_1+1$ with the order topology. This is compact Hausdorff, but not separable. (That is, take the space of all countable ordinals, together with the first uncountable ordinal, with the order topology. This is not first countable either. As a comment suggest, perhaps the author meant that only metrizable sp... |
2,098,693 | <p>Full Question: Five balls are randomly chosen, without replacement, from an urn that contains $5$
red, $6$ white, and $7$ blue balls. What is the probability of getting at least one ball of
each colour?</p>
<p>I have been trying to answer this by taking the complement of the event but it is getting quite complex. A... | callculus42 | 144,421 | <p>The idea of taking the converse probability sounds good to me. Let $r,b,w$ the events where <strong>no</strong> red, <strong>no</strong> black and <strong>no</strong> white balls are drawn.</p>
<p>Then it is asked for</p>
<p>$1-P(r\cup w\cup b)=1-\left[P(r)+P(w)+P(b)-P(r,w)-P(r,b)-P(w,b)+P(r,w,b)\right]$</p>
<p>... |
4,635,416 | <p>Let <span class="math-container">$X$</span> be a symmetric random variable, that is <span class="math-container">$X$</span> and <span class="math-container">$-X$</span> have the same distribution function <span class="math-container">$F$</span>. Suppose that <span class="math-container">$F$</span> is continuous and ... | grand_chat | 215,011 | <p>Define <span class="math-container">$M:=\{x\in {\mathbb R}\mid F(x)\ge\frac12\}$</span>. You've shown that <span class="math-container">$0\in M$</span>. It remains to show that no number less than zero is a member of <span class="math-container">$M$</span>. To do this:</p>
<ol start="0">
<li><p>You've already shown ... |
215,333 | <p>There are many symbols for understanding internet-related properties: <code>$NetworkConnected</code>, <code>PingTime</code>, <code>NetworkPacketTrace</code>, <code>NetworkPacketRecording</code>, etc.</p>
<p>But is there any convenient way of testing your network's upload speed from within Mathematica?</p>
| Rohit Namjoshi | 58,370 | <p>Here is another way. Install the speedtest cli application for your OS from <a href="https://www.speedtest.net/apps/cli" rel="nofollow noreferrer">here</a>.</p>
<pre><code>SetEnvironment["PATH" -> Environment["PATH"] <> "path to install dir"]
output = RunProcess[{"speedtest", "-fcsv", "--output-header"}, ... |
1,779,965 | <p>Given the numbers $x = 123$ and $y = 100$ how to apply the Karatsuba algorithm to multiply these numbers ?</p>
<p>The formula is </p>
<pre><code>xy=10^n(ac)+10^n/2(ad+bc)+bd
</code></pre>
<p>As I understand $n = 3$ (number of digits) and I tried writing the numbers as </p>
<pre><code>x = 10*12+3 , y = 10*10 +0 t... | Haseeb Saeed | 648,375 | <p><span class="math-container">$xy=123\cdot 100$</span></p>
<p><span class="math-container">$(12\cdot 10+3)(10\cdot 10+0)$</span></p>
<p><span class="math-container">$xy=10^n(ac)+10^n/2(ad+bc)+bd$</span>, where <span class="math-container">$n=2
a=12, b=3, c=10, d=0$</span></p>
<p>subcategory 1:</p>
<p><span class=... |
2,286,749 | <p>My question is about the general solution for the following differential equation:
$$ \frac{dx}{dt} = x^a(1-x)^b,\quad a,b\gt 0~~~~~~~~~~~~~~~(1)~. $$</p>
<p>Obviously, if $a=b=1$ then (1) reduces to
$$ \frac{dx}{dt} = x(1-x) $$ which has as solution $$ x(t) = \frac{1}{1 + A e^{-t}}\,,$$ for some constant, $A$.... | projectilemotion | 323,432 | <p>Well, your differential equation is separable:</p>
<p><span class="math-container">$$\int \frac{1}{x^a(1-x)^b}~dx=\int dt \tag{1}$$</span></p>
<p>The left hand side cannot be integrated in terms of elementary functions.</p>
<blockquote>
<p>However, one can directly apply the definition of the <a href="https://en.wik... |
1,030,335 | <blockquote>
<p>Let <span class="math-container">$n$</span> and <span class="math-container">$r$</span> be positive integers with <span class="math-container">$n \ge r$</span>. Prove that:</p>
<p><span class="math-container">$$\binom{r}{r} + \binom{r+1}{r} + \cdots + \binom{n}{r} = \binom{n+1}{r+1}.$$</span></p>
</bloc... | David | 119,775 | <p>As mentioned above, this has been answered before (actually very recently). But for something different, here is a pictorial proof.
$$\def\r{\color{red}{\bullet}}\def\b{\color{blue}{\bullet}}\def\u{\circ}\def\w{\bullet}\def\s{\ \ \ \ }
\eqalignno{
\matrix{\w\cr \w\s\w\cr \w\s\w\s\w\cr \w\s\w\s\w\s\w\cr \w\s\w... |
1,334,527 | <p>The integral in hand is
$$
I(n) = \frac{1}{\pi}\int_{-1}^{1} \frac{(1+2x)^{2n}}{\sqrt{1-x^2}}\, dx
$$
I dont know whether it has closed-form or not, but currently I only want to know its asymptotic behavior. Setting $x=\cos\theta$, then
$$
I(n) = \frac{1}{\pi}\int_{0}^{\pi/2} \Big[(1+2\cos\theta)^{2n}+(1-2\cos\theta... | achille hui | 59,379 | <p>To compute the asymptotic expansion of the integral, we split it into two pieces</p>
<p>$$\frac{1}{\pi}\int_{-1}^{1} \frac{(1+2x)^{2n}}{\sqrt{1-x^2}}dx
= \frac{1}{\pi}\left(\int_{-1}^0 + \int_0^1\right)\frac{(1+2x)^{2n}}{\sqrt{1-x^2}}dx
$$
Over the interval $[-1,0]$, we have $|1+2x|\le 1$, so the contribution there... |
1,658,577 | <p>I'm an electrical/computer engineering student and have taken fair number of engineering math courses. In addition to Calc 1/2/3 (differential, integral and multivariable respectfully), I've also taken a course on linear algebra, basic differential equations, basic complex analysis, probability and signal processing... | USA | 626,318 | <p>Your goal is slant to applied math.</p>
<p>Try “Real Variables with ..metric space and topology” by Robert Ash. The book will also teach you proof writing.
Prof Ash is an electrical engineer who later became a Math prof.
The book has answers to all problems that helps you to gain confidence quickly. Prof Ash also h... |
1,658,577 | <p>I'm an electrical/computer engineering student and have taken fair number of engineering math courses. In addition to Calc 1/2/3 (differential, integral and multivariable respectfully), I've also taken a course on linear algebra, basic differential equations, basic complex analysis, probability and signal processing... | Ziqi Fan | 851,072 | <p>There is no "engineering math" separate from "math". To learn good math as an engineer, you have no other way but to understand math from its nature.</p>
<p>The first course you learn is probably real analysis. In my opinion, no one masters real analysis when the topic is met the first time. Don'... |
2,659,448 | <p>The following question is an exercise from Munkres' Analyis on Manifolds (Chapter 4 - Section 20):</p>
<p>Consider the vectors $a_i$ in $R^3$ such that:</p>
<p>$[a_1\ a_2\ a_3\ a_4] = \begin{bmatrix} 1 & 0 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 2 & 0 \end{bmatrix}$</p>
<p>Let $V$ ... | Tashi Walde | 509,746 | <p>Recall that two frames $(a_1,\dots,a_n)$ and $(v_1,\dots,v_n)$ have the same orientation if the linear isomorphism which sends $a_i\mapsto v_i$ (which exists uniquely) has positive determinant. </p>
<p>The subspace $V$ has two bases $a=(1,1,1),b=(0,0,1)$ and $v=(1,1,2),w=(1,1,0)$. If you change from one basis to th... |
19,261 | <p>Every simple graph $G$ can be represented ("drawn") by numbers in the following way:</p>
<ol>
<li><p>Assign to each vertex $v_i$ a number $n_i$ such that all $n_i$, $n_j$ are coprime whenever $i\neq j$. Let $V$ be the set of numbers thus assigned. <br/></p></li>
<li><p>Assign to each maximal clique $C_j$ a unique p... | Benoît Kloeckner | 4,961 | <p>This topic being quite large, I cannot insist enough to recommand you to take a look to Marcel Berger's <em>Panoramic view of Riemannian geometry</em>. The Bonnet-Myers theorem, the sphere theorems (for the recent development on this one, I think the web page of Simon Brendle contains a survey) are two celebrated ex... |
3,261,846 | <blockquote>
<p>What is the solution to the IVP <span class="math-container">$$y'+y=|x|, \ x \in \mathbb{R}, \ y(-1)=0$$</span></p>
</blockquote>
<p>The general solution of the above problem is <span class="math-container">$y_{g}(x)=ce^{-x}$</span>.</p>
<p>How to find the particular solution? As <span class="math-c... | A.Γ. | 253,273 | <p>Hint: using the <a href="https://en.wikipedia.org/wiki/Integrating_factor" rel="nofollow noreferrer">integrating factor</a> the equation can be rewritten as
<span class="math-container">$$
(e^{x}y(x))'=e^{x}|x|.
$$</span>
Thus, you are left with writing down the solution to <span class="math-container">$w'(x)=f(x)$<... |
2,352,811 | <p>Why it's not enough for the partial derivatives to exist for implying differentiability of the function?
Why is the continuity of the partial derivatives needed?</p>
| krirkrirk | 221,594 | <p><strong>Hint</strong> : </p>
<p>consider $f(x,y) = \frac{xy}{x^2+y^2}$, $f(0,0) = 0$</p>
|
452,306 | <p>I am trying to be able to find the radius of a cone combined with a cylinder.
see my other question
(Solving for radius of a combined shape of a cone and a cylinder where the cone is base is concentric with the cylinder? part2 )</p>
<p>I have a volume calculation that Has been reduced as far as I know how to.</p>
... | Daniel Fischer | 83,702 | <p>It is sufficient to show that $g^{-1} \in \overline{A}$, then $\langle g\rangle \subset \overline{A}$, and $\overline{A}$ is the closure of a subgroup, hence a subgroup.</p>
<p>If $g$ has finite order, it is trivial that $g^{-1} \in A = \overline{A}$, so let's suppose that $g^n \neq 1$ for $n \neq 0$.</p>
<p>If $g... |
452,306 | <p>I am trying to be able to find the radius of a cone combined with a cylinder.
see my other question
(Solving for radius of a combined shape of a cone and a cylinder where the cone is base is concentric with the cylinder? part2 )</p>
<p>I have a volume calculation that Has been reduced as far as I know how to.</p>
... | zudumathics | 265,160 | <p>I really like Fischer's solution. I would like to post an answer based on Fischer's answer in the "filling the blank" spirit - certainly helpful for beginners like me.</p>
<p>We need to get help from the two theorems below (refer to section 1.15 in Bredon's <em>Topology and Geometry</em>):</p>
<ul>
<li>In a topolo... |
1,187,713 | <p>How would I go about proving that if $a_n$ is a real sequence such that $\lim_{n\to\infty}|a_n|=0$, then there exists a subsequence of $a_n$, which we call $a_{n_k}$, such that $\sum_{k=1}^\infty a_{n_k}$ is convergent.</p>
<p>I think that I can choose terms $a_{n_k}$ such that they are terms of a geometric series,... | Dimitris | 37,229 | <p>Your idea is good. You can pick $a_{k_1}$ such that $|a_{k_1}|<1/2$. Then pick $k_2>k_1$ so that $|a_{k_2}|<1/4$, and inductively pick ${k_n}>{k_{n-1}}$ such that $|a_{k_n}|<1/2^n$. </p>
|
127,086 | <p>I am struggling with an integral pretty similar to one already resolved in MO (link: <a href="https://mathoverflow.net/questions/101469/integration-of-the-product-of-pdf-cdf-of-normal-distribution">Integration of the product of pdf & cdf of normal distribution </a>). I will reproduce the calculus bellow for the ... | Did | 4,661 | <p><a href="http://www.youtube.com/watch?v=aNUr__-VZeQ">The horror, the horror</a>... :-)</p>
<p>Recall that $\Phi(x)=P[X\leqslant x]$ for every $x$, where the random variable $X$ is standard normal, and that, for every suitable function $u$,
$$
\int_{-\infty}^{+\infty}u(x)\phi(x)\mathrm dx=E[u(Y)],
$$
where the rando... |
117,024 | <p>The trivial approach of counting the number of triangles in a simple graph $G$ of order $n$ is to check for every triple $(x,y,z) \in {V(G)\choose 3}$ if $x,y,z$ forms a triangle. </p>
<p>This procedure gives us the algorithmic complexity of $O(n^3)$.</p>
<p>It is well known that if $A$ is the adjacency matrix o... | 27rabbit | 978,635 | <p>In Competitive Programming, we have a particularly popular algorithm for a rather sparse graph running in <span class="math-container">$O(m\sqrt{m})$</span> time complexity where <span class="math-container">$m$</span> is the number of the edges in the graph. Although it is certainly not the fastest, as Listing ment... |
4,637,565 | <p>I am thinking of positive sequences whose sum is infinite but whose sum of squares is not?</p>
<p>One representative sequence is <span class="math-container">$$x[n] = \frac{a}{n+b},$$</span> where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are given real numbers such that <sp... | Gareth Ma | 948,125 | <p>Essentially copying off <a href="https://en.wikipedia.org/wiki/Sequence_space" rel="nofollow noreferrer">wikipedia</a>, the property you are asking for is related to something called the <span class="math-container">$\ell^p$</span> sequence space. Specifically, for some base field, say the reals, for <span class="ma... |
4,637,565 | <p>I am thinking of positive sequences whose sum is infinite but whose sum of squares is not?</p>
<p>One representative sequence is <span class="math-container">$$x[n] = \frac{a}{n+b},$$</span> where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are given real numbers such that <sp... | Adam Latosiński | 653,715 | <p>Let's first notice, that if
<span class="math-container">$$\limsup_{n\to\infty} \frac{x_{n+1}}{x_n} < 1$$</span>
then both series <span class="math-container">$\sum_{n=1}^\infty x_n$</span> and <span class="math-container">$\sum_{n=1}^\infty x_n^2$</span> converge. On the other hand, if
<span class="math-containe... |
1,039,563 | <p>Whether the graphs G and G' given below are isomorphic?</p>
<p><img src="https://i.stack.imgur.com/0evn6.jpg" alt="enter image description here"></p>
| Arthur | 15,500 | <p>$f(t)$ goes to either positive or negative infinity as $t$ grows, assuming it has degree at least $1$. That means that there is some $T$ such that $|f(t)|>10^9$ as long as $t>T$.</p>
<p>Choose an $n\in \Bbb N$ such that $2\pi n>T$. Then compare
$$
\int_0^{2\pi n} f(t) \sin t\: dt
$$
and
$$
\int_0^{2\pi (n+... |
4,319,590 | <blockquote>
<p>Let <span class="math-container">$H$</span> be a subgroup of <span class="math-container">$G$</span> and <span class="math-container">$x,y \in G$</span>. Show that <span class="math-container">$x(Hy)=(xH)y.$</span></p>
</blockquote>
<p>I have that <span class="math-container">$Hy=\{hy \mid h \in H\}$</s... | Servaes | 30,382 | <p>You are entirely correct; there isn't much to be shown, and you've shown it.</p>
|
1,699,752 | <p>Does $ 1 + 1/2 - 1/3 + 1/4 +1/5 - 1/6 + 1/7 + 1/8 - 1/9 + ...$ converge? </p>
<p>I know that $(a_n)= 1/n$ diverges, and $(a_n)= (-1)^n (1/n)$, converges, but given this pattern of a negative number every third element, I am unsure how to determine if this converges. </p>
<p>I tried to use the comparison test, but ... | carmichael561 | 314,708 | <p>Let $s_n$ be the $n$th partial sum of the series. If the series converges, then the sequence $\{s_n\}$ is bounded.</p>
<p>However, observe that $s_4>1+\frac{1}{4}$, $s_7> 1+\frac{1}{4}+\frac{1}{7}$, and in general
$$ s_{3m+1}>\sum_{k=0}^m\frac{1}{3k+1} $$
Since $\sum_{k=0}^{\infty}\frac{1}{3k+1}$ diverges,... |
2,030,547 | <p>The following expression came up in a proof I was reading, where it is said "It is easily shown: $$\lim_{x\to\infty} x(1-\frac{\ln (x-1)}{\ln x})=0."$$</p>
<p>Unfortunately I'm not having an easy time showing it. I guess it should come down to showing that the ratio $\frac{\ln (x-1)}{\ln x}$ converges to 1 superlin... | Brian M. Scott | 12,042 | <p>The displayed line <em>is</em> the definition of $v_0^*$. Each vector $v_0$ in $V$ determines a linear functional $v_0^*$ (which I read ‘vee-nought-star’) on $X^*$, i.e., an element $v_0^*$ of $V^{**}$. This $v_0^*$ is therefore a linear function from $V^*$ to $\Bbb R$, and it’s defined by</p>
<p>$$v_0^*(f)=f(v_0)\... |
2,359,621 | <p>Consider $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ where</p>
<p>$$f(x,y):=\begin{cases}
\frac{x^3}{x^2+y^2} & \textit{ if } (x,y)\neq (0,0) \\
0 & \textit{ if } (x,y)= (0,0)
\end{cases} $$</p>
<p>If one wants to show the continuity of $f$, I mainly want to show that </p>
<p>$$ \lim\limits_... | Dave | 334,366 | <p>$\lim_\limits{(x,y)\to 0}$ likely means $\lim_\limits{(x,y)\to(0,0)}$, which means that $x$ and $y$ are both tending to $0$. One could use polar coordinates where $x=r\cos(\theta)$ and $y=r\sin(\theta)$ to obtain:
$$\lim_{(x,y)\to(0,0)}\frac{x^3}{x^2+y^2}=\lim_{r\to 0}\frac{r^3\cos^3(\theta)}{r^2}=\lim_{r\to 0} r\co... |
1,007,533 | <p>Prove that if $v$ is an eigenvector for the matrix $A$, then $A^2v=c^2v$</p>
<p>Pretty much all I have is:</p>
<p>$Av=cv$ where $v$ is a nonzero vector</p>
| The Artist | 154,018 | <p>$$Av=cv$$</p>
<p>$$AAv=Acv$$</p>
<p>c is a scalar you can factor it out</p>
<p>$$AAv=c(Av)$$</p>
<p>$$A^2v=c(Av)$$</p>
<blockquote>
<p>But you know that $Av=cv$ So </p>
</blockquote>
<p>$$A^2v=c(cv)=c^2v$$</p>
|
2,797,709 | <p>How is $\; 4 \cos^2 (t/2) \sin(1000t) = 2 \sin(1000t) + 2\sin(1000t)\cos t\,$? This is actually part of a much bigger physics problem, so I need to solve it from the LHS quickly. Is there an easy method by which I can do this?</p>
| José Carlos Santos | 446,262 | <p>Just use the fact that$$\cos(t)=\cos^2\left(\frac t2\right)-\sin^2\left(\frac t2\right)=2\cos^2\left(\frac t2\right)-1.$$</p>
|
4,003,948 | <p>In the Book that I'm reading (Mathematics for Machine Learning), the following para is given, while listing the properties of a matrix determinant:</p>
<blockquote>
<p>Similar matrices (Definition 2.22) possess the same determinant.
Therefore, for a linear mapping <span class="math-container">$Φ : V → V$</span> all ... | Trevor Gunn | 437,127 | <p>Suppose that <span class="math-container">$V$</span> is the vector space of quadratic polynomials and <span class="math-container">$\Phi$</span> is multiplication by <span class="math-container">$(1 + x)$</span> mod <span class="math-container">$x^3$</span>.</p>
<p>Now let us consider two different bases for <span c... |
686,361 | <p>Given if we know $P(S)$ and $P(C|S)$ and $P(D|S)$, how do you compute $E[C|D=d]$? One way that I thought of is to find the conditional probability of $P(C|D)$ by computing the joint probability $P(C,D,S)$ and marginalizing it over $S$. But, $P(D|S)$ is a binomial distribution with parameter $q$ and $S$. Finding the ... | Did | 6,179 | <p>$$
E(C\mid D=d)=\frac{E(C;D=d)}{P(D=d)}
$$
$$
E(C;D=d)=\sum_{c,s}c\,P(C=c\mid S=s)\,P(D=d\mid S=s)\,P(S=s)
$$
$$
P(D=d)=\sum_{s}P(D=d\mid S=s)\,P(S=s)
$$</p>
|
4,273,026 | <p>Let <span class="math-container">$\Omega\subset\mathbb{R}^n$</span> be a bounded open set, <span class="math-container">$n\geq 2$</span>. For <span class="math-container">$r>0$</span>, denote by <span class="math-container">$B_r(x_0)=\{x\in\mathbb{R}^n:|x-x_0|<r\}$</span> whose closure is a proper subset of <s... | user378654 | 970,339 | <p>No: let <span class="math-container">$u(x) = |x|^a$</span> and <span class="math-container">$x_0 = 0$</span>. I do this with <span class="math-container">$q = 1$</span> to simplify computation, but you can repeat the argument with any (or all) <span class="math-container">$q$</span>. The first inequality (up to modi... |
3,553,975 | <p>I fear that this is a stupid question, but I want to have a go anyway. </p>
<p>Let <span class="math-container">$k$</span> be a field, and let <span class="math-container">$f(x,y)$</span> be an irreducible homogeneous quadratic polynomial in <span class="math-container">$k[x,y]$</span>. </p>
<p><em>Question</em>: ... | Eugaurie | 382,410 | <p>A function with the property <span class="math-container">$x < y \Rightarrow f(x) < f(y) $</span> can be called <span class="math-container">$\textbf{strictly monotone increasing}$</span>. </p>
|
1,051,372 | <p>If $|z_1|=1,|z_2|=1$, how can one prove $|1+z_1|+|1+z_2|+|1+z_1z_2|\ge2$</p>
| Raclette | 196,274 | <p>You have that $|1+w| \geq |\Re(1+w)| = |1 + \Re(w)|$. Since you only consider $|w| = 1$, you have that $|1 + Re(w)| = 1 + Re(w)$. Thus, it is enough to show that
$$1 + \cos(t_1) + 1 + \cos(t_2) + 1 + \cos(t_1+t_2) \geq 2,$$
where $z_1 = e^{it_1}$ and $z_2 = e^{it_2}$.
If $t_1$ is fixed, then the LFH obtains its extr... |
724,900 | <p>Assuming $y(x)$ is differentiable. </p>
<p>Then, what is formula for differentiation ${d\over dx}f(x,y(x))$?</p>
<p>I examine some example but get no clue....</p>
| Cameron Williams | 22,551 | <p>There are two pieces to this: $f$ is a function of $x$ and a function of $y$ which suggests use of the chain rule. The multivariate chain rule says</p>
<p>$$\frac{d}{dx}f(x,y(x)) = \frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}\frac{dy}{dx}.$$</p>
<p>More generally, if you have a function $f(x,y,\ldot... |
1,334,719 | <p>Given $F=A^TA$, with $A$ is a $m\times n$ matrix. Then what is the derivative w.r.t. $A$ ?</p>
<p>I know when $A$ is a $m\times 1$ vector, the derivative is $$\frac{\partial F}{\partial A} = 2A$$.</p>
<p>Does this equation still hold when A is extended to $m\times n$ matrix?</p>
| mathcounterexamples.net | 187,663 | <p><strong>Hint.</strong></p>
<p>$F$ can be written by function composition of the functions:</p>
<p>$$R: A \longmapsto A^{T}$$ and
$$S: (A,B) \longmapsto A.B$$ The first one is linear, so its derivative is itself. The second one is bilinear and its derivative is $S^\prime(A,B)(h,k)=A.h+k.B$</p>
<p>At the end, you h... |
1,334,719 | <p>Given $F=A^TA$, with $A$ is a $m\times n$ matrix. Then what is the derivative w.r.t. $A$ ?</p>
<p>I know when $A$ is a $m\times 1$ vector, the derivative is $$\frac{\partial F}{\partial A} = 2A$$.</p>
<p>Does this equation still hold when A is extended to $m\times n$ matrix?</p>
| Mark Joshi | 106,024 | <p>$F(A+H) = (A+H)^{t}(A+H) = (A^t + H^t)(A+H)=A^t A + A^tH + H^t A + H^t H.$</p>
<p>So $$F(A+H) - F(A) = A^t H + H A^t + O(||H||^2).$$
So the derivative is the map
$$
H \mapsto A^t H + H A^t.
$$</p>
|
3,120,729 | <p>I came across this exercise:</p>
<blockquote>
<p>Prove that
<span class="math-container">$$\tan x+2\tan2x+4\tan4x+8\cot8x=\cot x$$</span></p>
</blockquote>
<p>Proving this seems tedious but doable, I think, by exploiting double angle identities several times, and presumably several terms on the left hand side ... | J.G. | 56,861 | <p>Defining <span class="math-container">$t:=\tan y$</span>, <span class="math-container">$$2\cot 2y+\tan y-\cot y=\frac{1-t^2}{t}+t-\frac{1}{t}=0.$$</span>This will work for any <span class="math-container">$y$</span>, so I guess I've answered your questions in reverse order (but to both I say yes). </p>
|
1,521,124 | <p>What will be the value of $3/1!+5/2!+7/3!+...$?</p>
<p>I'm trying to bring it in terms of $e$.Is it possible?</p>
<p>I used taylor series for e.</p>
| the_candyman | 51,370 | <p>I guess you can write it as follow:</p>
<p>$$\sum_{n=1}^{+\infty}\frac{2n+1}{n!} = \sum_{n=1}^{+\infty}\left(\frac{2}{(n-1)!}+\frac{1}{n!}\right) = \\
= \sum_{n=1}^{+\infty}\left(\frac{2}{(n-1)!}\right) +\sum_{n=1}^{+\infty}\left(\frac{1}{n!}\right) = \\
= 2\sum_{n=0}^{+\infty}\left(\frac{1}{n!}\right) +\sum_{n=0}^... |
3,027,925 | <p>Just for my own understanding of how exactly integration works, are these steps correct:</p>
<p><span class="math-container">$$\begin{align}\int x\,d(x^2) \qquad &\implies x^2 = u \\ & \implies x= \sqrt{u}\end{align}$$</span> </p>
<p>Thus, it becomes <span class="math-container">$$\int\sqrt{u}\,du = \frac... | PrincessEev | 597,568 | <p>While fundamentally correct, aside from your lack of a <span class="math-container">$+C$</span> constant, notationally your work is a bit of a mess (before someone edited it for clarity's sake). </p>
<p>First, we start with</p>
<p><span class="math-container">$$\int x d(x^2)$$</span></p>
<p>From here, we make the... |
2,741,832 | <p>When one first learns measure theory, it is a small novelty to find out that
$$\bigcup_{n=0}^\infty B_{\epsilon/2^n}(r_n)$$
is not all of $\mathbb{R}$, where $\{r_n\}$ is an enumeration of the rationals and $\epsilon$ is an arbitrary positive number (notice this fact is equally impressive if $\epsilon$ is small or l... | Ross Millikan | 1,827 | <p>One approach is to let $c_n=\frac 1n$ and let $d_n=\frac 1n$ when the rational is outside $(-1,2)$ and $d_n=2^{-n}$ when the rational is inside. The interval $(0,1)$ then has the measure argument work because none of the balls centered outside $(-1,2)$ can reach there. We just need to make sure enough of the earl... |
1,024,794 | <p>I have this equation: $x+7-(\frac{5x}8 + 10) = 3 $</p>
<p>I've used step-by-step calculators online but I simply don't understand it. Here is how I've tried to solve the problem: </p>
<p>$$x+7-\left(\frac{5x}8+10\right) = x + 7 - \frac{5x}8 - 10 = 3$$</p>
<p>$$x + 7 - \frac{5x}8 - 10 + 10 = 3 + 10$$</p>
<p>$$x +... | mick | 39,261 | <p>$\xi(s)=\xi(1-s)$</p>
<p>You defined $k(s)=\xi(1-s)$.</p>
<p>So $k(s)=\xi(s)$.</p>
<p>Next you wonder if $\xi(\xi(s)) = 0 => \xi(s) = 0$ is true iff RH is true.</p>
<p>There is no reason to assume those 2 problems are related.</p>
<p>Notice that if $\xi(s) = 0$ this implies that $\xi(\xi(s)) = \xi(0) = \frac... |
634,890 | <blockquote>
<p><strong>Moderator Notice</strong>: I am unilaterally closing this question for three reasons. </p>
<ol>
<li>The discussion here has turned too chatty and not suitable for the MSE framework. </li>
<li>Given the recent pre-print of <a href="http://arxiv.org/abs/1402.0290" rel="noreferrer">T. Ta... | Stephen Montgomery-Smith | 22,016 | <p>This web page has Theorem 6.1. It is written in Spanish, but actually is rather easy to follow even if (like me) you don't know any Spanish. However it is not made clear on this web site that the statement of Theorem 6.1 is "<strong>If</strong> $\|A^\theta \overset 0u\| \le C_\theta\|$, <strong>then</strong> $\| \... |
634,890 | <blockquote>
<p><strong>Moderator Notice</strong>: I am unilaterally closing this question for three reasons. </p>
<ol>
<li>The discussion here has turned too chatty and not suitable for the MSE framework. </li>
<li>Given the recent pre-print of <a href="http://arxiv.org/abs/1402.0290" rel="noreferrer">T. Ta... | AlsoInAstana | 124,205 | <p>I read on the Russian side dxdy a comment from a mathematician in Almaty, KZ, that sheds some light on the process. It's the comment on page 13 by MAnvarbek.
Otelbaev presented his proof 1 year ago, and immediately they found large errors, and no new ideas. The problem was with all the parameters.
Otelbaev worked ... |
634,890 | <blockquote>
<p><strong>Moderator Notice</strong>: I am unilaterally closing this question for three reasons. </p>
<ol>
<li>The discussion here has turned too chatty and not suitable for the MSE framework. </li>
<li>Given the recent pre-print of <a href="http://arxiv.org/abs/1402.0290" rel="noreferrer">T. Ta... | Dr. Betty Rostro | 129,304 | <p>I favor the Terry Tao version he uses Von Neuman criteria, this is a bit better and more elegant in my opinion.</p>
<p><a href="http://terrytao.wordpress.com/2014/02/04/finite-time-blowup-for-an-averaged-three-dimensional-navier-stokes-equation/" rel="nofollow">http://terrytao.wordpress.com/2014/02/04/finite-time-b... |
3,104,706 | <p>Let <span class="math-container">$T$</span> be the left shift operator on <span class="math-container">$B(l^{2}(\mathbb{N}))$</span>. How to see that von Neumann algebra generated by <span class="math-container">$T$</span> is <span class="math-container">$B(l^{2}(\mathbb{N}))$</span>?</p>
| Jens Schwaiger | 532,419 | <p>It is known that every infinite set contains a countable subset (axiom of choice). So let <span class="math-container">$X$</span> contain the subset <span class="math-container">$A:=\{a_n\mid n\in\mathbb{N}\}$</span> with <span class="math-container">$a_n\not=a_m$</span> for <span class="math-container">$n\not=m$</... |
3,104,706 | <p>Let <span class="math-container">$T$</span> be the left shift operator on <span class="math-container">$B(l^{2}(\mathbb{N}))$</span>. How to see that von Neumann algebra generated by <span class="math-container">$T$</span> is <span class="math-container">$B(l^{2}(\mathbb{N}))$</span>?</p>
| Community | -1 | <p>Suppose <span class="math-container">$X$</span> is infinite. Let <span class="math-container">$C=\{x_1,x_2,x_3,\dots\}$</span> be a countably infinite subset. </p>
<p>Define <span class="math-container">$h:X\to X$</span> by <span class="math-container">$h(x)=x, x\in X\setminus C$</span> and <span class="math-cont... |
2,934,238 | <p>Let <span class="math-container">$a\in \mathbb{Q}$</span> such that <span class="math-container">$18a$</span> and <span class="math-container">$25a$</span> are integers, then we wish to prove that <span class="math-container">$a$</span> must be an integer itself. What that means is that <span class="math-container">... | Mo Pol Bol | 359,447 | <p>You can also prove this by contradiction:</p>
<p>Assume $a=\frac{m}{n}$, where $m,n$ are coprime and $n\gt1$. Then, if $18a=k_1\in\mathbb{Z}$, by the fundamental theorem of arithmetic $$n=2^b3^c.$$ However, assuming $$25a=k_2\in\mathbb{Z},$$ implies $n=5$ or $n=25$, which is obviously a contradiction.</p>
|
2,934,238 | <p>Let <span class="math-container">$a\in \mathbb{Q}$</span> such that <span class="math-container">$18a$</span> and <span class="math-container">$25a$</span> are integers, then we wish to prove that <span class="math-container">$a$</span> must be an integer itself. What that means is that <span class="math-container">... | B. Goddard | 362,009 | <p>Another way to look at this: <span class="math-container">$18a$</span> and <span class="math-container">$25a$</span> are integers. Therefore, so is <span class="math-container">$25a-18a = 7a$</span>.</p>
<p>Therefore so is <span class="math-container">$18a-2(7a) = 4a.$</span></p>
<p>Therefore so is <span class="... |
2,934,238 | <p>Let <span class="math-container">$a\in \mathbb{Q}$</span> such that <span class="math-container">$18a$</span> and <span class="math-container">$25a$</span> are integers, then we wish to prove that <span class="math-container">$a$</span> must be an integer itself. What that means is that <span class="math-container">... | Bill Dubuque | 242 | <p><em>Conceptually</em> <span class="math-container">$\, \dfrac{m}{18} = a = \dfrac{n}{25}\,$</span> so it's least denominator divides <em>coprimes</em> <span class="math-container">$18,25$</span> so is <span class="math-container">$1,\,$</span> so <span class="math-container">$\,a\in\Bbb Z$</span></p>
<p><strong>Rem... |
244,769 | <p>I am DMing a game of DnD and one of my players is really into fear effects, which is cool, but the effect of having monsters suffer from the "panicked" condition gets tedious to render via dice rolls.</p>
<p>The rule is, on the battle grid the monster will run for 1 square in a random direction, then from ... | Adam | 74,641 | <p>To clear up ambiguities, the following code finds the transformation from the points in xy space <span class="math-container">$(0,1),(1,0)$</span> and <span class="math-container">$(0,0)$</span> to the points in uv space <span class="math-container">$a,b$</span> and <span class="math-container">$c$</span> projected ... |
3,858,414 | <p>I need help solving this task, if anyone had a similar problem it would help me.</p>
<p>The task is:</p>
<p>Calculate using the rule <span class="math-container">$\lim\limits_{x\to \infty}\left(1+\frac{1}{x}\right)^x=\large e $</span>:</p>
<p><span class="math-container">$\lim_{x\to0}\left(\frac{1+\mathrm{tg}\: x}{... | Aryaman Maithani | 427,810 | <p>Suppose that <span class="math-container">$[x] \cap [y] \neq \emptyset$</span> <strong>and</strong> <span class="math-container">$[x] \neq [y]$</span>.</p>
<p>Since the intersection is nonempty, we may pick <span class="math-container">$z \in [x] \cap [y]$</span>. Since the classes are not equal, we may pick an elem... |
10,468 | <p>I know that many graph problems can be solved very quickly on graphs of bounded degeneracy or arboricity. (It doesn't matter which one is bounded, since they're at most a factor of 2 apart.) </p>
<p>From Wikipedia's article on the clique problem I learnt that finding cliques of any constant size k takes linear time... | Martin Vatshelle | 23,539 | <p>There is one more approach to solve problems like Max Clique on graphs of bounded degeneracy.
You can look at the complement graph of a graph $G$ (i.e. every edge is a non-edge and every non-edge is an edge).
Solving Max Clique on $G$ is the same as solving Max Independent set on the complement.</p>
<p>For the comp... |
124,660 | <p>I'm solving a fairly simple equation :</p>
<pre><code>w[p1_, p2_, xT_] :=
94.8*cv*p1*y[(p1 - p2)/p1, xT]*Sqrt[(p1 - p2)/p1*mw/t1];
y[r_, xT_] := 1 - (1.4 r)/(3 xT*γ) /. γ -> 1.28;
sol = NSolve[{w[p1, p2, 0.66] == 30, p2 == 1.07}, {p1, p2}] /. {cv ->
1.77, t1 -> 318, mw -> 38};
</code></pre>
<p>Ma... | ubpdqn | 1,997 | <p>My original answer was a misinterpretation. I post this in case it is helpful.</p>
<pre><code>y[r_, xT_, g_] := 1 - (1.4 r)/(3 xT*g)
w[p1_, p2_, xT_, cv_, mw_, t1_] :=
94.8*cv*p1*y[(p1 - p2)/p1, xT, 1.28]*Sqrt[(p1 - p2)/p1*mw/t1];
ans = p2 /. NSolve[w[1.07, p2, 0.66, 1.77, 38, 318] == 30, p2]
plot1 = Plot[{w[1.0... |
175,723 | <p>I am reading Goldstein's Classical Mechanics and I've noticed there is copious use of the $\sum$ notation. He even writes the chain rule as a sum! I am having a real hard time following his arguments where this notation is used, often with differentiation and multiple indices thrown in for good measure. How do I get... | paul garrett | 12,291 | <p>Also, since $\theta$ is an algebraic integer, to show $\theta(\theta-1)/2$ is an algebraic integer, it suffices to show that it is $2$-adically integral. </p>
<p>Thus, the plan is to prove that <em>either</em> $2|\theta$ or $2|(\theta-1)$, in $\mathbb Z_2$.</p>
<p>Hensel's lemma shows that $x^3+11x-4=0$ has soluti... |
1,343,722 | <p>Note: I am looking at the sequence itself, not the sequence of partial sums.</p>
<p>Here's my attempt...</p>
<p>Setting up:</p>
<p>$$\left\{\frac{2(n+1)}{2(n+1)-1}\right\} - \left\{\frac{2n}{2n-1}\right\}$$</p>
<p>Simplifying:</p>
<p>$$\left\{\frac{2n+2}{2n+1}\right\} - \left\{\frac{2n}{2n-1}\right\}$$</p>
<p>... | Joel | 85,072 | <p>Assuming as @Soke does, your series is $$\sum_{n=1}^\infty \frac{n^2}{2^n}$$ we can use the geometric series in order to find a closed form solution. The trick is to take derivatives and multiply by $x$.</p>
<p>Note that $$f(x) = \sum_{n=1}^\infty x^n = \frac{x}{1-x}$$ yields a derivative of $$\sum_{n=1}^\infty n x... |
3,600,633 | <p>As I was reading <a href="https://math.stackexchange.com/questions/1918673/how-can-i-prove-that-the-finite-extension-field-of-real-number-is-itself-or-the">this question</a>, I saw Ethan's answer. However, perhaps this is very obvious, but why does the degree of the polynomial be at most <span class="math-container"... | N. S. | 9,176 | <p>Let <span class="math-container">$P(X) \in \mathbb R[X]$</span> be the minimal polynomial of <span class="math-container">$\alpha$</span>.</p>
<p>If <span class="math-container">$\alpha \in \mathbb R$</span> then <span class="math-container">$X-\alpha |P(X)$</span>. Since <span class="math-container">$X-\alpha \in ... |
3,809,127 | <blockquote>
<p>Determine if the sequence <span class="math-container">$x_k \in \mathbb{R}^3$</span> is convergent when <span class="math-container">$$x_k=(2, 1, k^{-1})$$</span></p>
</blockquote>
<p>Our professor gave a hint that one should look at <span class="math-container">$||2k-a||$</span> and try to find a contr... | Diger | 427,553 | <p>Start with your setup for <strong>even</strong> <span class="math-container">$n$</span> of two lists
<span class="math-container">$$1: (1,n/2+1),(2,n/2+2),...,(n/2,n) \\
2: (n/2+1,1),(n/2+2,2),...,(n,n/2)$$</span>
for which the corresponding sum equals <span class="math-container">$n^2/2$</span>. Any permutation wit... |
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