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 |
|---|---|---|---|---|
4,531,998 | <p>As you increase the value of n, you will generate all pythagorean triples whose first square is even. Is there any visual proof of the following explicit formula and where does it come from or how to derive it?</p>
<p><span class="math-container">$(2n)^2 + (n^2 - 1)^2 = (n^2 + 1)^2$</span></p>
<div class="s-table-co... | poetasis | 546,655 | <p>All Pythagorean triple have a "leg" that is a multiple of
<span class="math-container">$\space4.\space$</span> Your formula generates a subset of these.</p>
<p>We begin with Euclid's formula shown here as
<span class="math-container">$$A=m^2-k^2\quad B=2mk \quad C=m^2+k^2$$</span>
If we let <span class="ma... |
81,887 | <p>The problem: Prove that for $n \in \mathbb N$:</p>
<p>$$ \left(1 + \frac{1}{n} \right)^n = 1 + \sum_{m=1}^{n} \frac{1}{m!} \left(1 - \frac{1}{n} \right) \left(1 - \frac{2}{n} \right) \cdots \left(1 - \frac{m-1}{n} \right). $$</p>
<p>The hint is to use the binomial theorem. So the left side can become:</p>
<p>$$ \... | yunone | 1,583 | <p>Take your second sum
$$
\sum_{m=0}^{n} \frac{n!}{m!(n - m)!} \left(\frac{1}{n} \right)^m
$$
and write it as
$$
1+\sum_{m=1}^n \frac{n!}{m!(n - m)!} \left(\frac{1}{n} \right)^m
$$
to get the indices to match.</p>
<p>In your first sum
$$
\sum_{m=1}^{n} \frac{1}{m!} \left(1 - \frac{1}{n} \right) \left(1 - \frac... |
3,464,383 | <p>I couldn't find any substantial list of 'strange infinite convergent series' so I wanted to ask the MSE community for some. By <em>strange</em>, I mean infinite series/limits that <strong>converge when you would not expect them to and/or converge to something you would not expect</strong>.</p>
<p>My favorite conver... | Axion004 | 258,202 | <p>Suppose <span class="math-container">$\sum_{n=1}^{\infty} a_n$</span> and <span class="math-container">$\sum_{n=1}^{\infty} b_n$</span> are both divergent. Then, one might assume that <span class="math-container">$\sum_{n=1}^{\infty} (a_n+b_n)$</span> also diverges.</p>
<p>This is false. Suppose <span class="math-c... |
3,386,696 | <p>I need to solve
<span class="math-container">$\int_{0}^{\frac{\pi}{2}} \cos^6xdx$</span>
I tried to use <span class="math-container">$cos(3x)=4\cos^3x-3\cos x$</span> but not succeeded</p>
| drhab | 75,923 | <p>Suppose that <span class="math-container">$P(a)<P(a\mid b)$</span> and <span class="math-container">$P(a)<P(a\mid b^c)$</span>.</p>
<p>Then because at least one of <span class="math-container">$P(b)$</span> and <span class="math-container">$P(b^c)$</span> is positive: <span class="math-container">$$P(a)=P(a\m... |
2,010,768 | <p>I have the question </p>
<p>Solve the simultaneous equations </p>
<p>$$\begin{cases}
3^{x-1} = 9^{2y} \\
8^{x-2} = 4^{1+y}
\end{cases}$$</p>
<p><a href="https://i.stack.imgur.com/xN9Ny.jpg" rel="nofollow noreferrer">source image</a></p>
<p>I know that $x-1=4y$ and $3X-6=2+2y$ .</p>
<p>However when I checked th... | Frank | 332,250 | <p>From $$\begin{align*} & 3^{x-1}=9^{2y}\\ & 8^{x-2}=4^{y+1}\end{align*}\tag1$$
We can rewrite $9$ as $3^2$, $8$ as $2^3$ and $4$ as $2^2$. Doing so and setting the powers equivalent, we get$$\begin{align*} & x-1=4y\\ & 3x-6=2+2y\end{align*}\tag2$$
To answer your question, look at the second equation. ... |
818,850 | <p>I want to generate numbers $1$ to $10$ with uniform probability distribution. So I write the numbers $1$ to $10$ in the natural order. I keep writing the next $10$ number block by permuting the first $10$ numbers. Will this long string can be considered random with uniform distribution? Is there a problem?</p>
| Damian Pavlyshyn | 154,826 | <p>Try explaining it with a number line:</p>
<p>Divide the interval between 1 and 2 into ten equal parts and number them 0-9. Note that these are tenths, and so numbers of the form 1.4... fall into division four.</p>
<p>Now divide the interval between 1.4 and 1.5 into ten equal parts and note that these are hundredth... |
2,857,508 | <p>I'm trying to prove this: $$0.99^n \le 1/2,\text{ for }n=100$$</p>
<p>I tried Bernoulli's inequality $(1-0.01)^n \geq 1-n\cdot0.01$ and it gave me LHS $\geq0$. </p>
<p>I also tried to do this: $((1-0.01)^n)^n \leq(1-1/2)^n$ and it gave me LHS $\geq-99$ and RHS $\geq49$. </p>
<p>Now I'm stuck.</p>
| farruhota | 425,072 | <p>Alternatively:
$$0.99^{100}<\frac12 \iff 2\cdot 99^{100}<100^{100} \iff \\
2\cdot 99^{100}<(1+99)^{100}=1+100\cdot 99+\cdots +100\cdot 99^{99}+99^{100} \iff \\
2\cdot 99^{100}<1+9900+\cdots+(1+99)\cdot 99^{99}+99^{100}\iff \\
2\cdot 99^{100}<1+9900+\cdots+99^{99}+2\cdot 99^{100}.$$</p>
|
174,676 | <p>When I am faced with a simple linear congruence such as
$$9x \equiv 7 \pmod{13}$$
and I am working without any calculating aid handy, I tend to do something like the following:</p>
<p>"Notice" that adding $13$ on the right and subtracting $13x$ on the left gives:
$$-4x \equiv 20 \pmod{13}$$</p>
<p>so that $$x \eq... | John Butnor | 185,327 | <p>9x = 7 mod 13 </p>
<p>9x = 7 + 13n</p>
<p>9x = 20 for n = 1</p>
<p>9x = 33 for n = 2</p>
<p>9x = 46 for n = 3 </p>
<p>9x = 59 for n = 4 </p>
<p>9x = 72 for n = 5</p>
<p>Then x = 8 mod 13</p>
<p>You arrive at the correct answer before n = 13.</p>
|
2,982,723 | <p>I need help in this exercise. What I need to prove that the function <span class="math-container">$f$</span> given is not continuous in the point <span class="math-container">$(0,0)$</span></p>
<p><span class="math-container">$$ f(x,y) = \begin {cases} \frac {x^3\times y} {x^6+y^2} & (x,y) \not = 0\\ 0 & (... | Parcly Taxel | 357,390 | <p>We consider the path <span class="math-container">$y=x^3$</span> to <span class="math-container">$(0,0)$</span>. Along this path, the function becomes
<span class="math-container">$$\frac{x^3\cdot x^3}{x^6+x^6}=\frac{x^6}{2x^6}=\frac12$$</span>
and so the limit along this path is <span class="math-container">$\frac1... |
79,868 | <p>Given a function $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ satisfying suitable conditions (exponential decay at infinity, continuous, and bounded variation) is good enough, its <em>Mellin transform</em> is defined by the function</p>
<p>$$M(f)(s) = \int_0^{\infty} f(y) y^s \frac{dy}{y},$$</p>
<p>and $f(y)$ can be r... | Tom Copeland | 12,178 | <p>Two equations that encapsulate the properties of the Fourier and Mellin transforms:</p>
<p><span class="math-container">$$\int^{\infty}_{-\infty}{\exp(2 \pi ifx)\exp(-2 \pi ify)df} = \delta(x-y)$$</span></p>
<p><span class="math-container">$$\frac{1}{2\pi i} \int_{\sigma - i \infty}^{\sigma + i \infty} x^{-s} y^{s} ... |
2,026,406 | <p>As written in the title, does there exists a $\mathbb C$-Banach space isometric to a Hilbert Space but whose norm is not induced by an inner product?</p>
<p>Since an inner product in the Hilbert space has to fulfill the Parallelogram Identity, how could it be that such a $\mathbb C$-Banach space exists?</p>
| Eric Wofsey | 86,856 | <p>No such space exists. Let $H$ be a Hilbert space, and suppose $X$ is a $\mathbb{C}$-Banach space with $T:X\to H$ an isometry. Then define an inner product on $X$ by $\langle x,y\rangle=\langle T(x),T(y)\rangle$ (where the right-hand side is the inner product of $H$). Then this inner product induces the norm of $X... |
588,725 | <p>Prove the following statement:</p>
<p>$$\frac{1}{x}<\ln(x)-\ln(x-1)<\frac{1}{x-1}$$</p>
<p>Proof:</p>
<p>$$\frac{-1}{x^2}<\frac{1}{x(x-1)}<\frac{-1}{(x-1)^2}$$</p>
<p>$$e^{(\frac{-1}{x^2})}<e^{(\frac{-1}{x(x-1)})}<e^{(\frac{-1}{(x-1)^2})}$$</p>
<p>$$\lim_{x\to\infty}e^{(\frac{-1}{x^2})}<\li... | Community | -1 | <p>There's a mistake in your calculus: if we have a strictly inequality
$$f(x)<g(x)$$
then by passing to the limit we have
$$\lim_{x\to a}f(x)\leq \lim_{x\to a}g(x),\qquad a\in \mathbb{R}\cup\{\infty\}$$</p>
|
1,879,346 | <p>Ok, so I have this question from my math book.</p>
<blockquote>
<p>The vertices A,B,C of a triangle are (3,5),(2,6) and (-4,-2)
respectively. Find the coordinates of the circum-centre and also the
radius of the circum-circle of the triangle.</p>
</blockquote>
<p>How can we solve this? Can we use the distance... | Carser | 132,859 | <p>So you can find the linear equations of the segment bisectors of each edge (really you only need two) which are
$$
y=-x+1
$$
$$
y=x+3
$$
$$
y=-\frac{3x}{4} + \frac{5}{4}
$$
Interestingly, these all coincide at the midpoint of segment $\overline{BC}$ at $(-1,2)$ as shown below (graph available here: <a href="https://... |
848,631 | <p>The function</p>
<p>$$f(y) = \displaystyle \frac{\sin(Ny)}{\sin y}$$</p>
<p>is periodic with period $2 \pi$ in general. But tracing the graphic of that function for $N$ odd it seems that for $0 \leq x < \pi$ it is the same as $\pi \leq x < 2 \pi$.</p>
<p>So can we state that the period of $f(y)$ with $N$ od... | Daniel Fischer | 83,702 | <p>Note that</p>
<p>$$\sin (x+\pi) = -\sin x$$</p>
<p>for all $x$. Hence</p>
<p>$$\sin (x + k\pi) = (-1)^k\sin x.$$</p>
<p>So we have</p>
<p>$$\sin (N(y+\pi)) = \sin (Ny + N\pi) = (-1)^N\sin (Ny).$$</p>
<p>Thus, whenever $N$ and $M$ have the same parity,</p>
<p>$$\frac{\sin (Ny)}{\sin (My)}$$</p>
<p>has period ... |
2,046,773 | <p>Given a function $f$ that is differentiable at a point $x_0$, if we define (using the Riemann integral)</p>
<p>$$F(x) = \int_a^x f$$</p>
<p>Can we necessarily say that $F^{\prime}(x)$ is continuous at $x_0$? Going back and forth between $f$ and $F$ confuses me a bit. I <em>think</em> that the Fundamental Theorem o... | operatorerror | 210,391 | <p>As stated in the problem, $F'(x)$ would be not only continuous but differentiable at $x_0$. </p>
<p>The FTC allows you to conclude that for $x_0\in [a,x]$,
$$
F'(x_0)=f(x_0)
$$
Which means that $F'(x)$ in its region of validity has the properties $f$ has.</p>
|
2,289,754 | <p>Let $f_t: \mathbb R^2 \to \mathbb R$ define by $f_t(x,y) = t \sin(t|x^2+y^2-1|)$.</p>
<p>For all $\phi\in D(\mathbb R^2-\{(0,0)\})$, we denote by $T$ the map given by
$$T(\phi)=\lim_{t\to +\infty}\int_{\mathbb R^2} f_t(x,y) \, \phi(x,y)\, \mathrm{d}x\, \mathrm{d}y$$</p>
<p>How can prove that $T$ is a distribution ... | Sangchul Lee | 9,340 | <blockquote>
<p><strong>Lemma.</strong> Let $\varphi \in C_c^1(\mathbb{R})$. Then</p>
<p>$$ \lim_{t\to\infty} \int_{\mathbb{R}} \frac{t \sin(t|u|)}{2} \varphi(u) \, du = \varphi(0). $$</p>
</blockquote>
<p><em>Proof of Lemma.</em> We have</p>
<p>\begin{align*}
\int_{\mathbb{R}} \frac{t \sin(t|u|)}{2} \varphi(u... |
419,730 | <p>$EX = \int xf(x)dx$, where $f(x)$ is the density function of some X variable. This is pretty understandable, but what if I had, for example an $Y = -X + 1$ function and I had to calculate $EY$? How should I do this?</p>
| Thomas Russell | 32,374 | <p>You can calculate the expected value of some function $g$ of $X$ using the following formula:</p>
<p>$$E[g(X)]=\int_{-\infty}^{\infty}g(x)f(x)\:dx$$</p>
<p>An explanation for this can be found on the <a href="http://en.wikipedia.org/wiki/Expected_value#General_definition" rel="nofollow">wikipedia page</a>.</p>
|
391,569 | <p>I've started the chapter in my book where we begin to integrate trig functions, so bear in mind I've only got started and that I do not have a handle on more advanced techniques.</p>
<p>$\eqalign{
& \int {{{\sin }^3}x} dx \cr
& = \int {\sin x({{\sin }^2}x} )dx \cr
& = \int {\sin x({1 \over 2... | vadim123 | 73,324 | <p>You have unnecessary middle steps, just use $\sin^2x = 1-\cos^2x$ after the second line.</p>
<p>To handle other odd powers:</p>
<p>$$\sin^{2k+1}x=(\sin^{2}x)^k(\sin x)=(1-\cos^2x)^k(\sin x)$$</p>
<p>then use $u=\cos x$.</p>
|
2,160,671 | <p>I got stuck with this problem.</p>
<p>\begin{matrix}
1 & 1 & 1 & 1\\
0 & 1 & 1 & 0\\
0 & 0 & 1 & 1\\
\end{matrix}</p>
<p>Consider the $3\times 4$ matrix $\bf A$ (above). Do the columns of $\bf A$ span $\mathbb R^3$? </p>
<p>Prove your answer. Als... | GNUSupporter 8964民主女神 地下教會 | 290,189 | <p>Denote the $j$-th columnn of $A$ by $a_j$.</p>
<ol>
<li><p>The columns of $A$ span $\Bbb R^3$ because the first <strong>three</strong> columns $a_1,a_2,a_3$ are clearly linearly independent, and $\dim(\Bbb R^3) = \bf 3$, so $\{a_1,a_2,a_3\}$ are a linearly independent spanning subset of $\Bbb R^3$.</p></li>
<li><p>... |
616,454 | <p>How can I calculate this integral?</p>
<p>$$ \int {\exp(2x)}{\sin(3x)}\, \mathrm{d}x$$ </p>
<p>I tried using integration by parts, but it doesn't lead me any improvement. So I made an attempt through the replacement $$ \cos(3x) = t$$ and it becomes $$\frac{1}{-3}\int \exp\left(2\left(\dfrac{\arccos(t)}{3}\right)... | Salech Alhasov | 25,654 | <p>Hint:</p>
<p>$$\displaystyle\int e^{2x}\sin(3x)\, \mathrm{d}x=Ae^{2x}\sin(3x)+Be^{2x}\cos(3x)$$</p>
<p>where $A$ and $B$ are constant. </p>
|
2,534,145 | <p>I am trying to determine whether the series $\sum_{n=2}^\infty \frac{1}{n(n-1)}$ converges or not.</p>
<p>I first broke the sequence up into partial fractions to obtain </p>
<p>$$\sum_{n=2}^\infty -\frac{1}{n} + \frac{1}{n-1}$$
$$= \sum_{n=2}^\infty -\frac{1}{n} + \sum_{n=2}^\infty \frac{1}{n-1} $$</p>
<p>The fi... | Paolo Leonetti | 45,736 | <p>$$
\lim_{k\to \infty}\sum_{n=2}^k\left(\frac{1}{n-1}-\frac{1}{n}\right)=\lim_{k\to \infty} \left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\cdots-\frac{1}{k}\right)=\lim_{k\to \infty}1-\frac{1}{k}=1.
$$</p>
|
2,534,145 | <p>I am trying to determine whether the series $\sum_{n=2}^\infty \frac{1}{n(n-1)}$ converges or not.</p>
<p>I first broke the sequence up into partial fractions to obtain </p>
<p>$$\sum_{n=2}^\infty -\frac{1}{n} + \frac{1}{n-1}$$
$$= \sum_{n=2}^\infty -\frac{1}{n} + \sum_{n=2}^\infty \frac{1}{n-1} $$</p>
<p>The fi... | Dr. Sonnhard Graubner | 175,066 | <p>use that $$\sum_{n=2}^m\frac{1}{n(n-1)}=1-\frac{1}{m}$$</p>
|
2,534,145 | <p>I am trying to determine whether the series $\sum_{n=2}^\infty \frac{1}{n(n-1)}$ converges or not.</p>
<p>I first broke the sequence up into partial fractions to obtain </p>
<p>$$\sum_{n=2}^\infty -\frac{1}{n} + \frac{1}{n-1}$$
$$= \sum_{n=2}^\infty -\frac{1}{n} + \sum_{n=2}^\infty \frac{1}{n-1} $$</p>
<p>The fi... | Karn Watcharasupat | 501,685 | <p>\begin{align}
\sum_{n=2}^\infty \frac{1}{n(n-1)}=&\lim_{N \to \infty}\sum_{n=2}^N \frac{1}{n-1}-\frac{1}{n}\\
&=\lim_{N \to \infty} \bigg(\frac{1}{1}-\frac{1}{2}\\
&+\frac{1}{2}-\frac{1}{3}\\
&+\frac{1}{3}-\frac{1}{4}\\
&\vdots\\
&+\frac{1}{N-1}-\frac{1}{N} \bigg)\\
&=\lim_{N \to \infty} ... |
2,534,145 | <p>I am trying to determine whether the series $\sum_{n=2}^\infty \frac{1}{n(n-1)}$ converges or not.</p>
<p>I first broke the sequence up into partial fractions to obtain </p>
<p>$$\sum_{n=2}^\infty -\frac{1}{n} + \frac{1}{n-1}$$
$$= \sum_{n=2}^\infty -\frac{1}{n} + \sum_{n=2}^\infty \frac{1}{n-1} $$</p>
<p>The fi... | Sri-Amirthan Theivendran | 302,692 | <p>Use the limit comparison test
$$
\frac{1}{n(n-1)}\bigg/n^{-2}\to1
$$
as $n\to\infty$. Since $\sum n^{-2}$ converges, your series converges as well.</p>
|
2,610,560 | <blockquote>
<p>Given that $x, y, z$ are positive integers and that $$3x=4y=7z$$ find
the minimum value of $x+y+z$. The options are:</p>
<p>A) 33</p>
<p>B) 40</p>
<p>C) 49</p>
<p>D) 61</p>
<p>E) 84</p>
</blockquote>
<p>My attempt:</p>
<p>$y=\frac{3}{4}x, z=\frac{3}{7}x$. </p>
<p>Substit... | For the love of maths | 510,854 | <p>Turns out you made a mistake.<br>
$x+y+z=x+\frac 34x +\frac 37x=\frac {61}{28}x$.<br>
Substituting $x=28$ gives us $\frac {61}{28}*28= 61$.<br>
Option $D$ $(61)$ is correct. </p>
<hr>
<p>You could do this in another way:<br>
$LCM (3, 4, 7)= 3*4*7 = 84$<br>
$\therefore 3*28=4*21=7*12$<br>
So, Let $f(x)=x+y+z$<br>
... |
2,888,464 | <p>Let $S$ be the set of polynomials $f(x)$ with integer coefficients satisfying </p>
<p>$f(x) \equiv 1$ mod $(x-1)$</p>
<p>$f(x) \equiv 0$ mod $(x-3)$</p>
<p>Which of the following statements are true?</p>
<p>a) $S$ is empty .</p>
<p>b) $S$ is a singleton.</p>
<p>c)$S$ is a finite non-empty set.</p>
<p>d) $S$... | Bernard | 202,857 | <p>Since $f(x)\equiv f(n)\mod x-n$, the second condition says $f(3)=0$ and $f(x)=(x-3)g(x)$ for some $g(x)\in\mathbf Z[x]$. But then $\;f(1)=-2g(1)$ is even, contrary to the first condition $f(1)=1$.</p>
|
3,566,997 | <p>When I go through examples of calculating the intersection of two planes, there seems to be a convention of choosing an arbitrary point in order to solve the linear equations in question and get a particular point on the intersection line.</p>
<p>Let's take an example from a <a href="https://math.stackexchange.com/... | user577215664 | 475,762 | <p>You can differentiate wrt <span class="math-container">$y$</span>:
<span class="math-container">$$\frac{d}{dx}[a^y]=\frac{d}{dy}[a^y] \frac {dy}{dx}=\frac{d}
{dy}[a^y] y'=\frac{d}{dy}[e^{y \ln a}] y'$$</span>
Note that <span class="math-container">$(e^{cx})'=e^{cx}c$</span>
<span class="math-container">$$\frac{d}{dx... |
951,104 | <p>Problem :
If $$T_r =\frac{r}{4r^4+1}$$ then the value of $$\sum^{\infty}_{r=1} T_r$$ is ? </p>
<p>How to start such problem I am not getting any clue on this please suggest thanks .</p>
| Jlamprong | 105,847 | <p>Note that \begin{align}
\sum_{r=1}^{\infty}T_r &=\sum_{r=1}^{\infty}\frac{r}{4r^4+1}\\
&=\frac14\sum_{r=1}^{\infty}\left(\frac{1}{2r^2-2r+1}-\frac{1}{2r^2+2r+1}\right)\\
&=\frac14\sum_{r=1}^{\infty}\left(\frac{1}{2r^2-2r+1}-\frac{1}{2(r+1)^2-2(r+1)+1}\right).\\
\end{align}</p>
<p>From this point, I am s... |
833,353 | <p>Let $a,b \geq 0$ and $0<p,q < 1$ s.t. $p + q = 1$.</p>
<p>Is it true that $a^p b^q \leq a+b$?</p>
| Paramanand Singh | 72,031 | <p>Well, the inequality is much stronger. In fact we have $a^{p}b^{q} \leq pa + qb$ and equality holds only when $a = b$. Since $0 < p, q < 1$ it follows that $pa < a, qb < b$ so that $pa + qb < a + b$ and therefore we get $a^{p}b^{q} \leq pa + qb < a + b$. The proof for the general inequality $a^{p}b... |
4,386,333 | <p>This is Exercise 1.6(iii) from Intro. to K-Theory for <span class="math-container">$C^*$</span>-Algebras by Rørdam et al.</p>
<p>If <span class="math-container">$A$</span> is a unital <span class="math-container">$C^*$</span>-algebra, and <span class="math-container">$a \in A$</span> is invertible, then I want to sh... | Just a user | 977,740 | <p>Note that if <span class="math-container">$a$</span> is invertible, so is <span class="math-container">$a^*a$</span>. And <span class="math-container">$(a^*a)^{-1}\in C^*(a^*a,1)\subset C^*(a, 1)$</span>, and <span class="math-container">$a^{-1} = (a^*a)^{-1}a^*\in C^*(a, 1)$</span>. This can be used to show that th... |
2,614,776 | <p><strong>Exercise :</strong></p>
<blockquote>
<p><strong>(A.1)</strong> Let G be a group. The subgroup <span class="math-container">$Z(G) = \{z \in G | zg =gz \space \forall \space g \in G\}$</span> is called the center of <span class="math-container">$G$</span>. Show that <span class="math-container">$Z(G)$</span> i... | BallBoy | 512,865 | <p>For A.2 your link will work, there's just a difference in notation - your $D_3$ is the link's $D_6$.</p>
<p>For A.3, Colescu's comment is a good start. The symmetries of an equilateral triangle can be reached via rotations and reflections ($D_3$) or permuting the vertices ($S_3$). Show that these give the same grou... |
3,489,896 | <p>It is clear to me that <span class="math-container">$\liminf X_n \le \limsup X_n$</span> but every time that I see that <span class="math-container">$\liminf X_n \le \sup X_n$</span> it sound to me not so obvious awkward. Could someone help me to understand this inequality?<span class="math-container">$ $</span> <sp... | drhab | 75,923 | <p>Hint:</p>
<p><span class="math-container">$\sup X_n$</span> is an upperbound of the set <span class="math-container">$\{X_n\mid n\in\mathbb N\}$</span> which means that for <strong>every</strong> <span class="math-container">$k\in\mathbb N$</span> we have <span class="math-container">$X_k\leq\sup X_n$</span>.</p>
|
4,287,450 | <p>I tried solving this by finding dy/dx and letting the denominator be zero, yielding an equation whose solution can only be approximated. But that approximated value is not right as seen on Desmos.</p>
| Glorious Nathalie | 948,761 | <p><span class="math-container">$r = k \theta $</span></p>
<p><span class="math-container">$ x = k \theta \cos \theta $</span> and <span class="math-container">$ y = k \theta \sin \theta $</span></p>
<p>Slope is <span class="math-container">$\dfrac{ \sin \theta + \theta \cos \theta }{ \cos \theta - \theta \sin \theta} ... |
3,682,514 | <blockquote>
<p>The hyperbola is given with the following equation: <span class="math-container">$$3x^2+2xy-y^2+8x+10y+14=0$$</span>
Find the asymptotes of this hyperbola. (<span class="math-container">$\textit{Answer: }$</span> <span class="math-container">$6x-2y+5=0$</span> and <span class="math-container">$2x+2y... | Quanto | 686,284 | <p>To justify the approach, note that asymptotically the quadratic terms of the curve
<span class="math-container">$f(x,y)=3x^2+2xy-y^2+8x+10y+14$</span> dominate, i.e.</p>
<p><span class="math-container">$$3x^2+2xy-y^2=(3x-y)(x+y)=0$$</span></p>
<p>which corresponds to the asymptotic behaviors of the asymptotes and... |
3,682,514 | <blockquote>
<p>The hyperbola is given with the following equation: <span class="math-container">$$3x^2+2xy-y^2+8x+10y+14=0$$</span>
Find the asymptotes of this hyperbola. (<span class="math-container">$\textit{Answer: }$</span> <span class="math-container">$6x-2y+5=0$</span> and <span class="math-container">$2x+2y... | bruceyuan | 534,973 | <p>Just what I thought, asymptotes for quadratic curve can be reformatted as the ratio of <span class="math-container">$y/x$</span> approach to a constant as <span class="math-container">$x$</span> goes up.</p>
<p>as pointed out by other the slope of asymptotes is <span class="math-container">$\frac{m}{l}$</span>, let... |
145,156 | <p>I need to fetch some journals issues, in example like this:
www.sciencedirect.com/science/journal/00963003/124/1</p>
<p>You may check this part to be constant: </p>
<pre><code>string = "https://www.sciencedirect.com/science/journal/00963003"
</code></pre>
<p>I read <a href="http://mathematica.stackexchange.com/qu... | garej | 24,604 | <p>Is it what you are after?</p>
<pre><code>issues = Flatten@
Table[(ToString[i] <> "/" <> #) & /@ {"", "1", "2", "2-3", "4"}, {i, 1, 310}]
</code></pre>
<p>Then one could StringJoin, smth like:</p>
<pre><code>links = StringJoin[{string, "/", #}] & /@ issues;
</code></pre>
<p>"XMLObject" is a b... |
3,556,490 | <p>Can someone explain me how to solve the equation <span class="math-container">$\frac{x-n}{x+n} = e^{-x}$</span>, where <span class="math-container">$n$</span> is a non-zero natural number ? Unfortunately, I have not even an idea how to start. Any hint is much appreciated. </p>
<p>Many thanks in advance. </p>
| The_Sympathizer | 11,172 | <p>As @Jam mentions in the comments, this is similar to equations for which the Lambert W-function would be applicable, but a bit too complex - and it is in fact equations like this that are suggested to motivate a <em>generalized Lambert W-function</em> which, in this case, is pretty much equivalent to your equation: ... |
3,315,558 | <p>I am having a very frustrating time with the back book that says my answer is way off but to me everything looks fine:</p>
<p><span class="math-container">\begin{align*}
(xy)y'&= x^2+3y^2\\
y' &= \frac{x^2}{xy} + \frac{3y^2}{xy}\\
y' &= \frac{x}{y} + \frac{3y}{x}\\
y' &= \frac{1}{v} + 3v\\
y' &=... | Cesareo | 397,348 | <p>Hint.</p>
<p>A way to solve this DE is by making <span class="math-container">$z = y^2$</span> and then</p>
<p><span class="math-container">$$
\frac 12 x z' - 3z = x^2
$$</span></p>
<p>which is a linear DE. Solving for <span class="math-container">$z$</span> we have</p>
<p><span class="math-container">$$
z = y^2... |
3,315,558 | <p>I am having a very frustrating time with the back book that says my answer is way off but to me everything looks fine:</p>
<p><span class="math-container">\begin{align*}
(xy)y'&= x^2+3y^2\\
y' &= \frac{x^2}{xy} + \frac{3y^2}{xy}\\
y' &= \frac{x}{y} + \frac{3y}{x}\\
y' &= \frac{1}{v} + 3v\\
y' &=... | Allawonder | 145,126 | <p>Your solution simplifies to the cubical parabola <span class="math-container">$$y=kx^3.$$</span> Substituting this into the original equation with <span class="math-container">$k=1$</span> for simplicity shows something must be wrong despite your confidence to the contrary, for we have on the one hand <span class="m... |
1,451,308 | <p>$Y$ is a non-negative random variable, not necessary integrable. How to show
$$\lim_{n\rightarrow \infty }\frac{1}{n}\int_{Y<n}Y \, dP=0$$</p>
<p>My idea is to find a good upper bound that converges to zero. However, I didn't find a good one. My upper bound is
$$\frac{1}{n}\int_{Y<n}Y \, dP\leq \frac{1}{n}\i... | Dominik | 259,493 | <p>Note that the sequence of random variables $X_n = \frac{1}{n} I\{Y < n\} Y$ is bounded by $1$ and converges pointwise to $0$. The assertion now follows from the dominated convergence theorem.</p>
|
1,451,308 | <p>$Y$ is a non-negative random variable, not necessary integrable. How to show
$$\lim_{n\rightarrow \infty }\frac{1}{n}\int_{Y<n}Y \, dP=0$$</p>
<p>My idea is to find a good upper bound that converges to zero. However, I didn't find a good one. My upper bound is
$$\frac{1}{n}\int_{Y<n}Y \, dP\leq \frac{1}{n}\i... | saz | 36,150 | <p>Here is a direct proof (which doesn't require any additional theorems such as the dominated convergence theorem):</p>
<p>Fix $\epsilon>0$. Since $A_n := \{n \leq Y < \infty\}$ is a sequence of decreasing sets satisfying $\bigcap_{n \in \mathbb{N}} A_n = \emptyset$, the continuity of the (probability) measure ... |
153,473 | <p>How does one prove that if a $X$ is a Banach space and $x^*$, a continuous linear functional from $X$ to the underlying field, then $x^*$ attains its norm for some $x$ in $X$ and $\Vert x\Vert = 1$?</p>
<p>My teacher gave us a hint that we should use the statement that if $X$ is a reflexive Banach space, the unit b... | t.b. | 5,363 | <p>Suppose that $\varphi \neq 0$. Let $x_n$ be a sequence in the unit sphere such that $|\varphi(x_n)| \geq \lVert \varphi\| - \frac{1}{n}$ which exists by the definition of the operator norm. Choose a subsequence such that $x_{n_k} \to x$ weakly with $\|x\| \leq 1$ by weak sequential compactness of the unit ball. Dedu... |
2,859,276 | <p>Given a measurable space $(X,\scr{A})$ where $\scr{A}$ is a sigma algebra on $X$, I would like to know if $f:X \to \mathbb{R}$ is measurable then $cf$ where $c \in \mathbb{R}$ is also measurable.</p>
<p>A function $f:X \to \mathbb{R}$ is measurable if $[f > \alpha] = \{x \in X \:{:}\: f(x) > \alpha \} \in \ma... | drhab | 75,923 | <p>In general if $(X,\mathscr A)$, $(Y,\mathscr B)$ and $(Z,\mathscr C)$ are measurable spaces and $f:X\to Y$ and $g:Y\to Z$ are both measurable then also $g\circ f:X\to Z$ is measurable.</p>
<p>This because: $$(g\circ f)^{-1}(\mathscr C)=f^{-1}(g^{-1}(\mathscr C))\subseteq f^{-1}(\mathscr B)\subseteq\mathscr A$$</p>
... |
2,590,551 | <p>Suppose we have $A,B$ which are $R$-submodules of the $R$-module M. Then , there exists an exact sequence :</p>
<p>$0\rightarrow \frac{M}{A\cap B}\rightarrow \frac{M}{A}\times \frac{M}{B}\rightarrow \frac{M}{A+B} \rightarrow 0$</p>
<p>Where $A+B=\langle A\cup B\rangle$</p>
<p>I believe that that the first functio... | Dan | 500,478 | <p>You can use Kirchhoff's theorem. Let $D$ be the diagonal matrix whose entries are the degrees of the vertices of G. If you delete any row of $A-D$ (the Laplacian matrix) and its corresponding column, then the determinant of the resulting minor is the number of spanning trees for the graph.</p>
|
2,590,551 | <p>Suppose we have $A,B$ which are $R$-submodules of the $R$-module M. Then , there exists an exact sequence :</p>
<p>$0\rightarrow \frac{M}{A\cap B}\rightarrow \frac{M}{A}\times \frac{M}{B}\rightarrow \frac{M}{A+B} \rightarrow 0$</p>
<p>Where $A+B=\langle A\cup B\rangle$</p>
<p>I believe that that the first functio... | Rebecca J. Stones | 91,818 | <p>We throw away the loop, since spanning trees don't have loops. Also, since all edges have weight 1, a "minimum spanning tree" is just a "spanning tree". A $n$-vertex spanning tree has $n-1$ edges; in this case that's $4$ edges. There are $\binom{7}{4}=35$ subgraphs with $4$ edges (ignoring the loop), which I drew... |
115,477 | <p>Let us say that a finite set $A$ in the plane is $1$-separated if:</p>
<p><strong>1)</strong> it has an even number of points;</p>
<p><strong>2)</strong> no open ball of diameter $1$ contains more than $|A|/2$ points.</p>
<p>For a $1$-separated set $A$ define $G(A)$ to be a graph where two points $x,y$ in $A$ are... | domotorp | 955 | <p>No, there is a counterexample. Take a circle, whose diameter is slightly larger than 1 and put |A|-1 points evenly around its boundary and the last point to its center. This set will be 1-separated, but no matter how you partition it, the part containing the center will not be 1-separated.</p>
|
45,800 | <p>Let's say I want to calculate the following scalar-by-matrix derivative</p>
<p>$$\frac{\partial}{\partial A} \text{tr} \left[(\vec X^T A)^T (\vec X^T A)\right],$$</p>
<p>with $\vec X$ and $A$ being a $n \times 1$ and a $n \times m$ matrix, respectively.
Is there a way in Mathematica to get the result</p>
<p>$$2 \... | swish | 2,490 | <p>Your expression simplifies to this $$\vec X \vec X^T A + A^T \vec X \vec X^T$$
using just these rules</p>
<pre><code>Unprotect[D, Transpose, Dot];
(*Derivative rules*)
D[Tr[A_], X_] := Tr[D[A, X]]
D[Transpose[A_], X_] := D[A, X]\[Transpose]
D[A_ .B_, X_] := D[A, X].B + A.D[B, X]
(*Tranpose rules*)
0\[Transpose] :=... |
2,384,444 | <p>Determine the equation of the line the portion of which, intercepted by the axes, is divided by the point $(-5,4)$ in the ratio of $1:2$.</p>
<p>My Attempt:
Let the equation of straight line be
$$ax+by+c=0$$
It passes through the point $(-5,4)$.
$$-5a+4b+c=0$$</p>
| velut luna | 139,981 | <p>The particular solution is
$$y=\frac{1}{2}x^2 e^{-\sin x}$$</p>
<p>Then solve for the homogeneous solution.</p>
<p>Can you take it from here?</p>
|
1,300,391 | <p>For example, if I wanted to get two of a kind, I would need seven dice. This is because even if the first six were 1, 2, 3, 4, 5, 6, the next one would have to make a pair out of the previous dice.</p>
<p>So is there a formula for getting n of a kind 100% of the time?</p>
| Mark Joshi | 106,024 | <p>13 for three of a kind = 2 * 6 + 1
in general
6(n-1) + 1</p>
<p>since will $6(n-1)$ the only way you can't have n of a kind is if you have n-1 copies of each number from 1 to 6. </p>
|
319,288 | <p>Let <span class="math-container">$\mathcal{M}$</span> be the field of meromorphic functions of one (complex) variable and <span class="math-container">$w = w(z)$</span> an analytic function satisfying a polynomial equation</p>
<p><span class="math-container">$P(w; z) := w^n + a_{n-1}(z) w^{n-1} + \cdots + a_1(z) w ... | reuns | 84,768 | <p>This is not a complete answer but I think this might help you :</p>
<p>If <span class="math-container">$w(z)$</span> is meromorphic on a disk <span class="math-container">$\{|z-z_0| < r\}\subset \mathbb{C}$</span> then it is algebraic (over <span class="math-container">$\mathbb{C}(z)$</span>) iff </p>
<p><span ... |
2,912,043 | <p>How do I approach this problem? My book gives answer as $[\frac{3-\sqrt{5}}{2},\frac{3+\sqrt{5}}{2}]$. I tried forming an equation in $y$ and putting discriminant greater than or equal to zero but it didn't work. Would someone please help me?</p>
<p>I get $x^2 (y-1) + 2x (y+1) + (5y-5) =0$ and discriminant gives $2... | Can Yaylali | 524,466 | <p>The values of sinus function from $\pi$ to $2\pi$ are the same as minus the sinus function from $0$ to $\pi$. Or to be a bit more precise, consider a substitution $t\rightarrow t-\pi$. </p>
<p>Now what is sin$(t-\pi)$ ?</p>
|
2,819,652 | <p>I know that my function $f$ must satisfy the following condition for $x \geq 0:$
$$\frac{d\left(xf(x)-x\right)}{dx} \geq 0.$$
What can I say about $f?$ I am curious about its possible sign or variations with respect to $x$.</p>
<p>I have investigated Grönwall's inequality to find an upper bound for $f$ but it seems... | mvw | 86,776 | <p>We go to polar coordinates, keep track of the cyclic nature of the complex exponential function and pull the root there:
\begin{align}
z^3 &= 2 \iff \\
(r e^{i\phi})^3 &= 2 \, e^{i\cdot (0 + 2\pi k)} \iff \\
r e^{i \phi} &= \sqrt[3]{2} \, e^{i \cdot 2\pi k / 3}
\end{align}
Here $k \in \mathbb{Z}$. </p>
... |
290,507 | <p>Every metric space is associated with a topological space in a canonical way. According to <a href="http://topospaces.subwiki.org/wiki/Induced_topology_from_metric_is_functorial" rel="nofollow">this</a> source, this amounts to a <a href="http://en.wikipedia.org/wiki/Full_and_faithful_functors" rel="nofollow">full fu... | Berci | 41,488 | <p>Partial answer for your 'alien' question.</p>
<p><em>If</em> we assume that aliens also think about open sets as the complements of <em>closed</em> sets, <em>and</em> if we also agree with them that being closed means being closed under <em>limits</em> of sequences, and that $\lim(x_n)=x$ in metric space is defined... |
4,502,175 | <p>I want to create a simple demo of moving an eye (black circle) inside a bigger circle with a black stroke when moving a mouse. I have cursor positions mouseX and mouseY on a canvas and I need to map the value of mouse position into a circle so the eye is moving inside the circle.</p>
<p>This should be trivial but I ... | CyclotomicField | 464,974 | <p>Call <span class="math-container">$p$</span> the position of the cursor and <span class="math-container">$c$</span> the center of the large circle. First, we move the origin to be the center of the circle by considering the vector <span class="math-container">$v=p-c$</span> so that <span class="math-container">$v$</... |
4,651,215 | <p>Three Cowboys have a shoot out. A shoots with 1/3 accuracy, B 1/2, C one (never misses). A goes first. The cowboys attempt to shoot the cowboy with the best shooting ability. What’s the probability that A is still alive?</p>
<p>Edit: We assume that after A, B goes first and then C</p>
<p>My approach:</p>
<p>Since it... | Robert Shore | 640,080 | <p>Here's another way to look at it. If <span class="math-container">$a$</span> is even, then <span class="math-container">$a=2k, k \in \Bbb Z$</span> so <span class="math-container">$a^2=4k^2 \equiv 0 \pmod 4$</span>.</p>
<p>If <span class="math-container">$a$</span> is odd, then <span class="math-container">$a=2k+1,... |
128,009 | <p>Hi,
Is there any known sequence such that the sum of a combination of one subsequence never equals another subsequence sum. The subsequences should have elements only from the parent sequence.</p>
<p>Thanks
Sundi</p>
| Marc Palm | 10,400 | <p>Rather then only considering invariant PSD operators, you might want to consider all $G$ invariant operators, i.e., $G$-intertwiner. I describe their functional calculus below. Their functional calculus can be realized/studied via convolution products and representation theory. What I describe is in the realm of the... |
1,902,939 | <p>I was going through the problem attached herewith. the fact is I could not understand 2 things. </p>
<p>Firstly, how did the author came up with the idea of choosing the event sets the way he has described? How is this intuitive?</p>
<p>Secondly, I tried to find the $P(A_1)$ directly, and I came up with the follow... | Community | -1 | <p><strong><em>Edited</em></strong> to make the argument a bit more efficient.</p>
<p>I think this is more or less what the first hint is suggesting:</p>
<p>$aHbH = cH$, so $ab = aebe \in aHbH = cH$, where $e$ is the identity element. Multiplying on the left by $a^{-1}$ gives us $b \in a^{-1}cH$. Thus the left coset ... |
2,796,711 | <p>I have the following question in hand. </p>
<p>If $\lambda_1,\cdots,\lambda_n$ are the eigenvalues of a given matrix $A \in M_n$, then prove that the matrix equation $AB - BA = \lambda B$ has a nontrivial solution $B \neq 0 \in M_n$, if and only if $\lambda = \lambda_i - \lambda_j$ for some $i,j$.</p>
| user550103 | 550,103 | <p>Here is my attempt. Does this make sense to you experts?</p>
<p>If we vectorize such that
\begin{align}
AB - BA &= \lambda B \\
&\Downarrow \\
\mbox{vec}\left(AB - BA \right) &= \mbox{vec}(\lambda B) \\
\mbox{vec}\left(ABI - IBA \right) &= \mbox{vec}(\lambda B) \\
\left(\left(I \otimes A\right) - \... |
3,293,233 | <p>My robot has a laser time-of-flight distance sensor. Holding the distance and target constant: When I take single set of readings, three standard deviations usually is 3% to 6% of the mean.</p>
<p>If I take multiple sets of readings, each set will show three standard deviations to be 3-6% of the mean, BUT, averagi... | Empy2 | 81,790 | <p>There is natural fluctuation in the measurements, and the standard deviation <span class="math-container">$(\sigma)$</span> measures the size of those fluctuations. </p>
<p>The averages fluctuate less, by <span class="math-container">$\sigma/\sqrt n$</span>. That is the point of taking a mean. So you can divide ... |
3,898,904 | <p>I'm currently making a start on group theory and have hit a roadblock with a relatively basic theorem on finite cyclic groups. The specific relation killing me is:
<span class="math-container">$$\mathbb{Z}_n \times \mathbb{Z}_m \cong \mathbb{Z}_{mn} \Leftrightarrow \text{gcd}(m,n) = 1$$</span>
So, the most straightf... | Parcly Taxel | 357,390 | <p>The given biconditional, taken as a whole, extends to the infinite cyclic group <span class="math-container">$\mathbb Z$</span> when the latter is treated as <span class="math-container">$\mathbb Z_0$</span>. Let <span class="math-container">$m=0$</span>:
<span class="math-container">$$\mathbb Z×\mathbb Z_n\iff\gcd(... |
3,898,904 | <p>I'm currently making a start on group theory and have hit a roadblock with a relatively basic theorem on finite cyclic groups. The specific relation killing me is:
<span class="math-container">$$\mathbb{Z}_n \times \mathbb{Z}_m \cong \mathbb{Z}_{mn} \Leftrightarrow \text{gcd}(m,n) = 1$$</span>
So, the most straightf... | Community | -1 | <p><span class="math-container">$\Bbb Z\times\Bbb Z_n\not\cong\Bbb Z_n$</span>, since the former has a nontrivial free part , whereas the latter is just torsion.</p>
|
2,985,962 | <p>I was asking my friends a riddle about identifying hats. Each person has to correctly identify the colour of their own hat that was put on their head randomlly.
There is no defined number of either colour. So they could be all white or all black or any combination in between.</p>
<p>They gave an answer that gets 50... | Moko19 | 618,171 | <p>If I understand the riddle correctly, for a set of N people, there is an algorithm by which all but one person are guaranteed to get the correct answer, assuming all the players are perfect mathematicians and can arrange a strategy before being assigned hats.</p>
<p>The person in back counts the number of black hat... |
114,909 | <p>Consider this simple code:</p>
<pre><code>Grid[RandomInteger[{0, 1}, {10, 10}], Background -> LightBlue]
</code></pre>
<p><a href="https://i.stack.imgur.com/PfWwb.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PfWwb.png" alt="output"></a></p>
<p>Is it possible to modify the code so that the... | kglr | 125 | <pre><code>Grid[Item[#, Background -> If[# == 1, Blue]] & /@ # & /@ RandomInteger[{0, 1}, {10, 10}]]
</code></pre>
<p><img src="https://i.stack.imgur.com/iq0kd.png" alt="Mathematica graphics"></p>
<pre><code>Grid[RandomInteger[{0, 1}, {10, 10}]] /. 1 -> Item[1, Background -> Blue]
</code></pre>
<p><... |
1,913,394 | <p>I need to prove that rank($\mathrm{A}$) is not continuous everywhere but is lower semi-continuous everywhere, where $\mathrm{A}\in \mathbb{C}^{n\times m} $</p>
| am70 | 365,982 | <p>Let <span class="math-container">$A\in{\mathbb C}^{n×m}$</span>; we use the symbol <span class="math-container">$m$</span> to denote a square minor (a selection of <span class="math-container">$k$</span> rows and <span class="math-container">$k$</span> columns). For any minor <span class="math-container">$m$</span>... |
1,406,462 | <p>I've been studying an introductory book on set theory that uses the <a href="https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory">ZFC</a> set of axioms, and it's been really exciting to construct and define numbers and operations in terms of sets. But now I've seen that there are other alternatives to ... | Asaf Karagila | 622 | <p>Well. That depends on whom you might ask this.</p>
<ol>
<li><p>Set theory might be inconsistent. In particular $\sf ZFC$ and its extension by large cardinal axioms. It's a nontrivial thing, to feel safe with these theories, and it takes a lot of practice and time until you understand that $\sf ZFC$ is self-evident ... |
2,616,604 | <p>Why it is true that every odd number r that is not equivlant to p mod p has a odd number $s$ s.t</p>
<p>$rs\equiv 2p-1 \mod 2p $</p>
<p>My first thought is that i should have an inverse for $r$ in the ring so i can write.
$s\equiv r^{-1} (-1) \mod 2p $ but im not sure what in number theory tells me that $r^{-1} $ ... | Community | -1 | <p>Assuming $p$ is a prime, and that not just $r\ne p$ but $p\not\mid r$...</p>
<p>$r^{-1}$ exists because $r$ (being odd) is coprime with $2p$. Thus, $1=rs+2pt$ for some $s,t\in\mathbb Z$ (<a href="https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity" rel="nofollow noreferrer">Bézout's identity</a>), or, reducing m... |
3,537,549 | <p>In the game of poker, five cards from a standard deck of 52 cards are dealt to each player.</p>
<p>Assume there are four players and the cards are dealt five at a time around the table until all four players have received five cards.</p>
<p>a. What is the probability of the first player receiving a royal flush (th... | Eric Towers | 123,905 | <p>Line 1: <span class="math-container">$\cos \theta = -\sin(\theta + \pi/2)$</span>. Equivalently, make the substitution <span class="math-container">$\theta = \phi + \pi/2$</span> so that <span class="math-container">$\cos \theta = \cos(\phi + \pi/2) = - \sin \phi$</span> using a <a href="https://en.wikipedia.org/wi... |
29,114 | <p>In <a href="https://puzzling.stackexchange.com/questions/72494/use-2-0-1-and-9-to-make-34">this</a> question on the <a href="https://puzzling.stackexchange.com/"><em>Puzzling Stack Exchange</em></a>, I wrote <a href="https://puzzling.stackexchange.com/questions/72494/use-2-0-1-and-9-to-make-34/72504#72504">this</a> ... | TheSimpliFire | 471,884 | <p>The braces are meant to suit the size of the expression you want them under. If you look at the code, you see the following:</p>
<p><code>\underbrace{blah blah blah}</code></p>
<p>which means that everything you have put from <code>{</code> to <code>}</code> will be on top of the brace. On the contrary, it would ... |
1,484,880 | <p>i messed up an exam yesterday. given</p>
<p>$$\mathcal{B}:=\{(a,b]:a,b \in\mathbb{R}, a \le b\},$$</p>
<p>i was able to show that $\mathcal{B}$ is the base of a topology $\mathcal{T}$, and that $\mathcal{T}$ is finer than the standard topology. however, i couldn't find a countable dense subset in $(\mathbb{R},\mat... | DanielWainfleet | 254,665 | <p>This is known as the Sorgenfrey line.Although it's usually given in reverse ,with the base of intervals $[a,b)$ .(Not the same space but its mirror image, homeomorphic to it by the function $f(x)=-x$.) I will stick to your def'n.Let $C$ be a base. For each real $r$ there exists $C_r\in C$ such that $r\in C_r\subset... |
3,931,067 | <p>Let <span class="math-container">$Y$</span> be a Geometric random variable with <span class="math-container">$p\leq \frac{1}{2}$</span>. I know that <span class="math-container">$f(y) = (1-p)^y p$</span> for <span class="math-container">$y=0,1,2,...$</span>Therefore,</p>
<p><span class="math-container">$$E(e^{r(Y-1)... | user854214 | 854,214 | <p>Note that <span class="math-container">$a = 0$</span> doesn't work. For <span class="math-container">$a \neq 0$</span>, the left hand side has minimum period <span class="math-container">$2\pi$</span>, while the right hand side has minimum period <span class="math-container">$2\pi / a$</span>. Hence, <span class="ma... |
3,931,067 | <p>Let <span class="math-container">$Y$</span> be a Geometric random variable with <span class="math-container">$p\leq \frac{1}{2}$</span>. I know that <span class="math-container">$f(y) = (1-p)^y p$</span> for <span class="math-container">$y=0,1,2,...$</span>Therefore,</p>
<p><span class="math-container">$$E(e^{r(Y-1)... | player3236 | 435,724 | <p>From <span class="math-container">$0 \le b^2 = \cos(b^2)-1 \le 1-1$</span> we see that <span class="math-container">$b^2 = 0$</span>. Hence <span class="math-container">$b=0$</span>.</p>
<p>Now we have <span class="math-container">$a(\cos x-1) = \cos (ax)-1$</span>.</p>
<p>Substituting <span class="math-container">$... |
3,716,439 | <p>Uhm I'm having trouble solving for x in these type of equations: <span class="math-container">$a^x + b^x = c^x$</span> (where x is the only unknown)</p>
<p>I saw an instance by Presh Talwalkar in MindYourDecisions. I tried solving for x and used an example with a known answer. <span class="math-container">$3^x + 4... | Community | -1 | <p>The second equation is</p>
<p><span class="math-container">$$\dot x+\dot y=x+y$$</span> and <span class="math-container">$x+y=ce^t$</span>.</p>
<p>Then the first says</p>
<p><span class="math-container">$$ce^t+2y=\sin t.$$</span></p>
<p>Done.</p>
|
3,716,439 | <p>Uhm I'm having trouble solving for x in these type of equations: <span class="math-container">$a^x + b^x = c^x$</span> (where x is the only unknown)</p>
<p>I saw an instance by Presh Talwalkar in MindYourDecisions. I tried solving for x and used an example with a known answer. <span class="math-container">$3^x + 4... | Manjoy Das | 568,656 | <p>Multiplying the <span class="math-container">$1$</span>st equation by <span class="math-container">$(D-1)$</span> and the <span class="math-container">$2$</span>nd one by <span class="math-container">$D$</span> and then subtracting, we get<br>
<span class="math-container">\begin{align}
\{(D-1)(D+2)-D(D-1)\}y&=\s... |
241,494 | <p>Who has hint how to prove:
$\sum_{n=0}^N \sum_{k=1}^N g_k y_n = \sum_{n=1}^N \sum_{k=1}^n g_k y_{n-k}+\sum_{n=1}^N\sum_{k=1}^n g_n y_{N-k+1}$</p>
<p>Thank in advance!</p>
| preferred_anon | 27,150 | <p>It's almost always helpful to expand both sides to see where all the terms come from: </p>
<p>LHS: We can factor out $g_{k}$ from the inner sum, since it is independent of both $n$ and $N$. Thus, we have $g_{1}(y_{0}+y_{1}+\ldots+y_{N})+g_{2}(y_{0}+\ldots+y_{N})+\ldots+g_{N}(y_{0}+\ldots +y_{N})$, which we can eas... |
4,225,583 | <p>At my university, we came across following exercise:</p>
<p>Let <span class="math-container">$G$</span> be a finite group of odd order.</p>
<p>Show that every irreducible representation of <span class="math-container">$G$</span> over <span class="math-container">$\mathbb{C}$</span> with real valued character is triv... | Aphelli | 556,825 | <p>It’s well known (over <span class="math-container">$\mathbb{C}$</span>) that there is a perfect pairing <span class="math-container">$C \times F \rightarrow \mathbb{C}$</span>, where <span class="math-container">$C$</span> is the vector space generated by the conjugation classes (denote as <span class="math-containe... |
4,225,583 | <p>At my university, we came across following exercise:</p>
<p>Let <span class="math-container">$G$</span> be a finite group of odd order.</p>
<p>Show that every irreducible representation of <span class="math-container">$G$</span> over <span class="math-container">$\mathbb{C}$</span> with real valued character is triv... | David A. Craven | 804,921 | <p>Since <span class="math-container">$\chi$</span> is real, we have <span class="math-container">$\langle \chi,\chi\rangle=1$</span>. Notice that
<span class="math-container">$$\langle \chi,\chi\rangle=\frac{1}{|G|} \sum_{g\in G} \chi(g)\overline{\chi(g)}=\frac{1}{|G|}\left(\chi(1)+2\sum_{g\in I} \chi(g)^2\right),$$</... |
55,853 | <p>Given a function $f:X\to X$, let $\text{Fix}(f)=\{x\in X\mid x=f(x)\}$.</p>
<p>In a recent <a href="https://math.stackexchange.com/questions/55846/is-the-set-of-fixed-points-in-a-non-hausdorff-space-always-closed/55847#55847">comment</a>, I wondered whether
$X$ is Hausdorff $\iff$ $\text{Fix}(f)\subseteq X$ is clos... | LostInMath | 9,174 | <p>While I think that Sam's counterexample is fine, I would like to elaborate on ccc's comment to Sam's answer.</p>
<p>Recall that a topological space $X$ is called <em>Fréchet</em> if for every $A\subseteq X$ and every point $x\in \bar{A}$ there exists a sequence of points of $A$ converging to $x$.</p>
<p>Observe th... |
3,986,812 | <p>Given <span class="math-container">$U_n = (1 + 1/n)^n$</span> , <span class="math-container">$n = 1,2,3,\ldots\;.$</span></p>
<blockquote>
<p>Show that <span class="math-container">$2 \le U_n \le 3$</span> for all n</p>
</blockquote>
<p>This is what I've done. Can anyone help?</p>
<p><span class="math-container">$$\... | Arash | 866,952 | <p>Here is another way to look at it that might be helpful, <em>if the question is why for all <span class="math-container">$n$</span> it holds</em>. Consider <span class="math-container">$f(x)=(1+\frac{1}{x})^x$</span>. It is an increasing function of <span class="math-container">$x$</span>:</p>
<p><span class="math-c... |
1,356,585 | <p>I am reading Hajime Sato's: <em>Algebraic Topology, an Intuitive Approach</em>. His Sample Problem 1.3 is: <em>Show that the topological spaces X and I are not homeomorphic.</em> (Note that this requires a font where I appears as a simple straight vertical line.) </p>
<p>Sato argues by contradiction. Brief sketc... | Mathslover shah | 253,752 | <p>If I is disconnected by removing the point f(xo) and X by removing the point of conjunction xo ... there are two connected components of I while 4 connected components of X ... so if f is homeomorphism images of connected components are also connected ... is it possible here ? Can you map the 2 connected components ... |
1,356,585 | <p>I am reading Hajime Sato's: <em>Algebraic Topology, an Intuitive Approach</em>. His Sample Problem 1.3 is: <em>Show that the topological spaces X and I are not homeomorphic.</em> (Note that this requires a font where I appears as a simple straight vertical line.) </p>
<p>Sato argues by contradiction. Brief sketc... | John Gowers | 26,267 | <p>A <em>homeomorphism</em> is an <em>isomorphism</em> between topological spaces. That is to say, it preserves the topological structure. </p>
<p>Let's forget the definition of a homeomorphism in terms of continuous maps for now, and try and construct it ourselves. If $X$ is a topological space, and $Y$ is a topol... |
973,268 | <p>Today in highschool we were doing a chapter called "Roots of polynomials" where we learnt something new and interesting which is :</p>
<blockquote>
<p><span class="math-container">$ax^2+bx+c=0$</span> Has roots <span class="math-container">$\alpha$</span> , <span class="math-container">$\beta$</span> Then:... | MJD | 25,554 | <p>As the comments say, these are called <a href="http://en.wikipedia.org/wiki/Vieta%27s_formulas" rel="nofollow">Vieta's formulas</a>, and the plus or minus sign is not difficult to remember, because it always alternates. In general, if you have an $n$th-degree polynomial $$p(x) = a_0x^n + a_{1}x^{n-1} + a_2x^{n-2} +... |
973,268 | <p>Today in highschool we were doing a chapter called "Roots of polynomials" where we learnt something new and interesting which is :</p>
<blockquote>
<p><span class="math-container">$ax^2+bx+c=0$</span> Has roots <span class="math-container">$\alpha$</span> , <span class="math-container">$\beta$</span> Then:... | M.S.E | 153,974 | <p>For the fourth power</p>
<p>$$ax^4+bx^3+cx^2+dx+e=0$$ has roots $\alpha$ , $\beta$ , $\gamma$ , $\delta$</p>
<p>Then :</p>
<p>$$ \begin{align}
\alpha+\beta+\gamma+\delta & = -\frac ba \\
\alpha\beta + \alpha\gamma + \alpha\delta +
\beta\gamma + \beta\delta+\gamma\delta & = \hphantom{-}\frac ca \\
\alpha\... |
973,268 | <p>Today in highschool we were doing a chapter called "Roots of polynomials" where we learnt something new and interesting which is :</p>
<blockquote>
<p><span class="math-container">$ax^2+bx+c=0$</span> Has roots <span class="math-container">$\alpha$</span> , <span class="math-container">$\beta$</span> Then:... | Parth Thakkar | 70,311 | <p>I'll give examples, then you'll understand the general pattern better.</p>
<p>Take a degree four polynomial. I'll denote a general fourth-degree polynomial by <span class="math-container">$P_4$</span>. A general <span class="math-container">$5^{\text{th}}$</span> degree polynomial as <span class="math-container">$P... |
438,744 | <p>Let $A$ and $B$ be real square matrices of the same size. Is it true that
$$\det(A^2+B^2)\geq0\,?$$</p>
<p>If $AB=BA$ then the answer is positive:
$$\det(A^2+B^2)=\det(A+iB)\det(A-iB)=\det(A+iB)\overline{\det(A+iB)}\geq0.$$</p>
| Seirios | 36,434 | <p>If $A= \left( \begin{matrix} 1 &1 \\ 0 &1 \end{matrix} \right)$ and $B=\left( \begin{matrix} 1&0 \\ n&1 \end{matrix} \right)$, then $\det(A^2+B^2)=4(1-n)$.</p>
|
146,952 | <p>I have revised my original post. The questions I asked there were not well-put or even thought through. I don't want to delete, however, since some of the comments may be of interest to other MO users. </p>
<p>It is well known that the sums of the reciprocals of the primes diverges. But suppose we decompose the set... | Hugh Thomas | 468 | <p>The following is rather more simple-minded that what you are suggesting. </p>
<p>Let's say we have a deck of cards, with the cards labelled by the primes in order. We are going to think of constructing the partition by dealing out the cards into piles; the piles will be the parts of the partition. </p>
<p>The d... |
146,952 | <p>I have revised my original post. The questions I asked there were not well-put or even thought through. I don't want to delete, however, since some of the comments may be of interest to other MO users. </p>
<p>It is well known that the sums of the reciprocals of the primes diverges. But suppose we decompose the set... | Aaron Meyerowitz | 8,008 | <p>There is nothing special going on related to primes or even to integers (see below.) But let me start with the positive integers. What you want can't be harder than partitioning the positive integers similarly. Consider partitioning the positive integers in sequences as follows:</p>
<p>$\ 1,\ 3,\ 6,10,15,21\cdots$... |
146,952 | <p>I have revised my original post. The questions I asked there were not well-put or even thought through. I don't want to delete, however, since some of the comments may be of interest to other MO users. </p>
<p>It is well known that the sums of the reciprocals of the primes diverges. But suppose we decompose the set... | Stefan Kohl | 28,104 | <p>A very simple way to obtain such partition of the primes is to put the $n$-th prime into
the $k$-th set in the partition, where $k$ is the number of 1's in the binary representation of $n$.</p>
|
1,332,433 | <blockquote>
<p>Let <span class="math-container">$f:\mathbb R\to \mathbb R$</span> be such that for every sequence <span class="math-container">$(a_n)$</span> of real numbers,</p>
<p><span class="math-container">$\sum a_n$</span> converges <span class="math-container">$\implies \sum f(a_n)$</span> converges</p>
<p>Prov... | zhw. | 228,045 | <p>A start: There exists $\delta > 0$ such that $\frac{f(x)}{x}$ is bounded in $\{0<|x|<\delta\}.$ Proof: Otherwise there is a sequence of nonzero terms $x_n \to 0$ such that</p>
<p>$$\left|\frac{f(x_n)}{x_n}\right|> n$$</p>
<p>for all $n.$ We can pass to a subsequence $n_k$ such that $|x_{n_k}| < 1/k^... |
190,604 | <p><strong>Background</strong></p>
<p>I am currently trying to solve exercise 1.1.18 in Hatcher's Algebraic Topology. The part of the exercise I am interested in is the following:</p>
<blockquote>
<p>Using the technique in the proof of Proposition 1.14, show that if a space $X$ is obtained from a path-connected sub... | Holdsworth88 | 22,437 | <p><strong>Proof with Yoyo's suggestion</strong></p>
<p>Since $X$ is defined to be the result of attaching an $n$-cell to $A$ via some attaching map $\varphi:\partial D^n\rightarrow X$, it has the form $X=A \amalg e^n/\sim$, where $x\sim \varphi(x)$ for all $x \in \partial D^n$. Note first that since $A$ and $e^n$ are... |
3,140,384 | <p>Assume you're stranded on an island, and you've got some really important maths to do. Obviously you don't have a calculator with you, so you need to find another way.</p>
<p>I know this sounds absurd, but it quite annoys me that, apart from maybe addition to division, computing any other value always ends up at "P... | Community | -1 | <p>Like any algorithm, it will depend on your math skills, and knowledge.<a href="https://math.stackexchange.com/questions/3116326/is-there-a-faster-method-to-calculate-1-x-x-an-integer-than-this">Here</a> is an algorithm for <span class="math-container">$\frac{1}{x}$</span> for example. sqrt can be done by babylonian ... |
1,783,439 | <p>How will the basis vectors of the subspace $\mathbb{R}^n$ consisting of those vectors $A=(a_1,\cdots,a_n)$ such that $a_1+\cdots+a_n=0$ look like?</p>
<p>The initial problem was "what is the dimension of the subspace $\mathbb{R}^n$ consisting of those vectors $A=(a_1,\cdots,a_n)$ such that $a_1+\cdots+a_n=0$ ?</p>
... | user324795 | 324,795 | <p>Let $e_i$ denote the $i$th coordinate vector in $\mathbb{R}^n$. I claim that a basis for this space is given by
$$
e_1 - e_n, e_2 - e_n, \dots, e_{n-1} - e_n.
$$
It remains for you to prove that this is a linearly independent set which spans your space. If you're willing to use the fact that this space has dimensio... |
628,254 | <p>I am searching for some good book which section is devoted to modular arithmetic. I am self learner so I strongly prefer that book has exercises best with answers or solutions. I have CS background and has taken course on discrete mathematics but besides some basic facts on modulo operation it lacked some introducti... | Brian Spiering | 153,929 | <p><a href="http://rads.stackoverflow.com/amzn/click/0486284247" rel="nofollow">Concepts of Modern Mathematics</a> by Ian Stewart is a good introduction to many pure mathematical concepts, including modular arithmetic.</p>
|
3,723,183 | <p>I am working through a text book by <a href="http://www.hds.bme.hu/%7Efhegedus/Strogatz%20-%20Nonlinear%20Dynamics%20and%20Chaos.pdf" rel="nofollow noreferrer">Strogatz Nonlinear dynamics and chaos</a> . In chapter 2 question 2.2.1 , I am looking for an analytical solution. I have the question's answer but would lik... | Community | -1 | <p>An algebraic expression needs to have all its terms defined to have a meaning.</p>
<p>As</p>
<p><span class="math-container">$$\lim_{x\to0}\log x$$</span> is undefined, the whole expression <span class="math-container">$0\cdot\lim_{x\to0}\log x$</span> is undefined.</p>
<p>And we also have <span class="math-containe... |
96,518 | <p>Some time ago I had a physics test where I had the following integral: $\int y'' \ \mathrm{d}y$. The idea is that I had a differential equation, and I had acceleration (that is, $y''$) given as a function of position ($y$). The integral was actually equal to something else, but that's not the point. I neede... | user18063 | 18,063 | <p>When dealing with differentials, we generally have a "manipulate first, ask questions later" attitude. Differentials have a way of giving out correct results when manipulated formally but it's a mistake to think that something meaningful is going on. Okay, this is not totally true: there are non-traditional approach... |
1,405,637 | <p>Given that $ U_n=[x(1-x)]^n$ and $n\geq2$,$V_n=\int_{0}^{1}e^xU_ndx$,prove that $V_n+2n(2n-1)V_{n-1}-n(n-1)V_{n-2}=0$<br></p>
<p>I tried to solve it by integration by parts,taking $U_n$ as first function and $e^x$ as second function.<br></p>
<p>$V_n=-\int_{0}^{1}n[x(1-x)]^{n-1}(1-2x)e^x dx=-\int_{0}^{1}nU_{n-1}(1-... | Nate Eldredge | 822 | <p>Hint: use the Borel-Cantelli lemma to show that
$$P(X_n \ne 2 \text{ i.o.}) = 0.$$
(Independence is not needed,)</p>
|
3,962,427 | <p>I have proven that the sequence is convergent.</p>
<p>Let <span class="math-container">$x_n=\frac{(2n)!!}{(2n+1)!!}$</span></p>
<p><span class="math-container">$\frac{x_{n+1}}{x_n}$</span> <span class="math-container">$=\frac{2n+2}{2n+3}<1$</span> Therefore the sequence decreases. On the other hand
<span class="m... | B. Goddard | 362,009 | <p>See that</p>
<p><span class="math-container">$$x_n = \prod_{k=1}^n \frac{2k}{2k+1}, $$</span></p>
<p>so that</p>
<p><span class="math-container">$$\ln x_n = \sum_{k=1}^n \ln \left(\frac{2k}{2k+1}\right). $$</span></p>
<p>So we work with this series. First argue that</p>
<p><span class="math-container">$$\sum_{k=1}... |
2,846,928 | <blockquote>
<p>If I have the a complex number $z \in \mathbb{C}$ with absolute value $|z| = 1$, how do I show that $-i \frac {z-1}{z+1}$ is real? </p>
</blockquote>
| Math Lover | 348,257 | <p>Note that $$-i\frac{e^{i\theta}-1}{e^{i\theta}+1}\times \frac{e^{-i\theta/2}}{e^{-i\theta/2}}=-i\frac{e^{i\theta/2}-e^{-i\theta/2}}{e^{i\theta/2}+e^{-i\theta/2}}=\tan(\theta/2).$$</p>
|
2,846,928 | <blockquote>
<p>If I have the a complex number $z \in \mathbb{C}$ with absolute value $|z| = 1$, how do I show that $-i \frac {z-1}{z+1}$ is real? </p>
</blockquote>
| WimC | 25,313 | <p>If $\lvert z \rvert = 1$ then $\overline{z}=z^{-1}$. So complex cojugation of your expression results in $$i\frac{z^{-1}-1}{z^{-1}+1} = -i \frac{z-1}{z+1}.$$</p>
|
1,800,611 | <p>I've setup a recurrence relation as part of a numerical analysis problem, and found that $$x_{n+1} = \frac{x_n+1}{2}$$</p>
<p>The notes then say that for $x_0=0$, it is easy to show that $$x_k = 1 - 2^{-k}$$</p>
<p>Plugging in a few numbers shows this is clearly correct, and equating $x_{n+1}=x_n$ shows it converg... | ekkilop | 284,417 | <p>Another fun way of doing it is by means of a generating function. Define
$$
A(t) = \sum_{n\geq 0} x_nt^n = x_0 + x_1 t + x_2 t^2 + \dots \quad (1)
$$
for real $t<1$. Then multiply your recursion formula by $t$ and sum to get
$$
\sum_{n\geq 0} x_{n+1} t^n = \frac{1}{2} \sum_{n\geq 0} x_n t^n + \frac{1}{2} \sum_{n... |
503,808 | <p>Let $\mathbb{K} = \mathbb{Q}[\sqrt[3]{5}] \ $, and let $\mathbb{L}$ be the normal closure of $\mathbb{K}$. </p>
<p>Let $\mathbb{O}_{\mathbb{K}} \ $ be the integral closure of $\mathbb{Z}$ in $K$ and $\mathbb{O}_{\mathbb{L}} \ $ be the integral closure of $\mathbb{Z}$ in $\mathbb{L}$. I want to find the factorizatio... | user8268 | 8,268 | <p>A usual method is to look at $f=x^3-5$ in $\mathbb F_p[x]$ ($p$ the prime you want to factorize). If $f$ has no multiple factors then the factorization of $f$ gives you the factorization of $p$; $p$ is unramified. Namely, $f=f_1f_2...f_k$ means $p=P_1P_2...P_k$, the norm of $P_i$ is $p$ to the degree of $f_i$, and $... |
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