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 |
|---|---|---|---|---|
1,130,820 | <p>I've got a permutation $S$ and I need to find out all the permutations $R$ with: $R \circ R = S$.
How can I solve it using its product of disjoint cycles?
I know how to solve such an equation,but using letters and trying every case. But this permutation $S$ has $12$ elements,so it's impossible to solve it this way.<... | hmakholm left over Monica | 14,366 | <p>No, it is not intuitionistically valid. Here is a <a href="https://en.wikipedia.org/wiki/Kripke_semantics#Semantics_of_intuitionistic_logic">Kripke model</a> where it doesn't hold:</p>
<p>There are two frames, $w$ and $u$ with $w<u$.</p>
<p>In $w$ the universe is $\{\mathtt a\}$ and $P(\mathtt a)$ is true.</p>
... |
3,627,704 | <p>The pmf of a negative binomial distribution is</p>
<p><span class="math-container">$$p_X(x)= {x-1 \choose r-1}~ p^r~ (1-p)^{x-r}\quad x=r,r+1,\cdots$$</span></p>
<p>I want to verify that </p>
<p><span class="math-container">$$\sum \limits_{x=r}^{\infty} p_X(x)= 1$$</span></p>
<p>I start with</p>
<p><span class=... | heropup | 118,193 | <p>Taking your boxed equation <span class="math-container">$$(a+1)^{-n} = \sum_{k=0}^\infty (-1)^k \binom{n+k-1}{k} a^k$$</span> and choosing <span class="math-container">$a = p-1$</span>, <span class="math-container">$n = r$</span>, we obtain <span class="math-container">$$p^{-n} = (1 + (p-1))^{-n} = \sum_{k=0}^\infty... |
196,024 | <p>I am an undergrad and I know that the conjecture may have been proven recently. But in reading about it, I am entirely confused as to what it means and why it is important. I was hoping some of you kind people could help me.</p>
<p>I know there are several formulations of the conjecture.</p>
<p>Wolfram says:</p>
... | Hans-Peter Stricker | 1,792 | <p>Erica Klarreich gives a very concise and readable explanation what the ABC conjecture is about in this <a href="https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/" rel="nofollow noreferrer">article in the Quanta magazine</a>.</p>
<p>You may start reading (but don'... |
3,500,889 | <p>Given natural numbers <span class="math-container">$a,b,c,d$</span>, let <span class="math-container">$a,b$</span> be coprime with <span class="math-container">$b>a$</span> and let <span class="math-container">$c,d$</span> be coprime with <span class="math-container">$d>c$</span>. Define a function <span class... | Bram28 | 256,001 | <p>This is a classic. The one I always use is 'All fruits and vegetable are nutritious'</p>
<p>So yes, given the 'and', you might be inclined to symbolize "All teachers and students are here" as:</p>
<p><span class="math-container">$$\forall x ((T(x) \land S(x)) \to H(x))$$</span></p>
<p>but that would mean that "An... |
3,298,545 | <p>We say that the function <span class="math-container">$f(x, y)$</span> is <em>differentiable</em> at the point <span class="math-container">$(a, b)$</span> if, and only if,</p>
<p><span class="math-container">$$\lim_{(h,k)\rightarrow(0,0)} \frac{f(a+h,b+k)-f(a,b)-hf_1(a,b)-kf_2(a,b)}{\sqrt{h^2 + k^2}}=0.$$</span></... | David C. Ullrich | 248,223 | <p>If <span class="math-container">$f_1$</span> and <span class="math-container">$f_2$</span> exist in a neighborhood of <span class="math-container">$(a,b)$</span> and are <em>continuous at</em> <span class="math-container">$(a,b)$</span> then it's not hard to show <span class="math-container">$f$</span> is differenti... |
3,755,509 | <p>Let <span class="math-container">$A \in \mathbb R^{n\times n}$</span> be an invertible block anti-diagonal matrix (with <span class="math-container">$d$</span> blocks), i.e.
<span class="math-container">$$
A = \begin{pmatrix} & & & A_1 \\ & & A_2 & \\ & \cdot^{\textstyle \cdot^{\textstyle... | enedil | 126,823 | <p>If a function attains a minimum at point <span class="math-container">$(x_0, y_0)$</span> then both partial derivatives at this point are zero.
In this case, derivative with respect to <span class="math-container">$x$</span> is
<span class="math-container">$$
f(x, y)=8 x -12
$$</span>
and the derivative with respect... |
2,572,000 | <p>Consider function $y = \sqrt{\frac{1-2x}{2x+3}}$. To find the domain of this function we first find the domain of denominator in fraction:<br>
$2x+3 \neq 0$<br>
$2x \neq -3$<br>
$x \neq - \frac{3}{2}$<br>
So, the domain of $x$ (for fraction to be valid) is $x \in \left(- \infty, - \frac{3}{2}\right) \cup \left(- \fr... | A. Goodier | 466,850 | <p>If $x<-\frac{3}{2}$, then $2x+3<0$ and $1-2x>0$, so $\frac{1-2x}{2x+3}<0$, which means you can't take the square root and get a real answer.</p>
|
2,572,000 | <p>Consider function $y = \sqrt{\frac{1-2x}{2x+3}}$. To find the domain of this function we first find the domain of denominator in fraction:<br>
$2x+3 \neq 0$<br>
$2x \neq -3$<br>
$x \neq - \frac{3}{2}$<br>
So, the domain of $x$ (for fraction to be valid) is $x \in \left(- \infty, - \frac{3}{2}\right) \cup \left(- \fr... | fleablood | 280,126 | <p>$\frac ab > 0$ means either 1) !!BOTH!! $a > 0; b>0$ <strong><em>OR</em></strong> 2) !!BOTH!! $a < 0; b< 0$.</p>
<p>And as $\frac ab$ existing means $b \ne 0$ then $\frac ab > 0$ means either 1) both $a \ge 0; b < 0$ or both $a \le 0; b < 0$.</p>
<p>So $\frac{1-2x}{2x+3} \ge 0$ means </p>
... |
4,524,456 | <p>So for this question:</p>
<p>"Derive the equation of the surface obtained when the parabola <span class="math-container">$x=y^2$</span> is rotated about the x-axis. Identify this surface."</p>
<p>I'm not sure why the answer is <span class="math-container">$x=y^2+z^2$</span> (a circular paraboloid). Is it j... | Philip Speegle | 1,085,225 | <p>Taking <span class="math-container">$\log_a$</span> of both sides, we get</p>
<p><span class="math-container">\begin{align*}
&\log_a(a^{\log_b n}) = \log_a(n^{\log_b a})\\
\implies & \log_b n=\log_b a\log_a n\\
\implies & \frac{\log_b n}{\log_b a} = \log_a n,
\end{align*}</span></p>
<p>so the problem is ... |
571,445 | <p>Let ${a_n}$ be a bounded sequence of real numbers. Prove that ${a_n}$ has a subsequence that converges to lim sup $a_n$.</p>
<p>Thanks!</p>
| user99680 | 99,680 | <p>$\limsup a_n$ is, by definition, the largest limit point of the sequence $a_n$. When you work with Real numbers (or, more generally, when you work with 1st-countable spaces), for every limit point, $a$ , there is a sequence of points that converge to $a$.</p>
|
571,445 | <p>Let ${a_n}$ be a bounded sequence of real numbers. Prove that ${a_n}$ has a subsequence that converges to lim sup $a_n$.</p>
<p>Thanks!</p>
| Clayton | 43,239 | <p><strong>Hint:</strong> What is the definition of $\limsup$? Try to use the definition and a sequence involving something like $1/n$ to construct such a subsequence.</p>
|
1,811,907 | <p>I tried finding the maxima of $f(x)=\frac{3}{4}-x-x^2$ by taking the derivative and so on and use the fact that $\displaystyle\int_{a}^{b}f(x)\,dx \leq M(b-a)$ where $M$ is the global maximum, but then the maximum value depends on the values of $a$ and $b$.</p>
| André Nicolas | 6,312 | <p>Hint: Over what interval is the integrand $\frac{3}{4}-x-x^2$ non-negative?</p>
<p>It may be useful to sketch the curve $y=\frac{3}{4}-x-x^2$ to see what's going on.</p>
|
1,811,907 | <p>I tried finding the maxima of $f(x)=\frac{3}{4}-x-x^2$ by taking the derivative and so on and use the fact that $\displaystyle\int_{a}^{b}f(x)\,dx \leq M(b-a)$ where $M$ is the global maximum, but then the maximum value depends on the values of $a$ and $b$.</p>
| user1892304 | 137,930 | <p>Evaluating the integrand and solving the optimisation problem (as in marwalix and juantheron's answers) is quite unnecessary. Simply notice that the <a href="http://www.wolframalpha.com/input/?i=%C2%BE+-+x+-+x%5E2" rel="nofollow">integrand is a parabola</a> which is positive on $[-3/2, 1 /2]$ and negative everywhere... |
2,482,449 | <p>In the process of constructing a highway across a certain region in which there are many hills and valleys. the engineer will be certain that</p>
<p>There is some level in between the elevations of the highest hill and the
lowest valley at which the surface of the highway can be laid using the
tops of the hills as ... | Rupsa | 264,612 | <p>The characteristic equation of $A$ is
$$(x-1)(x-2)(x-3)=0$$</p>
<p>Then use Cayley-Hamilton theorem.</p>
<p>Then you can easily find $A^{-1}$.</p>
<p>The answer is $6$</p>
|
2,297,706 | <p>Say you're given the following SVD:</p>
<p>$B=
\begin{bmatrix}
1 & -2 & -2 \\
-6 & 3 & -6 \\
\end{bmatrix}
=
\begin{bmatrix}
0 & d \\
1 & e\\
\end{bmatrix}
\begin{bmatrix}
9 & 0 & 0 \\
0 & f & g \\
\end{bmatrix}
\begin{bmatrix}
-2/3 & -1/3 &... | Andre Gomes | 259,584 | <p>As Dietrich has said, the Killing form <span class="math-container">$\kappa$</span> defines an positive-definite inner product on the Lie algebra <span class="math-container">$L$</span>, which is also <span class="math-container">$ad$</span>-invariant. A Lie algebra is a compact Lie algebra iff it admits an <span cl... |
77,143 | <p>I have plotted function $(1+\frac{1}{x})^{x}$ using Maple and got following graph:</p>
<p><img src="https://i.stack.imgur.com/F00hg.jpg" alt="enter image description here"></p>
<p>So it seems that function isn't defined on $(-1,0)$ interval , but if I take that $x=\frac{-1}{3}$ I can write:</p>
<p>$$y=(1+\frac{1}... | Valerio Capraro | 18,539 | <p>I guess here there is kind o a philosophical problem: for instance, if you take $x=-\frac{1}{2}$, you face the problem $\sqrt{-1}$. So one can say that there are <em>some</em> points in $(-1,0)$ where the function is defined and some other where is not. In general, it's quite difficult to say <em>a priori</em> which... |
77,143 | <p>I have plotted function $(1+\frac{1}{x})^{x}$ using Maple and got following graph:</p>
<p><img src="https://i.stack.imgur.com/F00hg.jpg" alt="enter image description here"></p>
<p>So it seems that function isn't defined on $(-1,0)$ interval , but if I take that $x=\frac{-1}{3}$ I can write:</p>
<p>$$y=(1+\frac{1}... | GEdgar | 442 | <p>Another way to solve this is to allow complex values, and choose the principal value in the usual way. Then your function IS defined and continuous on $(-1,0)$, but has non-real values everywhere there! </p>
<p>Here it is, real part in green, imaginary part in red:</p>
<p><img src="https://i.stack.imgur.com/40qf... |
77,143 | <p>I have plotted function $(1+\frac{1}{x})^{x}$ using Maple and got following graph:</p>
<p><img src="https://i.stack.imgur.com/F00hg.jpg" alt="enter image description here"></p>
<p>So it seems that function isn't defined on $(-1,0)$ interval , but if I take that $x=\frac{-1}{3}$ I can write:</p>
<p>$$y=(1+\frac{1}... | Tigran Hakobyan | 18,140 | <p>The expression $x^y$ is defined for $x>0$ if $y\not \in Z $ in the case of real numbers. Otherwise we could write, for example, $-\sqrt[3]{2}=\sqrt[3]{-2}=(-2)^{1/3}=(-2)^{2/6}=\sqrt[6]{(-2)^2}=\sqrt[6]{2^2}=\sqrt[3]{2}$. Contradiction.</p>
<p>Sincerely,</p>
<p>Tigran</p>
|
302,162 | <p>How can we determine if any pair of the following graphs are isomorphic to each other? Is there an efficient way to know for sure? The obvious things to check for (number of edges, vertices, degrees) aren't fruitful because all three graphs have the same of each. Any suggestion appreciated.<img src="https://i.stack.... | draks ... | 19,341 | <p>You can check if two graphs are <strong>not</strong> isomorphic by looking at the spectrum of the adjacence matrix...</p>
|
1,417,155 | <p>Let $X_0=1$, define $X_n$ inductively by declaring that $X_{n+1}$ is uniformly distributed over $(0,X_n)$.
Now I can't understand how does $X_{n}$ gets defined. If someone would just derive the distribution of $X_2$ that would be helpful. I saw this in a problem and I can't really start trying it. Thanks for any he... | confused_dragon | 265,428 | <p>The distribution of $X_1$ is $U(0,X_0) =U(0,1) $ which has density
\begin{equation}
f_{X_1}(x) = \begin{cases}
1 \quad \mbox{for } x \in (0,1) \\
0 \quad \mbox{otherwise}
\end{cases}
\end{equation}
Now, $X_2 | X_1 = x_1$ follows a $U(0, x_1) $
which has density
\begin{equation}
f_{X_2 | X_1 = x_1}(x) = \begin{... |
1,004,801 | <p>I have a question which I'm deeply confused about. I was trying to do some problems my professor gave us so we could practice for exam, one of them says:</p>
<p>Give a partition of ω in ω parts, everyone of them of cardinal ω.</p>
<p>I know that $ ω=\left \{ 1,2,3,...,n,n+1,.... \right \}$ , but I thought that... | André Nicolas | 6,312 | <p>For example, the part $P_n$ could consist of all numbers of the form $2^{n-1} q$, where $q$ ranges over the odd numbers. So $P_1$ consists of all odd natural numbers, $P_2$ consists of all natural numbers that are divisible by $2$ but not by $4$, $P_3$ consists of all natural numbers that are divisible by $4$ but no... |
1,460,913 | <p>Could anyone derive or explain why the formula $p ( x , y | z ) = p ( x | z ) p ( y | x , z )$ is true?</p>
| Antitheos | 261,163 | <p><strong>Take the defintion</strong> (for $P(B) \neq 0$):
$$P(A|B) := \frac{P(A\cap B)}{P(B)}$$
Just use that for every term and you will see that the equation holds.</p>
<p><strong>Edit</strong><br>
I'll make it clearer and use your notation:<br>
$$p(x,y|z)=\frac{p(x,y,z)}{p(z)}$$
and
$$p(y|x,z)=\left(\frac{p(y,x,z... |
3,060,767 | <p>I have an expression I am to simplify:</p>
<p><span class="math-container">$$\frac{15\sqrt[4]{125}}{\sqrt[4]{5}}$$</span></p>
<p>I arrived at <span class="math-container">$15\sqrt[4]{25}$</span>. My textbook tells me that the answer is in fact <span class="math-container">$15\sqrt{5}$</span>. Here is my thought pr... | Alekos Robotis | 252,284 | <p>It turns out that <span class="math-container">$\sqrt[4]{25}=\sqrt{5}$</span>. This is because <span class="math-container">$25=5^2$</span>, so that <span class="math-container">$\sqrt[4]{25}=\sqrt[4]{5^2}=(5^2)^{1/4}=5^{1/2}=\sqrt{5}.$</span> So, you are correct, as is the book.</p>
|
649,169 | <p>I know this has to do with Euclidean division, I just can't prove the => direction.
Any tips would be appreciated!</p>
| Community | -1 | <p>By <a href="http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity" rel="nofollow">Bézout's identity</a> we have</p>
<p>$$\gcd(a,n)=1\iff \exists \alpha,\beta\in \mathbb Z\;|\; \alpha a+\beta n=1=(\alpha+\beta)a+\beta(n-a)=1\\\iff\gcd(n-a,a)=1$$</p>
|
2,632,668 | <p>It is well known that gamma function is not defined at negative integers , but my question is to know how i take the value of $\binom{n}{p}$ for $p>n$ then is this make a sense or it is $0$ by convention ? </p>
| Jack D'Aurizio | 44,121 | <p>If you directly define $\binom{n}{p}$ as $\frac{\Gamma(n+1)}{\Gamma(p+1)\Gamma(n-p+1)}$ and recall that $\Gamma(x)$ ha simple poles at $0,-1,-2,\ldots$, it is no wonder that $\binom{n}{p}=0$ for $p,n\in\mathbb{N}$ and $p>n$. It is usually introduced by convention, but it is also the unique convention which agrees... |
2,347,121 | <p>In set theory I read that the sets are either finite or infinite. If they are infinite then there are also two categories countably infinite or uncountable. Natural numbers $\Bbb{N}$, integers $\Bbb{Z}$, rational numbers $\Bbb{Q}$ etc. are examples of countably infinite sets , whereas real numbers $\Bbb{R}$, irratio... | Riccardo.Alestra | 24,089 | <p>Energy levels of a free particle </p>
|
411,268 | <p>Please, somebody can help me with this problem?</p>
<p><hr>
Let $V$ and $W$ be two closed subspaces of a Hilbert $(H, \langle \cdot,\cdot\rangle)$, and let $P:H\rightarrow V$ and $Q:H\rightarrow W$ the orthogonal projectors respectively. Show that
$$\langle\ (Q-P)(x),\ x\ \rangle\ \geq\ 0,\ \forall\ x\in H\quad \mb... | André Nicolas | 6,312 | <p>Use the Binomial Theorem to write down the expansion of
$$\frac{(1+x)^n-1}{x}.$$
Then differentiate twice, and set $x=1$. You will get a very close relative of your sum. </p>
|
411,268 | <p>Please, somebody can help me with this problem?</p>
<p><hr>
Let $V$ and $W$ be two closed subspaces of a Hilbert $(H, \langle \cdot,\cdot\rangle)$, and let $P:H\rightarrow V$ and $Q:H\rightarrow W$ the orthogonal projectors respectively. Show that
$$\langle\ (Q-P)(x),\ x\ \rangle\ \geq\ 0,\ \forall\ x\in H\quad \mb... | Brian M. Scott | 12,042 | <p>Note that</p>
<p>$$\binom{n}k\binom{k-1}2=\frac12\binom{n}k(k-1)(k-2)\;,$$</p>
<p>where the $(k-1)(k-2)$ looks like the coefficient of the second derivative of $x^{k-1}$. That suggests looking at something like</p>
<p>$$g(x)=\sum_{k=3}^n\binom{n}kx^{k-1}$$</p>
<p>and differentiating twice with respect to $x$ to ... |
4,033,610 | <p>Maybe my problem is the English, but I cannot see why Gallian’s statement of the First Principle of Mathematical Induction is true. In the 7th edition of Contemporary Abstract Algebra (p13) he states:</p>
<blockquote>
<p>Let <span class="math-container">$S$</span> be a set of integers containing <span class="math-c... | David K | 139,123 | <p>I think you have overlooked the fact that if <span class="math-container">$a = 7,$</span> then <span class="math-container">$7$</span> is an integer greater than or equal to <span class="math-container">$a.$</span></p>
<p>If <span class="math-container">$a = 7$</span> and if <span class="math-container">$S$</span> h... |
4,033,610 | <p>Maybe my problem is the English, but I cannot see why Gallian’s statement of the First Principle of Mathematical Induction is true. In the 7th edition of Contemporary Abstract Algebra (p13) he states:</p>
<blockquote>
<p>Let <span class="math-container">$S$</span> be a set of integers containing <span class="math-c... | Community | -1 | <p>The statement is meant to be read as <span class="math-container">$$\forall S\subseteq \Bbb Z, ((a\in S\land \forall n\ge a, (n\in S\Rightarrow n+1\in S))\Rightarrow (\forall m\ge a, m\in S))$$</span></p>
<p>Id est, that if <span class="math-container">$S$</span> is a subset of <span class="math-container">$\Bbb Z$<... |
2,063,742 | <p>Hi I'm just curious as I prepare for a final, I cooked up this problem and wanted to know the answer. Suppose all $f_n>0$ and $f_n \leq g$ for all n and x with g integrable and $f_n \to f$. Then does that imply $\int f \leq \int g$?</p>
<p>I know by dominated convergence that $\int f_n \to \int f$ and $\int f_n ... | parsiad | 64,601 | <p>$$\int f = \lim_n \int f_n \leq \lim_n \int g = \int g$$</p>
|
482,061 | <p>Given the continuous Probability density function $f(x)=\begin{cases} 2x-4, & 2\le x\le3 \\ 0 ,& \text{else}\end{cases}$</p>
<p>Find the cumulative distribution function $F(x)$.</p>
<p>The formula is $F(x)=\int _{ -\infty }^{ x }{ f(x) } $</p>
<hr>
<p><h3>My Solution</h3> <br>
The first case is when $2... | blondy | 92,383 | <p>when x<2, it is $\int_{-\infty}^{x}0du=0$ </p>
<p>when x>3, it is obviously 1. You should research the properties of $F(x)$</p>
|
2,263,622 | <p>It is a known fact that (220, 284) is the smallest pair of amicable numbers.
That proves that I don't understand at least one part of finding amicable pairs...</p>
<p>Please explain to me where I fall down:</p>
<p>Suppose I have the numbers 2 and 3.
The proper divisors of 2 is 1.
The proper divisors of 3 is 1 t... | Especially Lime | 341,019 | <p>Amicable pairs aren't pairs where each number's sum of proper divisors is the same. An amicable pair is where each number's proper divisors sum to <em>the other number</em>. So the sum of proper divisors for 220 is 284, and vice versa.</p>
|
2,711,447 | <p>Let $n$ (unknown) real numbers $x_i$ be given. Suppose all Vieta's coefficient equations are positive, i.e.
$$
a_1 = \sum_{i=1}^n x_i > 0\\
a_2 =\sum_{(i>j)} x_i x_j > 0\\
a_3 =\sum_{(i>j>k)} x_i x_j x_k> 0\\
\dots \\
a_n =\prod_{i=1}^n x_i > 0
$$
where the sums go over all possible indicate... | quasi | 400,434 | <p>Based on your definition of $a_1,...,a_n$, each of $x_1,...,x_n$ is a root of
$$x^n -a_1 x^{n-1} + a_2 x^{n-2} + \cdots + (-1)^na_n=0$$
Note the alternating signs.
<p>
Now suppose $a_1,...,a_n > 0$.
<p>
Then none of the roots can be zero (since the constant term is nonzero).
<p>
Moreover, none of the roots can be... |
201,845 | <p>I'm trying to do a homework assignment on my graphing calculator app for my ipad (Quick graph+).</p>
<p>(<em>Edit by Willie</em>: The images are originally posted here <a href="https://imgur.com/a/XRj4g" rel="nofollow noreferrer">http://imgur.com/a/XRj4g</a> I've included them below.)</p>
<hr>
<p><img src="https:... | Community | -1 | <p>You used the <em>if</em> operator incorrectly. The format is </p>
<pre><code>if( condition, what to do if True, what to do if False)
</code></pre>
<p>But you put another condition in the place of "what to do if false".</p>
<p>Here's an example: I want a function which is equal to $3x$ between $x=1$ and $x=4$, is... |
201,845 | <p>I'm trying to do a homework assignment on my graphing calculator app for my ipad (Quick graph+).</p>
<p>(<em>Edit by Willie</em>: The images are originally posted here <a href="https://imgur.com/a/XRj4g" rel="nofollow noreferrer">http://imgur.com/a/XRj4g</a> I've included them below.)</p>
<hr>
<p><img src="https:... | Allan Henriques | 666,324 | <ol>
<li>Just use desmos which is more efficient and functional </li>
</ol>
<p><a href="https://support.desmos.com/hc/en-us/articles/203192385-Piecewise" rel="nofollow noreferrer">https://support.desmos.com/hc/en-us/articles/203192385-Piecewise</a></p>
<ol start="2">
<li>wtf are you doing the graphing calculator ques... |
1,259,383 | <p>I have a distribution with literally an infinite number of potential data points. I need the standard deviation. I generate about a hundred points and take the standard deviation of the points. This gives a hopefully good approximation of the true standard deviation, but it won't, of course, be exact. How do I e... | user5832178 | 311,046 | <p>If you want to find out the uncertainty or standard error (SE) in the standard deviation of a chosen sample, then you can simply use $SE(\sigma) = \frac{\sigma}{\sqrt{2N - 2}}$, where $N$ is the number of data points in your sample. </p>
<p>Hope that helps!</p>
|
2,431,287 | <p>Suppose there is a coin toss game where quarters are thrown onto a checkerboard. Management keeps all of the quarters; however, if a quarter lands entirely within one square of the checkerboard the management pays a dollar. Assume that the edge of each square is twice the diameter of the quarter, and that outcomes a... | Xander Henderson | 468,350 | <p>Let's start with one square. If the radius of the coin is 1 unit, then you want your coin to land so that its center is at least 1 unit from any edge. In the figure below, this region is shown in blue (these are all of the points that are at least 1 unit from an edge).</p>
<p><a href="https://i.stack.imgur.com/Vo... |
54,634 | <p>As you know, <em>Mathematica</em> V10 has just been released and I think a lot of updates will be released. However, I don't see any command in the help menu to update (which is present in most other software).</p>
<p>How would we know if an update is available or not?</p>
<p>I am using trial version of V10.</p>
... | Mr.Wizard | 121 | <p>I know link-only answers are discouraged, but in this case I think it is the best answer available:</p>
<ul>
<li><a href="http://www.wolfram.com/mathematica/quick-revision-history.html">http://www.wolfram.com/mathematica/quick-revision-history.html</a></li>
</ul>
|
2,669,617 | <p>The bilinear axiom is:</p>
<pre><code> <cu + dv,w> = c<u,w> + d<v,w>
<u,cv + dw> = c<u,v> + d<u,w>
</code></pre>
<p>Where c and d are scalars and u, v, and w are vectors.</p>
<p>Can this be extended to something like</p>
<pre><code> <cu + dv, ew + fx> = ?
</code></pre>
| zwim | 399,263 | <p>Stirling formula may be difficult to remember, but the simpler one below is extremely useful and allows you to solve most asymptotic results with factorials: </p>
<p><a href="https://math.stackexchange.com/questions/144176/factorial-inequality-problem-left-frac-n2-rightn-n-left-frac-n3-righ?noredirect=1&lq=1">F... |
874,859 | <p>let $R$ be a finite boolean ring.<br>
prove that $|R|=2^n$ for some $n\in\mathbb N$.
<br><br>
I know that $R$ is commutative and for every element $a\in R\space a+a=0$ and $a^2=a$</p>
| vociferous_rutabaga | 164,345 | <p>Cauchy's theorem -- given a group $G$ and a prime $p$ such that $p$ divides $|G|$, there exists an element of $G$ of order $p$. Considering $(R,+)$ as an abelian group, this should give you a proof. (You actually only need the abelian case of Cauchy's theorem for this problem, which is quite easy to prove).</p>
|
2,537,031 | <p>This is a question in Geometry by Hartshorne Exercise 3.3 </p>
<p>The goal is using Ruler and compass and a given triangle ABC and given a segment DE, construct a rectangle with content equal to the triangle ABC, and with one side equal to DE. Any propositions in Euclid book I-IV are usable but its likely going to... | Nij | 226,212 | <p>For completeness' sake, as with much mathematics, what we might assume in theory doesn't always hold in reality; a model can only go so far in telling us what can or cannot happen.</p>
<p>A4 paper has dimensions defined as 210mm by 297mm. These are both divisible by 3, so that the ratio is 70:99.</p>
<p>70 lengths... |
4,575,800 | <p>Could maybe someone help me here?</p>
<blockquote>
<p>For <span class="math-container">$p\in [1,2)$</span> I need to show that <span class="math-container">$l^p$</span> has empty interior in <span class="math-container">$l^2$</span>.</p>
</blockquote>
<p>I know that I need to show that there is no open ball <span cl... | Ryszard Szwarc | 715,896 | <p>If <span class="math-container">$Y$</span> is a proper linear subspace of a normed space <span class="math-container">$X,$</span> then the interior of <span class="math-container">$Y$</span> is empty. Indeed if <span class="math-container">$Y$</span> contains a ball in <span class="math-container">$X,$</span> then b... |
3,712,934 | <p>Let <span class="math-container">$f:\mathbb{R}^n\rightarrow\mathbb{R}$</span> be a continuously differentiable strongly convex function with a globally <span class="math-container">$L$</span>-Lipschitz continuous gradient. The normal gradient method for computing the unconstrained minimizer <span class="math-contain... | AsAnExerciseProve | 697,762 | <p>Unfortunately, it is not possible. What you are looking for is a function <span class="math-container">$g(\alpha,\mu, L)$</span> that is dependent only on the Lipschitz constant <span class="math-container">$L$</span>, the strong convexity constant <span class="math-container">$\mu$</span>, and the set size <span cl... |
2,510,712 | <blockquote>
<p>If A is an integral domain, we have seen that in $A[x]$, $\text{deg }a(x)b(x)=\text{deg }a(x)+\text{deg }b(x)$. Show that if $A$ is not an integral doamin we can find polynomails $a(x), b(x)$ such that $$\text{deg}\ a(x)b(x) <\text{ deg}\ a(x)+ \text{deg}\ b(x).$$</p>
</blockquote>
<p>So if we don... | Arthur | 15,500 | <p>Let $r,s\in A$ be non-zero such that $rs=0$, and consider, say, $a(x)=rx^3$ and $b(x)=sx^7+1$.</p>
|
1,710,589 | <p><a href="https://i.stack.imgur.com/UWVRL.png" rel="nofollow noreferrer">Question cropped from textbook</a>
(Apologies for the link- I don't have enough rep to post the actual image.) [Now pasted below. Ed.]</p>
<p><a href="https://i.stack.imgur.com/mW13Q.png" rel="nofollow noreferrer"><img src="https://i.stack.imgu... | BruceET | 221,800 | <p>You have received excellent advice in the Comments and an earlier Answer. Maybe it will help you to understand what they are suggesting and doing, if you see this done in a more familiar format. </p>
<p>I have chosen to use Minitab software because it has
clearly labeled output. To start I put amount (y) and time (... |
2,768,608 | <p>$$\lim_{n\to\infty}\sum_{i=1}^n \frac{1}{\sqrt i}$$</p>
<p>This question was asked in an entrance test for an undergraduate program in India. I want to know how to approach such questions.</p>
<p>I was thinking of finding the partial sum till n and applying limit concepts to get an answer, but couldn't find such a... | marty cohen | 13,079 | <p>For an elementary proof of divergence,
note that</p>
<p>$\begin{array}\\
\sqrt{i+1}-\sqrt{i}
&=(\sqrt{i+1}-\sqrt{i})\dfrac{\sqrt{i+1}+\sqrt{i}}{\sqrt{i+1}+\sqrt{i}}\\
&=\dfrac{1}{\sqrt{i+1}+\sqrt{i}}\\
&<\dfrac{1}{2\sqrt{i}}\\
\end{array}
$</p>
<p>so
$\dfrac{1}{\sqrt{i}}
\gt 2(\sqrt{i+1}-\sqrt{i})
$... |
2,768,608 | <p>$$\lim_{n\to\infty}\sum_{i=1}^n \frac{1}{\sqrt i}$$</p>
<p>This question was asked in an entrance test for an undergraduate program in India. I want to know how to approach such questions.</p>
<p>I was thinking of finding the partial sum till n and applying limit concepts to get an answer, but couldn't find such a... | Hayk | 558,859 | <p>Simply observe that
$$
\sum_{i=n}^{2n} \frac{1}{\sqrt{i}} \geq (2n - n) \frac{1}{\sqrt{2n}} = \frac{\sqrt{n}}{\sqrt{2}} \to \infty,
$$
hence the series ( your sum ) diverges (is infinite).</p>
|
3,576,365 | <h2>CONTEXT</h2>
<p>Currently I am reading a series of book by Martin Gardner, the one I am working on is <strong>"The colossal book of Mathematics"</strong>. Knowing that this man is hail as the greatest Math-Magician of the 20th Century, I am still surprised by his <em>rather magical tricks</em>.</p>
<p>The... | h-squared | 728,189 | <p>Since</p>
<p><span class="math-container">$2019\equiv 1$</span> (mod <span class="math-container">$2$</span>)</p>
<p>There are a total of <span class="math-container">$2019$</span> nodes and atleast <span class="math-container">$1$</span> connected component must have odd nodes. Hence the husband must pick that is... |
3,161,662 | <p>i want to make a python program that ask from the user a number and the program should sums up the first n squares as long as the sum is smaller (not smaller than or equal to) than the numb er that the user has choose before
Like this line </p>
<pre><code>1+2**2 +3**2+4**2 +....+n2 < the number that the user cho... | Henno Brandsma | 4,280 | <p>To make it more about maths: <span class="math-container">$$\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$$</span></p>
<p>which allows us to make an estimate for <span class="math-container">$n$</span> given the entered number <span class="math-container">$m$</span>: <span class="math-container">$n$</span> is about <sp... |
3,161,662 | <p>i want to make a python program that ask from the user a number and the program should sums up the first n squares as long as the sum is smaller (not smaller than or equal to) than the numb er that the user has choose before
Like this line </p>
<pre><code>1+2**2 +3**2+4**2 +....+n2 < the number that the user cho... | aman | 656,814 | <p><strong>Approach 1:</strong>
<strong>Direct approach</strong></p>
<p>n=int(input(" Enter a number"))</p>
<p>sum=0</p>
<p>sumlist=[]</p>
<p>i=1</p>
<p>while <span class="math-container">$sum<n$</span>:</p>
<pre><code>sum=sum+i**2
sumlist.append(sum)
i=i+1
</code></pre>
<p>print(sumlist[-2])</p>
<p><stro... |
3,161,662 | <p>i want to make a python program that ask from the user a number and the program should sums up the first n squares as long as the sum is smaller (not smaller than or equal to) than the numb er that the user has choose before
Like this line </p>
<pre><code>1+2**2 +3**2+4**2 +....+n2 < the number that the user cho... | Claude Leibovici | 82,404 | <p>Using the same idea as Henno Brandsma, consider that you look for the zero of function
<span class="math-container">$$f(n)=\frac{n(n+1)(2n+1)}{6}-m$$</span> and use Newton method with <span class="math-container">$n_0=\sqrt[3]{3m}$</span>. Since
<span class="math-container">$$f(n_0)>0\qquad \text{and} \qquad f''(... |
3,235,131 | <p>Is there a real <span class="math-container">$3 \times 3$</span> matrix <span class="math-container">$A$</span>, such that: <span class="math-container">$A^3 = I_3$</span> and has at most one zero entry? If so, how can I find it?</p>
| Ethan Bolker | 72,858 | <p>You would never write the set <span class="math-container">$\{2,3,3,4,4,4,5,5,6\}$</span> in real (mathematical) life but you might construct it inadvertently. For example,
<span class="math-container">$$
\{ n \quad | \quad n = x+y \text{ where } 1 \le x,y \le 3 \}.
$$</span>
That set has just five elements.</p>
|
2,841,983 | <p>I was doing this question from an RMO Practice Paper, and I have been unable to solve it.</p>
<blockquote>
<p>Let $P(x)$ be a polynomial of degree $2015$. $P(k)=2^k$ for $k=0,1,2,\dots,2015$. Find $P(2016)$</p>
</blockquote>
<p>My attempt:</p>
<p>Let $Q(x)=P(x)-2^x$.</p>
<p>Then its zeroes are $0,1,2,\dots,201... | Dr. N.Padmanabhan | 271,578 | <p>Your answer is wrong because $Q(x)$ is not a polynomial due to the presence of $2$ to the power $x$.</p>
|
3,240,162 | <p>so I got this question that I am stuck on:
So:
Consider the set <span class="math-container">$$S = \{0,\pm1,\pm2,\pm3,\pm4,\pm5,\pm6,\pm7,\pm8\}$$</span>
Consider the relation <span class="math-container">$R$</span> on <span class="math-container">$S$</span> defined by <span class="math-container">$(a,b)$</span> par... | ArsenBerk | 505,611 | <p>In this case, for symmetry and transitivity, we can actually say that symmetry and transitivity are <em>inherited from equality</em>. Because, if we assume <span class="math-container">$[a]_4 = [b]_4$</span>, then clearly <span class="math-container">$[b]_4 = [a]_4$</span> (this is symmetry of equality) so symmetry ... |
1,925,896 | <p>Let $G$ be a family of subsets of $[n] = \{1,2,...,n\}$ such that for every
$C \in G$, its complement $[n]\setminus C$ is also in $G$ and for every
$A,B \in G$, $A \cap B \in G \wedge A \cup B \in G.$ I want to figure out the
number of possible sizes of $G$ assuming that $G$ is non-empty.</p>
<p>To me, this feels l... | Bernard | 202,857 | <p>Working with the augmented matrix:
\begin{align*}
\left[\begin{array}{rrr|r}
-2&0&1&3\\3&1&-2&1\\-1&-1&1&-2
\end{array}\right]&\rightsquigarrow
\left[\begin{array}{rrr|r}
1&1&-1&2\\-2&0&1&3\\3&1&-2&1
\end{array}\right]\rightsquigarrow
\left[... |
948,854 | <p>There is a test to find out if a person has cancer.</p>
<p>1.) 95% of people who will take this test AND have cancer will get a positive ID that they have cancer.</p>
<p>2.) 2% of people who take this test who DON'T have cancer will also get a positive ID that they have cancer.</p>
<p>3.) The chances that a perso... | Anatoly | 90,997 | <p>Point $1$ describes the true positive rate. Point $2$ describes the false positive rate. Point $3$ describes the overall prevalence. </p>
<p>The simplest way to solve this problem is to build a $2\times 2$ table. For instance, let us assume that we are screening a population of $2500$ subjects (this number can be t... |
1,647,356 | <blockquote>
<ol start="2">
<li>A universe contains the three individuals $a,b$, and $c$. For these individuals, a predicate $Q(x,y)$ is defined, and its truth values are given by the following table:
\begin{array}{c|ccc}
x\backslash y&a&b&c\\\hline
a&T&F&T\\
b&F&T&F\\
c&F&... | egreg | 62,967 | <p>No substitutions:
$$
\int\left(\frac{1}{\sqrt{1-x^2}}-\frac{-x}{\sqrt{1-x^2}}\right)\,dx
=\arcsin x-\sqrt{1-x^2}+c
$$</p>
<p>You can also do that way; continue with $u=\sqrt{t}$, so $t=u^2$ and $dt=2u\,du$; so you get
$$
\int\frac{4u^2}{(u^2+1)^2}\,du=
\int 2u\cdot\frac{2u}{(u^2+1)^2}\,du
$$
Noticing that $2u$ is t... |
1,876,381 | <p>I need to prove that $p^{m/n}$, $m$ and $n$ naturals, $n > 1$, with $p$ prime and $\gcd(m, n)=1$, is irrational. It's suggested that this proof should be by contradiction or contraposition. </p>
| marty cohen | 13,079 | <p>Since you want to prove that
$p^{m/n}$
is irrational,
you have to assume that
it is rational
and derive a contradiction.</p>
<p>So,
assume that
$p^{m/n}
= \frac{a}{b}
$
where
$(a, b) = 1$.</p>
<p>Then
$p^{m}
= \frac{a^n}{b^n}
$
or
$b^np^{m}
= a^n
$.</p>
<p>$p$ must divide $a$,
so let
$a = p^kq$
where
$p \not\mid ... |
1,876,381 | <p>I need to prove that $p^{m/n}$, $m$ and $n$ naturals, $n > 1$, with $p$ prime and $\gcd(m, n)=1$, is irrational. It's suggested that this proof should be by contradiction or contraposition. </p>
| marshal craft | 167,793 | <p>Assume the negation of the statement and set out for a contradiction. That is $$\frac{k}{l}^{\frac{n}{m}}=p$$</p>
<p>But the greatest common denominator is one so</p>
<p>$$\frac{k^{\frac{n}{m}}}{l^{\frac{n}{m}}}$$</p>
<p>Which must be a multiple of $l^{\frac{n}{m}}$. So $p=j^{\frac{n}{m}}$.</p>
<p>$p=({g^{\frac{... |
1,433,580 | <p>I'm stuck in this question, how can I find the limit below?
$$\lim\limits_{n\to \infty}{2n^n\over (n+1)^{n+1}}$$</p>
| 5xum | 112,884 | <p><strong>Hint</strong></p>
<p>$$\frac{2n^n}{(n+1)^{n+1}} = \frac{2}{n+1}\cdot \frac{n^n}{(n+1)^n}$$</p>
|
3,132,736 | <p>Good evening,</p>
<p>I'm struggling with understanding a proof:</p>
<p>I know, that a solution of <span class="math-container">$y'=c \cdot y$</span> is <span class="math-container">$y=a \cdot e^{ct}$</span> and it's clear how to calculate this.</p>
<p>I want to proof, that all solutions of a function describing a... | Matthew Leingang | 2,785 | <blockquote>
<p>It's fine to me how they show it's zero. But where does <span class="math-container">$(\frac{g}{e^{ct}})'$</span> come from and why are they using it here, what does it mean?</p>
</blockquote>
<p>I'm not sure what it <span class="math-container">$g(t)/e^{ct}$</span> <em>means</em> in and of itself. ... |
351,170 | <p>The <em>genus <span class="math-container">$g$</span> handlebodies</em> are building blocks of <span class="math-container">$3$</span>-manifolds. They are constructed from <span class="math-container">$3$</span>-ball <span class="math-container">$B^3$</span> by adding <span class="math-container">$g$</span>-copies o... | Marco Golla | 13,119 | <p>All Brieskorn spheres are <em>small Seifert fibred spaces</em> (small SFS, in brief), i.e. they admit a fibration <span class="math-container">$S^1 \to \Sigma(p,q,r) \to S^2$</span> with three multiple fibres.
This is easier to see when <span class="math-container">$p,q,r$</span> are pairwise coprime: the fibration ... |
338,549 | <p><strong>Question:</strong> We select an element of $[100]$ at random. Let $A$ be the event that this integer is divisible by $3$ and let $B$ be that event that this integer is divisble by $7$. So are $A$ and $B$ independent? </p>
<p>I think they're no, which make sense because $21$ is divisible by both $3$ and $... | Community | -1 | <p>There are $33$ integers in $[100]$ that's divisible by $3$ and $14$ integers that's divisible by $7$. Therefore as computer nerd has said:</p>
<p>$P(A)=\frac{33}{100}$ </p>
<p>$P(B)=\frac{14}{100}$</p>
<p>On the other hand, there're $4$ integers in $[100]$ that's divisible by both $3$ and $7$ (which basically me... |
3,811,012 | <p>I wish to find <span class="math-container">$\displaystyle \lim_{n \rightarrow \infty}\frac{n+1}{\sqrt{n}}$</span>.</p>
<p>Here is what I did:</p>
<p><span class="math-container">$1.$</span> Rewrite <span class="math-container">$\frac{n+1}{\sqrt{n}}$</span> to <span class="math-container">$(n+1) \cdot \frac{1}{\sqrt... | Alessio K | 702,692 | <p>You've used the product rule for limits incorrectly as mentioned in the comments.</p>
<p>Moreover we have <span class="math-container">$\lim_{x\rightarrow\ 0}\sin(x)=0$</span>, but <span class="math-container">$\lim_{x\rightarrow\ 0}\frac{\sin(x)}{x}=1.$</span></p>
<p>Since <span class="math-container">$\frac{n+1}{\... |
1,826,221 | <p>I just read about Continuum Hypothesis which states that there is no set $S$ with the cardinality of $S$ is strictly larger than $\mathbb{N}$ and strictly smaller than $\mathbb{R}$.</p>
<p>I recall that in a math class we proved that the power set $P(\mathbb{N})$ have the same cardinality with $\mathbb{R}$. So I wo... | Noah Schweber | 28,111 | <p>This is a great question! Unfortunately, the answer is (at first glance) a bit unsatisfying:</p>
<blockquote>
<p>It is undecidable from the axioms of ZFC (= standard set theory) whether such an $S$ exists.</p>
</blockquote>
<p>Specifically, the statement that no such $S$ exists is called the <strong>generalized ... |
2,436,634 | <p>The limit is $$ \lim_{(x,y)\to(0,0)} \frac{x\sin(y)-y\sin(x)}{x^2 + y^2}$$</p>
<p>My calculations: I substitute $y=mx$</p>
<p>\begin{align}\lim_{x\to 0} \frac{x\sin(mx)-mx\sin(x)}{x^2 + (mx)^2} &= \lim_{x\to 0} \frac{x(\sin(mx)-m\sin(x)}{x^2(1 + m^2)}\\ &= \lim_{x\to 0} \frac{1}{1+m^2}\bigg[\frac{\sin(mx)}... | Peter | 82,961 | <p>No, we have $$\lim_{x\rightarrow 0} \frac{\sin(mx)}{x}=\lim_{x\rightarrow 0}m\cdot \frac{\sin(mx)}{mx}=m=\lim_{x\rightarrow 0}\frac{m\sin(x)}{x}$$ , so the limit you want to calculate is $0$ , no matter what $m$ is.</p>
|
239,187 | <p>I need help constructing a plane function, <span class="math-container">$z = f(x,y)$</span> that goes through three points, (05, 22, 20). (89, 0, 89) and (-1, -1, 10). I have tried to input it, but I dont know how.</p>
<p><a href="https://i.stack.imgur.com/UNg8A.png" rel="nofollow noreferrer"><img src="https://i.sta... | Suba Thomas | 5,998 | <pre><code>LinearModelFit[{{5, 22, 20}, {88, 0, 88}, {-1, -1, 10}}, {x, y}, {x, y}]
</code></pre>
<blockquote>
<p><a href="https://i.stack.imgur.com/iJGoS.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/iJGoS.png" alt="enter image description here" /></a></p>
</blockquote>
|
633,150 | <p>What kind of topological questions does algebraic topology answer where point set topology is not enough?</p>
<p>Phrased differently:</p>
<p>Where is the line (or maybe intersection) between point set topology and algebraic topology? What is the distinction between the problems each deals with?</p>
<p>I want to u... | Ronnie Brown | 28,586 | <p>In order to "understand the motivation behind introducing the algebraic machinery to topology" you need to go back to the history of the subject, and how it developed out of problems in complex analysis, as did general topology too. If you can get hold of it, I recommend "History of Topology", edited I M James, Noth... |
2,840,452 | <p>Find a closed form for the following recurrence relation:</p>
<p>$\begin{cases}
C_n=3C_{n-1}+n+2\\
C_0=0\\
\end{cases}$</p>
<p>So we start with a guess $D_n=C_n+an+b\iff C_n=D_n-an-b$</p>
<p>substituting to the equations gives</p>
<p>$D_n-an-b=3(D_{n-1}-a(n-1)-b)+n+2\iff \\ \iff D_n=3D_{n-1}+n(a-3a+1)+(3a-3b+b+2... | nonuser | 463,553 | <p>Put $C_n=D_n-\frac{1}{2}n-\frac{7}{4}$ in to $$C_n=3C_{n-1}+n+2$$ and solve it on $D_n$...</p>
|
3,324,375 | <p>What axiom or definition says that mathematical operations like +, -, /, and * operate on imaginary numbers?</p>
<p>In the beginning, when there were just reals, these operations were defined for them. Then, <em>i</em> was created, literally a number whose value is undefined, like e.g. one divided by zero is undefi... | Community | -1 | <p>The "necessary and sufficient" axioms to define the complex numbers are</p>
<p><span class="math-container">$$(a,b)+(a',b')=(a+a',b+b')$$</span></p>
<p><span class="math-container">$$(a,b)\cdot(a',b')=(aa'-bb',ab'+a'b).$$</span></p>
<p>(Subtraction and division can be defined as the inverses of addition and multi... |
3,324,375 | <p>What axiom or definition says that mathematical operations like +, -, /, and * operate on imaginary numbers?</p>
<p>In the beginning, when there were just reals, these operations were defined for them. Then, <em>i</em> was created, literally a number whose value is undefined, like e.g. one divided by zero is undefi... | Allawonder | 145,126 | <p>We can always use the usual rules for doing arithmetic with real numbers on complex numbers too, provided we always substitute <span class="math-container">$-1$</span> for <span class="math-container">$i^2$</span> whenever we encounter it. It follows that once we allow the imaginary unit, we have a consistent algebr... |
3,324,375 | <p>What axiom or definition says that mathematical operations like +, -, /, and * operate on imaginary numbers?</p>
<p>In the beginning, when there were just reals, these operations were defined for them. Then, <em>i</em> was created, literally a number whose value is undefined, like e.g. one divided by zero is undefi... | hunter | 108,129 | <p>At the risk of sounding like a postmodernist: all numbers are imaginary.</p>
<p>Long ago, someone abstracted: what is the thing that this collection of sheep has in common with the number of fingers on my left hand, and called that thing "five." No inconsistencies were introduced and there were great simplification... |
1,129,070 | <p>In a recent examination this question has been asked, which says:</p>
<p>$a^2+b^2+c^2 = 1$ , then $ab + bc + ca$ gives = ?</p>
<p>What should be the answer? I have tried the formula for $(a+b+c)^2$, but gets varying answer like $0$ or $0.25$, on assigning different values to variables.</p>
<p><em>How to approach... | avz2611 | 142,634 | <p>$(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)=1+2(ab+bc+ca)$
therefore $$(ab+bc+ca)=((a+b+c)^2-1)/2$$ so the value is not fixed</p>
|
1,129,070 | <p>In a recent examination this question has been asked, which says:</p>
<p>$a^2+b^2+c^2 = 1$ , then $ab + bc + ca$ gives = ?</p>
<p>What should be the answer? I have tried the formula for $(a+b+c)^2$, but gets varying answer like $0$ or $0.25$, on assigning different values to variables.</p>
<p><em>How to approach... | prashanth rao | 184,275 | <p>I believe your answers should be in terms of a,b,c</p>
<p>(a+b+c-1)(a+b+c+1)/2</p>
|
1,940,784 | <p>Express $\sqrt[3]{(7+5\sqrt{2})}$ in the form $x+y\sqrt{2}$ with $x$ and $y$ rational numbers.</p>
<p>I.e. Show that it is $1+\sqrt{2}$.</p>
| Nick | 359,825 | <p>$$(1+\sqrt{2})^3=(1+2\sqrt{2}+2)(1+\sqrt{2})=(3+2\sqrt{2})(1+\sqrt{2})=3+3\sqrt{2}+2\sqrt{2}+4$$
$$\therefore (1+\sqrt{2})^3=7+5\sqrt{2}$$
$$\Rightarrow 1+\sqrt{2}=\sqrt[3]{7+5\sqrt{2}}$$</p>
|
3,534,075 | <p>Let </p>
<p><span class="math-container">$$
Y = \{ x = (x_n) \in l^2: x_{2n} = 0, \, n \in \mathbb{N}\}
$$</span></p>
<blockquote>
<p>If <span class="math-container">$x = (x_n) \in Y^{\perp}$</span> then <span class="math-container">$$ x_{2n + 1} = \langle x, e_{2n +
1} \rangle = 0, \, \forall n \in \mathbb{N}... | Nitin Uniyal | 246,221 | <p><span class="math-container">$t^3-1$</span> is anihilating polynomial for <span class="math-container">$A$</span>. I assume that <span class="math-container">$A$</span> is complex matrix and <span class="math-container">$1$</span> is not an eigenvalue, so possible minimal polynomials will be any of the <span class="... |
187,145 | <p>It is asserted in this <a href="http://www.math.shimane-u.ac.jp/memoir/39/D.Buhagiar.pdf" rel="nofollow">article on "Connective Spaces"</a> that in topological spaces, if $A \subseteq B$ are both connected and $C$ is maximally connected in $B - A$, then $B - C$ is connected (page 4, proposition (a)). I have not been... | paulb | 151,536 | <p>In the last paragraph, it is asked whether $A\cup C$ is connected when $A\subseteq B$ are connected sets and $C$ is a component of $B\setminus A$. This is generally false; consider the planar space $B=\{c\}\cup\bigcup_{n=1}^{\infty}A_n$, where $c=\langle 1,0\rangle$, and $A_n=\{\langle x,\frac{x}{n}\rangle:0\leq x... |
3,548,378 | <p>Usually, in the integration, <span class="math-container">$\int_Xf(x) \, d\mu(x)$</span>, people assume by default that <span class="math-container">$X$</span> is infinite. </p>
<p>If <span class="math-container">$X$</span> is finite, then people usually write:
<span class="math-container">$$\sum_{x\in X}f(x)p(x)$... | Kavi Rama Murthy | 142,385 | <p>Any convergent sum <span class="math-container">$\sum_n a_n$</span>, finite or infinite, can be thought of as <span class="math-container">$\int f \, d\mu$</span> where <span class="math-container">$\mu $</span> is the counting measure on the set of natural numbers (with the power set as the sigma algebra) and <spa... |
1,817,609 | <p>Here is a list of other systems:</p>
<ul>
<li>Babylonian numerals</li>
<li>Egyptian numerals</li>
<li>Aegean numerals</li>
<li>May numerals</li>
<li>Chinese numerals</li>
</ul>
<p>These system are far older than the current system. How did it get to be known and used internationally by nearly every cultures these ... | Robert Israel | 8,508 | <p>For some history of the Hindu-Arabic system, see e.g. <a href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_numerals.html" rel="nofollow">this article</a> and <a href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Arabic_numerals.html" rel="nofollow">this article</a>.
This system was introduced to ... |
390,438 | <p>Suppose X,Y are sets with at least 2 elements. Show that
$X\cup Y\le X\times Y$</p>
<p>So my first thought was that cardinality $|X|\ge 2$ and the same for $|Y|\ge 2$ but by the inclusion-exclusion principle we have $|X\cup Y|=|X|+|Y|-|X\cap Y|$ but the problem does not say if they are disjoint or not.
If we assum... | Community | -1 | <p>HINT: max$(|X\cup Y|)$=$|X|+|Y| \leq |X|.|Y|$ f0r the given conditions.</p>
|
3,316,970 | <p>I have been having some difficulties with this question. </p>
<p>How to find the maximum without the help of a calculator or graphing device?</p>
| Feng | 624,428 | <p>Hint: <span class="math-container">$$d^2=\dfrac {(m+1)^2} {m^{2}+1}=1+2\frac{m}{m^2+1}.$$</span>
Note that for <span class="math-container">$m>0$</span>,
<span class="math-container">$$\frac{m}{m^2+1}=\frac{1}{m+\frac1m}\leq \frac{1}{2}$$</span>
since <span class="math-container">$m+\frac1m\geq2$</span> for <spa... |
3,316,970 | <p>I have been having some difficulties with this question. </p>
<p>How to find the maximum without the help of a calculator or graphing device?</p>
| nonuser | 463,553 | <p>By inequality beetwen quadratic and arithmetic mean we have <span class="math-container">$$\sqrt{m^2+1\over 2}\geq {|m|+1\over 2}$$</span> we have <span class="math-container">$${|m|+1\over \sqrt{m^2+1}} \leq \sqrt{2}$$</span></p>
<p>But <span class="math-container">$|m+1|\leq |m|+1$</span> so <span class="math-con... |
5,628 | <p>In <a href="http://www.ams.org/notices/200501/fea-grossman.pdf" rel="noreferrer">this 2005 Notices article</a>, Jerold Grossman tracks the proportion of papers in Math Reviews with 1, 2, 3, and >3 authors over time. His data set ends in 1999. I seem to recall reading that in 200k, for some value of k, the number o... | John D. Cook | 136 | <p>I imagine you'll find more collaborative papers in applied math than in pure math. In fact, I'd speculate that the number of authors increases steadily as you move along the continuum from most pure (e.g. category theory) to most applied (e.g. applied statistics).</p>
|
251,771 | <p>Durrett has a theorem that says: if $X_1, X_2, ..., X_n$ are random variables then $X_1 + X_2 + ... + X_n$ are also random variables.</p>
<p>My issue is how to show that $F((x_1, x_2, ... , x_n)) = \sum_{i = 1}^{n} x_i$ is measurable?</p>
<p>Durrett says ${x_1 + x_2 + ... + x_n < a}$ is an open set $(-\infty, a... | Elchanan Solomon | 647 | <p>We have</p>
<p>$$\frac{(\sqrt{3} + \sqrt{5})^3 - 14(\sqrt{3} + \sqrt{5})}{4} = \frac{18\sqrt{3} + 14\sqrt{5} - 14\sqrt{3} -14 \sqrt{5}}{4} = \frac{4\sqrt{3}}{4} = \sqrt{3}$$</p>
<p>So $\sqrt{3} \in \mathbb{Q}(\sqrt{3} + \sqrt{5})$, and hence so is $\sqrt{5}$.</p>
<hr>
<p>The general trick in these situations is ... |
251,771 | <p>Durrett has a theorem that says: if $X_1, X_2, ..., X_n$ are random variables then $X_1 + X_2 + ... + X_n$ are also random variables.</p>
<p>My issue is how to show that $F((x_1, x_2, ... , x_n)) = \sum_{i = 1}^{n} x_i$ is measurable?</p>
<p>Durrett says ${x_1 + x_2 + ... + x_n < a}$ is an open set $(-\infty, a... | Thomas Andrews | 7,933 | <p>Note that $$\frac{1}{\sqrt{5}+\sqrt{3}}=\frac{\sqrt{5}-\sqrt{3}}2$$
So $\sqrt{5}-\sqrt{3}\in\mathbb Q[\sqrt{5}+\sqrt{3}]$ and therefore $\sqrt{3},\sqrt 5\in\mathbb Q[\sqrt{5}+\sqrt{3}]$</p>
|
214,087 | <blockquote>
<p>Determine whether the given set of invertible n x n matrices with real number entries is a subgroup of <span class="math-container">$GL(n,\mathbb{R})$</span></p>
<p>The n x n matrices with determinant 2</p>
</blockquote>
<p>The key said<img src="https://i.stack.imgur.com/UGaTX.png" alt="enter image desc... | Gerry Myerson | 8,269 | <p>"Subgroup of $G$" means "subgroup under the operation inherited from $G$". The operation on the big group is multiplication, not addition. </p>
|
1,041,684 | <p>How can you determine which one of these numbers is bigger (without calculating):</p>
<p>$\left(\frac{1}{2}\right)^{\frac{1}{3}}$ , $\left(\frac{1}{3}\right)^{\frac{1}{2}}$</p>
| barak manos | 131,263 | <p>Raise them both to the power of $6$.</p>
<p>Since they are both positive, their order will be preserved and you will get:</p>
<p>$$\left({\dfrac{1}{2}}\right)^2=\frac{1}{4} > \frac{1}{27}=\left({\dfrac{1}{3}}\right)^3$$</p>
|
1,041,684 | <p>How can you determine which one of these numbers is bigger (without calculating):</p>
<p>$\left(\frac{1}{2}\right)^{\frac{1}{3}}$ , $\left(\frac{1}{3}\right)^{\frac{1}{2}}$</p>
| Ross Millikan | 1,827 | <p>If you don't want to depend on the "trick" of raising to the sixth power, you can compare the logs: $\frac 13 \log \frac 12=\frac {- \log 2}3$ and $\frac 12 \log \frac 13=\frac {-\log 3}2$ Now $\frac 12 \gt \frac 13$ and $\log 3 \gt \log 2$, so $\frac {\log 3}2 \gt \frac {\log 2}3, \frac {-\log 3}2 \lt \frac {-\log... |
1,041,684 | <p>How can you determine which one of these numbers is bigger (without calculating):</p>
<p>$\left(\frac{1}{2}\right)^{\frac{1}{3}}$ , $\left(\frac{1}{3}\right)^{\frac{1}{2}}$</p>
| Abraham Zhang | 112,045 | <p>$$\left(\frac{1}{2}\right)^{\frac{1}{3}}=\frac{\sqrt[3]1}{\sqrt[3]2}=\frac1{\sqrt[3]2}$$
$$\left(\frac{1}{3}\right)^{\frac{1}{2}}=\frac{\sqrt1}{\sqrt3}=\frac1{\sqrt3}$$
Now it is obvious that
$$\sqrt[3]2<\sqrt3$$
Thus
$$\frac1{\sqrt[3]2}>\frac1{\sqrt3}$$</p>
|
1,045,571 | <p>Find the value of vertical asymptotes.</p>
<p>$$ y = \pm2\sqrt\frac{x}{x-2}$$</p>
<p>There are two fuctions right? so,</p>
<p>$$ y = f_1(x)= +2\sqrt\frac{x}{x-2}$$
$$ y = f_2(x)= -2\sqrt\frac{x}{x-2}$$</p>
<p>I'm good until here.</p>
<p>Now, the book says
$$\lim_{x\to2^+}f_1(x) = \lim_{x\to2^+} 2\sqrt\frac{x}{x... | Milo Brandt | 174,927 | <p>One fairly obvious fact about any <em>reasonable</em> extension of $2^n$ to real numbers is that it ought to be strictly increasing; then, if $n+1>x>n$ so must $2^x>2^n$ so if $2^n\geq n+1$, then $$2^x>2^n\geq n+1>x$$ as desired. So you only need $2^n\geq n+1$, which holds for all naturals (and 0)</p>... |
1,045,571 | <p>Find the value of vertical asymptotes.</p>
<p>$$ y = \pm2\sqrt\frac{x}{x-2}$$</p>
<p>There are two fuctions right? so,</p>
<p>$$ y = f_1(x)= +2\sqrt\frac{x}{x-2}$$
$$ y = f_2(x)= -2\sqrt\frac{x}{x-2}$$</p>
<p>I'm good until here.</p>
<p>Now, the book says
$$\lim_{x\to2^+}f_1(x) = \lim_{x\to2^+} 2\sqrt\frac{x}{x... | marty cohen | 13,079 | <p>Some elementary analysis.</p>
<p>If
$f(x)
=2^x-x
$,
then
$f'(x)
=2^x \ln 2 - 1
$
and
$f''(x)
=2^x (\ln 2)^2
$.</p>
<p>From this,
$f''(x)
> 0
$
for all $x$.</p>
<p>$f'(x) = 0$
when
$0 = 2^x \ln 2 - 1$
or
$2^x = 1/\ln 2$
or
$x
=(\ln(1/\ln 2))/\ln 2
=(-\ln(\ln 2)/\ln 2
\approx 0.5287663729
= x_0
$.</p>
<p>Theref... |
240,669 | <p><strong>Bug introduced in version 12.0.0, and persisting through 13.2.0 on Windows. Doesn't reproduce on ARM Mac versions 13.0.0 and above.</strong></p>
<hr />
<p>Calculating the integral <span class="math-container">$$\int\limits_0^1 \frac{x^2\log(1-x^4)} {1+x^4}\,dx$$</span> symbolically</p>
<pre><code>Integrate[x... | Michael E2 | 4,999 | <p>One way:</p>
<pre><code>Integrate[x^2*Log[1 - x^4]/(1 + x^4), {x, 0, I, 1}]
N@%
(* -0.162858 - 4.44089*10^-16 I *)
</code></pre>
<p>Another way:</p>
<pre><code>Integrate[x^2*Log[1 - x^4]/(1 + x^4), {x, 0, 1/2, 1}]
N[%]
(* -0.162858 - 2.28333*10^-16 I *)
</code></pre>
|
3,449,799 | <blockquote>
<p>Find all positive integer solutions to <span class="math-container">$24x+18y=6420$</span>. </p>
</blockquote>
<p>Here's my work.</p>
<p>Simplifying the equation gives <span class="math-container">$4x+3y=1070$</span>. Note that this equation has solutions because <span class="math-container">$\gcd (4... | John Hughes | 114,036 | <p><span class="math-container">$x = 266; y = 2$</span> gives a solution, so evidently your current solution (to the most weirdly edited problem ever!) is wrong. </p>
|
393,122 | <p>I want to prove
$$\sum_{n=0}^{\infty} \frac{1}{n!}$$ is a converging series. So I want to compare it with $\sum_{n=0}^{\infty} \frac{1}{n^2}$. I want to do direct comparison test.</p>
<blockquote>
<p>How to prove $n^2 < n! $ ?</p>
</blockquote>
| Dominic Michaelis | 62,278 | <p>Most likely this is proved first time with induction. You could also take the root test, or see that
\[ \sum_{n=1}^\infty \frac{1}{n!} = e-1 \]</p>
|
1,508,340 | <p>Prove that if $G$ is a group and $a\in G$, then we have $\forall m,n\in\mathbb{Z} $ that
$$a^{m} a^{n} = a^{m+n}.$$</p>
<p>I've proved the case when $m,n>0$ but I'm stuck on how to prove the case when one or both are negative without assuming that $(a^{-1})^n = a^{-n} $. </p>
| Joseph R. Heavner | 90,146 | <p>You have to consider a few cases to prove this theorem. In that way, I find it a bit annoying. Perhaps there is a better way, but I do not know of one. </p>
<p><em>Edit: There actually is a straightforward way of shortening this; see A.P.'s comment.</em> </p>
<p>You have considered one such case ($m,n > 0$). He... |
471,017 | <p>I'm reading linear programming and I bumped into the following:</p>
<p><img src="https://i.stack.imgur.com/8VJgx.png" alt="enter image description here"></p>
<p>I'm having trouble getting grasp on the proof of proposition 2. Could someone perhaps explain it to me in other terms? For some reason the proof is unclea... | Qiaochu Yuan | 232 | <p>No. In fact, if $f$ is an irreducible polynomial of degree at least $2$ then there are infinitely many primes $p$ such that $f$ does not have a root $\bmod p$. </p>
<p>The argument is standard and goes as follows. A theorem of Dedekind asserts that if $f$ factors as a product $\prod f_i(x) \bmod p$ of irreducible p... |
906,103 | <h1>Context:</h1>
<p>I'm trying to <strong>algebraically</strong> prove that an <strong>open interval</strong> is an <strong>open set</strong>. If I sketch it, as suggested by @rschwieb in this <a href="https://math.stackexchange.com/a/301381/688539">answer</a>, then it seems quite obvious that this is indeed true. But... | Clive Newstead | 19,542 | <p>Note that $x \in V_{\varepsilon}(a)$ if and only if $x > a-\varepsilon \ge c$ and $x < a+\varepsilon \le d$.</p>
|
1,731,497 | <p>I bring this sample in order to ilustrate</p>
<p>$$x! = 2^x + 8$$</p>
<p>I know the answer is $x=4$ but I dunno how to prove it. I mean, if i put the number 4 by observation, tryal and error, I can get the results, but I dunno how to solve it isolating x like this:</p>
<p>(1) $x (x-1)! = 2^x +8$</p>
<p>(2) $x = ... | Ethan Alwaise | 221,420 | <p>I think trial and error is the way to go. You can guess that the solution should be small since factorial dominates exponentials.</p>
|
1,731,497 | <p>I bring this sample in order to ilustrate</p>
<p>$$x! = 2^x + 8$$</p>
<p>I know the answer is $x=4$ but I dunno how to prove it. I mean, if i put the number 4 by observation, tryal and error, I can get the results, but I dunno how to solve it isolating x like this:</p>
<p>(1) $x (x-1)! = 2^x +8$</p>
<p>(2) $x = ... | Will Jagy | 10,400 | <p>Nothing much changes, even if you ask
$$ x! = 2^y + 8. $$
As soon as $x \geq 6,$ we have $x!$ divisible by $16.$ As soon as $y \geq 4,$ we know $2^y + 8$ is not divisible by $16.$ Since $6! = 720,$ we would need $y \geq 9,$ guaranteed failure.</p>
<p>So $x \leq 5.$ </p>
<p>As pointed out by @marty the same reasoni... |
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