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 |
|---|---|---|---|---|
2,400,336 | <p>My first try was to set the whole expression equal to $a$ and square both sides. $$\sqrt{6-\sqrt{20}}=a \Longleftrightarrow a^2=6-\sqrt{20}=6-\sqrt{4\cdot5}=6-2\sqrt{5}.$$</p>
<p>Multiplying by conjugate I get $$a^2=\frac{(6-2\sqrt{5})(6+2\sqrt{5})}{6+2\sqrt{5}}=\frac{16}{2+\sqrt{5}}.$$</p>
<p>But I still end up w... | N. F. Taussig | 173,070 | <p>Let $\sqrt{6 - \sqrt{20}} = \sqrt{a} - \sqrt{b}$, where $a$ and $b$ are rational numbers. Squaring both sides of the equation
$$\sqrt{a} - \sqrt{b} = \sqrt{6 - \sqrt{20}}$$
yields
\begin{align*}
a - 2\sqrt{ab} + b & = 6 - \sqrt{20}\\
a - 2\sqrt{ab} + b & = 6 - 2\sqrt{5}
\end{align*}
Matching rational and i... |
144,375 | <p>We know that every $2\times 2$ matrix in $PGL(2, \mathbb{Z})$ of order $3$ is conjugate to the matrix $$ \left( \begin{array}{cc} 1 & -1 \\ 1 & 0 \end{array} \right) $$. </p>
<p>I am interested in finding out to what extent this holds for $3\times 3$ integer invertible matrices.</p>
<p>In other words how m... | tj_ | 18,571 | <p>The finite subgroups of $GL_3(\mathbb{Z})$ are known in the literature: </p>
<p>$\qquad$ Tahara: On the finite subgroups of $GL(3,\mathbb{Z})$. Nagoya Math. J. 41(1971), 169-209. </p>
<p>In particular Proposition 3 states that there are exactly two non-conjugate subgroups of order three. Representants are
$$U_1=... |
3,210,791 | <p>The question is as follows:</p>
<p>Consider the following partial differntial equation (PDE)</p>
<p><span class="math-container">$2\frac{\partial^2u}{\partial x^2}+2\frac{\partial^2u}{\partial y^2} = u$</span></p>
<p>where <span class="math-container">$u=u(x,y)$</span> is the unknown function.</p>
<p>Define the ... | PrincessEev | 597,568 | <p>Same as you would an ordinary differential equation.</p>
<p>For each of <span class="math-container">$u_1,u_2,u_3$</span>, do the following for this equation:</p>
<ul>
<li><p>Calculate the relevant partial derivatives: that is, find <span class="math-container">$(u_1)_{xx},(u_1)_{yy},(u_2)_{xx},(u_2)_{yy},(u_3)_{x... |
163,917 | <p>Suppose $B_{\epsilon}$ are closed subsets of a compact space and $B_{\epsilon} \supset B_{\epsilon'} \quad \forall \epsilon > \epsilon'$. Furthermore, $B_0 = \bigcap_{\epsilon>0} B_{\epsilon}$. For a continuous function $f$ can we conclude that
$$f(B_0) = \bigcap_{\epsilon>0} f(B_{\epsilon})?$$</p>
<p>I ... | Asaf Karagila | 622 | <p>Suppose that $f\colon X\to Y$, then we assume that $X$ is compact and $f$ is continuous, however we need to assume that $Y$ is $T_1$ or that $X$ is Hausdorff.</p>
<p>First observe that since $B_\epsilon$ is closed it is compact too. Second we observe that the continuous image of a compact set is compact. (These two... |
266,285 | <p>I am trying to place my plot legend inside the graph. Currently it sits outside the plot and gets covered up when I place another plot next to it.</p>
<p><a href="https://i.stack.imgur.com/A0mK5.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/A0mK5.png" alt="enter image description here" /></a></... | Steven Buehler | 85,935 | <p>Not pretty, but I got it to work by creating a new Table with the required data points:</p>
<pre><code>url = StringTemplate[
"https://api.foursquare.com/v2/users/self/checkins?afterTimestamp=\
`a`&oauth_token=`b`&limit=250&v=20220406"][<|"a" -> startTime,
"b" ... |
266,285 | <p>I am trying to place my plot legend inside the graph. Currently it sits outside the plot and gets covered up when I place another plot next to it.</p>
<p><a href="https://i.stack.imgur.com/A0mK5.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/A0mK5.png" alt="enter image description here" /></a></... | Edmund | 19,542 | <p>Most of the functionality you seek can be found in the <a href="https://resources.wolframcloud.com/FunctionRepository/" rel="nofollow noreferrer">Wolfram Function Repository</a> functions <a href="https://resources.wolframcloud.com/FunctionRepository/resources/ToAssociations" rel="nofollow noreferrer"><code>Resource... |
3,401,260 | <p>I am supposed to find the intersection of :
<span class="math-container">$$\begin{cases} 2^{x}=y \\ 31x+8y-94=0 \end{cases}$$</span>
When I substitute the first equation into the second one:
<span class="math-container">$$\frac{94-31x}{8}=2^{x}$$</span> and I do not know how to continue. </p>
<p>Can anyone help me... | MafPrivate | 695,001 | <p><span class="math-container">$$\dfrac{94-31x}{8}=2^x \\ 94-31x=8(2^x) \\ 8(2^x)+31x-94=0$$</span>
As <span class="math-container">$8(2^x)+31x-94$</span> is strictly increasing, there is only one answer. Then we can try to find the answer by testing.</p>
<p>When <span class="math-container">$x=0$</span>, <span class... |
40,241 | <p>Let $N$ be a prime number. Let $J(N)$ be the jacobian of $X_\mu(N)$, the moduli space of elliptic curves with $E[N]$ symplectically isomorphic to $Z/NZ \times \mu_N$. Over complex numbers we get that J(N) is isogeneous to product of bunch of irreducible Abelian varieties. Is there a way of describing these Abelian v... | François Brunault | 6,506 | <p>The decomposition of $J(11)$ was known (at least over $\mathbf{C}$) to Hecke. It turns out that the Jacobian of the compactification of $\Gamma(11) \backslash \mathfrak{h}$ is isogenous to a product of 26 elliptic curves. All this is very well explained in the following article :</p>
<p>MR0463118 (57 #3079) Ligoza... |
2,247,798 | <p><strong>Question:</strong> If $\alpha$ is an angle in a triangle and $\tan{\alpha}=-2$, then one of the following is true:</p>
<p>a) $0<\alpha < \frac{\pi}{2}$</p>
<p>b) $\frac{\pi}{2}<\alpha < \pi$</p>
<p>c) Can't be decided.</p>
<p>d) There exist no such angle $\alpha$.</p>
<p>My reasoning was tha... | DonAntonio | 31,254 | <p>Without referring to the diagram, which is very misleading, the answer is (b), since:</p>
<p>(1) the angle must be between $\;0\;$ and $\;\pi\;$ radians as it belongs to a <em>triangle</em> (not written "a straight triangle" !), and</p>
<p>(2) It must such that the signs of sine and cosine as opposite, since </p>
... |
512,591 | <p>It is always confusing to prove with $\not\equiv$. Should I try contrapositive?</p>
| Shobhit | 79,894 | <p><strong>HINT:</strong></p>
<p>if $a$ is not divisible by $3$, then $a=3n+1$ or $a=3n+2$.</p>
<p>Calculate $a^2$, and take $\text{mod}3$ to conclude.</p>
|
4,090,970 | <p>Let <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> be independent exponential random random variables with common parameter <span class="math-container">$\lambda$</span> and let <span class="math-container">$Z = X + Y$</span>. Find <span class="math-container">$f_Z(z)$</span>.</p... | Community | -1 | <p>The mistake is in the upper bound in the very last integral. It should be <span class="math-container">$z$</span> because <span class="math-container">$x+y\leq z$</span> and since <span class="math-container">$Y$</span> is exponentially distributed it is nonnegative.
Then we have</p>
<p><span class="math-container">... |
609,845 | <blockquote>
<p>$5$ Integers are paired in all possible ways and each pair of integers
is added. The $10$ sums obtained are $1,4,7,5,8,9,11,14,15,10$. What are the
$5$ integers?</p>
</blockquote>
<p>This is what I got so far:</p>
<p>To get all possible pairs, each integer must be paired with the other $4$ integ... | Ross Millikan | 1,827 | <p>Adding your sums gives $84$. As each number participates in four pairs, the sum of the integers is $21$. The top two sum to $15$ and the bottom two to $1$, so the middle one is $5$. The top two are then $7,8$ or $6,9$, but $7,8$ would produce $12$, which is absent, so the top three are $5,6,9$, which account for ... |
1,042,212 | <p>If $\mathbb{Z}_p \leq K$ an algebraic extension, then $K$ has the identity $$\forall a \in K, \exists b \in K \text{ with } a=b^p$$</p>
<p>The proof is the following:</p>
<p>Let $a \in K$.</p>
<p>We take $\mathbb{Z}_p \leq \mathbb{Z}_p(a)$, $a$ algebraic over $\mathbb{Z}_p$.</p>
<p>So, $[\mathbb{Z}_p(a) : \mathb... | spatially | 124,358 | <p>I think @PhoemueX's answer will lead you to a general situation. It works, of course. But here let me provide you a quick and insight example on $R^1$.</p>
<p>Take $I=(0,1)$ and define $u_n$ in following way:</p>
<p>For each fixed $n$, we partition $I$ into $n$'s small subinterval with length $1/n$. Let's name tho... |
2,828,472 | <p>This question is regarding property of little o notation given in Apostol Calculus. The property is given on page 288 and stated as:</p>
<blockquote>
<p>Theorem 7.8 (c) As $x\to a$ we have $f(x)\cdot o (g(x)) = o(f(x)g(x))$.</p>
</blockquote>
<p>Here say $h(x) = o(g(x))$ then we have $f(x) \lim_{x\to a} \frac{h(... | user | 505,767 | <p>By definition</p>
<p>$$f(x) o (g(x)) = f(x)\cdot \omega(x)g(x)$$ </p>
<p>with</p>
<p>$$\omega(x)\to 0 \quad x\to a$$</p>
<p>therefore</p>
<p>$$f(x) o (g(x)) = \omega(x)\cdot f(x)g(x)=o(f(x)g(x))$$</p>
|
1,116,022 | <p>I've always had this doubt.
It's perfectly reasonable to say that, for example, 9 is bigger than 2.</p>
<p>But does it ever make sense to compare a real number and a complex/imaginary one?</p>
<p>For example, could one say that $5+2i> 3$ because the real part of $5+2i
$ is bigger than the real part of $3$? Or i... | Timeless | 546,112 | <p>I think that you should look at > or < relations like this:
a. they categorise unequal items
b. they represent some order e.g. 1<2<3<4 of the set of items
A complex number is always a pair of numbers a real number is one number. So it makes no sense to compare a pair against a single item like to ask i... |
2,651,394 | <p>I am attempting to create a function in Matlab which turns all matrix elements in a matrix to '0' if the element is not symmetrical. However, the element appears to not be reassigning.</p>
<pre><code>function [output_ting] = maker(a)
[i,j] = size(a);
if i ~= j
disp('improper input!')
else
end
c = 1;
b = a.';
... | shere | 524,467 | <p>Instead of dividing the top part by $\sin x$, take $\sin h$ out. So you have $$\ln(\cot h+\cot x)\over h$$
Now use L'Hôpital</p>
|
2,909,244 | <p>I have a homework, about calculate the limit of a series:
$$
\lim\limits_{n \to +\infty} \dfrac{\sqrt[n] {n^3} + \sqrt[n] {7}}{3\sqrt[n]{n^2} + \sqrt[n]{3n}}
$$
Solution is $\frac{1}{2}$. I am trying use the unequality:
$$
\dfrac{\sqrt[n] {n^3} }{3\sqrt[n]{n^2} + \sqrt[n]{3n}} \le \dfrac{\sqrt[n] {n^3} + \sqrt[n] {7... | Good Morning Captain | 220,841 | <p>It is true that $\lim_{n\to\infty} \sqrt[n]{n} \to 1$ and $\lim_{n\to\infty} \sqrt[n]{i} \to 1$ for some natural number $i$.</p>
<p>Rearranging, we see that $\sqrt[n]{n^2} = \sqrt[n]{n}\cdot\sqrt[n]{n}$</p>
<p>Thus $\lim_{n\to\infty} \sqrt[n]{n^2} = 1$ </p>
<p>Applying similar rules and the algebra of limits we s... |
1,722,226 | <p>How many solutions are there to the inequality
$x_1 + x_2 + x_3 ≤ 11$,
where $x_1, x_2$ and $x_3$ are non-negative integers? [Hint: Introduce
an auxiliary variable $x_4$ such that $x_1 + x_2 + x_3$ +
$x_4$ = 11.]</p>
<p>Would my reasoning be correct if I let $x_4 = x_1 + x_2 + x_3, x_4 = 11$</p>
<p>Then proceeded... | robjohn | 13,854 | <p>The "brute force" approach is given in ashleyde's answer: count the solutions to
$$
x_1+x_2+x_3=11-x_4\tag{1}
$$
for $x_4\in\{0,1,2,\dots,11\}$; that is,
$$
\sum_{x_4=0}^{11}\binom{13-x_4}{2}\tag{2}
$$
$(2)$ counts all solutions to
$$
x_1+x_2+x_3\le11\tag{3}
$$
Another way of looking at $(1)$ is to count the solutio... |
549,299 | <p>I'm searching(I searched this site first) for example of fields $F \subseteq K \subseteq L$ where $L/K$ and $K/F$ are normal but $L/F$ is not normal. Presenting some fields just for $F$ or $L$, instead of all three fields will help me too. Thanks for your attention.</p>
| Community | -1 | <p>Even before going for normal extensions, It would be useful to see if you can find a chain of subgroups $G_1\subset G_2 \subset G_3$ such that :</p>
<p>$G_3$ is normal over $G_2$ and $G_2$ is normal over $G_1$ but $G_3$ is not normal over $G_1$.</p>
<p>make sure your $G_3$ is galois group of some known polynomial.... |
4,417,901 | <p>In the first chapter of "Differential Equations, Dynamical Systems and an Introduction to Chaos" by Hirch, Smale and Devaney, the authors mention the first-order equation <span class="math-container">$x'(t)=ax(t)$</span> and assert that the only general solution to it is <span class="math-container">$x(t)=... | Quanto | 686,284 | <p>Differentiate under the integral sign <span class="math-container">$n$</span> times as follows
<span class="math-container">$$\int_0^\infty x^n e^{-\lambda x} dx=(-1)^n\frac{d^n}{d\lambda^n} \int_0^\infty e^{-\lambda x} dx=
(-1)^n\frac{d^n}{d\lambda^n}\frac1\lambda=\frac{n!}{\lambda^{n+1}}
$$</span></p>
|
135,159 | <p>Slow morning.
Can someone help me figure it out? I have a feeling it is trivially easy and not worthy of a thread.
$$
3^{n+1} + 3^n = 4\cdot3^n
$$</p>
<p>Thanks.</p>
| anon | 11,763 | <p><strong>Hint</strong>: Write $3^{n+1}$ as $3\cdot 3^n$, then factor $3^n$ out of the sum.</p>
<p>(I assume the question is about $3^{n+1}+3^n$.)</p>
|
33,582 | <p>My code finding <a href="http://en.wikipedia.org/wiki/Narcissistic_number">Narcissistic numbers</a> is not that slow, but it's not in functional style and lacks flexibility: if $n \neq 7$, I have to rewrite my code. Could you give some good advice?</p>
<pre><code>nar = Compile[{$},
Do[
With[{
n = 1000... | chyanog | 2,090 | <p>A fast method：</p>
<pre><code>nar[n_] := Pick[#, #~BitXor~Range[10^(n - 1), 10^n - 1], 0] &@
Flatten[Outer[Plus, ##] & @@ Array[Range[Boole[# == 1], 9]^n &, n]]
nar /@ Range[7] // AbsoluteTiming
</code></pre>
<blockquote>
<p>{0.314020, {{1, 2, 3, 4, 5, 6, 7, 8, 9}, {}, {153, 370, 371,
407}, ... |
4,159,341 | <p>There are <span class="math-container">$4$</span> coins in a box. One is a two-headed coin, there are <span class="math-container">$2$</span> fair coins, and the fourth is a biased coin that comes up <span class="math-container">$H$</span> (heads) with probability <span class="math-container">$3/4$</span>.</p>
<p>If... | Thomas Andrews | 7,933 | <p>Let the dice be <span class="math-container">$\{D,U,F_1,F_2\}$</span> where <span class="math-container">$D$</span> is double-sided, <span class="math-container">$U$</span> is the unfair die, and <span class="math-container">$F_i$</span> are the fair dice.</p>
<p>Then there are <span class="math-container">$6$</span... |
626,958 | <p>I know that $E[X|Y]=E[X]$ if $X$ is independent of $Y$. I recently was made aware that it is true if only $\text{Cov}(X,Y)=0$. Would someone kindly either give a hint if it's easy, show me a reference or even a full proof if it's short? Either will work I think :) </p>
<p>Thanks.</p>
<p>Edit: Thanks for the great... | Alecos Papadopoulos | 87,400 | <p><span class="math-container">$\operatorname{Cov}(X,Y)$</span> can be <span class="math-container">$0$</span> and the variables can still be dependent (exhibiting a purely non-linear dependence), and so <span class="math-container">$E(X\mid Y) \neq E(X)$</span>. In narrower terms, "mean independence" implies zero cov... |
4,014,554 | <p>A simple heuristic of the first million primes shows that no prime number can be bigger than the sum of adding the previous twin primes.</p>
<p>Massive update:
@mathlove made a comment that leaves me completely embarrassed. <span class="math-container">$13 > 7 + 5$</span> I don’t know how I missed it and I deeply... | rtybase | 22,583 | <p><strong>Futher to my comments ...</strong></p>
<p>For any <span class="math-container">$3$</span> consecutive primes (<strong>regardless</strong> of if they contain twin primes or not) <span class="math-container">$p_{n},p_{n+1},p_{n+2}$</span>, <a href="https://math.stackexchange.com/questions/413163/do-3-consecuti... |
644,057 | <p>I am having trouble with this problem:</p>
<p>Let $a_n$ be sequence of positive terms with $$\frac{a_{n+1}}{a_n}\lt \frac{n^2}{(n+1)^2}.$$
Then is the series $\sum a_n$ convergent?</p>
<p>Thanks for any help.</p>
| GEdgar | 442 | <p>Or <a href="http://mathworld.wolfram.com/GausssTest.html" rel="nofollow">Gauss's Test</a></p>
<p>$$
\frac{a_{n}}{a_{n+1}} > \frac{(n+1)^2}{n^2} = 1 + \frac{\color{red}2}{n}+\frac{1}{n^2},
$$
and $\color{red}2>1$ so we have convergence.</p>
|
82,770 | <p>I know of two places where $K_{*}(\mathbb{Z}\pi_{1}(X))$ (the algebraic $K$-theory of the group ring of the fundamental group) makes an appearance in algebraic topology. </p>
<p>The first is the Wall finiteness obstruction. We say that a space $X$ is finitely dominated if $id_{X}$ is homotopic to a map $X \righta... | Tom Goodwillie | 6,666 | <p>To add to Tim Porter's excellent answer: </p>
<p>The story of what we now call $K_1$ of rings begins with Whitehead's work on simple homotopy equivalence, which uses what we now call the Whitehead group, a quotient of $K_1$ of the group ring of the fundamental group of a space.</p>
<p>On the other hand, the story ... |
376,600 | <p>$$\lim_{n\to\infty} \int_{-\infty}^{\infty} \frac{1}{(1+x^2)^n}\,dx $$</p>
<p>Mathematica tells me the answer is 0, but how can I go about actually proving it mathematically?</p>
| Community | -1 | <p>You can also compute $\displaystyle \int_{-\infty}^{\infty}\dfrac{dx}{(1+x^2)^n}$ exactly and argue out what the limit is.
Let $x = \tan(t)$. We then get that
$$I_n = \int_{-\infty}^{\infty}\dfrac{dx}{(1+x^2)^n} = \int_{-\pi/2}^{\pi/2} \dfrac{\sec^2(t) dt}{\sec^{2n}(t)} = \int_{-\pi/2}^{\pi/2} \cos^{2n-2}(t) dt = \p... |
595,280 | <p>Let $V$ be a vector space over $\Bbb F$, and let $x\not=0,y\not=0 $ be two elements in $V$. </p>
<p>I want to show that $x\otimes_{_F} y=y\otimes_{_F} x$ iff $x=ay$ where $a\in \Bbb F$.</p>
<p>I know the second direction, so only want to see the first direction (If case).</p>
| rschwieb | 29,335 | <p>So for such $x,y\neq 0$, you want to show $x\otimes y=y\otimes x$ in $V\otimes_F V$ <strong>if</strong> $x=ay$ for some $a\in F$.</p>
<p>Then $x\otimes y=ay\otimes y=y\otimes ay=y\otimes x$.</p>
<p>Since this seems by far to be the easier half of the problem, I am beginning to wonder if you meant to ask about the ... |
2,929,804 | <p>My attempt</p>
<p>First I wanted to show <span class="math-container">$<3,x^2+1>$</span> is maximal
So, I supposed another maximal <span class="math-container">$A$</span> which contain <span class="math-container">$<3,x^2+1>$</span> properly, and choose element <span class="math-container">$a$</span> i... | Mathematician 42 | 155,917 | <p>Let <span class="math-container">$P(x)=\sum_{i=0}^na_ix^i\in \mathbb{Z}[x]$</span>. By the division algorithm, we can write <span class="math-container">$$P(x)=Q(x)(x^2+1)+r(x)$$</span>
where <span class="math-container">$Q(x),r(x)\in \mathbb{Z}[x]$</span> are defined uniquely such that <span class="math-container">... |
2,436,167 | <p>I appear to be misunderstanding a basic probability concept. The question is: you flip four coins. At least two are tails. What is the probability that exactly three are tails? </p>
<p>I know the answer isn't 1/2, but I don't know why that's so. Isn't the probability of just getting 1 tail in the remaining two coin... | Parcly Taxel | 357,390 | <p>"At least two are tails" does not specify <em>which</em> coins are tails – or heads for that matter. The $\frac12$ answer <em>assumes</em> that two specific coins are tails first, but either or both may be heads.</p>
|
129,912 | <p>Does
$$\frac{1}{N^2}\sum _{d=1}^N \log d \sum _{n=1}^{N/d} \frac{\phi(n)}{\log (dn)},$$</p>
<p>converges or not when $N$ goes to infinity? </p>
| Greg Martin | 5,091 | <p>The expression converges to $0$, even when $\phi(n)$ is replaced by the larger $n$. The contribution from $n\le\sqrt N/d$ can be given by ignoring the logarithm in the denominator:
\begin{align*}
\frac1{N^2} \sum_{d=1}^N \log d \sum_{n=1}^{\sqrt N/d} \frac{\phi(n)}{\log dn} &\le \frac1{N^2} \sum_{d=1}^N \log d \... |
129,912 | <p>Does
$$\frac{1}{N^2}\sum _{d=1}^N \log d \sum _{n=1}^{N/d} \frac{\phi(n)}{\log (dn)},$$</p>
<p>converges or not when $N$ goes to infinity? </p>
| Eric Naslund | 12,176 | <blockquote>
<p>For a precise asymptotic, we have that $$\sum_{d=1}^{N}\log d\sum_{n=1}^{N/d}\frac{\phi(n)}{\log(nd)}=-\frac{\zeta'(2)}{\zeta(2)}\text{li}(N^2)+O\left(N\right),$$ where $\text{li}(N)$ is the <a href="http://en.wikipedia.org/wiki/Logarithmic_integral_function" rel="nofollow">logarithmic Integral.</a> ... |
1,619,292 | <p>Let $\mathbf C$ be an abelian category containing arbitrary direct sums and let $\{X_i\}_{i\in I}$ be a collection of objects of $\mathbf C$. </p>
<p>Consider a subobject $Y\subseteq \bigoplus_{i\in I}X_i$ and put $Y_i:=p_i(Y)$ where $p_i:\bigoplus_{i\in I}X_i\longrightarrow X$
is the obvious projection. </p>
<p>I... | davidlowryduda | 9,754 | <p>Gleick's <a href="http://rads.stackoverflow.com/amzn/click/0143113453">Chaos: Making a New Science</a> is a beautiful book that can be read without pencil and paper.</p>
|
2,581,135 | <blockquote>
<p>Find: $\displaystyle\lim_{x\to\infty} \dfrac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}}.$</p>
</blockquote>
<p>Question from a book on preparation for math contests. All the tricks I know to solve this limit are not working. Wolfram Alpha struggled to find $1$ as the solution, but the solution process pre... | Higurashi | 204,434 | <p>Note that
$$\frac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}} = \frac{1}{\sqrt{1+\sqrt{\frac{1}{x}+\sqrt{\frac{1}{x^3}}}}}$$</p>
<p>You can also look at it as
$$\sqrt{\frac{x}{x+\sqrt{x+\sqrt{x}}}}$$ In case dividing by $\sqrt{x}$ bothers you.</p>
|
1,204,745 | <p>Let $(\Omega, A, \mathbb{P} )$ be a probability space. Let $f: \Omega \rightarrow [-\infty, \infty]$ an $A$-measurable function. </p>
<p>If $f$ is bounded on the positive side and unbounded on the negative side. Is it possible that $\mathbb{E}[f]$ (the expectation with probability measure $\mathbb{P}$ ) is finite?... | shalop | 224,467 | <p>Here is a strong example. Let $\Omega = [0,1]$, and $\mathcal{A}$ is the Borel sigma algebra. Consider $P$ to be Lebesgue measure on $\Omega$. Define</p>
<p>$f(\omega)=
\begin{cases}
q & \text{if } \omega = \frac{p}{q} \text{ in reduced form and $q$ is odd} \\
-q &\text{if } \omega = \frac{p}{q} \te... |
316,699 | <p>If $A,B,C$ are sets, then we all know that $A\setminus (B\cap C)= (A\setminus B)\cup (A\setminus C)$. So by induction
$$A\setminus\bigcap_{i=1}^nB_i=\bigcup_{i=1}^n (A\setminus B_i)$$
for all $n\in\mathbb N$.</p>
<p>Now if $I$ is an uncountable set and $\{B_i\}_{i\in I}$ is a family of sets, is it true that:
$$A\s... | Ittay Weiss | 30,953 | <p>De Morgan's laws are most fundamental and hold for all indexed families, no matter the cardinalities involved. So, $$A-\bigcap _{i\in I}A_i=\bigcup _{i\in I}(A-A_i)$$ and dually $$A-\bigcup_{i\in I}A_i=\bigcap _{i\in I}(A-A_i).$$ The proof is a very good exercise. </p>
|
4,330,547 | <p>I am trying to find an efficient way of computing the intersection point(s) of a circle and line segment on a spherical surface.</p>
<p>Say you have a sphere of radius R. On the surface of this sphere are</p>
<ol>
<li>a circle with center (<span class="math-container">$\theta_c$</span>,<span class="math-container">$... | Community | -1 | <p>The edge of the circle is on a plane that cuts the sphere normal to the radius, call it the cplane.</p>
<p>The line segment is on a circle disk that passes through the two points on the sphere and passes through the origin. This circle is on a plane, call it the lplane.</p>
<p>The circle and line segment are on the ... |
4,330,547 | <p>I am trying to find an efficient way of computing the intersection point(s) of a circle and line segment on a spherical surface.</p>
<p>Say you have a sphere of radius R. On the surface of this sphere are</p>
<ol>
<li>a circle with center (<span class="math-container">$\theta_c$</span>,<span class="math-container">$... | Andrew D. Hwang | 86,418 | <p>If everything is converted from geographic coordinates to Cartesian vectors, the calculation can be done by linear algebra and trigonometry.</p>
<p>Let <span class="math-container">$P_{0}$</span> denote the center of the circle <span class="math-container">$C$</span>; <span class="math-container">$P_{1}$</span> and ... |
1,046,066 | <p>Is this series $$\sum_{n\geq 1}\left(\prod_{k=1}^{n}k^k\right)^{\!-\frac{4}{n^2}} $$ convergent or divergent?</p>
<p>My attempt was to use the comparison test, but I'm stuck at finding the behaviour of $\displaystyle \prod_1^n k^k$ as $n$ goes to infinity. Thanks in advance.</p>
| Olivier Oloa | 118,798 | <p>Here is an elementary approach, without using the <a href="http://en.wikipedia.org/wiki/Glaisher%E2%80%93Kinkelin_constant">Glaisher-Kinkelin constant</a>.
Observe that
$$
\ln \left(\left(\prod_{k=1}^{n}k^k\right)^{\!-\frac{4}{n^2}}\right)= -\frac{4}{n^2} \sum_{k=1}^{n}k\ln k.
$$
Let $k\geq 1$ and let $x \in [k,k+1]... |
1,296,420 | <p>I was trying to find an example such that $G \cong G \times G$, but I am not getting anywhere. Obviously no finite group satisfies it. What is such group?</p>
| ಠ_ಠ | 169,780 | <p>As others have mentioned, the trivial group satisfies this property for reasons that are mostly unrelated to group theory. If a category has binary products and a terminal object $1$ then $A \times 1 \cong A$ in a canonical way. Of course, we also have $A \times B \cong B \times A$ canonically, so in fact a terminal... |
2,658,195 | <p>I have the following problem with which I cannot solve. I have a very large population of birds e.g. 10 000. There are only 8 species of birds in this population. The size of each species is the same.</p>
<p>I would like to calculate how many birds I have to catch, to be sure in 80% that I caught one bird of each s... | kludg | 42,926 | <p>Probably this is a version of <a href="https://en.wikipedia.org/wiki/Penney%27s_game" rel="nofollow noreferrer">Penney's game</a>. The second player has advantage over the first player.</p>
|
4,154,298 | <p>Suppose that<span class="math-container">$ f(x, y)$</span> given by <span class="math-container">$\sum_{i=0}^{a}\sum_{j=0}^{b}c_{i,j}x^iy^j$</span> is a polynomial in two variables with real coefficients such that among its coefficients there is a non-zero one. Prove that there is a point <span class="math-container... | Ian | 83,396 | <p>In principle, a sequence can start at any integer you want. But usually the reason to not have it start at <span class="math-container">$0$</span> or <span class="math-container">$1$</span> is to avoid shifting around somewhere else in the notation. For a possibly familiar example from calculus, you may have encount... |
2,414,492 | <p>Check the convergence of $$\sum_{k=0}^\infty{2^{-\sqrt{k}}}$$
I have tried all other tests (ratio test, integral test, root test, etc.) but none of them got me anywhere.
Pretty sure the way to do it is to check the convergence by comparison, but not
sure how.</p>
| Angina Seng | 436,618 | <p>There are $(n+1)^2-n^2=2n+1$ distinct $k$ with $n\le \sqrt k<n+1$.
For these $k$, $2^{-\sqrt k}\le 2^{-n}$. So, your sum compares to
$\sum_n (2n+1)2^{-n}$.</p>
|
3,986,831 | <p>the question is:
true or false: if <span class="math-container">$f_n(x)'$</span> converges uniformly to <span class="math-container">$f(x)'$</span> then <span class="math-container">$f_n(x)$</span> converges uniformly to <span class="math-container">$f(x)$</span>. I tried many examples and they all confirmed the sta... | Community | -1 | <p>To elaborate <a href="https://math.stackexchange.com/a/3986836/872872">Yuval's answer</a>, in general, you have the following theorem.</p>
<p>Suppose <span class="math-container">$(f_n)$</span> is a sequence of differentiable functions on <span class="math-container">$[a,b]$</span> such that <span class="math-contai... |
3,755,355 | <p>I wanted to prove that every group or order <span class="math-container">$4$</span> is isomorphic to <span class="math-container">$\mathbb{Z}_{4}$</span> or to the Klein group. I also wanted to prove that every group of order <span class="math-container">$6$</span> is isomorphic to <span class="math-container">$\ma... | AT1089 | 758,289 | <p><span class="math-container">$\bullet$</span> Let <span class="math-container">$G$</span> be a group of order <span class="math-container">$4$</span>. By Lagrange's theorem, <span class="math-container">$o(g)=\{1,2,4\}$</span> for each <span class="math-container">$g \in G$</span>. Note that <span class="math-contai... |
279,277 | <p>I have been told multiple times that the logarithmic function is the inverse of the exponential function and vice versa. My question is; what are the implications of this? How can we see that they're the inverse of each other in basic math (so their graphed functions, derivatives, etc.)?</p>
| Džuris | 43,535 | <p>Actually the logarithm function is defined as the inverse of exponent function. It's not a property of these functions, it's how the logarithm is introduced.</p>
<p>If $a^b=c$ then the power by which you raise a to obtain c is the logarithm: $b=log_a c$.</p>
|
476,147 | <p>I am working on a problem and I need help getting started. Any pointers would be greatly appreciate it</p>
<p>My problem: Given a $50,000 purse and 20/20 hindsight, and a particular stock, what are the best buying and selling points if the the only requirement is to maximize net profit. The stock is a daily chart g... | Carl Feynman | 91,620 | <p>You can use dynamic programming for this. See <a href="http://en.wikipedia.org/wiki/Dynamic_programming" rel="nofollow">http://en.wikipedia.org/wiki/Dynamic_programming</a>. In the terminology of the section entitled "Dynamic programming in mathematical optimization", the decision step is whether to buy or sell at... |
1,203,922 | <p>Show that it is possible to divide the set of the first twelve cubes $\left(1^3,2^3,\ldots,12^3\right)$ into two sets of size six with equal sums.</p>
<p>Any suggestions on what techniques should be used to start the problem?</p>
<p>Also, when the question is phrased like that, are you to find a general case that ... | David | 119,775 | <p>To start, you should work out the sum of all the cubes, preferably using the formula
$$1^3+2^3+\cdots+12^3=\frac{12^2\times13^2}{4}=6084\ .$$
So you need to find six of your twelve cubes which add up to $3042$. I should think it's trial and error from here, but would be glad to see if anyone has a smarter solution.... |
231,773 | <p>Let $G=(V,E)$ be an undirected graph. We form a graph $H=(V',E')$ from $G$ such that </p>
<ul>
<li>$V' = V \cup \{ w_e \mid e \in E \}$, and </li>
<li>$E' = \{ aw_e, bw_e \mid ab = e \in E \} \cup \{ w_e w_f \mid e,f \text{ are adjacent edges in }G \}$. </li>
</ul>
<p>Informally, $H$ is built from $G$ by subdividi... | Will Jagy | 3,324 | <p>dunno. few_reps found that all the other forms of discriminant $37^2$ fail. Go figure. Here are all the examples from <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/nipp.html#form" rel="nofollow">Nipp's extended tables</a> at <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/" rel="nof... |
231,773 | <p>Let $G=(V,E)$ be an undirected graph. We form a graph $H=(V',E')$ from $G$ such that </p>
<ul>
<li>$V' = V \cup \{ w_e \mid e \in E \}$, and </li>
<li>$E' = \{ aw_e, bw_e \mid ab = e \in E \} \cup \{ w_e w_f \mid e,f \text{ are adjacent edges in }G \}$. </li>
</ul>
<p>Informally, $H$ is built from $G$ by subdividi... | Will Jagy | 3,324 | <p>It turns out that the OP will be satisfied with just one form for each of these square discriminants that maps to itself. I should already say that this thing reminds me of Watson transformations. However, few_reps has shown that some of the forms of the genus interchange. It seems this mapping permutes the genus, a... |
33,387 | <p>I was told the following "Theorem": Let $y^{2} =x^{3} + Ax^{2} +Bx$ be a nonsingular cubic curve with $A,B \in \mathbb{Z}$. Then the rank $r$ of this curve satisfies</p>
<p>$r \leq \nu (A^{2} -4B) +\nu(B) -1$</p>
<p>where $\nu(n)$ is the number of distinct positive prime divisors of $n$.</p>
<p>I can not find a n... | Pete L. Clark | 299 | <p>A rational elliptic curve $E_{/\mathbb{Q}}$ can be put in the form you gave if and only if it has a rational point of order $2$: in the given equation, $(0,0)$ has order $2$, and in general any point of order $2$ can be "moved" to $(0,0)$ by a change of variables.</p>
<p>Therefore you are in the general situation o... |
2,009,557 | <p>I am pretty sure this question has something to do with the Least Common Multiple. </p>
<ul>
<li>I was thinking that the proof was that every number either is or isn't a multiple of $3, 5$, and $8\left(3 + 5\right)$.</li>
<li>If it isn't a multiple of $3,5$, or $8$, great. You have nothing to prove.</li>
<li>But if... | RGS | 329,832 | <p>It has been shown that if you have two coins of values $a $ and $b$, then you can make any value greater than $ab - a - b $ with them. For your case, this yields $7$.</p>
<p>As suggested in the comments, for you it suffices to show you can make $8$, $9$ and $10$ cents explicitly. Then any other number will be a giv... |
298,481 | <p>Find the general solution of </p>
<p>$y'' + \dfrac{7}{x} y' + \dfrac{8}{x^2} y = 1, x > 0$</p>
<p>I don't even know how to solve the homogeneous version because it involves variables...</p>
<p>Does anyone know how to solve it?</p>
| Maesumi | 29,038 | <p>$y=x^n$, $y'=nx^{n-1}$, $y''=n(n-1)x^{n-2}$. So $x^2y''+7xy'+8y=x^n[n(n-1)+7n+8]=0$ solve this equation for $n$.</p>
<p>So that gives $n^2+6n+8=0$ or $n=-4,-2$. The solution of homogeneous eq is $y_h=c_1x^{-4}+c_2x^{-2}$.</p>
<p>The particular solution here can be obtained by undetermined coefficient and guessing ... |
1,533,646 | <p>Dying someone appointed in the will the following: If his pregnant wife giving birth to a son , then she will inherit 1/3 of the estate and his son 2/3 . If giving birth to daughter , then she would inherit 2/3 of the property and the daughter 1/3 . The woman gave birth to twins after the death of her husband , a b... | Stefan Perko | 166,694 | <p>Additionally to Henning's answer:</p>
<p>Usually you have at least <em>some</em> expression in your logic expressing "contradiction", for example $0=1$ in Heyting Arithmetic. Then you immediately get all of intuitionistic logic by setting $\neg a :\equiv a \Rightarrow (0 = 1)$, so removing $\neg$ usually does not c... |
1,533,646 | <p>Dying someone appointed in the will the following: If his pregnant wife giving birth to a son , then she will inherit 1/3 of the estate and his son 2/3 . If giving birth to daughter , then she would inherit 2/3 of the property and the daughter 1/3 . The woman gave birth to twins after the death of her husband , a b... | Doug Spoonwood | 11,300 | <p>I use Polish notation.</p>
<p>The formation rules go:</p>
<ol>
<li>All lower case letters of the Latin alphabet qualify as significant expressions.</li>
<li>If $\alpha$ and $\beta$ qualify as significant expressions, then so do N$\alpha$, C$\alpha$$\beta$, A$\alpha$$\beta$, and K$\alpha$$\beta$.</li>
</ol>
<p>The... |
2,291,540 | <p>Is it possible to have a sequence of continuous functions $\{f_n\}_{n=1}^\infty$ on $[a,b]$ that converges uniformly to a function $f$ but $f$ is not bounded on $[a,b]$?</p>
| Arpan1729 | 444,208 | <p>If a sequence of continuous functions converges uniformly to $f$ then $f$ is continuous and a continuous function in a compact interval $[a,b]$ is always bounded, hence it's not possible.</p>
|
9,696 | <p>I am tutoring a Grade 2 girl in arithmetic. She has demonstrated an ability to add two-digit numbers with carrying. For example: </p>
<p>$$\;\;14\\
+27\\
=41$$ </p>
<p>I asked her to write this out horizontally, and this is what she produced. </p>
<p>$$12+47=41$$ </p>
<p>She evidently is failing to see the n... | Tamara Reynolds | 5,679 | <p>I agree that you should ask her to read the orginal problem aloud. Does she understand that the problem reads "fourteen plus twenty-seven?" If you write a problem both "horizontally" and "vertically" and then ask her to read both versions aloud, does she read them the same? Can she demonstrate that problem with unif... |
3,473,944 | <p>So i have an object that moves in a straight line with initial velocity <span class="math-container">$v_0$</span> and starting position <span class="math-container">$x_0$</span>. I can give it constant acceleration <span class="math-container">$a$</span> over a fixed time interval <span class="math-container">$t$</s... | boojum | 882,145 | <p>(This is too long for a comment, so I'm posting it here.)</p>
<p>There is <em>nothing</em> wrong with your formula: it is dimensionally correct and your derivation is handled properly algebraically. The issue that is causing you concern is because the formula <em>obscures</em> the relationship between initial velo... |
4,424,668 | <p>Suppose <span class="math-container">$V$</span> is a complex vector space with <span class="math-container">$n=\dim V=10$</span> and <span class="math-container">$N∈L(V)$</span> is nilpotent. What are possible values for <span class="math-container">$\dim\ker(N^3)-\dim\ker(N)$</span>? The only two things I know that... | angryavian | 43,949 | <p>If <span class="math-container">$k_1, \ldots, k_m \ge 1$</span> are the sizes of the Jordan blocks of <span class="math-container">$N$</span>, then <span class="math-container">$\dim \ker(N^3) = \sum_{i=1}^m \min\{1, k_i\} = \sum_{i=1}^m 1 = m$</span> and <span class="math-container">$\dim \ker(N) = \sum_{i=1}^m \mi... |
3,009,543 | <p>I am having great problems in solving this:</p>
<p><span class="math-container">$$\lim\limits_{n\to\infty}\sqrt[3]{n+\sqrt{n}}-\sqrt[3]{n}$$</span></p>
<p>I am trying to solve this for hours, no solution in sight. I tried so many ways on my paper here, which all lead to nonsense or to nowhere. I concluded that I h... | John_Wick | 618,573 | <p><span class="math-container">$\sqrt[3]{n+\sqrt{n}}-\sqrt[3]{n}=\frac{n+\sqrt{n}-n}{(n+\sqrt{n})^{2/3}+(n+\sqrt{n})^{1/3}n^{1/3}+n^{2/3}}\leq \frac{\sqrt{n}}{n^{2/3}+n^{1/3}n^{1/3}+n^{2/3}}=\frac{\sqrt{n}}{3n^{2/3}}\rightarrow 0$</span></p>
|
60,810 | <p>I am looking for a proof of the fact that if $f:\mathbb{R}\to \mathbb{R}$ is a group automorphism of $(\mathbb{R},+)$ that also preserves order, then there exists a positive real number $c$ s.t. $f(x)=cx$ for all $x\in \mathbb{R}$. If anyone can point to a reference, that will be great.</p>
| Steven Stadnicki | 785 | <p>IIRC this has been covered a handful of times on this site; the proof, in any case, is pretty straightforward. Obviously $f(0)=0$ and $f(1)\gt 0$ by the order-preserving property; say $f(1)=c$. Then $f(n)=cn$ for all $n\in\mathbb{N}$ ($f(1+1+\ldots+1) = f(1)+f(1)+\ldots+f(1)$) and $f(q) = cq$ for all $q\in\mathbb{... |
894,764 | <p>I am having problems understanding how to solve this question. </p>
<p>Find a linear function that satisfies both of the given conditions.</p>
<p>$f(-1) = 5, f(1) = 6$</p>
<p>Thanks,
<i>Note: i have the answer, just need help understanding</i></p>
| 5xum | 112,884 | <p><strong>Hint:</strong> What is the general form of a linear function? </p>
<p>($f(x) =$ something something $x$ something something...)</p>
|
376,484 | <p>My questions are motivated by the following exercise:</p>
<blockquote>
<p>Consider the eigenvalue problem
$$
\int_{-\infty}^{+\infty}e^{-|x|-|y|}u(y)dy=\lambda u(x), x\in{\Bbb R}.\tag{*}
$$
Show that the spectrum consists purely of eigenvalues. </p>
</blockquote>
<p>Let $A:L^2({\Bbb R})\to L^2({\Bbb R})$ be ... | vadim123 | 73,324 | <p>The statement is false. Suppose $f(p^m)=\begin{cases}p & m=1\\ 0 & m>1\end{cases}$. <br> For any fixed prime the limit is 0 as soon as $m>1$, however as $n\rightarrow \infty$ you keep getting primes and so the limit is not 0.</p>
|
1,989,291 | <p>What is the closed form of the following:</p>
<p>$$\sum_{j=1}^n 3^{j+1}$$</p>
<p>I'm new to summations. Is it this?</p>
<p>$$\sum_{j=1}^n 3^{j} + \sum_{j=1}^n 3$$</p>
<p>Then using the closed form formula:</p>
<p>$$\frac{3^{n+1} - 1}{2} + 3n$$</p>
| E. Joseph | 288,138 | <p>No, $3^{j+1}=3^j\times 3$.</p>
<p>So multiply by $3$ the sum $$\sum_{j=1}^n 3^j.$$</p>
<p>Plus you got this summation wrong, because </p>
<p>$$\sum_{j=1}^n 3^j=\frac{3^n-1}{3-1}=\frac{3^n-1}{2}$$</p>
<p>instead of $n+1$.</p>
<p>The result is then:</p>
<p>$$\frac {3^{n+1}-3}2.$$</p>
|
1,176,435 | <p>Consider $G(4,p)$ - the random graph on 4 vertices. What is the probability that vertex 1 and 2 lie in the same connected component?</p>
<p>So far, I have considered the event where 1 and 2 do not lie in the same component. Then vertex 1 must lie in a component of order 1, 2 or 3 that doesn't contain vertex 2. Howe... | drhab | 75,923 | <p><strong>Hint</strong>:</p>
<p>Let $X$ denote the number of edges that will be included. </p>
<p>Let $E$ be the event that vertex $1$ and $2$ lie in the same connected
component.</p>
<p>Then $P\left(E\right)=\sum_{k=0}^{6}P\left(E\mid X=k\right)P\left(X=k\right)$ </p>
<p>Here $X$ is binomially distributed with pa... |
975,034 | <p>I am given a quadric equation such that $ax^2 + bx +c=0$ whose roots are $\alpha$ and $\beta$ then what would be value $$\lim\limits_{x \to \alpha} \frac{1-\cos( ax^2 + bx +c) }{(x-\alpha)^2}$$
Now since $x$ is tending to root of input in $\cos$ so my limits become $0/0$ form so I applied L'Hospital Rule hence my li... | Community | -1 | <p>The mistake is</p>
<p>$$ax^2+bx+c=a(x-\alpha)(x-\beta)$$</p>
|
975,034 | <p>I am given a quadric equation such that $ax^2 + bx +c=0$ whose roots are $\alpha$ and $\beta$ then what would be value $$\lim\limits_{x \to \alpha} \frac{1-\cos( ax^2 + bx +c) }{(x-\alpha)^2}$$
Now since $x$ is tending to root of input in $\cos$ so my limits become $0/0$ form so I applied L'Hospital Rule hence my li... | mvggz | 167,171 | <p>Or faster : $ax^2+bx+c=a(x−α)(x−β)$ </p>
<p>1-cos(u) ~ $\frac{u^2}{2}$ , when u->0 </p>
<p>=> your expression is equivalent to : $\frac{a^2*(x-α)^2(x-β)^2}{2*(x-α)^2}$ when x-> α</p>
<p>So the final equivalent is : $\frac{a^2*(x-β)^2}{2}$ which is (I hope) the answer</p>
|
1,791,673 | <p>I was wondering about this, just now, because I was trying to write something like:<br>
$880$ is not greater than $950$. <br>
I am wondering this because there is a 'not equal to': $\not=$ <br>
Not equal to is an accepted mathematical symbol - so would this be acceptable: $\not>$? <br>
I was searching around but ... | Avery Church | 646,506 | <p>Saying "not less than" is different from saying "greater or equal to" because there is a chance it is not greater than and only equal to, meaning it would be false to list it as greater than if it is only possibly equal, and in any case not less than.</p>
<p>I would like to point out that the not less than sign wou... |
874,946 | <p>What is the remainder when the below number is divided by $100$?
$$
1^{1} + 111^{111}+11111^{11111}+1111111^{1111111}+111111111^{111111111}\\+5^{1}+555^{111}+55555^{11111}+5555555^{1111111}+55555555^{111111111}
$$
How to approach this type of question? I tried to brute force using Python, but it took very long time.... | hola | 154,508 | <p>These may help you:</p>
<p>Noting that <span class="math-container">$a^{40} = 1 \pmod {100}~\forall a$</span> coprime to <span class="math-container">$100$</span>, (follows directly from <a href="https://en.wikipedia.org/wiki/Euler%27s_theorem" rel="nofollow noreferrer">Euler's theorem</a>)
<span class="math-contai... |
1,067,131 | <p>I'm reading a analysis book for fun and I got stuck on a problem.</p>
<p>The task is to find the function $f$ if
$$f(x-y,x+y) = \frac{x^2 + y^2}{2xy}$$</p>
<p>Since I can see the solution $\frac{x^2 + y^2}{y^2 - x^2}$ from the book (it's given in the back), I can backwards engineer the solution:</p>
<p>$$ \frac{(... | Arthur | 15,500 | <p>Hint: We have $\Bbb Q \leq \Bbb Q(\omega_7 + \omega_7^5) \leq \Bbb Q(\omega_7)$.</p>
|
2,252,317 | <p>I have a rather challenging question on my assignment and I have put in my best effort for now. I think I just need a tiny nudge to set me in the right direction to finish this proof. If you could have a look, that would be great!</p>
<hr>
<p><strong>Background on Cosets and Operations Defined on $V/W$</strong></p... | levap | 32,262 | <p>Your proof is correct and can be finished by noting that if $w \in W$ then $w + W = W$. The reason is that</p>
<p>$$ w + W = \{ w + w' \, | \, w' \in W \} $$</p>
<p>and since $W$ is a subspace, it is closed under addition so we have $w + W \subseteq W$. On the other hand, if $w' \in W$ then $w' = w + (w' - w)$ (wh... |
3,011,862 | <p>Test the convergence <span class="math-container">$$\int_0^1 \frac{x^n}{1+x}dx$$</span></p>
<p>I have used comparison test for improper integrals..by comparing with <span class="math-container">$1/(1+x)$</span>...
so I found it convergent ..
But the solution set says that it is convergent if <span class="math-conta... | Bram28 | 256,001 | <p>You started out correctly by first focusing on the last digit, for which there are indeed <span class="math-container">$3$</span> options. So, after you have picked one of those, there are <span class="math-container">$4$</span> digits left to pick from for the first digit, and then there are <span class="math-conta... |
3,546,615 | <p>Why do we take thickness be differential of distance apart of elemental mass when calculating volume and be differential length of arc when calculating area of the sphere when integrating in terms of angle.</p>
<p><a href="https://i.imgur.com/Mw8oW85m.jpg" rel="nofollow noreferrer"><img src="https://i.imgur.com/Mw8... | Community | -1 | <p>Think of these computations as bases on the piling of <em>cone</em> frustra.</p>
<p>The volume of a single frustrum is the base area by the height, <span class="math-container">$\pi r^2\,dh$</span>.</p>
<p>The lateral area is the circumference of the base times the <em>slant height</em>, <span class="math-containe... |
1,483,489 | <p>What I have trying is:</p>
<p>Suppose that $f(x)$ has at least one zero $\alpha$ such that $f(x) = (x - \alpha)^sq(x)$, $s > 1$ in some extension. Then I guess that $(x-\alpha)^{s-1} \mid f(x)$.
So, $f(x)$ is not irreducible, where $f(x) = (x-a)^{s-1}h(x)$.
But is seems wrong once I neither used the hypothesis $... | Adam Hughes | 58,831 | <p>This isn't quite right, first of all $(x-\alpha)^{s-1}$ might not be a polynomial with coefficients in $F$ when $\alpha\not\in F$. However, you do know in characteristic $0$ that the derivative of a non-constant polynomial is not $0$. If</p>
<p>$$f(x)=(x-\alpha)^s\prod_{i=1}^n (x-\alpha_i)^{e_j}, s>1, e_j\ge 1$$... |
152,467 | <p>Can you please explain to me how to get from a nonparametric equation of a plane like this:</p>
<p>$$ x_1−2x_2+3x_3=6$$</p>
<p>to a parametric one. In this case the result is supposed to be </p>
<p>$$ x_1 = 6-6t-6s$$
$$ x_2 = -3t$$
$$ x_3 = 2s$$</p>
<p>Many thanks.</p>
| Robert Mastragostino | 28,869 | <p>Welcome to math.stackexchange!</p>
<p>A plane can be defined by three things: a point, and two non-colinear vectors in the plane (think of them as giving the plane a grid or coordinate system, so you can move from your first point to any other using them).</p>
<p>So first, we need an initial point: since there are... |
4,463 | <p>It seems that most authors use the phrase "elementary number theory" to mean "number theory that doesn't use complex variable techniques in proofs." </p>
<p>I have two closely related questions.</p>
<ol>
<li>Is my understanding of the usage of "elementary" correct?</li>
<li>It appears that advanced techniques fro... | Dan Piponi | 1,233 | <p>This question is formalised to some extent by reverse mathematicians who seek to understand precisely what parts of the foundations of mathematics are required to prove any given result. There is some interesting discussion in <a href="http://www.math.psu.edu/simpson/papers/hilbert.pdf" rel="nofollow">this</a> paper... |
2,452,547 | <p>I am given the initial value problem </p>
<blockquote>
<p>\begin{array}{l}
y' = \dfrac{7 x\,y}{7 x^{2}+2 y^{2}} \\
y(1)=1 \end{array}</p>
</blockquote>
<p>where I must answer in the form of $F(x,y)=\frac{7}{4}$.</p>
<p>Here, I am also asked to use the substitution $y=xu$ to transform this differential equa... | Tengu | 58,951 | <p>I found one, borrowing idea of sum of square from Maman's comment.</p>
<blockquote>
<p>If $p \equiv 3 \pmod{4}$ is a prime and $p \mid a^2+b^2$ then $p\mid a,p \mid b$.</p>
</blockquote>
|
2,452,547 | <p>I am given the initial value problem </p>
<blockquote>
<p>\begin{array}{l}
y' = \dfrac{7 x\,y}{7 x^{2}+2 y^{2}} \\
y(1)=1 \end{array}</p>
</blockquote>
<p>where I must answer in the form of $F(x,y)=\frac{7}{4}$.</p>
<p>Here, I am also asked to use the substitution $y=xu$ to transform this differential equa... | Community | -1 | <p>It could be hard to find solution of that problem without some special constraints, because, we can have that $a$ divides $\sum_{i=1}^n c_ib_i$ but that $a$ divides as many as we want numbers in the set $\{b_1,...b_n\}$.</p>
<p>To see that suppose that we have $\sum_{i=1}^n c_ib_i=da$ for some integer $d$.</p>
<p>... |
459,579 | <blockquote>
<p>Find the value of $3^9\cdot 3^3\cdot 3\cdot 3^{1/3}\cdot\cdots$</p>
</blockquote>
<p>Doesn't this thing approaches 0 at the end? why does it approaches 1?</p>
| Community | -1 | <p>Hint: $3^9\cdot3^3\cdot3^1\cdot\dots=3^{9+3+1+\cdots}$</p>
|
2,619,907 | <p>Let $\lambda$ be a partition of length $n$ and suppose its largest diagonal block, the Durfee square of $\lambda$, has size $r$. By this I mean that $\lambda = (\lambda_1,\ldots,\lambda_n)$ is a non-increasing sequence of numbers, which I depict by the following diagram</p>
<p>\begin{align*}
&\square \cdots \sq... | Peter Taylor | 5,676 | <p>The concept here, although I'm not sure how far I can formalise it, is to extend the diagonal as far as the bottom of the Ferrers diagram. Then consider only the lower triangle. So your example</p>
<p>$$\begin{align}
&\blacksquare\square\square\square\square\\
&\square\blacksquare\square\square\\
&\squa... |
3,516,754 | <p>I got stuck while doing exercise of the Apostol's Calculus, the exercise 28 of Section 5.5.</p>
<p>Here's the question</p>
<hr>
<p>Given a function <span class="math-container">$f$</span> such that the integral <span class="math-container">$A(x) = \int_a^xf(t)dt$</span> exists for each <span class="math-container... | Paramanand Singh | 72,031 | <p>You start your argument correctly that <span class="math-container">$f'(c) $</span> exists and hence <span class="math-container">$f$</span> is continuous at <span class="math-container">$c$</span> and therefore by FTC <span class="math-container">$A'(c) =f(c) $</span>. But beyond that you can't conclude anything.</... |
261,031 | <p>i hope some of you can support to solve my problem, i need to work on data in the following way, where the length of each of the lists or sublists is equal. As an example i want to share the data-pattern with you:</p>
<pre><code>list1={a,b,c};
list2={{d,e,f},{g,h,i},......} (in reality the number of sublists in list... | kglr | 125 | <pre><code>list1 = {a, b, c};
list2 = {{d, e, f}, {g, h, i}};
Map[Thread[{list1, #}] &] @ list2
</code></pre>
<blockquote>
<pre><code>{{{a, d}, {b, e}, {c, f}}, {{a, g}, {b, h}, {c, i}}}
</code></pre>
</blockquote>
|
2,475,507 | <blockquote>
<p>Find $f$ and $g$ such that domain $(f\circ g)=\mathbb{R}$ and domain $(g\circ f)=\emptyset$</p>
</blockquote>
<p>That's it, I can't think of any. </p>
<p>I've thought of $f(x)=-1$ and $g(x)=\sqrt{x}$, and then: $$f\big(g(x)\big)=-1$$ $$g\big(f(x)\big)=\sqrt{-1}$$ </p>
<p>Which would in principle sa... | user334639 | 221,027 | <p>That's indeed impossible.</p>
<p>Assuming you are talking about real functions defined on subsets of the real line.</p>
<p>Suppose the domain of $f\circ g$ is $\mathbb R$. Then $g$ is defined for all real numbers, and $f$ is defined at least for $x_0 = g(17) \in \mathbb R$. Now since $g$ is defined for all real nu... |
465,999 | <p>I'm not sure of this, can I have a constraint like this in a linear programming problem to be solved with simplex algorithm?</p>
<p>$$n_1t_1 + n_2t_2 > 200$$</p>
<p>where $n_1$ and $t_1$, $n_2$ and $t_2$ are different variables.</p>
| elbeardmorez | 15,263 | <p>It's non linear. It looks like separation of variables via substitution may work, introducing additional constraints, see p<span class="math-container">$15$</span> of: <a href="http://web.mit.edu/15.053/www/AMP-Chapter-13.pdf" rel="nofollow noreferrer">http://web.mit.edu/15.053/www/AMP-Chapter-13.pdf</a></p>
|
3,463,293 | <p>I've run into a problem that I can't explain to my class.
We are looking at the derivative for the equation <span class="math-container">$\frac{x}{y}+\frac{y}{x}=3y$</span>. We calculated it to be <span class="math-container">$\frac{y(x^2-y^2)}{x(3xy^2+x^2-y^2)}$</span> and we also verified it with Wolfram Alpha.</... | Narasimham | 95,860 | <p>I got the second derivative like:
<span class="math-container">$$ \frac{x}{y}+\frac{y}{x}= 3y,\,$$</span>
or
<span class="math-container">$$ \frac{x}{y^2}+\frac{1}{x}= 3$$</span>
Differentiate with quotient rule to obtain
<span class="math-container">$$ \frac{y^2-2xyy'}{y^4}= \frac{1}{x^2}\,$$</span>
Simplifying
<sp... |
3,463,293 | <p>I've run into a problem that I can't explain to my class.
We are looking at the derivative for the equation <span class="math-container">$\frac{x}{y}+\frac{y}{x}=3y$</span>. We calculated it to be <span class="math-container">$\frac{y(x^2-y^2)}{x(3xy^2+x^2-y^2)}$</span> and we also verified it with Wolfram Alpha.</... | md2perpe | 168,433 | <p>Along a curve <span class="math-container">$f(x,y) = 0$</span> the derivative <span class="math-container">$y'$</span> is given by
<span class="math-container">$$
y_f'(x,y)
= -\frac{\partial_x f(x,y)}{\partial_y f(x,y)}
.
$$</span></p>
<p>Given some smooth function <span class="math-container">$\varphi$</span> whi... |
3,548,064 | <p>I have two equalities:
<span class="math-container">$$ \alpha x^{2} + \alpha y^{2} - y = 0 $$</span>
<span class="math-container">$$ \beta x^{2} + \beta y^{2} - x = 0 $$</span></p>
<p>Where <span class="math-container">$$ \alpha, \beta $$</span> are both known constants.</p>
<p>How can I solve for <span class="ma... | Community | -1 | <p>The two curves are circles through the origin and they will intersect in at most one other point.</p>
<p>By eliminating <span class="math-container">$x^2+y^2$</span>, we have</p>
<p><span class="math-container">$$\beta y=\alpha x,$$</span></p>
<p>then</p>
<p><span class="math-container">$$\beta(\beta x^2+\beta y... |
180,296 | <p>I need an algorithm to decide quickly in the worst case if a 20 digit integer is prime or composite.</p>
<p>I do not need the factors.</p>
<p>Is the fastest way still a prime factorization algorithm? Or is there a faster way given the above relaxation?</p>
<p>In any case which algorithm gives the best worst case... | N. S. | 9,176 | <p>The standard proof is listed in the above comments, here is an alternate self contained complete proof.</p>
<p>Suppose $\overline{a}\cdot \overline{c}=1$. Then $n|ac-1$. Let $d=\gcd(a,n)$. Then</p>
<p>$d|n|ac-1$ and $d|c|ac$ thus $d|ac$ and $d|ac-1$. This implies that $d| (ac)-(ac-1)$, and hence $d|1$.</p>
<p>Now... |
3,534,985 | <blockquote>
<p>What is <span class="math-container">$\displaystyle\lim_{x\to\infty}(\left \lfloor{-\dfrac{1}{x}}\right \rfloor )$</span>?</p>
</blockquote>
<p>Why is it not -1? Book says 1.</p>
| Parcly Taxel | 357,390 | <p>As <span class="math-container">$x\to+\infty$</span>, <span class="math-container">$-\frac1x$</span> approaches <span class="math-container">$0$</span> from the <em>negative</em> side; it is negative for any sufficiently large finite <span class="math-container">$x$</span>. Thus we must conclude that the limit of it... |
3,534,985 | <blockquote>
<p>What is <span class="math-container">$\displaystyle\lim_{x\to\infty}(\left \lfloor{-\dfrac{1}{x}}\right \rfloor )$</span>?</p>
</blockquote>
<p>Why is it not -1? Book says 1.</p>
| Rajan | 745,962 | <p><span class="math-container">$$\lim_{x \to \infty} \lfloor -\frac{1}{x} \rfloor $$</span>
<span class="math-container">$$\lim_{x \to \infty} (-\frac{1}{x}-\{-\frac{1}{x}\})$$</span>
<span class="math-container">$$=0-1=-1$$</span></p>
|
2,579,572 | <blockquote>
<p>In an election, $10\%$ voters did not participate and $1200$ votes are found invalid. The winner gets $68\%$ of total voting list and he won by $56400$ votes. Find the votes polled in favor of losing candidate.</p>
</blockquote>
<p>I can't understand what should I do with the number of invalid votes.... | Community | -1 | <p>Let total number of voters be $x$. Then, we have: </p>
<p>Total votes = $\frac{9x}{10}$ and total number of valid votes = $\frac{9x}{10}-1200$</p>
<p>Because, the winner got $68\%$ votes <strong>of total list</strong>, the number of votes he got = $\frac{34x}{50}$ and thus, number of votes the loser got = $\frac{... |
1,127,596 | <p>I am trying to show that the value of $\int^\infty_0$$\int^\infty_0$ sin($x^2$+$y^2$) dxdy is $\frac{\pi}{4}$ using Fresnel integrals. I'm having trouble splitting apart the integrand in order to actually be able to use the Fresnel integrals. Any help is appreciated. </p>
<p>Answer:
$\int^\infty_0$ $\int^\infty_0$... | Sangchul Lee | 9,340 | <p>In real-world situation, often it is too ideal to consider all the interactions from arbitrarily long distance. So we may first suppress long-distance interactions and then let the suppression disappear. In mathematical terms, it means that we may understand the integral</p>
<p>$$ \int_{0}^{\infty}\int_{0}^{\infty}... |
3,015,596 | <p>A short introduction: The independence number <span class="math-container">$\alpha(G)$</span> of a graph <span class="math-container">$G$</span> is the cardinality of the largest independent vertex set. Independent vertex set is made only of vertices with no edges between them. </p>
<p><a href="https://i.stack.imgu... | mathworker21 | 366,088 | <blockquote>
<p>Can we replace the probabilistic part of the proof with something that is "more basic"?</p>
</blockquote>
<p>Yes, it is a shame that it is never explained that no actual probability is being used in these types of proofs. Below, <span class="math-container">$\pi$</span> represents a permutation, <spa... |
1,888,187 | <p>I am working on algebraic functions and I am stuck on this problem:</p>
<p>$f(x) = a * r^x$<br>
$(2,1),(3,1.5)$<br></p>
<p>This would be a simple problem if it weren't for that $1.5$ -</p>
<p>So, I have plugged in $2$ and $1$ into the function and this is what I got:
<a href="https://i.stack.imgur.com/MVbxv.png" ... | GoodDeeds | 307,825 | <p>$$f(2)=ar^2=1$$
$$a=\frac{1}{r^2}$$
$$f(3)=ar^{3}=1.5$$
$$r=1.5$$
$$a=\frac{1}{(1.5)^2}$$</p>
|
779,987 | <p>I want to understand intuitively why it is that the gradient gives the direction of steepest ascent. (I will consider the case of $f:\mathbb{R}^2\to\mathbb{R}$)</p>
<p>The standard proof is to note that the directional derivative is $$D_vf=v\cdot \nabla f=|\nabla f|\,\cos\theta$$ which is maximized at $\theta=0$. T... | Andrew D. Hwang | 86,418 | <p>$\newcommand{\R}{\mathbf{R}}$Let $U$ be an open set in $\R^{2}$ and $f:U \to \R$ a differentiable function. If $x_{0} \in U$, then by definition there exists a linear function $Df(x_{0}):\R^{2} \to \R$ such that
$$
\lim_{x \to x_{0}} \frac{|f(x) - f(x_{0}) - Df(x_{0})(x - x_{0})|}{\|x - x_{0}\|} = 0.
$$</p>
<p>If $... |
123,269 | <p>Consider the following differential equation:</p>
<p>$$\begin{align*}&\rho C_p\left(\frac{\partial T}{\partial t}\right)=k\left[\frac{\partial^2 T}{\partial x^2}\right]+\dot{q}\\
&\text{at }x=0,\;\frac{\partial T}{\partial x}=0\\
&\text{at }x=1,\frac{\partial T}{\partial x}=C_1(T(t,1)-C_2)\\
&\text{... | Young | 41,016 | <p>Changed the Derivative boundaries to <a href="http://reference.wolfram.com/language/ref/NeumannValue.html" rel="noreferrer"><code>NeumannValue</code></a> and FEA needed to be specified.</p>
<pre><code>c1 = -10;
c2 = 10;
c3 = 20;
q[t_, x_] := 100000;
heat = NDSolveValue[{
1591920 D[u[t, x], t] - ((87/100) D[u[t,... |
337,518 | <p>When it comes to numbering results in a mathematical publication, I'm aware of two methods: </p>
<ol>
<li><p>Joint numbering: <em>Thm. 1, Prop. 2, Thm. 3, Lem. 4, etc.</em></p></li>
<li><p>Separate numbering: <em>Thm. 1, Prop. 1, Thm. 2, Lem. 1, etc.</em></p></li>
</ol>
<p>Every piece of writting advice I have enc... | Bjørn Kjos-Hanssen | 4,600 | <p>If the paper contains three main theorems, each generalizing the previous, it is nice to be able to discuss them like this:</p>
<blockquote>
<p>While the extension of Theorem 1 to Theorem 2 uses only complex analysis, in Theorem 3 we will have to employ some Ramsey theory. </p>
</blockquote>
|
337,518 | <p>When it comes to numbering results in a mathematical publication, I'm aware of two methods: </p>
<ol>
<li><p>Joint numbering: <em>Thm. 1, Prop. 2, Thm. 3, Lem. 4, etc.</em></p></li>
<li><p>Separate numbering: <em>Thm. 1, Prop. 1, Thm. 2, Lem. 1, etc.</em></p></li>
</ol>
<p>Every piece of writting advice I have enc... | Timothy Chow | 3,106 | <p>This is a slight elaboration of François Dorais's comment. If you have a small number of theorems/lemmas/propositions—let's say, small enough that readers can reasonably be expected to hold all the theorems in their head at once—then the second method of numbering can help readers grasp the flow of the ... |
82,765 | <p><strong>Bug introduced in 9.0 and persisting through 12.2</strong></p>
<hr />
<p>I get the following output with a fresh Mathematica (ver 10.0.2.0 on Mac) session</p>
<pre><code>FullSimplify[Exp[-100*(i-0.5)^2]]
(* 0. *)
Simplify[Exp[-100*(i-0.5)^2]]
(* E^(-100. (-0.5+i)^2) *)
</code></pre>
<p><code>FullSimplif... | ilian | 145 | <p>This is certainly an undesirable result, although <code>FullSimplify</code> isn't doing anything wrong. </p>
<p>The transformations performed are</p>
<pre><code>Map[ExpandAll, ExpToTrig[E^(-100 (-0.5 + x)^2)]]
(* Cosh[25. - 100. x + 100 x^2] - Sinh[25. - 100. x + 100 x^2] *)
Map[TrigExpand, %]
(* 0. *)
</code><... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.