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 |
|---|---|---|---|---|
1,254,189 | <p>I know that I have to study the order of every element in $\mathbb{Q/Z}$.
But what do I do?
I've been struggling of what to do for this question</p>
| k1.M | 132,351 | <p>In the group $\mathbb Q/\mathbb Z$ we have the element $\frac 12+\mathbb Z$ which is of order two. But non of the elements of additive group $\mathbb Q$ is of order two. This shows that there is no such isomorphism. </p>
|
950,485 | <p>I have been trying to solve the following limit but am completely stuck.</p>
<p>$$\lim_{\alpha \rightarrow \infty} 1-\left( \frac{y+\alpha}{\alpha-1} \right)^{-\alpha}$$</p>
<p>I have tried inverting the ratio and came up with the following expression:</p>
<p>$$ 1 - \lim_{\alpha \rightarrow \infty} \left( 1-\frac... | Timbuc | 118,527 | <p>If you're already a little advanced in functions and their limits, you could try</p>
<p>$$\lim_{\alpha\to\infty}\frac{\log\left(1-\frac{y+1}{y+\alpha}\right)}{\frac1\alpha}\stackrel{\text{l'Hospital}}=\lim_{\alpha\to\infty}\frac{\frac{y+\alpha}{\alpha-1}\cdot\frac{y+1}{(y+\alpha)^2}}{-\frac1{\alpha^2}}=-\lim_{\alph... |
1,992,256 | <p>I have to prove</p>
<p>$\sqrt{1} + \sqrt{2} +...+\sqrt{n} \le \frac{2}{3}*(n+1)\sqrt{n+1}$</p>
<p>by using math induction. </p>
<p>First step is to prove that it works for n = 1 , which is true.
Next step is to prove it for n + 1. We can rewrite the formula using</p>
<p>$\sum_{i=1}^{n+1} \sqrt{i}= \sum_{i=1}^{n... | Jam | 161,490 | <p>You started off well and almost got it. Here's a hint, where LHS and RHS are respectively the left and right hand sides of the formula we want:</p>
<p>$$\begin{align}\mathrm{LHS}^2&=\left(\frac23\right)^2(n+1)\left(n+\frac52\right)^2\\
&=\left(\frac23\right)^2(n^3+6 n^2+11.25n+6.25)
\\
&<\left(\frac2... |
2,804,495 | <p>I was asked to solve this double integral:
Compute the area between $y=2x^2$ and $y=x^2$ and the hyperbolae $xy=1$ and $xy=2$ in </p>
<p>$$ \iint dx \,dy$$</p>
<p>I tried to solve it starting with considering that </p>
<p>$$x^2 \leq y \leq 2x^2 $$</p>
<p>suitabile for integration interval in $y$, obtaining the i... | SlipEternal | 156,808 | <p>Plot the four functions. See where they intersect. Once you do, you will find that you can rewrite this integral as:</p>
<p>$$\displaystyle \int_{ \tfrac{1}{ \sqrt[3]{2} } }^1\left( 2x^2-\dfrac{1}{x}\right)dx + \int_1^{\sqrt[3]{2}}\left(\dfrac{2}{x}-x^2\right)dx$$</p>
|
3,653,979 | <p>Let A = <span class="math-container">$\begin{bmatrix}r_1 & r_2 & r_3 & r_4 & r_5\end{bmatrix}^T$</span> have rows <strong><span class="math-container">$r_1$</span></strong>, <strong><span class="math-container">$r_2$</span></strong>, <strong><span class="math-container">$r_3$</span></strong>, <strong... | Ben Grossmann | 81,360 | <p><strong>Hint:</strong> Note that the two matrices under consideration can be written as <span class="math-container">$M_1A$</span> and <span class="math-container">$M_2A$</span> respectively, where
<span class="math-container">$$
M_1 =
\pmatrix{2&3&4&4&0\\ 1&2&0&0&0\\ 0&1&3&a... |
2,653,483 | <p>Let $a =111 \ldots 1$, where the digit $1$ appears $2018$ consecutive times.</p>
<p>Let $b = 222 \ldots 2$, where the digit $2$ appears $1009$ consecutive times.</p>
<p>Without using a calculator, evaluate $\sqrt{a − b}$.</p>
| hamam_Abdallah | 369,188 | <p>$$a=\sum_{i=0}^{2017}10^i $$
$$=\frac {10^{2018}-1}{10-1} $$
$$b=2\sum_{i=0}^{1008}10^i $$
$$=2\frac {10^{1009}-1}{10-1}$$</p>
<p>$$9 (a-b)=10^{2018}-2.10^{1009}+1$$
$$=(10^{1009}-1)^2$$</p>
<p>then</p>
<p>$$\sqrt {a-b}=\frac {10^{1009}-1}{3} $$</p>
<p>$=333...333.$ (1009 consecutive times).</p>
|
3,705,539 | <p>I am trying to learn more about probability and came across an interesting question that I am stuck on and can no longer find online. There are 20 numbered balls and 10 bins. Someone is trying to assign the balls to the bins, but does it with replacement on accident.</p>
<p>So they did the following: Place a ball i... | Phicar | 78,870 | <p>The way I understand from the comments is that you are modeling this is by a function <span class="math-container">$f:\{\text{bins}\}\longrightarrow \{\text{balls}\}$</span> in which you take a bin and you assign a ball to it, they can have the same ball(replacement). so there are indeed <span class="math-container"... |
2,209,438 | <p>I am trying to find this limit,</p>
<blockquote>
<p>$$\lim_{x \rightarrow 0} \frac{1}{x^4} \int_{\sin{x}}^{x} \arctan{t}dt$$</p>
</blockquote>
<p>Using the fundamental theorem of calculus, part 1,
$\arctan$ is a continuous function, so
$$F(x):=\int_0^x \arctan{t}dt$$
and I can change the limit to
$$\lim_{x \righ... | Nick Peterson | 81,839 | <p>Power series can help, if you know them.</p>
<p>We know that
$$
\arctan t=\sum_{n=0}^{\infty}(-1)^n\frac{t^{2n+1}}{2n+1},\qquad \lvert t\rvert<1.
$$
Therefore an antiderivative for $\arctan t$ is
$$
F(t):=\sum_{n=0}^{\infty}(-1)^n\frac{t^{2n+2}}{(2n+1)(2n+2)},\qquad \lvert t\rvert<1.
$$
So, as $x\to0$,
$$
F(... |
2,209,438 | <p>I am trying to find this limit,</p>
<blockquote>
<p>$$\lim_{x \rightarrow 0} \frac{1}{x^4} \int_{\sin{x}}^{x} \arctan{t}dt$$</p>
</blockquote>
<p>Using the fundamental theorem of calculus, part 1,
$\arctan$ is a continuous function, so
$$F(x):=\int_0^x \arctan{t}dt$$
and I can change the limit to
$$\lim_{x \righ... | davidlowryduda | 9,754 | <p>Here's one way to do this. The Taylor expansion for $\arctan(x) = x - x^3/3 + x^5/5 + \cdots$. So then
$$ \int_{\sin(x)}^x \arctan(t)dt \sim (x^2/2 - x^4/12 + x^6/30) - \frac{(\sin x)^2}{2} + \frac{(\sin x)^4}{12} - \frac{(\sin x)^6}{30}.$$</p>
<p>Using another Taylor expansion, $\sin x = x - x^3/3! + x^5/5! + \cdo... |
9,918 | <p>I recently flagged as "rude or offensive" the comment </p>
<blockquote>
<p>As the tone should suggest, he’s a crank. It’s a hysterical screed with a few nuggets of fact surrounded by a great deal of nonsense. E.g., he may find that set theory ‘doesn’t make sense’, but a great many of us have no trouble making sen... | Community | -1 | <p>I'll answer this from a bit of an outside perspective. I'm a moderator on <a href="https://skeptics.stackexchange.com/">Skeptics</a> where this issue was a significant concern. The topics on that site often evoke very strong reactions and it is not unusual for comments to get rather heated. </p>
<p>My observations ... |
1,368,988 | <p>I was thinking about different ways of finding $\pi$ and stumbled upon what I'm sure is a very old method: dividing a circle of radius $r$ up into $n$ isosceles triangles each with radial side length $r$ and central angle $$\theta=\frac{360^\circ}{n}$$ Use $s$ for the side opposite to $\theta$.</p>
<p><img src="htt... | Jack D'Aurizio | 44,121 | <p>$$\sqrt{2-2\cos x}=2\left|\sin\frac{x}{2}\right|$$
hence everything boils down to:
$$ \lim_{x\to 0}\frac{\sin x}{x}=1.$$</p>
|
2,934,028 | <blockquote>
<p>A particle moves along the top of the
parabola <span class="math-container">$y^2 = 2x$</span> from left to right at a constant speed of 5 units
per second. Find the velocity of the particle as it moves through
the point <span class="math-container">$(2, 2)$</span>. </p>
</blockquote>
<p>So I is... | Community | -1 | <p>in cartesian co-ordinate sysytem :</p>
<p><span class="math-container">$(speed)^2=\left(\dfrac{dx}{dt}\right)^2+\left(\dfrac{dy}{dt}\right)^2=25$</span> also, </p>
<p><span class="math-container">$\left(\dfrac{dy}{dt}\right)=\dfrac{1}{y}\left(\dfrac{dx}{dt}\right) $</span> put it in above equation </p>
<p>to get ... |
2,934,028 | <blockquote>
<p>A particle moves along the top of the
parabola <span class="math-container">$y^2 = 2x$</span> from left to right at a constant speed of 5 units
per second. Find the velocity of the particle as it moves through
the point <span class="math-container">$(2, 2)$</span>. </p>
</blockquote>
<p>So I is... | user | 505,767 | <p>In that case is useful to use parametric equation that is</p>
<p><span class="math-container">$$y^2=2x \implies \vec s(t)=(2c^2t^2,2ct) \implies \vec v(t)=s'(t)=(4c^2t,2c)$$</span></p>
<p>and since the point <span class="math-container">$(2,2)$</span> is reached at <span class="math-container">$t=1/c$</span>, that... |
2,609,537 | <p>Is the following Proof Correct? In particular please comment on the correctness of the given formulas.</p>
<p><strong>Theorem.</strong> Given that $x$ is a real number, $x\neq 0$, and $x + \frac{1}{x}$ is an integer. For all $n\ge 1$, $x^n+\frac{1}{x^n}$ is an integer.</p>
<p><strong>Proof.</strong> We construct t... | Atmos | 516,446 | <p>I think it is easier to see that</p>
<blockquote>
<p>$$
x^{n+2}+\frac{1}{x^{n+2}}=\left(x+\frac{1}{x}\right)\left(x^{n+1}+\frac{1}{x^{n+1}}\right)-\left(x^n+\frac{1}{x^n}\right)
$$</p>
</blockquote>
|
3,497,420 | <p>Consider the function <span class="math-container">$$f(x,y)=x^6-2x^2y-x^4y+2y^2.$$</span> The point <span class="math-container">$(0,0)$</span> is a critical point. Observe,
<span class="math-container">\begin{align*}
f_x & = 6x^5-4xy-4x^3y, f_x(0,0)=0\\
f_y & = 2x^2-x^4+4y. f_y(0,0)=0\\
f_{xx} & = 30x... | Cesareo | 397,348 | <p>With <span class="math-container">$g(x) = x^2$</span> we have</p>
<p><span class="math-container">$$
f(g(x),y) = g(x)^3-2g(x)y-g(x)^2y +2y^2
$$</span></p>
<p>or</p>
<p><span class="math-container">$$
f(g,y)=g^3-2g y-g^2y+2y^2
$$</span></p>
<p>and the hessian of <span class="math-container">$f(g,y)$</span> is</p>... |
3,826,994 | <p>I would like to find <span class="math-container">$z$</span> which minimizes the below, when <span class="math-container">$x$</span> is held at a specific value.</p>
<p><span class="math-container">$f(x,z) =\sqrt{\sqrt{x^2 + z^2} - 0.25}$</span></p>
<p>For example; I would like to find the value of <span class="math... | Soumyadwip Chanda | 823,370 | <p>The expression means that z subtends an angle <span class="math-container">$\frac{\pi}{2}$</span> at the points <span class="math-container">$2i$</span> and <span class="math-container">$6$</span></p>
<p>Ponder upon the following visual</p>
<p><a href="https://i.stack.imgur.com/BaDQy.png" rel="noreferrer"><img src="... |
1,572,045 | <p>This is maybe a stupid question, but I want to find the roots of:</p>
<blockquote>
<p>$$2(x+2)(x-1)^3-3(x-1)^2(x+2)^2=0$$</p>
</blockquote>
<p>What that I did:</p>
<p>$$\underbrace{2(x+2)(x-1)(x-1)(x-1)}_{A}-\underbrace{3(x-1)(x-1)(x+2)(x+2)}_{B}=0$$</p>
<p>So the roots are when $A$ and $B$ are both zeros when... | Leox | 97,339 | <p><strong>Ніnt:</strong>
$$2(x+2)(x-1)^3-3(x-1)^2(x+2)^2=- \left( x+8 \right) \left( x+2 \right) \left( x-1 \right) ^{2}.$$</p>
|
1,572,045 | <p>This is maybe a stupid question, but I want to find the roots of:</p>
<blockquote>
<p>$$2(x+2)(x-1)^3-3(x-1)^2(x+2)^2=0$$</p>
</blockquote>
<p>What that I did:</p>
<p>$$\underbrace{2(x+2)(x-1)(x-1)(x-1)}_{A}-\underbrace{3(x-1)(x-1)(x+2)(x+2)}_{B}=0$$</p>
<p>So the roots are when $A$ and $B$ are both zeros when... | SchrodingersCat | 278,967 | <p>$$2(x+2)(x-1)^3-3(x-1)^2(x+2)^2=0$$
$$(x-1)^2(x+2)\left[2(x-1)-3(x+2)\right]=0$$
$$(x-1)^2(x+2)(-x-8)=0$$
$$(x-1)^2(x+2)(x+8)=0$$</p>
<p>So this polynomial of degree $4$ has $4$ real roots i.e. $1,1,-2,-8$.</p>
|
1,044,507 | <p>Sample:
$$∀x ∈ R+,∃y ∈ R+, x < y ⇒ x > y$$</p>
<p>Say I tried <code>y = 5</code>. Do I need to check if the consequent is true for just the x values less than 5?</p>
<p>Secondly, Since there is no value y that makes the antecedent true, is this statement true since there are no counter examples? The implicat... | Dan Christensen | 3,515 | <p>I'm not sure what you mean by "check if the consequent is true," but we can prove </p>
<p>$∀x ∈ R+:∃y ∈ R+: [x < y \implies P(x,y)]$ </p>
<p>for <em>any</em> binary predicate $P$. As Henning pointed out, the required $y$ could be just $x$ itself.</p>
<p>The proof makes use of the following facts:</p>
<p>$\for... |
11,178 | <p>As far as I know, one way to take a homotopy colimit in a model category is to replace (up to acyclic fibration) all arrows in the diagram with cofibrations, and take the strict colimit of the resulting diagram.</p>
<p>In Top with the model structure given by Serre fibrations, cofibrations, and weak equivalences, if... | Charles Rezk | 437 | <p>(I'll assume that in a general model category $\mathcal{C}$, $\mathrm{Cyl}(X)$ really means: a factorization of $A\to X$ into a cofibration $A\to \mathrm{Cyl}(X)$ followed by a trivial fibration $\mathrm{Cyl}(X)\to X$.)</p>
<p>A sufficient condition on objects for the map in question 1 to be a weak equivalence, is ... |
1,354,490 | <p>Prove this function is injective $f(x)=x+\mod(x,7)$.</p>
<p><strong>Attempt:</strong></p>
<p>I tried separating in two cases: $x \equiv y \pmod 7$ and $x \not \equiv y \pmod 7 $:</p>
<p>First case:
$$f(x)=f(y) \iff x+\mod(x,7)=y+ \mod (y,7)\implies x= y
$$</p>
<p>But I couldn't prove the second case.</p>
| Steven Alexis Gregory | 75,410 | <p>We have $f:\mathbb Z \to \mathbb Z$ defined by
$f(x) = x + \operatorname{mod}(x,7)$</p>
<p>Define $g:\mathbb Z \to \mathbb Z$ by $g(y) = y - \operatorname{mod}(4y,7)$</p>
<p>Suppose $x = 7a + b$ where $0 \le b \lt 7$. Then $f(x) = 7a + b + b = 7a + 2b$.</p>
<p>\begin{align}
g(f(x))
&= g(7a + 2b)\\
&a... |
4,227,800 | <p>Let <span class="math-container">$p=10007$</span> which is prime. I want to find the number of matrices X of <span class="math-container">$2\times2$</span> dimension with elements from <span class="math-container">$\mathbb{Z}_p$</span> for which <span class="math-container">$X^2\equiv I$</span> (mod <span class="mat... | Alan Abraham | 823,763 | <p>Let's say <span class="math-container">$X=\begin{bmatrix} a&b\\c&d\end{bmatrix}$</span>, after computing <span class="math-container">$X^2$</span>, we get the following system of congruences
<span class="math-container">$$\begin{cases} a^2+bc\equiv 1\mod p\\b(a+d)\equiv 0\mod p\\c(a+d)\equiv 0\mod p\\d^2+bc\... |
4,227,800 | <p>Let <span class="math-container">$p=10007$</span> which is prime. I want to find the number of matrices X of <span class="math-container">$2\times2$</span> dimension with elements from <span class="math-container">$\mathbb{Z}_p$</span> for which <span class="math-container">$X^2\equiv I$</span> (mod <span class="mat... | paul garrett | 12,291 | <p>As in @ThomasAndrews' comment, there is indeed a systematic (not-so-computational) way to approach this, maybe less lending itself to computational errors and such.</p>
<p>So, for a two-by-two matrix with <span class="math-container">$x^2=1$</span>, certainly <span class="math-container">$x^2-1=0$</span>, so either ... |
177,144 | <p>If G is a finite group, I understand that the category of RO(G)-graded spectra, when rationalized, becomes Quillen equivalent to the category of Mackey functors valued in chain complexes of rational vector spaces.</p>
<p>How does RO(G) act on the category of Mackey functors? For instance, if F = F(G/H) is a Mackey... | Peter May | 14,447 | <p>The term ``RO(G)-graded spectrum'' is a misnomer, not to be used.
It is never used in the literature I'm familiar with, and it never
should be. There are several Quillen equivalent models for the
category of genuine $G$-spectra, and such animals represent
$RO(G)$-graded cohomology theories, but it is meaningless t... |
2,353,142 | <p>Solve:
$$(\cot^{-1} (x))^2 - 4\cot^{-1} (x) + 3 \geq 0$$.</p>
<p>My Attempt:
$$(\cot^{-1} (x))^2 - 4\cot^{-1} (x) + 3 \geq 0$$.
Let $\cot^{-1} (x)=t$. then</p>
<p>$$t^2-4t+3\geq 0$$
$$(t-3)(t-1)\geq 0$$
Either, $\cot^{-1} (x) \leq 1$</p>
<p>Or, $\cot^{-1} (x) \geq 3$</p>
<p>I solved till here, but couldn't get ... | Davide Giraudo | 9,849 | <p>Assume that $p\gt 1$ is such that inequality $2|x||y| \leqslant x^p + y^p$ holds for any $x,y\in\geqslant 0$. In particular, it holds for $x=y\geqslant 0$ hence we have for any positive $x$ the following inequality:
$$\tag{*} x^2 \leqslant x^p .$$
If $p\gt 2$, then divide in (*) by $x^2$ and let $x$ go to zero (... |
211,427 | <p>Can't seem to figure this one out. Could anyone help me out and explain it to me?<br>
Thank you.</p>
<p>Let $P$ and $Q$ be relations on $Z$ by x$P$y iff x + 1 <= y and a$Q$b iff a + 2 <= b. Prove that P $\circ$ Q = {(p,q) belonging to ZxZ | p + 3 <= q} </p>
| Yury | 35,791 | <p>Every bi-linear form $\sigma:V\times V \to F$ defines a linear operator $A$ from $V$ to $V^*$; $A$ maps vector $v\in V$ to linear functional $l_v$ such that $l_v(u) = \sigma(u,v)$. The correspondence between $\sigma$ and $A$ is “canonical” — it doesn't depend on the basis. </p>
<p>Now suppose that we fix a basis $e... |
3,411,081 | <p>A group of 12 people are going out to a concert on Saturday night. The group will take three cars with four people in each car. If they distribute themselves at random, what is the probability that A and B will be in the same car?</p>
<p>I tried (12C2*10C2*8C4*4C4)/(12C4*8C4*4C4) because you're choosing two first a... | amd | 265,466 | <p>Write down <span class="math-container">$1+1+\cdots+1=X$</span>. Now group runs of adjacent ones by going from left to right and for each <span class="math-container">$+$</span> sign you encounter deciding whether or not a group boundary occurs there. Parenthesize each group to get a partition of <span class="math-c... |
2,592,007 | <blockquote>
<p>Let $V$ be a vector space and $W,U\subseteq V$ subspaces s.t $W\not \subseteq U$
$\dim(V)=5, \dim(W)=2, \dim(U)=4$ </p>
<p>Prove\Disprove: $\dim(U\cap W)=1$</p>
</blockquote>
<p>So I started with \begin{align} & \dim(W+U)=\dim(U)+\dim(W)-\dim(U\cap W) \\[10pt]
\iff & \dim(W+U)=4+2-\di... | angryavian | 43,949 | <p>It remains to prove $\dim(U \cap W) \le 1$.
You already know $\dim(U \cap W) \le \dim(W) = 2$.
If you can show that $W \not\subseteq U$ implies $\dim(U \cap W) \ne 2$, then you are finished.</p>
<hr>
<p>Proving the unproved claim above (via contrapositive): if $\dim(U \cap W) = 2$, then $U \cap W = W$ because $\di... |
1,270,042 | <p>$$(a+5)(b-1)=ab-a+5b-5=20-5=15.$$</p>
<p>So, both $a + 5$ and $b-1$ divide $15$. </p>
<p>Then, $a + 5$ is one of $15, -15, 3, -3, 5, -5, 1, -1$, so $a$ is one of $10, -20, -2, -8, 0, -10, -4, -6$ and $b – 1$ is one of $15, -15, 3, -3, 5, -5, 1, -1$, so $b = 14, -14, 4, -2, 6, -4, 2, 0$.</p>
<p>Could all possibili... | Edwin Gray | 231,072 | <p>From ab = a -5b +20, we have b(a + 5) = a + 5 + 15.
Therefore a + 5 divides 15 and a must be positive.Hence a = 10.
Then 15b = 15 + 15, and b = 2. Edwin Gray</p>
|
872,889 | <p>What determinant is zero? What equation does this give for the plane?</p>
<p>I need some help here, am pretty stuck</p>
| GEdgar | 442 | <p>Five years ago or so I worked on some "tree-type" continued fractions...
$$
T =
1+\frac{\displaystyle
1+\frac{\displaystyle
1+\frac{\displaystyle
1+\frac{1+\cdots\;}{2+\cdots\;}
}{\displaystyle
2+\frac{2+\cdots\;}{4+\cdots\;}
}
}{\displaystyle
2+\frac{\displaystyle
2+\frac{2+\cdots\;}{4+... |
2,512,363 | <h2>Defining the barycentre and finding its variance</h2>
<p>I have a set of $N$ points at the locations $x_i$ which has weights $W_i$, $i=1,\ldots, N$ and want to find the barycenter (or center of gravity) </p>
<p>$$ B = {\sum_{i=1}^N W_i x_i\over \sum_{i=1}^N W_i}.$$</p>
<p>I want to find the variance of $B$.
I a... | Mixopteryx | 187,956 | <p>Through <a href="https://tel.archives-ouvertes.fr/tel-00783671" rel="nofollow noreferrer" title="Ladroit PhD thesis">this PhD thesis</a> (chapter 3.2.3) i found the following solution.</p>
<p>For simplicity, define
$$F_i = {W_i\over\sum W_i}, \text{ so that } B=\sum_{i=1}^N F_i x_i.$$
(As Deans answer suggest)</p... |
1,919,912 | <blockquote>
<p>Let $D$ be the Integral Domain with characteristic $m>0$. Prove that $m$ is prime.</p>
</blockquote>
<p>My Proof: </p>
<p>Since the characteristic of $D$ is $m$, $m\cdot b=0$ for all $b\in D$ and if $n\cdot b=0$ for all $b\in D$, then $m\leq n$. </p>
<p>Assume that $m$ is composite number. Then ... | egreg | 62,967 | <p>If your definition of integral domain requires an identity, it's easy: the unique ring homomorphism (preserving identities)
$$
\chi\colon\mathbb{Z}\to D
$$
has kernel $m\mathbb{Z}$, so $\mathbb{Z}/m\mathbb{Z}$ is isomorphic to a subring of $D$ and therefore is a domain. Hence either $m=0$ or $m$ is prime.</p>
<p>If... |
1,919,912 | <blockquote>
<p>Let $D$ be the Integral Domain with characteristic $m>0$. Prove that $m$ is prime.</p>
</blockquote>
<p>My Proof: </p>
<p>Since the characteristic of $D$ is $m$, $m\cdot b=0$ for all $b\in D$ and if $n\cdot b=0$ for all $b\in D$, then $m\leq n$. </p>
<p>Assume that $m$ is composite number. Then ... | QED | 91,884 | <p>As an integral domain has no zero divisors, if it has a finite characteristic it is itself finite. So it is a field and contains an unit element, $1$. Let $m=a.b$ if possible, for some integers $a,b>1$. Then $a.1=a\in D$. Let all the elements of $D$ be $x_1,\cdots,x_n$. If $a.x_i=a.x_j$ for $i\neq j$ then as $D$ ... |
817,386 | <p>For a function $f$ I know that: $$\int{f'(r)dr}=f(r)$$ where $f(r)$ is known. knowing the result of this integral how can i calculate $$\int{(f'(r))^2dr}$$ Is there any relation between these integrals?</p>
| Robert Bryant | 84,371 | <p>While there is no explicit formula of the exact kind you desire, if one is willing to reparametrize, there is an integral-free parametrization of the curves of the form $\bigl(x,f(x),g(x)\bigr)$ for which $g'(x) = (f'(x))^2$: As long as $f''(x)$ is non-vanishing, one can reparametrize such a curve in the form
$$
\bi... |
1,800,233 | <blockquote>
<p>If $n$ is composite and $\phi{(n)} | (n - 1)$ then prove that $n$ has at least four distinct prime factors.</p>
</blockquote>
<p><strong>Attempt:</strong></p>
<p>Since $n$ is not a prime, let's first take the case that $n$ is squarefree. Then $n = a_1 \cdot a_2 \cdots a_r$ where $a_i$ are the prime ... | alphacapture | 334,625 | <p>This solves the 2 prime factor case, which together with user5713492's answer, solves the problem:</p>
<p>Suppose we have primes $p$ and $q$ such that $\phi(pq)=(p-1)(q-1)|pq-1$. Then we have:</p>
<p>$$pq\equiv1\pmod{(p-1)(q-1)}$$</p>
<p>so</p>
<p>$$pq\equiv1\pmod{p-1}$$</p>
<p>and also</p>
<p>$$p\equiv1\pmod... |
3,310,038 | <p>If <span class="math-container">$U$</span> is an open set of <span class="math-container">$\mathbb{R}^{m}$</span>, do we have that <span class="math-container">$U\times \mathbb{R}^{n-m}$</span> is an open set of <span class="math-container">$\mathbb{R}^{n}$</span>? </p>
<p>Here <span class="math-container">$\mathbb... | Community | -1 | <p>Yes, we do. Because <span class="math-container">$U×\Bbb R^{n-m}$</span> is the product of open sets. Those sets are by definition open in <span class="math-container">$\Bbb R^n$</span>.</p>
<p>You may wish to look up <a href="https://en.m.wikipedia.org/wiki/Product_topology" rel="nofollow noreferrer">product top... |
3,006,952 | <p><strong>Find coordinate in first quadrant which tangent line to <span class="math-container">$x^3-xy+y^3=0$</span> has slope 0</strong></p>
<p>First, I do implicit differentiation:</p>
<p><span class="math-container">$\frac{3x^2-y}{x-3y^2}=y'$</span></p>
<p>so I look at the numerator and go hmmm if i put in (1,3)... | user | 505,767 | <p>We have that</p>
<p><span class="math-container">$$x^3-xy+y^3=0\implies 3x^2dx-dx-xdy+3y^2dy=0 \implies \frac{dy}{dx}=\frac{3x^2-y}{x-3y^2}=0$$</span></p>
<p>that is</p>
<p><span class="math-container">$$3x^2=y \implies (x,y)=(t,3t^2)$$</span></p>
<p><span class="math-container">$$x^3-xy+y^3=0 \iff t^3-3t^3+27t^... |
3,006,952 | <p><strong>Find coordinate in first quadrant which tangent line to <span class="math-container">$x^3-xy+y^3=0$</span> has slope 0</strong></p>
<p>First, I do implicit differentiation:</p>
<p><span class="math-container">$\frac{3x^2-y}{x-3y^2}=y'$</span></p>
<p>so I look at the numerator and go hmmm if i put in (1,3)... | Andrei | 331,661 | <p>You solved only half of the problem. You have that the derivative is <span class="math-container">$0$</span>, but you also need to use the fact that the point is on the graph of your line. You have two equations with two unknowns. Since they are not linear equations, you might have multiple solutions.</p>
<p>Just p... |
1,999,834 | <p>Let $\varphi : G \rightarrow H$ be a group homomorphism with kernel $K$ and let $a,b \in \varphi(G)$. Let $X = \varphi^{-1}(a)$ and $Y = \varphi^{-1}(b)$. Fix $u \in X$. Let $Z=XY$. Prove that for every $w \in Z$ that there exists $v \in Y$ such that $uv=w$. This is Dummit and Foote exercise 3.1.2.</p>
<p>My attemp... | Math Helper | 374,223 | <p>You are also supposing that $w\in Z$ correct? That is, $w\in XY$ so that $w=mn$ where $m\in X$ and $n\in Y$ and so $\phi(w)=\phi(mn)=\phi(m)\phi(n)=ab$.</p>
|
3,371,922 | <p>The definition of the limit states that limit of <span class="math-container">$f(x)$</span> when <span class="math-container">$x$</span> approaches <span class="math-container">$c$</span> is <span class="math-container">$L$</span> iff for every <span class="math-container">$\epsilon > 0$</span> there exists <spa... | José Carlos Santos | 446,262 | <p>The idea is that if, for instance,<span class="math-container">$$f(x)=\begin{cases}1&\text{ if }x=0\\2&\text{ otherwise,}\end{cases}$$</span>then <span class="math-container">$\lim_{x\to0}f(x)=2$</span>. The fact that <span class="math-container">$f(0)=1$</span> is not relevant here. What matters is that whe... |
2,578,444 | <blockquote>
<p><span class="math-container">$\tan x> -\sqrt 3$</span></p>
</blockquote>
<p>How do I solve this inequality?</p>
<p>From the <a href="https://www.desmos.com/calculator/qb8bg1vbsf" rel="nofollow noreferrer">graph</a> it is evident that <span class="math-container">$\tan x>-\sqrt 3$</span> for <span ... | Roger Figueroa Quintero | 253,300 | <p>There is an error when analyzing the function at $\pi/2$, since at that point there is a discontinuity. The correct way to proceed is analyzing the graph between $-\pi/2$ and $\pi/2$, in this way you get that the solution of $\tan(x)>-\sqrt{3}$ is $(-\pi/3,\pi/2)$ and adding the period $\pi$ you get $(n\pi-\pi/3,... |
3,691,255 | <p>Pierre runs a game at a fair, where each player is guaranteed to win $10. </p>
<p>Players pay a certain amount each time they roll an unbiased die, and must keep rolling until a ‘6’ occurs. </p>
<p>When a ‘6’ occurs, Pierre gives the player $10 and the game concludes. </p>
<p>On average, Pierre wishes to make a p... | Alexey Burdin | 233,398 | <p>The probability of not rolling <span class="math-container">$6$</span> for <span class="math-container">$k$</span> times is <span class="math-container">$\left(\frac{5}{6}\right)^k$</span>.<br>
The probability of not rolling <span class="math-container">$6$</span> for <span class="math-container">$k-1$</span> first ... |
661,771 | <p>I am stuck on the following problem that says : </p>
<blockquote>
<p>Which of the following is a solution to the differential equation $y'=|y|^{\frac12},y(0)=0\,$ where square root means the
positive square root ? </p>
<ol>
<li><p>$y(t)=\frac{t^2}{4}$ </p></li>
<li><p>$y(t)=-\frac{t^2}{4}$ </p></li>... | Community | -1 | <p>No I do not think this is correct. The idea seems correct, but the execution was poor. You should specify that $y\in X\backslash D$. I am also not sure how you justify your last inequality. If $t$ is arbitrary in $X\backslash D$ we cannot conclude $d(z,t)<r_1$ and $-d(z,t)>-r_1.$ </p>
<p>Here is how I would ... |
2,067,097 | <p>Given are two points on a line with coordinates. How do we calculate the third forming a perfect 60 degree triangle? So we have X,Y, but need Z...</p>
<p>X: 0,0    ( 0,0 i.e. horizontal, vertical )<br>
Y: 50, 0 <br>
Z: 25, ??</p>
<p>How to calculate the missing horizontal coordinate for Z? Formi... | Zin | 400,692 | <h2>With the video explanation</h2>
<p>You are asking for the intersection of two circles (this is <strong>not</strong> creating an <em>equilateral</em> triangle). You can take the equations of your circles $c_1: x^2+y^2=50^2$ and $c_2: (x-50)^2+y^2=50^2$.</p>
<p>$$x = \frac{d^2-r^2+R^2}{2d}$$
For your question the d... |
427,564 | <p>I'm supposed to give a 30 minutes math lecture tomorrow at my 3-grade daughter's class. Can you give me some ideas of mathemathical puzzles, riddles, facts etc. that would interest kids at this age?</p>
<p>I'll go first - Gauss' formula for the sum of an arithmetic sequence.</p>
| AD - Stop Putin - | 1,154 | <p>Let them try to create maps that need as few colours as possible. The rule is that two counties are neighbours if their borders meet in more than a finite number of points and neighbours should not have the same colour. Hopefully, this might lead to an interesting discussion about the 4-colour theorem. </p>
<p><a h... |
427,564 | <p>I'm supposed to give a 30 minutes math lecture tomorrow at my 3-grade daughter's class. Can you give me some ideas of mathemathical puzzles, riddles, facts etc. that would interest kids at this age?</p>
<p>I'll go first - Gauss' formula for the sum of an arithmetic sequence.</p>
| ZeerakW | 83,653 | <p>I think Map Colouring and Probability are a great ideas, as mentioned above.</p>
<p>I would also suggest finding the highest prime, it requires multiplication/division, which is always good to practice.</p>
<p>Otherwise you could consider having them enact Bubble Sort[1], based on their height or birthday.</p>
<p... |
427,564 | <p>I'm supposed to give a 30 minutes math lecture tomorrow at my 3-grade daughter's class. Can you give me some ideas of mathemathical puzzles, riddles, facts etc. that would interest kids at this age?</p>
<p>I'll go first - Gauss' formula for the sum of an arithmetic sequence.</p>
| Yuri | 83,680 | <p>I think most posters overestimate what you can learn a bunch of nine year olds in 30 minutes. Remember you are not 1-on-1 with the brightest student, but you need to have the full class being able to follow you. They maybe have had division and such in class but I imagine it isn't their second nature yet, so you can... |
2,196,936 | <p>How to prove that $7p + 3^p -4$ is not a perfect square? </p>
<p>I calculated: $\left(\frac{7p+3^p-4}{p}\right) = \left(\frac{-1}{p}\right)$. So if $p \equiv 3 \mod 4$, the result is $-1$. So in that case, $7p+3^p -4$ can't be a square. But what about the case $p \equiv 1 \mod 4$? Any hints? Thanks in advance.</p>
| Alex R. | 22,064 | <p>If $p=4k+1$, then:</p>
<p>$$7p+3^p-4\equiv -1+(-1)\pmod{4}=-2\pmod{4}.$$</p>
<p>But perfect squares must be 0,1 mod 4. </p>
|
3,779,589 | <p>Let the metric <span class="math-container">$d$</span> be defined as
<span class="math-container">$$
d(f,g) =\sup_{x\in[0,1]}|f(x)-g(x)|,
$$</span>
and let<br />
<span class="math-container">$$
H(x) = \begin{cases} 0 \text{ if } x \leq \frac{1}{2}\\ 1 \text { if } x > \frac{1}{2} \end{cases}.
$$</span>
Is <span c... | PrincessEev | 597,568 | <p>I wouldn't call this approach "by contradiction;" rather, if anything, it is typically referred to as the complementary approach. Want to find a quantity satisfying a condition? The complementary approach is to find the overall quantity disregarding such a condition, and then subtract off the ones <em>not<... |
3,041,907 | <p>I am unable to isolate the variable <span class="math-container">$x$</span> of this inequality <span class="math-container">$y \leq \sqrt{2x-x^2}$</span> ( where <span class="math-container">$0 \leq y \leq 1 $</span>)</p>
<p>Is it correct doing this: <span class="math-container">$y^2 \leq 2x-x^2$</span>?
I found ... | hamam_Abdallah | 369,188 | <p>Let <span class="math-container">$f(x)=\sqrt{2x-x^2}$</span></p>
<p>its domain is <span class="math-container">$[0,2]$</span>.</p>
<p><span class="math-container">$$f'(x)=\frac{1-x}{f(x)}$$</span></p>
<p>the maximum is <span class="math-container">$f(1)=1$</span>.</p>
<p>Let <span class="math-container">$y\in[0,... |
3,041,907 | <p>I am unable to isolate the variable <span class="math-container">$x$</span> of this inequality <span class="math-container">$y \leq \sqrt{2x-x^2}$</span> ( where <span class="math-container">$0 \leq y \leq 1 $</span>)</p>
<p>Is it correct doing this: <span class="math-container">$y^2 \leq 2x-x^2$</span>?
I found ... | egreg | 62,967 | <p>Since you're assuming <span class="math-container">$y\ge0$</span>, the given inequality is equivalent to <span class="math-container">$y^2\le2x-x^2$</span>, that is, <span class="math-container">$x^2-2x+y^2\le0$</span>.</p>
<p>The discriminant of <span class="math-container">$x^2-2x+y^2$</span> is <span class="math... |
240,461 | <p>What's the mathematica command to get the <strong>numerical value</strong> of :</p>
<p><span class="math-container">$$PV\int_0^\infty \frac{\tan x}{x}\text{d}x?$$</span></p>
<p>where <span class="math-container">$PV$</span> is the principal value.</p>
| Michael E2 | 4,999 | <p>The method <code>"PrincipalValue"</code> does not work on an integral with infinitely many poles, since each pole must be specified. Confining to a finite region of integration is problematic on integrals like <span class="math-container">$\int_0^\infty (\tan x/x)\,dx$</span> that converge slowly. <code>... |
1,319,476 | <p>This is a question related to another posted question:</p>
<p>The answer to the following question "Find all solutions to: $e^{ix}=i$" is as follows: </p>
<p>"Euler's formula: $e^{ix}=\cos(x)+i\sin(x)$,</p>
<p>so: $ \cos x+i\sin x=0+1⋅i$</p>
<p>compare real and imaginary parts
$\sin(x)=1$
and
$\cos(x)=0$</p>
<p... | Lucian | 93,448 | <p><strong>Hint:</strong></p>
<ul>
<li><p>$i$ is <strong>a point on the unit circle</strong>.</p></li>
<li><p>$e^{ix}$ is <em>also</em> <strong>a point on the unit circle</strong>, lying <em>x</em> radians away from $(1,0)$, in trigonometric or counterclockwise direction.</p></li>
</ul>
<p>So, to answer a question wi... |
2,507,613 | <p>I am trying to teach myself group theory and I recently came across the topic of Isomorphisms.
I know that 2 groups are isomorphic if there is a one-on-one correspondence between their elements.
So if the groups have a different order, does that mean they are not isomorphic? Such as a group $S$ and its permutation g... | José Carlos Santos | 446,262 | <p>The answer is “yes”, but your definition of isomorphism is not correct. An isomorphism between groups is a bijection <span class="math-container">$\varphi$</span> which preserves the product, that is, such that <span class="math-container">$\varphi(x.y)=\varphi(x).\varphi(y)$</span> for every <span class... |
2,445,655 | <p>Challenge: A Good Deal</p>
<p>You are currently learning some important aspects of collusion and cartels. This challenge puts you in the position of a bad guy, namely a price-fixing sales manager. Suppose that you find yourself in a so-called “smoke-filled room” to fix prices for the upcoming year with the sales ma... | sharpe | 217,520 | <p>Define $t:=-x$. Then,
\begin{align*}
&\lim_{x \to -\infty}2x+\sqrt{4x^2+x}=\lim_{t \to \infty} \left( -2t+\sqrt{4t^2-t} \right)\\
&=\lim_{t \to \infty} \left( -2t+\sqrt{4t^2-t} \right)\cdot\frac{2t+\sqrt{4t^2-t}}{2t+\sqrt{4t^2-t}} =\lim_{t \to \infty}\frac{-t}{2t+\sqrt{4t^2-t}} \\
&=\lim_{t \to \infty}\... |
2,445,655 | <p>Challenge: A Good Deal</p>
<p>You are currently learning some important aspects of collusion and cartels. This challenge puts you in the position of a bad guy, namely a price-fixing sales manager. Suppose that you find yourself in a so-called “smoke-filled room” to fix prices for the upcoming year with the sales ma... | farruhota | 425,072 | <p>Note that for $x\to -\infty$, $4x^2+x=(2x+\frac14)^2-\frac{1}{16} \sim (2x+\frac14)^2.$ Hence:
$$\lim_{x\to-\infty} \left(2x+\sqrt{4x^2+x}\right)=\lim_{x\to-\infty}\left(2x-\left(2x+\frac14\right)\right)=-\frac14.$$</p>
|
4,135,472 | <p><strong>What is the clearest and simplest way of proving that <span class="math-container">$[x]+[x+1/2]=[2x]$</span>? (Where <span class="math-container">$[x]$</span> is the greatest integer function)</strong></p>
<p>According to Bartleby, if <span class="math-container">$x=m$</span> for <span class="math-container"... | WhatsUp | 256,378 | <p>For <span class="math-container">$x > 0$</span>:</p>
<p><span class="math-container">$[2x]$</span> is the number of positive integers <span class="math-container">$\leq 2x$</span>. Count them by separating the even and odd numbers.</p>
<p>For general <span class="math-container">$x$</span>: add a sufficiently lar... |
3,145,896 | <h1>Solve for <span class="math-container">$x$</span></h1>
<p>I have an equation that I have been working on solving; I know the solution, but I cannot get to it myself. Almost every simplification I do reverts back to a previous step. Can anyone show me how to solve for <span class="math-container">$x$</span> in this... | Community | -1 | <p>Taking the base-<span class="math-container">$6$</span> antilogarithm,</p>
<p><span class="math-container">$$(2x-3)(x+5)=6^{\log_3x}=e^{\ln6\ln x/\ln3}=x^{\ln6/\ln3}.$$</span></p>
<p>Because of the irrational exponent, there is no closed-form solution and you need to use a numerical method.</p>
|
3,242,553 | <p>I got two sequences of stochastic process <span class="math-container">$(X_n(t))_{t \in [0,1]}$</span> and <span class="math-container">$(Y_{n}(t))_{t \in [0,1]}$</span>, defined on a probability space <span class="math-container">$(\Omega, \mathcal{F},P)$</span>, and know that their distance in the sup-norm on <spa... | kimchi lover | 457,779 | <p>What you want is <a href="https://en.wikipedia.org/wiki/Slutsky%27s_theorem" rel="nofollow noreferrer">Slutsky's theorem</a>. If, for some <span class="math-container">$t$</span>, the sequence <span class="math-container">$X_n(t)$</span> converges in distribution, and if <span class="math-container">$Y_n(t)-X_n(t)$<... |
3,917,255 | <p>Why does Chebyshev's inequality demand that <span class="math-container">$\mathbb{E(}X^2) < \infty$</span>?</p>
| Emmanuel C. | 619,398 | <p>Indeed, <span class="math-container">$f$</span> is a closed map. Take a closed subset <span class="math-container">$F\subseteq \mathbb S^1\times\mathbb S^1$</span>. Since <span class="math-container">$\mathbb S^1\times \mathbb S^1$</span> is compact, it follows that <span class="math-container">$F$</span> is compact... |
2,903,163 | <p>I don't really know whether to put this in Physics forums since it is relating to Mechanics, or Math since the question is actually about the math being done. Don't criticize me over it.</p>
<p>So for the question: I was doing some review problems on Lagrange's equations, KE+PE, and I found <a href="http://wwwf.impe... | Robert Lewis | 67,071 | <p>To convert the cartesian expression for kinetic energy,</p>
<p>$T = \dfrac{m}{2}(\dot x^2 + \dot y^2 + \dot z^2) \tag 1$</p>
<p>into sperical coordinates $r,\phi, \theta$ such that</p>
<p>$x = r \sin \theta \cos \phi, \tag 2$</p>
<p>$y = r\sin \theta \sin \phi, \tag 3$</p>
<p>$z = r\cos \theta, \tag{4}$</p>
<p... |
477,477 | <p>Prove that $e^x=-x^2+2x+5 $ have exactly two solutions.</p>
<p>Is it enoguht that Vertex of the parabola is over $y=e^x$ and arms of it looks down</p>
| Ross Millikan | 1,827 | <p>The basic idea is that quadrilaterals average $90^\circ$ angles. If four of them meet at every corner, you wont have the required $720^\circ$ deficit to make a closed sphere. You need eight three-way vertices to get it to close. This is like the requirement that using hexagons and pentagons you need 12 pentagons ... |
2,868,595 | <p>A Vitali set is a subset $V$ of $[0,1]$ such that for every $r\in \mathbb R$ there exists one and only one $v\in V$ for which $v-r \in \mathbb Q$. Equivalently, $V$ contains a single representative of every element of $\mathbb R / \mathbb Q$.</p>
<p>The proof I read is in this short article on Wikipedia: <a href="h... | DanielWainfleet | 254,665 | <p>Notation: $x+S=S+x=\{x+s:s\in S\}$ when $x\in \Bbb R$ and $S\subset \Bbb R.$</p>
<p>Let $q_0$ be the unique member of $V\cap \Bbb Q.$ Let $V_0=V+(-q_0).$ Let $W_0=\cup_{n\in \Bbb Z}(n+V_0).$ For $q\in (0,1)\cap \Bbb Q$ let $W_q=q+W_0.$</p>
<p>Observe that $(-q)+W_q=W_0$ when $q\in \Bbb Q\cap [0,1)$ and that if $q,... |
2,476,181 | <p>Where Ω = {1,2,...,p}, all Ω are equally likely and p is prime how would I show that if A and B are independent events then at least one of A and B is either ∅ or Ω?</p>
| Bram28 | 256,001 | <p>What you calculated is all the different arrangements of $9$ consonants and $4$ vowels, and you did this correctly, but you forgot to arrange the vowels <em>among</em> the consonants, so you need to multiply all of this by ${13 \choose 4}$</p>
|
2,197,790 | <h3>Question</h3>
<blockquote>
<p>A sequence $\{a_n\}$ of real numbers is said to be a Cauchy sequence of for
each $\epsilon$ > 0 there exists a number $N > 0$ such that m, $n > N$ implies
that $|a_n − a_m| <\epsilon$.</p>
<p>Prove that every convergent sequence is a Cauchy sequence</p>
</blockquot... | Jacob Wakem | 117,290 | <p>If it is not Cauchy, it is obviously not convergent: for |a_n -a_m| is greater than some epsilon for arbitrarily large n,m. This assures a_n and a_m do not both get within epsilon/2 of some a_infinity. </p>
|
25,260 | <h2>TL;DR:</h2>
<hr />
<p>Tell me which topics should i study the most, based on this three tests:</p>
<p>Mathematics (A):
<a href="https://www.studyinjapan.go.jp/ja/_mt/2021/06/2020_ga_math_a.pdf" rel="nofollow noreferrer">2020</a>
<a href="https://www.studyinjapan.go.jp/ja/_mt/2021/06/2019_ga_math_a.pdf" rel="nofollo... | Alexander Woo | 4,991 | <p>I would say the assumption is that people heading to mathematics graduate school know about the Wronskian, but this assumption isn't universally true.</p>
<p>Certainly, anyone who has studied a semester of differential equations (and is heading to graduate school) should know it.</p>
<p>However, there is a substanti... |
74,271 | <p>Hello, all!</p>
<p>I have a big sum of log-normal (with location parameter $\mu$ and scale parameter $\sigma$) random variables $X_i$ $\sum_{i=1}^N X_i$ with $N \gg 1$.
How could I estimate convergence rate to a gaussian distribution relative to $\mu$ and $\sigma$?</p>
<p>Thank you.</p>
| Igor Rivin | 11,142 | <p>Log normal distribution has finite variance, so if you subtract the mean, the magic words are "Berry-Esseen theorem". If you don't subtract the mean, the sum diverges.</p>
|
393,378 | <p>Let <span class="math-container">$K=\mathbb{Q}(x)$</span> be the rational functions in one variable <span class="math-container">$x$</span> and let the automorphisms <span class="math-container">$\phi,\psi$</span> of <span class="math-container">$K$</span> be defined as <span class="math-container">$\phi(x)=-\frac{1... | GNiklasch | 49,003 | <p>As hinted in a comment, this is a special case of <a href="https://en.wikipedia.org/wiki/L%C3%BCroth%27s_theorem" rel="nofollow noreferrer">Lüroth's theorem</a>, but it's not hard to find an explicit generator for <span class="math-container">$K_0$</span>.</p>
<p>Note that <span class="math-container">$$x+\phi(x)+\p... |
1,568,233 | <p>$$\sum_{n=1}^{\infty}n^210^{-n} = \frac{110}{3^6}$$
I noticed this while playing around on my calculator. Is it true and how come?</p>
| Bernard | 202,857 | <p>This is because:
\begin{align*}
\sum_n n^2x^n&=x^2\sum_n n(n-1)x^{n-2}+x\sum_n nx^{n-1}\\
&=x^2\Bigl(\frac 1{1-x}\Bigr)''+x\Bigl(\frac 1{1-x}\Bigr)'\\
&=\frac{2x^2}{(1-x)^3}+\frac x{(1-x)^2}=\frac{x^2+x}{(1-x)^3}=\color{red}{\frac{x(x+1)}{(1-x)^3}}.
\end{align*}
Then set $x=\dfrac1{10}$.</p>
|
4,638,170 | <p>I'm trying to write a proof to show that a tree structure of finite nodes terminate.</p>
<p>Suppose we can say that either a node is a parent of another node (<span class="math-container">$Pqp$</span>: <span class="math-container">$q$</span> is the parent of <span class="math-container">$p$</span>), or it is a termi... | Samuel Adrian Antz | 1,045,826 | <p>As requested by ronno, here is my comment as a full answer with more details (like both directions) and references (as a more general view on the concepts might be helpful). For completeness, I have also included the backwards direction:</p>
<p><strong>Lemma</strong>: <span class="math-container">$X$</span> connecte... |
1,898,207 | <blockquote>
<p>A man speaks the truth $8$ out of $10$ times. A fair die is thrown. The man says that the number on the upper face is $5$. Find the probability that the original number on the upper face is $5$.</p>
</blockquote>
<p>While solving I find two ways (shown in the image). I think one of them is correct an... | Patrick Stevens | 259,262 | <p>Your first one is false. $P(X \mid T)$ is $1$, not $\frac{1}{6}$. Given that the man speaks the truth, the die definitely showed $5$.</p>
<p>Your second method is much more natural, and it is correct (assuming you plugged the numbers in correctly).</p>
|
255,761 | <p>Given a graph of n vertices, it is possible to plot a discrete signal (or function) of n samples on the vertices of the graph, so that one can visualize the features of the signal on the graph (see the attached image)?</p>
<p><a href="https://i.stack.imgur.com/4s3Qz.png" rel="noreferrer"><img src="https://i.stack.im... | flinty | 72,682 | <pre><code>g = PetersenGraph[];
(* get the 3d coordinates of the graph vertices *)
coords = Append[#, 0] & /@ GraphEmbedding[g];
points = Point[coords];
(* create lines between the edges *)
connections = EdgeList[g];
lines = Line[coords[[#]]] & /@ (connections /. UndirectedEdge -> List);
(* generate some ... |
255,761 | <p>Given a graph of n vertices, it is possible to plot a discrete signal (or function) of n samples on the vertices of the graph, so that one can visualize the features of the signal on the graph (see the attached image)?</p>
<p><a href="https://i.stack.imgur.com/4s3Qz.png" rel="noreferrer"><img src="https://i.stack.im... | kglr | 125 | <pre><code>SeedRandom[1]
signals = RandomReal[1, 10];
</code></pre>
<p>To get a 3D graph that looks like the one in OP, we can use <code>signals</code> to specify a custom <code>VertexShapeFunction</code> and use it with <code>PetersenGraph</code>:</p>
<pre><code>PetersenGraph[
EdgeStyle -> Directive[Dashed, Red],
... |
3,242,844 | <p><span class="math-container">$$\int_0^{\pi/6} \frac{x\cos x}{1+2\cos x}dx$$</span></p>
<p>Does it have a closed solution? <a href="https://www.wolframalpha.com/input/?i=int%20%5Cfrac%7Bxcos%20x%7D%7B1%2B2cos%20x%7Ddx%20from%200%20to%20%5Cpi%2F6" rel="nofollow noreferrer">WA</a> outputs this result.</p>
| clathratus | 583,016 | <p><strong>(Basically) Complete Answer</strong></p>
<p>We define
<span class="math-container">$$f(t)=\int_0^t \frac{\cos x}{1+2\cos x}dx$$</span>
Then from integration by parts,
<span class="math-container">$$J=\int_0^{\pi/6}\frac{x\cos x}{1+2\cos x}dx=\frac{\pi}{6}f\left(\frac\pi6\right)-\int_0^{\pi/6}f(t)dt\, .$$</... |
42,301 | <p>everyone! I am sorry, but I am an abcolute novice of Mathematica (to be more precise this is my first day of using it) and even after surfing the web and all documents I am not able to solve the following system: </p>
<pre><code>Solve[{y*(((y*x)/(beta*b))^(1/(beta - 1)) - v) - c*alpha ==
0, ((x/alpha))*(((y*x)... | sakra | 68 | <p>An alternate solution using <code>Fold</code>:</p>
<pre><code>Fold[If[Last[#2] < Last[#1], #2, #1] &, {0, 0, Infinity}, mya]
</code></pre>
<p>If the list is known to be non-empty, the following solution is faster:</p>
<pre><code>Fold[If[Last[#2] < Last[#1], #2, #1] &, First[mya], Rest[mya]]
</code><... |
1,835,158 | <p>I wont to choose three random integer point in origin $|x|\leq r, |y|\leq r$ at plane as $(a_{1},b_{1}),(a_{2},b_{2}),(a_{3},b_{3})$.
What the probability that this three point create a right triangle ( it is depend to r? what about isosceles triangle?
I think that its zero but I cant proof. Thank you.</p>
| Jamal Farokhi | 244,152 | <p>My attempt:
If $r\in \mathbb{N}$ then there are $(2r+1)^{2}$ integer pair point and also there are $(2r+1)^{6}$ way to chose pair three integer number. So lets that $N(r)=(2r+1)^{6}$. We try to find all points that they create a right triangle. We denoted them by $N^{\prime}(r)$. For instance in this two figure, for... |
14,458 | <p>I want to be able to plot several numerical solutions of an ODE, plus its analytical solution in one plot, in order to see how the numerical solutions converge towards the analytical one with respect to the number of steps. The method I'm using is Euler's method for the equation
$ y'(t) = 1-t +4y(t), y(0)=1$</p>
<p... | J. M.'s persistent exhaustion | 50 | <p>Sascha showed you how to use the built-in <code>"ExplicitEuler"</code> option. You mention</p>
<blockquote>
<p>I want to be able to plot several numerical solutions of an ODE, plus its analytical solution in one plot, in order to see how the numerical solutions converge...</p>
</blockquote>
<p>Here's one way to ... |
4,174 | <p>I'm developing a course that focuses on the transistion from arithmetic to algebraic thinking, particularly in grades 5-8. We will do this through focus on the common core. I'm also putting together a collection of suggested readings from the math education literature. I would be interested to hear your suggestio... | JPBurke | 759 | <p>Early algebra research necessarily deals with the development of algebraic reasoning and questions like "what is algebra" and "what counts as algebraic thinking and reasoning?" And my own readings on early algebra have helped me to focus on what about algebraic thinking are students developing, apart from the manipu... |
3,795,072 | <p><strong>QUESTION</strong></p>
<blockquote>
<p>What is the sign of <span class="math-container">$$\frac{d}{dx} \bigg(\frac{x+1}{x}+\bigg(\frac{2x}{x+1}\bigg)^{\dfrac{\ln(6/5)}{\ln(31/25)}}\bigg)$$</span> when <span class="math-container">$x > {10}^{500}$</span>?</p>
</blockquote>
<p><strong>MY ATTEMPT</strong></p>... | Jean Marie | 305,862 | <p>This is not a matter of computational power.</p>
<p>Consider your function as a composition :</p>
<p><span class="math-container">$$f(u)=u+\left(\frac{2}{u}\right)^{\alpha}$$</span></p>
<p><span class="math-container">$$\text{with} \ u:=\frac{x+1}{x}=1+\frac{1}{x}\tag{1}$$</span></p>
<p>where <span class="math-conta... |
3,795,072 | <p><strong>QUESTION</strong></p>
<blockquote>
<p>What is the sign of <span class="math-container">$$\frac{d}{dx} \bigg(\frac{x+1}{x}+\bigg(\frac{2x}{x+1}\bigg)^{\dfrac{\ln(6/5)}{\ln(31/25)}}\bigg)$$</span> when <span class="math-container">$x > {10}^{500}$</span>?</p>
</blockquote>
<p><strong>MY ATTEMPT</strong></p>... | gt6989b | 16,192 | <p>Define <span class="math-container">$a = \frac{\ln(6/5)}{\ln(31/25)} \approx 0.847$</span> and let
<span class="math-container">$$
f(x)
= \frac{x+1}{x}+\left(\frac{2x}{x+1}\right)^a
= 1 + \frac1x + 2^a \left(1 - \frac{1}{x+1}\right)^a.
$$</span>
Then,
<span class="math-container">$$
\begin{split}
f'(x)
&= -\f... |
587,217 | <p>I know that the units of 2 by 2 matrices with integer entries must have a determinant of 1 or -1, and I have proved that if the determinant is zero then the matrix is not a unit, however I am wondering how you would go about proving that matrices with determinants other than 1 and -1 are not units?</p>
| DonAntonio | 31,254 | <p>An element $\;r\;$ in a ring is a unit if there exists another element $\;x\;$ there s.t. $\;rx=1\;$ .</p>
<p>If $\;A,B \;$ are square integer matrices ,then</p>
<p>$$AB=1\implies \det A=\frac1{\det B}\;$$</p>
<p>But the rightmost number is <em>not</em> an integer if $\;\det B\neq\pm 1\;$ ...</p>
|
207,185 | <p>How would I go about proving this without a truth table?</p>
<p>$[(p \lor q) \implies r ] \implies [ \neg r \implies (\neg p \land \neg q)]$</p>
| Blue | 409 | <p><strong>Fact.</strong> If $x$, $y$, $z$ are points on the complex plane, then the circumcenter of $\triangle xyz$ is the point</p>
<p>$$ i \frac{x(z\overline{z} - y\overline{y})+ y(x\overline{x}-z\overline{z})+z(y\overline{y}-x\overline{x})}{x(\overline{z} - \overline{y})+ y(\overline{x}-\overline{z})+z(\overline{y... |
2,643,099 | <p>Can someone help me explain why it is true that</p>
<p>$$\sin(\pi/2-\theta)=\sqrt{1-\sin^2\theta}$$</p>
<p>When answering please explain the different relation which is used</p>
<p>Thanks</p>
| MrYouMath | 262,304 | <p>You should be asking why </p>
<p>$$\sin^2(\pi/2-\theta) = 1- \sin^2(\theta).$$</p>
<p>By the trigonometric Pythagorean theorem, we know that</p>
<p>$$\cos^2(\theta)+\sin^2(\theta)=1 \implies \cos^2(\theta)=1-\sin^2(\theta)$$</p>
<p>is valid.</p>
<p>If you additionally use the complementary formula for trigonome... |
3,161,371 | <p>For <span class="math-container">$p, q$</span> prime, if <span class="math-container">$q$</span> divides an integer <span class="math-container">$n$</span> but <span class="math-container">$p$</span> does not, show that <span class="math-container">$\text{gcd}(n, pq) = q$</span></p>
<p>This statement sort of remind... | Robert Israel | 8,508 | <p>Hint: there are only <span class="math-container">$4$</span> divisors of <span class="math-container">$p\cdot q$</span>. Which could be <span class="math-container">$\gcd(n,p\cdot q)$</span>?</p>
|
2,352,821 | <p>So, in order to obtain the required answer, I tried to apply some Taylor expansions, which led me to nowhere actually.
After a while I tried to use the summation theorem </p>
<p>$\sum_{n=-\infty}^{+\infty}{f\left(n\right)}=-\sum_{i=1}^{m}{Res_{z=z_i}{\pi\cot\left(\pi z\right)f\left(z\right)}}$ at $f\left(z\right)$... | Jack D'Aurizio | 44,121 | <p>Since $\frac{1}{4n^2-1}=\frac{1}{2}\left(\frac{1}{(2n-1)}-\frac{1}{(2n+1)}\right)$ by squaring we get
$$\sum_{n\geq 1}\frac{1}{(4n^2-1)^2} = \frac{1}{4}\sum_{n\geq 1}\frac{1}{(2n-1)^2}+\frac{1}{4}\sum_{n\geq 1}\frac{1}{(2n+1)^2}-\frac{1}{4}\sum_{n\geq 1}\left(\frac{1}{2n-1}-\frac{1}{2n+1}\right)$$
where the first tw... |
4,613,982 | <p>Calculate the integral <span class="math-container">$$\int_{-\infty }^{\infty } \frac{\sin(\Omega x)}{x\,(x^2+1)} dx$$</span> given <span class="math-container">$$\Omega >>1 $$</span></p>
<p><a href="https://i.stack.imgur.com/mPVqq.jpg" rel="nofollow noreferrer">I tried but couldn't find C1</a></p>
| Lai | 732,917 | <p><span class="math-container">$$
\begin{aligned}
\int_{-\infty}^{\infty} \frac{\sin (\Omega x)}{x\left(x^2+1\right)} d x =&\int_{-\infty}^{\infty} \frac{x \sin (\Omega x)}{x^2\left(x^2+1\right)} d x \\
= & \underbrace{\int_{-\infty}^{\infty} \frac{\sin (\Omega x)}{x} d x}_{=\pi}- \underbrace{ \int_{-\infty}^... |
4,291,880 | <p>I don't understand how you would take the conjugate of a quadratic equation and how it would be useful to solve this question.</p>
<p>I would normally show it by saying if <span class="math-container">$b$</span> is real, then it is equal to <span class="math-container">$\alpha$</span> times <span class="math-contain... | Mohammad Riazi-Kermani | 514,496 | <p>Note that the conjugate of <span class="math-container">$z^2$</span> is <span class="math-container">$\bar z^2$</span></p>
<p>For real numbers <span class="math-container">$a$</span> and <span class="math-container">$b$</span> the conjugate of <span class="math-container">$z^2 + az + b$</span> is <span class="math-c... |
7,715 | <p>I am starting on a Phd program and am supposed to read Colliot Thelene and Sansuc's article
on R-equivalence for tori. I find it very difficult and although I have some knowledge over schemes , I am completely baffled by this scalar restriction business of having a field extension $K/k$ , a torus over $K$ and "restr... | Pete L. Clark | 1,149 | <p>As said, the sought after concept is also known as Weil restriction. In a word, it is the algebraic analogue of the process of viewing an <span class="math-container">$n$</span>-dimensional complex variety as a <span class="math-container">$(2n)$</span>-dimensional real variety.</p>
<p>The setup is as follows: let ... |
4,401,460 | <p>I'm wondering what other tools there are aside from radicals can be used to extend fields in the context of solving polynomials. Since <span class="math-container">$S_5$</span> isn't solvable, constructing a field with a Galois group of <span class="math-container">$S_5$</span> with respect to <span class="math-cont... | MangoPizza | 956,791 | <p>Let <span class="math-container">$f(x)$</span> denote the expected number of runs till we get remainder <span class="math-container">$0$</span> with <span class="math-container">$x$</span> denoting the current remainder mod <span class="math-container">$3$</span>.</p>
<p>Note that each remainder has <span class="mat... |
4,401,460 | <p>I'm wondering what other tools there are aside from radicals can be used to extend fields in the context of solving polynomials. Since <span class="math-container">$S_5$</span> isn't solvable, constructing a field with a Galois group of <span class="math-container">$S_5$</span> with respect to <span class="math-cont... | JMP | 210,189 | <p>The probability that the game is a success in the <span class="math-container">$k^{th}$</span> roll is given by the coefficient of <span class="math-container">$x^k$</span> in</p>
<p><span class="math-container">$$\frac12\sum_{k=1}^\infty \left(\frac{2x}3\right)^k \tag{1}$$</span></p>
<p>The expected number of throw... |
4,401,460 | <p>I'm wondering what other tools there are aside from radicals can be used to extend fields in the context of solving polynomials. Since <span class="math-container">$S_5$</span> isn't solvable, constructing a field with a Galois group of <span class="math-container">$S_5$</span> with respect to <span class="math-cont... | drhab | 75,923 | <p>If <span class="math-container">$n$</span> denotes an arbitrary integer and <span class="math-container">$D$</span> is the result of throwing a fair die then:<span class="math-container">$$\Pr(3\text{ divides }n+D)=\frac13$$</span></p>
<p>This because for <em>every</em> integer <span class="math-container">$n$</span... |
918,689 | <p>of 5 be selected that contain /at least/ 1 of the broken bulbs?</p>
<p>So far, I have tried only 1 method, as it's the only one I've been taught, but I don't know if I am doing it right.
I tried doing C(100,1)/C(100,5) but it just doesn't seem right. Is it? If it isn't, what am I doing wrong?</p>
| user84413 | 84,413 | <p>Using subtraction (as described in another answer) is probably the best way to do this, but you could also break it up into two cases: </p>
<p>where you get exactly 1 defective bulb, and where you get both of the defective bulbs.</p>
<p>This approach gives an answer of $\displaystyle\binom{2}{1}\binom{98}{4}+\bino... |
2,114,446 | <p>But, just to get across the idea of a generating function, here is how a generatingfunctionologist might answer the question: the nth Fibonacci number, $F_{n}$, is the coefficient of $x^{n}$ in the expansion of the function $\frac{x}{(1 − x − x^2)}$ as a power series about the origin.</p>
<p>I am reading a book abo... | angryavian | 43,949 | <p>If you consider the power series
$$f(x) = F_0 + F_1 x + F_2 x^2 + \cdots$$
then the relation $F_{n+2} = F_n + F_{n+1}$ implies
$$x^2 f(x) + x f(x) - F_0 x = f(x) - F_0 - F_1 x.$$
(Write out each term as a power series, and combine terms.)
Rearranging and plugging in $F_0=0$ and $F_1=1$ gives
$$(1-x-x^2) f(x) = x.$$<... |
1,579,371 | <p>I'm studying for my number theory test tomorrow, and these are the last questions in my study guide. I think I understand Fermat's factorization, however, I can't tell how my professor wants us to answer these questions. One of them is going to be on the exam.</p>
<ol>
<li><p>Set <span class="math-container">$n= 874... | Raymond Manzoni | 21,783 | <p>I'll try to prove Tito Piezas III's (+1) neat conjecture :
$$\tag{1}32\;\Re\operatorname{Li}_3\left(i\,\phi^k\right)\stackrel{\color{blue}?}=\operatorname{Li}_3\left(\phi^{-4k}\right)-4\operatorname{Li}_3\left(\phi^{-2k}\right)+10\,k\operatorname{Li}_3\left(\phi^{-2}\right)-\frac{4k^3+5k}{6}\,\ln^3(\phi^2)-8k\,\zeta... |
2,491,577 | <p>What is the meaning of phrase,<strong>"Compactness and Connectedness are intrinsic properties of a topological space"</strong>?</p>
| Pedro | 70,305 | <p>According to the <a href="https://books.google.com.br/books?id=ehmMCgAAQBAJ&pg=PA197&lpg=PA197&dq=%22intrinsic+properties%22+topology&source=bl&ots=B-udXETF1Z&sig=651ngQs6klMPghpzBFnF1PdyQog&hl=pt-BR&sa=X&ved=0ahUKEwiF7uzorY_XAhVIvZAKHaeOC74Q6AEITjAG#v=onepage&q=%22intrinsic%2... |
413,778 | <p>Let <span class="math-container">$G$</span> be a finite abelian group, <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> be two non-empty subsets of <span class="math-container">$G$</span> of equal size. Suppose that for each irreducible character <span class="math-container">$\chi$... | Absol | 91,692 | <p><span class="math-container">$\DeclareMathOperator\Irr{Irr}$</span>You can see this from the fact that for abelian groups, irreducible characters form a <span class="math-container">$\mathbb{C}$</span>-basis of the space of functions from <span class="math-container">$G$</span> to <span class="math-container">$\math... |
2,543,169 | <p>The question is pretty self explanatory, but I’ve encountered situations where, for the length of some vector $\vec{a}$, to denote the length (or magnitude, which ever you prefer) as either $\| \vec{a}\|= \sqrt{a_1^2+a_2^2+\ldots+a_n^2}$ or $|\vec{a}|= \sqrt{a_1^2+a_2^2+\ldots+a_n^2}$ and I was wondering which notat... | user | 505,767 | <p>The first one is the more correct, in the sense that is preferable when you are dealing both with vectors and numbers and it is necessary to avoid misunderstanding.</p>
<p>The notation $|\cdot|$ indicates the absolute value for numbers, but it is also frequently and widely used for vectors when it is clear from the... |
2,543,169 | <p>The question is pretty self explanatory, but I’ve encountered situations where, for the length of some vector $\vec{a}$, to denote the length (or magnitude, which ever you prefer) as either $\| \vec{a}\|= \sqrt{a_1^2+a_2^2+\ldots+a_n^2}$ or $|\vec{a}|= \sqrt{a_1^2+a_2^2+\ldots+a_n^2}$ and I was wondering which notat... | Fly by Night | 38,495 | <p>You see both sets of notation. For vectors you see ${\bf v}$, $\vec{v}$ and $\underline{v}$, perhaps others. For the norm you see $|\cdot|$ and $\| \cdot \|$. It depends if you're in high-school or university, do physics or pure maths. </p>
<p>As a mathematician, I prefer $\|{\bf v}\|$ for the norm of a vector. By ... |
4,118 | <p>I've recently dipped my toes into the world of number theory; and I've bought a book that to me is quite unconventional: R. P. Burn, <em>A Pathway into Number Theory</em>. I've yet to put the book through its paces, but it seems agreeable enough to me. The book is unique in that it poses a sequence of questions to y... | Michael Joyce | 1,397 | <p><em><a href="https://math.dartmouth.edu/news-resources/electronic/kpbogart/" rel="nofollow noreferrer">Combinatorics Through Guided Discovery</a></em> by the late Kenneth Bogart is a great introduction to combinatorics through a guided set of problems and is freely available for download at the link given above.</p>... |
513,779 | <p>If $a,b\in\mathbb{N}$ are odd</p>
<p>then demonstrate:
$$ {\sqrt{a^2 + b^2}} \not\in \mathbb{Q}$$ </p>
<p>I try to guess that $$ {\sqrt{a^2 + b^2}} \in\mathbb{Q}.$$ Then i write $$ {\sqrt{a^2 + b^2}= m/n}.$$ After that: $$ {n\sqrt{a^2 + b^2}= m}$$ , I raised at squared and i have like $$ n^2(a^2+ b^2)=m^2 $$ and ... | André Nicolas | 6,312 | <p><strong>Hint:</strong> The square of an odd number is of the shape $4k+1$, indeed $8k+1$. </p>
|
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