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 |
|---|---|---|---|---|
989,118 | <p>The sum is $$\sum_{n>0} \mathrm{i}^n\frac{J_n(x)}{n}\sin\frac{2\pi n}{3}\stackrel{?}{=}0$$ and $$\sum_{n>0} (\mathrm{-i})^n\frac{J_n(x)}{n}\sin\frac{2\pi n}{3}\stackrel{?}{=}0$$</p>
<p>I suspect they are zero because I am working on a project which from symmetry consideration they should be zero. </p>
<p><st... | Hagen von Eitzen | 39,174 | <p>Note that $i^2=-i^4=i^6=-i^8=\ldots$ so that the sum simplifies <em>significantly</em>.</p>
|
851,647 | <p>I have
$$
X_i, i=1,...,n_1,
$$
$$
Y_j, j=1,...,n_2,
$$
$$
Z_k, k=1,...,n_3,
$$
whereat $X_i, Y_j$ and $Z_k$ are all independent. Does then follow that
$$
\bar{X}=\frac{1}{n_1}\sum_{i=1}^{n_1}X_i, \bar{Y}=\frac{1}{n_2}\sum_{=1}^{n_2}Y_j, \bar{Z}=\frac{1}{n_3}\sum_{k=1}^{n_3}Z_k
$$
independent, too?</p>
| vadim123 | 73,324 | <p>$$V-FA^M=P\left(\frac{G}{B}\right)A^M + P\left(\frac{H}{B}\right)=P\left(\left(\frac{G}{B}\right)A^M + \left(\frac{H}{B}\right)\right)$$</p>
<p>$$P=\frac{V-FA^M}{\left(\frac{G}{B}\right)A^M + \left(\frac{H}{B}\right)}$$</p>
|
160,392 | <p>Did I solve the following quadratic equation correctly.</p>
<p>$$W(W+2)-7=2W(W-3)$$</p>
<p>I got.</p>
<p>$$W^2-8W+7$$
Then for my solution I got.</p>
<p>$$(W-1)(W-7)$$</p>
| Dinuki Seneviratne | 250,288 | <p>$$W (W+2) -7 = 2W (W-3)$$
Expanding the brackets :
$$W^2+2W -7 = 2W^2-6W$$
Then shift the right-hand side part of the equation to the left-hand side of the equation:
$$W^2+2W-7-(2W^2-6W) = (2W^2-6W)-(2W^2-6W)$$
$$W^2+2W-7-(2W^2-6W)= 0$$
Expand the brackets:
$$W^2+2W-7-2W^2+6W= 0$$
$$W^2-2W^2+2W+6W-7=0$$
$$-W^2+8W... |
3,219,187 | <p>The cancellation law for the multiplication of natural numbers is:</p>
<p><span class="math-container">$$\forall m, n\in\mathbb N, \forall p\in\mathbb N-\{0\}, m\cdot p=n\cdot p\Rightarrow m=n.$$</span></p>
<p>Is it possible to show this using induction?</p>
<p>I tried to define <span class="math-container">$$X=\... | hmakholm left over Monica | 14,366 | <p>If you have proved trichotomy <em>and</em> enough laws for addition, then you could do something like:</p>
<p>Assume <span class="math-container">$n\ne m$</span>, then without loss of generality <span class="math-container">$m>n$</span> which is to say, <span class="math-container">$m=n+k$</span> for some <span ... |
1,660,361 | <blockquote>
<p>$$x^4+x^2-2$$</p>
</blockquote>
<p>How do I write in a that can be used for partial fraction? taking out $x$ does not help, what is the process? </p>
<p>using Wolfram I got to $(x^2+2)(x-1)(x+1)$</p>
| Austin Mohr | 11,245 | <p>Writing a polynomial as the product of smaller multiplicative factors is called "factoring".</p>
<p>You might try the substitution $z = x^2$ to get $z^2 + z - 2$. Can you factor this polynomial?</p>
|
481,673 | <p>Find all functions $g:\mathbb{R}\to\mathbb{R}$ with $g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y)$ for all $x,y$.</p>
<p>I think the solutions are $0, 2, x$. If $g(x)$ is not identically $2$, then $g(0)=0$. I'm trying to show if $g$ is not constant, then $g(1)=1$. I have $g(x+1)=(2-g(1))g(x)+g(1)$. So if $g(1)=1$, we can show i... | Andrey Sokolov | 30,727 | <p>Substituting an infinitesimal $y=dx$ gives</p>
<p>$g(x)+g(x)g(0)=g(0)+g(x)+g(0)$ or $g(x)g(0)=2 g(0)$</p>
<p>and</p>
<p>$g'(x)+g(x)g'(0)=g'(0)x+g'(0)$,</p>
<p>for a smooth function $g(x)$.
The first equation implies either $g(0)=0$ or $g(x)=2$.
Solving the differential equation with the condition $g(0)=0$ gives<... |
481,673 | <p>Find all functions $g:\mathbb{R}\to\mathbb{R}$ with $g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y)$ for all $x,y$.</p>
<p>I think the solutions are $0, 2, x$. If $g(x)$ is not identically $2$, then $g(0)=0$. I'm trying to show if $g$ is not constant, then $g(1)=1$. I have $g(x+1)=(2-g(1))g(x)+g(1)$. So if $g(1)=1$, we can show i... | Calvin Lin | 54,563 | <p>Let <span class="math-container">$P(x,y)$</span> denote <span class="math-container">$g(x+y) + g(x) g(y) =g(xy) + g(x) + g(y) $</span>. The important statements that we'd consider are</p>
<p><span class="math-container">$$ \begin{array} { l l l }
P(x,y) & : g(x+y) + g(x) g(y) =g(xy) + g(x) + g(y) & (1) \\
... |
399,564 | <p>Using Mathematical induction on $k$, prove that for any integer $k\geq 1$,</p>
<p>$$(1-x)^{-k}=\sum_{n\geq 0}\binom{n+k-1}{k-1}x^n$$</p>
<p>How should I proceed? The tutorial teacher attempted this question and forgot halfway through... <em>facepalm</em></p>
| digital-Ink | 22,728 | <p>The idea is that you prove it for $k=1$ (which is exactly the sum of a geometrical progression), then assuming it is true for $k$, differentiate both members of the equality and obtain the result for $k+1$. The computation must be quite elementary.</p>
|
2,517,601 | <p>I tried to find the residue of the function $$f(z) = \frac{1}{z(1-\cos(z))}$$ at the $z=0$. So I did:</p>
<p>\begin{align}
\hbox{Res}(0)&=\lim_{x \to 0}(z)(f(z))\\
\hbox{Res}(0)&= \lim_{x \to 0} \frac{1}{1-\cos(z)}
\end{align}</p>
<p>and I got that the residue to be $\infty$, and it seems to be wrong.</p>
| robjohn | 13,854 | <p>Since $1-\cos(z)=2\sin^2(z/2)$,
$$
\begin{align}
\frac1{z(1-\cos(z))}
&=\frac2{z^3}\left(\frac{z/2}{\sin(z/2)}\right)^2\\
&=\frac2{z^3}\left(\frac1{1-\frac1{24}z^2+O\!\left(z^4\right)}\right)^2\\
&=\frac2{z^3}\left(1+\frac1{12}z^2+O\!\left(z^4\right)\right)\\[3pt]
&=\frac2{z^3}+\color{#C00}{\frac1{6z... |
3,663,267 | <p>in a linear algebra class i was given a theorem: If <span class="math-container">$\left\{\boldsymbol{u}_{1}, \boldsymbol{u}_{2}, \ldots, \boldsymbol{u}_{n}\right\}$</span> is a linearly independent subset of <span class="math-container">$\mathbb{R}^{n}$</span> then
<span class="math-container">$
\mathbb{R}^{n}=\oper... | ancient mathematician | 414,424 | <p>It is clear that all your matrices act as the identity on all basis vectors except those labelled <span class="math-container">$i,j,k$</span>; so we need only compute what happens to these three vectors. It is also clear that all the matrices send these three vectors to linear combinations of these three. So we can ... |
1,136,458 | <p>Suppose $G$ is an abelian group. Then the subset $H= \big\{ x\in G\ | \ x^3=1_G \big\}$ is a subgroup of $G$.</p>
<p>I was able to show the statement is correct but the weird thing is that I didn't use the fact $G$ is abelian. <strong>Is it possible that the fact $G$ is abelian is redundant?</strong></p>
<p>Here ... | nbubis | 28,743 | <p>The problem is your calculator or computer, not the mathematical limit. Because of the way floating point numbers work, they store a limited amount digits as well as the power. So even though $1\times 10^{-11}$ is valid, you can't store the number $1+10^{-11}$ accurately, since this would require too many digits of ... |
1,147,773 | <p>Do you have any explicit example of an infinite dimensional vector space, with an explicit basis ?</p>
<p>Not an Hilbert basis but a family of linearly independent vectors which spans the space -any $x$ in the space is a <strong>finite</strong> linear sum of elements of the basis.</p>
<p>In general the existence ... | Aaron Maroja | 143,413 | <p>What about $$\beta=\{1, x, x^2, \ldots, x^n,\ldots \}$$</p>
<p>basis of $K[x]$ polynomial ring, where $K$ is any field. In this case we have $[K[x]: K] = \infty$.</p>
<p>Another example is </p>
<p>$$\gamma = \{1 , \cos x, \sin x, \ldots , \cos n x, \sin n x, \ldots \}$$</p>
<p>is a basis for the Euclidian Space ... |
1,154,690 | <p>We have instituted random drug testing at our company. I was charged with writing the code to generate the weekly random list of employees. We've gotten some complaints because of people getting picked more frequently than they expected and our lawyer wants some evidence that these events fall within the bell curve ... | Litho | 197,288 | <blockquote>
<p>So after 10 weeks, you have 5 times as many opportunities for someone to be picked 5 times as you do in 5 weeks (500% increase in odds over increase 100% in time).</p>
</blockquote>
<p>Actually, the difference in odds is much higher. After 5 weeks, you have one possibility to be picked on 5 weeks: th... |
289,405 | <p>Consider an elementary class $\mathcal{K}$. It is quite common in model theory that a structure $K$ in $\mathcal K$ comes with a closure operator $$\text{cl}: \mathcal{P}(K) \to \mathcal{P}(K), $$ which establishes a <a href="https://en.wikipedia.org/wiki/Pregeometry_(model_theory)" rel="nofollow noreferrer">pregeo... | Alex Kruckman | 2,126 | <p>Here's an example showing that in general, for pregeometries arising in model theory, you can't characterize the dimension of a union of a chain of models just in terms of the dimensions of the models. In other words, it matters how the models embed into each other.</p>
<p>Consider the theory of a single equivalenc... |
2,038,498 | <p>How do you prove this formula
$$\lim_{x \to\infty}\frac{x^{x^2}}{2^{2^x}}$$</p>
<p>Since both top and bottom approaches infinity, I assume it is L'Hospital's rule to solve it, but after the first step I'm stuck
$$\lim_{x \to\infty}\frac{x^2 logx}{2^xlog2}$$
So how can I solve this problem, it seems the answ... | Ronnie Brown | 28,586 | <p>You need to go back to the intuitive idea of the join. </p>
<p>Suppose $X,Y$ are two subsets of Euclidean space of sufficiently high dimensions that the line segments joining points $x \in X$ to points $y \in Y$ do not meet. The union of these line segments is then the <strong>join</strong> $X * Y$. Its points are... |
1,866,801 | <p>Let $A$ be an infinite set, $B\subseteq A$ and $a\in B$. Let $X\subseteq \mathcal{P}(A)$ be an infinite family of subsets of $A$ such that $a\in \bigcap X$.</p>
<p>Suppose $\bigcap X\subseteq B$. Is it possible that, for every non-empty finite subfamily $Y\subset X$, $\bigcap Y \not\subseteq B$ ? </p>
<p>Thanks fo... | gt6989b | 16,192 | <p><strong>HINT</strong></p>
<p>Squaring both equations you get
$$
\sin^2 x + \sin^2 y + 2\sin x \sin y = 1\\
\cos^2 x + \cos^2 y + 2\cos x \cos y = 0
$$
Now add them together to get
$$
2 + 2 \sin x \sin y + 2 \cos x \cos y = 1
$$
or in other words
$$
\frac{-1}{2} = \cos x \cos y + \sin x \sin y = \cos (x-y)
$$</p>
|
1,866,801 | <p>Let $A$ be an infinite set, $B\subseteq A$ and $a\in B$. Let $X\subseteq \mathcal{P}(A)$ be an infinite family of subsets of $A$ such that $a\in \bigcap X$.</p>
<p>Suppose $\bigcap X\subseteq B$. Is it possible that, for every non-empty finite subfamily $Y\subset X$, $\bigcap Y \not\subseteq B$ ? </p>
<p>Thanks fo... | Ken Duna | 318,831 | <p>There are identities for sum of sin and cos:</p>
<p>$$\sin(x)+\sin(y) = 2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)_.$$
$$\cos(x) + \cos(y) = 2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right).$$</p>
<p>Using the first equation tells us that $\cos\left(\frac{x-y}{2}\right)\neq 0.$ Ther... |
1,866,801 | <p>Let $A$ be an infinite set, $B\subseteq A$ and $a\in B$. Let $X\subseteq \mathcal{P}(A)$ be an infinite family of subsets of $A$ such that $a\in \bigcap X$.</p>
<p>Suppose $\bigcap X\subseteq B$. Is it possible that, for every non-empty finite subfamily $Y\subset X$, $\bigcap Y \not\subseteq B$ ? </p>
<p>Thanks fo... | Ovi | 64,460 | <p>I will only look for solutions on $[0, 2\pi)$</p>
<p>We need to make $2$ observations:</p>
<p>$1)$ The maximum value of $\sin \theta$ is $1$. Therefore, if any of $\sin x$, $\sin y$ is negative, $\sin x + \sin y < 1$. It follows that both $x$ and $y$ must be $\in [0, \pi]$. </p>
<p>$2)$ $\cos x= -\cos y \iff \... |
903,117 | <p>I am trying to evaluate
$$\int_{-\infty}^{\infty} \frac{\sin(x)^2}{x^2} dx $$
Would a contour work? I have tried using a contour but had no success.
Thanks.</p>
<p>Edit: About 5 minutes after posting this question I suddenly realised how to solve it. Therefore, sorry about that. But thanks for all the answers anywa... | Jack D'Aurizio | 44,121 | <p>Why not to try integration by parts? This just gives:</p>
<blockquote>
<p>$$\int_{\mathbb{R}}\frac{\sin^2 x}{x^2}\,dx=\int_{\mathbb{R}}\frac{\sin(2x)}{x}\,dx=\pi.$$</p>
</blockquote>
<p>With the same approach you can also find the values of
$$I_m = \int_{\mathbb{R}}\frac{\sin^m(x)}{x^m}\,dx.$$</p>
|
988,581 | <p>Our teacher challenge us a question and it goes like this:
The derivative of a function is define such as </p>
<p>$$\begin{cases} 1 & \text{if } x>0 \\ 2 & \text{if } x=0 \\ -1 &\text{if } x<0\end{cases} $$</p>
<p>Now he said that "$2$, if $x=0$" is not possible.He asked us to proved that the d... | JessicaK | 102,435 | <p>Let $a_{n} = \frac{\sqrt{n^{3}+2}}{n^{4}+3n^{2} + 1}$ and let $b_{n} = \frac{1}{n^{\alpha}}$ for some positive $\alpha$. Consider</p>
<p>$$\lim_{n\rightarrow\infty} \frac{a_{n}}{b_{n}} = \lim_{n\rightarrow \infty} \frac{\frac{\sqrt{n^{3}+2}}{n^{4}+3n^{2} + 1}}{\frac{1}{n^{\alpha}}} = \lim_{n\rightarrow \infty} \fra... |
381,309 | <p>Let <span class="math-container">$T\in B(\mathcal{H} \otimes \mathcal{H})$</span> where <span class="math-container">$\mathcal{H}$</span> is a Hilbert space. We can define operators
<span class="math-container">$$T_{[12]}= T \otimes 1;\quad T_{[23]}= 1 \otimes T$$</span>
and if <span class="math-container">$\Sigma: ... | Jesse Peterson | 6,460 | <p>The proof for the reals can be generalized to any non-discrete locally compact group <span class="math-container">$G$</span>. We let <span class="math-container">$K \subset G$</span> be any compact set with positive Haar measure <span class="math-container">$\lambda(K) > 0$</span> (e.g., <span class="math-contain... |
2,010,329 | <p>The function under consideration is:</p>
<p>$$y = \int_{x^2}^{\sin x}\cos(t^2)\mathrm d t$$</p>
<p>Question asks to find the derivative of the following function. I let $u=\sin(x)$ and then $\tfrac{\mathrm d u}{\mathrm d x}=\cos(x)$. Solved accordingly but only to get the answer as
$$ \cos(x)\cos(\sin^2(x))-\cos(... | Claude Leibovici | 82,404 | <p><strong>Hint</strong></p>
<p>The fundamental theorem of calculus write $$\frac d {dx}\int_{a(x)}^{b(x)} f(t)\,dt=f(b(x)) b'(x)-f(a(x)) a'(x)$$ </p>
|
4,180,344 | <p>I started learning from Algebra by Serge Lang. In page 5, he presented an equation<br><i></p>
<blockquote>
<p>Let <span class="math-container">$G$</span> be a commutative monoid, and <span class="math-container">$x_1,\ldots,x_n$</span> elements of <span class="math-container">$G$</span>. Let <span class="math-contai... | Suzu Hirose | 190,784 | <p>Here <span class="math-container">$\psi(v)$</span> is a different ordering of the actions of multiplication. The <span class="math-container">$\Pi x_{\psi(v)}$</span> refers to a different order of the terms in the product than <span class="math-container">$\Pi{x_v}$</span>. In words, what that means is "Regard... |
2,298,855 | <p>Let $u,v\in H^1(\Omega )$ where $\Omega =B_2\backslash B_1$. How can I have continuity and coercivity of
$$a(u,v)=\int_{\Omega }\nabla u\cdot \nabla v-\int_{B_2\backslash B_1}3^{-d}uv\ ?$$</p>
<p>The best thing I get is $$|a(u,v)|\leq \|\nabla u\|_{L^2}\|\nabla v\|_{L^2}+3^{-d}\|u\|_{L^2}\|v\|_{L^2},$$
but I can't ... | Surb | 154,545 | <p>By definition, $\|u\|_{H^1}=\|u\|_{L^2}+\|\nabla u\|_{L^2}$, therefore $\|u\|_{L^2},\|\nabla u\|_{L^2}\leq \|u\|_{H^1}$ and same for $v$. You can then conclude for continuity. For coercivity,
$$a(u,u)=\int_\Omega |\nabla u|^2-3^{-d}\int_\Omega |u|^2\geq \int_\Omega |\nabla u|^2-3^{-d}\int_\Omega |u|^2=\|u\|_{H^1}^2-... |
2,983,519 | <p>I have to calculate the limit of this formula as <span class="math-container">$n\to \infty$</span>.</p>
<p><span class="math-container">$$a_n = \frac{1}{\sqrt{n}}\bigl(\frac{1}{\sqrt{n+1}}+\cdots+\frac{1}{\sqrt{2n}}\bigl)$$</span></p>
<p>I tried the Squeeze Theorem, but I get something like this:</p>
<p><span cla... | Will Jagy | 10,400 | <p>for a decreasing function such as <span class="math-container">$1/\sqrt x$</span> with <span class="math-container">$x$</span> positive, a simple picture shows
<span class="math-container">$$ \int_a^{b+1} \; f(x) \; dx < \sum_{k=a}^b f(k) < \int_{a-1}^{b} \; f(x) \; dx $$</span>
<span class="math-container">... |
1,372,779 | <p>Given that $x$ is a positive integer, find $x$ in $(E)$.</p>
<p>$$\tag{E} j-n=x-n\cdot\left\lceil\frac{x}{n}\right\rceil$$
All $n, j, x$ are positive integers.</p>
| Khosrotash | 104,171 | <p>1st step<br>
$n|x$ then ,not solvable for x $$x=nk+0\\j-n=x-n.k\\j-n=x-x=0 \rightarrow j=n \\x=nk \\x=jk \\$$so $x$ is an integer that divide by n<br>
2nd step :<br>
now take $$x=nk+r\\0<r<n \\\frac{x}{n}=\frac{nk+r}{n}=k+\frac{r}{n}\\\left \lceil \frac{x}{n} \right \rceil=k+1\\j-n=x-n(k+1)\\j-n=(nk+r)-n(k... |
827,072 | <p>How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$
using the formula
$(x-q)^2 + (y-p)^2 = r^2$.</p>
<p>I know i need to use that formula but have no idea how to start, I have tried to start but don't think my answer is right.</p>
| Erk Ekin | 238,077 | <p>This method that I wrote creates a circle object which has a radius and a centre point. (written in <em>Swift</em> but can be port to many other languages.)</p>
<pre><code>
func circle(p1: (x: Float,y: Float), p2: (x: Float,y: Float), p3: (x: Float,y: Float)) -> (r: Float, c: (x: Float,y: Float)) {
let x1 ... |
2,365,839 | <p>My textbook claims that, for small step size $h$, Euler's method has a global error which is at most proportional to $h$ such that error $= C_1h$. It is then claimed that $C_1$ depends on the initial value problem, but no explanation is given as to how one finds $C_1$.</p>
<p>So if I know $h$, then how can I deduce... | KittyL | 206,286 | <p>Given an IVP:
$$\frac{dy}{dt}=f(t,y), y(a)=y_0, t\in [a,b].$$
Here is a Theorem from Numerical Analysis by Sauer:</p>
<blockquote>
<p>Assume that $f(t,y)$ has a Lipschitz constant $L$ for the variable $y$ and that the solution $y_i$ of the initial value problem at $t_i$ is approximated by $w_i$, using Euler's met... |
2,365,839 | <p>My textbook claims that, for small step size $h$, Euler's method has a global error which is at most proportional to $h$ such that error $= C_1h$. It is then claimed that $C_1$ depends on the initial value problem, but no explanation is given as to how one finds $C_1$.</p>
<p>So if I know $h$, then how can I deduce... | Lutz Lehmann | 115,115 | <p>Consider the IVP <span class="math-container">$y'=f(t,y)$</span>, <span class="math-container">$y(t_0)=y_0$</span>. Let <span class="math-container">$t_k=t_0+kh$</span>, <span class="math-container">$y_k$</span> computed by the Euler method. We know that the error order of the Euler method is one. Thus the iterates ... |
3,407,474 | <p>I can understand that The set <span class="math-container">$M_2(2\mathbb{Z})$</span> of <span class="math-container">$2 \times 2$</span>
matrices with even integer entries is an infinite non commutative ring. But why It doesn't have any unity? I think <span class="math-container">$I(2\times2)$</span> is a unity for ... | rschwieb | 29,335 | <p>As everyone pointed out right away, your candidate is not an element of your ring.</p>
<p>Here is a slightly different way to prove that <span class="math-container">$M_2(2\mathbb Z)$</span> can't have an identity. </p>
<p>Notice that because your entries are always sums of products of things in <span class="math... |
1,975,199 | <blockquote>
<p>In this problem, construct a bijection to show the identity. Use the definitions of these quantities, not formulas for them.</p>
<p>For <span class="math-container">$0≤j≤n$</span>
<span class="math-container">$$\binom{n}{j}=\binom{n}{n-j}$$</span></p>
</blockquote>
<p>I know how to prove this by inducti... | Parcly Taxel | 357,390 | <p>Consider the number of ways to arrange $n$ bits, $j$ of which are ones; by definition there are $\binom nj$ ways of doing so. For every such way we can flip the bits to obtain an arrangement of $n$ bits where $n-j$ of them are ones; there are $\binom n{n-j}$ arrangements here. For example:</p>
<pre><code>0010011010... |
675,857 | <p>I am trying to figure out easy understandable approach to given small number of $n$, list all possible is with proof, I read this paper but it is really beyond my level to fathom,</p>
<p>attempt for $\phi(n)=110$, </p>
<p>$$\phi(n)=110=(2^x)\cdot(3^b)\cdot(11^c)\cdot(23^d)\quad\text{ since }\quad p-1 \mid \phi(n)=... | vadim123 | 73,324 | <p>Suppose that $\phi(n)=110=2\cdot 5\cdot 11$. We factor $n=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$. Since $\phi$ is multiplicative, $\phi(n)=\phi(p_1^{a_1})\phi(p_2^{a_2})\cdots \phi(p_k^{a_k})$. We also can evaluate the totient at any prime power, so $$\phi(n)=\frac{n}{p_1p_2\cdots p_k}(p_1-1)(p_2-1)\cdots(p_k-1)$$</... |
595,865 | <p>A real number $r$ belonging to $[0,1]$ is said to be computable if there is a simple TM such that for each binary encoding of $n$ ($n$ is a natural number), returns the $n$th bit in the binary expansion of $r$. Give a proof that there are non-computable real numbers.</p>
| rewritten | 43,219 | <p>It's a diagonal argument. You have to assume (or read it somewhere) that there are strictly more real numbers than natural numbers. It is a theorem, and it can be proved in several ways, but it's not the point here. The point is that you have to know that any function from $\mathbb{N}$ to $\mathbb{R}$ (or any interv... |
819,805 | <p>The polynomial remainder theorem states that when a polynomial $P(x)$ of degree $> 0$ is divided by $x-r$ ($r$ being some constant) the remainder is equal to $P(r)$, that is:<br>
$$\begin{array}l If & \quad P(x) = (x-r)Q(x)+R \\ then & \quad P(r) = R \end{array}$$
The algebra and the graphic representatio... | Bill Dubuque | 242 | <p>It is clearer in congruence language, $\ {\rm mod}\ \,x\!-\!r\!:\,\ x\equiv r\,\Rightarrow\ P(x)\equiv P(r).\ $ This is true because polynomials are composed of sums and products, and congruences are equivalence relations that are compatible with sums and products, i.e. $\, A\equiv a,\,B\equiv b\,\Rightarrow\, A+B\e... |
1,629,592 | <p>I tried to prove this question by first considering the possible last digit of $p$ when $p=n^2+5$, but that reasoning got me nowhere. Then I tried to prove it by contrapositive, and however I just couldn't really find where to start.<br>
Hence I'm here asking for some hints (only hints, no solution please).</p>
<p>... | Tsemo Aristide | 280,301 | <p>$n^2=0,1, 4,9,6,5$ mod $10$ so $n^2+5= 6,9,4,1,0,5$ mod 10. If $n^2+5$ prime, it is odd and not divisible by 5 the result follows. so you can only have $n=1, 9$ mod 10</p>
|
3,921,335 | <p>how can I formally proof that for a > 1: <span class="math-container">$$ a> \sqrt a > \sqrt[3]a > \sqrt[4]a ... $$</span>?
Can someone help? ;)</p>
| Peter Szilas | 408,605 | <p>Set <span class="math-container">$a:=e^b,$</span> where <span class="math-container">$b>0;$</span></p>
<p>Then</p>
<p><span class="math-container">$e^b>e^{b/2}>.....e^{b/n}>e^{b/(n+1)}$</span> since <span class="math-container">$f(x)=e^x$</span> is an increasing function.</p>
|
1,595,080 | <p>I am doing a question out of H. E. Rose's A Course In Number Theory (Chapter 2, problem 2) which I have been struggling with for some time. However I found a solution which strays from the way advised by the author, and I would just like some validation as to whether it is correct. </p>
<p>The question starts by as... | Dietrich Burde | 83,966 | <p>Your estimate somehow arrives at the bound $n\log(n)+O(1)$ which is certainly not $\le 1$ for large $n$ - where you say, "This is where I make an assumption", it is not true.</p>
|
1,595,080 | <p>I am doing a question out of H. E. Rose's A Course In Number Theory (Chapter 2, problem 2) which I have been struggling with for some time. However I found a solution which strays from the way advised by the author, and I would just like some validation as to whether it is correct. </p>
<p>The question starts by as... | Wojowu | 127,263 | <p>There are quite a few mistakes happening here.</p>
<ol>
<li><p>You get $|\sum_{i\leq n}\mu(n)|\leq n$ and you multiply it by $\sum_{i \leq n} \frac{1}{i}$, getting on the left hand side $|\sum_{i \leq n} \frac{\mu(i)}{i}|$. You seem to be using a rule of a sort $(\sum_{i\leq n}a_n)(\sum_{i\leq n}b_n)=\sum_{i\leq n}... |
3,379,899 | <p>We have two coins. The first one is a fair coin and has <span class="math-container">$50\%$</span> chance of landing on head and <span class="math-container">$50\%$</span> of landing on tails. for the other one it is <span class="math-container">$60\%$</span> for head and <span class="math-container">$40\%$</span> f... | Mark Fischler | 150,362 | <p>The answer, giving two terms corresponding to the actual coin being fair, then the actual coin being biased, will be
<span class="math-container">$$P(T) = \frac{\left(\frac12\right)^3}{\left(\frac12\right)^3+\frac25\left(\frac35\right)^2} \left(\frac12\right)
+ \frac{\frac25\left(\frac35\right)^2}{\left(\frac12\righ... |
2,776,388 | <p>I tried this problem and I first found the lim of $x^x$ as $x$ approaches zero from right to be 1 (I did this by re-writing $x^x$ as an exponential) and when I repeated the same process to find lim of $x^{(x^x)}$, I found 1 again but the final answer should be ZERO. Could I have an explanation on why it's a zero?</p... | user | 505,767 | <p>Note that</p>
<p>$$x^{x^x}=e^{x^x\log x}\to0$$</p>
<p>indeed</p>
<p>$$x^x\log x\to -\infty$$</p>
|
4,371,541 | <p>The series representation <span class="math-container">$\sum1/n^s$</span> of the Riemann zeta function <span class="math-container">$\zeta(s)$</span> is known to converge for <span class="math-container">$\sigma>1$</span>, where <span class="math-container">$s=\sigma+it$</span>.</p>
<p>In order to show an infinit... | davidlowryduda | 9,754 | <p>It is not true that the Dirichlet series for <span class="math-container">$\zeta(s)$</span> converges uniformly for <span class="math-container">$\mathrm{Re} s > 1$</span>. This is also not how the "easy" arguments go to show that <span class="math-container">$\zeta(s)$</span> is holomorphic for <span c... |
1,383,613 | <p>Give recursive definition of sequence $a_n = 2^n, n = 2, 3, 4... where $ $a_1 = 2$</p>
<p>I'm just not sure how to approach these problems. </p>
<p>Then it asks to give a def for:</p>
<p>$a_n = n^2-3n, n = 0, 1, 2...$</p>
<p>Thanks for all the help!</p>
| Terra Hyde | 253,768 | <p>For the first one, write the term $a_{n+1}$ and compare it to $a_n$:
$$a_{n+1}=2^{n+1}=2\cdot2^n=2a_n$$</p>
<p>For the second one, repeat the process:
$$\begin{align}a_{n+1}&=(n+1)^2-3(n+1)\\
&=n^2+2n+1-3n-3\\
&=n^2-3n+2n-2\\
&=a_n+2(n-1)
\end{align}$$</p>
|
1,686,012 | <p>Let $T$: $\mathbb{R}^3 \to \mathbb{R}$ be a linear transformation.</p>
<p>Show that either $T$ is surjective, or $T$ is the zero linear transformation.</p>
<p>My approach:</p>
<p>First we start off by supposing T is not surjective and we want to show that $\forall \overrightarrow v \in \mathbb{R}^3, T(\overrighta... | Arthur | 15,500 | <p>Assume that $T$ is <em>not</em> the zero transformation (this means that there is a $v$ such that $T(v) = r \neq 0$), and deduce that it is surjective (by using $v$ and linearity of $T$).</p>
|
2,158,981 | <p>How should i solve : $$\sum_{r=1}^n (2r-1)\cos(2r-1)\theta $$</p>
<p>I can solve $\sum_{r=1}^n cos(2r-1)\theta $ by considering
$\Re \sum_{r=1}^n z^{2r-1} $ and using summation of geometric series, but I can't seem to find a common geometric ratio when $ 2r-1 $ is involved in the summation.</p>
<p>Visually : $\su... | S.C.B. | 310,930 | <p>Note that $$\sum_{r=1}^{n} (2r-1) \cos (2r-1) \theta=\sum_{r=1}^{n} \frac{\mathrm{d}}{\mathrm{d}\theta}\sin (2r-1) \theta= \frac{\mathrm{d}}{\mathrm{d}\theta}\sum_{r=1}^{n} \sin (2r-1) \theta$$
Now calculate $$\sum_{r=1}^{n} \sin (2r-1) \theta=\frac{ \sin^2(n\theta) }{ \sin(\theta) }$$ Through <a href="https://m... |
244,489 | <p>Given:</p>
<p>${AA}\times{BC}=BDDB$</p>
<p>Find $BDDB$:</p>
<ol>
<li>$1221$</li>
<li>$3663$</li>
<li>$4884$</li>
<li>$2112$</li>
</ol>
<p>The way I solved it:</p>
<p>First step - expansion & dividing by constant ($11$):
$AA\times{BC}$=$11A\times{BC}$</p>
<ol>
<li>$1221$ => $1221\div11$ => $111$</li>
<li>$3... | Christian Blatter | 1,303 | <p>Given two nonoverlapping squares in a large square $Q$ there is a line $g$ separating the two squares. Assume $g$ is not parallel to one of the sides of $Q$ (otherwise we are done). Let $A$ and $C$ be the two vertices of $Q$ farthest away from $g$ on the two sides of $g$. The two edges of $Q$ meeting at $A$ togethe... |
3,470,703 | <p><span class="math-container">$\int \left(\frac{1}{2}t+\frac{\sqrt{2}}{4} \right)e^{t^2-\sqrt{2}t}\ dt$</span></p>
| 19aksh | 668,124 | <p>Hint <span class="math-container">$$\text{Let} \ t^2-\sqrt2 t = x \implies (2t - \sqrt2)dt=dx \implies \left(\frac{t}{2}-\frac{\sqrt2}{4}\right)dt = \frac {dx}4$$</span></p>
|
16,247 | <p>Given a map $f:B^n \to S^n$, where $B^n$ is the unit ball and $S^n$ is the unit sphere, is it true that the degree of $f|_{S^n}$ is always 0, where $f_{S^n}$ is the restriction of $f$ to $S^n$? If so, why? </p>
<p>Thanks!</p>
| Mariano Suárez-Álvarez | 274 | <p>The degree is zero because the restriction of $f$ to the boundary of $B^n$ (which most humans write $S^{n-1}$! :) ) is homotopic to to a constant map.</p>
|
3,554,555 | <p>Let <span class="math-container">$S_n=1-2^{-n}$</span>. Prove that the sequence <span class="math-container">$S_n$</span> converges to <span class="math-container">$1$</span> as <span class="math-container">$n$</span> approaches infinity. </p>
<p>I let <span class="math-container">$N = \lceil\log_2 \varepsilon\rcei... | Anguepa | 196,351 | <p>By definition of convergence you must show that, for every <span class="math-container">$\varepsilon>0$</span>, there exists <span class="math-container">$N>0$</span> such that, for every <span class="math-container">$n\geq N$</span>, it holds that
<span class="math-container">$$
|1-(1-2^{-n})|=|2^{-n}|=2^{-n... |
3,554,555 | <p>Let <span class="math-container">$S_n=1-2^{-n}$</span>. Prove that the sequence <span class="math-container">$S_n$</span> converges to <span class="math-container">$1$</span> as <span class="math-container">$n$</span> approaches infinity. </p>
<p>I let <span class="math-container">$N = \lceil\log_2 \varepsilon\rcei... | The 2nd | 751,538 | <p>From the following limit properties:</p>
<blockquote>
<p><span class="math-container">$$\lim_{x \to \infty} \frac{1}{n^{x}} = 0, \forall n>1$$</span></p>
<p><span class="math-container">$$\lim_{x \to \infty} f(x) \pm g(x) = \lim_{x \to \infty} f(x) \pm \lim_{x \to \infty} g(x)$$</span></p>
<p><span class="math... |
332,001 | <p><a href="https://en.wikipedia.org/wiki/Cauchy%27s_theorem_(geometry)#Generalizations_and_related_results" rel="nofollow noreferrer">Wikipedia states</a> that A. D. Alexandrov generalized <em>Cauchy's rigidity theorem for polyhedra</em> to higher dimensions. </p>
<p>The relevant statement in the article is not linke... | j.c. | 353 | <p>The following is Theorem 27.2 of Igor Pak's book <a href="http://www.math.ucla.edu/~pak/book.htm" rel="noreferrer">Lectures on Discrete and Polyhedral Geometry</a> (which in general is a very nice resource for these sorts of questions):</p>
<blockquote>
<p>Let <span class="math-container">$P,Q\subset\mathbb{R}^d$... |
19,672 | <p>The <a href="http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind" rel="nofollow noreferrer">Stirling number of the second kind</a> is the number of ways to partition a set of $n$ objects into $k$ non-empty subsets. In Mathematica, this is implemented as <code>StirlingS2</code>. How can I enumerate all t... | Dr. belisarius | 193 | <pre><code><< Combinatorica`
KSetPartitions[{a, b, c}, 2]
(*
{{{a}, {b, c}}, {{a, b}, {c}}, {{a, c}, {b}}}
*)
StirlingS2[3, 2]
(* 3 *)
</code></pre>
|
1,115,389 | <p>Given: </p>
<p>$$\sum_{n = 0}^{\infty} a_nx^n = f(x)$$</p>
<p>where:</p>
<p>$$a_{n+2} = a_{n+1} - \frac{1}{4}a_n$$</p>
<p>is the recurrence relationship for $a_2$ and above ($a_0$ and $a_1$ are also given).</p>
<p>Is there a nice closed form to this pretty recurrence relationship?</p>
| user203856 | 203,856 | <p>Don't have enough points for a comment, but you seem to be contradicting yourself when you first say that for the first rook you have 9^2 , for the second you have 8^2 and then for the total options you cite 81, 80, etc.</p>
|
4,412,700 | <p>I've been working on a problem but have been stuck for several hours finishing it.</p>
<p>The problem is to show that
<span class="math-container">$$
\frac{x}{(e+x) \ln(e+x)} \leq \ln(\ln(e+x)) \leq \frac{x}{e}
$$</span>
for all <span class="math-container">$x > 0$</span>.</p>
<p>I proceeded by first integratin... | Átila Correia | 953,679 | <p><strong>HINT</strong></p>
<p>Let <span class="math-container">$u = \ln(e + x)$</span> (observe that <span class="math-container">$u\geq 1$</span>). Then one has that
<span class="math-container">\begin{align*}
\frac{x}{(e + x)\ln(e + x)} \leq \ln(\ln(e + x)) & \Longleftrightarrow \frac{e^{u} - e}{e^{u}u} \leq \l... |
47,492 | <p>Consider a continuous irreducible representation of a compact Lie group on a finite-dimensional complex Hilbert space. There are three mutually exclusive options:</p>
<p>1) it's not isomorphic to its dual (in which case we call it 'complex')</p>
<p>2) it has a nondegenerate symmetric bilinear form (in which case ... | Faisal | 430 | <p>This was a comment on Torsten's answer, but it got too long.</p>
<p>Suppose $G$ is connected and semisimple. Fixing a choice $\Phi^+$ of positive roots for $G$, we can describe $w_0$ as the unique element of the Weyl group of $G$ that takes $\Phi^+$ to the negative roots $\Phi^- = -\Phi^+$. Now, $-w_0$ is an involu... |
3,297,110 | <p>Evaluate <span class="math-container">$$\int\frac{1}{x^{\frac{25}{25} }\cdot x^{\frac{16}{25}}+x^{\frac{9}{25}}}dx$$</span></p>
<p>I start by factoring <span class="math-container">$$\int\frac{1}{x^{\frac{25}{25} }\cdot x^{\frac{16}{25}}+x^{\frac{9}{25}}}dx=\int\frac{1}{x^{\frac{9}{25}}\left(x^{\frac{32}{25}}+1\rig... | Claude Leibovici | 82,404 | <p>Let <span class="math-container">$$x=y^{25}\implies dx=25\,y^{24}\,dy$$</span>
<span class="math-container">$$\int\frac{dx}{x^{\frac{25}{25} }\cdot x^{\frac{16}{25}}+x^{\frac{9}{25}}}=25\int\frac{ y^{15}}{y^{32}+1}\,dy=\frac {25}{16}\int\frac{16\, y^{15}}{y^{32}+1}\,dy=\frac {25}{16}\int\frac{ \left(y^{16}\right)'}{... |
654,263 | <p>My intention is neither to learn basic probability concepts, nor to learn applications of the theory. My background is at the graduate level of having completed all engineering courses in probability/statistics -- mostly oriented toward the applications without much emphasis on mathematical rigor.</p>
<p>Now I am ve... | user901823 | 123,198 | <p>Williams, Probability with Martingales</p>
|
654,263 | <p>My intention is neither to learn basic probability concepts, nor to learn applications of the theory. My background is at the graduate level of having completed all engineering courses in probability/statistics -- mostly oriented toward the applications without much emphasis on mathematical rigor.</p>
<p>Now I am ve... | Jyotirmoy Bhattacharya | 1,195 | <p>Billingsley's <em>Probability and Measure</em> covers many interesting topics and has excellent problems. It has an unusual approach: building up the mathematical structure of subject not at once but in a number of iterative steps of increasing complexity. That takes up time but I have still found this repetition he... |
629,887 | <p>I was going through one of the topic "Introduction to Formal proof". In one example while explaining "Hypothesis" and "conclusion" got confused.</p>
<p>The example is as follows:</p>
<blockquote>
<p>If $x\geq4$, then $2^x \geq x^2$.</p>
</blockquote>
<p>While deriving conclusion article said "As $x$ grows large... | Pedro | 23,350 | <p>Consider $f(x,y)=(x+y)^n$. Expand, differentitate w.r.t to $x$, multiply by $x$ and plug $x=t$, $y=1-t$. If you know about Berstein Polynomials, you shouldn't be surprised the result is just $t$. </p>
<p><strong>Spoiler</strong> Differentiate $$n(x+y)^{n-1}=\sum_{k=1}^nk\binom nkx^{k-1}y^{n-k}$$ Multiply by $x$, di... |
321,255 | <p>I've been looking for the definition of projective ideal but haven't found anything, all I've seen is the definition of projective module (but I don't know how these are related, if they are ¿?). Does anyone know a book where I can look up basic facts about projective ideals?</p>
| Jyrki Lahtonen | 11,619 | <p>I don't know of any other meaning of a <em>projective ideal</em> other than the one suggested by Boris Novikov, i.e. an ideal of a ring <span class="math-container">$R$</span> that is also projective as an <span class="math-container">$R$</span>-module. I want to emphasize that such an ideal <span class="math-contai... |
271,785 | <p>I have to prove that $f(x)=\log(1+x^2)$ is Uniform continuous in $[0,\infty)$ (with $\epsilon ,\delta$ formulas...)</p>
<p>I wrote the definition: (what I have to prove):</p>
<p>$\forall \epsilon>0 \quad \exists \delta>0 \quad s.t. \quad \forall x,y\in [0,\infty) : \quad |x-y|<\delta \quad \Rightarrow \qu... | mrf | 19,440 | <p>By the mean value theorem,
$$f(x)-f(y) = f'(c)(x-y) = \frac{2c}{1+c^2}(x-y),$$
for some $c$ between $x$ and $y$.</p>
<p>Show that $$ \left|\frac{2c}{1+c^2}\right| \le 1,$$
for example using the AM-GM inequality.</p>
|
3,688,151 | <blockquote>
<p>A necklace is made up of <span class="math-container">$3$</span> beads of one sort and <span class="math-container">$6n$</span> of another, those of each sort being similar. Show that the number of arrangement of the beads is <span class="math-container">$3n^2+3n +1$</span>.</p>
</blockquote>
<p><str... | Olius | 791,850 | <p>This is open to interpretation, but I understand the statement that "it is a necklace" to mean that rotating the beads cyclically yield equivalent arrangements (but not flipping the necklace). I assume this in what follows.</p>
<p>Observe that in a given arrangement, the <span class="math-container">$3$</span> bea... |
1,728,662 | <p>Each set has bijective ordinal and cardinality is defined as the least one of such ordinals. I know that a set of ordinals is well-ordered by $\subseteq$ (inclusion) and thus has $\subseteq$-least element. However, I wonder which axiom guaranteed that the bijective ordinals above really construct to be a set. I worr... | Asaf Karagila | 622 | <p>The power set axiom is what allows us to ensure that there are arbitrarily large ordinals, this is true even without assuming the axiom of choice, as witnessed by Hartogs' theorem. </p>
<p>Well, to be accurate, power set ensures there are very large well ordered sets, and replacement gives us the ordinals. </p>
<p... |
1,170,708 | <p>What functions satisfy $f(x)+f(x+1)=x$?</p>
<p>I tried but I do not know if my answer is correct.
$f(x)=y$</p>
<p>$y+f(x+1)=x$</p>
<p>$f(x+1)=x-y$</p>
<p>$f(x)=x-1-y$</p>
<p>$2y=x-1$</p>
<p>$f(x)=(x-1)/2$</p>
| Milo Brandt | 174,927 | <p>Your conclusion isn't quite right, and isn't found correctly either; your definition of $f(x)=y$ is a bit misleading because we <em>have</em> to interpret it as applying to a particular $x$ and not to <em>all</em> $x$. So when you step from
$$f(x+1)=x-y$$
to
$$f(x)=x-1-y$$
by "shifting" the argument by one (which be... |
1,734,819 | <p>I think I'm on the right track with constructing this proof. Please let me know.</p>
<p>Claim: Prove that there exists a unique real number $x$ between $0$ and $1$ such that
$x^{3}+x^{2} -1=0$</p>
<p>Using the intermediate value theorem we get
$$r^{3}+r^{2}-1=c^{3}+c^{2}-1$$
......
$$r^{3}+r^{2}-c^{3}-c^{2}=0$$</... | Community | -1 | <p>Existence follows from the intermediate value theorem, and uniqueness from the fact that the derivative has constant sign on this interval. But let's prove that the zero is actually unique on the entire real line.</p>
<hr>
<p>Suppose the polynomial $p$ has real zeros at $\alpha$ and $\beta$. Take $\beta$ to be the... |
174,075 | <p>What is the difference when a line is said to be normal to another and a line is said to be perpendicular to other?</p>
| VISHAL YASH | 196,457 | <p>A normal is an object such as a line or vector that is perpendicular to a given object. For example, in the two-dimensional case, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point.</p>
|
1,307,280 | <p>I m in the point X. I m 2 blocks up from a point A and 3 blocks down from my home H. Every time I walk one block i drop a coin.</p>
<p>H
.
.
.
X
.
.
A</p>
<p>If the coin is face I go one block up and if it is not face I go one block down.</p>
<p>Which is the probability of arriving home before the point A?</p>
... | Michael Hardy | 11,667 | <blockquote>
<blockquote>
<p>when $x$ gets closer to $a$, $f(x)$ gets closer to $L$</p>
</blockquote>
</blockquote>
<p>That is wrong. Consider two examples:</p>
<ul>
<li><p>Let $f(x) = 6-(x-4)^2$. Clearly $f(x)$ never gets bigger than $6$, so the limit cannot be $7$, but $f(x)$ gets closer to $7$ (and to al... |
165,487 | <p>After I use Simplify on an expression I get$\dfrac{1}{2}\sqrt{-\dfrac{\sqrt{(-b^2+16|c|^2)(4|c|^2+b\Im(c))^2}}{4a(4|c|^2+b\Im(c)])}}$. This expression can clearly be simplified further by noticing that the square bracket term in the numerator cancels the other bracket term in the denominator so $\dfrac{1}{2}\sqrt{-\... | Alexei Boulbitch | 788 | <p>You have given contradictive assumptions. In Mma the condition that a variable, say, c, is positive (<code>c>0</code>) automatically means that it belongs to Reals. Thus, when you fix <code>c ∈ Complexes && (a | b | c) > 0</code> you mislead Mma. </p>
<p>According to your initial expressions the para... |
1,679,044 | <blockquote>
<p>Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle. Then find the area.</p>
<p>$$y = 2x^2, y = 8x^2, 3x + y = 5, x ≥ 0$$ </p>
</blockquote>
<p>Here is my drawing: <a href="http://www.webassign.net/waplots/d/a... | 3SAT | 203,577 | <p>You can use double integral, </p>
<p>the intersection points: </p>
<p>between $y=8x^2$ and $y=5-3x$ are at $x=-1$ and $x=5/8$</p>
<p>between $y=2x^2$ and $y=5-3x$ are at $x=-5/2$ and $x=1$</p>
<p>let's assume that we want $\text{d$y$d$x$}$ </p>
<p>for $I_1$: when $y$ goes from $y=2x^2$ to $y=8x^2$ then $x$ goes... |
1,906,013 | <blockquote>
<p>Let $f$ be a smooth function such that $f'(0) = f''(0) = 1$. Let $g(x) = f(x^{10})$. Find $g^{(10)}(x)$ and $g^{(11)}(x)$ when $x=0$.</p>
</blockquote>
<p>I tried applying chain rule multiple times:</p>
<p>$$g'(x) = f'(x^{10})(10x^9)$$</p>
<p>$$g''(x) = \color{red}{f'(x^{10})(90x^8)}+\color{blue}{(... | benguin | 121,903 | <p>Try applying the general Leibniz rule after you take the first derivative: <a href="https://en.m.wikipedia.org/wiki/General_Leibniz_rule" rel="nofollow">https://en.m.wikipedia.org/wiki/General_Leibniz_rule</a></p>
<p>From there, you should be able to identify which terms in the sum go to zero when $x$ is evaluated ... |
2,861,443 | <p>Let $C_c(\mathbb{R})$ be the following:</p>
<p>$$C_c = \{ f \in C(\mathbb{R}) \mid \exists \text{ } T > 0 \text{ s.t. } f(t) = 0 \text{ for } |t| \geq T\}$$</p>
<p>Let $T_n \in L(C_c(\mathbb{R}))$ be a linear operator such that:</p>
<p>$$T_n u = \delta_n *u, \forall u \in C_c(\mathbb{R}),$$</p>
<p>where</p>
... | Paul Frost | 349,785 | <p>You know that each chart $\phi : U \to V \subset \mathbb{R}^n$ for $M$ induces a canonical bijection $\tilde{\phi} : TM \mid_U = p^{-1}(U) \to V \times \mathbb{R}^n$. We define $W \subset TM$ to be open if $\tilde{\phi}(W \cap p^{-1}(U))$ is open in $V \times \mathbb{R}^n$ for all $\phi$.</p>
<p>It is an easy exerc... |
3,605,368 | <p>Imagine a <span class="math-container">$9 \times 9$</span> square array of pigeonholes, with one pigeon in each pigeonhole. Suppose that all at once, all the pigeons move up, down, left, or right by one hole. (The pigeons on the edges are not allowed to move out of the array.) Show that some pigeonhole winds up with... | fleablood | 280,126 | <p>Label each pigeon as <span class="math-container">$(a,b)$</span> where <span class="math-container">$a$</span> is the column number and <span class="math-container">$b$</span> is the row number. Each pigeon must move to a square that has the same column or row number but the row or column number is off by <span cla... |
3,176,155 | <blockquote>
<p>Let <span class="math-container">$ABCD$</span> be a parallelogram. I proved that the angle bisectors of <span class="math-container">$A$</span>, <span class="math-container">$B$</span>, <span class="math-container">$C$</span>, <span class="math-container">$D$</span> form a rectangle. How can I prove t... | Vasili | 469,083 | <p><a href="https://i.stack.imgur.com/iRtKy.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/iRtKy.png" alt="enter image description here" /></a></p>
<p>Without loss of generality, we can assume that <span class="math-container">$AB>BC$</span>. Let's start with proving that <span class="math-contai... |
3,857,698 | <p>Let <span class="math-container">$D_1, ..., D_n$</span> be arbitrary <span class="math-container">$n$</span> sets where <span class="math-container">$D_i \cap D_j \neq \emptyset$</span>. In the simplified case where <span class="math-container">$n = 2$</span>, we have that
<span class="math-container">$$
\begin{spli... | Vercassivelaunos | 803,179 | <p>We can start way earlier to get a geometric interpretation, at the real numbers. Multiplication by a real number is a combination of scaling and mirroring. Multiplying by a a positive number is scaling the real line, multiplying by <span class="math-container">$-1$</span> is mirroring it at the origin. On an abstrac... |
2,459,169 | <p>Let $A$ be an $n \times n$ matrix. Show that if $A^2 = 0$, then $I − A$ is nonsingular and $(I − A)^{−1} = I + A$.</p>
<p>(Matrix Algebra)</p>
| Clive Newstead | 19,542 | <p>Nonsingularity follows from existence of an inverse. Note that
$$(I-A)(I+A) = I^2-AI+IA-A^2$$
Now do some (basic) matrix algebra, using the fact that $A^2=0$.</p>
|
2,459,169 | <p>Let $A$ be an $n \times n$ matrix. Show that if $A^2 = 0$, then $I − A$ is nonsingular and $(I − A)^{−1} = I + A$.</p>
<p>(Matrix Algebra)</p>
| Robert Lewis | 67,071 | <p>Note that, just as with numbers</p>
<p>$(1 + x)(1 -x) = (1 - x) + x(1 - x) = 1 - x + x - x^2 = 1 - x^2, \tag 1$</p>
<p>so with matrices</p>
<p>$(I - A)(I + A) = I - A^2; \tag 2$</p>
<p>the algebraic maneuvers are essentially the same in each case. Now with</p>
<p>$A^2 = 0, \tag 3$</p>
<p>(2) becomes</p>
<p>$... |
203,138 | <p>I'm starting to take calculus in college (the first offered math class) and I was shown this problem. Find $f(x)$ when$$-\frac{df(x)}{dx}\cdot\frac{x}{f(x)}=1$$Rearranging this to make$$\frac{d\,f(x)}{dx}=-\frac{f(x)}{x}$$gives me an interesting relationship, and very strongly suggests to me a polynomial function, b... | EuYu | 9,246 | <p>Let us write $y=f(x)$. The trick is to notice that
$$xy^{\prime} + y = (xy)^{\prime}$$
which is a reverse application of the product rule. Therefore
$$\frac{d}{dx}(xy) = 0 \implies xf(x) = C$$
for some constant $C$. The general solution is then given by
$$f(x) = \frac{C}{x}$$
There is a systematic method for attack... |
88,880 | <p>In a short talk, I had to explain, to an audience with little knowledge in geometry or algebra, the three different ways one can define the tangent space $T_x M$ of a smooth manifold $M$ at a point $x \in M$ and more generally the tangent bundle $T M$:</p>
<ul>
<li>Using equivalent classes of smooth curves through ... | Nick Salter | 960 | <p>Klaus Jänich's undergraduate-level book "Vector Analysis" includes a section showing the equivalence between the three descriptions of the tangent space that you mention. He gives rigorous proofs of everything, and also provides a fair amount of motivation. </p>
|
1,653,806 | <p>This is a but of a more mathematically juvenile question but I'm trying to get all my intuition in order. When taking a limit we can cancel things that might be zero because in taking a limit, we allow ourselves to avoid trouble spots. If the limit exists, it is very much a number/function which could be zero. So, i... | Jack Lee | 1,421 | <p>The divergence of a vector field $V = V^a \partial _a$ on a (pseudo-)Riemannian manifold is given by $\operatorname{div} V = V^a{}_{;a}$. In words, this is obtained by taking the trace of the total covariant derivative. In the special case of $\mathbb R^n$ with a flat Euclidean or pseudo-Euclidean metric, this yield... |
3,105,482 | <p>For the formula:</p>
<p><span class="math-container">$$1 = \sqrt{x^2 + y^2 + z^2 + w^2}$$</span></p>
<p>How to rewrite it to find <span class="math-container">$w$</span>?</p>
| Andrei | 331,661 | <p>Square the equation, move <span class="math-container">$x^2+y^2+z^2$</span> to the other side, then take the square root. Notice that you can take either the square root or the negative of that.</p>
|
1,173,643 | <p>I saw the following statement written, but I can't understand why it is true.</p>
<p>$$
\dfrac {P(A \text{ and } B)}{P(B)} = \dfrac{P(A)-P(A \text{ and }B^c)}{ 1-P(B^c)}
$$</p>
<p>Any help understanding why these are equivalent would be appreciated.</p>
| Michael Burr | 86,421 | <p>First, the denominators $P(B)$ and $1-P(B^c)$ are equal because $P(B)+P(B^c)=1$ (either $B$ happens or it doesn't). </p>
<p>For the numerator, $P(A\text{ and }B)$ is the probability that both $A$ and $B$ happen. For $P(A)-P(A\text{ and }B)$, consider $P(A)$ first. $P(A)$ is the probability that $A$ happens. Whe... |
2,761,658 | <blockquote>
<p>Two numbers $x$ and $y$ are chosen at random from the numbers $1,2,3,4,\ldots,2004$. The probability that $x^3+y^3$ is divisible by $3$ is?</p>
</blockquote>
<p>The correct answer is $\dfrac13$ while mine is $\dfrac{445}{2003}$</p>
<p>My attempt:</p>
<p>For $x^3+y^3$ to be divisible by $3$, EITHER ... | Rebecca J. Stones | 91,818 | <h3>With replacement</h3>
<p>As you basically note $x^3+y^3 \equiv x+y \pmod 3$, which follows from <a href="https://en.wikipedia.org/wiki/Fermat%27s_little_theorem" rel="nofollow noreferrer">Fermat's Little Theorem</a>.</p>
<p>We note that $2004 \equiv 0 \pmod 3$, so there's exactly $2004/3=668$ numbers equivalent t... |
108,110 | <p>How can I use <em>Mathematica</em> to expand such a product (only need a finite number of terms):</p>
<p>$$\prod^{\infty}_{n=1}\frac{({1-yq^{n+1}})({1-y^{-1}q^n})}{(1-q^n)^2}$$</p>
| JimB | 19,758 | <p>If you follow @J.M. 's hint and assume that <code>0 < q < 1</code>, you can get a value for the infinite product:</p>
<pre><code>Product[(1 - y q^(n + 1)) (1 - q^n/y)/(1 - q^n)^2,
{n, 1, ∞},
Assumptions -> 0 < q < 1]
</code></pre>
<p>with result</p>
<pre><code>-((y QPochhammer[1/y, q] QPochhamm... |
963,125 | <p>I'm not exactly sure where to start on this one. Any help would be greatly appreciated.</p>
<p>Show that $4$ does not divide $12x+3$ for any $x$ in the integers.</p>
<p>Here's what I have so far:</p>
<p>There exist c in the integers such that 4 | 12x+3. Then 12x+3 = 4y for some y in the integers. This is a contra... | NovaDenizen | 109,816 | <p>Saying $4 | 12x + 3$ is equivalent to saying there exists an integer $k$ such that $4k = 12x + 3$.</p>
<p>If this is the case, then also $4k - 12x = 3$, or $4(k - 3x) = 3$. $4 \cdot 0 = 0$ and $4 \cdot 1 = 4$, so this would imply $0 < k - 3x < 1$, which is impossible. So it must not be the case that $4 | 12... |
1,160,076 | <p>Consider $f:\left[a,b\right]\rightarrow \mathbb{R}$ continuous at $\left[a,b\right]$ and differentiable at $\left(a,b\right)$.<br>
$\forall x\:\ne \frac{a+b}{2}$, $f'\left(x\right)\:>\:0$, but at $x\:=\frac{a+b}{2}$, $f'\left(x\right)\:=\:0$.<br>
Show that $f$ is <strong>strictly increasing</strong> at $\left[a,b... | rafalpw | 217,690 | <p>1) $f$ is strictly increasing in $[a,c]$ and in $[c,b]$</p>
<p>Let $x,y \in [a,c]$ and $ x<y $ Then, from the mean value theorem, we have: $f(y)-f(x)=f'(\xi)(y-x) $ where $\xi \in (x,y) $ Since $\xi \in (a,c)$ we have $f'(\xi)>0$ and $(y-x)>0$ , so $f(y)-f(x)>0$</p>
<p>Proof for $[c,b]$ is identical.</... |
1,682,961 | <p>I was making a few exercises on set proofs but I met an exercise on which I don't know how to start:</p>
<blockquote>
<p>If $A \cap C = B \cap C $ and $ A-C=B-C $ then $A = B$</p>
</blockquote>
<p>Where should I start? Should I start from $ A \subseteq B $ or should I start from this $ ((A\cap C = B\cap C) \land... | Arthur | 15,500 | <p>You show $A\subseteq B$ and $B \subseteq A$, as one usually would when showing that two sets are equal. Since the conditions are symmetric in $A$ and $B$, the two proofs are completely analoguous, so I will only do one of them.</p>
<p>To show $A \subseteq B$, take an $a \in A$, and note that either $a \in C$ or $a ... |
235,940 | <p>Consider a sphere $S$ of radius $a$ centered at the origin. Find the average distance between a point in the sphere to the origin.</p>
<p>We know that the distance $d = \sqrt{x^2+y^2+z^2}$. </p>
<p>If we consider the problem in spherical coordinates, we have a 'formula' which states that the average distance $d_... | Ross Millikan | 1,827 | <p>If $\rho$ is the radius, the volume is in fact $\iiint dV$ without the $\rho$. The average distance is then as you have said. This is an example of the general formula for the average value of a variable $X$ over a probability distribution $V$, which is $\bar X= \tfrac {\int X dV}{\int dV}$</p>
|
24,912 | <p>I have the category-theoretic background of the occasional stroll through MacLane's text, so excuse my ignorance in this regard. I was trying to learn all that I could on the subject of tensor algebras, and higher exterior forms, and I ran into the notion of cohomological determinants. Along this line of inquiry, I ... | user234212323 | 429,204 | <p><a href="https://agrothendieck.github.io/divers/ps.pdf" rel="nofollow noreferrer">Pursuing stacks</a> should fit here.</p>
<p>There are plenty of uses of Picard categories (see the appendix), and in fact, the appendix list some examples of Picard 2, 3-categories.</p>
|
293,234 | <p>I recently asked a question about <a href="https://math.stackexchange.com/questions/287116/proof-that-mutual-statistical-independence-implies-pairwise-independence">pairwise versus mutual independence</a> (also related to <a href="https://math.stackexchange.com/questions/281800/example-relations-pairwise-versus-mutu... | JSchlather | 1,930 | <p>Using Sage I was able to calculate $[L: \mathbb Q]=64$. Basically you can create number fields in Sage and then create relative extensions of these number fields. I was able to factor the minimum polynomial of $\alpha$ over each extension and then adjoin a root until I reached a point at which the minimum polynomial... |
2,067,553 | <p>I'm currently working on another problem: let $x_1,x_2,x_3$ be the roots of the polynomial: $x^3+3x^2-7x+1$, calculate $x_1^2+x_2^2+x_3^2$. Here is what i did: $x^3+3x^2-7x+1=0$ imply $x^2=(7x-x^3-1)/3$. And so $x_1^2+x_2^2+x_3^2= (7x_1-x_1^3-1)+7x_2-x_2^3-1+7x_3-x_3^3-1)/3= 7(x_1+x_2+x_3)/3+(x_1^3+x_2^3+x_3^3)-1$.... | Arnaldo | 391,612 | <p><strong>Hint</strong></p>
<p>$$(x_1)^2+(x_2)^2+(x_3)^2=(x_1+x_2+x_3)^2-2(x_1x_2+x_1x_3+x_2x_3)$$</p>
<p>You just have to find $x_1+x_2+x_3$ and $x_1x_2+x_1x_3+x_2x_3$ from the coeficients.</p>
<p>Check here: <a href="https://en.wikipedia.org/wiki/Vieta%27s_formulas" rel="nofollow noreferrer">https://en.wikipedia.... |
785,844 | <p>I've got the following limit to solve:</p>
<p>$$\lim_{s\to 1} \frac{\sqrt{s}-s^2}{1-\sqrt{s}}$$</p>
<p>I was taught to multiply by the conjugate to get rid of roots, but that doesn't help, or at least I don't know what to do once I do it. I can't find a way to make the denominator not be zero when replacing $s$ fo... | user148623 | 148,623 | <p>$$\frac{\sqrt{s} - s^2}{1 - \sqrt{s}} = \frac{\sqrt s \left(1 - s^{3/2}\right)}{\left(1 - s^{3/2}\right)\left(1 + s^{3/2}\right)} = \frac{\sqrt s}{1 + s^{3/2}}$$</p>
|
146,075 | <p>Within my limited experience, I have only known free groups to occur through two mechanisms: as fundamental groups of trees (graphs) and ping-pong. And sometimes only through one way: the fact that sufficiently high-powers of hyperbolic elements in a Gromov-hyperbolic group generate a free group arises via ping-pong... | Misha | 21,684 | <p>This is a very extended comment to your question. </p>
<p>Your question does not have "obvious yes or no" answer; in part, this is because you asked 3 somewhat different questions: </p>
<ol>
<li><p>"Can one explicitly describe all finitely generated free subgroups..." In this form, the question has no answer since... |
1,270,584 | <p>I tried googling for simple proofs that some number is transcendental, sadly I couldn't find any I could understand.</p>
<p>Do any of you guys know a simple transcendentality (if that's a word) proof?</p>
<p>E: What I meant is that I wanted a rather simple proof that some particular number is transcendental ($e$ o... | hmakholm left over Monica | 14,366 | <p>It's possible to argue fairly elementarily that <em>Liouville's number</em>
$$ L = \sum_{k=1}^\infty 10^{-k!} $$
is transcendental, by showing directly that for every integer polynomial
$$p(x) = a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$$
there is a decimal position in $p(L)$ that must be nonzero.</p>
<p>The powers of ... |
3,291,303 | <p>this very same problem appeared in a different thread but the questions was slightly different. In my case, I'm looking precisely for the answer.</p>
<p>This is how I solved it, I only need confirmation of whether this actually is correct:</p>
<p>So I assumed that 1111... (100 ones) is going to be exactly divided ... | J. W. Tanner | 615,567 | <p>Mod <span class="math-container">$10^7-1: 10^7\equiv1\implies10^{98}\equiv1\implies10^{100}\equiv100\implies10^{100}-1\equiv99.$</span></p>
<p>I.e., <span class="math-container">$10^7-1|(10^{100}-1)-99\implies \dfrac{10^7-1}9|\dfrac{10^{100}-1}9-\color{blue}{11}$</span></p>
|
3,657,106 | <blockquote>
<p>Sam was adding the integers from <span class="math-container">$1$</span> to <span class="math-container">$20$</span>. In his rush, he skipped one of the numbers and forgot to add it. His final sum was a multiple of <span class="math-container">$20$</span>. What number did he forget to add?</p>
</block... | Bram28 | 256,001 | <p>Here is where your approach goes wrong:</p>
<p>The sum of all numbers <em>with the exception of the one skipped one</em> is a multiple of <span class="math-container">$20$</span>.</p>
<p>So, if <span class="math-container">$k$</span> is the skipped number, what you have is: <span class="math-container">$210-k = 20... |
581,958 | <p>I am working on a project about Mathematics and Origami. I am working on a section about how origami can be used to solve cubic equations. This is the source I am looking at:<br>
<a href="http://origami.ousaan.com/library/conste.html">http://origami.ousaan.com/library/conste.html</a><br>
I understand the proof they ... | Guest | 201,847 | <p>Thre's a good video showing the trisection bit here:</p>
<p><a href="https://www.youtube.com/watch?v=SL2lYcggGpc&feature=em-subs_digest" rel="nofollow">https://www.youtube.com/watch?v=SL2lYcggGpc&feature=em-subs_digest</a></p>
|
1,602,392 | <p>I think I have a proof using Pythagoras for $\sqrt{a_1^2} + \sqrt{a_2^2} > \sqrt{a_1^2 + a_2^2}$.</p>
<p><em>I'm interested in whether there's a way to use that proof with Pythagoras to prove the general $a_n$ case (for this, hints are appreciated rather than complete proofs), and <strong>also</strong> in other ... | user236182 | 236,182 | <p>Both sides of the inequality are non-negative (for all $a_i\in\mathbb R$), therefore the following equivalences hold:</p>
<p>$$\sqrt{a_1^2} + \sqrt{a_2^2} +\cdots + \sqrt{a_n^2} \ge \sqrt{a_1^2 + a_2^2 +…+a_n^2}$$</p>
<p>$$\iff \left(\sqrt{a_1^2} + \sqrt{a_2^2} +\cdots + \sqrt{a_n^2}\right)^2 \ge \left(\sqrt{a_1^2... |
1,909,422 | <blockquote>
<p>let $A=\left(\begin{array}{ccc} -5 & -1 & 6 \\ -2 & -5 & 8 \\ -1 & -1 & 1 \end{array}\right)$ and $B=\left(\begin{array}{ccc} -9 & 3 & -3 \\ -14 & 4 & -7 \\ -2 & 1 & -4 \end{array}\right)$ matrices above $\mathbb{C}$ are they similar?</p>
</blockquote>
... | H. H. Rugh | 355,946 | <p>Indeed they are not similar. But $A$ is in fact not diagolizable.
It has a size 3 Jordan block:</p>
<p>$(A+3E)^2 \neq 0$ (whereas $(B+3E)^2=0$). This is also what you got using the $m_\lambda$ formulation.</p>
|
1,909,422 | <blockquote>
<p>let $A=\left(\begin{array}{ccc} -5 & -1 & 6 \\ -2 & -5 & 8 \\ -1 & -1 & 1 \end{array}\right)$ and $B=\left(\begin{array}{ccc} -9 & 3 & -3 \\ -14 & 4 & -7 \\ -2 & 1 & -4 \end{array}\right)$ matrices above $\mathbb{C}$ are they similar?</p>
</blockquote>
... | InsideOut | 235,392 | <p>A well known theorem state that: <em>two matrices are similar if and only if they have the same Jordan normal form</em> . In your case, the Jordan normal forms are different, so they cannot be similar.</p>
|
3,714,418 | <p>I need to calculate the angle between two 3D vectors. There are plenty of examples available of how to do that but the result is always in the range <span class="math-container">$0-\pi$</span>. I need a result in the range <span class="math-container">$\pi-2\pi$</span>.</p>
<p>Let's say that <span class="math-conta... | Siong Thye Goh | 306,553 | <p>If <span class="math-container">$C \subset D$</span>, then we have <span class="math-container">$A - D \subset A-C$</span>.</p>
<p>We have <span class="math-container">$B - A \subset B$</span>, hence <span class="math-container">$A- B \subset A - (B-A)$</span>.</p>
|
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Douglas S. Stones | 2,264 | <p>You have a glass of red wine and a glass of white wine (of equal volume). You take a teaspoon of the red wine and put it in the glass of white wine and stir. You then take a teaspoon of the white wine (which now has a teaspoon of the red wine in it) and put it in the glass of red wine and stir.</p>
<blockquote>
... |
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