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 |
|---|---|---|---|---|
1,228,290 | <p>We have the system $y''=-7y'-12y-u'-2u$</p>
<p>If we choose $x_1=y,x_2=y'$ we can write the system as</p>
<p>$x'=Ax + Bu \\ y= Cx$</p>
<p>Finding A is easy, but how do I find expressions for $B$ and $C$ when we have derivatives of the input in the expression?</p>
| Edda Stefano Toro | 637,360 | <p>"Suppose if the physical meaning of <span class="math-container">$y =$</span> position and <span class="math-container">$u =$</span> force. What is the physical meaning of <span class="math-container">$x_2$</span>? We know that the <span class="math-container">$x_1$</span> will be position, what is the physical mean... |
3,245,517 | <p>As by <a href="https://arxiv.org/abs/0803.3787" rel="nofollow noreferrer">Landau's proof</a>
<span class="math-container">$$\sum_{n=1}^{\infty} \mu(n)/n = 0$$</span>
Therefore for any <span class="math-container">$N \in \mathbb{N}$</span>,
<span class="math-container">$$ \sum_{n=1}^{N} \mu(n)/n = -\sum_{n=N+1}^{\in... | Hagen von Eitzen | 39,174 | <p>By definition, <span class="math-container">$$\sum_{k=1}^\infty a_k=L $$</span>
means that for every <span class="math-container">$\epsilon>0$</span>, there exists <span class="math-container">$n_0$</span> such that
<span class="math-container">$$\left|\sum_{k=1}^n a_k-L\right|<\epsilon \text{ for all }n>n... |
1,869,768 | <p>I need to do the following limit without using L'Hopital and I have not been able, please help</p>
<blockquote>
<p>$$\lim\limits_{x \to 3} \left(\frac{x-1}{2x-4}\right)^{\frac{1}{x-3}}$$</p>
</blockquote>
| haqnatural | 247,767 | <p>Since $\lim _{ x\rightarrow 0 }{ { \left( 1+x \right) }^{ \frac { 1 }{ x } } } =e$ (or $\\ \lim _{ x\rightarrow \infty }{ { \left( 1+\frac { 1 }{ x } \right) }^{ x } } =e\\ $)</p>
<blockquote>
<p>$$\lim _{ x\to 3 } \left( \frac { x-1 }{ 2x-4 } \right) ^{ \frac { 1 }{ x-3 } }=\lim _{ x\to 3 } \left( 1+\frac... |
686,536 | <p>Suppose $L$ is a field extension of $K$ and $\alpha$ an element in a field extension of $L$. Can we say $[K\colon L(\alpha)]=[K\colon K(\alpha)]$? I tried to prove this, but I couldn't come up with a proof. I need hints. Thank you.</p>
| Jay | 230,257 | <p>There is a disproof of this statement in general. Take $ L^2([-1,1],\mathbb{C})$, and $a_n(x)= \exp(2\pi i n x)$ and $b_n (x) = \exp(-2\pi i n x)$. In this case, both $a_n \rightharpoonup a = 0$ and $b_n\rightharpoonup b=0$, but $a_n (x)\cdot b_n(x) =1$ for all $x$. Clearly $c_n=c =1$. Then
$$\lim\limits_{n \to \in... |
3,256,509 | <p>I need to show that <span class="math-container">$ \sum_{k=0}^n (-1)^{k} {{m+1}\choose k }{{m+n-k}\choose m }= 0 $</span> if <span class="math-container">$n>0$</span>. Here <span class="math-container">$m$</span> is a non negative integer.</p>
<p>I am thinking induction, but do I apply it on <span class="math-c... | G Cab | 317,234 | <p>Also, we can write it as
<span class="math-container">$$
\eqalign{
& \sum\limits_{0\, \le \,k\, \le \,n} {\left( { - 1} \right)^{\,k} \left( \matrix{
m + 1 \cr
k \cr} \right)\left( \matrix{
m + n - k \cr
m \cr} \right)} \quad \left| {\,0 \le m} \right.\quad = \cr
& = \sum\limits_{\left(... |
1,469,695 | <p><a href="https://i.stack.imgur.com/fLWrM.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fLWrM.png" alt="enter image description here"></a></p>
<p>I have the following graph, and i'm trying to determine the maximum. When I use conventional methods, I end up with the ordered pair (2.35, 1.28) roug... | André Nicolas | 6,312 | <p>By the result with caps that they quote, from $k$ divides $r$ and $d$ divides $a$ we conclude that $kd$ divides $ra$. Similarly, $kd$ divides $sb$. So $kd$ divides the sum $ra+sb$, and therefore $kd$ divides $d$. For positive $k$ this is only possible if $k=1$.</p>
|
3,755,709 | <p>How do you prove that the derivative of <span class="math-container">$\tan^{-1}(x)$</span> is equal to <span class="math-container">$\frac{1}{1+x^2}$</span> geometrically?</p>
<p>I figured it out by working it out using implicit differentiation.</p>
<p>I also found how to plot a semi-circle using <span class="math-c... | marty cohen | 13,079 | <p>Another proof comes from
the atan addition/subtraction formula
<span class="math-container">$\arctan(a)\pm\arctan(b)
=\arctan(\frac{a\pm b}{1\mp ab}
$</span>.</p>
<p>Then,
with a little hand-waving
as <span class="math-container">$h \to 0$</span>
and assuming that
<span class="math-container">$\lim_{z \to 0} \dfrac{... |
578,938 | <p>Induction step: We assume that P(k) is true and then we need to show that P(k+1) is true as well.</p>
<p>If k is arbitrary and we assume it's correct, then how come one can't say, </p>
<p>j = (k+1)</p>
<p>and assume p(j) is true because j is arbitrary just like k and it has the same form of p(k).</p>
<p>That log... | HJ_beginner | 477,906 | <p>I'm a beginner so the following is something that helps my intuition.</p>
<hr>
<p>By the CLT, if <span class="math-container">$Y_1, ..., Y_n$</span> are iid Bernoulli with parameter <span class="math-container">$p$</span> then</p>
<p><span class="math-container">$$ \sqrt{n} \left( \frac{\bar{Y}_n - p}{\sqrt{pq}} ... |
3,460,766 | <p>I have a function:
<span class="math-container">$$r=\sqrt{x^2+y^2+z^2}$$</span>
and wish to calculate:
<span class="math-container">$$\frac{d^2r}{dt^2}$$</span></p>
<hr>
<p>so far I have said:
<span class="math-container">$$\frac{d^2r}{dt^2}=\frac{\partial^2r}{\partial x^2}\left(\frac{dx}{dt}\right)^2+\frac{\parti... | Oscar Lanzi | 248,217 | <p>Let <span class="math-container">$t=a/b$</span> be a positive number other than <span class="math-container">$1$</span>. Then</p>
<p><span class="math-container">$(bt)^{bt}=b^b$</span></p>
<p><span class="math-container">$(bt)^t=b$</span></p>
<p><span class="math-container">$bt=b^{1/t}$</span></p>
<p><span clas... |
932,907 | <p>So from what I understand $\langle w | v \rangle=\vec w^* \cdot \vec v$. Ok. I'm fine with that notation. But then I've seen $\langle x | y \rangle=\delta(x-y)$ and $\langle x | p \rangle=e^{-ixp/\hbar}$. I can see that these are the eigenfunctions of position and momentum respectively, but I don't see how they'... | John | 176,379 | <p>In mathematics, $\sin (x)^2 = \sin x^2 = \sin (x^2) $ because $\sin$ comes after exponentiation and multiplication in order of operations. The $sin^2$ construct was created to alleviate the pain of performing exponentiation on the result of $sin$. </p>
<p>In computer programming, however, $ \sin(x)^2 = \sin^2 x $ b... |
444,517 | <p>Consider the functional equation
$$f(x+y) = f(x)g(y)+f(y)g(x)$$
valid for all complex $x,y$. The only solutions I know for this equation are $f(x)=0$, $f(x)=Cx$, $f(x)=C\sin(x)$ and $f(x)=C\sinh(x)$.</p>
<p>Question $1)$ Are there any other solutions ?</p>
<hr>
<p>If we set $x=y$ we can conclude that if there exi... | leshik | 15,215 | <p>You can take $g=1,$ then your equation reduces to $f(x+y)=f(x)+f(y).$ Without any additional assumption on $f,$ this functional equation has plenty of discontinuous solutions. See <a href="http://en.wikipedia.org/wiki/Cauchy%27s_functional_equation">here</a> </p>
<p>Of course, once you add some regularity condition... |
444,517 | <p>Consider the functional equation
$$f(x+y) = f(x)g(y)+f(y)g(x)$$
valid for all complex $x,y$. The only solutions I know for this equation are $f(x)=0$, $f(x)=Cx$, $f(x)=C\sin(x)$ and $f(x)=C\sinh(x)$.</p>
<p>Question $1)$ Are there any other solutions ?</p>
<hr>
<p>If we set $x=y$ we can conclude that if there exi... | celtschk | 34,930 | <p>Be $B$ a basis of $\mathbb R$ as vector space over $\mathbb Q$, then it is immediately clear that for any $b,b'\in B$, we can independently choose $f(b)$ and $f(b')$ because the addition theorem doesn't relate values in different $\mathbb Q$-subspaces of $\mathbb R$.</p>
<p>We can remove this ambiguity by additiona... |
444,517 | <p>Consider the functional equation
$$f(x+y) = f(x)g(y)+f(y)g(x)$$
valid for all complex $x,y$. The only solutions I know for this equation are $f(x)=0$, $f(x)=Cx$, $f(x)=C\sin(x)$ and $f(x)=C\sinh(x)$.</p>
<p>Question $1)$ Are there any other solutions ?</p>
<hr>
<p>If we set $x=y$ we can conclude that if there exi... | Malper | 78,841 | <p>As leshik pointed out, this equation has plenty of discontinuous solutions (e.g. for $g=1$ it becomes <a href="http://en.wikipedia.org/wiki/Cauchy%27s_functional_equation" rel="nofollow noreferrer">Cauchy's functional equation</a>), so let's just consider continuous solutions. $f=0$ is the trivial solution; from now... |
1,182,429 | <p>Take for instance the following problem. You have two beakers of the same height. One has tick marks that break it into thirds. The other has tick marks that separate it into fourths. The water levels are 1/3 and 1/4 respectively. If I did not know about the concept of LCDs, how would I figure out how much water the... | GFauxPas | 173,170 | <p>If you are trying to figure out how much water there is total, pour the contents of one beaker into the other. You can then use any sort of interpolation method, such as a permanent marker and a shoelace, to make the scale of the beaker precise enough to measure how much water there is total.</p>
|
4,426,896 | <p>Let <span class="math-container">$R$</span> be a ring <span class="math-container">$I\subset R$</span> an ideal.
Are the following true or false conditionals?</p>
<ol>
<li>If <span class="math-container">$R$</span> is a field then so is <span class="math-container">$R/I$</span>.</li>
<li>If <span class="math-contain... | supersuper | 790,617 | <p>If <span class="math-container">$F$</span> is a field, then its only ideals are <span class="math-container">$0$</span> and <span class="math-container">$F$</span>.
So, <span class="math-container">$F/I$</span> is <span class="math-container">$F$</span> or <span class="math-container">$0$</span>, the first is a fiel... |
2,734,257 | <p>I have tried to show that this limit :
$$\lim\limits_{n\to \infty }\frac{n}{n!^{\frac 1 n}}=e$$</p>
<p>using $ \lim (1+\frac 1 n)^{\frac 1 n} , n \to \infty $ , I don't find any equivalence , however wolfram alpha says that is $e$ as shown <a href="https://www.wolframalpha.com/input/?i=lim+(n+%2F+((n!)%5E(1%2Fn))... | user | 505,767 | <p>By <a href="https://en.wikipedia.org/wiki/Stirling%27s_approximation" rel="nofollow noreferrer">Stirling's approximation</a></p>
<p>$$n! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n$$</p>
<p>then</p>
<p>$$\frac{n}{n!^{\frac 1 n}}\sim \frac{n}{(\sqrt{2 \pi n})^\frac1n\left(\frac{n}{e}\right)}\to e$$</p>
|
3,847,369 | <p>I was given the following definition of the cross product:</p>
<blockquote>
<p>The vector product <span class="math-container">$\underline{a}\times\underline{b}$</span> is defined as the vector with magnitude <span class="math-container">$\lvert\underline{a} \times \underline{b}\rvert = \vert\underline{a}\rvert\lver... | J.G. | 56,861 | <p>The quoted definition is careless: <span class="math-container">$|a\times b|=|a||b|(\sin\theta)c$</span> where <span class="math-container">$c\cdot c=1$</span> and <span class="math-container">$a,\,b,\,c$</span> form a right-handed system. The result is parallel to <span class="math-container">$c$</span> if the sine... |
1,026,066 | <p>Problem:
It is researched that 60% of people in city goes to cinema on daily basis, 40% of people goes to theater on daily basis. It is also known that 20% simultaneously goes to both theater and cinema.</p>
<p>What is</p>
<ol>
<li>
Probability that chosen person does not attend both.</li>
<li>Probability that ch... | Vilsen | 193,284 | <p>First a note to your solution of item (2), it should be $P(A)+P(B)-P(A\cap B)$. Regarding item (3)-(4), you are in fact not looking for the joint probability, but the conditional probability i.e. $P(B|A)$ and $P(A|B)$ respectively, where the conditional probability is defined as usual:
\begin{equation}
P(X|Y) = \fra... |
1,026,066 | <p>Problem:
It is researched that 60% of people in city goes to cinema on daily basis, 40% of people goes to theater on daily basis. It is also known that 20% simultaneously goes to both theater and cinema.</p>
<p>What is</p>
<ol>
<li>
Probability that chosen person does not attend both.</li>
<li>Probability that ch... | ImOnlyListening | 828,915 | <p>On your answer (1), you are implying that the two events are independent which does not seem to be the case.
I would have use the De Morgan's Law to solve it:</p>
<p><span class="math-container">$$ \Pr \big( ( A \cup B ) ^ { \mathrm c } \big) = 1 - \Pr ( A \cup B ) = 0.2 $$</span></p>
<p><span class="math-container"... |
2,800,441 | <p>Anyone can give me the path for this, could not figure out which theory/method to use for this...
$$\lim_{n\to\infty}\left(\frac{1}{1\cdot4}+\frac{1}{4\cdot7} + \frac{1}{7\cdot10}+.....+ \frac {1}{(3n-2)\cdot(3n+1)}\right)=?$$</p>
| José Carlos Santos | 446,262 | <p><strong>Hint:</strong> $\displaystyle\frac1{1\times4}=\frac13\left(1-\frac14\right)$, $\displaystyle\frac1{4\times7}=\frac13\left(\frac14-\frac17\right)$, $\displaystyle\frac1{7\times10}=\frac13\left(\frac17-\frac1{10}\right)$, …</p>
|
2,187,929 | <p>A speaks truth $3$ times out of $4$ and $B$ $7$ times out of $10$ . they both agree that a white ball has been drawn out from a bag containing $6$ balls of different color . find the probability that the statement is true . </p>
<p>my try .</p>
<p>probability when they say false and agree = $\left(\dfrac56\right)\... | Cato | 357,838 | <p>if a ball is drawn and they are asked if it is white true/false and they both say true - then </p>
<p>P(White | say true) = P(white and say it's true) / P(say true) =
(1/6)(7/10)(3/4) / ((1/6)(7/10)(3/4) + (5/6)(3/10)(1/4)) </p>
<p>= (7/80) (7/80 + 1 / 16) = 7 / 12</p>
<p>one in 16 times - it isn't white, but t... |
3,472,345 | <p>Problem: calculate <span class="math-container">$\int _0^{2\pi }e^{\cos \left(x\right)}\cos \left(\sin \left(x\right)\right)dx$</span></p>
<p>This is a problem which post in <a href="https://math.stackexchange.com/questions/409171/how-to-evaluate-int-02-pie-cos-theta-cos-sin-theta-d-theta?noredirect=1&lq=1">How... | Infiniticism | 628,759 | <p>Via Fourier expansion the integrand equals to <span class="math-container">$\sum _{n=0}^{\infty } \frac{\cos (n x)}{n!}$</span> According to orthogonality of trig functions one has <span class="math-container">$I=\frac{\int_0^{2 \pi } \cos (0 x) \, dx}{0!}=2\pi$</span>.</p>
|
1,715,683 | <p>I am stuck in the following problem. Can someone show me how shall I finish the track ?</p>
<p>the problem is : if $f:\mathbb{R}\rightarrow \mathbb{R}$ twice differentiable function and $f(0)=f'(0)=1$ and $f''\geqslant f$. Then show that $f$ is non-negative.</p>
<p>Thank you in advance.</p>
| S.C.B. | 310,930 | <p><strong>EDIT</strong></p>
<p>The answer below does not fully answer the question. </p>
<p><strong>HINT</strong></p>
<p>Let us first prove this for $x \ge 0$. </p>
<p>If there exists such $x \ge 0$ that $f(x)<0$, we get that there exists such a $c$ that $f(c)=0$. </p>
<p>Let $c_1$ be the smallest such $c$.</p... |
1,715,683 | <p>I am stuck in the following problem. Can someone show me how shall I finish the track ?</p>
<p>the problem is : if $f:\mathbb{R}\rightarrow \mathbb{R}$ twice differentiable function and $f(0)=f'(0)=1$ and $f''\geqslant f$. Then show that $f$ is non-negative.</p>
<p>Thank you in advance.</p>
| C. Ding | 320,080 | <p>Let
$$g(x)=(f'(x)-f(x))e^x,$$
then $g'(x)=(f''(x)-f(x))e^x\geq 0,$
and we know that $g(0)=(f'(0)-f(0))e^0=0,$
thus $g(x)\begin{cases}
\leq 0,x\leq 0;\\
\geq 0,x>0.
\end{cases}$</p>
<p>Let $$h(x)=f(x)e^{-x},$$
then $h'(x)=(f'(x)-f(x))e^{-x}=g(x)e^{-2x}.$
Thus $h(x)\geq h(0)=1,$and $f(x)=h(x)e^x\geq e^x\geq... |
1,721,584 | <p>The image attached below is a problem on induction, the proof has been included.
I am enquiring if anyone could explain line for line what the proof states with its notation ( the notation is new to me). (I have a bit of experience with proof by induction, but is stumped by this problem)</p>
<p><a href="https://i.s... | Aditya Agarwal | 217,555 | <p>So you are evidently confused in how the derivative of $x^2-x$ is computed.
We know that $\frac{d}{dx}k f(x)=k\frac d{dx}f(x)$ and $\frac d{dx}(f(x)+g(x))=\frac d{dx}f(x)+\frac d{dx}g(x)$. <br> So
$$\frac d{dx}(x^2-x)=\frac d{dx}x^2+\frac d{dx}(-x)=2x+(-1)\frac d{dx}(x)=2x+(-1)1=2x-1$$</p>
|
835,536 | <p>Can you help me with this limit? What do I have to do? I'm lost.</p>
<p>$$\lim_{n\to\infty}n\left(\sum_{i=1}^{n}\dfrac{1}{(n+i)^2}\right)$$</p>
<p>The solution given is $\dfrac{1}{2}$.</p>
| Paul | 16,158 | <p>Note that
$$n\left(\sum_{i=1}^{n}\dfrac{1}{(n+i)^2}\right)=\frac{1}{n}\sum_{i=1}^{n}\dfrac{1}{(1+\frac{i}{n})^2}.$$
By <a href="http://en.wikipedia.org/wiki/Riemann_sum">Riemann sum</a>, we have
$$\lim_{n\to\infty}n\left(\sum_{i=1}^{n}\dfrac{1}{(n+i)^2}\right)=\int_0^1\frac{dx}{(1+x)^2}=-\frac{1}{1+x}\Big|_0^1=-\f... |
2,197,561 | <p>If $f(x)$ is an irreducible cubic, then $\operatorname{Gal}(f(x))\cong S_3$ or $A_3$. But what about the converse? That is, if $\operatorname{Gal}(K/F)\cong S_3$, is it necessarily true that $K$ is the splitting field of some irreducible cubic in $F[x]$?</p>
| Charith | 321,851 | <p>A quick addition to the post that I thought will be useful for someone else who need a little more clarification: </p>
<p><span class="math-container">$K$</span> being a Galois extension implies that it is Separable.<br>
Since <span class="math-container">$p(x)$</span> has a root, <span class="math-container">$\th... |
1,510,693 | <p>Suppose $T\in(V)$ and $(T-2I)(T-3I)(T-4I) = 0$. Suppose $\lambda$ is an eigenvalue of $T$. Prove $\lambda = 2$ or $\lambda = 3$ or $\lambda = 4$</p>
<p>What properties of polynomials will prove this?</p>
| John Joy | 140,156 | <p>Another approach is
$$-\sin x-\cos x = 0 \implies \sin^2 x+2\sin x\cos x+\cos^2 x=0\implies 2\sin x \cos x=-1$$
$$\implies \sin 2x=-1\implies\dots$$</p>
|
1,942,110 | <p>Let $x,y,z>0$,prove or disprove
$$\sqrt[4]{\dfrac{(xy+yz+xz)(x^2+y^2+z^2)}{9}}\ge\sqrt[3]{\dfrac{(x+y)(y+z)(z+x)}{8}}$$</p>
<p>I tried many times ,use
$$x^2+y^2+z^2\ge\dfrac{1}{3}(x+y+z)^2$$
and kown $$9(x+y)(y+z)(z+x)\ge 8(x+y+z)(xy+yz+xz)$$</p>
| Michael Rozenberg | 190,319 | <p>Let $x+y+z=3u$, $y+xz+yz=3v^2$ and $xyz=w^3$.</p>
<p>Hence, it's obvious that our inequality is equivalent to $f(w^3)\geq0$, where $f(w^3)=w^3+A(u,v^2)$, </p>
<p>which says that it's enough to prove our inequality for the minimal value of $w^3$.</p>
<p>But $x$, $y$ and $z$ are positive roots of the equation
$$(X-... |
3,113,186 | <p>Is this a power series? My book defines power series <span class="math-container">$\sum\limits_{n=1}^{+\infty} a_n (x-x_0)^n$</span></p>
| xpaul | 66,420 | <p>Let
<span class="math-container">$$ a_m=\left\{\begin{array}{ll}
n^n,\text{ if }m=n! \text{ for }n=1,2,3,\cdots,\\
0,\text{ else.}
\end{array}\right. $$</span>
Then
<span class="math-container">$$ \limsup_{m\to\infty}\sqrt[m]{a_m}=\limsup_{n\to\infty}\sqrt[n!]{n^n}=1 $$</span>
and hence
<span class="math-container"... |
717,882 | <p>The $x^{2/2}$ can be represented by these ways:
$$\begin{align}
x^{2\over2}=\sqrt{x^2} = |x|\\
\end{align}
$$
And<br>
$$\begin{align}
x^{2\over2}=x^{1} = x\\
\end{align}
$$
Which one is correct?
And what is the domain of $x^{2 \over 2}$?</p>
| kmitov | 84,067 | <p>$(x^2)^{1/2}$ is defined for every real $x$, but $(x^{1/2})^2$ is defined for $ x \ge 0$</p>
|
717,882 | <p>The $x^{2/2}$ can be represented by these ways:
$$\begin{align}
x^{2\over2}=\sqrt{x^2} = |x|\\
\end{align}
$$
And<br>
$$\begin{align}
x^{2\over2}=x^{1} = x\\
\end{align}
$$
Which one is correct?
And what is the domain of $x^{2 \over 2}$?</p>
| Anonymous Computer | 128,641 | <p>When you put an exponent in fractional form, you run into some problems. $x^{2/2}=|x|$, but $x^1=x$. You cannot simply cancel out fractions in exponents, because you may forget the restrictions.</p>
<p>A fake proof that $\sqrt{-1}=1$ uses this. The proof goes like so:
$$\sqrt{-1}=(-1)^{1/2}$$
$$=(-1)^{2/4}$$
$$=\sq... |
717,882 | <p>The $x^{2/2}$ can be represented by these ways:
$$\begin{align}
x^{2\over2}=\sqrt{x^2} = |x|\\
\end{align}
$$
And<br>
$$\begin{align}
x^{2\over2}=x^{1} = x\\
\end{align}
$$
Which one is correct?
And what is the domain of $x^{2 \over 2}$?</p>
| Emanuele Paolini | 59,304 | <p>The rule
$$
x^{pq} = (x^p)^q
$$
is not always valid if $p$ or $q$ are not integers and $x<0$. As you have noticed
$$
-1 = (-1)^1 = (-1)^{\frac 2 2} \neq ((-1)^2)^{\frac 1 2} = 1^{\frac 1 2} = 1.
$$</p>
<p>So, unfortunately, $x^{\frac 2 2} \neq \sqrt{x^2}$ for $x<0$.
For the last question: $x^{\frac 2 2} ... |
717,882 | <p>The $x^{2/2}$ can be represented by these ways:
$$\begin{align}
x^{2\over2}=\sqrt{x^2} = |x|\\
\end{align}
$$
And<br>
$$\begin{align}
x^{2\over2}=x^{1} = x\\
\end{align}
$$
Which one is correct?
And what is the domain of $x^{2 \over 2}$?</p>
| ColinK | 121,252 | <p>I'm amazed to ctrl-f 'order of operations' and come up with nothing here!</p>
<p>Other answers do well to explaining what's going on here, but recognize that the confusion is about perceived ambiguity with respect to order of operations. In this case, there is an invisible (but understood) set of brackets in the ex... |
2,650,454 | <p>I am able to prove the binomial theorem (see $(1))$ by induction. I have also seen the binomial theorem written in another way (see $(2)$) where the summation changes a little bit. I understand how to compute both theorems, but I am having a difficult time proving the second theorem from the first. </p>
<p>Obviousl... | David Stanley | 135,244 | <p>The actual equations are the same, the only thing that's different is the index set you're summing over. So you only need to prove that these two sets are equal:</p>
<p>$$ \left \{ (i, n - i) | i \in \mathbb N, i \leq n \right \} $$</p>
<p>and </p>
<p>$$ \left \{ (i_1, i_2) | i_1, i_2 \in \mathbb N, i_1 + i_2 = ... |
907,076 | <p>Axiom of specification is schema because it talks about definite condition(or wff) which use notion of finite but this again we define from sets. But in logic we defined wff using consept of tuple and finite like thing. I have just started studying logic. And I am confused about using n-tuple or n-ary relation. Is n... | user132181 | 132,181 | <p><a href="http://en.wikipedia.org/wiki/Axiom_schema_of_specification" rel="nofollow noreferrer">From Wikipedia</a>:</p>
<blockquote>
<p>One instance of the schema is included for each formula <span class="math-container">$φ$</span> in the language of set theory with free variables among <span class="math-container">$... |
907,076 | <p>Axiom of specification is schema because it talks about definite condition(or wff) which use notion of finite but this again we define from sets. But in logic we defined wff using consept of tuple and finite like thing. I have just started studying logic. And I am confused about using n-tuple or n-ary relation. Is n... | Nagase | 117,698 | <p>I think I may get some of the confusion behind the OP. In set theory, we generally use first-order logic in order to formulate the axioms (this is more visible with regards to the axiom schema of comprehension and replacement). So, in a sense, set theory "depends" (I'm using the word rather loosely) on a background ... |
2,968,241 | <blockquote>
<p>Let</p>
<p><span class="math-container">$$f(z)=\frac{e^{\sin{z}}-1}{z^3}.$$</span></p>
<p>a) Determine if <span class="math-container">$f$</span> has a pole at <span class="math-container">$0$</span> and determine its order.</p>
<p>b) If <span class="math-container">$f(z)=\sum_{n=-\inft... | José Carlos Santos | 446,262 | <p>Let <span class="math-container">$f(z)=e^{\sin z}$</span>. Then:</p>
<ul>
<li><span class="math-container">$f(0)=1$</span>;</li>
<li><span class="math-container">$f'(0)=1$</span>;</li>
<li><span class="math-container">$f''(0)=1$</span>;</li>
<li><span class="math-container">$f'''(0)=0$</span>,</li>
</ul>
<p>you ha... |
113,362 | <blockquote>
<p>Is there some inherent quality of a mathematical object that marks it as being "naturally" a thingie or a cothingie?</p>
</blockquote>
<p>Suppose, for example, that two mathematical concepts, say, doodad and doohickey, originally defined far, far away from the reaches of category theory, are later di... | Qiaochu Yuan | 232 | <p>An algebraic category like $\text{Grp}$ or $\text{Ring}$ often has a forgetful functor to $\text{Set}$ which has a left adjoint (the free object functor) but generally not a right adjoint. It follows that the forgetful functor preserves limits but generally not colimits. That's one reason you might consider limits m... |
1,422,140 | <p>Suppose we have a Hilbert space $X$, a weakly convergent sequence $u_k\rightharpoonup u$ and a convergent operator $T_k \rightarrow T$ in the norm of $\mathcal{L}(X)$ (bounded, linear operators).
Is the assertion $T_k u_k \rightharpoonup Tu$ correct?</p>
<p>Thanks in advance!</p>
| R.N | 253,742 | <p>ok suppose $f\in{X^*}$ then $foT\in{X^*}$ since $u_k$ is weakly convergent we have
$foT_k u_k \rightharpoonup foTu$</p>
|
1,422,140 | <p>Suppose we have a Hilbert space $X$, a weakly convergent sequence $u_k\rightharpoonup u$ and a convergent operator $T_k \rightarrow T$ in the norm of $\mathcal{L}(X)$ (bounded, linear operators).
Is the assertion $T_k u_k \rightharpoonup Tu$ correct?</p>
<p>Thanks in advance!</p>
| Svetoslav | 254,733 | <p>As noted by @Razieh Noori, if $f\in X^*$, then $f\circ T\in X^*$. Since $u_k\rightharpoonup u \Rightarrow \exists M:\|u_k\|\leq M\quad \forall k\quad$. Also $T_k\rightarrow T \Rightarrow$ for $\epsilon>0\quad\|T_k-T\|_{op}<\frac{\epsilon}{2 M \|f\|_*}$ for $k>K_1$.</p>
<p>Then</p>
<p>$|f\circ T_ku_k-f\ci... |
1,194,181 | <p>Let $G:\mathbb{R}^n \to\mathbb{R}^n$ be transformation such that $G(x):=Ax+b$ where $A\in\mathcal{M}_{nxn}(\mathbb{R})$ and $b\in\mathbb{R}^n$ such that $det(A-I)\neq0$ .</p>
<p>How would you prove G has a unique fixed point $p\in\mathbb{R}^{n}$ ? </p>
| Daniel Valenzuela | 156,302 | <p>Having a fix point means $x=Ax+b$, i.e. $0=(A-I)x+b$. But now $A-I$ is invertible, in particular there exists one and only one element which hits $-b$.</p>
|
1,636,979 | <p>The question says:
<a href="https://i.stack.imgur.com/NEDTp.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/NEDTp.png" alt="enter image description here"></a></p>
<p>The solution set was posted and there are a few things I don't quite understand from it.</p>
<p><a href="https://i.stack.imgur.com... | André Nicolas | 6,312 | <p>$a^2$ is an informal abbreviation for $a\cdot a$.</p>
<p>As to your later question about $a^2\cdot a^{-1}$, it means $(a\cdot a)\cdot a^{-1}$. By associativity this is $a\cdot (a\cdot a^{-1})$, which is $a\cdot 1$, which is $a$.</p>
|
2,347,775 | <p>Suppose we have arbitrary predicates are $P$ and $Q$.</p>
<p>Let the statements be defined as follows:</p>
<p>$F1:$ [for all x, P(x)] is false OR [for some x, Q(x)] is true</p>
<p>$F2:$ [for some x, P(x)] is false OR [for all x, Q(x)] is true</p>
<p>Prove that $F1 \neq\implies F2$ (does not apply)</p>
<p>We hav... | chi | 207,328 | <p>Rephrasing using sets instead of properties, on your domain $\mathbb{N}$:</p>
<p>F1: $P\neq \mathbb{N}$ OR $Q\neq \emptyset$</p>
<p>F2: $P=\emptyset$ OR $Q=\mathbb{N}$</p>
<p>Can you find a counterexample to $F1 \implies F2$, i.e. a way to make $F1$ true but $F2$ false?</p>
|
65,355 | <p>In "Tilting Theory and Cluster Combinatorics" Buan, Marsh, Reineke, Reiten, and Todorov constructed cluster categories for mutation finite cluster algebras (without coefficients), and Amiot gives a construction of a cluster category given a quiver with potential whose Jacobian algebra is finite dimensional (in parti... | Jan Grabowski | 13,215 | <p>There are several different questions in here and what follows are only partial answers to some of these, mostly consisting of pointers to pieces of the literature.</p>
<ul>
<li>"In what other instances have cluster
categories been constructed?"</li>
</ul>
<p>For surveys on cluster categories, I would recommend:</... |
296,737 | <p>Im trying to show that the ring of polynomials in one variable over the complex numbers is not isomorphic to the ring over $\mathbb C$ with two variables $x$ and $y$ modulo $\langle x^2-y^3\rangle$. I've shown previously that if the relationship $p^2=q^3$ holds for some $p$ and $q$ in one variable, there exists $r$ ... | Mariano Suárez-Álvarez | 274 | <p>The algebra $\def\CC{\mathbb C}\CC[T]$ is integrally closed in its fraction field; this follows from Gauss's lemma, for example.</p>
<p>On the other hand, the element $t=x/y$ of the fraction field $F$ of $A=\CC[X,Y]/(X^2-Y^3)$, which is not if $A$, satisfies the polynomial $f(T)=T^2-Y\in F[T]$. This means that $A$ ... |
11,405 | <p>A user named "FeelingSick" posted an "answer" with some <em>extremely</em> unsuitable content at the question <a href="https://math.stackexchange.com/questions/538536/how-do-proofs-work-in-mathematics-down-to-the-basic-level">How do "proofs" work in mathematics, down to the basic level?</a>. [Warning: the ... | Mariano Suárez-Álvarez | 274 | <p>Done. ${}{}{}{}{}{}{}{}{}{{}}$</p>
|
11,405 | <p>A user named "FeelingSick" posted an "answer" with some <em>extremely</em> unsuitable content at the question <a href="https://math.stackexchange.com/questions/538536/how-do-proofs-work-in-mathematics-down-to-the-basic-level">How do "proofs" work in mathematics, down to the basic level?</a>. [Warning: the ... | Asaf Karagila | 622 | <p>People should flag such posts as offensive posts, and not downvote the answer at all. For two main reasons:</p>
<ol>
<li>Every such flag automatically downvotes the post.</li>
<li>After six flags the post is deleted immediately. </li>
</ol>
<p>If one feels that deletion is not enough, and some action needs to be t... |
1,361,478 | <p>A function of a single variable is denoted $f(x)$, of two variables if denoted $f(x,y)$</p>
<p>What about infinite variables? How do we denote such thing and do such things exist?</p>
| lulu | 252,071 | <p>If you want an example in which each variable can change the value of the function, try:</p>
<p>$$f(\vec x) = \sum \frac{1}{(n^2 + |x_n|)}$$</p>
<p>That sum converges (by comparison) no matter what the $x_n$ may be.</p>
|
2,131,709 | <p><span class="math-container">$$F=\left(y\cos \left(xy\right)+e^{x+y}\right)i+\left(x\cos \left(xy\right)+e^{x+y}\right)j$$</span></p>
<p>also show that <span class="math-container">$$∫_cF ⋅ dr = e^2-e^{-2}$$</span>
where c is the straight line from <span class="math-container">$\left(-1,-1\right)$</span> to <span c... | Kuifje | 273,220 | <p>By definition, the potential is a function $f: \mathbb{R}^2\rightarrow \mathbb{R}$ such that
$$
\nabla f = \pmatrix{\frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y}}= \vec{F}
$$</p>
<p>Can you solve this equation ?</p>
<p>Once you have found $f(x,y)$, the fundamental theorem of line integrals states ... |
1,359,624 | <p>How to find derivative of</p>
<p>$$f(x)=|\sin^{-1}(2x^2-1)|$$</p>
<p>Please provide stepwise mechanism.</p>
<p>The original question was to find domain of derivative of y=|arc sin(2x^2−1)|.</p>
<p>My METHOD- My attempt was to break y into intervals ,i.e., where \sin^{-1}(2x^2-1)>=0 and where \sin^{-1}(2x^2-1)<... | Jan Eerland | 226,665 | <p>$$|a\cdot\sin(2x^2-1)|\Longrightarrow$$</p>
<p>$$\frac{d}{dx}(|a\cdot\sin(2x^2-1)|)=$$
$$\frac{d}{dx}(|a\cdot\sin(1-2x^2)|)=$$
$$\frac{d|u|}{du}\cdot\frac{du}{dx}=$$</p>
<p>(With $u=a\cdot(1-x^2)$ and $\frac{d}{du}(|u|)=\frac{u}{|u|}$)</p>
<p>$$\frac{a(\frac{d}{dx}(a\cdot\sin(1-2x^2)))\sin(1-2x^2)}{|a\cdot\sin(2x... |
356,033 | <p>This question is similar to <a href="https://math.stackexchange.com/questions/171643/is-every-forest-with-more-than-one-node-a-bipartite-graph">Is every forest with more than one node a bipartite graph?</a>, but requires a proof by induction.</p>
<p>This was a past exam question.</p>
<p>-</p>
<p>Let P(G) be the p... | joriki | 6,622 | <p>I think this is all unnecessarily complicated. You can use induction over the number of edges, without removing edgeless nodes.</p>
<p>Basis step: A forest without edges is bipartite.</p>
<p>Inductive step: Assume all forests with $k$ edges are bipartite. Given a forest $F$ with $k+1$ edges, remove an edge $e$ and... |
356,033 | <p>This question is similar to <a href="https://math.stackexchange.com/questions/171643/is-every-forest-with-more-than-one-node-a-bipartite-graph">Is every forest with more than one node a bipartite graph?</a>, but requires a proof by induction.</p>
<p>This was a past exam question.</p>
<p>-</p>
<p>Let P(G) be the p... | Douglas S. Stones | 139 | <p>This seems to be an attempt at a "two dimensional" induction. This means that we no longer have a single base case.</p>
<p>To be consistent with your method, I think it'd be best to choose and arbitrary number of vertices $n \geq 0$, then induct on the number of edges. The base case is the $n$-vertex null graph, ... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Steve Huntsman | 1,847 | <p><a href="https://books.google.com/books?id=eJp330pIvQ4C" rel="nofollow noreferrer">Volume I of <em>Diffusions, Markov processes and martingales</em></a> by Rogers and Williams has a probabilistic proof of the fundamental theorem (<a href="https://books.google.com/books?id=eJp330pIvQ4C&lpg=PP1&pg=PA41#v=onepa... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Felipe Voloch | 2,290 | <p>Gauss's first proof goes more or less like this. Let $p(z)$ be a polynomial of degree $n$ and complex coefficients. Write $p(x+iy) = a(x,y) + ib(x,y)$, where $a,b$ have real coefficients. The crucial observation is that the branches of $a=0$ and $b=0$ as real curves interlace at infinity (as can be seen from the deg... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | KConrad | 3,272 | <p>Here is a translation into English of a second "real" proof from the journal Ilya mentioned in his answer. This proof is due to Petya Pushkar', in the 1997 paper titled <em>О некоторых топологических доказательствах основной теоремы алгебры</em>; this is on mathnet.ru <a href="http://mi.mathnet.ru/eng/mp/... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Timothy Chow | 3,106 | <p>A recent and very important contribution to the literature on the fundamental theorem of algebra is Joe Shipman's article "Improving the Fundamental Theorem of Algebra," <em>Math. Intelligencer</em> 29 (2007), 9-14, <a href="https://doi.org/10.1007/BF02986170" rel="nofollow noreferrer">doi:10.1007/BF029861... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | jasomill | 9,352 | <p>I'm partial to Milnor's proof in <em>Topology from the Differentiable Viewpoint</em>, a slightly simpler variant of the "every complex non-constant polynomial $p$ is surjective" proof given above, published somewhat earlier (1965). In brief:</p>
<p><em>Definition:</em> Let $f: M \to N$, $M,N \subset \mathbb{R}^n$, ... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Autumn Kent | 1,335 | <p>Gersten and Stallings gave a proof using free groups:</p>
<p><a href="https://www.ams.org/journals/proc/1988-103-01/S0002-9939-1988-0938691-3/home.html" rel="nofollow noreferrer">On Gauss's first proof of the fundamental theorem of algebra, Proc. Amer. Math. Soc. 103 (1988), 331-332</a></p>
<p>of course giving all o... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Victor Miller | 2,784 | <p>I'm surprised that no-one's mentioned the proof using Roueche's theorem:</p>
<p>Given $f,g$ holomorphic and $C$ a closed contour if $|g(z)|< |f(z)|$ on $C$ then $f$ and $f+g$ have the same number of zeros (counting multiplicity) in the interior of $C$. There's an easy proof of this using the Cauchy integral for... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Todd Trimble | 2,926 | <p>This is one of the proofs currently on the <a href="https://ncatlab.org/nlab/show/fundamental+theorem+of+algebra" rel="nofollow noreferrer">nLab</a>. My goal in writing it was to see how elementary I could make it, that if you squint a little it <em>might</em> have been a proof from the late <span class="math-contai... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Peter Scholze | 6,074 | <p>Here is a variant of d'Alembert's argument using the minimum of <span class="math-container">$|p(z)|$</span>. It has the advantage that it proves more generally the Gelfand-Mazur theorem (usually proved by complex analysis): Any Banach field <span class="math-container">$K$</span> over <span class="math-container">$... |
2,251,501 | <p>How can I calculate the number of perfect cubes among the first $4000$ positive integers?</p>
<p>Is there any trick to solving such questions?</p>
| John Doe | 399,334 | <p>If you didn't have a calculator and needed to work out what the largest integer $x$ was such that $x^3\leq4000$ without just computing $\sqrt[3]{4000}$, then you could estimate it - $10^3=1000$, $20^3=8000$ so $10<x<20$. Then keep narrowing it down, e.g. go halfway $15^3=225\cdot15=2250+1125=3375$, and $16^3=2... |
2,384 | <p>I am tutoring several talented students, middle school level and early high school level, in mathematics. I am always looking for new sources from which to draw questions. Can anyone recommend books, web-sites, etc. with a interesting questions?</p>
<p>I know of the ArtofProblemSolving.com. I am currently using the... | Larry Wang | 73 | <p><a href="http://www.mathcounts.org/" rel="nofollow">Mathcounts</a> is aimed at talented middle school students. If you are affiliated with a school, you can register as a coach with them, and they can provide you with many helpful resources. If not, they still have problem sets available for sale from their catalog,... |
2,384 | <p>I am tutoring several talented students, middle school level and early high school level, in mathematics. I am always looking for new sources from which to draw questions. Can anyone recommend books, web-sites, etc. with a interesting questions?</p>
<p>I know of the ArtofProblemSolving.com. I am currently using the... | Community | -1 | <p>I'm working on <a href="http://www.buzzmath.com/" rel="nofollow">www.BuzzMath.com</a>, could fit your needs.</p>
|
2,384 | <p>I am tutoring several talented students, middle school level and early high school level, in mathematics. I am always looking for new sources from which to draw questions. Can anyone recommend books, web-sites, etc. with a interesting questions?</p>
<p>I know of the ArtofProblemSolving.com. I am currently using the... | Isaac | 72 | <p><a href="http://arml.com/books.php" rel="nofollow">Books of old ARML contests</a></p>
|
3,730,256 | <p>I have a real orthogonal matrix so the column vectors form an orthogonal system and thus the vectors have length one.</p>
<p>I now want to show that for an arbitrary column vector <span class="math-container">$v_k \in \mathbb{R^n}$</span> the absolute value of the greatest entry <span class="math-container">$|v_{k_i... | lisyarus | 135,314 | <p>Let <span class="math-container">$v \in \mathbb R^n$</span> be a unit vector. That is, <span class="math-container">$\sum |v_i|^2=1$</span>. Let <span class="math-container">$k$</span> be the index of the coordinate with largest absolute value. Then,</p>
<p><span class="math-container">$$|v_k|^2 \leq \sum |v_i|^2 = ... |
121,678 | <p>While teaching number theory this quarter, I have come across a phenomenon which was already addressed <a href="https://mathoverflow.net/questions/16141">in another MO posting</a>, but I have new questions. Let $p$ be a prime congruent to 3 mod 4. Then an elementary refinement of Wilson's theorem says that $\frac{... | François Brunault | 6,506 | <p>Regarding your question (1), here are two articles I know dealing with the problem of computing factorials :</p>
<p>Crandall, Dilcher, Pomerance. <em>A search for Wieferich and Wilson primes.</em> Math. Comp. 66 (1997), no. 217, 433--449. (MR1372002)</p>
<p>Costa, Gerbicz, Harvey. <em><a href="http://arxiv.org/... |
1,548,314 | <p>Does there exist a continuous function $f:\mathbb R^2\to \mathbb R$ such that $\displaystyle \frac{\partial f}{\partial x}$ <strong>does not exists</strong> but $\displaystyle \frac{\partial^2 f}{\partial x\partial y}$ exists.</p>
<p>I think yes. But I am unable to find an example of such function. Can anyone help ... | Siddharth Joshi | 288,487 | <p>The function $f: R^2 \to R$ where $f(x, y) = |x|+xy$ is an example</p>
<p>See that for the following function $$f_{yx} = 1$$ for every $(x, y) \ \epsilon \ R^2$ </p>
<p>But $f_x$ doesn't exist at the following set of points $\{ (0, y) \ \epsilon \ R^2 \ | \ y \ \epsilon \ R \} $ </p>
<p>Thus for the set $\{ (0, y... |
1,828,024 | <p>If each ticket for a lottery has a 1 in 258,890,850 chance... what happens if you buy 10 tickets? </p>
<p>Is it:</p>
<ul>
<li>A 10 in 258,890,850 chance?</li>
<li>A 1 in 258,890,840 chance?</li>
<li>A 1 in 25,889,085.0 chance?</li>
</ul>
<p>What is the best way to represent the chances? This is very simple, but I... | Mark | 76,963 | <p>Let me divide it into two bits: (1) what are the upper bounds of $S$ and (2) is there a least one?</p>
<p>First part: $u$ is an upper bound of $S$ if $s \propto u$ for all $s$ in $S$. Given $s$, the expression $s \propto u$ means that $u$ lies in $\{s, 4s, 4s+1, 4s+2, \ldots\}$. Call this set $U(s)$. For instance, ... |
328,263 | <p>Let $\ S\ $ be a non-empty Set, and suppose s$\ \in S $.
Assuming $\ S\ $ is finite, what can we deduce about the relationship between $\ |\mathcal P(S\ \setminus \{s\} )| $ and $\ | \mathcal P(S)|?$</p>
<p>May I know if the question is looking for $\ |\mathcal P(S\ \setminus \{s\} )| =$ $\ | \mathcal P(S)| \over ... | Boris Novikov | 62,565 | <p>$|\mathcal P(S \setminus \{s\} )| =\frac { | \mathcal P(S)|}{2}$</p>
|
655,185 | <blockquote>
<p>Let $G$ be a non-trivial finite group. Let $\chi$ be an irreducible character of the group $G$. Find $$\frac{1}{|G|}\sum_{g \in G} {\chi( g)}$$</p>
</blockquote>
<p>I try. But I think that I am wrong. $$G=C_{i_1}\oplus C_{i_2}\oplus\ldots C_{i_k},$$
as $C_{i}$ is a cyclic group
If $\chi$ is 1-dimen... | Tobias Kildetoft | 2,538 | <p>Here is an extension of the comment by mt_:</p>
<p>Consider the definition of the inner product on the characters $\langle \chi,\psi\rangle = \frac{1}{|G|}\sum_{g\in G}\chi(g)\overline{\psi(g)}$.</p>
<p>Can you get the sum you are looking as an inner product between suitable characters?</p>
|
3,274,496 | <p>Please correct my thinking, if anything not make sense to you.</p>
<p>A vector in <span class="math-container">$R^n$</span> is nothing but an assemblage of its co-ordinate w.r.t. some basis in the form of <span class="math-container">$n \times 1$</span> matrix.</p>
<p>Statement: If set of vectors(w.r.t. some basi... | PaleBlueDot | 488,358 | <p>You needn't constrict your question to <span class="math-container">$\mathbb{R}^n$</span>, this works in every finite-dimensional vector space <span class="math-container">$V$</span>. Like said in the comments, every <span class="math-container">$n$</span>-dimensional vector space <span class="math-container">$V$</s... |
2,980,446 | <p>I have this problem and I really don't know, how to edit it to get some solution.
<span class="math-container">$\lim_{n \to \infty} \sqrt{n} * [\sqrt{n+1} - \sqrt{n}]$</span></p>
<p>So my question Is what to do with <span class="math-container">$\sqrt{n} * [\sqrt{n+1} - \sqrt{n}]$</span>? </p>
| user | 505,767 | <p><strong>HINT</strong></p>
<p><span class="math-container">$$\sqrt n(\sqrt{n+1}-\sqrt n)=\sqrt n(\sqrt{n+1}-\sqrt n)\frac{\sqrt{n+1}+\sqrt n}{\sqrt{n+1}+\sqrt n}=$$</span></p>
<p><span class="math-container">$$=\frac{\sqrt n}{\sqrt{n+1}+\sqrt n}=\frac{\sqrt n}{\sqrt n}\frac{1}{\sqrt{1+\frac1n}+1}$$</span></p>
|
2,416,718 | <p>Prove that for all real numbers $x,y,z$:<br>
$$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{x+y+z}\implies \frac{1}{x^5}+\frac{1}{y^5}+\frac{1}{z^5}=\frac{1}{(x+y+z)^5}.$$</p>
| Dietrich Burde | 83,966 | <p>Hint: We have
$$
0=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}-\frac{1}{x+y+z}=\frac{(x + y)(x + z)(y + z)}{(y + z + x)xyz}.
$$
Hence we have
$$(x+y)(x+z)(y+z)=0.$$</p>
|
239,622 | <p>For what values of $1\le p \le \infty$ does $f(x,y)=\frac{1}{1+|x|+|y|}$ with $(x,y) \in \mathbb{R}^2$ belong to $L^p(\mathbb{R}^2)$?</p>
<p>Using Wolfram Alpha I've found that the answer should be $p > 2$, but I don't know how to begin proving it. I've thought about doing a change of variable, after considering... | Lukas Geyer | 43,179 | <p>You started promising, you just need to integrate. To start, for $y>0$ and $p>1$,
$$
\int_0^\infty \frac{dx}{(1+x+y)^p} = -\frac{1}{p-1} \left[\frac{1}{(1+x+y)^{p-1}}\right]_{x=0}^\infty = \frac{1}{(p-1)(1+y)^{p-1}}.
$$
Then
$$
\int_0^\infty \frac{dy}{(p-1)(1+y)^{p-1}}
$$
converges iff $p>2$. (And you can e... |
2,379,873 | <p>In a text I was reading; this was included in a section on set operations...</p>
<p>(-∞, 0) ∪ (0, ∞)</p>
<p>Is that a valid set operation if the typical "{}" are missing?</p>
<p>If it is valid; I assume this would also be valid...</p>
<p>[1, 3] = {x: 1 <= x <= 3}</p>
| 5xum | 112,884 | <ul>
<li>$(-\infty, 0)$ is a validly defined set.</li>
<li>$(0, \infty)$ is a validly defined set.</li>
<li>if $A$ and $B$ are sets, then $A\cup B$ is a validly defined set.</li>
</ul>
<p>In fact, $$(-\infty, 0)\cup (0, \infty) = \{x| x<0\}\cup \{x| x>0\} = \{x| x<0\lor x>0\} = \{x| x\neq 0\} = \mathbb R\s... |
339,585 | <p>Show that the Function $$g(x):=\begin{cases} \left| x \right| \sin{( \cot{x} )} & \text{for }x\notin \left\{ 0,\frac{1}{42} \right\}, \\ 0 & \text{for }x=0, \\ 10^{42} & \text{for }x=\frac{1}{42} \end{cases}$$ </p>
<p>is not continuous, but in the Point $\xi=0$ is continuous. You can use that $\sin,\co... | DonAntonio | 31,254 | <p>Hint: Try to prove the easy</p>
<p><strong>Lemma:</strong> If for two functions $\,f(x)\,,\,g(x)\,$ we have that there exist $\,\epsilon\,,\, M\in\Bbb R_+\,$ s.t.:</p>
<p>$$\lim_{x\to x_0}f(x)=0\;\;\wedge\;\;|g(x)|\le M\,\,\,,\;\;\forall\,x\in (x_0-\epsilon,x_0+\epsilon)\;,\;\;\text{then also}\;\;\lim_{x\to x_0}f(... |
667,781 | <p>There are many curves that extend integer exponentiation to larger domains, so why was this one chosen?</p>
| Michael Hoppe | 93,935 | <p>It's economic. You define exponentiation to one base, say $e$, rigorously and reduce all other exponentiation to that case.</p>
<p>Edit: The reason why mathematicians chose just $e$ instead of $42$ for that purpose has reasons which go way beyond the scope of the OP's question.</p>
|
667,781 | <p>There are many curves that extend integer exponentiation to larger domains, so why was this one chosen?</p>
| Christian Chapman | 17,622 | <p>$f(x)=e^{\ln(x)\cdot y}$ is the unique form satisfying natural exponentiation:</p>
<p>\begin{equation}\tag{1}f(x,0) = 1\end{equation}</p>
<p>\begin{equation}\tag{2} x\cdot f(x,y) = f(x,y+1)\end{equation}</p>
<p>and differentiability:
\begin{equation}\tag{3}\frac{d}{dx}f(x,y)=y\cdot f(x,y-1)\end{equation}</p>
|
3,813,434 | <p>I am trying to prove the following :</p>
<blockquote>
<p>Let <span class="math-container">$R$</span> be a commutative ring with unity, <span class="math-container">$a,b\in R$</span>; suppose <span class="math-container">$(a)+(b)$</span> is principal. Show that <span class="math-container">$(a)\cap(b)$</span> is prin... | Bill Dubuque | 242 | <p>The trick is to effectively "cancel" <span class="math-container">$d$</span> at the <em>start</em> (vs. end) of proof. More precisely, since <span class="math-container">$\,d\mid a\,$</span> there exists <span class="math-container">$\,\frac{a}d\in R\,$</span> with <span class="math-container">$\,\frac{a}d... |
3,649,613 | <p>Let <span class="math-container">$\phi: U \rightarrow \mathbb{R}^m$</span>, <span class="math-container">$U\subseteq \mathbb{R}^n$</span> open, such that <span class="math-container">$\phi(U)$</span> is open and <span class="math-container">$\phi$</span> is a homeomorphism onto its image.</p>
<blockquote>
<p>Let ... | Noob mathematician | 779,382 | <p>Theorem:<strong>Continuous image of a compact set is compact.</strong> See the proof is <a href="https://math.stackexchange.com/q/2699919/779382">here</a></p>
<p>Since <span class="math-container">$\phi:U\to \phi(U)$</span> is a homeomorphism, hence <span class="math-container">$\phi^{-1}:\phi(U) \to U (\subseteq ... |
1,334,557 | <p>Suppose that an entire function $f$ has uncountably many zeros. Is it true that $f=0$?</p>
<p>I have no idea how to proceed with this. Perhaps some theorem that I am not aware of. I have done an undergraduate course in complex analysis.</p>
| Joe | 107,639 | <p>As John already pointed out, an uncountable subset $A\subseteq\Bbb C$ has necessarely an accumulation point. Then the identity principle for holomorphic functions, says that a function which is $0$ on $A$, is necessarely $0$ on its whole domain (which is $\Bbb C$ if your function is entire).</p>
|
1,628,839 | <p>For what values of $p$ does the following integral converge:</p>
<p>$\sum_{n=2}^{\infty} \frac{1}{n(\ln\ n)^p}.$</p>
<p>Ans. (Integral Test) $\int\limits_{n=2}^{n=\infty}\frac{1}{n(\ln n)^p} = \frac{1}{(-p+1)(ln\ n)^{p-1}}$</p>
<p>I know that $p \neq 1$, but I do not understand why the answer is $p > 1$ </p>
| Claude Leibovici | 82,404 | <p><strong>Hint</strong></p>
<p>$$\int (x-1)\sin (4x)\,dx=\int x \sin (4x)\,dx-\int \sin (4x)\,dx$$ $$\int x \sin (4x)\,dx=\frac 14\int (4x)\sin(4x)\,dx=\frac 1{16}\int y\sin(y)\,dy $$ Now, integrate by parts.</p>
|
105,378 | <p>I want to ask you if can it be so simple to prove that $\lim _{x \to \infty}\sum_{1}^{\infty}\frac{x^2}{1+n^2x^2}=\sum_{1}^{\infty}\frac{1}{n^2}$ by divide the numerator and denominator with $x^2$ and that's it? </p>
<p>If it this simple indeed you can write a comment and I'll delete the question after I'll read it... | Sasha | 11,069 | <p>Use
$$
\frac{1}{n^2} \frac{x^2}{1+x^2} \leqslant \frac{1}{n^2} \frac{x^2}{x^2 + n^{-2}} < \frac{1}{n^2}
$$
Thus
$$
\frac{x^2}{1+x^2} \sum_{n=1}^\infty \frac{1}{n^2} \leqslant \sum_{n=1}^\infty \frac{x^2}{n^2+x^2} < \sum_{n=1}^\infty \frac{1}{n^2}
$$
Both upper and the lower bounds have the same limi... |
2,781,867 | <p>$$
a+\frac{b}{2}+\frac{c}{3}=7 \left(1+\frac{1}{2}+\frac{1}{3} \right)
$$
Find the number of positive integral solution.</p>
| farruhota | 425,072 | <p>Use the method given in this <a href="https://math.stackexchange.com/questions/2114321/prove-sum-k-1n-frac1nk-geq-frac712/2114328#2114328">answer</a> by Jack D'Aurizio. </p>
<p>Note: $H_n=1+\frac12+\frac13+\cdots+\frac1n=\sum_{k=1}^n \frac1n$ is called <a href="https://en.wikipedia.org/wiki/Harmonic_number" rel="no... |
4,625,085 | <p>Let <span class="math-container">$D$</span> be a digraph as follows:
<a href="https://i.stack.imgur.com/sN32Z.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sN32Z.png" alt="enter image description here" /></a></p>
<p>I want to compute a longest simple path of it.</p>
<p>For an acyclic digraph, <a... | Sangchul Lee | 9,340 | <p>Here is a weaker result which is still enough to show that the limit is not <span class="math-container">$e$</span>:</p>
<blockquote>
<p><strong>Claim.</strong> We have
<span class="math-container">$$ \lim_{n\to\infty} \frac{f(n)}{n} = \sqrt{3} + 2\log(1+\sqrt{3}) - \log 2 \approx 3.04901 $$</span></p>
</blockquote>... |
3,446,693 | <p>A highway patrol plane is flying <span class="math-container">$1$</span> mile above a long, straight road, with constant ground speed of <span class="math-container">$120$</span> m.p.h. Using radar, the pilot detects a car whose distance from the plane is <span class="math-container">$1.5$</span> miles and decreasin... | David K | 139,123 | <p>The problem statement is ambiguous.</p>
<p>Assuming both the airplane and the car are moving along parallel, perfectly horizontal straight lines
(one along the road, which is assumed to be level, the other parallel to and above the road)
and that those parallel lines are exactly <span class="math-container">$1$</sp... |
296,138 | <p>My son, who is 16, is doing some independent research. A lower bound depending on $n$ for $\left\{ \left( \frac{4}{3} \right)^n \right\}=\left( \frac{4}{3} \right)^n-\left\lfloor \left(\frac{4}{3} \right)^n \right\rfloor$, the fractional part of $\left( \frac{4}{3} \right)^n$, might help him improve his results. Not... | Sylvain JULIEN | 13,625 | <p>I'll give a refined version of the idea expressed in my comments as an answer as it seems to capture the real thing.</p>
<p>Let $ r_{k}(n) $ be the representant of $ \binom{n}{k} $ in $ \mathbb{Z}/3^{k}\mathbb{Z} $ whose absolute value is minimal. Then the fractional part of $ (4/3)^n $ is $ \sum_{k=1}^{n}\dfra... |
296,138 | <p>My son, who is 16, is doing some independent research. A lower bound depending on $n$ for $\left\{ \left( \frac{4}{3} \right)^n \right\}=\left( \frac{4}{3} \right)^n-\left\lfloor \left(\frac{4}{3} \right)^n \right\rfloor$, the fractional part of $\left( \frac{4}{3} \right)^n$, might help him improve his results. Not... | Seva | 9,924 | <p>It may be interesting to note that, subject to the <a href="https://en.wikipedia.org/wiki/Abc_conjecture" rel="noreferrer">ABC conjecture</a>, you have the fantastically good estimate
$$ \left\{ \left( \frac43 \right)^n\right\} \gg_\delta \delta^n,\quad \delta\in(0,1). $$</p>
<p>The proof goes as follows. </p>
<... |
791,157 | <p>I am stuck on the following question.</p>
<blockquote>
<blockquote>
<p>Suppose $f = u+iv$ is entire and there exists $M > 0$ such that $|u(z)| \leq M$ for all $z\in C$. Show that $f$ is constant.</p>
</blockquote>
</blockquote>
<p>I would figure we would have to use Liouville's Theorem to show this is t... | AsdrubalBeltran | 62,547 | <p>Ok in this case, then the first theorem fundamental of calculus say that:</p>
<p>$$F(x)=\int_0^x{f(t)\,dt}\Longrightarrow F'(x)=f(x)$$
how $F'(x)=(1-x)e^{-x}+a=f(x)$ then how $f(1)=1\Longrightarrow F'(1)=(1-1)e^{-1}+a=f(1)=1 $
then $$a=1$$ therefore $f(x)=(1-x)e^{-x}+1$, $a=1$ and $b=0$
.</p>
|
2,135,151 | <p>Prove that, If $r$ is a real number such that $r^2 = 2$, $r$ is irrational.</p>
<hr>
<p><strong>Proposition:</strong> If $r$ is a real number such that $r^2 = 2$, then $r$ is irrational.</p>
<p><strong>Hypothesis:</strong> If $r$ is a real number such that $r^2 = 2$.</p>
<p><strong>Conclusion:</strong> $r$ is ir... | Sachchidanand Prasad | 249,258 | <p>Let us suppose that $r$ is a rational number then $r=\frac{p}{q}$ where $\gcd(p,q)=1.$ Now,
\begin{align*}
r^2 & = \frac{p^2}{q^2}\\
\implies 2& = \frac{p^2}{q^2}\implies p^2=2q^2\implies p^2\ \text{is an even number$\implies p$ is an even number. }\\
\implies p^2=(2k)^2 &= 2q^2 \implies q^2=2k^2\implie... |
2,135,151 | <p>Prove that, If $r$ is a real number such that $r^2 = 2$, $r$ is irrational.</p>
<hr>
<p><strong>Proposition:</strong> If $r$ is a real number such that $r^2 = 2$, then $r$ is irrational.</p>
<p><strong>Hypothesis:</strong> If $r$ is a real number such that $r^2 = 2$.</p>
<p><strong>Conclusion:</strong> $r$ is ir... | Eric Wofsey | 86,856 | <p>Most of the answers here are at least somewhat misleading. The only step of your proof that is definitely an error is that $r^2=2$ does <em>not</em> imply $r=\sqrt{2}$. It implies either $r=\sqrt{2}$ or $r=-\sqrt{2}$, and so for your argument to work you would need to separately get a contradiction in both of thes... |
41,832 | <p>I'm trying to teach myself algebra and derivatives. I learned the derivative for $f(x) = x^2$ from a lesson, and now I thought I would see if I could figure out the derivative of $f(x) = x+x$ on my own. </p>
<p>I know the formula for derivatives is: $$\lim_{\delta\rightarrow 0}\frac{f(x+\delta)
- f(x)}{\... | Najib Idrissi | 10,014 | <p>Add the $\delta$s and simplify, you get $\displaystyle \frac{2\delta}{\delta} = 2$. The derivative is then $2$.</p>
|
3,375,366 | <blockquote>
<p>How do we find the latus rectum of parabola when the equation is given in this polar form?
<span class="math-container">$$1/r = 1 + \cos t$$</span></p>
</blockquote>
<p>This curve cuts the <span class="math-container">$x$</span> axis on <span class="math-container">$1/2$</span> and <span class="mat... | Rishi | 691,870 | <p>Mostly it depends how fast you in calculation. It's same formula but different ways to write
<span class="math-container">$$(a+b)^3=a^3+b^3+3ab(a+b)$$</span>
<span class="math-container">$$(x+1)^3=x^3+1+3x(x+1)$$</span>
<span class="math-container">$$(x-1)^3=x^3-1-3x(x-1)$$</span>
<span class="math-container">$$..... |
387,542 | <p>e.g. The function $e^x$ reflected through $y=x$ is $\ln x$. Is this always true OR just in some cases?</p>
| Federica Maggioni | 49,358 | <p>Reflection with respect to the diagonal $y=x$ means substitution of $x$ with $y$. So, start from $x$, apply $f$ and get $y$, then substitute $y$ with $x$, obtaining $x$. Conversely, start with $y$, apply substitution and get $x$, then apply $f$ to obtain $y$. As you can see, substitution is right and left inverse of... |
33,430 | <p>Using <kbd>ctrl</kbd><kbd>/</kbd> you can make a fraction. If you have selected something it will appear in the numerator.</p>
<p>Does there exist a shortcut to make the selected text appear in the denominator instead?
If not, is it possible to create a shortcut that does this?</p>
| N.J.Evans | 11,777 | <p>I can't imagine any case where this would be the best option, but no one has mentioned <code>SectorChart</code> yet - where I use equal theta bins, and radius indicates percentage. The only benefit I see is you can compare percentages within a sublist easily. </p>
<pre><code>GraphicsGrid[
Partition[
SectorChart[... |
163,934 | <p>Suppose $F : [0,1]^n \to [0,1]^n$ is continuously differentiable and $0 < \frac{\partial F_1}{\partial x_i} \leq \dots \leq \frac{\partial F_n}{\partial x_i} < \beta < 1$ for all $i =1,\dots,n$. Conjecture: there exists unique $x^* = F(x^*)$, and moreover, $x_1^* \leq \dots \leq x_n^*$.</p>
<p>Proof of the... | Aaron Meyerowitz | 8,008 | <p>So I see that you are assuming that the $n^2$ first order partial derivatives are all strictly between $0$ and $1$. The fixed point can be found by repeatedly applying $F$ starting (almost?) anywhere. It would be sufficient then that the property $x_1 \le x_2 \le \cdots \le x_n$ is preserved by application of $F$. O... |
3,786,472 | <p><strong>An elevator ascends from rest with an acceleration of <code>0.6 m/s^2</code>, before slowing down with a deceleration of <code>0.8 m/s^2</code> for the next stop. The total time taken is 10 seconds. Find the distance between the stops.</strong></p>
<p>I have tried this problem over multiple spells over and o... | Alex R. | 22,064 | <p>Let <span class="math-container">$T$</span> denote the time at which the elevator stops accelerating and starts decelerating. Then, since the final velocity is 0, find the time:</p>
<p><span class="math-container">$$0.6(m/sec^2) T-0.8(m/sec^2)(10sec-T)=0.$$</span></p>
<p>Solve for <span class="math-container">$T$</s... |
3,786,472 | <p><strong>An elevator ascends from rest with an acceleration of <code>0.6 m/s^2</code>, before slowing down with a deceleration of <code>0.8 m/s^2</code> for the next stop. The total time taken is 10 seconds. Find the distance between the stops.</strong></p>
<p>I have tried this problem over multiple spells over and o... | mjw | 655,367 | <p><span class="math-container">$$a_1 = \phantom{-}0.6 {\text{ m}}/{\text{s}^2}$$</span>
<span class="math-container">$$a_2 = -0.8 {\text{ m}}/{\text{s}^2}$$</span></p>
<p>The basic equations of motion that we will use are:</p>
<p><span class="math-container">$$\begin{aligned}
v & = a t + v_0 \\
x & = \frac{1}... |
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