qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
2,496,114 | <p>Show that $ a \equiv 1 \pmod{2^3 } \Rightarrow a^{2^{3-2}} \equiv 1 \pmod{2^3} $</p>
<p>Show that$ a \equiv 1 \pmod{2^4 } \Rightarrow a^{2^{4-2}} \equiv 1 \pmod{2^4} $</p>
<p><strong>Answer:</strong></p>
<p>$ a \equiv 1 \pmod{2^3} \\ \Rightarrow a^2 \equiv 1 \pmod{2^3} \\ \Rightarrow a^{2^{3-2}}=a^{2^1} \equiv... | Community | -1 | <p>Hint : Combine the <a href="https://en.wikipedia.org/wiki/Dirichlet_kernel" rel="nofollow noreferrer">Dirichlet kernel</a> to the following inequality :</p>
<p>$$\frac{1}{2n+1}\geq\frac{1}{(2n+1)^2}|\frac{sin((n+0.5)x)}{sin(0.5x)}|\geq \frac{1}{(2n+1)^2}|\frac{sin((n+0.5)x)}{0.5x}|$$</p>
<p>And use the same induct... |
1,173,002 | <p>Do I have to use the diagonalization of A?</p>
| Brian M. Scott | 12,042 | <p>$10$ dominoes:</p>
<p>$$\begin{array}{|c|c|c|} \hline
\;\;&&&&5&&8&\\ \hline
&1&1&&5&&8&\\ \hline
&&&&&&&\\ \hline
&2&&4&&7&&10\\ \hline
&2&&4&&7&&10\\ \hline
&&&&a... |
908,083 | <p>I'd like to know what methods can I apply to simplify the fraction $\frac{4x + 2}{12 x ^2}$ </p>
<p>Is it valid to divide above and below by 2? (I didn't know it but Geogebra's Simplify aparantly does this)</p>
<p>Thanks in advance</p>
| recursive recursion | 118,924 | <p>You can factor out a two from the numerator and denominator to get $$\frac{2x+1}{6x^2}.$$ multiplying the numerator and the denominator by the same number will never affect the value of the fraction. </p>
|
1,982,216 | <p>Consider the operator $B: L^1\left(\mathbb{R^+} \right)\to L^1\left(\mathbb{R^+} \right)$ defined for each $f\in L^1\left(\mathbb{R^+} \right)$ by
$$(Bf)(t)=\int_0^\infty\alpha (t,s)f(s)ds, \ \ \ \text{for} \ \ t\geq 0$$
where $\alpha:\mathbb{R^+} \times \mathbb{R^+}\to\mathbb{R}$ is a real function satisfying
$$\le... | PhoemueX | 151,552 | <p>In general, such an operator need not be compact. Indeed, let $g \in L^1 ((0,\infty)) $ be arbitrary and define
$$
\alpha (t,s) = g (t) \sum_{n=1 }^\infty 1_{[n,n+1)}(s) e^{2\pi i n t} $$
and consider the sequence $f_n = 1_{[n,n+1)} $ which is bounded in $L^1$. Then $B f_n (t) = g (t) e^{2\pi i n t} $, which conve... |
4,394,676 | <p>Solve</p>
<p><span class="math-container">$$\frac{dy}{dx}=\cos(x-y)$$</span></p>
<p>So I know I need to make the substitution <span class="math-container">$u=x-y$</span> but then what's <span class="math-container">$du$</span>, is it <span class="math-container">$du=dx-dy$</span>?</p>
<p>Or do I rewrite <span class=... | PrincessEev | 597,568 | <p>Don't overthink things. In this context, <span class="math-container">$y$</span> is a function of <span class="math-container">$x$</span>; finding <span class="math-container">$du$</span> amounts to first finding <span class="math-container">$u'$</span>, and clearly</p>
<p><span class="math-container">$$u(x) = x - y... |
238,970 | <p>Recently I've come across 'tetration' in my studies of math, and I've become intrigued how they can be evaluated when the "tetration number" is not whole. For those who do not know, tetrations are the next in sequence of iteration functions. (The first three being addition, multiplication, and exponentiation, while ... | Douglas Shamlin | 453,618 | <p>This may be a possible answer.</p>
<p>Let $f(x)=\log_{10}(x+1)$ and $g(x)=10^x-1$ (inverse function of $f$).<br>
Then let $f^n(x) = f(f(\cdots(f(x))\cdots))$ with n $f$'s, similarly for $g$.</p>
<p>$$^{x+2}10\approx\lim_{n\to \infty} g^n(f^n(10^{10})\cdot(\ln10)^x)$$</p>
<p>The values behave fairly well for posit... |
238,970 | <p>Recently I've come across 'tetration' in my studies of math, and I've become intrigued how they can be evaluated when the "tetration number" is not whole. For those who do not know, tetrations are the next in sequence of iteration functions. (The first three being addition, multiplication, and exponentiation, while ... | Vessel | 714,488 | <p>We want to find a solution for <span class="math-container">$z$</span> within the following equation:
<span class="math-container">$$
_{}^yx=z
$$</span>
Taking the super logarithm (slog) of base <span class="math-container">$x$</span> on both sides gives:
<span class="math-container">$$
y=\text{slog} _x(z)
$$</span>... |
1,072,639 | <p>A function is <b>bijective</b> if it is both <b>surjective</b> and <strong>injective</strong>. Is there a term for when a function is both <strong>not surjective</strong> and <strong>not injective</strong>?</p>
| Ilmari Karonen | 9,602 | <p>As far as I know, there isn't. The concept of a "non-surjective and non-injective function" just doesn't generally arise often enough to need a special term.</p>
|
2,329,751 | <p>I have this definition: $f:R^n → R^m$ is differentiable at $a∈R^n$, if there exists a linear transformation $μ:R^n→R^m$ such that</p>
<p>$\lim_{h \to 0} \frac{|f(a+h)-f(a)-\mu(h)|}{|h|} = 0$.</p>
<p>My questions are what's the linear transformation $μ(h)$ for? What does it mean and where does it come from? Why is ... | Horstenson | 60,224 | <p>The linear map $\mu$ is the derivative of $f$ at $a$. It's the best linear approximation of $f$ near $a$.</p>
<p>For a detailed answer look <a href="https://math.stackexchange.com/questions/1784262/how-is-the-derivative-truly-literally-the-best-linear-approximation-near-a-po">here</a>.</p>
|
2,329,751 | <p>I have this definition: $f:R^n → R^m$ is differentiable at $a∈R^n$, if there exists a linear transformation $μ:R^n→R^m$ such that</p>
<p>$\lim_{h \to 0} \frac{|f(a+h)-f(a)-\mu(h)|}{|h|} = 0$.</p>
<p>My questions are what's the linear transformation $μ(h)$ for? What does it mean and where does it come from? Why is ... | Christian Blatter | 1,303 | <p>One can talk about these things in a denominator-free way. The essential point is the following: </p>
<p>Assume that a point of interest $p$ in the domain of $f$ is given, and you want to know how the values of $f$ behave in the immediate neighborhood of $p$. If $f$ is a nice function then moving away by $h$ from ... |
109,922 | <p>Let $B_n$ denote the group of signed permutations on $n$ letters. Is there a good explanation or understandable way to see why
$$
\sum_{w\in B_n}q^{\text{inv}(w)}=(2n)_q(2n-2)_q\cdots(2)_q?
$$</p>
<p>I've been thinking about it on and off while reading through Taylor's <em>Geometry of the Classical Groups</em>, b... | Tommy | 59,433 | <p>There are multiple ways to generalize inv to $B_n$, and the literature is not always consistent. Here is one such generalization (denoted $\mathrm{inv_B}$ for clarity), and a corresponding proof. As in hoyland's answer, the approach is to show a bijection between permutations and inversion sequences, where the corre... |
1,820,690 | <p>Find two numbers, $A$ and $B$, both smaller than $100$, that have a lowest common multiple of $450$ and a highest common factor of $15$.</p>
<p>I know that this involves the formula of </p>
<p>$A × B = LCM × HCF$</p>
<p>But I don't quite understand the above formula so I rather memorise it and that is why I can't... | Claude Leibovici | 82,404 | <p><strong>Hint</strong></p>
<p>$$z^{10}-z^5-992=0\implies Z^2-Z-992=0\implies(Z+31)(Z-32)=0$$ using $Z=z^5$.</p>
<p>I am sure that you can take it from here.</p>
|
1,820,690 | <p>Find two numbers, $A$ and $B$, both smaller than $100$, that have a lowest common multiple of $450$ and a highest common factor of $15$.</p>
<p>I know that this involves the formula of </p>
<p>$A × B = LCM × HCF$</p>
<p>But I don't quite understand the above formula so I rather memorise it and that is why I can't... | Jack's wasted life | 117,135 | <p>The equation is quadratic in $z^5$ with one positive and one negative root.
$$
z^5=a\implies z=a^{1\over5}e^{2\pi i{k\over5}}, 0\le k<5\\
\Re z=a^{1\over5}\cos(2\pi k/5)
$$
If $a>0,$ we have to make the cosine part negative which leaves us with $k=2,3$.</p>
<p>If $a<0,$ we have to make the cosine part pos... |
1,424,913 | <p>I am trying to solve the following problem.</p>
<blockquote>
<p>Let $G$ be a group. If $M, N \subset G$ are such that $x^{-1} M x = M$
and $x^{-1} N x = N$ for all $x \in G$ and $M \cap N = \{1\}$, prove
that $m n = n m$ for all $m \in M, n \in N$.</p>
</blockquote>
<p>I have already proven it for the specif... | David Quinn | 187,299 | <p>The minimum of $\cosh x$ occurs when $x=0$, so in this case you require $\frac{xy}{2}=0$ so you can conclude that either $x=0$ or $y=0$ or both</p>
|
88,159 | <p>Define $\omega=e^{i \pi /4}$. Is there an elegant way of showing that $20^{1/4} \omega^3$ is not inside $\mathbb{Q}(20^{1/4} \omega)$?</p>
<p>The way i am doing it is by observing that $20^{1/4} \omega$ is algebraic over $\mathbb{Q}$ with minimal polynomial $x^4+20$ and i assume that $20^{1/4} \omega^3$ is in the s... | pki | 17,464 | <p>Here's one way that I can think of. We want to show that $\omega^2\not\in \mathbb{Q}(20^{1/4}\omega)$. It follows that it's sufficient to show that $x^4+20$ is irreducible over $\mathbb{Q}(i)=\mathbb{Q}(\omega^2)$.</p>
<p>Any factorization in $\mathbb{Q}(i)$ will actually be a factorization in the Gaussian integers... |
550,441 | <p>Say I roll a 6-sided die until its sum exceeds $X$. What is E(rolls)?</p>
| mjqxxxx | 5,546 | <p>Let $L(x)$ be the expected number of rolls to reach $x$. Clearly $L(x)=0$ for $x\le 0$, and for $x\ge 1$ the expected number is one more than the expected number remaining after the next roll. That is,
$$
L(x)=I_{+}(x) + \frac{1}{6}\sum_{k=1}^{6}L(x-k),
$$
where $I_{+}(x)$ is $1$ for positive $x$ and zero otherwis... |
3,931,246 | <p>I want to find the range of <span class="math-container">$x$</span> on which <span class="math-container">$f$</span> is decreasing, where
<span class="math-container">$$f(x)=\int_0^{x^2-x}e^{t^2-1}dt$$</span></p>
<p>Let <span class="math-container">$u=x^2-x$</span>, then <span class="math-container">$\frac{du}{dx}=2... | Z Ahmed | 671,540 | <p><span class="math-container">$f(x)=\int_{0}^{x^2-x} e^{t^2-1} dt \implies f'(x)= (2x-1) e^{(x^2-x)^2-1} >0 ~if~ x>1/2$</span>. Hence <span class="math-container">$f(x)$</span> in increasing for <span class="math-container">$x>1/2$</span> and decreasing for <span class="math-container">$x<1/2$</span>. Ye... |
4,463,559 | <p>Let <span class="math-container">$K \subset \mathbb{R}^n\times [a,b]$</span> a compact subset. For each <span class="math-container">$t \in [a,b]$</span>, let <span class="math-container">$K_t= \{x \in \mathbb{R}^n : (x,t) \in K\}$</span>. Suppose that, for all <span class="math-container">$t \in [a,b]$</span>, <spa... | Mason | 752,243 | <p>By Tonelli's theorem,
<span class="math-container">$$m(K) = \int_{\mathbb{R}}\int_{\mathbb{R}^n}1_K(x, t)\,dx\,dt = \int_{\mathbb{R}}m(K_t)\,dt.$$</span>
Hence <span class="math-container">$m(K) = 0$</span> if and only if <span class="math-container">$m(K_t) = 0$</span> for almost every <span class="math-container">... |
947,290 | <p>In a cyclic group of order 8 show that element has a cube root. So for some $a\in G$ there is an element $x \in G$ with $x^3=a.$</p>
<p>Also show in general that if $g=<a>$ is a cyclic group of order m and $(k,m)=1$ then each element in G has a $k$th root. What element will $a^k$ generate? Use this to expre... | Timbuc | 118,527 | <p>Since $\;\text{gcd}\;(3,8)=1\;$ , if $\;G=\langle z\rangle\;$ then also $\;G=\langle z^3\rangle\;$ , and from here for each $\;a\in G\;$ there exists $\;k\in\Bbb Z\;$ so that we have</p>
<p>$$a=(z^3)^k= (z^k)^3$$</p>
<p>Take just $\;x=z^k\;$</p>
|
57,281 | <p><strong>Bug introduced in 10.0 and fixed in 10.3</strong></p>
<hr>
<p>I'm having trouble calculating the median of a <code>Dataset[]</code> in <em>Mathematica</em> 10.</p>
<p>The situation is as follows. Consider a dataset that was defined as follows:</p>
<pre><code>dataset = Dataset[{<|"a"->1,"b"->2|&g... | Taliesin Beynon | 7,140 | <p>Median itself doesn't work on associations of vectors:</p>
<pre><code>In[9]:= Median[{<|"a" -> 1, "b" -> 2|>, <|"a" -> 3, "b" -> 4|>}]
During evaluation of In[9]:= Median::rectn: Rectangular array of real numbers is expected at position 1 in Median[{<|a->1,b->2|>,<|a->3,b-&... |
317,753 | <p>I am taking real analysis in university. I find that it is difficult to prove some certain questions. What I want to ask is:</p>
<ul>
<li>How do we come out with a proof? Do we use some intuitive idea first and then write it down formally?</li>
<li>What books do you recommended for an undergraduate who is studying ... | Community | -1 | <p><a href="http://rads.stackoverflow.com/amzn/click/0471317160" rel="nofollow">Real Analysis, by Gerald B. Folland (Author).</a></p>
|
223,631 | <p>I'm using NeumannValue boundary conditions for a 3d FEA using NDSolveValue. In one area I have positive flux and in another area i have negative flux. In theory these should balance out (I set the flux inversely proportional to their relative areas) to a net flux of 0 but because of mesh and numerical inaccuracies... | Alex Trounev | 58,388 | <p>We can use mesh of first order for 3D visualization and short time for visibility. We also change boundary conditions:</p>
<pre><code>Needs["NDSolve`FEM`"]; a =
ImplicitRegion[True, {{x, -1, 1}, {y, -1, 1}, {z, 0, 1}}];
b = Cylinder[{{0, 0, -1/5}, {0, 0, 0}}, (650/1000)/2];
c = Cylinder[{{1, 1, -1/5}, {1, 1, 0}},... |
176,340 | <p>I am running an iterative routine that I want to export to a file while each iteration is computed, instead of storing everything in memory and then exporting to a file. </p>
<p>My solution is to write to an "m" file that saves the values in the usual array format that mathematica understands (e.g. {{2,1},{3,1}} fo... | Bill | 18,890 | <p>Assuming one row will fit in memory but the matrix will not, use <code>Table</code> to construct each row and supply the necessary delimiters with the minimum calculation, storage and <code>If</code>.</p>
<pre><code>sm = 3;
rm = 3;
SetDirectory[NotebookDirectory[]];
DeleteFile["test.m"];
stream = OpenAppend["test.m... |
526,837 | <p>Let $(\Omega, {\cal B}, P )$ be a probability space, $( \mathbb{R}, {\cal R} )$ the usual
measurable space of reals and its Borel $\sigma$- algebra, and $X : \Omega \rightarrow \mathbb{R}$ a random variable.</p>
<p>The meaning of $ P( X = a) $ is intuitive when $X$ is a discrete random variable, because it's the d... | Madrit Zhaku | 34,867 | <p>$$
\int\frac{-4x}{2x+1}dx=-4\int\frac{x}{2x+1}dx=-4\int\frac{\frac{2x+1}{2}-\frac{1}{2}}{2x+1}dx=-4\int\frac{\frac{1}{2}(2x+1-1)}{2x+1}dx=-2\int\frac{2x+1-1}{2x+1}dx=-2\int(\frac{2x+1}{2x+1}-\frac{1}{2x+1})dx=-2\int{(1-\frac{1}{2x+1})}dx=-2\int dx+2\int\frac{1}{2x+1}dx=-2x+\ln|2x+1|+C.
$$</p>
<p>$C$ is a integrabil... |
22,101 | <p>The general rule used in LaTeX doesn't work: for example, typing <code>M\"{o}bius</code> and <code>Cram\'{e}r</code> doesn't give the desired outputs.</p>
| quid | 85,306 | <p>It would be a misconception to consider the source of a post as somehow analogous to the source of a LaTeX document (not only diacritics but quite literally <em>nothing</em> works <em>except</em> of course the math-environment). Instead it is better to think of the source of a HTML page. The basic formatting is done... |
3,371,302 | <p>trying to find all algebraic expressions for <span class="math-container">${i}^{1/4}$</span>.</p>
<p>Using. Le Moivre formula , I managed to get this : </p>
<blockquote>
<p><span class="math-container">${i}^{1/4}=\cos(\frac{\pi}{8})+i \sin(\frac{\pi}{8})=\sqrt{\frac{1+\frac{1}{\sqrt{2}}}{2}} + i \sqrt{\frac{1-... | Community | -1 | <p>Let this first root be <span class="math-container">$z$</span>, and the other ones <span class="math-container">$zw$</span>. Then</p>
<p><span class="math-container">$$i=(zw)^4=z^4w^4=iw^4$$</span> and <span class="math-container">$w^4=1$</span>. Hence you multiply <span class="math-container">$z$</span> by the fou... |
4,272,964 | <p>I want to solve the equation following in a set of complex numbers:</p>
<p><span class="math-container">$$z^2 + \bar z = \frac 1 2$$</span></p>
<p><strong>My work so far</strong></p>
<p>Apparently I have a problem with transforming equation above into form that will be easy to solve. I tried to multiply sides by <sp... | dxiv | 291,201 | <p>Taking conjugates <span class="math-container">$\,z^2 + \bar z = \frac 1 2 = \bar z ^2 + z\,$</span>, then eliminating <span class="math-container">$\bar z = \frac{1}{2}-z^2$</span> between the two:</p>
<p><span class="math-container">$$
\begin{align}
\left(\frac{1}{2}-z^2\right)^2+z &= \frac{1}{2}
\\ \iff\;\;\;... |
2,849,017 | <p>\begin{align}
dA & = 2RR\,dv = 2R^2\,dv \\[8pt]
A & = \int_0^\pi 2R^2\,dv \\[8pt]
\text{arclength} & = R\,dv
\end{align}</p>
<p><a href="https://i.stack.imgur.com/YBIX5.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/YBIX5.png" alt="enter image description here"></a></p>
<p>Area of a... | user | 505,767 | <p>It is not a correct set up to calculate the area since the width of the rectangle varies with $v$.</p>
<p>As an alternative we can use</p>
<p>$$A=2\int_0^R 2\sqrt{R^2-y^2}\,dy$$</p>
<p><a href="https://i.stack.imgur.com/gBlGK.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/gBlGK.jpg" alt="enter... |
1,865,364 | <p>After having seen a lengthy and painful calculation showing
$\operatorname{Gal}(\mathbb Q[e^\frac{2\pi i}3, \sqrt[\leftroot{-2}\uproot{2}3]{2}]/\mathbb Q)\cong S_3$, I'm wondering whether there's a slick proof $\operatorname{Gal}(\mathbb Q[e^\frac{2\pi i}p, \sqrt[\leftroot{-2}\uproot{2}p]{2}]/\mathbb Q)\cong S_p$ fo... | DonAntonio | 31,254 | <p>The field $\;K:=\Bbb Q\left(\zeta:=e^{2\pi i/p},\,\sqrt[p]2\right)\;$ is the splitting field of $\;f(x):=x^p-2\in\Bbb Q[x]\;$ , and since this is an irreducible polynomial (why?) then $\;G:=Gal(K/\Bbb Q)\;$ acts transitively over its roots, which are $\;\alpha_i:=\sqrt[p]2\,\zeta^k\;,\;\;k=0,1,2,...,p-1\;$ .</p>
<p... |
2,943,461 | <p>I'm stumped on a math puzzle and I can't find an answer to it anywhere!
A man is filling a pool from 3 hoses. Hose A could fill it in 2 hours, hose B could fill it in 3 hours and hose C can fill it in 6 hours. However, there is a blockage in hose A, so the guy starts by using hoses B and C. When the blockage in hose... | Bram28 | 256,001 | <p>Let's use meta-logic for this problem:</p>
<p>Assuming there is an answer to this problem at all, it must be true that it doesn't make a difference as to whether the guy uses both hoses B and C, or just hose A, for if there was a difference, then given that we are not told how long the blockage lasted, the problem ... |
3,433,277 | <blockquote>
<p>It is given
<span class="math-container">$f:\mathbb R \rightarrow \mathbb R$</span>
<span class="math-container">$$f(x):=\tan^{-1}(x+1)+ \cot^{-1}(x)$$</span>
<span class="math-container">$\mathcal R_f=?$</span></p>
</blockquote>
<p>So far, I've learned <span class="math-container">$\tan$</span... | Mani khurana | 724,374 | <p>Note that <span class="math-container">$\cot^{-1}(\pm \infty)= 0$</span> and <span class="math-container">$\tan^{-1} (\pm \infty) =\pm \pi/2$</span>, <span class="math-container">$\cot^{-1}(0^+)= \pi/2$</span> so the least value taken by the given function <span class="math-container">$$f(x)= \cot^{-1}x +\tan^{-1} (... |
686,981 | <p>So I have to solve the equation $$y^2=4\tag{1.9.88 unit 3*}$$</p>
<p>I did this: $$y^2=4 \text{ means } \sqrt{y^2}=\sqrt{4}=>y=2$$</p>
<p>But I have a problem, $y$ can be either negative or positive so I need to do: $$\sqrt{y^2}=|y|=2=>y=2- or- y=-2$$</p>
<p>Is it right?</p>
| user2369284 | 91,771 | <p>Yes, it is right. I'll recommend a better way to approach this. Just factorize it.</p>
<p>$(y-2)(y+2) = 0$</p>
<p>$y = 2,-2$</p>
|
995,159 | <p>I have a matrix $(a_{j,k})_{j,k\in\mathbb{N}}$ given by:</p>
<p>$ a_{j,k} = \dfrac{1 -e^{-jk}}{jk + 1}$</p>
<p>and I need to show that this induces a bounded operator on $\ell^2$. I'm pretty sure Schur's test is inconclusive. So my guess is to use the Hilber-Schmidt test, which states that if,</p>
<p>$\sum\limits... | Mustafa Said | 90,927 | <p>$|\frac{1-e^{-jk}}{jk+1}|^2 \leq \frac{1}{j^2k^2}$ for large $j,k$.</p>
|
2,171,237 | <p>$f(x)$ is continuous on $[0,\pi]$ and $\int_0^\pi{f(x)\sin xdx} = \int_0^\pi{f(x)\cos xdx} = 1.$</p>
<p>Find $\min\int_0^\pi {f^2(x)dx}.$</p>
<p>I try to solve this problem by this:
$$\begin{array}{l}
{\left( {\int\limits_0^\pi {f(x)\sin xdx} } \right)^2} \le \left( {\int\limits_0^\pi {{f^2}(x){{\sin }^2}xdx} } ... | Martin R | 42,969 | <p>Define
$$
g(x) = f(x) - \frac 2\pi \sin x - \frac 2\pi \cos x \, .
$$
Then
$$
\int_0^\pi g(x) \cos x \, dx= \int_0^\pi f(x) \cos x \, dx
- \frac 2\pi \int_0^\pi \sin x \cos x \, dx
- \frac 2\pi \int_0^\pi \cos^2 x \, dx \\
= 1 - 0 - 1 = 0
$$
and similarly,
$$
\int_0^\pi g(x) \sin x \, dx= 0 \, .
$$
Therefore
... |
152,295 | <p>What is the definition of picture changing operation?
What is a standard reference where it is defined - not just used?</p>
| José Figueroa-O'Farrill | 394 | <p>Although it’s behind an Elsevier pay-wall, there is one paper which explains in cohomological terms the picture-changing operator in the context of string field theory. If I remember correctly it is a kind of connecting homomorphism. The paper in question is “<a href="http://dx.doi.org/10.1016/0370-2693%2888%29913... |
2,657,053 | <blockquote>
<p>Suppose I know that
$$\sum_{i=1}^n i^2=\frac{n(n+1)(2n+1)}{6}\,\,\,\, \tag{1} $$
How can I prove the the following?
$$
\sum_{i=0}^{n-1} i^2=\frac{n(n-1)(2n-1)}{6}
$$</p>
</blockquote>
<hr>
<p>I have looked up the solution to the other problem but it seems to be a bit confusing to me. Is it pos... | user | 505,767 | <p>Simply note that</p>
<p>$$\sum_{i=0}^{n-1} i^2=\frac{n(n-1)(2n-1)}{6}=\left(\sum_{i=1}^n i^2\right)-n^2$$</p>
<p>indeed</p>
<p>$$\sum_{i=0}^{n-1} i^2=\sum_{i=1}^{n-1} i^2=\left(\sum_{i=1}^n i^2\right)-n^2=\frac{n(n+1)(2n+1)}{6}-n^2=\frac{n(n-1)(2n-1)}{6}$$</p>
|
3,520,327 | <p>Currently in Calculus II and I was introduced to hyperbolic trigonometric functions and it threw me for a loop. I’m really confused on their MEANING... and what they represent. I can use the formulas for them easily but it doesn’t actually make sense to me. Can someone please help me out? Are there any good books yo... | Arthur | 15,500 | <p>That is the gist of it, yeah.</p>
<p>There are, in practice, several ways to do this, and here is a short summary. A direct proof uses intermediate, already-known implications chained together like this.
<span class="math-container">$$
p\to p_1\\
p_1\to p_2\\
\vdots\\
p_n\to q
$$</span></p>
<p>A contrapositive pro... |
1,281,967 | <p>This is a dumb question I know.</p>
<p>If I have matrix equation $Ax = b$ where $A$ is a square matrix and $x,b$ are vectors, and I know $A$ and $b$, I am solving for $x$.</p>
<p>But multiplication is not commutative in matrix math. Would it be correct to state that I can solve for $A^{-1}Ax = A^{-1}b \implies x =... | Stefan Perko | 166,694 | <p>Usually, if you are solving for $x$, the easiest way which is also safe to "work", is Gaussian Elimination. (<a href="http://en.wikipedia.org/wiki/Gaussian_elimination" rel="nofollow">http://en.wikipedia.org/wiki/Gaussian_elimination</a>)
It also tells you, if there are no or infinitely many solutions.</p>
<p>Inver... |
1,944,628 | <p>Let $(X,{\mathcal T}_X)$ and $(Y,{\mathcal T}_Y)$ be topologiclal spaces, and let $f,g:(X,{\mathcal T}_X)\to(Y,{\mathcal T}_Y)$ be continuous maps. </p>
<p>Define the equality set as $$E(f,g) = \{x\in X \ | \ f(x) = g(x) \}$$</p>
<p>I have worked out that if $(Y,{\mathcal T}_Y)$ is Hausdorff, then $E(f,g)$ is $... | DanielWainfleet | 254,665 | <p>An example where $Y$ is a $T_1$ space but not Hausdorff . Let $T_R$ be the usual topology on the reals $R$. Let $Q$ be the rationals.Let $Y=Q\cup ((R$ \ $Q)\times \{0,1\}).$</p>
<p>For $b\subset R$ let $b^*=(b\cap Q)\cup ((b$ \ $Q)\times \{0,1\}).$ Let $B=\{b^*$ \ $c : b\in T_R$ and $c$ is finite $\}.$ Then $B$ is ... |
1,082,390 | <p>$$\lim_{x \to \infty} \left(\sqrt{4x^2+5x} - \sqrt{4x^2+x}\ \right)$$</p>
<p>I have a lot of approaches, but it seems that I get stuck in all of those unfortunately. So for example I have tried to multiply both numerator and denominator by the conjugate $\left(\sqrt{4x^2+5x} + \sqrt{4x^2+x}\right)$, then I get $\di... | Thomas Andrews | 7,933 | <p>Note that $$\sqrt{4x^2+5x}-\sqrt{4x^2+x} = \frac{4x}{\sqrt{4x^2+5x}+\sqrt{4x^2+x}}$$</p>
<p>And the denominator is between $2x+2x=4x$ and $(2x+\frac{5}{4})+(2x+\frac{1}{4})=4x+\frac{3}{2}$. So the limit be $1$.</p>
<p>Alternatively, show that:</p>
<p>$$\lim \left(2x+\frac{5}{4}-\sqrt{4x^2+5x}\right) = 0$$</p>
<p... |
2,426,897 | <p>Let $\mathbb{H}$ be the ring of real quaternions and $Z(\mathbb{H})$ be its center. Of course $Z(\mathbb{H})=\mathbb{R}$. </p>
<p>Suppose $a+bi+cj+dk$, $x+yi+zj+wk \in \mathbb{H}$ such that $(a+bi+cj+dk)(x+yi+zj+wk) \in Z(\mathbb{H})$. </p>
<p>Does it imply $(x+yi+zj+wk)(a+bi+cj+dk) \in Z(\mathbb{H})$?</p>
| Jack D'Aurizio | 44,121 | <p>$\sqrt{2017}\approx\sqrt{2000}=20\sqrt{5}\approx 20\cdot 2.236 \approx 45$ and
$$44^2 = 1936,\qquad 45^2=2025$$
hence $\sqrt{2017}\in\color{red}{\left(44,45\right)}$.</p>
|
3,282,400 | <p>I would like to illustrate my confusion about this topic by building up the issue from more or less first principles. Let <span class="math-container">$U \subseteq \mathbb{R}^n$</span> be an open subset and let <span class="math-container">$f:U \to \mathbb{R}^m$</span>. We say that <span class="math-container">$f$</... | Daniel Kawai | 466,883 | <p>I will use the result in <a href="https://math.stackexchange.com/questions/66253/proving-ab-afbfa-fb-f/4038289#4038289">this page</a>. Let <span class="math-container">$\pi:A\rightarrow A/B$</span> be the canonical surjection.</p>
<p>It is sufficient to prove:</p>
<p><span class="math-container">$$[A_f:A^g][B_g:B^f]... |
317,160 | <p>If $ f(x) = \frac{1}{(1+x)\sqrt x} $ how to find all $ p > 0 $ such that
$$ \int^{\infty}_0 |f(x)|^p dx < \infty $$
The integral is with respect to lebesgue measure. Any solution or hints would be helpful. The answer is the integral converges iff $ p\in (\frac{2}{3}, 2) $.</p>
| Hanul Jeon | 53,976 | <p><strong>Hint:</strong> Since
$$\int_0^{\infty} f(x)^p dx =\int_0^1 f(x)^p dx+\int_1^{\infty} f(x)^p dx$$</p>
<p>So if $0<p<1$, we can only consider the integral $\int_1^{\infty} f(x)^p dx$, and
$$\int_1^{\infty} f(x)^p dx < \int_1^\infty \left(\frac{1}{x\sqrt{x}}\right)^p dx$$ </p>
<p>And
$$\int_1^\infty ... |
317,160 | <p>If $ f(x) = \frac{1}{(1+x)\sqrt x} $ how to find all $ p > 0 $ such that
$$ \int^{\infty}_0 |f(x)|^p dx < \infty $$
The integral is with respect to lebesgue measure. Any solution or hints would be helpful. The answer is the integral converges iff $ p\in (\frac{2}{3}, 2) $.</p>
| Julien | 38,053 | <p>At $+\infty$, you have
$$
|f(x)|^p\sim \frac{1}{x^{3p/2}}
$$
which converges if and only if $3p/2>1$.</p>
<p>At $0$,
$$
|f(x)|^p\sim\frac{1}{x^{p/2}}
$$
which converges if and only if $p/2<1$.</p>
<p>So your integral converges if and only if
$$
\frac{2}{3}<p<2.
$$</p>
|
4,500,163 | <blockquote>
<p>Take two positive integers <span class="math-container">$a$</span> and <span class="math-container">$b$</span> that are not multiples of <span class="math-container">$5.$</span> Then, construct a list in the following fashion: let the first term be <span class="math-container">$5,$</span> and starting w... | Bill Dubuque | 242 | <p>Though you've accepted an answer a day prior, it's worth strong emphasis that problems like this can be solved more simply (and more generally) using basic ideas about permutation cycles (if these are unfamiliar then see the alternative direct proof in a Remark below). Bringing to the fore the innate <span class="ma... |
3,393,466 | <p>I am in final year of my undergraduate in mathematics from a prestigious institute for mathematics. However a thing that I have noticed is that I seem to be slower than my classmates in reading mathematics. As in, how muchever I try, I seem to finish my works at the last moment and I rarely find any time for extra r... | Piquito | 219,998 | <p>It seems to me that over time you will know more about mathematics than your fellow students. What should happen is that you have a certain cultural sense of mathematics and that you probably read much more than what you need to do your homework while the other students focus on what they are asked for and nothing m... |
3,393,466 | <p>I am in final year of my undergraduate in mathematics from a prestigious institute for mathematics. However a thing that I have noticed is that I seem to be slower than my classmates in reading mathematics. As in, how muchever I try, I seem to finish my works at the last moment and I rarely find any time for extra r... | hal4math | 699,910 | <p>The question and information given is maybe a bit vague to give a satisfying and meaningful answer. But that shall me not stop from still trying: </p>
<p>I think everyone of us knows that there these phrases in math like "easy to see" or similar ones that can occupy ones attention for hours and clearly will lead to... |
1,994,922 | <p>Given $B_1, B_2,\ldots$ are independent and bounded variables with $E(B_i) = 0$ for all $i=1,2,\ldots$. Define $S_n = B_1+ B_2+\ldots + B_n$ with variance $s_n^2\rightarrow \infty$. Prove that $\frac{S_n}{s_n}$ has a central limit.</p>
<p><strong>My attempt:</strong> Due to the given condition, without i.i.d proper... | grand_chat | 215,011 | <p>You are on the right track. Notice that (given $\varepsilon>0$) there exists $N$ such that
$$
I\left( |B_j|^2>\varepsilon s_n^2\right) = 0
$$
for <strong>all</strong> $j$ and all $n\ge N$. (The reason is that $s_n$ is a deterministic sequence tending to infinity, while the $B$'s are bounded, so eventually the... |
327,750 | <p>$$\bigcup_{n=1}^\infty A_n = \bigcup_{n=1}^\infty (A_{1}^c \cap\cdots\cap A_{n-1}^c \cap A_n)$$</p>
<p>The results is obvious enough, but how to prove this</p>
| Andreas Blass | 48,510 | <p>The $n$'s on the right side should be $i$'s (or the $i$ should be $n$).</p>
<p>For any $x$, the statement "there is an $i\in\mathbb N$ such that $x\in A_i$" is equivalent to "there is a smallest $i\in\mathbb N$ such that $x\in A_i$".</p>
|
201,381 | <p>I have basic training in Fourier and Harmonic analysis. And wanting to enter and work in area of number theory(and which is of some interest for current researcher) which is close to analysis. </p>
<blockquote>
<p>Can you suggest some fundamental papers(or books); so after reading these I can have, hopefully(p... | Community | -1 | <p>I think it is remarkable that nobody mentioned André Weil's <em>Basic Number Theory</em> until now. André Weil made both marvellous contributions to harmonic analysis on locally compact Abelian groups and to number theory.</p>
<p>In <em>Basic Number Theory</em>, familiarity with number theory is not a prerequisite.... |
528,456 | <p>I have a question regarding L'hospital's rule. </p>
<p>Why can I apply L'hospital's rule to $$\lim_{x\to 0}\frac{\sin 2x}{ x}$$ and not to $$\lim_{x\to 0} \frac{\sin x}{x}~~?$$</p>
| amWhy | 9,003 | <p>You <strong>can</strong> apply l'Hôpital's rule in <em>both</em> cases! </p>
<p>You can apply l'Hôpital's rule whenever you have an <a href="http://en.wikipedia.org/wiki/Indeterminate_form#List_of_indeterminate_forms">indeterminate form</a>.</p>
|
528,456 | <p>I have a question regarding L'hospital's rule. </p>
<p>Why can I apply L'hospital's rule to $$\lim_{x\to 0}\frac{\sin 2x}{ x}$$ and not to $$\lim_{x\to 0} \frac{\sin x}{x}~~?$$</p>
| Arthur | 15,500 | <p>The reason you cannot use L'Hopital on the $\sin(x)/x$ limit has nothing to do with calculus, and more with logic, and the problem is subtle.</p>
<p>To use L'Hopital you need to know the derivative of $\sin(x)$. What is that derivative? You'd say $\cos(x)$ on reflex, and then you'd miss the problem. See, to calcula... |
2,062,398 | <p>May you tell me if my translation to symbolic logic is correct? </p>
<p>Thank you so much! Here is the problem:</p>
<p>To check that a given integer $n > 1$ is a prime, prove that it is enough to show that $n$ is not divisible by any prime $p$ with $p \le \sqrt{n}$.</p>
<p>$$\forall p \in P ~\forall n \in N ~(... | marwalix | 441 | <p>Assume it is rational. So there exists $p,q\in \Bbb{Z}^*$ such that</p>
<p>$${p\over q}=\sqrt{2}+\sqrt[3]{5}$$</p>
<p>Putting $\sqrt{2}$ on the L.H.S and cubing one gets</p>
<p>$$\left(p-q\sqrt{2}\right)^3=5q^3$$</p>
<p>And this leads to</p>
<p>$$p^3+6pq^2-5q^3=\left(3p^2q+2q^3\right)\sqrt{2}$$</p>
<p>And this... |
3,224,475 | <p>Let <span class="math-container">$\mathbb{Z}_8$</span> be the ring containing elements integer modulo 8 with operation <span class="math-container">$+$</span> and <span class="math-container">$.$</span> being addition and multiplication modulo 8 resp. I want to find <span class="math-container">$a$</span> for every ... | Ethan Bolker | 72,858 | <p>Here's an answer to the question</p>
<blockquote>
<p>How come the time complexity of Binary Search is <span class="math-container">$\log n$</span>?</p>
</blockquote>
<p>that describes informally what's going on in the binary tree in the question and in the video (which I have not watched).</p>
<p>You want to kn... |
3,224,475 | <p>Let <span class="math-container">$\mathbb{Z}_8$</span> be the ring containing elements integer modulo 8 with operation <span class="math-container">$+$</span> and <span class="math-container">$.$</span> being addition and multiplication modulo 8 resp. I want to find <span class="math-container">$a$</span> for every ... | Ariel Serranoni | 253,958 | <p>First, it is important to note that the running time of an algorithm is usually represented as function of the input size. Then, we 'measure' the complexity by fitting this function into a class of functions. For instance, if <span class="math-container">$T(n)$</span> is the function describing your algorithm's runn... |
2,875,907 | <p>There are set of rods of length <span class="math-container">$1,2,3,4 \dots N$</span>.
Two players take turns to chose 3 rods and compose triangle with non-zero area. After that this particular 3 rods are removed.
If it is not possible to compose triangle then player looses.</p>
<p>Who has winning strategy?</p>
<... | Kaban-5 | 580,885 | <p>This is a partial answer that explains that the first player
wins if $n \!\!\! \mod \!\! 6 \in \{0, 4, 5 \}$.</p>
<p>First thing to note is that $1$ can never participate in any triangles, so we can
pretend that it does not exist and game is player on rods of length $2, 3, \ldots, n$.
The main idea of first play... |
1,968,978 | <p>Let $f=(f_0,f_1,f_2...)$ and $g=(g_0,g_1,g_2,...)$ be sequences in $F^{\infty}$. We define multiplication $fg$ by expressing the $n$-th component $(fg)_n=\sum_{i=0}^ng_if_{n-i}$. If $h=(h_0,h_1,h_2,...)$ is also in $F^{\infty}$, we want to show multiplication is associative. Hoffman and Kunze give the following calc... | E.H.E | 187,799 | <p>$$y^2+19y=216$$
$$y(y+19)=(8)(27)$$
so the $$y=8$$</p>
|
2,431,548 | <p>Okay, so, my teacher gave us this worksheet of "harder/unusual probability questions", and Q.5 is real tough. I'm studying at GCSE level, so it'd be appreciated if all you stellar mathematicians explained it in a way that a 15 year old would understand. Thanks!</p>
<p>So, John has an empty box.
He puts some red cou... | drhab | 75,923 | <p>I preassume that the first counter taken by Linda is not put back in the box.</p>
<p>Suppose that there are $n$ red counters in the box. </p>
<p>Then there are $4n$ blue counters in the box, and $5n$ counters in total.</p>
<p>Then the probability that the first counter taken by Linda is red is $\frac{n}{5n}=\frac... |
2,431,548 | <p>Okay, so, my teacher gave us this worksheet of "harder/unusual probability questions", and Q.5 is real tough. I'm studying at GCSE level, so it'd be appreciated if all you stellar mathematicians explained it in a way that a 15 year old would understand. Thanks!</p>
<p>So, John has an empty box.
He puts some red cou... | joshuaheckroodt | 464,094 | <p>The rule of thumb in probability is that the word <strong>and</strong> implies multiplication, and <strong>or</strong> implies addition. Seeing as Linda is picking one red counter <strong>and</strong> one red counter, you know that its going to be the two probabilities of a red counter being picked multiplied by eac... |
328,670 | <p>Suppose I is an ideal of a ring R and J is an ideal of I, is there any counter example showing J need not to be an ideal of R? The hint given in the book is to consider polynomial ring with coefficient from a field, thanks</p>
| Thomas Andrews | 7,933 | <p>Consider $R=\mathbb Q[x]$, and $I=xR$ be the most obvious ideal of $R$.</p>
<p>Note that we can define $J$ as a subset of $I$ to be an ideal of $I$ if $J$ is a subgroup of $(I,+)$ and $IJ\subseteq J$. Find a $J$ that is a super-set of $x^2R$ but does not contain all of $I=xR$.</p>
|
1,437,287 | <p>On <a href="https://en.wikipedia.org/wiki/Geometric_series#Geometric_power_series" rel="nofollow">Wikipedia</a> it is stated that by differentiating the following formula holds:</p>
<p>$$ \sum_n n q^n = {1\over (1-q)^2}$$</p>
<p>Does this not require a proof? It seems to me because the series is infinite it is not... | AnthonyCaterini | 152,553 | <p>I think that since the series is absolutely convergent, taking a limit and differentiating commute</p>
|
3,344,728 | <p>Let <span class="math-container">$S$</span> be the set of all real numbers except <span class="math-container">$-1$</span>. Define <span class="math-container">$*$</span> on <span class="math-container">$S$</span> by
<span class="math-container">$$a*b=a+b+ab.$$</span></p>
<p>Goal: Show that <span class="math-contai... | Luiz Cordeiro | 58,818 | <p>We do it by contradiction: Let <span class="math-container">$a,b\in S$</span>, i.e., <span class="math-container">$a,b\neq -1$</span>, but suppose that <span class="math-container">$a*b=-1$</span>. This means that
<span class="math-container">\begin{align*}
a+b+ab&=-1\\
b+ab&=-1-a\\
b(1+a)&=-(1+a)\tag{$\... |
4,467,841 | <p>For a complex number <span class="math-container">$z=a+bi$</span> and a positive real value <span class="math-container">$R$</span>, we have <span class="math-container">$e^{Rbi}=\cos(Rb)+i\sin(Rb)$</span>. I am struggling to understand this since no matter how large <span class="math-container">$b$</span> or <span ... | Integral fan | 977,478 | <p>Based on what you have given, <span class="math-container">$Rb$</span> is a real number. I'll call it <span class="math-container">$\theta$</span>. I may be interpreting the question incorrectly, but then all that is happening is the fact that <span class="math-container">$e^{i\theta}$</span> represents a number on ... |
88,788 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/51292/relation-of-this-antisymmetric-matrix-r-beginpmatrix-01-10-endpmatr">Relation of this antisymmetric matrix $r = \begin{pmatrix} 0&amp;1\\ -1&amp;0 \end{pmatrix}$ to $i$</a> </p>
</blockquote>
... | Deven Ware | 14,334 | <p>Our matrices are of the form $$\left(\begin{smallmatrix} a & -b \\ b & a\end{smallmatrix}\right)$$ While our complex numbers are of the form $a + bi$ </p>
<p>Both of these depend on $a, b$ </p>
<p>Can you see a way to map from $\mathbb{C} \rightarrow H$ ? </p>
<p>Hint: For the harder one, multiplication $... |
1,255,970 | <p>What is
$$\int_{K} e^{a \cdot x+ b \cdot y} \mu(x,y)$$
where $K$ is the Koch curve and $\mu(x,y)$ is a uniform measure <a href="http://wwwf.imperial.ac.uk/~jswlamb/M345PA46/%5BB%5D%20chap%20IX.pdf" rel="nofollow noreferrer">look here</a>.</p>
<p><strong>Attempt:</strong>
I can evaluate the integral numerically and... | GEdgar | 442 | <p>not an answer yet, just some thoughts. </p>
<p>Say our Koch curve $K$ starts at $(0,0)$, ends at $(1,0)$ and the midpoint is at $(1/2, 1/(2\sqrt{3}\;))$. Mark seems to have used this, since his computation with $a=b=1$ agrees with mine.</p>
<p>Self-similarity is described by two maps of the plane to itself:
$$
L... |
1,157,007 | <p>I know that $f$ and $g$ have a pole or order $k$ in $z=0$.
$f-g$ is holomorph in $\infty$.</p>
<p>I need to prove that:</p>
<p>$$\oint_{|z|=R} (f-g)' dz = 0$$</p>
<p>Any help?</p>
<p>Note: $f$ and $g$ only have a singularity in $z=0$</p>
| Jason | 195,308 | <p>As Git Gud alluded to in their comment, simply parameterise the curve and calculate the integral directly. The usual parameterisation is $\gamma:[0,2\pi]\rightarrow\mathbb C:\ t\mapsto Re^{it}$. Letting $h=f-g$, we have
$$\oint_Ch'(z)\ \mathrm dz=\int_0^{2\pi}h'(\gamma(t))\gamma'(t)\ \mathrm dt=\int_0^{2\pi}(h\circ\... |
3,457,277 | <p>why Pi is transcendental number if <span class="math-container">$\pi$</span> also have algebraic equation like below which have root at <span class="math-container">$x =\pi/3$</span> as <span class="math-container">$n$</span> tends to infinity.
<span class="math-container">$$\Biggl(\biggl(\Bigl(\bigl((x^2-2^{(n2... | Matt Samuel | 187,867 | <p>Algebra for the most part is not concerned with sequences and limits. A real or complex number is said to be algebraic if it is a root of a single polynomial equation of finite degree with integer coefficients. This is something that <span class="math-container">$\pi$</span> is not.</p>
<p>If you are willing to use... |
1,569,543 | <blockquote>
<p>Prove or disprove: $(\ln n)^2 \in O(\ln(n^2)).$ </p>
</blockquote>
<p>I think I would start with expanding the left side. How would I go about this?</p>
| Marco Bellocchi | 169,978 | <p>You can say something about trees on at least 3 vertices with $\lambda_2=1$.</p>
<p>In particular if your graph T is a tree on at least 3 vertices, then $\lambda_2=1$ if and only if T is a star</p>
|
2,808,159 | <p><strong>The question is:</strong> </p>
<blockquote>
<p>A half cylinder with the square part on the $xy$-plane, and the length $h$ parallel to the $x$-axis. The position of the center of the square part on the $xy$-plane is $(x,y)=(0,0)$.
<img src="https://i.stack.imgur.com/fB5le.jpg" alt="Image description"> ... | mr_e_man | 472,818 | <p>"the boundaries of $y$ are $0$ and $r$"</p>
<p>There's your problem.</p>
<p>It should be $-r$ and $r$.</p>
|
2,736,323 | <blockquote>
<p>Given that $Y \sim U(2, 5)$ and $Z = 3Y - 4$, what is the distribution for $Z$?</p>
</blockquote>
<p>I've worked out that for $Y \sim N(2, 5)$, $Z \sim N(2, 45)$ since </p>
<p>$$\mu=3\cdot2 - 4 = 2$$</p>
<p>and </p>
<p>$$\sigma^2=3^2 \cdot 5 = 45$$</p>
<p>I'm wondering how the working differs whe... | drhab | 75,923 | <p>Let $a,b\in\mathbb R$ with $a<b$, and let $U$ be uniformly distributed over $(0,1)$.</p>
<hr>
<p>In general a random variable $X$ is uniformly distributed over interval $(a,b)$ if and only if its CDF can be prescribed by:</p>
<ul>
<li>$x\mapsto0$ if $x\leq a$</li>
<li>$x\mapsto\frac{x-a}{b-a}$ if $a<x\leq b... |
1,850,418 | <p>An argument has two parts, the set of all premises, and the conclusion drawn from said premise. Now since there's only 1 conclusion, it would be weird to choose a name for the 'second' part of the argument. However, what is the first part called? I used to think that this was actually called the premise, however tha... | user21820 | 21,820 | <p>In natural language arguments, you have finitely many premises. This means that you can put them all together in a single premise that is the conjunction of all the premises.</p>
<p>However, in general, you can treat the premises as a <strong>collection</strong> (which may not be finite). This collection is in firs... |
1,242,075 | <p>I have some function $g$. I know that $g \in C^1[a,b]$, so $g'(x)$ exists. I want to know, if $g: [a,b] \to [a,b]$ is onto. How can I find out if this is true or not?</p>
<p>P.S. I am not saying all $g$ have the said property, I want to have some kind of test to distinguish functions with this property from functio... | Andrew D. Hwang | 86,418 | <p>Not sure this is the type of criterion you're seeking, but if $g:[a, b] \to [a, b]$ is continuously-differentiable (continuous in particular), then by the Intermediate Value Theorem, $g$ is surjective if and only if $g$ achieves the values $a$ and $b$, i.e., there exist numbers $x_{\min}$ and $x_{\max}$ in $[a, b]$ ... |
2,916,306 | <p>Let $f(x,y) = x\ln(x) + y\ln(y)$ be defined on space </p>
<p>$S = \{(x,y) \in \mathbb{R}^2| x> 0, y > 0, x + y = 1\}$.</p>
<p>My question is, how do I take the partial derivative for this function, given that the parameters are coupled through $x+y = 1$.</p>
<p>A first idea would be to do it ignoring the co... | Hans Lundmark | 1,242 | <p>Taking the partial of $f(x,y)$ with respect to $x$ means that you vary $x$ while holding $y$ constant, but you can't do that if you insist on fulfilling the constraint $x+y=1$, so it's doesn't really make sense to talk about partial derivatives in this situation.</p>
<p>(You can eliminate $y$ and get a one-variable... |
3,848,179 | <blockquote>
<p>The velocity <span class="math-container">$v$</span> of a freefalling skydiver is modeled by the differential equation</p>
<p><span class="math-container">$$ m\frac{dv}{dt} = mg - kv^2,$$</span></p>
<p>where <span class="math-container">$m$</span> is the mass of the skydiver, <span class="math-container... | Sage Stark | 745,622 | <p>Hint: Remember that the integral is equivalent to the signed area under a curve in the two-dimensional case. Also remember that <span class="math-container">$x^2+y^2=r^2$</span> is the formula for a circle of radius <span class="math-container">$r$</span> centered at the origin.</p>
|
809,499 | <p>The no. of real solution of the equation $\sin x+2\sin 2x-\sin 3x = 3,$ where $x\in (0,\pi)$.</p>
<p>$\bf{My\; Try::}$ Given $\left(\sin x-\sin 3x\right)+2\sin 2x = 3$</p>
<p>$\Rightarrow -2\cos 2x\cdot \sin x+2\sin 2x = 3\Rightarrow -2\cos 2x\cdot \sin x+4\sin x\cdot \cos x = 3$</p>
<p>$\Rightarrow 2\sin x\cdot ... | DonAntonio | 31,254 | <p>$$\begin{cases}\sin 2x=2\sin x\cos x\\{}\\\sin 3x=\sin2x\cos x+\sin x\cos2x=2\sin x\cos^2x+\sin x(1-2\sin^2x)\end{cases}$$</p>
<p>Thus we get</p>
<p>$$0=\sin x+4\sin x\cos x-2\sin x\cos^2x-\sin x+2\sin^3x$$</p>
<p>Divide al through by $\;\sin x\;$ (why can we?):</p>
<p>$$4\cos x-2\cos^2x+2\sin^2x=0\iff2\cos x-\c... |
1,928,259 | <p>I have the following problem: </p>
<blockquote>
<p>The function $f(x)$ is odd, its period is $5$ and $f(-8) = 1$. What is $f(18)$?</p>
</blockquote>
<p>So, $f(-8) = f(-8 + 5) = 1$. I also know that you could replace $(-8)$ with $(-3)$ and still get the same result of $1$.</p>
<p>I'm just learning about periods.... | Gordon | 169,372 | <p>$\left(\frac{\partial f}{\partial y} \right)_x = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y} \right)$. It is similar to that $\frac{\partial f}{\partial x}=f_x$.</p>
|
20,972 | <p>Find the values of $x \in \mathbb{Z}$ such that there is no prime number between $x$ and $x^2$. Is there any such number?</p>
| Ross Millikan | 1,827 | <p>Despite the comments about Bertrand's postulate, there is still the range $-\sqrt{2} \le x \le \sqrt{2}$. If you want $x$ a natural number, there is $1$ and maybe $0$.</p>
|
20,972 | <p>Find the values of $x \in \mathbb{Z}$ such that there is no prime number between $x$ and $x^2$. Is there any such number?</p>
| Fixee | 7,162 | <p>Given the current wording of the question, you can set $x$ to any integer in $\{-1, 0, 1\}$ and there will be no prime between $x$ and $x^2$. For any other integer $x$, there will always be a prime between $x$ and $x^2$ (as noted in the comments to your question).</p>
|
2,061,547 | <p>I am solving for the zeroes of the function:</p>
<blockquote>
<p>$$\frac{\cos(x)(3\cos^2(x)-1)}{(1+\cos^2(x))^2}$$</p>
</blockquote>
<p>The zeroes of the function I found were done by setting $\cos(x)=0$, and $3\cos^2(x)-1=0$</p>
<p>For the $3\cos^2(x)-1=0$ I solved it and got $x=\cos^{-1}(\frac{\sqrt3}{3})$ bu... | layman | 131,740 | <p>This is one of those "follow your nose" proofs where you write one step down and see where you can go from there, and do it for each step. Here is what I came up with:</p>
<p>We want to show $\{1/n \}_{n=1}^{\infty}$ has $0$ as its only limit point. Assuming you can prove $0$ is a limit point, let's show that the... |
3,264,693 | <p>For context, I have been relearning a lot of math through the lovely website Brilliant.org. One of their sections covers complex numbers and tries to intuitively introduce Euler's Formula and complex exponentiation by pulling features from polar coordinates, trigonometry, real number exponentiation, and vector space... | David K | 139,123 | <p>The failure of duality is that there was never really duality there in the first place.</p>
<p>It's true that in <em>most cases,</em> the vector from the origin to <span class="math-container">$c^{a+bi} z$</span> is rotated and stretched (or shrunk) relative to the vector from the origin to <span class="math-contai... |
3,232,296 | <ol>
<li><p>For , ∈ ℝ, we have ‖−‖≤‖+‖. </p></li>
<li><p>The dot product of two vectors is a vector. </p></li>
<li><p>For ,∈ℝ, we have ‖−‖≤‖‖+‖‖. </p></li>
<li><p>A homogeneous system of linear equations with more equations than variables will always have at least one parameter in its solution. </p></li>
<li><p>Given a... | Kyle | 647,271 | <p>Number 3 is incorrect. Why? Because of the well-known fact that <span class="math-container">$|\bf{x} + \bf{y}| \le |\bf{x}| + |\bf{y}|$</span> (the Triangle Inequality).</p>
<p>In particular, <span class="math-container">$|\bf{u} - \bf{v}| = |\bf{u} + (- \bf{v})| \le |\bf{u}| + |- \bf{v}| = |\bf{u}| + |\bf{v}|$</s... |
4,156,482 | <blockquote>
<p>Can every continuous function <span class="math-container">$f(x)$</span> from <span class="math-container">$\mathbb{R}\to \mathbb{R}$</span> be continuously "transformed" into a differentiable function?</p>
</blockquote>
<p>More precisely is there always a continuous (non constant) <span cla... | Troposphere | 907,303 | <p>Inspired by <a href="https://math.stackexchange.com/a/4157185/907303">Frank's</a> notion of "corner", let's define</p>
<ul>
<li>The function <span class="math-container">$f$</span> has a <strong>"flat"</strong> at <span class="math-container">$x$</span> iff for every <span class="math-container">... |
2,483,188 | <p>I am facing this problem: </p>
<p><strong>Turn into cartesian form:</strong></p>
<p>$$\dfrac{1-e^{i\pi/2}}{1 + e^{i\pi/2}}$$</p>
<p>I've tried to operate and I've come up to this:</p>
<p>$$\dfrac{1-2e^{i\pi/2} + e^{i\pi}}{1 - e^{i\pi}}$$</p>
<p>I do not know how to go on, and I've tried to operate with the cart... | zhw. | 228,045 | <p>Hint: Let $h(x) = x^{-1/4}.$ Consider the $F\in X^*$ given by $F(f) = \langle f,h\rangle.$</p>
|
1,376,981 | <p>The context is as follows: I am asking this question because I would like feedback; I am a beginner to mathematical proofs.</p>
<p>We wish to show $\sum\limits_{k=1}^{n}kq^{k-1} = \frac{1-(n+1)q^{n} + nq^{n+1}}{(1-q)^{2}}$ for arbitrary $q>0$. My attempt:</p>
<p>Let as first consider the trivial base case $n=1$... | David Quinn | 187,299 | <p>Start with the standard formula for the sum of the first $n$ terms of a geometric series and differentiate both sides. This will give you the formula you have</p>
|
1,376,981 | <p>The context is as follows: I am asking this question because I would like feedback; I am a beginner to mathematical proofs.</p>
<p>We wish to show $\sum\limits_{k=1}^{n}kq^{k-1} = \frac{1-(n+1)q^{n} + nq^{n+1}}{(1-q)^{2}}$ for arbitrary $q>0$. My attempt:</p>
<p>Let as first consider the trivial base case $n=1$... | tired | 101,233 | <p>The finite geometric series is given by</p>
<p>$$
G(q,n)=\sum_{k=0}^nq^k=\frac{1-q^{n+1}}{1-q}
$$
for constant $q$ such that $|q|<1$.</p>
<p>Now $\frac{d}{dq}G(q,n)$ is the series you are looking for...</p>
|
2,705,980 | <p>I have the following problem:
\begin{cases}
y(x) =\left(\dfrac14\right)\left(\dfrac{\mathrm dy}{\mathrm dx}\right)^2 \\
y(0)=0
\end{cases}
Which can be written as:</p>
<p>$$ \pm 2\sqrt{y} = \frac{dy}{dx} $$</p>
<p>I then take the positive case and treat it as an autonomous, seperable ODE. I get $f(x)=x^2$ as my ... | Arian | 172,588 | <p>Since $\{Av_1,...,Av_n\}$ is linearly independent then $\{v_1,...,v_n\}$ is linearly independent. Suppose other wise. Let $a_1,...,a_n\in\mathbb{R}$ not all zero such that $a_1v_1+...+a_nv_n=0$ then by linearity of $A$ one gets
$$0=A(0)=A(a_1v_1+...+a_nv_n)=a_1Av_1+...+a_nAv_n$$
thus a contradiction. Having establis... |
848,229 | <p>Two teams take part at a KO-tournament with n rounds. Assuming, that the teams
win all their games until they are paired together, what is the probability
that they both meet in the final ?</p>
<p>I figured out that the solution is </p>
<p>$P_n=\prod_{j=2}^n (1-\frac{1}{2^j-1})=\frac{2^{n-1}}{2^n-1}$</p>
<p>So,... | Claude Leibovici | 82,404 | <p>What you have done is very good. As commented by G. H. Faust, using exponentials make things much simpler. Consider $$\int e^{a x}e^{i b x}~dx=\int e^{(a+i b) x}~dx=\frac{e^{(a+i b)x}}{a+i b}=\frac {e^{a x}}{a^2+b^2}(a-ib)e^{i b x}$$ Expanding, we then get $$\int e^{a x}e^{i b x}~dx=\frac {e^{a x}}{a^2+b^2}\Big([a... |
1,581,456 | <p>Given functions $g,h: A \rightarrow B$ and a set C that contains at least two elements, with $f \circ g = f \circ h$ for all $f:B \rightarrow C$. Prove that $g = h$. </p>
<p>My logic is to take C = B and h(x) =x for all x in particular and the result follows immediately. But, I don't see the use of the condition on... | will | 253,085 | <p>Given positive integer $n$ that defines $z := \exp(\pi i/n),$ we can generalize the oddly periodic sum with
$$
S_n(x) := \sum_{k=0}^{n-1}\frac{2k+1}{1-xz^{2k+1}}.
$$
The power series when $|x| < 1$ is
$$
S_n(x) = \sum_{E=0}^{\infty}x^E\sum_{k=0}^{n-1}(2k+1)z^{2kE+E}.
$$
This simplifies with the weighted sum,
$$
\... |
77,311 | <p>I am a first-time user pf <em>Mathematica</em> (V10). I know it's easy to install palettes, but uninstalling them drives me crazy. I want to delete one. Who can help me to do that? </p>
| Anton Antonov | 34,008 | <p>Open the folder:</p>
<pre class="lang-mathematica prettyprint-override"><code>SystemOpen[FileNameJoin[{$UserBaseDirectory, "SystemFiles", "FrontEnd", "Palettes"}]]
</code></pre>
<p>and delete the palettes you want.</p>
<p>If you do not find the palette you want to remove, try <code>$Use... |
211,175 | <p>In Gradshteyn and Ryzhik, (specifically starting with the section 3.13) there are several results involving integrals of polynomials inside square root. These are given in terms of combinations of elliptic integrals. See for instance: <a href="https://i.stack.imgur.com/6Cqyb.jpg" rel="nofollow noreferrer"><img src=... | bbgodfrey | 1,063 | <p><strong>Important Edit Made</strong></p>
<p>Mathematica can perform the 3.31 integral, if <code>Assumptions</code> is changed from <code>{ a > b > c >= u}</code> to <code>{a > b > c > u}</code>.</p>
<pre><code>s = Integrate[1/Sqrt[(a - x) (b - x) (c - x)], {x, -Infinity, u},
Assumptions ->... |
4,499,058 | <blockquote>
<p>Let <span class="math-container">$A:L_2[0,1]\to L_2[0,1]$</span> be defined by<span class="math-container">$$Ax(t)=\int \limits _0^1t\left (s-\frac{1}{2}\right )x(s)\,ds\quad \forall t\in [0,1].$$</span>Compute the adjoint and the norm of <span class="math-container">$A$</span></p>
</blockquote>
<p>This... | Ryszard Szwarc | 715,896 | <p>The operator is of the form <span class="math-container">$$ Ax=\langle x, g\rangle \,f$$</span> where <span class="math-container">$f(t)=t$</span> and <span class="math-container">$g(t)=t-{1\over 2}.$</span> Thus the range is one-dimensional. Its norm is equal <span class="math-container">$$\|A\|=\|f\|_2\|g\|_2={1\o... |
313,025 | <p>I got two problems asking for the proof of the limit: </p>
<blockquote>
<p>Prove the following limit: <br/>$$\sup_{x\ge 0}\ x e^{x^2}\int_x^\infty e^{-t^2} \, dt={1\over 2}.$$</p>
</blockquote>
<p>and, </p>
<blockquote>
<p>Prove the following limit: <br/>$$\sup_{x\gt 0}\ x\int_0^\infty {e^{-px}\over {p+1}} \,... | robjohn | 13,854 | <p>After a suitable substitution, both limits are easily handled by <a href="http://en.wikipedia.org/wiki/Dominated_convergence_theorem" rel="nofollow">Dominated Convergence</a> or <a href="http://en.wikipedia.org/wiki/Monotone_convergence_theorem#Lebesgue.27s_monotone_convergence_theorem" rel="nofollow">Monotone Conve... |
2,650,628 | <p>The equation $\log_e(x) + \log_e(1+x) =0$ can be written as:</p>
<p>a) $x^2+x-e=0$</p>
<p>b) $x^2+x-1=0$</p>
<p>c) $x^2+x+1=0$</p>
<p>d) $x^2+xe-e=0$</p>
<p>I tried differentiating both sides, then it becomes $\frac{1}{x}+\frac{1}{1+x}=0$, but I dont get any of the answers.</p>
| PM. | 416,252 | <p>Since $\log(A) + \log(B) = \log(AB)$
$$
\begin{align}
\log(x) + \log(1+x) &=0\\
\Rightarrow \log{\left( x(1+x)\right)} &=0 \\
\Rightarrow x(1+x) &= 1 \\
\Rightarrow x^2 + x -1 &=0
\end{align}
$$</p>
|
941,709 | <p><strong>Question:</strong> Let $X$ be any set with at least two elements. Assume that the only open
subsets of $X$ are the empty set $\emptyset$ and $X$ itself.
- Which subsets of $X$ are closed?
- Which subsets of $X$ are compact?</p>
<p><strong>My thoughts:</strong> Thus also $\emptyset$ and $X$ have to be also c... | Asaf Karagila | 622 | <p>The equivalence between compactness and bounded+closed is only true in metric spaces. In fact not even that, just in a particular class of metric spaces, not even in all of them. For general metric spaces we need to strengthen bounded to "totally bounded".</p>
<p>And in general for topological spaces we don't have ... |
2,750,931 | <p>I'm in the process of exploring Bra Ket notation. In it, I often find operators in the form $\lvert a\rangle\langle b\rvert$, which can be thought of as multiplying a row vector $a$ with a column vector $b$.</p>
<p>This strikes me as a construction which should probably have a name that I can research to understan... | Misha Lavrov | 383,078 | <p>The class of matrices are precisely the matrices with rank $1$, and the matrix $a b^{\mathsf T}$ specifically is called the <em>outer product</em> of $a$ and $b$ (by analogy with the inner product $a^{\mathsf T}b$, which gives a scalar).</p>
<p>Especially in the context of quantum states, it is also common to ident... |
2,247,498 | <p>Imagine a circle of radius R in 3D space with a line l running threw it's center C in a direction perpendicular to the plane of the circle. Basically, like the axel of a wheel. </p>
<p>From a given point P that is not on the circle or on l, a ray extends to intersect both l and the circle. What would be the equatio... | Joseph O'Rourke | 237 | <p>
<a href="https://i.stack.imgur.com/hqGN7.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/hqGN7.jpg" alt="DiskVec"></a>
<hr />
Rotate everything so that the disk lies in the $xy$-plane with its center at the origin, and the line $L$ coincides with the $z$-axis. O... |
586,112 | <p>Consider following statment $\{\{\varnothing\}\} \subset \{\{\varnothing\},\{\varnothing\}\}$</p>
<p>I think above statement is false as $\{\{\varnothing\}\}$ is subset of $\{\{\varnothing\},\{\varnothing\}\}$ but to be proper subset there must be some element in $\{\{\varnothing\},\{\varnothing\}\}$ which is not i... | Community | -1 | <p>(J. Stewart. Calculus pp 391) I believe Stewart defines an antiderivative as an indefinite integral. </p>
<p><img src="https://i.stack.imgur.com/wy1kk.png" alt="enter image description here"></p>
|
159,585 | <p>This is a kind of a plain question, but I just can't get something.</p>
<p>For the congruence and a prime number $p$: $(p+5)(p-1) \equiv 0\pmod {16}$.</p>
<p>How come that the in addition to the solutions
$$\begin{align*}
p &\equiv 11\pmod{16}\\
p &\equiv 1\pmod {16}
\end{align*}$$
we also have
$$\begin{... | Community | -1 | <p>First note that $p$ has to be odd. Else, $(p+5)$ and $(p-1)$ are both odd.</p>
<p>Let $p = 2k+1$. Then we need $16 \vert (2k+6)(2k)$ i.e. $4 \vert k(k+3)$.</p>
<p>Since $k$ and $k+3$ are of opposite parity, we need $4|k$ or $4|(k+3)$.</p>
<p>Hence, $k = 4m$ or $k = 4m+1$. This gives us $ p = 2(4m) + 1$ or $p = 2(... |
159,585 | <p>This is a kind of a plain question, but I just can't get something.</p>
<p>For the congruence and a prime number $p$: $(p+5)(p-1) \equiv 0\pmod {16}$.</p>
<p>How come that the in addition to the solutions
$$\begin{align*}
p &\equiv 11\pmod{16}\\
p &\equiv 1\pmod {16}
\end{align*}$$
we also have
$$\begin{... | Theorem | 24,598 | <p>The first two solution can be seen easily , ie you have $p=-5,1=11,17 \pmod{16}$. To find the next two solution , as we know $p$ should satisfy $(p+5)(p-1)=16$ then the solution of this quadratic equation as $p=3,-7 =3,9\pmod {16}$</p>
|
2,878,412 | <p>I've been working on a problem that involves discovering valid methods of expressing natural numbers as Roman Numerals, and I came across a few oddities in the numbering system.</p>
<p>For example, the number 5 could be most succinctly expressed as $\texttt{V}$, but as per the rules I've seen online, could also be ... | Community | -1 | <p>Wikipedia explicitly enumerates the patterns for the units, the tens and the hundredths.</p>
<p><a href="https://en.wikipedia.org/wiki/Roman_numerals#Basic_decimal_pattern" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Roman_numerals#Basic_decimal_pattern</a></p>
<p>This doesn't leave room for extravagan... |
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