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 |
|---|---|---|---|---|
3,143,670 | <p>I'm not entire sure how to proceed on this question. I believe I am supposed to use a triangle inequality with epsilons and <span class="math-container">$m$</span>, <span class="math-container">$n \geq N$</span> to get <span class="math-container">$N_1$</span> and <span class="math-container">$N_2$</span> before set... | JoseSquare | 643,097 | <p><strong>Hint</strong></p>
<p>You have to prove that for every <span class="math-container">$\epsilon >0$</span> there exists <span class="math-container">$N \in \Bbb{N}$</span> such
if <span class="math-container">$n,m > N$</span> then <span class="math-container">$|x_n -y_n -x_m + y_m| < \epsilon$</span>... |
804,483 | <p>The following integrals look like they might have a closed form, but Mathematica could not find one. Can they be calculated, perhaps by differentiating under the integral sign?</p>
<p>$$I_1 = \int_{-\infty }^{\infty } \frac{\sin (x)}{x \cosh (x)} \, dx$$
$$I_2 = \int_{-\infty }^{\infty } \frac{\sin ^2(x)}{x \sinh (... | Graham Hesketh | 66,912 | <p>For the first one we need:
$$\int _{-1/2}^{1/2}\!{{\rm e}^{2\,iax}}{da}={\frac {\sin \left( x
\right) }{x}}\tag{1}$$
$$ \frac{1}{\cosh \left( x \right)}=-2\,\sum _{n=1}^{\infty
} \left( -1 \right) ^{n}{{\rm e}^{- \left| x \right| \left( 2\,n-1
\right) }}\tag{2}$$
$$\int _{-\infty }^{\infty }\!{{\rm e}^{2\,iax}}{... |
804,483 | <p>The following integrals look like they might have a closed form, but Mathematica could not find one. Can they be calculated, perhaps by differentiating under the integral sign?</p>
<p>$$I_1 = \int_{-\infty }^{\infty } \frac{\sin (x)}{x \cosh (x)} \, dx$$
$$I_2 = \int_{-\infty }^{\infty } \frac{\sin ^2(x)}{x \sinh (... | Random Variable | 16,033 | <p>First I'm going to evaluate $$\int_{-\infty}^{\infty} \frac{\cos ax}{\cosh x} \ dx .$$</p>
<p>Integrate the function $ \displaystyle f(z) = \frac{e^{iaz}}{\cosh z}$ around a rectangle on the complex plane with vertices at $z= R$, $ z= R + i \pi$, $z= -R + i \pi$, and $z= - R$.</p>
<p>As $R \to \infty$, $ \displays... |
42,040 | <p>Suppose the polynomial $t^k - a$ has a root (hence splits) in $\mathbb{Q}(\zeta_k)$. For which $k$ does it follow that one of the roots of $t^k - a$ is rational? In particular, are there infinitely many such $k$? </p>
<p>A counting argument shows this is true whenever $k$ has the property that $\varphi(k)$ is a pow... | Matt E | 221 | <p>This is an amplification of Gerry Myerson's answer, which may be helpful.</p>
<p>You are asking about the kernel of the map $\mathbb Q^{\times}/(\mathbb Q^{\times})^k \to L^{\times}/(L^{\times})^k,$ where $L =\mathbb Q(\zeta_k)$.</p>
<p>In general, for any field $K$ of char. prime to $k$, there is a natural isomor... |
1,977,588 | <p>In books like Calculus (Larson), in the theorems'definitions like Rolle's theorem, when they talk about continuity, they use closed intervals [a,b]. But when they talk about differentiability they use open brackets (a,b). </p>
<p>Why are closed intervals used for continuity and open intervals for differentiability... | Aloizio Macedo | 59,234 | <p>It is not that "closed intervals are used for continuity and open intervals for differentiability" (more on this one later). It is that, <strong>for Rolle's Theorem</strong> (and the Mean Value Theorem), we <em>need</em> those hypotheses. </p>
<p>In the proof, we use that a continuous function on $[a,b]$ attains a ... |
127,808 | <p>I have <a href="https://math.stackexchange.com/questions/356925/a-basis-of-the-symmetric-power-consisting-of-powers">asked this question on math.se</a>, but did not get an answer - I was quite surprised because I thought that lots of people must have though about this before:</p>
<p>Let $V$ be a complex vector spac... | Abdelmalek Abdesselam | 7,410 | <p>I would look up a book on the calculus of finite differences in a multivariate setting.
The claim here is to show that for any multi-index $\alpha=(\alpha_1,\ldots,\alpha_n)$
of length $k$ one can express the multiple derivative at zero
$$
\left(\frac{\partial}{\partial t}\right)^\alpha \
(t_1x_1+\cdots+ t_n x_n)^k... |
22,207 | <p>How to make a defined symbol stay in symbol form?</p>
<pre><code>w = 3; g = 4;
{w, g}[[2]]
</code></pre>
<blockquote>
<p><code>3</code></p>
</blockquote>
<p>I want the output to be <strong><code>g</code></strong> and not <code>3</code>. For example, if I want to save different definitions by <code>DumpSave</co... | Jacob Akkerboom | 4,330 | <p>I suppose this answer does not have much added value over that of Jens, but I'll post it anyway. As a remark about the part of the question about DumpSave, an alternative method to that of Jens is the following. I find that in cases where things get evaluated that you don't want to get evaluated, it helps to tempora... |
2,951,825 | <p>I want to show formally that </p>
<p><span class="math-container">$$M =\{(t, \vert t \vert) \text{ }\vert t \in \mathbb{R} \} $$</span> </p>
<p>is not a smooth <span class="math-container">$C^{\infty}$</span>-submanifold of <span class="math-container">$\mathbb{R}^2$</span>. </p>
<p>My attempts: Intuitively it's ... | Ernie060 | 592,621 | <p>It's not always easy to show that a subset isn't a smooth submanifold using the definition. May I suggest another approach?</p>
<p>Maybe you have seen a version of the Implicit Function Theorem like this:</p>
<blockquote>
<p>Let <span class="math-container">$S\subset\mathbb{R}^2$</span> be a submanifold (curve) ... |
3,800,521 | <p>Let <span class="math-container">$x=\tan y$</span>, then
<span class="math-container">$$
\begin{align*}\sin^{-1} (\sin 2y )+\tan^{-1} \tan 2y
&=4y\\
&=4\tan^{-1} (-10)\\\end{align*}$$</span></p>
<p>Given answer is <span class="math-container">$0$</span></p>
<p>What’s wrong here?</p>
| 19aksh | 668,124 | <p>We can't bluntly take <span class="math-container">$\sin^{-1}(\sin 2y) = 2y$</span> and so with <span class="math-container">$\tan^{-1}(\tan 2y)$</span>, because we don't know the value of <span class="math-container">$2y$</span> and the range in which it lies.</p>
<p><a href="https://i.stack.imgur.com/MijvP.png" re... |
2,667,230 | <p>Let (X, d) be a complete metric space. Let$ f : X → X$ be a function such that for all distinct$ x, y ∈ X$ ,</p>
<p>$ d(f^ k (x), f^ k (y)) < c · d(x, y)$, for some real number $c < 1$ and an integer $k > 1$. Show that f has a unique fixed point. </p>
<p>my attempt : i take $f(x) = x $and $f(y) = y$ .now... | epi163sqrt | 132,007 | <p>We interprete the problem as follows: Given is the alphabet $V=\{1,2\}$. Find the number of strings consisting of characters of $V$ of length $n\geq 0$ so that each occurrence of $1$ is followed by <em>at least</em> $d$ characters $2$. We do so by encoding the problem using generating functions.</p>
<blockquote>
... |
345,310 | <p>This is computed based on the following recursive formula <span class="math-container">$$w_n=\frac{\lambda_nw_{n+1}+\mu_nw_{n-1}+1}{\lambda_n+\mu_n}$$</span> where: <span class="math-container">$n$</span> is the inital state, State <span class="math-container">$0$</span> is absorbing, <span class="math-container">$\... | Honza | 141,969 | <p>A solution for <span class="math-container">$w_i$</span> can be built directly by defining <span class="math-container">$$\delta_i=w_{i+1}-w_i$$</span> where <span class="math-container">$\delta_i$</span> is clearly the expected time to reach State <span class="math-container">$i$</span> (for the first time) from S... |
896,940 | <p>i tried 9 D + (-10 D)</p>
<p>9= 0000 1001</p>
<p>10= 0000 1010</p>
<p>Reverse 10 = 1111 0101 and add 1 become 1111 0110</p>
<p>after that add up 9 D + (-10 D) == 0000 1001 + 1111 0110 but the answer is equal to 1111 1111 whch is 255 in decimal but the answer should be -1 right? anything goes wrong?</p>
<p>Thank... | cjferes | 89,603 | <p>In two's complement representation for binary numbers, the number 1111 1111 represents -1. You missinterpreted the result as a "normal" binary number.</p>
<p>In two's complement, binary numbers of $2^n$ bits represent values ranging from $-2^{n-1}$ to $2^{n-1}-1$.</p>
|
151,864 | <p>I would like to generate a random password of a defined length which can easily be typed in with a standard keyboard.</p>
<p>As a start I tried the following:</p>
<pre><code>SeedRandom["pass"];
StringJoin[RandomChoice[CharacterRange[33, 126], 10]
(* "=IP@7mbYcB" *)
</code></pre>
<p>Do you know other solutions?</p... | yohbs | 367 | <p>Here's something which is nice and might be easy to remember:</p>
<pre><code>StringJoin @@ RandomSample[#, Length@#] &@
Flatten@{IntegerString@RandomInteger[{10, 999}],
Capitalize /@ RandomWord[3],
RandomSample[Characters@"!@_%$^=+*.", 2]
}
</code></pre>
<p>Select examples:</p>
<pre><code>"Tearless+... |
771,959 | <p>Let $p$ be prime, $n \in \mathbb{N}$ and $p \nmid n$. </p>
<p>$\Phi_n$ is the $n$-th cyclotomic polynomial.</p>
<p>How can I find the maximum $n \in \mathbb{N}$ (with $p \nmid n)$ so that $\Phi_n$ splits into linear factors over $\mathbb{Z}/(p)$.</p>
| Kaj Hansen | 138,538 | <p>$a \equiv b \pmod{n} \iff n|(a-b)$. Knowing this, then certainly $n|r(a-b)$.</p>
<p>Hence, $n|(ra-rb) \iff ra \equiv rb \pmod{n}$.</p>
|
6,887 | <p>Let x_1, x_2, ... be iid draws from a laplace distribution with scale parameter b. Is there a relatively nice closed form for x_1+x_2+...x_n? I've seen a derivation floating around for when b=1, but I couldn't figure out a generalisation. </p>
| David Bar Moshe | 1,059 | <p>The distribution of the $n$-th convolution of the Laplace distribution can be computed from the characteristic function (see on <a href="https://en.wikipedia.org/wiki/Laplace_distribution" rel="nofollow">Wikipedia</a>):
$$\frac{\exp(i \mu t)}{1+b^2 t^2} \,.$$
The characteristic function of the $n$-th convolution bec... |
2,138,009 | <p>Let $f(z)=(1+i)z+1$. Then $f(z)=\sqrt 2 e^{i\pi/4}z+1$ and thus $f=t\circ h\circ r$ where $t$ is the translation of vector $1$, $r$ the rotation of center $0$ and angle $\pi/4$ and $h$ the homothetic of parameter $\sqrt 2$. I found a fix point $z=\frac{-1}{2}+\frac{i}{2}$.</p>
<p>1) What if $f\circ f\circ f\circ f\... | Intelligenti pauca | 255,730 | <p>The fixed point is $z_0=i$ and rewriting $f$ as
$$
f(z)-i=(1+i)(z-i)=\sqrt2 e^{i\pi/4}(z-i)
$$
you can see that $f$ is a rotation of $\pi/4$ and a homothetic transformation of ratio $\sqrt2$, both with center $z_0$.</p>
<p>It is then obvious that:
$$
f^n(z)-i=(1+i)^n(z-i)=2^{n/2}e^{in\pi/4}(z-i),
$$
that is $f^n$... |
427,835 | <p>Which website/journal/magazine would you recommend to keep up with advances in applied mathematics?
More specifically my interest are:</p>
<ul>
<li>multivariate/spatial interpolation</li>
<li>numerical methods</li>
<li>computational geometry</li>
<li>geostatistics</li>
<li>etc</li>
</ul>
<p>I am looking for a fair... | lhf | 589 | <p>Try the <a href="http://www.siam.org/journals/sirev.php" rel="nofollow">SIAM Review</a>. It features Survey and Review papers of wide interest. </p>
|
3,101,098 | <p>From 11, 12 in the book Logic in Computer Science by M. Ryan and M. Huth:</p>
<p>**</p>
<blockquote>
<p>"What we are saying is: let’s make the assumption of ¬q. To do this,
we open a box and put ¬q at the top. Then we continue applying other
rules as normal, for example to obtain ¬p. But this still depends o... | Mauro ALLEGRANZA | 108,274 | <p>In the calculus there are different types of rules; some allow us to "discharge" assumptions, like e.g. <span class="math-container">$\to$</span>-intro; others do not.</p>
<p>The "mechanism" is quite simple: we can made whatever assumption we want, but every "result" we get applying the rules to it will depend on t... |
2,834,195 | <p>Using the method of characteristics on a PDE system, I have gotten a parametric differential equation
$$
\frac{dy}{dx} = \frac{y - xy}{1 + xy - x}.
$$
where $x$ and $y$ are both functions of a third variable $t$. How could I use Mathematica to solve for the solution curve that $(x(t), y(t))$ follows? There is a simi... | Community | -1 | <p><a href="https://en.m.wikipedia.org/wiki/Stereographic_projection" rel="nofollow noreferrer">Stereographic projection </a> is easy to visualize for $S^2\setminus \{p\}$; and the notion can be extended to $S^n\setminus \{p\}$...</p>
<p>This "shows" that $S^n$ can be thought of as the one-point compactification of $\... |
2,834,195 | <p>Using the method of characteristics on a PDE system, I have gotten a parametric differential equation
$$
\frac{dy}{dx} = \frac{y - xy}{1 + xy - x}.
$$
where $x$ and $y$ are both functions of a third variable $t$. How could I use Mathematica to solve for the solution curve that $(x(t), y(t))$ follows? There is a simi... | Henno Brandsma | 4,280 | <p>If that last theorem is allowed to use, your statement is an immediate corollary of it. In your setup you only need note that $X=\mathbb{S}^n\setminus \{p\}$ is locally compact, and the identity is the homeomorphism, and then the conclusion is that $\mathbb{S}^n$ is homeomorphic to the one-point compactification of ... |
1,427,816 | <p>This is kinda of a philosophical question I guess. But are the elmements of the topological closure inside the linear space $X$ all the time? Or do they become apperent when we introduce the topology? And hence introduce the topolgy to control these elements of the space which are there but out of control when we on... | Ian | 83,396 | <p>One subtle difference between metric spaces and topological spaces is that the "completion of a topological space" is not a well-defined notion. </p>
<p>Of course the closure is well-defined at the level of topological spaces. But unlike the completion, the closure is not really a unary operation, it is a binary op... |
4,289,129 | <p>Let <span class="math-container">$H$</span> be a group with identity <span class="math-container">$1_H$</span> that is generated by 2 elements <span class="math-container">$a,b$</span> that commute (<span class="math-container">$ab=ba$</span>) and where each has at most order <span class="math-container">$3$</span>.... | Shaun | 104,041 | <p>Since, by definition of a presentation, the presentation</p>
<p><span class="math-container">$$P=\langle a,b\mid a^3,b^3, ab=ba\rangle$$</span></p>
<p>defines a group that maps onto <span class="math-container">$H$</span>, and that group defined by <span class="math-container">$P$</span> is <span class="math-contai... |
4,098,682 | <p>I am trying to prove this following theorem about multiplying left cosets.</p>
<blockquote>
<p>Let <span class="math-container">$H \subset G$</span> a subgroup and <span class="math-container">$G/H$</span> the set of left cosets of <span class="math-container">$H$</span> in <span class="math-container">$G$</span>. W... | Arturo Magidin | 742 | <p>I don’t much like your first argument, to be honest...</p>
<p>Proving that multiplication “is well defined” means proving that if <span class="math-container">$aH=a’H$</span> and <span class="math-container">$bH=b’H$</span>, then <span class="math-container">$abH = a’b’H$</span>. I’m not sure your argument establish... |
1,920,994 | <p>My calculus teacher gave us this interesting problem: Calculate</p>
<p>$$ \int_{0}^{1}F(x)\,dx,\ $$ where $$F(x) = \int_{1}^{x}e^{-t^2}\,dt $$</p>
<p>The only thing I can think of is using the Taylor series for $e^{-t^2}$ and go from there, but since we've never talked about uniform convergence and term by term in... | Claude Leibovici | 82,404 | <p>You could do it directly. Since $$\int e^{-t^2}\,dt=\frac{\sqrt{\pi }}{2} \text{erf}(t)$$ $$F(x) = \int_{1}^{x}e^{-t^2}\,dt=\frac{\sqrt{\pi }}{2} (\text{erf}(x)-\text{erf}(1))$$ Now, integrating by parts $$\int \text{erf}(x)\,dx=x \,\text{erf}(x)+\frac{e^{-x^2}}{\sqrt{\pi }}$$ </p>
<p>I am sure that you can take ... |
2,631,342 | <p>$$\lim_{x\rightarrow 14}\frac{\sqrt{x-5}-3}{x-14}$$</p>
<p>How do I evaluate the limit when I put x = 14 and I got 0/0?</p>
| Dr. Sonnhard Graubner | 175,066 | <p>write $$\frac{\sqrt{x-5}-3}{x-14}\cdot \frac{\sqrt{x-5}+3}{\sqrt{x-5}+3}$$</p>
|
2,631,342 | <p>$$\lim_{x\rightarrow 14}\frac{\sqrt{x-5}-3}{x-14}$$</p>
<p>How do I evaluate the limit when I put x = 14 and I got 0/0?</p>
| ajotatxe | 132,456 | <p>Hint:</p>
<p>$$x-14=(\sqrt{x-5}+3)(\sqrt{x-5}-3)$$</p>
|
2,631,342 | <p>$$\lim_{x\rightarrow 14}\frac{\sqrt{x-5}-3}{x-14}$$</p>
<p>How do I evaluate the limit when I put x = 14 and I got 0/0?</p>
| E.H.E | 187,799 | <p>By L'Hôpital's rule</p>
<p>$$\lim_{x\rightarrow 14}\frac{\sqrt{x-5}-3}{x-14}=\lim_{x\rightarrow 14}\frac{\frac{1}{2}}{\sqrt{x-5}}=\frac{1}{6}$$</p>
|
1,168,446 | <p>I have the following nonlinear differential equation (I am using $y$ as shorthand $f(x)$):</p>
<p>$$\sin(y - y') = y''$$</p>
<p>I have tried the following</p>
<p>$$\cos(y - y')(y'-y'') = y'''$$
$$-\sin(y - y')(y'-y'')^2 + \cos(y - y')(y''-y''') = y''''$$
$$-y''(y'-y'')^2 + \dfrac{y'''}{y'-y''}(y''-y''') = y''''$$... | abel | 9,252 | <p>i don't know how useful this is to you but here it is. we will make a change of variable $$y-y' = u.$$ then the differential equation $y'' = \sin (y-y')$ can be transformed into $$\sin u = y''= y'-u'=y-u-u'$$ now we have two first order equations </p>
<p>$$\begin{align}\frac{dy}{dx} &= y - u\\ \frac{du}{dx} &... |
251,705 | <p>I would like to find the residue of $$f(z)=\frac{e^{iz}}{z\,(z^2+1)^2}$$ at $z=i$. One way to do it is simply to take the derivative of $\frac{e^{iz}}{z\,(z^2+1)^2}$. Another is to find the Laurent expansion of the function.</p>
<p>I managed to do it using the first way, and the answer is $-3/(4e)$. However, I'm ou... | Ivan Lerner | 40,086 | <p>Use the formula for the first term of the Laurent series:$$a_{-1}=\frac{1}{(m-1)!}\frac{d^{m-1}}{dz^{m-1}}\left((z-z_0)^mf(z)\right)$$
Where m is the order of the pole.
You can get to this formula by taking the Taylor expansion of the function $f(z)(z-z_0)^m$ since it is holomorphic, and making the Laurent expansion... |
405,205 | <p>Some friends and I have a family of polynomials (in one variable) with rational coefficients and we would very much like a formula for them. Grasping at straws, we computed many examples and wrote them in the basis of binomial coefficients. Specifically, I mean the basis <span class="math-container">$\left\{\binom... | Per Alexandersson | 1,056 | <p>We conjecture that the coefficients of Jack polynomials can be expressed nicely in this basis, see <a href="https://arxiv.org/pdf/1810.12763.pdf" rel="nofollow noreferrer">https://arxiv.org/pdf/1810.12763.pdf</a></p>
<p>Also, there is a close connection with rook polynomials and hit polynomials, as well as the relat... |
1,112,081 | <p>Does $\int_0^\infty e^{-x}\sqrt{x}dx$ converge? Thanks in advance.</p>
| Vim | 191,404 | <p>Though I don't know how to calculate the exact value, it is quite easy to show it does converge.<br/>
One basic fact in improper integrals is that, whether it converges or not depends completely on how the integrated function behaves at the "bad" points (say, infinity or points where the function is not defined) and... |
7,237 | <p>this came up in class yesterday and I feel like my explanation could have been more clear/rigorous. The students were given the task of finding the zeros of the following equation $$6x^2 = 12x$$ and one of the students did $$\frac{6x^2}{6x}=\frac{12x}{6x}$$ $$x = 2$$ which is a valid solution but this method elimin... | Frank Newman | 5,104 | <p>It comes up frequently in solving trigonometric equations such as:</p>
<p><span class="math-container">$2\sin x\cos x=\sin x$</span></p>
<p>Students often divide by <span class="math-container">$\sin x$</span>. I find myself using this line a lot:</p>
<p>"<em>If you're ever tempted to divide both sides by a var... |
1,323,845 | <p>For a nonnegative integer $n$, a composition of $n$ means a partition in which the order of the parts matters.</p>
<p>Consider the generating function
$$C(x) = \sum_{n=0}^{\infty} c_nx^n,$$
where $c_n$ is the number of distinct compositions of $n$ (note that $c_0=1$ by convention).</p>
<p>What is the value of $C\l... | Matematleta | 138,929 | <p>In fact, $X$ is not even connected: </p>
<p>Since $B$ contains more than one point, choose $b_{0}, b_{1}\in B$ and let $d(b_{0}, b_{1})=r>0$. Choose $0<s<r$, such that $d(b_{0},x)\neq s$ for any $x\in X$. This is possible since $B$ is countable and since $\left \{ x\in X:d(x,b_{0})=s \right \}\subseteq B$<... |
2,638,679 | <p><a href="https://i.stack.imgur.com/S4p0Y.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/S4p0Y.jpg" alt="enter image description here"></a></p>
<p>Due apologies for this rustic image. But while drawing this lattice arrangement about the "square numbers" , I discovered a pattern here wherein if I ... | Fred | 380,717 | <p>Use the following nice and easy formula:
$$\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}^{-1}=\frac{1}{ad-bc}\begin{pmatrix}
d & -b\\
-c & a
\end{pmatrix}.$$</p>
|
121,450 | <p>I am trying to prove that the series <span class="math-container">$\sum \dfrac {1} {\left( m_{1}^{2}+m_{2}^{2}+\cdots +m_{r }^{2}\right)^{\mu} } $</span> in which the summation extends over all positive and negative integral values and zero values of <span class="math-container">$m_1, m_2,\dots, m_r$</span>, except ... | Eric Naslund | 6,075 | <p>Here is a way I like. We can rewrite your sum as $$\sum_{\boldsymbol{m}\in\mathbb{Z}^{m}\backslash\{\boldsymbol{0}\}}\frac{1}{\|\boldsymbol{m}\|_{2}^{r+\epsilon}}$$ where $\epsilon>0.$ Then since $$\|x\|_{2}\geq\max_{i}|x_{i}|,$$ by using the comparison test, we know that our original series will converge if $$\... |
122,546 | <p>There is a famous proof of the Sum of integers, supposedly put forward by Gauss.</p>
<p>$$S=\sum\limits_{i=1}^{n}i=1+2+3+\cdots+(n-2)+(n-1)+n$$</p>
<p>$$2S=(1+n)+(2+(n-2))+\cdots+(n+1)$$</p>
<p>$$S=\frac{n(1+n)}{2}$$</p>
<p>I was looking for a similar proof for when $S=\sum\limits_{i=1}^{n}i^2$</p>
<p>I've trie... | Henry | 6,460 | <p>You can use something similar, though it requires work at the end. </p>
<p>If $S_n = 1^2 +2^2 + \cdots + n^2$ then
$$S_{2n}-2S_n = ((2n)^2 - 1^2) + ((2n-1)^2-2^2) +\cdots +((n+1)^2-n^2)$$</p>
<p>$$=(2n+1)(2n-1 + 2n-3 + \cdots +1) = (2n+1)n^2$$ using the Gaussian trick in the middle. </p>
<p>Similarly $$S_{2n+1}... |
1,109,759 | <p>I.e, prove $\lVert f+g \rVert\ \le \lVert f \rVert + \lVert g \rVert$ for all $f,g$ in $C^\infty [0,1]$,
$$\lVert f \rVert =(\int_0^1 \lvert f(x) \rvert ^2 dx)^{1/2}$$</p>
<p>I think we're supposed to use Cauchy-Schwarz: $\lvert \int_0^1 f(x)g(x) dx \rvert \le \left( \int_0^1 \lvert f(x) \rvert ^2 dx \right)^{1/2... | Arch | 208,530 | <p>I think Alex actually intends to prove ||f|| is certainly a norm.</p>
<p>Just one comment: Use $||f+g||^2 $ to avoid square roots.</p>
<blockquote>
<p>$||f+g||^2 = ||f||^2 + ||g||^2 +2 |<f,g>| \leq ||f||^2 + ||g||^2 +2 ||f||||g|| = (||f||+||g||)^2$,</p>
</blockquote>
<p>and we are done.</p>
|
1,524,349 | <p>This is Problem 45 in Chapter 19 in Michael Spivak's book "Calculus".</p>
<ol start="45">
<li>(a) Suppose that $\frac {f(x)} x$ is integrable on every interval [a, b] for $0$ < a < b, and that $\lim_{x\to0}f(x)=A$ and $\lim_{x\to\infty}f(x)=B$. Prove that for all $\alpha$, $\beta$ > $0$ we have</li>
</ol>
<p... | Mark Viola | 218,419 | <p><strong>HINT:</strong></p>
<p>Since $\frac{f(x)}{x}$ is an arbitrary integrable function, it can be approximated in the $\ell^1$ norm by a compactly supported smooth function $\frac{g(x)}{x}$. So, for all $\epsilon>0$, </p>
<p>$$\int_a^b \left|\frac{f(x)}{x}-\frac{g(x)}{x}\right|\,dx<\epsilon$$</p>
<p>Then... |
2,359,292 | <p>I have been working on a problem in Quantum Mechanics and I have encountered a equation as given below.</p>
<p>$$\frac{d\hat A(t)}{dt} = \hat F(t)\hat A(t)$$</p>
<p>Where ^ denotes it is an operator </p>
<p>How will this differential equation be solved? Will the usual rules for linear homogeneous first order diff... | Fabian | 7,266 | <p>You can solve it by iteration (assuming convergence). Assuming that you are interested in the solution with the initial condition $\hat A(0)= I$, the iterative solution reads
$$\hat A(t) = I +\int_0^t\hat F(t_1)\,dt_1 + \int_0^t\int_0^{t_1}\hat F(t_1) \hat F(t_2)\,dt_1\,dt_2 + \cdots \tag{1}$$</p>
<p>For convenienc... |
4,045,074 | <p><strong>Let <span class="math-container">$X$</span> be the random variable whose cumulative distribution function is
<span class="math-container">$$
F_X (x) = \begin{cases}
0, & \text{for} \space x\lt 0 \\
\frac{1}{2}, & \text{for} \space 0\le x\le 1 \\
1, & \text{for} \space x\gt 1 \\
\end{c... | TravorLZH | 748,964 | <p>Without the knowledge of partial summation, we just use the traditional summation by parts:</p>
<p>Let <span class="math-container">$\pi(x)$</span> denote the number of prime numbers less than or equal to <span class="math-container">$x$</span>, so for all <span class="math-container">$n\in\mathbb Z^{>0}$</span><... |
1,932,961 | <p>Prove by mathematical induction that
$$\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$$
holds $\forall n\in\mathbb{N}$.</p>
<hr>
<p>(1) Assume that $n=1$. Then left side is $1^2 =1$ and right side is $6/6 = 1$, so both sided are equal and expression holds for $n = 1$.</p>
<p>(2) Let $k \in \mathbb{N}$ is given. As... | Sathasivam K | 355,833 | <p>Since,$xyz=1$,we have any two of x,y,z is negative or all must be positive and in both case all three are non zero.</p>
<p>CASE 1:
If all x,y,z are positive I hope that you may easily prove it.</p>
<p>CASE 2:
if x,y is negative then we have $$x>x-1$$
But $$ x^2≤(x-1)^2.$$since if we assume $x=\frac{1}{2} $ th... |
185,177 | <p>Let $X$ be a smooth finite type separated connected Deligne-Mumford stack over $\mathbb C$.</p>
<p>Does there exist a finite etale morphism $Y\to X$ with $Y$ a scheme?</p>
<p>What if $X$ is an algebraic space (i.e., trivial stabilizers)?</p>
<p>Edit: I changed the old question to a different question which should... | Niels | 11,682 | <p>To give a more simple example than Daniel's, you can just consider for X a projective line with a single orbifold point. By Riemann-Hurwitz X is simply connected and so there is no non-trivial finite étale morphism Y→X. This holds over an algebraically closed field of characteristic zero say (but would work in chara... |
104,375 | <p>How I am supposed to transform the following function in order to apply the laplace transform.</p>
<p>$f(t) = t[u(t)-u(t-1)]+2t[u(t-1) - u(t-2)]$</p>
<p>I know that it has to be like this</p>
<p>$L\{f(t-t_0)u(t-t_0)\} = e^{-st_0}F(s), F(s) = L\{f(t)\}$</p>
| Blah | 6,721 | <p>An exercise in set theory (if $k$ runs through $\mathbb Z$, then so does $-k$):</p>
<p>$[a] = \\
\{b \in \mathbb{Z} \text{ such that there exists }k \in \mathbb{Z} \text{ such that }a-b=3k\}=\\
\{b \in \mathbb{Z} \text{ such that there exists }k \in \mathbb{Z} \text{ such that }b=a-3k\}=\\
\{b \in \mathbb{Z} \te... |
837,570 | <p>Prove that $\arctan{x}=\frac{1}{x^2}$ has only one solution on the set of real numbers.</p>
<p>I need some help with it, would greatly appreciate it.</p>
| DSinghvi | 148,018 | <p>Infer from the graph plotting.
Draw the graph of $y=1/x^2$ on paper and then make the graph of $\arctan(x)$ wherever they intersect is your solution and number of points of intersection are your number of solution. This is answer is given on the assumption that you know the basic graphs of $1/x^2$ and $\arctan(x)$.... |
2,631,230 | <p>So, I'm studying mathematics on my own and I took a book about Proofs in Abstract Mathematics with the following exercise:</p>
<p>For each $k\in\Bbb{N}$ we have that $\Bbb{N}_k$ is finite</p>
<p>Just to give some context on what theorems and definitions we can use:</p>
<ol>
<li>Definition: $\Bbb{N}_k = \{1, 2, ..... | Ng Chung Tak | 299,599 | <p>\begin{align}
E(Z) &= \int_{0}^{1} \int_{0}^{1} Z f(x,y) \, dx \, dy \\
&= \int_{0}^{1} \int_{0}^{1} 4xy\sqrt{x^2+y^2} \, dx \, dy \\
&= \int_{0}^{1} 2y \left( \int_{0}^{1} 2x\sqrt{x^2+y^2} \, dx \right) dy \\
&= \int_{0}^{1} 2y
\left(
\int_{0}^{1} \sqrt{u+y^2} \, du
\righ... |
424,514 | <p>Suppose one has a generating function <span class="math-container">$$F(z) = \sum_{k\ge 0} f(k) z^k$$</span>
for some <span class="math-container">$f:\mathbb{Z}\rightarrow \mathbb{Z}$</span>. Is there a way to express an iteration of <span class="math-container">$f$</span> in terms of <span class="math-container">$F(... | Gerald Edgar | 454 | <p>That seems really unlikely.<br />
For example, <span class="math-container">$$F(z)=\sum_{k=0}^\infty 2^kz^k = \frac{1}{1-2z}$$</span> is a rational function, but <span class="math-container">$$G(z)=\sum_{k=0}^\infty 2^{2^k}z^k$$</span> has radius of convergence <span class="math-container">$0$</span>.</p>
|
1,793,231 | <p>Can you please help me on this question?
$\DeclareMathOperator{\adj}{adj}$</p>
<p>$A$ is a real $n \times n$ matrix; show that:</p>
<p>$\adj(\adj(A)) = (\det A)^{n-2}A$</p>
<p>I don't know which of the expressions below might help</p>
<p>$$
\adj(A)A = \det(A)I\\
(\adj(A))_{ij} = (-1)^{i+j}\det(A(i|j))
$$</p>
<p... | Ian | 83,396 | <p>I would discourage you from using the word "adjoint" in this context. This is an accepted usage of the word, but there is another concept in linear algebra which is <em>always</em> referred to by the word "adjoint". The two can be easily confused. An unambiguous word that can be used in this context is "adjugate", a... |
3,491,028 | <p>Problem:<br>
Suppose that <span class="math-container">$x_1$</span>, <span class="math-container">$x_2$</span> and <span class="math-container">$x_3$</span> are independent uniformly distributed on the interval <span class="math-container">$[1,3]$</span>. What is the probability that
<span class="math-container">$x_... | antkam | 546,005 | <p>I disagree with the other answer (and OP, and another commenter) that the <span class="math-container">$x_2$</span> limit has to be <span class="math-container">$\min(5-x_1, 3)$</span>. Why should it be that? <span class="math-container">$x_2$</span> can be the entire range <span class="math-container">$[1,3]$</sp... |
1,768,700 | <p>According to my knowledge, to prove that $24^{31}$ is congruent to $23^{32}$ mod 19, we must show that both numbers are divisible by 19 i.e. their remainders must be equal with mod 19. Please correct me if I'm wrong.</p>
<p>So, I was able to reduce $23^{32}$ and find its mod 19, which is 17 but I am having a bit of... | user5713492 | 316,404 | <p>With perhaps a little less arithmetic, $2^2=4\equiv23\pmod{19}$, and $4\times5=20\equiv1\pmod{19}$, so $24\equiv5\equiv4^{-1}\equiv2^{-2}\pmod{19}$. By Fermat's little theorem,
$$23^{32}=2^{2\times32}=2^{64}\equiv2^{64-7\times18}\equiv2^{-62}\equiv2^{-2\times31}\equiv24^{31}\pmod{19}$$</p>
|
823,055 | <p>This may be a naive question. I am reading the definition of differetiablity of a function $f:\mathbb{R^n}\rightarrow \mathbb{R^m}$ in the book Calculus Manifolds. I already know that all norms on $\mathbb{R}^n$ induce the same metric topology. If we change the norms in the definition (for example we can use the man... | Lee Mosher | 26,501 | <p>$M \times [0,1]$ is homeomorphic to the "solid Klein bottle", and its boundary is the ordinary 2-dimensional Klein bottle, which is nonorientable.</p>
|
1,837,356 | <p>I'm reading <a href="http://www.careerbless.com/aptitude/qa/permutations_combinations_imp8.php" rel="nofollow">this passage</a> and wondering why</p>
<p>Number of ways in which
k identical balls can be distributed into
n distinct boxes =</p>
<p>$$\binom {k+n-1}{n-1}$$</p>
<p>could someone explain it to me plea... | pancini | 252,495 | <p>Imagine you lay out $k$ balls in a straight line. Then you divide them up into boxes by setting out markers splitting them up. For example, if you have $10$ balls and $3$ boxes, you might do</p>
<p>$$\text{b, b, MARKER, b, b, b, MARKER, b, b, b, b, b}$$</p>
<p>and this sequence means two balls in the first box, th... |
1,837,356 | <p>I'm reading <a href="http://www.careerbless.com/aptitude/qa/permutations_combinations_imp8.php" rel="nofollow">this passage</a> and wondering why</p>
<p>Number of ways in which
k identical balls can be distributed into
n distinct boxes =</p>
<p>$$\binom {k+n-1}{n-1}$$</p>
<p>could someone explain it to me plea... | true blue anil | 22,388 | <p>This is what is called "stars and bars" combinatorics</p>
<p>Suppose there are $15$ balls, and $3$ boxes.</p>
<p>The balls could be variously distributed, e.g.</p>
<p>$\Large\bullet\bullet\bullet+\bullet\bullet\bullet\bullet\bullet+\bullet\bullet\bullet\bullet\bullet\bullet\bullet= 15$</p>
<p>I have used $+$ for... |
1,522,929 | <p>For every fixed $t\ge 0$ I need to prove that the sequence $\big\{n\big(t^{\frac{1}{n}}-1\big) \big\}_{n\in \Bbb N}$ is non-increasing, i.e.
$$n\big(t^{\frac{1}{n}}-1\big)\ge (n+1)\big(t^{\frac{1}{n+1}}-1\big)\;\ \forall n\in \Bbb N$$
I'm trying by induction over $n$, but got stuck in the proof for $n+1$:
<br/>
For ... | Max0815 | 595,084 | <p>Here is another solution with a different contour.</p>
<p>Let <span class="math-container">$$I=\int^{\infty}_{-\infty}e^{ix^2}\text{ d}x$$</span> and let <span class="math-container">$$f(z)=e^{iz^2}$$</span> Note that our function is even.</p>
<p>In user279043's answer, the contour they chose was (what I would presu... |
1,522,929 | <p>For every fixed $t\ge 0$ I need to prove that the sequence $\big\{n\big(t^{\frac{1}{n}}-1\big) \big\}_{n\in \Bbb N}$ is non-increasing, i.e.
$$n\big(t^{\frac{1}{n}}-1\big)\ge (n+1)\big(t^{\frac{1}{n+1}}-1\big)\;\ \forall n\in \Bbb N$$
I'm trying by induction over $n$, but got stuck in the proof for $n+1$:
<br/>
For ... | K.defaoite | 553,081 | <h2>Another method.</h2>
<p><span class="math-container">$$\int_{-\infty}^\infty\exp(ix^2)\mathrm dx=2\int_0^{\infty} \exp(ix^2)\mathrm dx\tag{1}$$</span></p>
<p>Let <span class="math-container">$-z=ix^2\implies x=(iz)^{1/2}\implies \mathrm dx=\frac{i^{1/2}}{2}z^{-1/2}\mathrm dz$</span> hence
<span class="math-containe... |
4,001,031 | <p>(For all those that it may concern, this is not a duplicate of my previous post, But starts in a similar way.)</p>
<p>A triangle with side lengths a, b, c with a height(h) that intercepts the hypotenuse(c) at (x , y) such that it is split into two side lengths, c = m + n, we can find Pythagoras theorem using the ar... | J.G. | 56,861 | <p>As with your previous question, you have given a valid proof of Pythagoras... if I've followed your argument correctly. First, I'll condense it, if only for my own benefit. (I also swap round <span class="math-container">$a,\,b$</span>, because traditionally these are respectively opposite <span class="math-containe... |
23,846 | <p>I'm stuck with this algebra question.</p>
<p>I try to prove that the exterior algebra $R$ over $k^d$, that is, the $k$-algebra that is generated by $x_1,\ldots,x_d$ and $x_ix_j=- x_jx_i$ for each $i,j$, has just one simple module which is not faithful.</p>
<p>I think the only simple module is $k$, but I am not rea... | Mariano Suárez-Álvarez | 274 | <p><em>(I will be assuming $k$ is a field; if it is not, then you will need some hypothesis on it for your statement to be true)</em></p>
<p>Suppose $R$ is a ring and that $S$ is a non-zero simple left $R$-module. Pick a non-zero element $s_0\in S;$ then the map $\phi:r\in R\mapsto sr_0\in S$ is a surjective map of le... |
1,685,895 | <blockquote>
<blockquote>
<p>Question: Find a value of $n$ such that the coefficients of $x^7$ and $x^8$ are in the expansion of $\displaystyle \left(2+\frac{x}{3}\right)^{n}$ are equal.</p>
</blockquote>
</blockquote>
<hr>
<p>My attempt:</p>
<p>$\displaystyle \binom{n}{7}=\binom{n}{8} $</p>
<p>$$ n(n-1)(n-... | Archis Welankar | 275,884 | <p>The general term of $(a+b)^n$ $$t_{r+1}={n\choose r}.a^r.b^{n-r}$$ plug in r as $6,7$ and you will get it</p>
|
1,685,895 | <blockquote>
<blockquote>
<p>Question: Find a value of $n$ such that the coefficients of $x^7$ and $x^8$ are in the expansion of $\displaystyle \left(2+\frac{x}{3}\right)^{n}$ are equal.</p>
</blockquote>
</blockquote>
<hr>
<p>My attempt:</p>
<p>$\displaystyle \binom{n}{7}=\binom{n}{8} $</p>
<p>$$ n(n-1)(n-... | Uri Goren | 203,575 | <p>The coefficient of $x^7$ is
$$\binom{n}{7}\frac{2^{n-7}}{3^7}$$
And the coefficient of $x^8$ is
$$\binom{n}{8}\frac{2^{n-8}}{3^8}$$
Comparing them we get:
$$\binom{n}{8}=\binom{n}{7}\frac{3}{2}$$</p>
|
401,002 | <p>$\forall x \neg A \implies \neg \exists xA$<br>
I won't ask you to solve this for me, but can you please give some guiding lines on how to approach a proof in NDFOL?<br>
There are many tricks that the TA shows in class, that I could not dream of...</p>
<p>P.S. I managed to proof $\neg \exists xA \implies \forall x ... | Dan Christensen | 3,515 | <p>Stated in a slightly different way from LF...</p>
<ol>
<li><p>$\forall x \neg A(x)$ (Assume)</p></li>
<li><p>$A(y)$ (Assume)</p></li>
<li><p>$\neg A(y)$ (Universal Specification, line 1)</p></li>
<li><p>$A(y) \wedge \neg A(y)$ (Join, lines 2, 3)</p></li>
<li><p>$\neg\exists x A(x) $ (Conclusion, line 2)</p></li>... |
2,677 | <p>If <em>G</em> is a group, its <strong>abelianization</strong> is the abelian group <em>A</em> and the map <em>G</em> → <em>A</em> such that any map <em>G</em> → <em>B</em> with <em>B</em> abelian factors through <em>A</em>. Abelianization is a functor, and in general a very lossy operation. The map <em>G... | Eric Wofsey | 75 | <p>I don't have anything to say about specific examples, but here are some general remarks. A way to construct the abelianization of any compact group is to consider its image under the product of all its 1-dimensional unitary representations. This is because a compact abelian group is characterized by its set of cha... |
2,555,861 | <p>I am reading up on <strong>Fraleigh's</strong> <em>A First Course in Abstract Algebra</em>, and he says ($H$ subgroup of $G$) $Hg=gH$ $iff$ $i_g[H]=H$ $iff$ $H$ is invariant under all inner automorphisms. I look up invariant and I find this definition:</p>
<p>"Firstly, if one has a group G acting on a mathematical... | openspace | 243,510 | <p>You could try this : let $\delta x = \frac{1}{n}$, then $x_{k} = \frac{k}{n}$. Now we can consider $\displaystyle \sum_{k=0}^{n-1}\frac{1}{n}(e^{x_{k+1}}-e^{x_{k}})$, then consider $\displaystyle\sum \frac{e^{k/n}(e^{1/n}-1)}{n}$. Now estimate $e^{1/n}$ and find the ''$\lim\sum$''</p>
|
838,690 | <p>True or false question</p>
<p>If B is a subset of A then {B} is an element of power set A. </p>
<p>I think this is true.</p>
<p>Because B is {1,2} say A {1,2,3} then power set of includes </p>
<p>$\{\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{3,2\},\{1,2,3\},\emptyset\}$</p>
<p>Unless {B} means $\{\{1,2\}\}$</p>
| Avraham | 91,378 | <p>The definition of a power set of a set $A$ is the set of all subsets of $A$ including $A$ itself and the null set. As $B$ is a subset of $A$ in your question, then yes $B$ is an element in the power set of $A$.</p>
|
2,357,272 | <p>Find out the sum of the following infinite series
$$\frac{3}{2^2(1)(2)} + \frac{4}{2^3(2)(3)} +\dots+\frac{r+2}{2^{r+1}(r)(r+1)}+\cdots
$$
up to $r\to\infty$.</p>
<p>MY TRY:- I tried to split $r+2$ as $[(r+1) +{(r+1)-r}]$ so that I can cancel one term from each terms in the numerator. Then I got an expression whic... | Dr. Sonnhard Graubner | 175,066 | <p>prove by induction that for your sum is hold $$\sum_{i=1}^n\frac{i+2}{2^{i+1}i(i+1)}=\frac{2^{-n-1} \left(2^n n+2^n-1\right)}{n+1}$$</p>
|
617,163 | <p>I need to find a proper definition of a quantile.
It says:
a p-th quantile $x_p$ is a number, that satisfies the following conditions:
$$
0<p<1
$$
and
$$
P(X \le x_{p}) \ge p
$$
and
$$
P(X \ge x_{p}) \ge 1-p
$$
is this definition right?</p>
| alexjo | 103,399 | <p>The profit is $\pi(p,q)=pq-c(q)$ where $p$ is the selling price, $q$ is the quantity selled and $c(q)$ the cost to produce the quantity $q$. So you have $\pi(p,250)=250p-c(250)$ and you know that $\pi(p,250)=50p$; then you have $c(250)=200p$ and finally
$$
\frac{\pi(p,250)}{c(250)}=\frac{250p-200p}{200p}=\frac{50}{2... |
617,163 | <p>I need to find a proper definition of a quantile.
It says:
a p-th quantile $x_p$ is a number, that satisfies the following conditions:
$$
0<p<1
$$
and
$$
P(X \le x_{p}) \ge p
$$
and
$$
P(X \ge x_{p}) \ge 1-p
$$
is this definition right?</p>
| okarin | 112,825 | <p>Total Money: Price of $250$ chairs </p>
<p>Gain: Price of $250 - 200 = 50$ chairs</p>
<p>Profit Percent: $\frac{\text{gain}}{\text{spent}} = \frac{250 - 200}{250 - (250 - 200)} = \frac{50}{200} = 25\%$</p>
|
1,038,198 | <p>How do you prove that
$8 \cos{(x)}\cos{(2x)}\cos{(3x)} - 1 = \dfrac{\sin{(7x)}}{\sin{(x)}}$?</p>
| Community | -1 | <p>We have</p>
<p>$$8 \sin x\cos{(x)}\cos{(2x)}\cos{(3x)} =4\sin(2x)\cos(2x)\cos(3x)=2\sin(4x)\cos(3x)$$
Moreover</p>
<p>$$\sin x+\sin(7x)=2\sin\left(\frac{x+7x}{2}\right)\cos\left(\frac{7x-x}{2}\right)=\cdots$$
and the result follows easily.</p>
|
62,000 | <p>Let $I,J,K$ be three non-void sets, and let $\gamma$:$I\times J\times K\rightarrow\mathbb{N}$.
Is there some nonempty set $X$, together with some functions {$\{ f_{i}:X\rightarrow X;i\in I\} $},
some subsets {$\{ \Omega_{j}\subset X;j\in J\} $}, and some
points {$\{p_{k}\in X;k\in K} $} s.t. $\mid f_{i}^{-1}\left(p_... | Gerhard Paseman | 3,402 | <p>Consider the following construction. Let $Y$ be a subset of
$X$ such that $Y$ is (equipollent to) $I \times K \times \omega$.
I think of it as $I$-many copies of an array with
$K$-many rows and each row
has countably many elements. The $k$th row in the $i$th array
is the preimage of $p_k$ under $f_i$. (For $h$ n... |
2,481,767 | <p>Let A={$3m-1|m\in Z$} and B={$4m+2|m\in Z$} and let $f:A\rightarrow B$ is defined by </p>
<p>$f(x)=\frac{4(x+1)}{3}-2$ . Is f surjective?</p>
<p>I'm not really sure how to prove this. By trying out certain values it seems it's surjective. This is my work so far:</p>
<p>$f(x)=y \iff \frac{4(x+1)}{3}-2 = y \iff x=\... | copper.hat | 27,978 | <p>Solve $f(3n-1) = 4m+2$ to get $n=m+1$.</p>
<p>In particular, for any $b \in B$ there is some $a \in A$ such that
$f(a) = b$.</p>
<p>In fact, it is unique.</p>
<p>In particular, it is not hard to compute $f^{-1}(b) = {1 \over 2} b$.</p>
|
3,657,075 | <p><a href="https://i.stack.imgur.com/ytcQ3.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ytcQ3.png" alt="enter image description here"></a></p>
<blockquote>
<p>In the given figure <span class="math-container">$\angle BAE, \angle BCD$</span> and <span class="math-container">$\angle CDE$</span> a... | marty cohen | 13,079 | <p>BD = 5.</p>
<p>Drawing a perpendicular from D to AE at G,
DG = 4 so EG = 3
and AG = 5 so AE = 8.</p>
<p>As to the angles,
EDG = CDB (both 3-4-5)
so CDE = AED = CBD
so BDG = 90
so ABD = 90.</p>
|
3,657,075 | <p><a href="https://i.stack.imgur.com/ytcQ3.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ytcQ3.png" alt="enter image description here"></a></p>
<blockquote>
<p>In the given figure <span class="math-container">$\angle BAE, \angle BCD$</span> and <span class="math-container">$\angle CDE$</span> a... | Calvin Lin | 54,563 | <p>The original question is under-defined. </p>
<p>E.g. We could have <span class="math-container">$ACB$</span> as a straight line, with the desired angles still right.<br>
Explicitly, <span class="math-container">$ A = (0,0), B = (0, -4), C = (0, -1), D = (4, -1), E = (4, 4) $</span> which gives us <span class="math-... |
1,768,100 | <p>I have started studying field theory and i have a question.somewhere i saw that a finite field with $p^m $ elements has a subfield of order $p^m $ where $m$ is a divisor of $n $.My question that if it is a field then how can it have a proper subfield.because since it is field it doesnt have any proper ideal.how can... | paf | 333,517 | <p>You should parametrize your line segment as
$$\gamma : t\mapsto t(1+i)+(1-t)(-i)$$
when $t\in [0;1]$. Then, you have to replace in your integral $z$ by $\gamma(t)$, $dz$ by $\gamma'(t)dt$ and $L$ by $[0;1]$ (the same as a standard change of variables for real integrals) and you should be able to compute the integra... |
66,670 | <p>I want to use:</p>
<pre><code>demand = {1.92,
2.07,
2.37,
2.72,
2.87}*10^6;
NSolve[SetV == demand[[1]]/(Cpf (1 - χ)), χ]
</code></pre>
<p>I want to make a vector of solutions for chi (χ) given each of the demand vector components.</p>
| Bob Hanlon | 9,362 | <p>Bounding the range of n resolves the issue with <code>Maximize</code></p>
<pre><code>Maximize[{(3 n + 4)/(2 n + 1), Element[n, Integers], -100 <= n <= 100}, n]
</code></pre>
<blockquote>
<p>{4, {n -> 0}}</p>
</blockquote>
<p>Or,</p>
<pre><code>Maximize[{(3 n + 4)/(2 n + 1), -100 <= n <= 100}, n, In... |
2,516,023 | <blockquote>
<p>Why does taking logarithms on both sides of $0<r<s$ reverse the inequality for logarithms with base $a$, $0<a<1$?</p>
</blockquote>
<p>I would like some intuition on why this works. I tried graphing $\log_{0.5}(x)$ on Desmos, for example, and if the graph were true this would be evident f... | marty cohen | 13,079 | <p>Because
$\log_a(b)
=\dfrac{\log_c b}{\log_c a}
$
for any
$a, b, c > 0$.</p>
<p>The usual thing
is to choose
$c=e$ or $c=10$;
the key point is that
in both cases
$c > 1$.</p>
<p>Therefore,
if $b > 1$ and
$0 < a < 1$
then
$\log_c b > 0$
and
$\log_c a < 0$
so that
$\log_a b < 0$
and the usual ... |
148,313 | <p>Someone has claimed that he has constructed a quaternion representation of the one dimensional (along the x axis) Lorentz Boost.</p>
<p>His quaternion Lorentz Boost is $v'=hvh^*+ 1/2( [hhv]^*-[h^*h^*v^*]^*)$ where h is (sinh(x),cosh(x),0,0). He derived this odd transform by substituting the hyperbolic sine and cosi... | Ronald | 27,884 | <p>Aye, this is <a href="https://en.wikipedia.org/wiki/Yao%27s_Millionaires%27_Problem">Yao's Millionaires Problem</a>!</p>
|
2,098,810 | <p>In a triangle,what is the ratio of the distance between a vertex and the orthocenter and the distance of the circumcenter from the side opposite vertex.</p>
| szw1710 | 130,298 | <p>Another solution could be given by the <a href="https://en.wikipedia.org/wiki/Hermite%E2%80%93Hadamard_inequality" rel="nofollow noreferrer">Hermite-Hadamard inequality</a>. </p>
<p>It is easy to verify that $f(x)=\sqrt{\strut 1+x^2}$ is convex (by $f''>0$). Then $$f(0)\leqslant \frac{1}{2}\int\limits_{-1}^1 f(x... |
1,805,615 | <p>I have one problem. I am sure it is not complicated, but I only need help to see am I, at least, on the right path.</p>
<p><strong>Problem: Let $S=Span\{(0,-2,3),(1,1,1),(2, -2, 8)\}\subseteq \mathbb R^3$. Find subspace $T$ of space $\mathbb R^3$ so that $\mathbb R^3=S \oplus T$.</strong></p>
<p>Here is what I hav... | DonAntonio | 31,254 | <p>An idea: indeed, $\;\dim\mathcal S=\dim\text{Span}\,S=2\;$ ,so why won't you reduce your matrix (say, by rows to make it easier) to check what vector to take out (the one lin. dep. in the other two) and begin to check what vector to add in order to make the whole thing linearly independent?:</p>
<p>$$\begin{pmatrix... |
349,309 | <p>I seem to be short on examples for $I$-adic completions of rings.</p>
<p>I know that a ring is $I$-adically complete if the canonical homomorphism into the inverse limit is an isomorphism. My thinking and searching on the internet has been surprisingly fruitless, though, for examples where the map is either surject... | rschwieb | 29,335 | <p>One thing I learned much later is: if <span class="math-container">$I$</span> is a nilpotent ideal, then <span class="math-container">$R$</span> is <span class="math-container">$I$</span>-adically complete.</p>
<p>Thinking of the completion as a subring of <span class="math-container">$\prod R/I^n$</span>, it's clea... |
3,903,774 | <p><span class="math-container">$30$</span> red balls and <span class="math-container">$20$</span> black balls are being distributed to <span class="math-container">$5$</span> kids, so that each kid gets at least one red ball. In how many ways can we distribute balls?</p>
<p>Circle the correct answers:</p>
<p>a) <span ... | Pietro Paparella | 414,530 | <p>That’s the square of the two-norm (aka the Euclidean norm).</p>
|
340,264 | <p>Given that</p>
<p>$L\{J_0(t)\}=1/(s^2+1)$</p>
<p>where $J_0(t)=\sum\limits^{∞}_{n=0}(−1)n(n!)2(t2)2n$,</p>
<p>find the Laplace transform of $tJ_0(t)$. </p>
<p>$L\{tJ_0(t)\}=$_<strong><em>_</em>__<em>_</em>__<em>_</em>___<em></strong>---</em>___?</p>
| azimut | 61,691 | <p><strong>Hint:</strong></p>
<p>Show that if $y^2\equiv 2\mod p^n$, there is a solution of $z^2\equiv 2\mod p^{n+1}$ with $z\equiv y\mod p^n$ (this technique is called Hensel lift).</p>
|
1,230,159 | <p>Where can I find a complete proof to the fact that the integral closure of $\mathbb{Z}$ in $\mathbb{Q}(i)$ is $\mathbb{Z}[i]$ (the Gaussian integers are the integral closure of $\mathbb{Z}$ in the Gaussian rationals)? For such a seemingly standard fact, I can not seem to find a complete proof of this anywhere. Yes, ... | user26857 | 121,097 | <p>If $z\in\mathbb Q[i]$ is integral over $\mathbb Z$, then it's integral over $\mathbb Z[i]$. But $\mathbb Z[i]$ is a UFD, so it's integrally closed. It follows $z\in\mathbb Z[i]$. (Recall or prove that $\mathbb Q[i]$ is the field of fractions of $\mathbb Z[i]$.)</p>
<p>Conversely, for $z\in\mathbb Z[i]$, $z=m+in$, $... |
1,211,978 | <p>I cannot find the roots of the characteristic equation to get a solution. I only know the basic way to solve these equations. I factored out an $r^2$.</p>
<p>$2r^5-7r^4+12r^3-8r^2 = 0$</p>
<p>$r^2(2r^3-7r^2+12r-8) = 0$</p>
| Pieter21 | 170,149 | <p>Check Wolfram alpha for further factorization.</p>
<p><a href="https://www.wolframalpha.com/input/?i=2x%5E3-7x%5E2%2B12x-8%3D0&lk=4&num=2" rel="nofollow">https://www.wolframalpha.com/input/?i=2x%5E3-7x%5E2%2B12x-8%3D0&lk=4&num=2</a></p>
<p>Are you sure you have all signs right? Also the other answe... |
729,444 | <p>Let be two lists $l_1 = [1,\cdots,n]$ and $l_2 = [randint(1,n)_1,\cdots,randint(1,n)_m]$ where $randint(1,n)_i\neq randint(1,n)_j \,\,\, \forall i\neq j$ and $n>m$. How I will be able to found the number of elements $x\in l_1$, to select, such that the probability of $x \in l_2$ is $1/2$?. I'm trying using the bi... | hmakholm left over Monica | 14,366 | <p>A proof sketch could be:</p>
<p><em>1. Every (nonzero) vector is an eigenvector.</em> Let $v\ne 0$ and suppose $Tv$ is not a multiple of $v$. Then $v$ and $Tv$ are linearly independent; extend $\langle v,Tv\rangle$ to a basis $\langle v, Tv, v_3,v_4,\ldots,v_n\rangle$. By assumption $T$ has the same matrix represen... |
2,532,280 | <p>If a N×N (N≥3) Hermitian matrix <strong>A</strong> meets the following conditions: </p>
<ol>
<li><strong>A</strong> is positive semi-definite (not positive definite, i.e. <strong>A</strong> has at least M zero eigenvalue, where M is a given paremeter with 1≤M≤N-1).</li>
<li>The sum of each off diagonal results in 0... | gen-ℤ ready to perish | 347,062 | <p>$$
Q(x) = 4x^2 + (5k+3)x + \left(2k^2-1\right) = 0 \\
Q(x) = Ax^2 + Bx + C = 0 \\
$$</p>
<p>For the zeroes of the quadratic $Q(x)$ to be the same in magnitude but opposite in sign, then $Q(x)$ must be symmetrical about the axis $x=0$. The key here is to know that the axis of symmetry of a parabola is $x=-b/(2a)$.</... |
293,341 | <p>My apologies if this question is more appropriate for mathisfun.com, but I can only get so far reading about combinatrics and set theory before the interlocking logic becomes totally blurred. If this is a totally fundamental concept, feel free just to name it so I can read and understand the math myself.</p>
<p>So ... | Red Banana | 25,805 | <p>It is a good thing to try different books, in my experience as a self-learner I found that a lot of traditionally aclaimed books are incredibly hard, there's always an author that can help you to grasp core ideas easily, for example, in calculus I read a little of the <a href="https://rads.stackoverflow.com/amzn/cli... |
2,155,652 | <p>I have a question regarding this proof my professor gave us. For the third property, I understand the proof up to the sentence "If $x \in E'$, i.e., x is a limit point of E."
Well, I also understand that if x is not in F, then x can't be a limit point since F is closed.
After that, I don't fully understand it. Could... | fleablood | 280,126 | <p>Before we even start, notice if $E\subset F $ means $E' \subset F'$ because....</p>
<p>If every neighborhood of $x $ contains a point of $E $ that very same point of E is also a point of $F $ so every neighborhood of $x $ contains a point of $F $.</p>
<p>Now that comment you don't understand (and to tell the truth... |
2,414,011 | <p>In my recent works in PDEs, I'm interested in finding a family of cut-off functions satisfying following properties:</p>
<p>For each $\varepsilon >0$, find a function ${\psi _\varepsilon } \in {C^\infty }\left( \mathbb{R} \right)$ which is a non-decreasing function on $\mathbb{R}$ such that:</p>
<ol>
<li>${\psi... | username | 948,485 | <p>Take the construction proposed <a href="https://math.stackexchange.com/a/4365567/948485">here</a> and use <span class="math-container">$\psi_\epsilon(x)=f(\frac x\epsilon)$</span>. Then, <span class="math-container">$\psi^\prime_\epsilon $</span> is supported on <span class="math-container">$(\epsilon,2\epsilon)$</s... |
809,516 | <p>I need to calculate </p>
<p>$$\lim_{x \to \infty} \frac{((2x)!)^4}{(4x)! ((x+5)!)^2 ((x-5)!)^2}.$$</p>
<p>Even I used Striling Approximation and Wolfram Alpha, they do not help.</p>
<p>How can I calculate this?</p>
<p>My expectation of the output is about $0.07$.</p>
<p>Thank you in advance.</p>
| Leucippus | 148,155 | <p>The limit as given in the problem is equal to zero. This is shown by the following. </p>
<p>Using $\Gamma(1+x) = x \Gamma(x)$ the expression to evaluate is seen as
\begin{align}
\phi_{n} &= \frac{\Gamma^{4}(2n+1) }{ \Gamma(4n+1) \Gamma^{2}(n+6) \Gamma(n-4)} \\
&= \frac{n^{2}(n-1)^{2}(n-2)^{2}(n-3)^{2}(n-4)^... |
324,385 | <p>I'm going through Wallace Clarke Boyden's <a href="http://books.google.com/books?id=OhMAAAAAYAAJ&pg=PA71#v=onepage&q&f=false" rel="noreferrer">A First Book in Algebra</a>, and there's a section on finding the square root of a perfect square polynomial, eg. <span class="math-container">$4x^2-12xy+9y^2=(2x... | Steven Alexis Gregory | 75,410 | <p>If you know that the polynomial is a perfect square, then the square root algorithm works. For example</p>
<hr>
<p>$$\sqrt{x^6 - 6x^5 + 17x^4 - 36x^3 + 52x^2 - 48x + 36}$$</p>
<hr>
<p>\begin{array}{lcccccccccccccc}
&&x^3 && -3x^2 && +4x && -6\\
&&---&---&---&--... |
3,376,443 | <p>A bin has 2 white balls and 3 black balls. You play a game as follows: you draw balls one at a time without replacement. Every time you draw a white ball , you win a dollar, but every time you draw a black ball , you loose a dollar . You can stop the game at any time.Devise a strategy for playing this game which res... | Sasha Kozachinskiy | 547,528 | <p>Assume that we have <span class="math-container">$a$</span> white balls and <span class="math-container">$b$</span> blacks balls. We can choose between two things: to play or not to play. In the first case our profit is <span class="math-container">$0$</span>. Now assume that we choose to play. With probability <sp... |
3,376,443 | <p>A bin has 2 white balls and 3 black balls. You play a game as follows: you draw balls one at a time without replacement. Every time you draw a white ball , you win a dollar, but every time you draw a black ball , you loose a dollar . You can stop the game at any time.Devise a strategy for playing this game which res... | A.J. | 654,406 | <p>There may be a more elegant and/or general way to do this, but here's a brute-force approach.</p>
<p>Consider the following strategy: Draw balls until you've drawn more white balls than black, or if that's no longer possible, until you've drawn both white balls.</p>
<p>Under this strategy, only the following outco... |
3,376,443 | <p>A bin has 2 white balls and 3 black balls. You play a game as follows: you draw balls one at a time without replacement. Every time you draw a white ball , you win a dollar, but every time you draw a black ball , you loose a dollar . You can stop the game at any time.Devise a strategy for playing this game which res... | leonbloy | 312 | <p>Draw the possible paths. The rectangles represent the number of white/black balls. Transitions to the left correspond to a white ball extracted (plus one). The numbers in blue are the probabilities. </p>
<p>It's clear that when we have zero white balls we should stop and when we have zero black balls we should cont... |
2,094,596 | <p>I'm questioning myselfas to why indeterminate forms arise, and why limits that apparently give us indeterminate forms can be resolved with some arithmetic tricks. Why $$\begin{equation*}
\lim_{x \rightarrow +\infty}
\frac{x+1}{x-1}=\frac{+\infty}{+\infty}
\end{equation*} $$</p>
<p>and if I do a simple operation,</... | Ben Grossmann | 81,360 | <p>Really, this has to do with the definition of continuity. The function $Q(x,y) = x/y$ is continuous except at $y = 0$. Thus, whenever $f(t) \to L_f$ and $g(t) \to L_g \neq 0$, we have
$$
\lim_{t \to a} \frac{f(t)}{g(t)} = \lim_{t \to a} Q(f(t),g(t)) = \lim_{(x,y) \to (L_f,L_g)}Q(x,y) = Q(L_f,L_g)
$$
However, $Q$ ... |
715,361 | <p>Let $\Omega$ be a bounded domain and $f_n\in L^2(\Omega)$ be a sequence such that
$$\int_\Omega f_nq\operatorname{dx}\leq C<\infty\qquad \text{for all}\quad q\in H^1(\Omega),\ \|q\|_{H^1(\Omega)}\leq1,\ n\in\mathbb{N}.\quad (1) $$
Is it then possible to conclude that
$$ \sup_{n\in\mathbb{N}}\|f_n\|_{L^2(\Omega)}... | Mark Bennet | 2,906 | <p>Here's another way to look at it. You can also recast the equation as follows, without cancelling anything or multiplying or dividing by anything which might be zero: $$0=\frac {x-4}{x-1}-\left(\frac {1-4}{x-1}\right)=\frac {x-1}{x-1} $$Now do you see what is going on?</p>
|
262,173 | <p>Consider $x^2 + y^2 = r^2$. Then take the square of this to give $(x^2 + y^2)^2 = r^4$. Clearly, from this $r^4 \neq x^4 + y^4$. </p>
<p>But consider: let $x=a^2, y = b^2 $and$\,\,r = c^2$. Sub this into the first eqn to get $(a^2)^2 + (b^2)^2 = (c^2)^2$. $x = a^2 => a = |x|,$ and similarly for $b.$</p>
<p>Now ... | Did | 6,179 | <p>The fact that $x=a^2$ is quite far to imply that $a=|x|$ (second paragraph).</p>
|
716,036 | <blockquote>
<p>Suppose that a curve $\mathbf\gamma$ in $\mathbb R^3$ has constant strictly positive curvature function $\mathbf\kappa(s)$, and constant non-zero torsion function $\mathbf\tau(s)$. Prove that the curve is a helix.</p>
</blockquote>
<p>I think it is easier to work backward here. First I can show that ... | Ted Shifrin | 71,348 | <p>You have it. If $\kappa=a/c$ and $\tau=b/c$, where $c=\sqrt{a^2+b^2}$, then your curve is congruent to (differs by a rigid motion from) the circular helix $\alpha(t)=(a\cos t, a\sin t, bt)$. Given $\kappa$ and $\tau$, you can determine $a$ and $b$ by algebra.</p>
|
1,001,320 | <p>I was wondering how to do an inequality problem involving QM-AM-GM-HM.</p>
<p>Question: For positive $a$, $b$, $c$ such that $\frac{a}{2}+b+2c=3$, find the maximum of $\min\left\{ \frac{1}{2}ab, ac, 2bc \right\}$.</p>
<p>I was thinking maybe apply AM-GM, however, I'm not sure what to plug in. Any help would be app... | Community | -1 | <p><strong>Hint:</strong></p>
<p>$$\frac{\frac{a}{2}+b}{2}\ge\sqrt{\frac{ab}{2}}\iff \left(\frac{\frac{a}{2}+b}{2}\right)^2\ge\frac{ab}{2}$$</p>
<p>$$\frac{2c+b}{2}\ge\sqrt{2bc}\iff \left(\frac{2c+b}{2}\right)^2 \ge 2bc$$</p>
<p>$$\frac{\frac{a}{2}+2c}{2}\ge\sqrt{ac}\iff \left(\frac{\frac{a}{2}+2c}{2}\right)^2\ge ac... |
59,828 | <p>Is there a way to display the variable name instead of its value? for example, I need something like<code>varname = 1; function[varname];</code> and the output is <code>varname</code> instead of <code>1</code></p>
| eldo | 14,254 | <pre><code>varname = 1;
SetAttributes[ShowName, HoldAll]
ShowName[name_] :=
Row[{"The name is ", HoldForm @ varname, " and its value is ", ReleaseHold @ varname}]
ShowName @ varname
</code></pre>
<blockquote>
<p>The name is varname and its value is 1</p>
</blockquote>
<p>Or simply</p>
<pre><code>HoldForm @ var... |
4,513,368 | <p>The following question seems to be quite simple, but I am having a hard time to prove it rigorously.</p>
<p>Consider <span class="math-container">$n\in\mathbb{N}$</span> vertices, for example <span class="math-container">$\{v_1,\ldots, v_n\}$</span>. I have some further information on these vertices, namely, that an... | User5678 | 632,875 | <p>A graph <span class="math-container">$G:=(E,V)$</span> is connected if there exists a path from any node to any other node in the graph.</p>
<p>The main property <span class="math-container">$P$</span>of the graph <span class="math-container">$G$</span> you are looking at is that every vertex is associated with at l... |
1,022,380 | <p>in below link, (formula (34)-(40)) there are some definition of Dirac delta function in terms of other functions such as Airy function, Bessel function of the first kind, Laguerre polynomial,....</p>
<p><a href="http://mathworld.wolfram.com/DeltaFunction.html" rel="nofollow noreferrer">http://mathworld.wolfram.com/D... | Ross Millikan | 1,827 | <p>The delta "function" $\delta(x)$ is supposed to be zero as $x$ gets large in either direction, so basing one on $\cosh(x)$ is hard because $\cosh(x) \to +\infty$ as $x$ gets large in either direction. That makes $\operatorname{sech} (x)$ a good candidate. Since $\int_{-\infty}^{+\infty} \operatorname{sech} x dx=\... |
494,227 | <blockquote>
<p>If I had $6$ feet of fencing could I fence a region that has area $3$
square feet?</p>
</blockquote>
<p>So, I must show that there is a curve in the plane of my fencing that has length $6$ feet that bounds the region of area. </p>
<p>How can I prove this? </p>
| user71352 | 71,352 | <p>Recall that the isoperimetric inequality states that the length of a closed curve $L$ and the enclosed area $A$ satisfy $4\pi A\le L^{2}$.</p>
<p>So if you could find such a curve in the plane then by the isoperimetric inequality $12\pi=4\pi(3)\le6^{2}=36$.</p>
<p>But $\pi>3$ so $12\pi>36$. Contradiction. H... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.