qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,502,309 | <p>The initial notation is:</p>
<p>$$\sum_{n=5}^\infty \frac{8}{n^2 -1}$$</p>
<p>I get to about here then I get confused.</p>
<p>$$\left(1-\frac{3}{2}\right)+\left(\frac{4}{5}-\frac{4}{7}\right)+...+\left(\frac{4}{n-3}-\frac{4}{n-1}\right)+...$$</p>
<p>How do you figure out how to get the $\frac{1}{n-3}-\frac{1}{n-... | Jeevan Devaranjan | 220,567 | <p>Factor $n^2 - 1$ as a difference of squares. Then you will end up with the fraction $\frac{8}{(n-1)(n+1)}$. You can now use partial fractions to solve the problem.</p>
|
430,654 | <p>Show that this sequence converges and find the limit.
$a_1 = 0$, $a_{n+1} = \sqrt{5+2a_{n} }$ </p>
| Samrat Mukhopadhyay | 83,973 | <p>To show that the sequence $\{a\}_{n\geq 1}$ does converge to some limit $L$, I shall show that actually, the function $f:\mathbb{R^+}\rightarrow \mathbb{R^+}$ such that $f(x)=\sqrt{5+2x}$, is a contraction on $\mathbb{R}^+$ and hence it has at least one fixed point $L>0$. Then by the argument @Brian M. Scott prov... |
5,897 | <p>The following creates a button to select a notebook to run. When the button is pressed it seems that Mathematica finds the notebook but cannot evaluate it. The following error occurs</p>
<blockquote>
<p>Could not process unknown packet "1"</p>
</blockquote>
<pre><code>Button["run file 1",
NotebookEvaluate[... | Mr.Wizard | 121 | <p>In the comments celtschk suggested <code>Button[..., Method -> "Queued"]</code> and Christina confirmed it as a solution.</p>
|
467,301 | <p>I'm reading Intro to Topology by Mendelson.</p>
<p>The problem statement is in the title.</p>
<p>My attempt at the proof is:</p>
<p>Since $X$ is a compact metric space, for each $n\in\mathbb{N}$, there exists $\{x_1^n,\dots,x_p^n\}$ such that $X\subset\bigcup\limits_{i=1}^p B(x_i^n;\frac{1}{n})$. Let $K=\frac{2p}... | Norbert | 19,538 | <p>Fix $p\in X$, then the function $f:X\to\mathbb{R}_+:x\mapsto d(x,p)$ is a continous function on a compact space $X$. Hence it is bounded, i.e. there exist $K>0$ such that for all $x\in X$ holds $d(x,p)\leq K/2$. Now take arbitrary $x,y\in X$, then
$$
d(x,y)\leq d(x,p)+d(p,y)\leq K
$$</p>
|
130,806 | <p><strong>Qusestion:</strong> Let f be a continuous and differentiable function on $[0, \infty[$, with $f(0) = 0$ and such that $f'$ is an increasing function on $[0, \infty[$. Show that the function g, defined on $[0, \infty[$ by $$g(x) = \begin{cases} \frac{f(x)}{x}, x\gt0& \text{is an increasing function.... | chemeng | 25,845 | <p>With differentiation we get:$$g'(x)=\frac{f'(x)*x-f(x)}{x^2}$$ We want to show that $g'(x)>0 \ \Leftrightarrow \ f'(x)*x-f(x) > 0 \ (1)\ \forall \ x > 0 $
Reforming (1) we got: $$f'(x) > \frac{f(x)}{x} \ (2)$$ Applying the mean value theorem at [0,x], we got $f'(c)=\frac{f(x)}{x}$, $ 0<... |
3,352,691 | <blockquote>
<p>Determine the greatest possible value of <span class="math-container">$$\sum_{i=1}^{10}{\cos 3x_i}$$</span>
for real numbers <span class="math-container">$x_1,x_2....x_{10}$</span> satisfying
<span class="math-container">$$\sum_{i=0}^{10}{\cos x_i}=0$$</span></p>
</blockquote>
<p><strong>My att... | Allawonder | 145,126 | <p><em>Hint:</em> Use the fact that <span class="math-container">$$\cos a + \cos b=2\cos\left(\frac12(a+b)\right)\cos\left(\frac12(a-b)\right).$$</span></p>
<p>If you pair the summands and apply above transformation, then the sum becomes a product with <span class="math-container">$10$</span> cosine factors, and a sca... |
3,352,691 | <blockquote>
<p>Determine the greatest possible value of <span class="math-container">$$\sum_{i=1}^{10}{\cos 3x_i}$$</span>
for real numbers <span class="math-container">$x_1,x_2....x_{10}$</span> satisfying
<span class="math-container">$$\sum_{i=0}^{10}{\cos x_i}=0$$</span></p>
</blockquote>
<p><strong>My att... | Community | -1 | <p><strong>Visualising the solution</strong></p>
<p>You have asked for help in visualising the solution. I think you will find it useful to have in mind the picture of <span class="math-container">$y=x^3$</span> for <span class="math-container">$-1\le x\le1$</span>.</p>
<p>Now consider the arrangement of the 10 numbe... |
4,638,490 | <p>Given functions f and g, as above, what exactly does it mean? Does it mean, for example, that g(n) is <em>exactly</em> equal to <span class="math-container">$2^{h(n)}$</span> for some function h contained in <span class="math-container">$O(f(n))$</span> - or does it rather mean that <span class="math-container">$g(n... | Zachary | 433,146 | <p>It means that for all large enough <span class="math-container">$N$</span>, there exists <span class="math-container">$C>0$</span> such that
<span class="math-container">$$g(n)\le 2^{Cf(n)}.$$</span>
Alternatively, it means that
<span class="math-container">$$\log g(n)=O(f(n)).$$</span></p>
|
280,156 | <p>I have a code as below:</p>
<pre><code>countpar = 10;
randomA = RandomReal[{1, 10}, {countpar, countpar}];
randomconst = RandomInteger[{0, 1}, {countpar, 1}];
For[i = 1, i < countpar + 1, i++,
If[randomconst[[i, 1]] != 0,
randomA[[All, i]] = 0.; randomA[[i, All]] = 0.;
randomA[[i, i]] = 1;
];
];
</code></pre... | Nasser | 70 | <blockquote>
<p>no no, I want 3/2 as the answer. I just want it to print the answers
3/2, 7/5, 17/12, 41/29...</p>
</blockquote>
<p>One of 10 possible ways</p>
<pre><code>y = 1;
x = 2;
n = 2;
a := (y^(x - 1) + n)/(y^(x - 1) + y^(x - 2))
Last@Reap@Do[
Sow[a];
y = a
, {m, 10}
]
</code></pre>
<p><img src="htt... |
280,156 | <p>I have a code as below:</p>
<pre><code>countpar = 10;
randomA = RandomReal[{1, 10}, {countpar, countpar}];
randomconst = RandomInteger[{0, 1}, {countpar, 1}];
For[i = 1, i < countpar + 1, i++,
If[randomconst[[i, 1]] != 0,
randomA[[All, i]] = 0.; randomA[[i, All]] = 0.;
randomA[[i, i]] = 1;
];
];
</code></pre... | Syed | 81,355 | <pre><code>Clear["Global`*"]
y = 1;
x = 2;
n = 2;
NestList[(#^(x - 1) + n)/(#^(x - 1) + #^(x - 2)) &, y, 6]
</code></pre>
<hr />
<p><strong>EDIT</strong></p>
<p>If the same definition must be used and kept, then define:</p>
<pre><code>Clear["Global`*"]
y = 1;
x = 2;
n = 2;
a[y_] := (y^(x - 1) +... |
483,442 | <p>I am trying to learn about velocity vectors but this word problem is confusing me.</p>
<p>A boat is going 20 mph north east, the velocity u of the boat is the durection of the boats motion, and length is 20, the boat's speed. If the positive y axis represents north and x is east the boats direction makes an angle o... | user84413 | 84,413 | <p>Select your first sock. Now you have 15 choices left for your 2nd sock, and 14 of them will allow you to avoid getting a pair. </p>
<p>For your 3rd sock, you have a total of 14 choices remaining, and you can choose any sock other than the first two chosen and their mates to avoid a pair, so you have 12 choices to... |
182,527 | <p>I have the following question:</p>
<p>Let $X$: $\mu(X)<\infty$, and let $f \geq 0$ on $X$. Prove that $f$ is Lebesgue integrable on $X$ if and only if $\sum_{n=0}^{\infty}2^n \mu(\lbrace x \in X : f(x) \geq 2^n \rbrace) < \infty $.</p>
<p>I have the following ideas, but am a little unsure. For the forward d... | Makoto Kato | 28,422 | <p>Let $E_n = \lbrace x \in X : 2^n \leq f(x) < 2^{n+1} \rbrace$.
Let $F = \lbrace x \in X : f(x) < 1 \rbrace$.
Then $X$ is a disjoint union of $F$ and $E_n$, $n = 0,1,\dots$.
Hence $\int_X f(x) d\mu = \int_F f(x) d\mu + \sum_{n=0}^{\infty} \int_{E_n} f(x) d\mu$.</p>
<p>Let $G_n = \lbrace x \in X : f(x) \geq 2^n... |
182,527 | <p>I have the following question:</p>
<p>Let $X$: $\mu(X)<\infty$, and let $f \geq 0$ on $X$. Prove that $f$ is Lebesgue integrable on $X$ if and only if $\sum_{n=0}^{\infty}2^n \mu(\lbrace x \in X : f(x) \geq 2^n \rbrace) < \infty $.</p>
<p>I have the following ideas, but am a little unsure. For the forward d... | Community | -1 | <p><strong>Hint:</strong> Switching the order of integration gives $$\int_X f(x)\,\mu(dx)=\int_X \int_0^{f(x)}\,dt\,\mu(dx)
=\int_0^\infty \mu(x: f(x)>t)\,dt$$
This equation is true whether the two sides are finite or infinite. </p>
<p>Since $G(t):= \mu(x: f(x)>t)$ is decreasing, we have
$$G(2^{n+1}) \int_{... |
182,527 | <p>I have the following question:</p>
<p>Let $X$: $\mu(X)<\infty$, and let $f \geq 0$ on $X$. Prove that $f$ is Lebesgue integrable on $X$ if and only if $\sum_{n=0}^{\infty}2^n \mu(\lbrace x \in X : f(x) \geq 2^n \rbrace) < \infty $.</p>
<p>I have the following ideas, but am a little unsure. For the forward d... | tomasz | 30,222 | <p>The fact that the summation is infinite does not mean that it does not converge, consider $f(x)=\frac{1}{\sqrt x}$ on $[0,1]$. It is integrable, yet clearly the summation is not finite in this case: it is $\sum_n 2^n\cdot 2^{-2n}=2<\infty$.</p>
<p>Put $A_n:=\{ x\vert f(x)\geq 2^n\},B_n:=A_n\setminus A_{n+1},a_n=... |
1,384,947 | <p>I'm trying to figure out when numbers reach "periodicity" given known values. I've included an example below with image:</p>
<p>I have known sizes (<em>100, 75, and 50</em>) that I would like to know how many times I would need to repeat each item for all the sizes to line up or be periodic. Does anyone know of a... | Narasimham | 95,860 | <p>Let's say you posted like this:</p>
<p>~~~~~</p>
<p>for which </p>
<p>$$\cos(\phi)=\frac{a}{\sqrt{a^2+b^2}};$$<br>
$$\sin(\phi)=\frac{b}{\sqrt{a^2+b^2}}$$
Using this, let's transform our equation to</p>
<p>$$\cos(\phi)\sin(x)+\sin(\phi)\cos(x)=\frac{c}{\sqrt{a^2+b^2}}$$</p>
<p>This will bring us to</p>
<p>$$ \... |
2,206,938 | <p>Context: <a href="http://www.hairer.org/notes/Regularity.pdf" rel="nofollow noreferrer">http://www.hairer.org/notes/Regularity.pdf</a>, section 4.1 (pages 15-16)</p>
<blockquote>
<p>Define
$$(\Pi_x\Xi^0)(y)=1 \qquad (\Pi_x\Xi)(y)=0 \qquad (\Pi_x\Xi^2)(y)=c$$
and
$$(\Pi^{(n)}_x\Xi^0)(y)=1 \qquad (\Pi^{(n)}_x... | zhw. | 228,045 | <p>Hint: Try something like</p>
<p>$$f(x) = \sum_{n=1}^{\infty}c_n x^{1/n}$$</p>
<p>for suitable positive constants $c_n.$</p>
|
2,555,463 | <p>Given a line $l$ and two points $p_1$ and $p_2$, identify the point $v$ which is equidistant from $l$, $p_1$, and $p_2$, assuming it exists.</p>
<p>My idea is to: (1) identify the parabolas containing all points equidistant from each point and the line, then (2) intersect these parabolas. As $v$ is equidistant from... | Community | -1 | <p>Without loss of generality, $l$ is the $x$ axis, $p_1$ is at $(-a,y_1)$ and $p_2$ at $(a,y_2)$.</p>
<p>One of the parabolas is $$y^2=(x+a)^2+(y-y_1)^2$$ and the other</p>
<p>$$y^2=(x-a)^2+(y-y_2)^2.$$</p>
<p>After elimination of $y$,</p>
<p>$$y_2(x+a)^2-y_1(x-a)^2+y_1y_2(y_1-y_2)=0$$ can give you two solutions.<... |
2,555,463 | <p>Given a line $l$ and two points $p_1$ and $p_2$, identify the point $v$ which is equidistant from $l$, $p_1$, and $p_2$, assuming it exists.</p>
<p>My idea is to: (1) identify the parabolas containing all points equidistant from each point and the line, then (2) intersect these parabolas. As $v$ is equidistant from... | Anatoly | 90,997 | <p>An approach that could be useful not only to provide the solutions, but also to discuss and understand their existence, is as follows. Given two points $P_1$ and $P_2$ and a line, we can set, without loss of generality, an $xy$ plane where the $x$-axis coincides with the line and the non-negative portion of the $y$... |
2,555,463 | <p>Given a line $l$ and two points $p_1$ and $p_2$, identify the point $v$ which is equidistant from $l$, $p_1$, and $p_2$, assuming it exists.</p>
<p>My idea is to: (1) identify the parabolas containing all points equidistant from each point and the line, then (2) intersect these parabolas. As $v$ is equidistant from... | smichr | 122,921 | <p>@anatoly has a nice answer which identifies conditions when a solution is expected. But there is an additional condition which, if not satisfied, prohibits a solution. @nominal-animal elucidates all conditions. I add an additional answer here only to show that using a different orientation of points and lines gives ... |
45,211 | <p>Instances of SAT induce a bipartite graph between clauses vertices and variable vertices, and for planar 3SAT, the resulting bipartite graph is planar. </p>
<p>It would be very convenient if there was a planar layout that had all the variable vertices in one line and all the clause vertices in a straight line. This... | David Eppstein | 440 | <p>Any planar graph can be drawn with curves for the edges and its vertices in any position in the plane.</p>
<p>But with straight line segment edges, it's not always possible, even for graphs in which every vertex in A has degree exactly two, and even if you relax the straight-line requirement for A and only require ... |
1,482,205 | <p>Show that $\sigma(AB) \cup \{0\} = \sigma(BA) \cup \{0\}$ in general, and that $\sigma(AB) = \sigma(BA)$ if $A$ is bijective. </p>
<p>I studied the associative statement of this somewhere but it did not include the zeroth element.
If you assume the bijection, how can you show the first part?</p>
<h2>My attempt</h... | fleablood | 280,126 | <p>"...that simplified is b/b+a/b. So then I added those two fractions together and got ab/b .."</p>
<p>That shouldn't make any sense and if I saw your paper I may understand why you did it. But it's simply wrong.</p>
<p>$1 + \frac ab$</p>
<p>$\frac{1b}{1b} + \frac ab$</p>
<p>$\frac{b}{b} + \frac ab$ You are righ... |
1,107,250 | <p><img src="https://i.stack.imgur.com/ILg7L.png" alt="enter image description here"></p>
<p>In above lemma, why $|a'| \leq 1$ still holds? I didn't see how it relates to "algebraic conjugate of a root of unity is also a root of unity", since $a$ is the sum of unity.</p>
<p>(definition of algebraic conjugate: <a href... | David Zhang | 80,762 | <p>@Michael has already found a very elegant answer, but I would like to contribute an alternate approach that I stumbled across. It turns out that the polynomials $p_n$ are precisely the degree $2n+1$ Bernstein polynomial approximants of $u(x-1/2)$, where $u$ is the unit step function (and $u(0) = 1/2$). These are wel... |
3,285,255 | <p>Show <span class="math-container">$D = \{f \in C^{2}[0,1]): f(x) > 0, \ \forall x \in [0,1], \|f'\|_{\infty}<1, |f''(0)| > 2\}$</span> is open w.r.t Sup Norm.</p>
<p>Sup Norm = <span class="math-container">$\|f\|_{2,\infty, [0,1]} = sup_{x \in[0,1]}|f(x)| + sup_{x\in[0,1]}|f'(x)| + sup_{x \in [0,1]}|f''(x)... | zhw. | 228,045 | <p>You wrote "This condition means there exists a <span class="math-container">$\delta_{3} > 0 $</span> s.t. <span class="math-container">$f''(0) > \delta_{3} > 2$</span>." You're missing the negative case, e.g., <span class="math-container">$f''(0)=-3$</span></p>
<p>I would let <span class="math-container">$... |
3,285,255 | <p>Show <span class="math-container">$D = \{f \in C^{2}[0,1]): f(x) > 0, \ \forall x \in [0,1], \|f'\|_{\infty}<1, |f''(0)| > 2\}$</span> is open w.r.t Sup Norm.</p>
<p>Sup Norm = <span class="math-container">$\|f\|_{2,\infty, [0,1]} = sup_{x \in[0,1]}|f(x)| + sup_{x\in[0,1]}|f'(x)| + sup_{x \in [0,1]}|f''(x)... | mechanodroid | 144,766 | <p>Alternatively, notice that <span class="math-container">$\phi, \psi : C^2[0,1] \to \mathbb{R}$</span> given by <span class="math-container">$\phi(f)
= \|f'\|_\infty$</span> and <span class="math-container">$\psi(f) = |f''(0)|$</span> are continuous functions w.r.t the given norm on <span class="math-container">$C^2... |
192,821 | <p>I am using <a href="https://reference.wolfram.com/language/ref/TransformedField.html" rel="noreferrer"><code>TransformedField</code></a> to convert a system of ODEs from Cartesian to polar coordinates:</p>
<pre><code>TransformedField[
"Cartesian" -> "Polar",
{μ x1 - x2 - σ x1 (x1^2 + x2^2), x1 + μ x2 - σ x2... | Bill Watts | 53,121 | <p>My slightly different method matches Mathematica.</p>
<pre><code>aCartToCyl[{ax_, ay_}] := {ax Cos[ϕ] + ay Sin[ϕ], ay Cos[ϕ] - ax Sin[ϕ]}
aCartToCyl[{μ x1 - x2 - σ x1 (x1^2 + x2^2),
x1 + μ x2 - σ x2 (x1^2 + x2^2)}] // Simplify;
% /. {x1 -> r Cos[ϕ], x2 -> r Sin[ϕ]} // Simplify
(*{μ r - r^3 σ, r}*)
</cod... |
3,429,623 | <p>Is the union of <span class="math-container">$\emptyset$</span> with another set, <span class="math-container">$A$</span> say, disjoint? Even though <span class="math-container">$\emptyset \subseteq A$</span>?</p>
<p>I would say, yes - vacuously. But some confirmation would be great.</p>
| Levi | 513,190 | <p>It might depend on what you mean by disjoint. I would say that the following definition is reasonable.</p>
<blockquote>
<p><strong>Definition.</strong> Sets <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are <em>disjoint</em> if <span class="math-container">$A \cap B = \emptys... |
545,634 | <p>Consider a function $f:\Omega\subset\mathbb{R}^n\rightarrow\mathbb{R}^m$, for which the Jacobian matrix </p>
<p>$J_f(x_1,...,x_n)= \left( \begin{array}{ccc}
\frac{\partial f_1}{\partial x_1} & ... & \frac{\partial f_1}{\partial x_n} \\
\vdots & & \vdots \\
\frac{\partial f_m}{\partial x_1} & .... | leonbloy | 312 | <p>Actually, the function is one-to-one on the relevant domain. And the image (and hence the range of $Y$) is $(1/4,1] \cup [0,1/4]=[0,1]$</p>
<p>Hence, you can simply apply the formula $$f_Y(y)= \frac{f_X(x)}{|g'(x)|}=\frac{1}{2 |x-1|}$$</p>
<p>But $|x-1|= \sqrt{y}$</p>
<p>Hence $$f_Y(y) = \frac{1}{2 \sqrt{y}} , \... |
1,872,136 | <p>Suppose $a$ and $b$ are integers. Is there a closed form for the following sum?</p>
<p>$$F(x,a,b)=\sum_{n=0}^{\infty} e^{-x (n+1/2+\sin(\frac{a}{b} \pi (n+1/2)))}$$</p>
| Robert Israel | 8,508 | <p>For each particular $b$, it will have a closed form: since the $\sin$ term depends only on $n \mod (2b)$, this reduces to the sum of $2b$ geometric series. I doubt that there is a closed form expression as a function of $a$ and $b$.</p>
|
1,872,136 | <p>Suppose $a$ and $b$ are integers. Is there a closed form for the following sum?</p>
<p>$$F(x,a,b)=\sum_{n=0}^{\infty} e^{-x (n+1/2+\sin(\frac{a}{b} \pi (n+1/2)))}$$</p>
| Simply Beautiful Art | 272,831 | <p>We can approximate in the following manner:</p>
<p>$$F(x)\approx\sum_{n=0}^\infty e^{-x(n+1/2)}=\frac{e^{-x/2}}{1-e^{-x}}$$</p>
|
2,131,679 | <p>Let $f: \mathbb{R} \to \mathbb{R}$ be continuous and $D \subset \mathbb{R}$ be a dense subset of $\mathbb{R}$. Furthermore, $\forall y_1,y_2 \in D \ f(y_1)=f(y_2)$. Should $f$ be a constant function?</p>
<p>My attempt:
Since $f$ is continuous
$$\forall x_0 \ \forall \varepsilon >0 \ \exists \delta>0 \ \forall... | quasi | 400,434 | <p>Let $F$ be the point where the line $AO$ intersects side $BC$.
<p>
Applying Ceva's Theorem,</p>
<p>\begin{align*}
&\frac{AE}{EB}\cdot\frac{BF}{FC}\cdot\frac{CD}{DA} = 1\\[6pt]
\implies\;&\frac{2}{1}\cdot\frac{BF}{FC}\cdot\frac{2}{1} = 1\\[6pt]
\implies\;&\frac{BF}{FC}=\frac{1}{4}
\end{align*}</p>
<p>Wi... |
865,293 | <blockquote>
<p>Prove $\ln[\sin(x)] \in L_1 [0,1].$</p>
</blockquote>
<p>Since the problem does not require actually solving for the value, my strategy is to bound the integral somehow. I thought I was out of this one free since for $\epsilon > 0$ small enough, $$\lim_{\epsilon \to 0}\int_\epsilon^1 e^{\left|\ln(... | Community | -1 | <p>A simpler approach would be to observe that the function $x^{1/2}\ln \sin x$ is bounded on $(0,1]$, because it has a finite limit as $x\to 0$ -- by L'Hôpital's rule applied to $\dfrac{\ln \sin x}{x^{-1/2}}$. This gives $|\ln \sin x|\le Mx^{-1/2}$.</p>
<hr>
<p>As Byron Schmuland noted, $e^{|\ln \sin x|} = 1/\sin... |
865,293 | <blockquote>
<p>Prove $\ln[\sin(x)] \in L_1 [0,1].$</p>
</blockquote>
<p>Since the problem does not require actually solving for the value, my strategy is to bound the integral somehow. I thought I was out of this one free since for $\epsilon > 0$ small enough, $$\lim_{\epsilon \to 0}\int_\epsilon^1 e^{\left|\ln(... | copper.hat | 27,978 | <p>Here is a proof that hides behind a theorem on swapping order of integration:</p>
<p>We have $0 \le {x \over 2} \le \sin x$, and $-\log$ is decreasing on $(0,1]$.</p>
<p>Then $\int_0^1 | \log(\sin x)| dx = \int_0^1 - \log( \sin x) dx \le \int_0^1 - \log( { x \over 2}) dx = \log 2 + \int_0^1 - \log( { x}) dx$.</p>... |
2,764,141 | <p>This is what I have tried so far: </p>
<p>Since $g(z)$ is bounded, then $\lim\limits_{z\rightarrow 0} zg(z)=0$ and hence $z=0$ is a removable singularity of $g(z)$. We can define $g(0) = \lim\limits_{z\rightarrow 0} f(z)f(\frac{1}{z})$ and make $g$ entire.</p>
<p>Then $g(z)$ is a bounded entire function and hence ... | mol3574710n0fN074710n | 55,485 | <p>Try $c\cdot f\left(z\right)^{-1} = f\left(z^{-1}\right)$:</p>
<p>$f$ is represented by its power series everywhere:
$f\left(z\right) = \sum_{n=0}^{\infty}\,c_n\cdot z^n$</p>
<p>From the first line we can conclude that only a single exponent occurs in that power series.</p>
|
3,858,962 | <p>given a rectangle <span class="math-container">$ABCD$</span> how to construct a triangle such that <span class="math-container">$\triangle X, \triangle Y$</span> and <span class="math-container">$\triangle Z$</span> have equal areas.i dont know where to start. .i tried some algebra with the area of the triangles an... | QED | 91,884 | <p><a href="https://i.stack.imgur.com/s1yGL.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/s1yGL.png" alt="enter image description here" /></a></p>
<p>Just make <span class="math-container">$$\frac{|DF|}{|FC|}=\frac{2}{\sqrt{5}+1},\ \frac{|AE|}{|EC|}=\frac{|FC|}{|CD|}=\frac{\sqrt{5}+1}{\sqrt{5}+3}$$... |
1,041,212 | <p>I want to show that the function $f(X) = -log \ det(X)$ is convex on the space $S$ of positive definite matrices. </p>
<p>What I have done:</p>
<p>It seems like this problem could be tackled by considering the restriction of $f(X)$ to a line through a given point $X \in S$ so that $g(\alpha) = f(X + \alpha V)$ for... | Community | -1 | <p>More generally, let $f(X)=-log(|\det(X)|)$, let $\Omega$ be a convex subspace of $GL_n(\mathbb{R})$ and, if $X\in\Omega$, let $T_X$ be the tangent space in $X$ to $\Omega$. If $f$ is convex over $\Omega$, then, for every $X\in\Omega,H\in T_X$, $f''_X(H,H)\geq 0$. The converse is true if $f''_X(H,H)>0$ when $H\not... |
1,660,794 | <p>Suppose $$a'(x)=b(x)$$ and $$b'(x)=a(x)$$</p>
<p>What is $$\int x \sin (x) a(x) dx$$</p>
<p>Thanks!</p>
| lhf | 589 | <p>You're talking about <a href="https://en.wikipedia.org/wiki/Polygonal_number" rel="nofollow">Polygonal numbers</a>.</p>
<p>The $n$-th $s$-gonal number is
$$
\frac{n^2(s-2)-n(s-4)}{2}
$$</p>
<p>If you have a number $N$ and want to see whether it is an $s$-gonal number, then you have to solve a quadratic equation in... |
1,994,277 | <p>I am studying linear representation theory for finite groups and came across the claim in title. When $n\geq 5$, $S_n$ does not have an irreducible, $2$- dimensional representation.But I am not sure where to begin with. </p>
<p>Although it seems that this result will follow from <a href="https://math.stackexchange.... | Dietrich Burde | 83,966 | <p>It is also possible to prove the following, using elementary methods of the above <a href="https://math.stackexchange.com/questions/69384/low-dimensional-irreducible-representations-of-s-n">link</a>:</p>
<blockquote>
<p>For $n \geq 5$, the only representations of $S_n$ of dimension $<n$ are direct sums of <br>... |
113,446 | <p>Suppose a simple equation in Cartesian coordinate:
$$
(x^2+ y^2)^{3/2} = x y
$$
In polar coordinate the equation becomes $r = \cos(\theta) \sin(\theta)$. When I plot both, the one in polar coordinate has two extra lobes (I plot the polar figure with $\theta \in [0.05 \pi, 1.25 \pi]$ so the "flow" of the curve is cle... | Narasimham | 19,067 | <p>$$ r = \pm \sqrt{x^2+y^2} =f(\theta)$$ </p>
<p>Basically when you entered into polar coordinates usage you had implicatively or unwittingly accepted that all radius vectors can be either positive or negative.</p>
<p>It is consequential to the above artefact, negative sign makes complete sense to <em>all</em> pola... |
72,201 | <p>Two people play a game. They play a series of points, each producing a winner and a loser, until one player has won at least 4 points and has won at least 2 more points than the other. Anne wins each point with probability p.
What is the probability that she wins the game on the kth point played for k=4,5,6,...</p>
... | joriki | 6,622 | <p>I think you've already got the right result for $k=4$. For higher $k$, note that the game has to go through a tie in order to continue. For instance, for someone to win 5:3, the score must have been 3:3 (since it must have been 4:3, and 4:2 would have ended the game). So you can calculate the probability for 3:3, wh... |
1,109,443 | <p>I'm currently learning for an algebra exam and I have some examples of questions from few last years. And I can't find a solution to this one:</p>
<blockquote>
<p>Give three examples of complex numbers where z = -z</p>
</blockquote>
<p>The only complex number I can think of is 0. Because it is a complex number, ... | VividD | 121,158 | <p>No, <strong>nobody can be $1/12th$ Cherokee</strong>.</p>
<p>I'll prove a stronger statement:</p>
<blockquote>
<p><em>For any natural $p$ and $q$, one can't be a $p/q$ Cherokee if $p$ <strong>is not</strong> divisible by $3$, and $q$ <strong>is</strong> divisible by $3$</em>.</p>
</blockquote>
<p>First, the sta... |
1,109,443 | <p>I'm currently learning for an algebra exam and I have some examples of questions from few last years. And I can't find a solution to this one:</p>
<blockquote>
<p>Give three examples of complex numbers where z = -z</p>
</blockquote>
<p>The only complex number I can think of is 0. Because it is a complex number, ... | gnasher729 | 137,175 | <p>You can get as close as you like to any fraction. For example, you can be between 1 / 12.5 and 1 / 11.5 Cherokee, and then claiming "I'm one 12th Cherokee" in every day language would be correct. </p>
<p>Also, the fractions that we usually use are approximations. If one parent is pure white and one parent is pure b... |
1,109,443 | <p>I'm currently learning for an algebra exam and I have some examples of questions from few last years. And I can't find a solution to this one:</p>
<blockquote>
<p>Give three examples of complex numbers where z = -z</p>
</blockquote>
<p>The only complex number I can think of is 0. Because it is a complex number, ... | Jay | 144,901 | <p>Assuming that we define your heritage as 1/2 (father + mother), then it is not possible to come up with exactly 1/12. Assuming we start with people who are full Cherokee or not Cherokee at all, i.e. 0 or 1, then all their descendants are going to be some integer over a power of 2. A power of 2 can never reduce to 12... |
1,109,443 | <p>I'm currently learning for an algebra exam and I have some examples of questions from few last years. And I can't find a solution to this one:</p>
<blockquote>
<p>Give three examples of complex numbers where z = -z</p>
</blockquote>
<p>The only complex number I can think of is 0. Because it is a complex number, ... | András Hummer | 62,202 | <p>Mathematically you can't. Let $a$ and $b$ mark the number of generations since the last time each of your parents had exactly one 100% Cherokee ancestor. Let's assume that</p>
<p>$\frac{\frac{1}{2^a} + \frac{1}{2^b}}{2} = \frac{1}{12}$</p>
<p>$\frac{1}{2^a} + \frac{1}{2^b} = \frac{1}{6}$</p>
<p>$\frac{6}{2^a} + \... |
4,469,733 | <p>When randomly selecting a kitten for adoption, there is a <span class="math-container">$23 \%$</span> chance of getting a black kitten, a <span class="math-container">$50 \%$</span> chance of getting a tabby kitten, a <span class="math-container">$7 \%$</span> chance of getting a calico kitten, and a <span class="ma... | River Li | 584,414 | <p>We have
<span class="math-container">\begin{align*}
\int_2^\infty \frac{\ln^2 t}{t(t - 1)}\,\mathrm{d} t
& \overset{t = 1/x} =
\int_0^{1/2} \frac{\ln^2 x}{1 - x} \,\mathrm{d} x \\
&\le \int_0^{1/2} \frac{\ln^2 x}{1 - 1/2} \,\mathrm{d} x\\
&= 2\int_0^{1/2} \ln^2 x \,\mathrm{d} x\\
&= \ln^2 2 + 2... |
97,261 | <p>This semester, I will be taking a senior undergrad course in advanced calculus "real analysis of several variables", and we will be covering topics like: </p>
<p>-Differentiability.
-Open mapping theorem.
-Implicit function theorem.
-Lagrange multipliers. Submanifolds.
-Integrals.
-Integration on surfaces.
-Stokes ... | Potato | 18,240 | <p>I like C.H. Edwards, "Advanced Calculus of Several Variables." It's cheap and contains many exercises and examples.</p>
|
526,627 | <p>What is the answer for factoring:</p>
<p>$$10r^2 - 31r + 15$$</p>
<p>I have tried to solve it. This was my prior attempt:</p>
<p>$$10r^2 - 31r + 15\\
= (10r^2 - 25r) (-6r + 15)\\
= -5r(-2r+5) -3 (2r-5) $$ </p>
| Omar | 95,182 | <p>$10r^2 -31r + 15$</p>
<p>what are the factors of 10: 1, 2,5, and 10. Then what are the factors of 15: 1, 3, 5, and 15. From here, there two ways factoring or using the foil method.</p>
<p>The factoring method.</p>
<p>$10r^2 -31r + 15$</p>
<p>$(10r^2 -25r)(-6r + 15)$</p>
<p>$5r(2r - 5) - 3(2r - 5)$ since we have... |
3,172,693 | <p>Can anybody help me with this equation? I can't find a way to factorize for finding a value of <span class="math-container">$d$</span> as a function of <span class="math-container">$a$</span>:</p>
<p><span class="math-container">$$d^3 - 2\cdot d^2\cdot a^2 + d\cdot a^4 - a^2 = 0$$</span></p>
<p>Another form:</p>
... | farruhota | 425,072 | <p>You can do it step-by-step:
<span class="math-container">$$\begin{align}\frac{1}{s^2(s+2)}&=\frac1s\cdot \frac{1}{s\cdot (s+2)}=\\
&=\frac1s\cdot \left(\frac As+\frac{B}{s+2}\right)=\\
&=\frac A{s^2}+\frac1s\cdot \frac B{s+2}=\\
&=\frac{A}{s^2}+\frac{C}{s}+\frac{D}{s+2}.\end{align}$$</span></p>
|
2,515,939 | <p>So, I just need a hint for proving
$$\lim_{n\to \infty} \int_0^1 e^{-nx^2}\, dx = 0$$ </p>
<p>I think maybe the easiest way is to pass the limit inside, because $e^{-nx^2}$ is uniformly convergent on $[0,1]$, but I'm new to that theorem, and have very limited experience with uniform convergence. Furthermore, I don... | Sangchul Lee | 9,340 | <p>There are many ways to prove the limit, and better inputs would lead to a better quantitative bounds.</p>
<ol>
<li><p>Let $\epsilon \in (0, 1)$. Since $x \mapsto e^{-nx^2}$ is decreasing on $[0, 1]$, we have</p>
<p>$$ 0 \leq \int_{0}^{1}e^{-nx^2} \,dx = \int_{0}^{\epsilon}e^{-nx^2} \,dx + \int_{\epsilon}^{1}e^{-nx... |
1,595,946 | <blockquote>
<p>Let $f:(a,b)\to\mathbb{R}$ be a continuous function such that
$\lim_\limits{x\to a^+}{f(x)}=\lim_\limits{x\to b^-}{f(x)}=-\infty$.
Prove that $f$ has a global maximum.</p>
</blockquote>
<p>Apparently, this is similar to the EVT and I believe the proof would be similar, but I cannot think anything... | Ruben | 299,730 | <p>By definition, $\lim_{x \rightarrow a} f(x) = -\infty$ means that $\forall m \in \mathbb{R}, \exists \epsilon_1 \ \text{s.t.}\ | x - a | < \epsilon_1 \implies f(x) < m$. Likewise, $\lim_{x \rightarrow b} f(x) = -\infty$ means that $\forall m \in \mathbb{R}, \exists \epsilon_2 \ \text{s.t.}\ | x - b | < \eps... |
2,329,542 | <p>I looked up wikipedia but honestly I could not make much sense of what I will basically study in Abstract Algebra or what it is all about .</p>
<p>I also looked up a question here :
<a href="https://math.stackexchange.com/questions/855828/what-is-abstract-algebra-essentially">What is Abstract Algebra essentially?</... | Pawel | 355,836 | <p>Abstract algebra "generalizes" the notion of numbers.</p>
<p>For example, we can add numbers. But we can also add apples. Also, the order of adding numbers does not matter. Similarly with adding apples. We can add zero to any number and nothing changes. We can also add zero apples and nothing changes.</p>
<p>In ot... |
2,329,542 | <p>I looked up wikipedia but honestly I could not make much sense of what I will basically study in Abstract Algebra or what it is all about .</p>
<p>I also looked up a question here :
<a href="https://math.stackexchange.com/questions/855828/what-is-abstract-algebra-essentially">What is Abstract Algebra essentially?</... | Jesse Madnick | 640 | <p>Abstract algebra is the study of <strong>operations</strong>.</p>
<p>As humans, we agree that $a + b = b + a$ makes intuitive sense. We take this rule as an axiom, building our human number system with this axiom as part of the foundation. But maybe <strong>aliens from Planet Zog</strong> believe that $a + b$ sho... |
3,828,003 | <blockquote>
<p>Show that <span class="math-container">$G=\{0,1,2,3\}$</span> over addition modulo 4 is isomorphic to <span class="math-container">$H=\{1,2,3,4\}$</span> over multiplication modulo 5</p>
</blockquote>
<p>My solution was to brute force check validity of <span class="math-container">$f(a+b)=f(a)f(b)$</spa... | Stephen Goree | 763,360 | <p>Both are cyclic groups of order 4. This is enough to say they are isomorphic because all cyclic groups of order <span class="math-container">$n\in\mathbb{N}$</span> are isomorphic to <span class="math-container">$\mathbb{Z}_n$</span>, but the general idea is that you're mapping one cyclic generator to another.</p>
<... |
3,518,285 | <p>I started studying the book of Daniel Huybrechts, Complex Geometry An Introduction. I tried studying <a href="https://mathoverflow.net/questions/13089/why-do-so-many-textbooks-have-so-much-technical-detail-and-so-little-enlightenme">backwards</a> as much as possible, but I have been stuck on the concepts of <a href=... | GreginGre | 447,764 | <p>Yes, they are. Note that 6. and 7. are clearly equivalent (if we have 6. take for <span class="math-container">$S$</span> and <span class="math-container">$U$</span> the images of <span class="math-container">$W\times \{0\}$</span> and <span class="math-container">$\{0\}\times W$</span> under an isomorphism <span cl... |
1,588,665 | <p>I have been reading up on finding the eigenvectors and eigenvalues of a symmetric matrix lately and I am totally unsure of <strong>how and why</strong> it works. Given a matrix, I can find its eigenvectors and values like a machine but the problem is, I have no intuition of how it works.</p>
<p>1) I understand that... | Andre | 231,643 | <p>In the step going from $9x^2=36x$ to $9x=36$, you divided by $x$.
As you cannot divide by 0, this step is only valid for $x \ne 0$. So you lost the solution $x=0$ with this transformation.</p>
|
232,672 | <p>Yesterday, I posted a question that was received in a different way than I intended it. I would like to ask it again by adding some context. </p>
<p>In ZF one can prove $\not\exists x (\forall y (y\in x)).$ This statement can be read in many ways, such as (1) "there is no set of all sets" (2) "the class of all sets... | Andreas Blass | 6,794 | <p>I regard ZF (or better ZFC) as a (partial) description of the behavior of actual sets. The theorem you quoted says, in that context, that there is no set containing everything. In the same context, I might sometimes talk about classes, but I would regard such talk as an abbreviation for statements that are only ab... |
58,631 | <p>I am (partly as an exercise to understand <em>Mathematica</em>) trying to model the response of a damped simple harmonic oscillator to a sinusoidal driving force. I can solve the differential equation with some arbitrarily chosen boundary conditions, and get a nice graph;</p>
<pre><code>params = {ν1 -> 1.0, ω1 -... | Dr. belisarius | 193 | <pre><code>params = {ν1 -> 1, ω1 -> 10, F -> 4};
system = {D[ x1[t], {t, 2}] == -ν1 D[x1[t], t] - ω1^2 x1[t] + F Cos[ω t], x1[0] == 1, x1'[0] == 0};
soln = DSolve[system /. params, x1[t], t][[1, 1]];
(* and the steady state is*)
lim = ((List @@ (Expand@soln[[2]])) /. x_ /; (Limit[x, t -> Infinity] == 0) :&... |
2,895,655 | <blockquote>
<p>Four coins of different colour are thrown. If three out of these show heads then find the probability that the remaining one shows tails. </p>
</blockquote>
<p>My approach:</p>
<p>$A$: The event in which 3 heads appear in 3 coins out of 4</p>
<p>$B$: The event in which the 4th coin shows tails</p>
... | WaveX | 323,744 | <p>Your mistake is in the event $A$.</p>
<blockquote>
<p>The event in which 3 heads appear in 3 coins out of 4</p>
</blockquote>
<p>What we mean by that is that we have flipped <em>at least</em> $3$ heads. Out of the $16$ ways we can flip four coins, $5$ of them have at least $3$ heads (THHH, HTHH, HHTH, HHHT, HHHH... |
279,985 | <p>How can I convert a Beta Distribution to a Gamma Distribution?
Strictly speaking, I want to transform parameters of a Beta Distribution to parameters of the corresponding Gamma Distribution. I have mean value, alpha and beta parameters of a Beta Distribution and I want to transform them to those of a Gamma Distribu... | rukhsar khan | 210,429 | <p>if x denotes beta distribution of 1st kind with parameters @ and 1 then -logx will follow gamma distribution with parameters @ and 1 </p>
|
1,652,701 | <p>Am I on the right track to solving this?</p>
<p>$$e^z=6i$$
Let $w=e^z$</p>
<p>Thus,</p>
<p>$$w=6i$$
$$e^w=e^{6i}$$
$$e^w=\cos(6)+i\sin(6)$$
$$\ln(e^w)=\ln(\cos(6)+i\sin(6))$$
$$w=\ln(\cos(6)+i\sin(6))$$
$$e^z=\ln(\cos(6)+i\sin(6))$$
$$\ln(e^z)=\ln(\ln(\cos(6)+i\sin(6)))$$
$$z=\ln(\ln(\cos(6)+i\sin(6)))$$</p>
<p>... | Eleven-Eleven | 61,030 | <p>$e^z=6i$.</p>
<p>Let $z=x+iy$. Note that $e^z=e^x\cdot e^{iy}$</p>
<p>Thus
$$e^z=e^x\cdot e^{iy}=6e^{i\left(\frac{\pi}{2}+2k\pi\right)}$$</p>
<p>So $e^x=6$ and so $x=\ln{6}$.</p>
<p>So $y=\frac{\pi}{2}+2k\pi$</p>
<p>Therefore you have as your solutions $z=\ln{6}+i\left(\frac{\pi}{2}+2k\pi\right)$ for integer $... |
1,652,701 | <p>Am I on the right track to solving this?</p>
<p>$$e^z=6i$$
Let $w=e^z$</p>
<p>Thus,</p>
<p>$$w=6i$$
$$e^w=e^{6i}$$
$$e^w=\cos(6)+i\sin(6)$$
$$\ln(e^w)=\ln(\cos(6)+i\sin(6))$$
$$w=\ln(\cos(6)+i\sin(6))$$
$$e^z=\ln(\cos(6)+i\sin(6))$$
$$\ln(e^z)=\ln(\ln(\cos(6)+i\sin(6)))$$
$$z=\ln(\ln(\cos(6)+i\sin(6)))$$</p>
<p>... | Michael Hardy | 11,667 | <p>Suppose $z=x+iy$ and $x$ and $y$ are real. Then
$$
6i = 6(0 + i) = e^z = e^{x+iy} = e^x e^{iy} = e^x(\cos y + i\sin y).
$$
So $e^x = 6$ and $0+1i=\cos y + i\sin y$. Thus $\cos y=0$ and $\sin y=1$. So $y = \pi/2$ or $\pi/2+ 2\pi n$ for some integer $n$.</p>
|
3,459,106 | <p>I have a function
<span class="math-container">$$
\frac{\ln x}{x}
$$</span> and I wonder, is <span class="math-container">$y=0$</span> an asymptote? I mean it is kinda strange that graph is in some place is going through that asymptote. I know it meets the criterium of asymptote, but its kinda strange if you unders... | K.Power | 306,685 | <p>You are correct to have doubts. For a counterexample to the statement just take each <span class="math-container">$f_n=f$</span>, where <span class="math-container">$f$</span> is any discontinuous function of your choice. Then clearly <span class="math-container">$f_n\to g$</span> uniformly, and <span class="math-co... |
43,355 | <p>I recently came across the following formula, which is apparently known as <em>Laplace's summation formula:</em></p>
<p>$$\int_a^b f(x) dx = \sum_{k=a}^{b-1} f(k) + \frac{1}{2} \left(f(b) - f(a)\right) - \frac{1}{12} \left(\Delta f(b) - \Delta f(a)\right) $$
$$+ \frac{1}{24} \left( \Delta^2 f(b) - \Delta^2 f(a) \... | SandeepJ | 5,372 | <blockquote>
<p>However, the usual suspects (the
arXiv, Wikipedia, MathWorld, Google)
aren't turning up much.</p>
</blockquote>
<p>You forgot google books!</p>
<p>There are references to the Laplace summation formula in two books. </p>
<ol>
<li><p>Page 248 in <em>The rise and development of the
theory of seri... |
43,355 | <p>I recently came across the following formula, which is apparently known as <em>Laplace's summation formula:</em></p>
<p>$$\int_a^b f(x) dx = \sum_{k=a}^{b-1} f(k) + \frac{1}{2} \left(f(b) - f(a)\right) - \frac{1}{12} \left(\Delta f(b) - \Delta f(a)\right) $$
$$+ \frac{1}{24} \left( \Delta^2 f(b) - \Delta^2 f(a) \... | Anixx | 10,059 | <p>You may be also interested in this formula for indefinite sum of $f(x)$:</p>
<p>$$\sum_x f(x)=-\sum_{k=1}^{\infty}\frac{\Delta^{k-1}f(x)}{k!}(-x)_k+C$$</p>
<p>where $(x)_k=\frac{\Gamma(x+1)}{\Gamma(x-k+1)} $ is a falling factorial.</p>
|
470,617 | <ol>
<li><p>Two competitors won $n$ votes each.
How many ways are there to count the $2n$ votes, in a way that one competitor is always ahead of the other?</p></li>
<li><p>One competitor won $a$ votes, and the other won $b$ votes. $a>b$.
How many ways are there to count the votes, in a way that the first competitor ... | Batominovski | 72,152 | <p>Consider the polynomial $$P(z):=z^6+z^5+z^4+z^3+z^2+z+1\,.$$ Let $t:=z+\dfrac{1}{z}$. Therefore, $P(z)=z^3\,Q(t)$, where
$$Q(t):=t^3+t^2-2t-1\,.$$</p>
<p>Let $\omega:=\exp(\text{i}\theta)$, where $\theta:=\dfrac{2\pi}{7}$. Then, $\omega$, $\omega^2$, $\omega^3$, $\omega^4$, $\omega^5$, and $\omega^6$ are all the... |
470,617 | <ol>
<li><p>Two competitors won $n$ votes each.
How many ways are there to count the $2n$ votes, in a way that one competitor is always ahead of the other?</p></li>
<li><p>One competitor won $a$ votes, and the other won $b$ votes. $a>b$.
How many ways are there to count the votes, in a way that the first competitor ... | Michael Rozenberg | 190,319 | <p>$$\cos\frac{2\pi}{7}+\cos\frac{4\pi}{7}+\cos\frac{6\pi}{7}=\frac{2\sin\frac{\pi}{7}\cos\frac{2\pi}{7}+2\sin\frac{\pi}{7}\cos\frac{4\pi}{7}+2\sin\frac{\pi}{7}\cos\frac{6\pi}{7}}{2\sin\frac{\pi}{7}}=$$
$$=\frac{\sin\frac{3\pi}{7}-\sin\frac{\pi}{7}+\sin\frac{5\pi}{7}-\sin\frac{3\pi}{7}+\sin\frac{7\pi}{7}-\sin\frac{5\pi... |
3,450,283 | <p>I confronted with a statement: </p>
<p>Given a ring homomorphism <span class="math-container">$f:A\to B$</span>, with commutative rings with identity <span class="math-container">$A,B$</span>. If <span class="math-container">$A,B$</span> are both subrings of a bigger commutative ring with identity <span class="mat... | Community | -1 | <p><span class="math-container">$A$</span> needs not be a subring of <span class="math-container">$B$</span>. Consider <span class="math-container">$R=\Bbb R[x]$</span>, <span class="math-container">$A=\Bbb R[x^2]\subseteq \Bbb R[x]$</span>, <span class="math-container">$B=\Bbb R[x^3]\subseteq \Bbb R[x]$</span> and <sp... |
4,219,614 | <p>In proving a change of basis theorem in linear algebra, our professor draw this diagram and simply stated that because all the outer squares in this diagram commute, the inner square (green) must also commute (I didn't write the exact mappings, because I think this question is more about diagram chasing and that it... | Zhen Lin | 5,191 | <p>Allow me to change notation.
Essentially, you are in this situation:
<span class="math-container">\begin{align}
f' \circ a & = b \circ f \\
g' \circ b & = d \circ g \\
h' \circ a & = c \circ h \\
k' \circ c & = d \circ k
\end{align}</span>
You want to know if <span class="math-container">$g \circ f =... |
130,306 | <p>I am trying to make a relatively complex 3D plot in order to show the variation of a curve with a parameter. Here is the code</p>
<pre><code>AnsNf[x_, nf_] = (2 \[Pi] x^4)/((11 - (2 (2 + nf))/3) (1 + 1/2 x^6 Log[4. x^2])) + (14.298 (1 + (1.81 - 0.292 nf) x^2 - 2.276 x^2 Log[x^2/(1 + x^2)]))/(1 + (9.926 + 1.795 nf) ... | Edmund | 19,542 | <p>Try <a href="http://reference.wolfram.com/language/ref/ParametricPlot3D.html" rel="nofollow noreferrer"><code>ParametricPlot3D</code></a> instead.</p>
<pre><code>img = ParametricPlot3D[
Evaluate@
Flatten[Function[{i}, {t, i, #} & /@
{AnsatzINf[t, i], AnsatzINfNoc[t, i]}] /@ Range[4], 1],
{t, 0, 5}... |
441,888 | <p>I should clarify that I'm asking for intuition or informal explanations. I'm starting math and never took set theory so far, thence I'm not asking about formal set theory or an abstract hard answer. </p>
<p>From Gary Chartrand page 216 Mathematical Proofs - </p>
<p>$\begin{align} \text{ range of } f & = \{f(x... | Asaf Karagila | 622 | <p>The first set is the collection of all the $x$ which are both elements in $S$ and satisfy the property $P$.</p>
<p>The second set is the collection of the objects "$P(x)$" for all $x\in S$, for example if $P(x)$ is the function $x^2$ and $S=\Bbb N$ then the result is the set of squares.</p>
|
441,888 | <p>I should clarify that I'm asking for intuition or informal explanations. I'm starting math and never took set theory so far, thence I'm not asking about formal set theory or an abstract hard answer. </p>
<p>From Gary Chartrand page 216 Mathematical Proofs - </p>
<p>$\begin{align} \text{ range of } f & = \{f(x... | egreg | 62,967 | <p>I've never seen notation such as $\{n\in\mathbb{N}:2n+1\}$ and the answer you refer to says that $\{g\in G:gG\}$ is incorrect.</p>
<p>Well, incorrectness is a relative concept. Before using a notation you should define its meaning; nothing prevents you from assigning a meaning to $\{x\in X: f(x)\}$, but this is usu... |
441,888 | <p>I should clarify that I'm asking for intuition or informal explanations. I'm starting math and never took set theory so far, thence I'm not asking about formal set theory or an abstract hard answer. </p>
<p>From Gary Chartrand page 216 Mathematical Proofs - </p>
<p>$\begin{align} \text{ range of } f & = \{f(x... | HTFB | 50,125 | <p>Mauro makes part of this point, but it's worth stressing: the two different ways of notating a set are used for instances of application of two different axioms (strictly, axiom schemas) in ZF:
$$
B = \{x \in A : \phi(x)\}
$$
is an application of the subset axiom (a.k.a. separation):
$$\forall y_1, \ldots, y_n \for... |
1,272,124 | <p>we know that $1+2+3+4+5.....+n=n(n+1)/2$</p>
<p>I spent a lot of time trying to get a formula for this sum but I could not get it :</p>
<p>$( 2 + 3 + . . . + 2n)$</p>
<p>I tried to write the sum of some few terms.. Of course I saw some pattern between the sums but still the formula I Got didn't give a correct sum... | Mark Bennet | 2,906 | <p>Hint: can you see a factor $2$ somewhere ...</p>
|
636,391 | <p>Evaluate the following indefinite integral.</p>
<p>$$\int { \frac { x }{ 4+{ x }^{ 4 } } }\,dx$$</p>
<p>In my homework hints, it says let $ u = x^2 $. But still i can't continue.</p>
| Wmmoreno | 102,299 | <p><strong>Solve:</strong>
\begin{eqnarray}
\int\frac{x}{4+x^{4}}dx&=&\frac{1}{2}\int\frac{du}{4+u^2}; \text{ if $u=x^{2}$}\\
&=& \frac{1}{2}(\frac{1}{2}\arctan{\frac{u}{2}})\\
&=&\frac{1}{4}\arctan{\frac{x^{2}}{2}+C}
\end{eqnarray}</p>
|
2,593,361 | <p>I’m trying to solve what I’ll call the p-Laplace Equation which is</p>
<p>$$\Delta_p u = 0$$</p>
<p>where $\Delta_p u$ is the p-Laplacian. It is defined as </p>
<p>$$\Delta_p u = \nabla \cdot (|\nabla|^{p-2} \nabla u).$$</p>
<p>Any ideas? I haven’t seen this in a book or anything. I just thought that by analogy,... | dxiv | 291,201 | <blockquote>
<p>Let $x=p+f$ where $p \in \mathbb{Z}$ and $0 \lt f \lt 1$</p>
</blockquote>
<p>Let that be $0 \color{red}{\le} f \lt 1$ in general. Next, let $p = k \cdot n + r$ with $k, r \in \mathbb{Z}$ and $0 \le r \le n-1\,$ by Euclidean division, so that $\,x=k \cdot n + r + f\,$. Then, using that $0 \le r \le n... |
2,445,693 | <p>I know that the derivative of $n^x$ is $n^x\times\ln n$ so i tried to show that with the definition of derivative:$$f'\left(x\right)=\dfrac{df}{dx}\left[n^x\right]\text{ for }n\in\mathbb{R}\\{=\lim_{h\rightarrow0}\dfrac{f\left(x+h\right)-f\left(x\right)}{h}}{=\lim_{h\rightarrow0}\frac{n^{x+h}-n^x}{h}}{=\lim_{h\right... | Peter Szilas | 408,605 | <p>$n^h = \exp((h \log n))$;</p>
<p>$\dfrac{n^h-1}{h} = \dfrac{\exp(h(\log n))-1}{h};$</p>
<p>$z: = h\log n$.</p>
<p>Then:</p>
<p>$\lim_{h \rightarrow 0}\dfrac{\exp(h(\log n))-1}{h} =$</p>
<p>$\lim_{z \rightarrow 0}$ $\log n \dfrac{\exp(z) -1}{z} =$</p>
<p>$\log n ×1= \log n$.</p>
<p>Used: $\lim_{z \rightarrow 0... |
1,581,756 | <p>Find the general solution of $$z(px-qy)=y^2-x^2$$ Let $F(x,y,z,p,q)=z(px-qy)+x^2-y^2$. This gives $$F_x=zp+2x$$
$$F_y=-zq-2y$$
$$F_z=px-qy$$
$$F_p=zx$$
$$F_q=-zy$$
By Charpit's method we have $$\frac{dx}{zx}=\frac{dy}{-zy}=\frac{dz}{z(px-qy)}=\frac{dp}{-zp-2x-p^2x+pqy}=\frac{dq}{zq+2y-pxy+q^2y}$$</p>
<p>By equating... | Asinomás | 33,907 | <p>If a number is product of distinct primes then it has a power of $2$ number of divisors (it has $2^{\omega(n)}$ divisors, where $\omega(n)$ is the number of prime divisors of $n$).</p>
<p>So if your number is product of distinct primes all of its divisors has a power of $2$ number of divisors.</p>
<p>From here we ... |
418,748 | <p>I tried to calculate, but couldn't get out of this:
$$\lim_{x\to1}\frac{x^2+5}{x^2 (\sqrt{x^2 +3}+2)-\sqrt{x^2 +3}}$$</p>
<p>then multiply by the conjugate.</p>
<p>$$\lim_{x\to1}\frac{\sqrt{x^2 +3}-2}{x^2 -1}$$ </p>
<p>Thanks!</p>
| amWhy | 9,003 | <p>You were right to multiply "top" and "bottom" by the conjugate of the numerator. I suspect you simply made a few algebra mistakes that got you stuck with the limit you first posted:</p>
<p>So we start from the beginning:</p>
<p>$$\lim_{x\to1}\frac{\sqrt{x^2 +3}-2}{x^2 -1}$$ </p>
<p>and multiply top and bottom by ... |
24,927 | <p>Does this mean that the first homotopy group in some sense contains more information than the higher homotopy groups? Is there another generalization of the fundamental group that can give rise to non-commutative groups in such a way that these groups contain more information than the higher homotopy groups? </p>
| Akhil Mathew | 536 | <p>Let $\mathcal{C}$ be a category with finite limits and a final object. In general, if $Y$ is an object in $\mathcal{C}$ such that $\hom(X, Y)$ is naturally a group for each $X \in \mathcal{C}$, then $Y$ is called a "group object" in $\mathcal{C}$; that is, there is a multiplication map $Y \times Y \to Y$ and an inve... |
4,489,675 | <p>When saying that in a small time interval <span class="math-container">$dt$</span>, the velocity has changed by <span class="math-container">$d\vec v$</span>, and so the acceleration <span class="math-container">$\vec a$</span> is <span class="math-container">$d\vec v/dt$</span>, are we not assuming that <span class... | mmesser314 | 294,354 | <p>You are right. Infinitesimals are an imprecise way of talking about limits. It is a little odd that mathemeticians use them, because math is all about rigor.</p>
<p>Physicists tend to be looser about tiny mathematical details because they are interested in modeling the behavior of the universe. They can be satisfied... |
2,135,717 | <p>Let $G$ be an Abelian group of order $mn$ where $\gcd(m,n)=1$. </p>
<p>Assume that $G$ contains an element of $a$ of order $m$ and an element $b$ of order $n$. </p>
<p>Prove $G$ is cyclic with generator $ab$.</p>
<hr>
<p>The idea is that $(ab)^k$ for $k \in [0, \dots , mn-1]$ will make distinct elements but do ... | David Hill | 145,687 | <p>Hint: (1) Part of the definition of $l=\mathrm{lcm}(m,n)$ is that if $m|k$ and $n|k$, then $l|k$. (2) if $\gcd(m,n)=1$ then $\mathrm{lcm}(m,n)=mn$.</p>
<p>So, if $(ab)^k=a^kb^k=1$, then $a^k=1$ and $b^k=1$. What can you conclude about $k$?</p>
|
855,227 | <p>I'm trying to understand Taylor's Theorem for functions of $n$ variables, but all this higher dimensionality is causing me trouble. One of my problems is understanding the higher order differentials. For example, if I have a function $f(x, y)$, then it's first differential is: </p>
<p>$$df = \frac{\partial f}{\part... | Chinny84 | 92,628 | <p>Lets denote $$df = \left(dx\frac{\partial}{\partial x} + dy\frac{\partial}{\partial y}\right)f$$
then
$$
d^2f = \left(dx\frac{\partial}{\partial x} + dy\frac{\partial}{\partial y}\right)\left(dx\frac{\partial}{\partial x} + dy\frac{\partial}{\partial y}\right)f = \left(dx\frac{\partial}{\partial x} + dy\frac{\parti... |
855,227 | <p>I'm trying to understand Taylor's Theorem for functions of $n$ variables, but all this higher dimensionality is causing me trouble. One of my problems is understanding the higher order differentials. For example, if I have a function $f(x, y)$, then it's first differential is: </p>
<p>$$df = \frac{\partial f}{\part... | Claude Leibovici | 82,404 | <p><strong>Hint</strong></p>
<p>Let us consider first the case of a function $f(x)$ and you expand it by a first order Taylor series, you represent locally the function by a straight line. If you go to the second order expansion, you represent locally the function by a parabola.</p>
<p>When you have a function $f(x,y... |
4,289,381 | <p>I am trying to answer a question about line integrals, I have had a go at it but I am not sure where I am supposed to incorporate the line integral into my solution.</p>
<p><span class="math-container">$$ \mathbf{V} = xy\hat{\mathbf{x}} + -xy^2\hat{\mathbf{y}}$$</span>
<span class="math-container">$$ \mathrm{d}\math... | Vince Vickler | 981,114 | <ul>
<li>One could also use a parametrization.</li>
</ul>
<p>Set <span class="math-container">$x(t)=t$</span> and and <span class="math-container">$y(t)= t^2/3$</span>.</p>
<ul>
<li>The parabola corresponds to the endpoint of the position vector :</li>
</ul>
<p><span class="math-container">$ \vec r (t)= t\vec i + \frac... |
3,479,144 | <p>Let <span class="math-container">$(X,M,\mu)$</span> be a measure space and <span class="math-container">$f \in L^{1}(X,\mu)$</span>. Then show that for <span class="math-container">$E \in M$</span>, <span class="math-container">$\lim_{k \rightarrow \infty} \int_{E} |f|^{1/k} = \mu(E)$</span>. I am able to show this ... | Ross Millikan | 1,827 | <p>Looking at the <a href="https://en.wikipedia.org/wiki/Square_root#Square_roots_of_positive_integers" rel="nofollow noreferrer">expansions of the small square roots</a>, it appears that if <span class="math-container">$k$</span> divides <span class="math-container">$2a$</span> the expansion of <span class="math-conta... |
2,987,994 | <p>I'm looking for definite integrals that are solvable using the method of differentiation under the integral sign (also called the Feynman Trick) in order to practice using this technique.</p>
<p>Does anyone know of any good ones to tackle?</p>
| FDP | 186,817 | <p>Maybe you can look at:</p>
<p><a href="https://math.stackexchange.com/a/2989801/186817">https://math.stackexchange.com/a/2989801/186817</a></p>
<p>Feynman's trick is used to compute:</p>
<p><span class="math-container">\begin{align}\int_0^{\frac{\pi}{12}}\ln(\tan x)\,dx\end{align}</span></p>
|
2,987,994 | <p>I'm looking for definite integrals that are solvable using the method of differentiation under the integral sign (also called the Feynman Trick) in order to practice using this technique.</p>
<p>Does anyone know of any good ones to tackle?</p>
| FDP | 186,817 | <p>The integral <span class="math-container">$\displaystyle \int_0^1 \frac{x-1}{\ln x}dx$</span> can be used to introduce Feynman's trick (Leibniz was already using this trick)</p>
|
48,865 | <blockquote>
<p>Suppose $f:X\to Y$ is a morphism of <em>smooth connected</em> schemes (over some base). Say $Z\subseteq Y$ is a closed subscheme with complement $U$ so that $f$ pulls back (restricts) to isomorphisms on $Z$ and $U$. Does it follow that $f$ is an isomorphism?</p>
</blockquote>
<p>If we drop the condit... | Steven Landsburg | 10,503 | <p>Reducing to the affine case, the question is this:</p>
<p>Given a ring homomorphism $R\rightarrow S$, and given an ideal $I\subset R$, suppose that
all of the following are isomorphisms:
$$R/I\rightarrow S/IS$$
$$R_f\rightarrow S_f\quad \hbox{for any $f\in I$}$$
Can we conclude that $R\rightarrow S$ is an isomorphi... |
48,865 | <blockquote>
<p>Suppose $f:X\to Y$ is a morphism of <em>smooth connected</em> schemes (over some base). Say $Z\subseteq Y$ is a closed subscheme with complement $U$ so that $f$ pulls back (restricts) to isomorphisms on $Z$ and $U$. Does it follow that $f$ is an isomorphism?</p>
</blockquote>
<p>If we drop the condit... | Anton Geraschenko | 1 | <p>This is essentially the argument in <a href="https://mathoverflow.net/users/986/bhargav">Bhargav</a>'s comment. <a href="https://mathoverflow.net/users/28/matt-satriano">Matt Satriano</a> showed me the separatedness argument.</p>
<p>We first apply the valuative criterion for separatedness to show that $f:X\to Y$ is... |
49,068 | <p>Given lists $a$ and $b$, which represent multisets, how can I compute the complement $a\setminus b$?</p>
<p>I'd like to construct a function <code>xunion</code> that returns the symmetric difference of multisets.
For example, if $a=\{1, 1, 2, 1, 1, 3\}$ and $b=\{1, 5, 5, 1\}$, then their symmetric difference is $\b... | wxffles | 427 | <p>Here's my version.</p>
<pre><code>Clear[multiComplement];
multiComplement[a_, b_] :=
Join @@ (ConstantArray[First@#, Max[Last@#, 0]] & /@ (Tally[a] /.
(Tally[b] /. {e_, c_Integer} :> {e, k_Integer} -> {e, k - c})));
</code></pre>
<p>In action:</p>
<pre><code>With[{a = {1, 1, 2, 1, 1, 3}, b = {1, 5... |
49,068 | <p>Given lists $a$ and $b$, which represent multisets, how can I compute the complement $a\setminus b$?</p>
<p>I'd like to construct a function <code>xunion</code> that returns the symmetric difference of multisets.
For example, if $a=\{1, 1, 2, 1, 1, 3\}$ and $b=\{1, 5, 5, 1\}$, then their symmetric difference is $\b... | Dr. belisarius | 193 | <p>Another way (slower than rasher's):</p>
<pre><code>Clear[simComplement];
simComplement[a_, b_] := Join @@ (Fold[DeleteCases[#1, #2, {1}, 1] &, #[[1]],
Join@#[[2]]] & /@ {{a, b}, {b, a}})
With[{a = {1, 1, 2, 1, 1, 3}, b = {1, 5, 5, 1}}, simComplement[a, b]]
(*
{2, 1,... |
300,753 | <p>Revision in response to early comments. Users of set theory need an <em>implementation</em> (in case "model" means something different) of the axioms. I would expect something like this:</p>
<blockquote>
<p>An <em>implementation</em> consists of a "collection-of-elements" $X$, and a relation
(logical pairing)... | Eric Wofsey | 75 | <p>This answer doesn't really have any ideas that are not already present in Noah Schweber's answer, but there are some points that I feel should be made more forcefully. In particular, I'd like to focus on a couple statements you've made which I think reflect a fundamental misunderstanding of the purpose of axiomatic... |
78,341 | <p>I believe I read somewhere that residually finite-by-$\mathbb{Z}$ groups are residually finite. That is, if $N$ is residually finite with $G/N\cong \mathbb{Z}$ then $G$ is residually finite.</p>
<p>However, I cannot remember where I read this, and nor can I find another place which says it. I was therefore wonderin... | Andreas Thom | 8,176 | <p>This is not true. The most prominent examples of non-residually finite central extensions of residually finite groups (by $\mathbb Z$) are certain lattices in non-linear Lie groups.</p>
<p>See for example</p>
<p>M. S. Raghunathan. Torsion in cocompact lattices in coverings of Spin(2, n). Math. Annalen 266, 403–419... |
78,341 | <p>I believe I read somewhere that residually finite-by-$\mathbb{Z}$ groups are residually finite. That is, if $N$ is residually finite with $G/N\cong \mathbb{Z}$ then $G$ is residually finite.</p>
<p>However, I cannot remember where I read this, and nor can I find another place which says it. I was therefore wonderin... | Andreas Thom | 8,176 | <p>The modified question has a positive answer if $N$ is finitely generated.</p>
<p>Consider an extension $1 \to N \to G \to \mathbb Z \to 1$ and take a lift $u \in G$ of the generator of $\mathbb Z$. If $N$ is finitely generated and $H' \subset N$ is a subgroup of finite index, then the intersection of all subgroups ... |
1,186,270 | <p>$k(x) = e^{-\frac{x^2}{2}}$ on $[-1,2]$</p>
<p>I think the derivative of that is $ -x e^{-\frac{x^2}{2}}$. I don't know how to find zero from that equation.</p>
| Tim Raczkowski | 192,581 | <p>Hint: $e^x\ne 0$ for any real $x$.</p>
|
1,186,270 | <p>$k(x) = e^{-\frac{x^2}{2}}$ on $[-1,2]$</p>
<p>I think the derivative of that is $ -x e^{-\frac{x^2}{2}}$. I don't know how to find zero from that equation.</p>
| Mathemagician1234 | 7,012 | <p>$k(x) = e^{-1/2*x^2}$ defined on [-1,2]$\subset R$. So $k'(x) = -xe^{-1/2*x^2}$ =0. Clearly k'(x)= 0 iff x= 0 since $e^{-1/2*x^2}\neq 0$ for all x in <strong>R</strong>. So the only non-endpoint critical point is at x= 0.We also have to check the endpoints on the bounded interval. At x=-1, k'(-1)= $\frac{1}{e^{1/2}... |
105,723 | <p>I am using MMA do do some algebra and I need to do some simplification. For example, I have the following expression
$$ x^2 \left( \frac{6 \left(2 c_1 x^6+c_2\right)}{x^4} \right)-x \left( 4 c_1 x^3-\frac{2 c_2}{x^3}\right)-8\left(c_1 x^4+\frac{c_2}{x^2}\right ) $$
Can I get intermediate steps to reach the final an... | jjc385 | 11,035 | <p>I generalize Jason B's answer to group by like terms in any number of coefficients, as well as in cases where you have polynomials multiplying one another:</p>
<p>It sounds like you mostly want <code>Plus</code> to keep from evaluating. You can do this by replacing <code>Plus</code> with <code>plus</code>, where <... |
4,528,059 | <p>The graph of <span class="math-container">$y = f(x)$</span> is as follows:</p>
<p><a href="https://i.stack.imgur.com/vprAu.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/vprAu.png" alt="enter image description here" /></a></p>
<p>Find <span class="math-container">$$\int_{-1}^{1} f(1-x^2) dx$$</sp... | Moko19 | 618,171 | <p>Take the case of <span class="math-container">$n=2$</span>. The quotes are <span class="math-container">$q_1$</span> and <span class="math-container">$q_2$</span>. Suppose we bet <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span>, and let <span class="math-container">$x'_1=\fra... |
3,572,967 | <p>Can I ask how to solve this type of equation:</p>
<blockquote>
<p><span class="math-container">$$\log_{yz} \left(\frac{x^2+4}{4\sqrt{yz}}\right)+\log_{zx}\left(\frac{y^2+4}{4\sqrt{zx}}\right)+\log_{xy}\left(\frac{z^2+4}{4\sqrt{xy}}\right)=0$$</span></p>
</blockquote>
<p>It is given that <span class="math-contain... | LHF | 744,207 | <p>Notice that we can use the following property <span class="math-container">$\log_a \frac{b}{c}=\log_a b-\log_a c$</span> to get:</p>
<p><span class="math-container">$$\log_{yz} \left(\frac{x^2+4}{4\sqrt{yz}}\right)=\log_{yz}\frac{x^2+4}{4}-\log_{yz}\sqrt{yz}=\log_{yz}\frac{x^2+4}{4}-\frac{1}{2}$$</span></p>
<p>The... |
3,572,967 | <p>Can I ask how to solve this type of equation:</p>
<blockquote>
<p><span class="math-container">$$\log_{yz} \left(\frac{x^2+4}{4\sqrt{yz}}\right)+\log_{zx}\left(\frac{y^2+4}{4\sqrt{zx}}\right)+\log_{xy}\left(\frac{z^2+4}{4\sqrt{xy}}\right)=0$$</span></p>
</blockquote>
<p>It is given that <span class="math-contain... | DeepSea | 101,504 | <p>Hint : First use: <span class="math-container">$a^2 + 4 \ge 4a$</span> for each term on the left. Then split the log and show next that the left is <span class="math-container">$\ge 0$</span>. Equality is at <span class="math-container">$ x = y = z = 2$</span>.</p>
|
1,814,216 | <p>I was trying to show that $\sin(x)$ is non-zero for integers $x$ other than zero and I thought that this result might emerge as a corollary if I managed to show that the result in question is true. </p>
<p>I think it's possible to demonstrate this by looking at the power series expansion of $\sin(x)$ and assuming t... | Karthik Vasu | 344,346 | <p>$ \mathbb{N} $ is countable and $ \mathbb{R} $ is uncountable, so there can never be a bijection</p>
|
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