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 |
|---|---|---|---|---|
3,344,766 | <p>It is too be proved that <span class="math-container">$3(p^2)+61$</span> is a perfect square only for <span class="math-container">$$p=1 ; p=N$$</span>
The question arises from a problem in arithmatic progression with a certain soft constraint of natural numbers. At the end a quadratic is formed and since our constr... | J. W. Tanner | 615,567 | <p><span class="math-container">$q^2-3p^2=61$</span> is a <a href="http://mathworld.wolfram.com/PellEquation.html" rel="nofollow noreferrer">Pell-like equation</a>.</p>
<p>(The classic Pell equation would have <span class="math-container">$1$</span> instead of <span class="math-container">$61.$</span>)</p>
<p>I belie... |
3,030,332 | <p>I have started complex analysis and I am stuck on one definition 'extended complex plane'.Book is saying 'To visualize point at infinity,think of complex plane passing through the equator of a unit sphere centred at 0.To each point z in plane there corresponds exactly one point P on surface of sphere which is obtain... | RandomMathGuy | 624,016 | <p>I think that description may be a typo... I think they want to say the following:</p>
<p>Imagine a unit sphere resting on top of the complex plane. For each point <span class="math-container">$z\in\mathbb{C}$</span> draw a line segment connecting that point to the "north pole" of the sphere. This line will intersec... |
231,403 | <p>Let $1 \le p < \infty$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, q)$ such that$$\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)} \text{ for all }u \in W^{1, 1}(0, 1)?$$</p>
| Fedor Petrov | 4,312 | <p>Of course. </p>
<p>For $u\in C^1$ denote $v=u-\int_0^1 u$, then $v(x_0)=0$ for some $x_0\in (0,1)$ and for all $x\in (0,1)$ we have $$|v(x)|=|v(x)-v(x_0)|=\left|\int_{x_0}^x v'(t)dt\right|\leqslant \|v'\|_{L^1(0,1)}=\|u'\|_{L^1(0,1)},$$
thus $M:=\|v\|_{L^{\infty}(0,1)}\leqslant \|u'\|_{L^1(0,1)}$. This extends to $... |
1,149,907 | <p>What are some different methods to evaluate</p>
<p>$$ \int_{-\infty}^{\infty} x^4 e^{-ax^2} dx$$</p>
<p>for $a > 0$.</p>
<p>This integral arises in a number of contexts in Physics and was the original motivation for my asking. It also arises naturally in statistics as a higher moment of the normal distributio... | Jack D'Aurizio | 44,121 | <p>Assuming $a>0$, we have:
$$ I = \frac{1}{a^{5/2}}\int_{0}^{+\infty}x^{3/2}e^{-x}\,dx = \frac{\Gamma\left(5/2\right)}{a^{5/2}}=\color{red}{\frac{3\sqrt{\pi}}{4\, a^{5/2}}}.$$</p>
|
1,149,907 | <p>What are some different methods to evaluate</p>
<p>$$ \int_{-\infty}^{\infty} x^4 e^{-ax^2} dx$$</p>
<p>for $a > 0$.</p>
<p>This integral arises in a number of contexts in Physics and was the original motivation for my asking. It also arises naturally in statistics as a higher moment of the normal distributio... | glS | 173,147 | <p>Let $b>0$ be any positive, even integer, and let $a>0$. Then you have
$$ \int_{-\infty}^\infty dx \, x^b e^{-ax^2}
= 2 \int_0^\infty dx \, x^b e^{-ax^2}
= a^{- (b+1)/2} \int_0^\infty e^{-t} t^{\frac{b-1}{2}}
= \color{red}{a^{-(b+1)/2} \Gamma \left(\frac{b+1}{2} \right) }. $$
The particular case $b=4$ gives yo... |
2,045,812 | <p>(gcd; greatest common divisor) I am pulling a night shift because I have trouble understanding the following task.</p>
<p>Fibonacci is defined by this in our lectures:<br>
I) $F_0 := 1$ and $F_1 := 1$</p>
<p>II) For $n\in\mathbb{N}$, $n \gt 1$ do $F_n=F_{n-1}+F_{n-2}$</p>
<p><strong>Task</strong><br>
Be for n>0 t... | S. Y | 364,140 | <p>Here you want to replace $a$ and $b$ with $a_n$ and $b_n$. We have
$$a_nF_{n} + b_n F_{n+1}=gcd(F_n, F_{n+1}) =gcd(F_{n-1}, F_n)=a_{n-1}F_{n-1}+ b_{n-1}F_n$$</p>
<p>Replace $F_{n+1}$ with $F_n+F_{n-1}$, we get
$$(a_n + b_n - b_{n-1})F_n + (b_n-a_{n-1})F_{n-1}=0$$</p>
<p>If we let $a_n + b_n - b_{n-1}=0$ and $ ... |
2,869,305 | <p>I need some help evaluating this limit... Wolfram says it blows up to infinity but I don't think so. I just can't prove it yet.</p>
<p>$$ \lim_{x\to\infty}(x!)^{1/x}-\frac{x}{e} $$</p>
| Claude Leibovici | 82,404 | <p>In the same spirit as in previous answers, considering $$A=(x!)^{\frac{1}{x}}-\frac{x}{e}=B-\frac{x}{e}$$
$$B=(x!)^{\frac{1}{x}}\implies \log(B)={\frac{1}{x}}\log(x!)$$ Now, using Stirling approximation
$$\log(B)={\frac{1}{x}}\left(x (\log (x)-1)+\frac{1}{2} \left(\log (2 \pi )+\log
\left({x}\right)\right)+\frac{... |
825,531 | <p>How many positive integers $n$ are there, such that both $2n$ and $3n$ are perfect squares? I tried to use modular arithmetic, but I'm stuck.</p>
| lab bhattacharjee | 33,337 | <p>If $x\ne0,$</p>
<p>$$2x=a^2,3x=b^2\implies \frac{a^2}{b^2}=\frac23\iff \left(\frac ab\right)^2=\frac23$$</p>
<p>or on multiplication, $$6x^2=a^2b^2\iff \left(\frac{ab}x\right)^2=6$$ </p>
<p>which is impossible as $a,b,x$ are integers</p>
|
825,531 | <p>How many positive integers $n$ are there, such that both $2n$ and $3n$ are perfect squares? I tried to use modular arithmetic, but I'm stuck.</p>
| lab bhattacharjee | 33,337 | <p>Let $x=r\cdot2^a\cdot3^b$ where $(r,6)=1$</p>
<p>As $2x$ is perfect square, $r\cdot2^{a+1}\cdot3^b$ must be, so $r$ must be perfect square $=s^2$(say) and $a+1,b$ each must be even $\implies a$ must be odd</p>
<p>As $3x$ is perfect square, $s^2\cdot2^a\cdot3^{b+1}$ must be, so $a,b+1$ each must be even $\implies ... |
57,232 | <p>I have huge matrices in the form of</p>
<pre><code>mtx1 = {{24+24 FF[6,9] GG[5,10]+24 FF[7,8] GG[5,10]+24 FF[5,10] GG[6,9]+24 FF[6,9] GG[6,9]+24 FF[7,8] GG[6,9]+24 FF[5,10] GG[7,8]+24 FF[6,9] GG[7,8]+24 FF[7,8] GG[7,8],24+24 FF[5,10] GG[5,10]+24 FF[6,9] GG[5,10]+24 FF[7,8] GG[5,10]+24 FF[6,9] GG[6,9]+24 FF[7,8] GG[... | Mr.Wizard | 121 | <p>For the example you gave <code>FF[___]</code> and <code>GG[___]</code> are the only non-number terms, therefore by polynomial sort order you could use simply:</p>
<pre><code>mtx1[[All, All, 1]]
</code></pre>
<blockquote>
<pre><code>{{24, 24, 24}, {24, 24, 24}, {24, 24, 24}}
</code></pre>
</blockquote>
<p>I shall ... |
1,955,509 | <p>There's this exercise in Hubbard's book:</p>
<blockquote>
<p>Let $ h:\Bbb R \to \Bbb R $ be a $C^1$ function, periodic of period $2\pi$, and define the function $ f:\Bbb R^2 \to \Bbb R $ by
$$f\begin{pmatrix}r\cos\theta\\r\sin\theta \end{pmatrix}=rh(\theta)$$</p>
<p>a. Show that $f$ is a continuous real-v... | dxiv | 291,201 | <p>Hint: for $|x| \lt 1$</p>
<p>$$f(x) = \sum_{n=0}^{\infty} x^{n+2} = \frac{x^2}{1-x} = -1 -x + \frac{1}{1-x}$$</p>
<p>Now consider $f''(\frac{1}{4})$.</p>
|
1,546,599 | <p>What method should I use for this limit?
$$
\lim_{n\to \infty}{\frac{n^{n-1}}{n!}}
$$</p>
<p>I tried ratio test but I ended with the ugly answer
$$\lim_{n\to \infty}\frac{(n+1)^{n-1}}{n^{n-1}}
$$
which would go to 1? Which means we cannot use ratio test. I do not know how else I could find this limit.</p>
| Sergio Adrián Lagunas Pinacho | 238,699 | <p>$$\lim_{n \to\infty} \left(\frac{n+1}{n}\right)^{n-1} = e$$ so it diverges.</p>
|
4,597,679 | <p>My textbook is <em>A Mathematical Introduction to Logic, 2nd Edition</em> by Enderton.</p>
<p>The question initially comes when I was trying to prove Exercise 4. on pg.99</p>
<p><span class="math-container">$$\text{Show that if }x \text{ does not occur free in }\alpha,\text{ then }\alpha \vDash \forall x \alpha$$</s... | Mauro ALLEGRANZA | 108,274 | <p><em>In "everyday-life" there are no free variables</em>.</p>
<p>In the language of predicate logic a free variable works like a pronoun in natural language: <span class="math-container">$\text{red}(x)$</span> is like "it is red". Its meaning depends on the context.</p>
<p>How we specify the "... |
3,440,732 | <p>How can I show that <span class="math-container">$P=\{\{2k-1,2k\},k\in \mathbb {N}\}$</span> is the basis of <span class="math-container">$\mathbb N$</span>.
Obviously we have to show that basis's orders do work.
But the problem is if I take two subsets that belong to <span class="math-container">$P$</span> , for ex... | Aldoggen | 607,154 | <p>For <span class="math-container">$P$</span> to be a basis of <span class="math-container">$\mathbb{N}$</span>, you need to show that </p>
<ol>
<li>Every natural number is contained in a basis element. Since <span class="math-container">$n\in\{n-1,n\}$</span> or <span class="math-container">$n\in\{n,n+1\}$</span>, d... |
2,576,801 | <p>$$\frac {|A-B|}{15} = \frac {|B-A|}{10} = \frac {|A\cap B|}{6}$$</p>
<ul>
<li>What is the minimal value of $A \cup B$?</li>
</ul>
<p>This question is from my exam. I've tried to solve it by giving values and added them up. </p>
<p>Regards</p>
| Community | -1 | <p><strong>Hint:</strong></p>
<p>Note that: $$|A \cup B| = |A-B| + |B-A | + |A \cap B|$$</p>
<p>If we have: $$\frac{|A-B|}{15}=\frac{|B-A|}{10}=\frac{|A \cap B|}{6}= k$$ we will get: $$|A \cup B| = 15k+10k+6k = 31k$$ Can $k$ be $\leq 0$?</p>
|
524,568 | <p>I'm working on a recursive function task which i'm a bit stuck at. I've tried to google it on how I can solve this task, but with no luck</p>
<p>Here is the task:</p>
<blockquote>
<p><em>Provide a recursive function $r$ on $A$</em>* <em>which gives the number of
characters in the string</em></p>
</blockquote>
... | Carlos Eugenio Thompson Pinzón | 99,344 | <p>This seems rather a programing question than a mathematical one, but the idea of recursiveness would be something like:</p>
<ol>
<li>if the string is empty, answer 0</li>
<li>otherwise, answer is 1 + the lenght of the substring after second position</li>
</ol>
<p>This recursiveness only make sense in implementatio... |
3,194,169 | <p>I am not a mathematician but I have a question.</p>
<p>I found </p>
<ul>
<li><span class="math-container">$1 / 3 = 0.33333....$</span></li>
<li><span class="math-container">$2 / 3 = 0.666666...$</span></li>
<li><span class="math-container">$3 / 3 = 1$</span> while it should be <span class="math-container">$0.9999... | E.H.E | 187,799 | <p><span class="math-container">$$0.9999999...=\frac{9}{10}(1+\frac{1}{10}+\frac{1}{10^2}+.....)=\frac{9}{10}\frac{1}{1-\frac{1}{10}}=1$$</span></p>
|
608,875 | <p>solve this equation
$$\sqrt{\sqrt{3}-\sqrt{\sqrt{3}+x}}=x$$</p>
<p>My try: since
$$\sqrt{3}-x^2=\sqrt{\sqrt{3}+x}$$
then
$$(x^2-\sqrt{3})^2=x+\sqrt{3}$$</p>
| Michael Hoppe | 93,935 | <p>You're on the right track. Square again, you'll get $x=0$ as a possible solution. Remembering that $x^3-x=x(x-1)(x+1)$ you should be able to factor the equation and get $x=-1$ as a possible solution. Now solve the remaining quadratic.</p>
|
2,319,119 | <p>Been given a question and find it to be too vague to know what's going on. </p>
<p>The question is: </p>
<p>$f(x) = 2x + 2$. Define $f(x)$ recursively.</p>
<p>I'm just quite puzzled as there is no $f(0)$, $f(1)$ or $f(x-1)$ function to go by other than the original function.</p>
<p>Supposed to be in the form of ... | Red shoes | 219,176 | <p><strong>Hint</strong>: there is no sequence in $S$ converging to $f=0 \in C[0,1]$ (uniformly convergence)</p>
|
3,789,873 | <p>Let <span class="math-container">$X, Y, X_n's$</span> be random variables for which <span class="math-container">$X_n+\tau Y\to_D X+\tau Y$</span> for every fixed positive constant <span class="math-container">$\tau$</span>. Show that <span class="math-container">$X_n \to_D X$</span>.</p>
<p>I dont think we can let ... | shalop | 224,467 | <p>Write</p>
<p><span class="math-container">$$\Bbb E[e^{itX_n}] - \Bbb E[e^{itX}] = \bigg(\Bbb E[e^{it X_{n}}] - \Bbb E[e^{it X_{n} + i t \epsilon Y}]\bigg) + \bigg(\Bbb E[e^{it X_{n} + i t \epsilon Y}] - \Bbb E[e^{it X + i t \epsilon Y}]\bigg)+ \bigg(\Bbb E[e^{it X + i t \epsilon Y}]-\Bbb E[e^{it X}]\bigg).$$</span... |
4,095,715 | <p>I know how to do these in a very tedious way using a binomial distribution, but is there a clever way to solve this without doing 31 binomial coefficients (with some equivalents)?</p>
| Igor Rivin | 109,865 | <p>Your expression equals <span class="math-container">$$(1+x^2)^n (1+x^4)^n.$$</span> You should be able to finish using the binomial theorem.</p>
|
4,064,353 | <blockquote>
<p>The number <span class="math-container">$1.5$</span> is special because it is equal to one quarter of the sum of its digits, as <span class="math-container">$1+5=6$</span> and <span class="math-container">$\frac{6}{4}=1.5$</span> .Find all the numbers that are equal to one quarter of the sum of their ... | Henry | 6,460 | <p>This is not much better than a brute force method:</p>
<ul>
<li><p>The sum of the digits <span class="math-container">$S$</span> is a non-negative integer, so a quarter of it <span class="math-container">$\frac{S}4$</span> is non-negative, of the form <span class="math-container">$x$</span> or <span class="math-cont... |
67,434 | <p>Let $f: \mathbb{R}^n \to L^2(\mathbb{R}^d) $ be a Bochner-integrable function (all measures are the Lebesgue measure). Does then $ \int_{\mathbb{R}^n} f(x) d\lambda^n (y) = \int_{\mathbb{R}^n} f(x)(y) d\lambda^n $ hold for $\lambda^d$-almost all $y \in \mathbb{R}^d$? I.e. can one compute such Bochner integrals just... | Gerald Edgar | 454 | <p>Answer: YES and NO. </p>
<p>YES: In any practical situation you are likely to meet, your formula is correct. You would prove it using Fubini's Theorem, pairing your two sides with an arbitrary $h \in L^2(\mathbb R^d)$ and getting the same answer on both sides. The catch is, you have to be able to apply Fubini... |
2,811,991 | <p>$$\int_{-3}^5 f(x)\,dx$$<br>
for
$$ f(x) =\begin{cases}
1/x^2, & \text{if }x \neq 0 \\
-10^{17}, & \text{if }x=0
\end{cases}
$$</p>
<p>I tried with Newton Leibniz formula, is this correct ?</p>
<p>$\int_{-3}^0 f(x)dx$ + $\int_{0}^5 f(x)dx$ =</p>
<p>$1/x^2 |_{-3}^{0} $ $ + $ $1/x^2 |_0^5$=</p>
<p>$3/(-... | Robert Israel | 8,508 | <p>The $-10^{17}$ is a red herring: the value of a function at a single point has no effect on the integral. On the other hand, this does signal that something funny is likely to be happening as $x \to 0$, and indeed it does: the integrand goes to
$\infty$ there. Since the integrand is unbounded, this is not an ordi... |
2,808,512 | <p>Show that if an $n\times n$ matrix $A$ satisfies $A^T=-A$, then $x^TAx=0$ for any $n\times 1$ vector $x$.</p>
<p>My attempt: Since matrix transpose won't affect the diagonal entries, so matrix $A$ has only zeros on its diagonal. </p>
<p>Then I tried to write $x$ in the form of $\begin{pmatrix}
x_1\\
x_2\\
\vdots\\... | Botond | 281,471 | <p>We have that
$$x^TA^Tx=\sum_{i,j}x_i(A^T)_{ij}x_j=\sum_{i,j}x_iA_{ji}x_j=\sum_{i,j}x_jA_{ji}x_i=x^TAx$$
But we also have that
$$x^TA^Tx=\sum_{i,j}x_i(-A)_{ij}x_j=-x^TAx$$
So
$$x^TAx=-x^TAx$$
So it must be $0$.</p>
|
851,459 | <p>hello $$$$ I am trying to find explanation how to derive cahn hilliard equation:</p>
<p>$$ u_t =\Delta (w'(u)-\epsilon ^2 \Delta u)$$ </p>
<p>as gradient flow of energy functional $$ : E[u]=\int w(u)+\epsilon ^2 |\nabla u|^2 ) $$
I tried to follow the definition of gradient flow from :
<a href="http://anhngq... | QA Ngô | 122,043 | <p>Let $H$ be a Hilbert space and $F :H \to \mathbb R$. Suppose that $F$ is Gâteaux differentiable at $u \in H$, then gradient of $F$ at $u$, denoted by $\text{grad}F(u)$, is given by the unique $w \in H$ which satisfies
$$\langle F'(u), v \rangle_H = \langle w, v \rangle_H \quad \forall v \in H.$$
When we restrict $v$... |
1,930,722 | <p>$$\lim_{x\to 0}\frac{x}{x+x^2e^{i\frac{1}{x}}} = 1$$</p>
<p>I've trying seeing that:</p>
<p>$$e^{\frac{1}{x^2}i} = 1+\frac{i}{x^2}-\frac{1}{x^42!}-i\frac{1}{x^63!}+\frac{1}{x^84!}+\cdots\implies$$</p>
<p>$$x^2e^{\frac{1}{x^2}i} = x^2+\frac{i}{1}-\frac{1}{x^22!}-i\frac{1}{x^43!}+\frac{1}{x^64!}+\cdots$$</p>
<p>bu... | Olivier Oloa | 118,798 | <p>We assume that $x \in \mathbb{R}$. One may write, as $x \to 0$,
$$
\left|\frac{x}{x+x^2e^{i\frac{1}{x}}} \right|=\frac{1}{\left|1+xe^{i\frac{1}{x}} \right|}\le \frac{1}{1-\left|xe^{i\frac{1}{x}} \right|}=\frac{1}{1-\left|x\right|} \to 1
$$ since
$$
||a|-|b||\le |a+b|.
$$</p>
|
1,130,645 | <p>I am asked to prove: if <span class="math-container">$|a|<\epsilon$</span> for all <span class="math-container">$\epsilon>0$</span>, then <span class="math-container">$a=0$</span></p>
<p>I can prove this as follows.</p>
<p>Assume <span class="math-container">$a \not= 0$</span></p>
<p>I want to show then that <... | Rob Arthan | 23,171 | <p>Infinitesimals aren't really going to help. What you are trying to prove is true in any ordered field. A proof along the lines you suggest requires the extra assumption that $a$ is "standard" (see Hagen von Eltzen's answer) and so proves something weaker than you want. If you accept a proof by contradiction as "dire... |
3,203,577 | <p>What is the computational complexity of solving a linear program with <span class="math-container">$m$</span> constraints in <span class="math-container">$n$</span> variables?</p>
| Jeff Linderoth | 669,913 | <p>The best possible (I believe) is by Michael Cohen, Yin Tat Lee, and Zhao Song:
Solving linear program in the current matrix multiplication time.
<a href="https://arxiv.org/abs/1810.07896" rel="noreferrer">https://arxiv.org/abs/1810.07896</a>
(STOC 2019)
Hope this helps.</p>
|
2,512,396 | <p>i am trying to compute inverse Laplace transform of function = $\tan ^{-1} (2/s)$</p>
| Math Lover | 348,257 | <p>Hint: $$\mathcal{L}(tf(t))=-\frac{d}{ds}(F(s)).$$</p>
|
2,441,793 | <p><strong>Questions.</strong> </p>
<p>(0) Is there a usual technical term in ring theory for the following kind of module?</p>
<blockquote>
<p>$M$ is a free $R$-module over a commutative ring $R$, and for each $R$-basis $B$ of $M$, each $b\in B$, and each <em>unit</em> $r$ of $R$, we have $r\cdot b=b$</p>
</blockq... | user334639 | 221,027 | <p>It's nice that you have this disquietude at this point.</p>
<p>Mathematical precision and clarity depends on the context, the writer, the intended reader and their common backgrounds.</p>
<p>Absolute precision using just symbols is possible in theory but in practice you will be much better using words instead.</p>... |
2,795,652 | <blockquote>
<p>Given a set of decimal digits. And given a set of primes $\mathbb{P}$, find some $p \in \mathbb{P}$ such that $p^n, n \in \mathbb{N} $ contained in itself all the digites from a given set, and it does not matter in what order.</p>
</blockquote>
| lhf | 589 | <p>More is true:</p>
<blockquote>
<p>For any finite sequence of decimal digits, there is a power of $2$ whose decimal expansion begins with these sequence.</p>
</blockquote>
<p>See <a href="https://math.stackexchange.com/questions/13131/starting-digits-of-2n">here</a> for a proof.
For algorithms, see <a href="https... |
661,542 | <p>Permutation without consecutive numbers, such as 1,2 or 5,4?</p>
| Caleb Stanford | 68,107 | <p><strong>Hint:</strong> Split into cases on the position of the number $3$.</p>
<ul>
<li><p>If the number $3$ is not first or last, then $1$ and $5$ must be on either side of it. Then what?</p></li>
<li><p>If the number $3$ is first or last, then $1$ or $5$ will be next to it. How many ways are to fill the other n... |
2,197,065 | <p>Baire category theorem is usually proved in the setting of a complete metric space or a locally compact Hausdorff space.</p>
<p>Is there a version of Baire category Theorem for complete topological vector spaces? What other hypotheses might be required?</p>
| fleablood | 280,126 | <p>Probably because average distance is considered too easy.</p>
<p>I'm not entirely sure what you mean be an "average time problem" but the one thing that makes average speed problems different is that speed is not an accumulative measured value but a ratio between two accumulative values. $\text{speed} = \dfrac {\t... |
4,131,208 | <p>Does the following converge to <span class="math-container">$0$</span> for <span class="math-container">$\theta>-1/2$</span>?</p>
<p><span class="math-container">$$\lim_{n\rightarrow\infty}\frac{\sum_{i=1}^ni^{3\theta}}{\left(\sum_{i=1}^ni^{2\theta}\right)^{3/2}}$$</span></p>
<p>I'd like to use the comparison tes... | Claude Leibovici | 82,404 | <p>Anothe solution using generalized harmonic nmbers
<span class="math-container">$$a_n=\frac{\sum_{i=1}^ni^{3\theta}}{\Big[\sum_{i=1}^ni^{2\theta}\Big]^{\frac 32}}=\frac{H_n^{(-3 \theta )}}{\Big[H_n^{(-2 \theta )}\Big]^{\frac 32}}$$</span> Using asymptotics
<span class="math-container">$$H_n^{(-3 \theta )}=n^{3 \theta... |
492,125 | <p>How many different equivalence relations can be defined on a set of five elements?</p>
| André Nicolas | 6,312 | <p>We have $5$ people, and want to find the number of ways to break them up into groups (equivalence classes). With a number significantly bigger than $5$, it would be useful to develop some theory (and there is such theory). With $5$, we just count in a systematic way.</p>
<p>(i) One big happy family, everybody equiv... |
4,496,772 | <p>Let <span class="math-container">$V$</span> be a vector space of dimension <span class="math-container">$n$</span> over a finite field <span class="math-container">$F$</span> with <span class="math-container">$q$</span> elements, and let <span class="math-container">$U\subseteq V$</span> be a subspace of dimension <... | Mike Earnest | 177,399 | <p>Cpc's answer is fantastic, but here is a more high-level presentation. <a href="https://en.wikipedia.org/wiki/Correspondence_theorem" rel="nofollow noreferrer">Noether's fourth isomorphism</a> says that the subspaces of <span class="math-container">$V$</span> which contain <span class="math-container">$U$</span> are... |
4,496,772 | <p>Let <span class="math-container">$V$</span> be a vector space of dimension <span class="math-container">$n$</span> over a finite field <span class="math-container">$F$</span> with <span class="math-container">$q$</span> elements, and let <span class="math-container">$U\subseteq V$</span> be a subspace of dimension <... | JBL | 1,080,305 | <p>The point of the hint is to count the elements of that set: given what you already know, it is easy to do it by first choosing <span class="math-container">$W$</span> then choosing <span class="math-container">$U$</span>. Doing it the other way is easy in terms of the number you're trying to compute. And the two e... |
3,372,952 | <blockquote>
<p>I know that all the subspaces of <span class="math-container">$\mathbb{R}^{3}$</span> over <span class="math-container">$\mathbb{R}$</span> are:</p>
<ul>
<li><p>Zero subspace.</p></li>
<li><p>Lines passing through origin.</p></li>
<li><p>Planes passing through origin.</p></li>
<li><p><spa... | user | 505,767 | <p>We have considered all the possible dimensions contained in <span class="math-container">$\mathbb{R^3}$</span> therefore there is not any other subspace to be considered.</p>
<p>It's a <a href="https://en.wikipedia.org/wiki/Proof_by_exhaustion" rel="nofollow noreferrer">proof by exhaustion</a>.</p>
|
4,381,737 | <p><span class="math-container">$\mathbb{Z}[i] = \{a + bi : a, b \in \mathbb{Z}\}$</span> are the Gaussian integers. I want to show that there is no ideal <span class="math-container">$I \subseteq \mathbb{Z}[i]$</span> such that the quotient <span class="math-container">$\mathbb{Z}[i]/I$</span> is a field of size <span... | lhf | 589 | <p>Your proof is fine.</p>
<p>Alternatively, <span class="math-container">$\mathbb{Z}[i]$</span> is a PID and so <span class="math-container">$I=\langle a+bi \rangle$</span>. Then <span class="math-container">$\mathbb{Z}[i]/I \cong \mathbb F_3$</span> implies <span class="math-container">$a^2+b^2=3$</span>, a contradic... |
2,532,327 | <p>Expand the following function in Legendre polynomials on the interval [-1,1] :</p>
<p>$$f(x) = |x|$$</p>
<p>The Legendre polynomials $p_n (x)$ are defined by the formula :</p>
<p>$$p_n (x) = \frac {1}{2^n n!} \frac{d^n}{dx^n}(x^2-1)^2$$</p>
<p>for $n=0,1,2,3,...$</p>
<p>My attempt :</p>
<p>we have using the fa... | PM. | 416,252 | <p>You are being asked to compute the coefficients $a_n$ in the expansion</p>
<p>$$
f(x)=\sum_{n=0}^\infty a_n P_n(x)
$$</p>
<p>See for example <a href="http://mathworld.wolfram.com/Fourier-LegendreSeries.html" rel="nofollow noreferrer">here</a> for more information.</p>
<p>Another very useful reference (in general)... |
769,272 | <p>So, I'm trying to solve the wave equation with the Fourier transform, and I'm struggling to figure out how to apply the BC's. Here's the problem I considered:</p>
<p>$$\frac{d^2u}{dt^2}=c^2\frac{d^2u}{x^2}$$
$$u(x,0)=g(x)$$
$$\frac{du}{dt}=0$$ at t = 0</p>
<p>Running through computations, I find that the Fourier ... | Ted Shifrin | 71,348 | <p>The first fundamental form determines the Gaussian curvature but definitely does <em>not</em> determine the second fundamental form. Indeed, the first fundamental form can be defined on an abstract surface (say, the hyperbolic plane) that does not even live inside $\Bbb R^3$.</p>
<p>You need the Gauss equations in ... |
72,876 | <p>A quiver in representation theory is what is called in most other areas a directed graph. Does anybody know why Gabriel felt that a new name was needed for this object? I am more interested in why he might have felt graph or digraph was not a good choice of terminology than why he thought quiver is a good name. (I r... | Qiaochu Yuan | 290 | <p>Here is some speculation. Some abstractions in mathematics can be used to study so many different things that even when you use the same abstraction you might want to give it a different name to indicate what sort of thing you're actually studying. </p>
<p>In other words, the name you use declares an <em>intention:... |
2,662,717 | <p>Let $(f_k)_{k=m}^\infty$ be a sequence of differentiable functions $f_k:[a,b]\rightarrow R$ whose derivatives are continuous. Suppose there exists a sequence $(M_k)_{k=m}^\infty$ in $R$ with $|f_k'|\le M_k$ for all $x\in X, k\geq m,$ and such that $\sum_{k=m}^\infty M_k$ converges. Assume also that there is some $x_... | Robert Stuckey | 440,720 | <p>The trick I've learned for double induction is to do your base case $m=0$ and $n=0$ and then fix $n$ to an arbitrary value (I usually start by fixing the second one), and then do induction on $m$. Then fix $m$ to an arbitrary value and do induction on $n$. Once you've done this you know that for any value of $n$ you... |
2,662,717 | <p>Let $(f_k)_{k=m}^\infty$ be a sequence of differentiable functions $f_k:[a,b]\rightarrow R$ whose derivatives are continuous. Suppose there exists a sequence $(M_k)_{k=m}^\infty$ in $R$ with $|f_k'|\le M_k$ for all $x\in X, k\geq m,$ and such that $\sum_{k=m}^\infty M_k$ converges. Assume also that there is some $x_... | N. Shales | 259,568 | <p>The process below is quite revealing if you bear in mind the Fibonacci sequence </p>
<p>$$\begin{array}{ccccccc}F_1&F_2&F_3&F_4&F_5&F_6&\ldots\\1&1&2&3&5&8&\ldots\end{array}$$ </p>
<p>and focus on the coefficients on the right hand sides:</p>
<p>$$\begin{align} F_k&... |
2,294,321 | <blockquote>
<p>Suppose $F(x)$ is a continuously differentiable (that is, first derivative exist and are continuous) vector field in $\mathbb{R}^3$ that satisfies the bound $$|F(x)| \leq \frac{1}{1 + |x|^3}$$
Show that $\int\int\int_{\mathbb{R}^3} \text{div} F dx = 0$.</p>
</blockquote>
<p>Attempted proof - Suppos... | Gio67 | 355,873 | <p>Apply the divergence theorem on a sequence of balls $B(0,n)$ where $n\to\infty$ and use the bound on $F$ to bound the boundary integral $\int_{\partial B(0,n)}F\cdot \nu\,dS$ </p>
|
702,879 | <p>I am studying exponentials from MacLane-Moerdijk's book, "Sheaves in geometry and Logic". I do not understand the following: Induced by the product-exponent adjunction, consider the bijection $$\hom(Y\times X,Z)\to\hom(Y,Z^X)\;\;\;\;\;\;(\star)$$
They say: The existence of the above adjunction can be stated in eleme... | ye23der | 133,737 | <p>Ok, I believe I have understood this. Can someone check if this is a correct way of viewing it?</p>
<p>Let <span class="math-container">$\phi_Y \colon \hom(Y, Z^X) \to \hom(Y \times X, Z)$</span> be the bijection. The identity <span class="math-container">$1 \colon Z^X \to Z^X$</span> goes to an arrow <span class="m... |
3,288,651 | <p>I have a problem understanding a proof about ideals, which states that every ideal in the integers can be generated by a single integer. And with that I realized that I also don't really understand ideals in general and the intuition behind them. </p>
<p>So let me start by the definition of an ideal. For <span clas... | Anurag A | 68,092 | <p>An ideal <span class="math-container">$I$</span> of <span class="math-container">$\Bbb{Z}$</span> generated by <span class="math-container">$a,b \in \Bbb{Z}$</span> consists of all possible integer linear combinations of <span class="math-container">$a$</span> and <span class="math-container">$b$</span> (<em>similar... |
3,537,297 | <p>I am asked the following question (<strong>non-calculator</strong>):</p>
<blockquote>
<p>Solve for <span class="math-container">$0 \leq x \leq 2 \pi$</span></p>
<p><span class="math-container">$$4 \sin x =\sqrt{3} \csc x + 2 - 2\sqrt{3} $$</span>
<strong>Answer:</strong> <span class="math-container">$x = \... | Quanto | 686,284 | <p>Hint:</p>
<p>Simplify your solutions with <span class="math-container">$\sqrt{4+2\sqrt3} = 1+\sqrt3$</span>. Therefore <span class="math-container">$\sin x= \frac12,\> -\frac{\sqrt3}2$</span>.</p>
|
221,667 | <p>I'm taking a second course in linear algebra. Duality was discussed in the early part of the course. But I don't see any significance of it. It seems to be an isolated topic, and it hasn't been mentioned anymore. So what's exactly the point of duality?</p>
| Tabes Bridges | 36,394 | <p>Expanding on Hew Wolff's comment: in algebraic topology, a major theme is <em>functoriality</em>, which roughly boils down to the idea that when we associate algebraic objects to topological spaces, we should similarly obtain any maps between the algebraic objects from maps between topological spaces so that the alg... |
2,479,305 | <p>I'm studying first-order logic and I saw the textbook put equality symbol $=$ ads a logical symbol.</p>
<p>It's not a logical connective symbol, so $=xy$ is an atomic formula.
<a href="https://en.wikipedia.org/wiki/Atomic_formula" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Atomic_formula</a></p>
<p>Ho... | Caleb Stanford | 68,107 | <p>This is a good question. Depending on who you ask, $=$ is either a special logical symbol, or a special predicate symbol. So it can be presented as either one.</p>
<p>Syntactically, it behaves like a two-place predicate. Two-place predicates, like $\equiv, <, \le$, etc. create a formula out of any two terms. $=$... |
777,691 | <p>I would like to prove if $a \mid n$ and $b \mid n$ then $a \cdot b \mid n$ for $\forall n \ge a \cdot b$ where $a, b, n \in \mathbb{Z}$</p>
<p>I'm stuck.<br>
$n = a \cdot k_1$<br>
$n = b \cdot k_2$<br>
$\therefore a \cdot k_1 = b \cdot k_2$</p>
<p>EDIT: so for <a href="http://en.wikipedia.org/wiki/Fizz_buzz" rel="... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\,\ a,b\mid n\iff {\rm lcm}(a,b)\mid n\!\!\!\overset{\ \ \ \large \times\, (a,b)}\iff ab\mid n(a,b),\, $ which is not equivalent to $\,ab\mid n\ $ </p>
|
1,279,570 | <p>I know that if $u\cdot v = 0$ then by definition, $u_1v_1 + u_2v_2 + \cdots + u_nv_n = 0$</p>
<p>I also know that if $u$ and $v$ are linearly independent and a matrix $A$ has $u$ and $v$ as its successive column vectors then the equation $Ax = 0$ will have only the trivial solution. </p>
<p>However I can't think ... | Peter Woolfitt | 145,826 | <p>You can use the fact that two vectors are linearly dependent iff one is a scalar multiple of the other:</p>
<p>For the sake of contradiction suppose $u$ is nonzero and $u=cv$ (i.e. the vectors are dependent). Then $0=u\cdot v=c(u_1\cdot u_1+\dots+u_n\cdot u_n)=c(u_1^2+\dots+u_n^2)$, but this can only be $0$ if each... |
1,279,570 | <p>I know that if $u\cdot v = 0$ then by definition, $u_1v_1 + u_2v_2 + \cdots + u_nv_n = 0$</p>
<p>I also know that if $u$ and $v$ are linearly independent and a matrix $A$ has $u$ and $v$ as its successive column vectors then the equation $Ax = 0$ will have only the trivial solution. </p>
<p>However I can't think ... | Math1000 | 38,584 | <p>Assume that $$c_1u + c_2v = 0$$ for some scalars $c_1, c_2$. Then $$0 = \langle u, c_1u + c_2v\rangle = \langle u,c_1u\rangle + \langle u,c_2v\rangle = c_1\langle u,u\rangle + c_2\langle u,v\rangle = c_1\langle u,u\rangle,$$
since $\langle u,v\rangle=0$. Since $u\ne0$, it follows that $c_1=0$. From there we see imme... |
1,387,454 | <p>What is the sum of all <strong>non-real</strong>, <strong>complex roots</strong> of this equation -</p>
<p>$$x^5 = 1024$$</p>
<p>Also, please provide explanation about how to find sum all of non real, complex roots of any $n$ degree polynomial. Is there any way to determine number of real and non-real roots of an ... | Américo Tavares | 752 | <ul>
<li>Since $x^{5}=1024$ implies that $x=\sqrt[5]{1024}=4=x_0$ is one of the roots of $x^{5}-1024=0$, it follows that
$$p(x)=x^{5}-1024=(x-4)q(x), $$
where $q(x)$ is a forth degree polynomial with real coefficients and leading term equal to $x^{4}$. Hence it is of the form
$$q(x)=x^{4}+bx^{3}+cx^{2}+dx+e. $$</li>... |
2,425,916 | <p>What is this set describing?</p>
<p>{$n∈\mathbb{N}|n\ne 1$ and for all $a∈\mathbb{N}$ and $b∈\mathbb{N},ab=n$ implies $a= 1$ or $b= 1$}</p>
<p>Is it describing a subset of natural numbers, excluding 1, that is the product of two other natural numbers, of which one must be 1? Isn't that just every natural number ex... | fleablood | 280,126 | <p>!</p>
<p>"Is it describing a subset of natural numbers, excluding 1, that is the product of two other natural numbers, of which one must be 1? Isn't that just every natural number except 1?"</p>
<p>Almost. It is saying that if $n = ab$, no matter which $a$ or $b$ you choose, it <em>must</em> be that either $a$ or... |
1,591,278 | <p>$$f(x) =\frac{ (x - 1) }{(x^2 - 1)}$$
$$g(x) = \frac{1}{(x + 1)}$$</p>
<p>Is $f = g$? why?</p>
<p>Solution: answer is no.
I don't get it. why can't it be same when it is the alt form?? Can anyone explain this further...</p>
<p>additional questions for further clarification:
$$f(x) = \lim_{x\to 1}\frac{ (x^2 - 1) ... | SchrodingersCat | 278,967 | <p>No, they are not the same because the <a href="https://en.wikipedia.org/wiki/Domain_of_a_function" rel="nofollow">domain of definition</a> of both the functions are different.</p>
<p>The domain of f $=\{x|x\in \mathbb{R} \, \land \, x \not = 1,-1\}$ whereas domain of g $=\{x|x\in \mathbb{R} \, \land \, x \not = -1 ... |
147,181 | <p>I can't figure out how to make ListDensityPlot with both logarithmic coloring and logarithmic scale on both x and y axis.</p>
<p><a href="https://mathematica.stackexchange.com/questions/36830/logarithmic-scale-in-a-densityplot-and-its-legend">Logarithmic scale in a DensityPlot and its legend</a></p>
<p>This quest... | Milad P. | 39,744 | <p>You can manually Plot the log of your function and specify <a href="https://reference.wolfram.com/language/ref/Ticks.html" rel="nofollow noreferrer">Ticks</a> (e.g. <code>Ticks -> {{{1, 10}, {2, 100}, {3, 1000}}, {{1, 10}, {2, 100}, {3, 1000}}}</code>) and <a href="https://reference.wolfram.com/language/ref/Color... |
147,181 | <p>I can't figure out how to make ListDensityPlot with both logarithmic coloring and logarithmic scale on both x and y axis.</p>
<p><a href="https://mathematica.stackexchange.com/questions/36830/logarithmic-scale-in-a-densityplot-and-its-legend">Logarithmic scale in a DensityPlot and its legend</a></p>
<p>This quest... | Carl Woll | 45,431 | <p>In M11.2 you can use <a href="http://reference.wolfram.com/language/ref/ScalingFunctions" rel="nofollow noreferrer"><code>ScalingFunctions</code></a> to make the ticks logarithmic, for example:</p>
<pre><code>DensityPlot[
Sin[x +y]^2 ,{x, 1, 20}, {y, 1, 20},
ScalingFunctions -> {"Log", "Log", "Linear"},
... |
2,486,095 | <p>Given an acute-angled triangle $\Delta ABC$ having it's Orthocentre at $H$ and Circumcentre at $O$. Prove that $\vec{HA} + \vec{HB} + \vec{HC} = 2\vec{HO}$</p>
<p>I realise that $\vec{HO} = \vec{BO} + \vec{HB} = \vec{AO} + \vec{HA} =\vec{CO} + \vec{HC}$ which leads to $3\vec{HO} = (\vec{HA} + \vec{HB} + \vec{HC}) +... | Lutz Lehmann | 115,115 | <p>You just need to iterate further. Using python this is quickly implemented as</p>
<pre><code>x,y,z = 0.,0.,0.; last=3.
for k in range(20):
x=(6-2*y+z)/5;
y=(x+13+2*z)/10;
z=(26+y-4*x)/8;
norm = max(abs(u) for u in [x-1,y-2,z-3])
print " %2d: (%.12f,%.12f,%.12f) (%+.8f, %+.8f, %+.8f), %.8f... |
4,221,545 | <p>Let <span class="math-container">$f$</span> be a twice-differentiable function on <span class="math-container">$\mathbb{R}$</span> such that <span class="math-container">$f''$</span> is continuous. Prove that
<span class="math-container">$f(x) f''(x) < 0$</span> cannot hold for all <span class="math-container">$x... | Hagen von Eitzen | 39,174 | <p>Assume wlog that <span class="math-container">$f''(0)>0$</span>. Then <span class="math-container">$f(0)<0$</span> and as neither factor can change sign, <span class="math-container">$f''(x)>0>f(x)$</span> for all <span class="math-container">$x\in\Bbb R$</span> and <span class="math-container">$f$</spa... |
1,969,934 | <p>Taken from <a href="https://en.wikipedia.org/wiki/E_(mathematical_constant)" rel="nofollow noreferrer">Wikipedia</a>:</p>
<blockquote>
<p>The number $e$ is the limit $$e = \lim_{n \to \infty} \left (1 + \frac{1}{n} \right)^n$$</p>
</blockquote>
<p><strong>Graph of $f(x) = \left (1 + \dfrac{1}{x} \right)^x$</stro... | Community | -1 | <p>Notice that</p>
<p>$$\left(1+\dfrac{1}{n}\right)^n=e^{n\ln\left(1+\frac{1}{n}\right)}$$</p>
<p>Can you solve the limit now?</p>
|
1,521,739 | <p>The following is the Meyers-Serrin theorem and its proof in Evans's <em>Partial Differential Equations</em>:</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/XnzXY.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XnzXY.png" alt="enter image description here"></a>
<a href="https://i.stack... | heropup | 118,193 | <p>Assuming you have done the coordinate transformation correctly, then the basic idea is that you calculate the vertex and focus of the transformed parabola, then perform the inverse transformation on those coordinates to recover the vertex and focus in the untransformed (original) coordinates.</p>
<p>So for example,... |
3,597,829 | <p>Edit: I am using the natural logarithm in what follows.</p>
<p>I want to figure out how to show by hand that the maximum of
<span class="math-container">$$\log(4)c+\log(3)a+\log(2)x$$</span>
when
<span class="math-container">$$a\geq 0, c\geq 0, x \geq 0, y \geq 0,$$</span>
<span class="math-container">$$a+c+x+y=1,... | mathlove | 78,967 | <p><span class="math-container">$\frac{\log(6)}{2}+\frac{\sqrt{1-4\gamma}}{2}\log(3/2)$</span> is not the maximum of <span class="math-container">$$f(c,a,x):=\log(4)c+\log(3)a+\log(2)x$$</span></p>
<p>For example, for <span class="math-container">$\gamma=4/25$</span>, we have
<span class="math-container">$$f(4/5,0,0)=... |
145,046 | <p>I'm a first year graduate student of mathematics and I have an important question.
I like studying math and when I attend, a course I try to study in the best way possible, with different textbooks and moreover I try to understand the concepts rather than worry about the exams. Despite this, months after such an in... | Brian Rushton | 51,970 | <p>When a newborn baby sees his mothers face</p>
<p>The first of all objects, the eyes first light</p>
<p>It may forget</p>
<p>It may pass on</p>
<p>But it will not forget all</p>
<p>Something remains, a hidden sight.</p>
<p>Seen again, again, again,</p>
<p>Her face becomes familiar</p>
<p>Then sought for</p>
... |
325,337 | <p>There's a continuous function $f: \mathbb{Q} \rightarrow \mathbb{R}$</p>
<p>Prove that $ \exists t \in \mathbb{R} \setminus \mathbb{Q} \ \ \exists g \in \mathbb{R} :\ \ \lim _{q \rightarrow t, \ q\in \mathbb{Q}}f(q)=g$.</p>
<p>Prove that there are $2^{\aleph _0}$ such numbers $t$.</p>
<p>I know that if a functio... | Brian M. Scott | 12,042 | <p>Let $\Bbb P=\Bbb R\setminus\Bbb Q$, and for $x\in\Bbb R$ let $B(x,\epsilon)=(x-\epsilon,x+\epsilon)$. Suppose that the first result is false. Then for each $x\in\Bbb P$ there is an $n(x)\in\Bbb N$ such that for each $\epsilon>0$ there are $p,q\in B(x,\epsilon)\cap\Bbb Q$ with $|f(p)-f(q)|\ge 2^{-n}$. For $k\in\Bb... |
898,034 | <blockquote>
<p>Solve $$(x-3)\left(\frac{\mathrm dy}{\mathrm dx}\right)+y=6e^x, x>0$$</p>
</blockquote>
<p>I have a very similar problem like this on my homework, and I have no clue how to set it up or even start. How could I set this up?</p>
| Sidharth Ghoshal | 58,294 | <p>This is a standard linear ordinary differential equation</p>
<p>To solve it we note:</p>
<p>$$ \frac{dy}{dx} + \frac{1}{x - 3}y = \frac{1}{x-3}6e^x$$</p>
<p>We now seek an integration factor $w(x)$ which can be multiplied on both sides to form (just observing left hand side)</p>
<p>$$w(x) \frac{dy}{dx} + \frac{w... |
898,034 | <blockquote>
<p>Solve $$(x-3)\left(\frac{\mathrm dy}{\mathrm dx}\right)+y=6e^x, x>0$$</p>
</blockquote>
<p>I have a very similar problem like this on my homework, and I have no clue how to set it up or even start. How could I set this up?</p>
| evinda | 75,843 | <p>$$(x-3) \frac{dy}{dx}+y=0 \Rightarrow \frac{dy}{dx}=\frac{-y}{x-3} \Rightarrow -\frac{dy}{y}=\frac{dx}{x-3} \Rightarrow \int \left ( \frac{-1}{y} \right)dy=\int \frac{1}{x-3} dx \\ \Rightarrow -\ln |y| =\ln |x-3|+c \Rightarrow e^{-\ln |y|}=e^{\ln |x-3|+c} \Rightarrow \frac{1}{|y|}=C |x-3| \Rightarrow |y|=\frac{c'}{|... |
898,034 | <blockquote>
<p>Solve $$(x-3)\left(\frac{\mathrm dy}{\mathrm dx}\right)+y=6e^x, x>0$$</p>
</blockquote>
<p>I have a very similar problem like this on my homework, and I have no clue how to set it up or even start. How could I set this up?</p>
| Mary Star | 80,708 | <p>$$(x-3)(\frac{dy}{dx})+y=6e^x \Rightarrow \frac{dy}{dx}+\frac{1}{x-3}y=\frac{6 e^x}{x-3} \Rightarrow y'(x)+\frac{1}{x-3}y(x)=\frac{6 e^x}{x-3} $$</p>
<p>The solution of the homogeneous problem is the following:</p>
<p>$$y'_h(x)+\frac{1}{x-3}y_h(x)=0 \Rightarrow \frac{dy_h}{dx}=-\frac{1}{x-3}y_h \Rightarrow \frac{d... |
898,034 | <blockquote>
<p>Solve $$(x-3)\left(\frac{\mathrm dy}{\mathrm dx}\right)+y=6e^x, x>0$$</p>
</blockquote>
<p>I have a very similar problem like this on my homework, and I have no clue how to set it up or even start. How could I set this up?</p>
| izœc | 83,639 | <p>There are plenty of good standard-type answers here about solving the general linear differential equation of the form you are presented with; I just want to make a small note that, from a calculus perspective, finding a specific solution you needn't bother with the high-powered approach of integrating factors, but ... |
4,031,633 | <p>Let <span class="math-container">$R$</span> be a commutative ring and <span class="math-container">$M$</span> is an <span class="math-container">$R$</span>-module. The following statement is well-known.</p>
<p>If <span class="math-container">$M$</span> is finitely presented flat module, <span class="math-container">... | Minseon Shin | 7,719 | <p>Set <span class="math-container">$I := J(R)$</span>. Suppose <span class="math-container">$M/IM$</span> has rank <span class="math-container">$n$</span> as a free <span class="math-container">$A/I$</span>-module; let <span class="math-container">$x_{1},\dotsc,x_{n} \in M$</span> be elements whose images in <span cla... |
1,834,756 | <p>The Taylor expansion of the function $f(x,y)$ is:</p>
<p>\begin{equation}
f(x+u,y+v) \approx f(x,y) + u \frac{\partial f (x,y)}{\partial x}+v \frac{\partial f (x,y)}{\partial y} + uv \frac{\partial^2 f (x,y)}{\partial x \partial y}
\end{equation}</p>
<p>When $f=(x,y,z)$ is the following true?</p>
<p>$$\begin{alig... | Fei PAN | 677,479 | <p>Let <span class="math-container">$f$</span> be an infinitely differentiable function in some open neighborhood around <span class="math-container">$(x,y,z)=(a,b,c)$</span>.</p>
<p><span class="math-container">$f( x,y,z) =f\left( a,b,c \right) +\left( x-a,y-b,z-c \right) \cdot \left( \begin{array}{c}
\frac{\partial... |
1,594,622 | <p>As an example, calculate the number of $5$ card hands possible from a standard $52$ card deck. </p>
<p>Using the combinations formula, </p>
<p>$$= \frac{n!}{r!(n-r)!}$$</p>
<p>$$= \frac{52!}{5!(52-5)!}$$</p>
<p>$$= \frac{52!}{5!47!}$$</p>
<p>$$= 2,598,960\text{ combinations}$$</p>
<p>I was wondering what the l... | Alex R. | 22,064 | <p>$n!$: the number of ways to select $n$ cards (from $n$ cards), such that order matters.</p>
<p>$\frac{n!}{(n-r)!}=n(n-1)(n-2)\cdots (n-r+1):$ the number of ways of selecting $r$ cards from $n$ cards such that order matters.</p>
<p>$r!$: The number of possible orderings of $r$ cards.</p>
<p>We <em>don't</em> want ... |
1,594,622 | <p>As an example, calculate the number of $5$ card hands possible from a standard $52$ card deck. </p>
<p>Using the combinations formula, </p>
<p>$$= \frac{n!}{r!(n-r)!}$$</p>
<p>$$= \frac{52!}{5!(52-5)!}$$</p>
<p>$$= \frac{52!}{5!47!}$$</p>
<p>$$= 2,598,960\text{ combinations}$$</p>
<p>I was wondering what the l... | Graham Kemp | 135,106 | <p>There are $52!$ distinct ways to shuffle a deck of $52$ cards.</p>
<p>However, we don't care about the order of the top $5$ or the bottom $47$ cards, we only wish to count ways to select distinct sets of cards for the hand.</p>
<p>Every deck is one of a group of $5!$ which differ only by the order of the top $5$ c... |
639,665 | <p>How can I calculate the inverse of $M$ such that:</p>
<p>$M \in M_{2n}(\mathbb{C})$ and $M = \begin{pmatrix} I_n&iI_n \\iI_n&I_n \end{pmatrix}$, and I find that $\det M = 2^n$. I tried to find the $comM$ and apply $M^{-1} = \frac{1}{2^n} (comM)^T$ but I think it's too complicated.</p>
| lennon310 | 120,707 | <p>$M^{-1} = \begin{pmatrix} 0.5I_n&-0.5iI_n \\-0.5iI_n&0.5I_n \end{pmatrix}$</p>
|
342,096 | <p>$$\left( \left( 1/2\,{\frac {-\beta+ \sqrt{{\beta}^{2}-4\,\delta\,
\alpha}}{\alpha}} \right) ^{i}- \left( -1/2\,{\frac {\beta+ \sqrt{{
\beta}^{2}-4\,\delta\,\alpha}}{\alpha}} \right) ^{i} \right) \left(
\sqrt{{\beta}^{2}-4\,\delta\,\alpha} \right) ^{-1}$$</p>
<p>What exactly is the connection between the above ... | xavierm02 | 10,385 | <p>Let $F_0=0$, $F_1=1$, $\forall n \in \Bbb N, F_{n+2}=F_{n+1}+F_n$</p>
<p>It's the Fibonacci sequence.</p>
<p>Now let $\forall n \in \Bbb N,V_n=\begin{pmatrix}
F_n\\F_{n+1}\end{pmatrix}$</p>
<p>Let $M=\begin{pmatrix}0 & 1\\ 1 & 1\end{pmatrix}$</p>
<p>Then $MV_n=\begin{pmatrix}0 & 1\\ 1 & 1\end{pma... |
342,096 | <p>$$\left( \left( 1/2\,{\frac {-\beta+ \sqrt{{\beta}^{2}-4\,\delta\,
\alpha}}{\alpha}} \right) ^{i}- \left( -1/2\,{\frac {\beta+ \sqrt{{
\beta}^{2}-4\,\delta\,\alpha}}{\alpha}} \right) ^{i} \right) \left(
\sqrt{{\beta}^{2}-4\,\delta\,\alpha} \right) ^{-1}$$</p>
<p>What exactly is the connection between the above ... | Will Jagy | 10,400 | <p>If you start with $x^2 - x - 1$ and make it homogeneous, $x^2 - x y - y^2,$ the result relates to Fibonacci numbers in that $$ F_{n+1}^2 - F_{n+1} F_n - F_n^2 = (-1)^n. $$ This uses numbering $$ F_0 = 0, \; F_1 = 1, \; F_2 = 1, \; F_3 = 2, \; F_4 = 3, \; F_5 = 5, \ldots $$ and $$ x = F_{n+1}, \; y = F_n. $$</... |
23,312 | <p>What is the importance of eigenvalues/eigenvectors? </p>
| Herb | 7,398 | <p>I think if you want a better answer, you need to tell us more precisely what you
may have in mind: are you interested in theoretical aspects of eigenvalues; do
you have a specific application in mind? Matrices by themselves are just arrays of
numbers, which take meaning once you set up a context. Without the c... |
23,312 | <p>What is the importance of eigenvalues/eigenvectors? </p>
| Doc | 86,414 | <p>I would like to direct you to an answer that I posted here: <a href="https://math.stackexchange.com/questions/357340/importance-of-eigenvalues/494168#494168">Importance of eigenvalues</a></p>
<p>I feel it is a nice example to motivate students who ask this question, in fact I wish it were asked more often. Persona... |
23,312 | <p>What is the importance of eigenvalues/eigenvectors? </p>
| Srijit | 389,895 | <p>Lets Go Back to the Historical Background to get the motivation!</p>
<p>Consider The Linear Transformation T:V->V.</p>
<p>Given a Basis of V, T is characterized beautifully by a Matrix A,which tells everything about T.</p>
<p>The Bad part about A is that " A changes with the change of Basis of V ".</p>
<p>Why it... |
885,129 | <p>I want to speed up the convergence of a series involving rational expressions the expression is $$\sum _{x=1}^{\infty }\left( -1\right) ^{x}\dfrac {-x^{2}-2x+1} {x^{4}+2x^{2}+1}$$
If I have not misunderstood anything the error in the infinite sum is at most the absolute value of the last neglected term. The formula ... | Brad | 108,464 | <p>Here is the first step toward making the series evaluate quicker. You may need just as many terms but it will be faster. Break the series up through partial fractions.</p>
<p>$$\dfrac {-x^{2}-2x+1} {x^{4}+2x^{2}+1} = -\frac{2 (x-1)}{\left(x^2+1\right)^2}-\frac{1}{x^2+1}$$</p>
<p>I will focus on the second term in ... |
1,918,144 | <p>Suppose $\sum\limits_{k=0}^{\infty} b_k $ converges absolutely and has the sum $b$. Suppose $a \in\mathbb R$ with $|a|<1$. What is the sum of the series $\sum\limits_{k=0}^{\infty}(a^kb_0 +a^{k-1}b_1 +a^{k-2}b_2 +...+b_k)$?</p>
| kobe | 190,421 | <p>It's $\frac{b}{1-a}$. Since the series $\sum\limits_{k = 0}^\infty a^k$ and $\sum\limits_{k = 0}^\infty b_k$ converge absolutely, the Cauchy product of the two series converges to $\sum\limits_{k = 0}^\infty a^k\cdot \sum\limits_{k = 0}^\infty b_k = \frac{1}{1-a}\cdot b = \frac{b}{1-a}$. The Cauchy product is </p>
... |
4,079,711 | <p>Calculate <span class="math-container">$$\iint_D (x^2+y)\mathrm dx\mathrm dy$$</span> where <span class="math-container">$D = \{(x,y)\mid -2 \le x \le 4,\ 5x-1 \le y \le 5x+3\}$</span> by definition.</p>
<hr />
<p>Plotting the set <span class="math-container">$D$</span>, we notice that it is a parallelogram. I tried... | D_S | 28,556 | <p>Suppose your definition of the product topology on <span class="math-container">$X \times Y$</span> is that whose open sets are unions of the sets of the form <span class="math-container">$U \times V$</span> for <span class="math-container">$U, V$</span> open in <span class="math-container">$X, Y$</span>.</p>
<p>Let... |
1,472,916 | <p>I am doing it in the following way. Is it correct?</p>
<p>Set $S = \{1,\pi,\pi^2,\pi^3,...,\pi^n\}$ is LI over $\mathbb Q$</p>
<p>Suppose
$a_0\times 1 + a_1\times\pi + ... a_n\times \pi^n = 0$ where all the a_i's are not $0$.</p>
<p>Then $\pi$ is a root of $a_0 + a_1x + ... + a_nx^n = 0$
which is imposible since... | Scientifica | 164,983 | <p>It seems correct to me, but you have to correct some few things:</p>
<ol>
<li><p>You said "Set $S=\{1,\pi,\dots,\pi^n\}$ is LI over $\mathbb Q$" and then you proved that it is LI and that's a bad way to write things. Instead you can say: "Set $S=\{1,\pi,\dots,\pi^n\}$. Let's prove that $S$ is LI over $\mathbb Q$".<... |
2,592,282 | <p>If $f$and $g$ is one to one then ${f×g}$ and ${f+g}$ is one to one </p>
<p>Is this true? If not, I need some clarification to understand </p>
<p>Thanks</p>
| quasi | 400,434 | <p>Let $f,g:\mathbb{R}\to\mathbb{R}$ be given by $f(x) = x$, and $g(x) = x$. Then $f,g$ are one-to-one, but $fg$ is not one-to-one.
<p>
Let $f,g:\mathbb{R}\to\mathbb{R}$ be given by $f(x) = x$, and $g(x) = -x$. Then $f,g$ are one-to-one, but $f+g$ is not one-to-one.
<p>
To the OP:
<p>
What are the simplest functions w... |
4,055,850 | <p>I was thinking it could be plugging in <span class="math-container">$x^2+x$</span> to the <span class="math-container">$f'(x)$</span>, then using the Chain Rule to solve it, but I'm not sure if it is right. Please help!</p>
| ryang | 21,813 | <p>Among</p>
<ul>
<li><span class="math-container">$r\,\mathrm{cis}(\pi+\theta),$</span></li>
<li><span class="math-container">$r\,\mathrm{cis}(\pi-\theta),$</span> and</li>
<li><span class="math-container">$r\,\mathrm{cis}(2\pi-\theta),$</span></li>
</ul>
<p>only the third one refers to the complex conjugate of <span ... |
2,929,887 | <p>Show that if a square matrix <span class="math-container">$A$</span> satisfies the equation <span class="math-container">$p(A)=0$</span>, where <span class="math-container">$p(x) = 2+a_1x+a_2x^2+...+a_kx^k$</span> where <span class="math-container">$a_1,a_2,...,a_k$</span> are constant scalars, then <span class="mat... | Fred | 380,717 | <p>Let <span class="math-container">$u \in ker(A)$</span>, then <span class="math-container">$0=p(A)u=2u+a_1Au+...+a_kA^ku=2u$</span>, hence <span class="math-container">$u=0$</span>. This shows that <span class="math-container">$A$</span> is invertble.</p>
<p>Can you proceed ?</p>
|
2,929,887 | <p>Show that if a square matrix <span class="math-container">$A$</span> satisfies the equation <span class="math-container">$p(A)=0$</span>, where <span class="math-container">$p(x) = 2+a_1x+a_2x^2+...+a_kx^k$</span> where <span class="math-container">$a_1,a_2,...,a_k$</span> are constant scalars, then <span class="mat... | P Vanchinathan | 28,915 | <p>You have a polynomial with non-zero constant term satisfied by <span class="math-container">$A$</span>.
What you have to do is write reformulate <span class="math-container">$p(A)=0$</span> as
<span class="math-container">$2I=-(a_1A+a_2A^2+\cdots +a_kA^k )$</span>.
This means <span class="math-container">$I = -\frac... |
4,084,486 | <p>I am studying different integral transform methods, and I am confused on why saying things such as
<span class="math-container">$$
\mathcal{F}^{-1}[1] = \delta(x)
$$</span>
is valid? If you actually plug this in,
<span class="math-container">$$
\mathcal{F}^{-1}[1] = \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{ikx}dx
$$<... | Mark Viola | 218,419 | <p>The Fourier Transform is a <a href="https://en.wikipedia.org/wiki/Fourier_transform#Tempered_distributions" rel="noreferrer">Tempered Distribution</a> (See <a href="https://en.wikipedia.org/wiki/Distribution_(mathematics)#Tempered_distributions_and_Fourier_transform" rel="noreferrer">THIS</a>). Hence, we interpret ... |
2,780,216 | <p>How many integer solutions are there to the equation $a + b + c = 21$ if $a \geq3$,
$b \geq 1$, and $c \geq 2b$?</p>
| farruhota | 425,072 | <p>Brute forcing:
$$\begin{align}(a,b,c)=&(3,1,17), (3,2,16), (3,3,15), (3,4,14), (3,5,13), (3,6,12),\\
&(4,1,16), (4,2,15), (4,3,14), (4,4,13), (4,5,12),\\
&(5,1,15), (5,2,14), (5,3,13), (5,4,12), (5,5,11),\\
&(6,1,14), (6,2,13), (6,3,12), (6,4,11),\\
&(7,1,13), (7,2,12), (7,3,11), (7,4,10),\\
&... |
1,500,156 | <p>Call a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ nondecreasing if $x,y \in \mathbb{R}^n$ with $x \geq y$ implies $f(x) \geq f(y)$. Suppose $f$ is a nondecreasing, but not necessarily continuous, function on $\mathbb{R}^n$, and $D \subset \mathbb{R}^n$ is compact. Show that if $n = 1$, $f$ always has a maximu... | user2566092 | 87,313 | <p>So if you define $f(x) = -\infty, x \leq b$ and $f(x) = 1/(x-b),b < x < a$ and $f(x) = + \infty, x \geq a$ then $f$ is increasing on $(b,a)$ but does not attain a finite maximum.</p>
|
4,490,664 | <p>I want to examine the number of non-trivial zeros of <span class="math-container">$\zeta(s)$</span> in the small substrip given by <span class="math-container">$a \leq \Re s \leq 1$</span>, where <span class="math-container">$a>0$</span> is an absolute constant, and <span class="math-container">$U \leq \Im s \leq... | davidlowryduda | 9,754 | <p>It is known that there are on the order of <span class="math-container">$T \log T$</span> zeros up to height <span class="math-container">$T$</span> lying <em>exactly on</em> the critical line <span class="math-container">$\mathrm{Re} s = \tfrac{1}{2}$</span>. It is possible to show that there are at most <span clas... |
2,370,716 | <p>Give the postive integer $n\ge 2$,and $x_{i}\ge 0$,such $$x_{1}+x_{2}+\cdots+x_{n}=1$$
Find the maximum of the value
$$x^2_{1}+x^2_{2}+\cdots+x^2_{n}+\sqrt{x_{1}x_{2}\cdots x_{n}}$$</p>
<p>I try
$$x_{1}x_{2}\cdots x_{n}\le\left(\dfrac{x_{1}+x_{2}+\cdots+x_{n}}{n}\right)^n=\dfrac{1}{n^n}$$</p>
| Community | -1 | <p>When the function is symmetric in several variables, I believe it can be shown the extrema occur at points where the $x_i $ are all equal. Then $x_i=\frac1n $ for each i.Then there's the matter of whether it's a max or a min. Putting that aside:
Well, we get $$n×\frac{1}{n^2}+\sqrt {\left(\frac1n \right)^n} $$ $$=... |
2,586,625 | <p>Consider the set $$S=\{x\mid x\in S\}.$$</p>
<p>For every element $x$, either $x\in S$ or $x\not\in S.$</p>
<p>If we know that $x$ is in fact element of $S$, then, by definition, $x\in S$ so it is true that $x\in S$.</p>
<p>If we know that $x\not\in S$, then, by definition, $x\not\in S$ so it is true that $x\not\... | Patrick Stevens | 259,262 | <p>You haven't actually defined a set there. You've written down some symbols and are hoping that they define a set, but they don't.</p>
<p>The axiom schema which lets you define sets as "the things which satisfy some property" is the Axiom Schema of Comprehension: roughly, if $\phi$ is a predicate and $S$ is a set, t... |
2,586,625 | <p>Consider the set $$S=\{x\mid x\in S\}.$$</p>
<p>For every element $x$, either $x\in S$ or $x\not\in S.$</p>
<p>If we know that $x$ is in fact element of $S$, then, by definition, $x\in S$ so it is true that $x\in S$.</p>
<p>If we know that $x\not\in S$, then, by definition, $x\not\in S$ so it is true that $x\not\... | Ethan Bolker | 72,858 | <p>This is an interesting question.</p>
<p>A set is determined by the elements it contains. That's the intuitive content of the assertion that two sets are equal if and only if they have the same elements.</p>
<p>So the set you're asking about is just $S$. It contains what it contains.</p>
<p>I think you are a littl... |
2,701,582 | <p>I thought I was doing this right until I checked my answer online and got a different one. I worked through the problem again and got my original answer a second time so this one is bothering me since the other similar ones I have done checked out okay. Please let me know if I'm doing something wrong, thanks!</p>
<... | Ewan Delanoy | 15,381 | <p>Answer : no, of course not. For example $\sigma=(1,2,\ldots,n)$ and $\tau=(1,2)$ together generate $S_n$, but the subsets of $A=\lbrace \sigma, \tau \rbrace$ only give four subgroups of $S_n$, and there are many more subgroups.</p>
|
4,362,221 | <p>Show that for all <span class="math-container">$z$</span>, <span class="math-container">$\overline{e^z} = e^\bar{z}$</span></p>
<p>I'm a little stuck with this one.</p>
<p>First defined the following to help solve it:
<span class="math-container">$$
z = a + bi
$$</span>
then plugging that in to the question gives
<s... | user2661923 | 464,411 | <p><span class="math-container">$\displaystyle \overline{e^{x + iy}} =
\overline {e^x \times e^{iy}} =
\overline {e^x \times [\cos(y) + i\sin(y)]}$</span></p>
<p><span class="math-container">$\displaystyle = e^x \times [\cos(y) - i\sin(y)].$</span></p>
<hr />
<p><span class="math-container">$\displaystyle e^{\overlin... |
556,423 | <blockquote>
<p>Determine using multiplication/division of power series (and not via WolframAlpha!) the first three terms in the Maclaurin series for $y=\sec x$.</p>
</blockquote>
<p>I tried to do it for $\tan(x)$ but then got kind of stuck. For our homework we have to do it for the $\sec(x)$. It is kind of tricky. ... | Community | -1 | <p>Let's write</p>
<p>$$1 = \cos{x} \sec{x} = \left(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots\right)(a_0 + a_1 x + a_2x^2 + \dots)$$</p>
<p>Expand the right hand side to find that</p>
<p>$$1 = 1 (a_0) + x (a_1) + x^2 \left(a_2 - a_0 \frac{1}{2!}\right) + x^3 \left(a_3 - a_1 \frac{1}{2!}\right) + x^4 \left(a_4 - a_... |
128,122 | <p>Original Question: Suppose that $X$ and $Y$ are metric spaces and that $f:X \rightarrow Y$. If $X$ is compact and connected, and if to every $x\in X$ there corresponds an open ball $B_{x}$ such that $x\in B_{x}$ and $f(y)=f(x)$ for all $y\in B_{x}$, prove that f is constant on $X$. </p>
<p>Here's my attempt:
Cover ... | Brian M. Scott | 12,042 | <p>You’ve not yet used the fact that $X$ is connected, and that’s essential. Otherwise, $X$ and $Y$ could both be the two-point discrete space, and $f$ could be the identity map; all of the hypotheses except connectedness of $X$ would be satisfied, but $f$ wouldn’t be constant.</p>
<p>Let’s go back to your finite cove... |
1,354,015 | <blockquote>
<p>If <span class="math-container">$a, b, c, d, e, f, g, h$</span> are positive numbers satisfying <span class="math-container">$\frac{a}{b}<\frac{c}{d}$</span> and <span class="math-container">$\frac{e}{f}<\frac{g}{h}$</span> and <span class="math-container">$b+f>d+h$</span>, then <span class="... | Keith | 244,951 | <p>This is false.</p>
<p>For example,
$$\frac{1}{3} < \frac{5}{12}, \quad \frac{52}{5} < \frac{11}{1}, \quad \frac{53}{8} > \frac{16}{13}.$$</p>
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.