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 |
|---|---|---|---|---|
3,148,803 | <p>I am reading a paper in which the following result is used, but I can’t see the proof of this.</p>
<blockquote>
<p>Let <span class="math-container">$R$</span> be a commutative ring with only two maximal ideals, say <span class="math-container">$M_1$</span> and <span class="math-container">$M_2$</span>. Suppose <s... | user647486 | 647,486 | <p>Take <span class="math-container">$R=\mathbb{Q}\times\mathbb{Q}$</span>, <span class="math-container">$M_1=\mathbb{Q}\times\{0\}$</span>, <span class="math-container">$M_2=\{0\}\times\mathbb{Q}$</span>, and <span class="math-container">$m_1=(2,0)\in M_1\setminus M_2$</span>. Then <span class="math-container">$(1,1)\... |
4,049,205 | <p>Let <span class="math-container">$\{a_n\}:\mathbb{N}\to\mathbb{R}$</span>, and we define <span class="math-container">$E$</span> to be the set of all sub-sequential limits of <span class="math-container">$\{a_n\}$</span> as well as possibly the symbols <span class="math-container">$\infty$</span> and <span class="ma... | Brian M. Scott | 12,042 | <p>Let <span class="math-container">$\sigma=\langle a_n:n\in\Bbb N\rangle$</span> be a sequence in <span class="math-container">$\Bbb R$</span>. If <span class="math-container">$\sigma$</span> is unbounded, then it has a subsequence converging to <span class="math-container">$\infty$</span> or to <span class="math-cont... |
1,117,081 | <p>In number theory we have so-called explicit formula's in terms of the Riemann zeta zero's.
For instance to count the sum of the logarithms of the primes below some given integer.</p>
<p>(second Chebyshev Function)</p>
<p>Consider the floor function : <a href="http://mathworld.wolfram.com/FloorFunction.html" rel="n... | e2theipi2026 | 46,264 | <p>I don't think your going to find an explicit formula for the floor function and here is why,</p>
<p>$nextprime(n)= 1+\sum_{j=1}^{2n}\lfloor\frac{n!^j}{j!}\rfloor-\lfloor\frac{n!^j-1}{j!}\rfloor$ </p>
<p>$prevprime(n)=n+1-\sum_{j=1}^{n}\lfloor\frac{j!^{n-1}}{(n-1)!}\rfloor-\lfloor\frac{j!^{n-1}-1}{(n-1)!}\rfloor$</... |
1,117,081 | <p>In number theory we have so-called explicit formula's in terms of the Riemann zeta zero's.
For instance to count the sum of the logarithms of the primes below some given integer.</p>
<p>(second Chebyshev Function)</p>
<p>Consider the floor function : <a href="http://mathworld.wolfram.com/FloorFunction.html" rel="n... | Steven Clark | 371,769 | <p>I've derived an explicit formula for $S(x)=Floor(x)$ from the explicit formula for $Q(x)$ based on the relationship between $S(x)$ and $Q(x)$. The $c(n)$ coefficient referenced in (3) below involves both a Dirichlet convolution and a Dirichlet inverse, but I believe it simplifies such that $c(n)=1$ when $n$ is a squ... |
12,239 | <p>I know this is a trivial problem but am stuck with this thing.</p>
<p>$A \in \mathbb{R}^{m \times m}$ is a symmetric positive definite matrix and $C \in \mathbb{R}^{m \times p}$
</p>
<p>How do I show that the matrix $$B = \begin{bmatrix} A & C \\ C^T & 0 \end{bmatrix}$$ is indefinite?</p>
<p>I can show th... | Parthe | 289,408 | <p>Consider $$z = \begin{bmatrix}x\\ y\end{bmatrix}^T$$ with $x \in R^m$ and $y \in R^p$</p>
<p>Then $$f(z) = z^TBz = x^TAx + 2x^TCy$$ which containts a bilinear term, which is linear as a function of $y$. Hence given an $x$, an appropriate $y$ can be chosen such that the $f(z) < 0$</p>
|
4,235,927 | <p>There is a game where you are asked to roll two fair six-sided dice. If the sum of the values equals 7, then win £21. However, must pay £5 to play each time both dice are rolled. Do you play this game?</p>
<p>One way to think about this is that getting a 7 comes with 1/6 chance, and to make money we need to get 7 at... | Adi | 578,455 | <p>There are 36 possible combinations when you roll two dice. How many of these add up to 7? (6+1, 5+2, 4+3, 3+4, 2+5, 1+6) --> 6. So there is a 1/6 chance that you will win.</p>
<p>Let <span class="math-container">$X$</span> be the random variable that denotes the amount you win. With probability 1/6, you win 21 an... |
2,430,378 | <p>Prove that $H = \{1, −1, i, −i\}$ is a subgroup of the group of nonzero complex number under multiplication.</p>
<p>Since $H$ is finite it is enough to show that if $H$ is closed under multiplication then it is a subgroup of $G$. I can easily verify Given any elements of $H$ they are closed. Or that multiplication ... | 5xum | 112,884 | <blockquote>
<p>Or that multiplication of $\pm1$ and $\pm i$ are by definition $\pm 1$ or $\pm i$</p>
</blockquote>
<p>Technically, you also need to show that that and that multiplication of $\pm 1$ and $\pm 1$ is $\pm 1$ or $\pm i$, and that multiplication of $\pm i$ and $\pm i$ is $\pm 1$ or $\pm i$.</p>
<hr>
<p... |
2,430,378 | <p>Prove that $H = \{1, −1, i, −i\}$ is a subgroup of the group of nonzero complex number under multiplication.</p>
<p>Since $H$ is finite it is enough to show that if $H$ is closed under multiplication then it is a subgroup of $G$. I can easily verify Given any elements of $H$ they are closed. Or that multiplication ... | rschwieb | 29,335 | <p>Well... $\mathbb C\setminus\{0\}$ is a group, and it has an element $i$.</p>
<p>This element $i$ generates a cyclic subgroup</p>
<p>$\langle i\rangle=\{i, i^2,i^3,\ldots\}=\{i, -1, -i, 1\}$.</p>
<p>Does anything more need to be said?</p>
<p>Looking at it as a cyclic subgroup at least makes it obvious it's closed... |
1,101,520 | <blockquote>
<p>Evaluate the integral
$$\int\frac{5x^3+3x-1}{(x^3+3x+1)^3}\,dx$$</p>
</blockquote>
<p><strong>My Attempt:</strong></p>
<p>Let $f(x) = \frac{ax+b}{(x^3+3x+1)^2}.$ Now differentiate both side with respect to $x$, and we get</p>
<p>$$
\begin{align}
f'(x) &= \frac{(x^3+3x+1)^2\cdot a-2\cdot (x^3+... | Américo Tavares | 752 | <p><strong>Ostrogradsky-Hermite method.</strong></p>
<p>To integrate a rational function $P(x)/Q(x)$ <em>without decomposing it into partial fractions and without finding the roots of the denominator</em>, we can use the <a href="https://www.encyclopediaofmath.org/index.php/Ostrogradski_method" rel="nofollow">Ostrogra... |
2,176,815 | <p>Find all collision points between the parametric curves $x_1=\cos t$, $y_1=2 \sin t + \sin 2t$ and $x_2=2t$, $y_2= t - \frac12$.</p>
<p>I have no idea of how to proceed from the starting step of setting the $x$ values equal to each other, and the $y$ values equal to each other.</p>
<p>I also don't know what the an... | Nosrati | 108,128 | <p>Let $t$ and $s$ be two values for which
\begin{cases}
\cos t=2s,\\
2 \sin t + \sin 2t=s - \dfrac12
\end{cases}
we have from first $s=\dfrac12\cos t$ and with the second $2 \sin t + \sin 2t=\dfrac12\cos t - \dfrac12$. After solve this we find $t=0$ and $\tan^3\frac{t}{2}+\tan \frac{t}{2}+8=0$. The last give $\tan \f... |
3,671,077 | <p>Hey im first year math student I need to find the limit of <span class="math-container">$(a_n) =\frac{2}{n^{2}}\sum_{k=1}^{n}[kx]$</span></p>
<p>what I tried to do was <span class="math-container">$ \frac{2}{n^{2}}\sum_{k=1}^{n}[kx]=\sum_{k=1}^{n}\frac{2}{n^{2}}[kx]=\frac{2}{n^{2}}[kx]+\frac{2}{n^{2}}[kx]+...+\frac... | Pavel Kozlov | 143,912 | <p><span class="math-container">$$\frac{2}{n^{2}}\sum_{k=1}^{n} (kx-1) \leq a_n \leq \frac{2}{n^{2}}\sum_{k=1}^{n} kx,$$</span>
or
<span class="math-container">$$\frac{2x}{n^{2}}\frac{n(n+1)}{2}-\frac{2}{n} \leq a_n \leq \frac{2x}{n^{2}}\frac{n(n+1)}{2},$$</span>
so the limit is equal to <span class="math-container">$x... |
154,573 | <p>How could we solve $$\sqrt{x} + \ln(x) -1 = 0$$ without using Mathematica? Obviously a solution is $x = 1$, but what are the other exact solutions?</p>
| Mathlover | 22,430 | <p>$\sqrt{x} + \ln(x) -1 = 0$</p>
<p>$x=e^u$</p>
<p>$e^{u/2} + u -1 = 0$</p>
<p>$1-u=e^{u/2}$</p>
<p>$(1-u)e^{-u/2}=1$</p>
<p>$\frac{(1-u)}{2}e^{-u/2}e^{1/2}=\frac{e^{1/2}}{2}$</p>
<p>$\frac{(1-u)}{2}e^{\frac{(1-u)}{2}}=\frac{e^{1/2}}{2}$</p>
<p>$\frac{(1-u)}{2}=p$</p>
<p>$pe^{p}=\frac{1}{2}e^{1/2}$</p>
<p>We ... |
1,027,775 | <p>I'm considering the function </p>
<p>$$\frac{1}{e^{\frac{1}{z}}+2}$$</p>
<p>Clearly, it has a unique singularity at $z=0$. I feel like its an essential but I can't find the Laurent expansion or any other way of proving it... Can you help me?</p>
<p>Thank you very much, I usually see these, but this time I'm stuck... | dantopa | 206,581 | <p>With some imagination, one can see the poles collecting near the origin.</p>
<p>Conformal map of
$$
f(z) = \frac{1}{\exp \left( 1 / z \right) + 2}.
$$</p>
<p>Blue: $\Re (f)$, Red: $\Im (f)$</p>
<p><a href="https://i.stack.imgur.com/MmMix.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/MmMix.... |
2,306,309 | <p>Induction on $n$: </p>
<p>If $n=1$, then $A_{n+2}=A_{3} \simeq \mathbb{Z}_3$ has subgroup $1 \simeq S_n = S_1$. Assume for some $n>1$, $S_n \simeq H$ for some $H \leq A_{n+2}$. I want to show that $S_{n+1} \simeq H'$ for some $H'\leq A_{n+3}$.</p>
<p>Note that $H \simeq S_n \leq S_{n+1}$, and by inductive hypot... | Zain Patel | 161,779 | <p>Send $\sigma \mapsto \sigma$ if $\sigma$ is an even permutation. Let $\tau = (n+1 \, \, \, n+2)$ and send $\sigma \mapsto \sigma \tau $ for odd $\sigma$. This injects $S_n \to A_{n+2}$ and is a homorphism. </p>
|
4,363,566 | <p>I am trying to prove Euler's equation for the case of <span class="math-container">$4x$</span>, that is:
<span class="math-container">$$e^{i4x} = \cos 4x+i \sin 4x.$$</span>
I can get the imaginary part in a form that is equivalent to <span class="math-container">$\sin 4x$</span>, but I am struggling with the real p... | Átila Correia | 953,679 | <p>Is this you are looking for?</p>
<p><span class="math-container">\begin{align*}
\cos(4x) & = 2\cos^{2}(2x) - 1\\\\
& = 2(2\cos^{2}(x) - 1)^{2} - 1\\\\
& = 2(4\cos^{4}(x) - 4\cos^{2}(x) + 1) - 1\\\\
& = 8\cos^{4}(x) - 8\cos^{2}(x) + 1
\end{align*}</span></p>
<p>If you prefer to make progress based on ... |
3,276,351 | <blockquote>
<p>A particle is allowed to move in the <span class="math-container">$\mathbb{Z}\times \mathbb{Z}$</span> grid by choosing any of the two jumps:</p>
<p>1) Move two units to right and one unit up</p>
<p>2) Move two units up and one unit to right.</p>
<p>Let <span class="math-container">$P=(... | Peter Green | 278,485 | <p>Let a be the count of the first move and b be the count of the second move.</p>
<p><span class="math-container">$$(x,y) = (2,1)a + (1,2)b$$</span></p>
<p>Where x an y are nonnegative integers representing our position and a position is reachable if a and b are both nonnegative integers.</p>
<p><span class="math-c... |
1,037,061 | <blockquote>
<p>Find the Consumer Surplus, given the demand and supply equations $$ D(x)=\frac{405}{\sqrt{x}} $$
$$ S(x)=5\sqrt{x} $$
The equilibrium point is $(81,45)$.</p>
</blockquote>
<p>I know the formula for <a href="http://en.wikipedia.org/wiki/Economic_surplus#Calculation_from_supply_and_demand" rel="nof... | boldbrandywine | 122,692 | <p>Just use $u$-substitution. The integral can be evaluated with that method. Let $u=x^2$.</p>
|
1,037,061 | <blockquote>
<p>Find the Consumer Surplus, given the demand and supply equations $$ D(x)=\frac{405}{\sqrt{x}} $$
$$ S(x)=5\sqrt{x} $$
The equilibrium point is $(81,45)$.</p>
</blockquote>
<p>I know the formula for <a href="http://en.wikipedia.org/wiki/Economic_surplus#Calculation_from_supply_and_demand" rel="nof... | Community | -1 | <p>Let $u=x^2\Rightarrow \frac{du}{2}=xdx$. Thus
$$\int xe^{-x^2}dx=\frac{1}{2}\int e^{-u}du=-\frac{1}{2}e^{-u}=-\frac{1}{2}e^{-x^2}$$</p>
<p>Then</p>
<p>$$\int_0^{\infty} xe^{-x^2}dx=-\frac{1}{2}\lim_{b\rightarrow\infty}(e^{-b^2}-e^0)=\frac{1}{2}$$</p>
|
3,854,400 | <p>I have a recurrence relation with a geometric series within it. I want to simplify the series to something more useful.</p>
<p><span class="math-container">$$\left(\frac{3}{2}\right)^0 + \left(\frac{3}{2}\right)^1 + \left(\frac{3}{2}\right)^2 + \dots + \left(\frac{3}{2}\right)^{n-2} + \left(\frac{3}{2}\right)^{n-1}$... | Joe | 107,639 | <p>HINT: compute
<span class="math-container">$$
(1+q+q^2+\cdots+q^{n-1})(1-q)
$$</span></p>
|
3,854,400 | <p>I have a recurrence relation with a geometric series within it. I want to simplify the series to something more useful.</p>
<p><span class="math-container">$$\left(\frac{3}{2}\right)^0 + \left(\frac{3}{2}\right)^1 + \left(\frac{3}{2}\right)^2 + \dots + \left(\frac{3}{2}\right)^{n-2} + \left(\frac{3}{2}\right)^{n-1}$... | Wz S | 677,974 | <p>set
<span class="math-container">$$
S =\left(\frac{3}{2}\right)^{0}+\left(\frac{3}{2}\right)^{1}+\left(\frac{3}{2}\right)^{2}+\cdots+\left(\frac{3}{2}\right)^{n-2}+\left(\frac{3}{2}\right)^{n-1}
$$</span>
then
<span class="math-container">$$\frac{3}{2}S=\left(\frac{3}{2}\right)^{1}+\left(\frac{3}{2}\right)^{2}+\left... |
3,854,400 | <p>I have a recurrence relation with a geometric series within it. I want to simplify the series to something more useful.</p>
<p><span class="math-container">$$\left(\frac{3}{2}\right)^0 + \left(\frac{3}{2}\right)^1 + \left(\frac{3}{2}\right)^2 + \dots + \left(\frac{3}{2}\right)^{n-2} + \left(\frac{3}{2}\right)^{n-1}$... | fleablood | 280,126 | <p><span class="math-container">$1 + a + a^2 + a^3 + .......+a^n = \frac {a^{n+1}-1}{a-1}$</span> whenever <span class="math-container">$a \ne 1$</span>. That's a formula you should know and love.[1]</p>
<p>So plug in <span class="math-container">$a = \frac 32$</span> (and replace <span class="math-container">$n$</spa... |
3,638,206 | <p>Is it true that if <span class="math-container">$m$</span> is a Lebesgue measure on <span class="math-container">$[0,1]$</span> and that <span class="math-container">$\lambda$</span> is the counting measure on <span class="math-container">$[0,1]$</span>, and that both defined on the Lebesgue <span class="math-contai... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$0=m(\{x\})=\int_{\{x\}} h d\lambda =h(x)$</span> so <span class="math-container">$h(x)=0$</span> for all <span class="math-container">$x$</span>. This implies <span class="math-container">$m(E)=0$</span> for all <span class="math-container">$E$</span>, a contradiction. </p>
|
221,867 | <p>For a Hermitian nonnegative-definite matrix $A$, if $Ax$ is always real for any real vector $x$, can we conclude that $A$ is also real?</p>
| Logan Stokols | 41,434 | <p>Imagine if $A$ had some non-real entries. Then pick the column which has those non-real entries, say it is the i'th column. $A e_i$ is equal to the i'th column of $A$, so it has non-real entries. </p>
|
3,075,987 | <p>So I am currently studying Calculus and Linear Algebra and I came across the same concepts that is being applied in a lot of the proofs that I read for Calculus and Linear algebra but not capable of fully understanding it.</p>
<p><strong>Claim:</strong> There exists a <span class="math-container">$\lambda \in \math... | Brian Borchers | 6,310 | <p>If </p>
<p><span class="math-container">$(\nabla F(a)) \cdot u= (\lambda \nabla G(a)) \cdot u$</span> </p>
<p><em>for all unit vectors <span class="math-container">$u$</span></em>, then in particular, we can consider the unit vector <span class="math-container">$u=e^{(i)}$</span> which has a one in the <span class... |
3,111,628 | <blockquote>
<p>Let <span class="math-container">$U \in M_{n,k}(\mathbb{R})$</span> such that : <span class="math-container">$^t UU = I_k$</span>. Then I would like to understand <strong>geometrically</strong> why <span class="math-container">$U ^t U$</span> is the orthogonal projection on <span class="math-container... | Aaron Montgomery | 485,314 | <p>To illustrate the problem with this approach, consider the following rearrangement:
<span class="math-container">$$\text{IINDVIISIIBLTY}$$</span>
You listed five possible events to consider:</p>
<blockquote>
<p>two 'I's are adjacent, three 'I's are adjacent, four 'I's are adjacent, five 'I's are adjacent, six 'I'... |
3,284,801 | <p>Let <span class="math-container">$D\subset \mathbb{R}^2$</span> be the closed unit disk, boundary of <span class="math-container">$D$</span> is <span class="math-container">$\mathbb{S}^1$</span>, let <span class="math-container">$a,b\in \mathbb{S}^1$</span> are two distinct points, <span class="math-container">$A,B\... | William Elliot | 426,203 | <p>No. Let <span class="math-container">$A = \{a\}$</span> and <span class="math-container">$B$</span> a circle inside <span class="math-container">$D$</span> tangent to S<span class="math-container">$^1$</span> at <span class="math-container">$b$</span>.<br>
Let <span class="math-container">$x$</span> be the center o... |
3,595,653 | <p>Assume that I have a scalar variable which takes values on <span class="math-container">$\mathbb{R}$</span> and prescribes polar coordinates (in radians) on, say, a unit circle <span class="math-container">$r=1$</span>. Assume now that I have two such (possibly different) variables <span class="math-container">$a$</... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$(\cos(a)r-\cos(b)r)^2-(\sin(a)r-\sin(b)r)^2$</span> inside the square root sign is not correct. It should<span class="math-container">$(\cos(a)r-\cos(b)r)^2+(\sin(a)r-\sin(b)r)^2$</span> be </p>
<p>Expanding the squares and using the fact that <span class="math-container">$\cos^{2} x+... |
436,316 | <p>Let <span class="math-container">$f\colon\mathbb R^2\to\mathbb R$</span> be a measurable function such that
<span class="math-container">\begin{equation*}
F(t):=\int_{\mathbb R}dx\,f(t,x)
\end{equation*}</span>
exists and is finite for all real <span class="math-container">$t$</span>. Suppose that
<span class="... | Iosif Pinelis | 36,721 | <p><span class="math-container">$\newcommand{\R}{\mathbb R}$</span>Let us give a counterexample with <span class="math-container">$f$</span> continuous and (as a bonus) <span class="math-container">$f_t(t,x)$</span> (infinitely) smooth in <span class="math-container">$t$</span>.</p>
<p>Let <span class="math-container">... |
2,856,568 | <p>In the example below I get the first part. May I know why they're dividing by 2 for the second part? I'm asking because I feel the answer for second part should also be $\binom{10}{5}$. </p>
<p>In the first part simply by choosing 5 people, we're already splitting the squad into two teams. So I don't really see a d... | uniquesolution | 265,735 | <p>Observe that the choice of choosing five people and the other five are counted separately in the first case, but yield the same separation into two groups in the second. That's why there is division by $2$.</p>
|
2,856,568 | <p>In the example below I get the first part. May I know why they're dividing by 2 for the second part? I'm asking because I feel the answer for second part should also be $\binom{10}{5}$. </p>
<p>In the first part simply by choosing 5 people, we're already splitting the squad into two teams. So I don't really see a d... | Bodhi | 143,793 | <p>The second part of your problem will be $\binom{10}{5}$ if the question is of the following form: </p>
<blockquote>
<p>How many ways to split $10$ people into two groups of $5$ people so that the first group is Team A and second group is Team B? </p>
</blockquote>
<p>Here, unlike the original question, we restri... |
1,480,103 | <p>I have an assignment, that beside other things, asks about a basis of a group, more precisely, it have been asked to "give an example of a basis for $T_d$" (tetrahedron group).</p>
<p>But I been on all lectures and searched internet for such a definition, but found nothing.</p>
<p>Any idea what "Group Basis" means... | algbr | 587,554 | <p>R. Lyndon and P. Schupp defined in their book "Combinatory Group Theory" a free group F with basis X in the following way: </p>
<p>"Let X be a subset of a group F. Then F is a free group with basis X
provided the following holds: if $\phi$ is any function from the set X into a group H,
then there exists a unique ex... |
4,270,745 | <blockquote>
<p>I am trying to find this limit:
<span class="math-container">$$\lim_{n \to \infty}\left[\,\sqrt{\,2\,}\,\frac{\Gamma\left(n/2 + 1/2\right)}{\Gamma\left(n/2\right)} - \,\sqrt{\,n\,}\right]$$</span></p>
</blockquote>
<p>I know that
<span class="math-container">$$\frac{\sqrt{2}\Gamma(\frac{n+1}{2})}{\Gamma... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
3,328,969 | <blockquote>
<p>Let <span class="math-container">$a,b \geq 0$</span> with <span class="math-container">$a+b \leq c$</span> for a small constant <span class="math-container">$c$</span> between <span class="math-container">$0$</span> and <span class="math-container">$1$</span>.</p>
<p>What is the minimum of <span class="... | Colliot | 80,507 | <p><span class="math-container">$\cos(a) + \cos(b) = \cos(a) + \cos(c_0 - a)$</span>. The derivative with respect to <span class="math-container">$a$</span> is <span class="math-container">$-\sin(a)+\sin(c_0-a)$</span>. For <span class="math-container">$0 < c_0 \le c < 1, 0 \le a \le c_0$</span>, the derivative i... |
116,196 | <p>Simple Question from a Stumped Rookie: </p>
<p>Using ListPlot, I can't get PlotStyle to look below the first level of a nested list to assign individual saturation values for each point; it assigns the first three saturation values to the first three groups of points. </p>
<pre><code>pts = {{{0.10, 485}, {0.22, 4... | Darrell Plank | 40,625 | <p>Okay - never contributed before so I hope I don't screw up this answer. This will, I believe, do what you're looking for. It just finds all the "junctions" and then repeatedly contracts the nodes of degree 2 around each such junction until they're all gone. reduce[g,v] removes the degree 2 vertices around vertex ... |
194,569 | <blockquote>
<p>How can we find an integer $n$ such that $\mathbb{Z}[\frac{1}{20},\frac{1}{32}]=\mathbb{Z}[\frac{1}{n}]$?</p>
</blockquote>
<p>Thanks in advance!</p>
| Brian M. Scott | 12,042 | <p>HINT: In order to have $\Bbb Z\left[\frac1{20},\frac1{32}\right]\subseteq\Bbb Z\left[\frac1n\right]$, having integers $a$ and $b$ such that $\frac1{20}=\frac{a}n$ and $\frac1{32}=\frac{b}n$ will work nicely. This implies that $n=20a$ and $n=32b$, so $n$ must be a common multiple of $20$ and $32$. On the other hand, ... |
194,569 | <blockquote>
<p>How can we find an integer $n$ such that $\mathbb{Z}[\frac{1}{20},\frac{1}{32}]=\mathbb{Z}[\frac{1}{n}]$?</p>
</blockquote>
<p>Thanks in advance!</p>
| Marc van Leeuwen | 18,880 | <p>Rapid answer: since $\frac1{32}=\frac1{2^5}=\frac{2\times5^3}{20^3}\in\mathbb Z[\frac1{20}]$, one can take $n=20$.</p>
<p>More generally the following considerations apply.</p>
<p>Let $R$ be a subring of $\mathbf Q$, and consider the set $D$ of positive integers $n$ such that $\frac1n\in R$. Then for all <em>reduc... |
1,161,089 | <p>Sorry if this may seem trivial - I just started studying Group Theory. This is the problem:</p>
<blockquote>
<p>Prove that <span class="math-container">$(g,h) \rightarrow hg$</span> does not define a group action with <span class="math-container">$g$</span> acting on <span class="math-container">$h$</span>. Prove in... | paw88789 | 147,810 | <p>It would be $\frac{1}{2^y}\frac{1}{2^{x-y}}$ for $y$ heads and $x-y$ tails. And this is equal to $\frac{1}{2^x}$</p>
|
1,904,786 | <p>How can I get a good approximation of the sum</p>
<p>$$\sum_{s=1}^\infty x^s\ln(s)$$</p>
<p>by hand ?</p>
<p>If I only consider the part of the function, where it is strictly decreasing, I can bound the sum by integrals. But how can I approximate the sum upto this point ?</p>
| H. H. Rugh | 355,946 | <p>You may at least improve convergence by the following manipulation:
With $f(x) =\sum_{s\geq 1} x^s \log(s)$ we have:
$$(1-x) f(x) = \sum_{s\geq 2} x^s \log(s) - \sum_{s\geq 1} x^{s+1} \log(s) = \sum_{s\geq 1} x^{s+1} \log(1+\frac{1}{s}) $$
You may then use $\log(1+\frac{1}{s}) = \frac{1}{s} - \frac{1}{2s^2} + ...$... |
43,326 | <p>Are there known sharp upper bounds (in terms of $k$ or $\omega(k)$, the number of distinct prime divisors of $k$) for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$ for $k > 1$ subject to the constraint $\sum_{p \mid k} \frac{1}{p+1} < 1$? (This would be a special case of the general result of O. Izhboldin a... | George Lowther | 1,004 | <p>I still have no idea what the answer to this question is. However, it is possible to attack the problem in several different ways, and there are various different (but logically equivalent) ways of stating it. I'll post some of these as an answer now, as it seems rather long to fit in the original statement of the q... |
84,310 | <p>Is the following problem still open? Let $S$ be a non-empty set of prime numbers such that whenever $p,q\in S$, all the prime factors of $pq+1$ are also elements of $S$. Is $S$ infinite?</p>
| Timothy Foo | 18,494 | <p>Not claiming to be a real number theorist or anything of that sort...Just wondering, is one possible approach to Mark Sapir's strong conjecture (assuming an answer isn't already known) to calculate
$$
\sum_{\substack{r,s < q\\ q| rs+1}}\Lambda(r)\Lambda(s)
$$</p>
<p>by</p>
<p>$$
\int_0^1 \left(\sum_{r,s < q... |
84,310 | <p>Is the following problem still open? Let $S$ be a non-empty set of prime numbers such that whenever $p,q\in S$, all the prime factors of $pq+1$ are also elements of $S$. Is $S$ infinite?</p>
| James Currie | 8,394 | <p>This was given as a problem in one of the MAA journals in the 1980's as a double-starred problem (no solution provided). I met it as a challenge problem on an assignment in John Poland's (Carleton University) modern algebra course. I worked on it for a few months, observing as Mark Sapir does that such a set contain... |
84,310 | <p>Is the following problem still open? Let $S$ be a non-empty set of prime numbers such that whenever $p,q\in S$, all the prime factors of $pq+1$ are also elements of $S$. Is $S$ infinite?</p>
| Shahrooz | 19,885 | <p>Dear Currie</p>
<p>The below problem is still open:</p>
<p>Do we have infinite prime number of the form $2p+1$, where $p$ is prime?</p>
<p>One of interesting way to examine this problem is %Sandram$ table.
For further information about Sandram table, you can see the book</p>
<p>" Ingenuity in Mathematics" by Ros... |
547,184 | <p>Determine values of the constants $A$ and $B$ such that $a_n=An+B$ is a solution of the recurrence relation $a_n=2a_{n-1}+n+5$.</p>
<p>I know that the characteristic equation is $r-2 = 0$ which has the root $r = 2$.
Usually I find constants $A$ and $B$ by $a_n=Ar^n+Br^n$. It
looks like I cannot apply this here beca... | DanielV | 97,045 | <p>You have $\forall n ~:~ a_n = 2~a_{n-1} + n + 5$ and $\forall n ~:~ a_n = A~n+B$. So construct the equations</p>
<p>$$\begin{array} {c|cc}
n & a_n = 2~a_{n-1} + n + 5 & a_n = A~n+B \\ \hline
1 & a_1 = 2~a_{0} + 6 & a_1 = A+B \\
2 & a_2 = 2~a_{1} + 7 & a_2 = 2~A+B \\
3 & a_3 = 2~a_{2} ... |
44,983 | <p>(I had no idea how to tag this, any help would be great) :)</p>
<p>I am a programmer with little math experience beyond high school - this will be demonstrated by my explanation. </p>
<p>I am developing an application that needs a formula. The formula involves calculating what amounts of food to eat based on facto... | Robert Israel | 8,508 | <p>It's basically a linear programming problem: if $x_1$, $x_2$, $x_3$, $x_4$ are the numbers of boxes ordered from the four companies, you would like</p>
<p>$$ \eqalign{ x_1 + x_2 + x_3 + x4 &= 240 \cr
86 x_1 + 79 x_2 + 15 x_3 + 10 x_4 &= 40 \times 240\cr
11 x_1 + 9 x_2 + 77 x_3 &... |
2,493,123 | <p>I am pretty much stuck, since smaller examples are impossible to find, nor does this seem like an induction question. I am thinking of using proof by contradiction, but don't know where to find the contradiction. I also struggle to recall the theorem that may help me answer this problem.</p>
| Jacob Manaker | 330,413 | <p>There are a couple problems with your proof. </p>
<p>The biggest one is simply that you have the definition of convex hull wrong. $\DeclareMathOperator{conv}{conv} \conv{(S)}$ is <em>not</em> simply $\bigcap{\{X:S\subseteq X\}}$; in fact, $$\bigcap{\{X:S\subseteq X\}}=S$$ Instead, $$\conv{(S)}=\bigca... |
3,070,292 | <p>I'm trying to prove or disprove the following statement:</p>
<blockquote>
<p>Number of even natural solutions of <span class="math-container">$x_1+x_2+x_3+x _4=14$</span> is equal to the number of odd natural solutions of <span class="math-container">$x_1+x_2+x_3+x _4=14$</span>.</p>
</blockquote>
<p>I think tha... | Myunghyun Song | 609,441 | <p>By letting <span class="math-container">$x_1'=x_1+1$</span>, <span class="math-container">$x_2'=x_2+1$</span>, <span class="math-container">$x_3'=x_1-1$</span> and <span class="math-container">$x_4'=x_4-1$</span>, we can produce an odd solution <span class="math-container">$(x_1',x_2',x_3',x_4')$</span> from an even... |
10,352 | <p>I teach high school geometry and see many of my students fall in to the trap of "it looks like it, so it's true" -- a <a href="https://en.wikipedia.org/wiki/Van_Hiele_model" rel="noreferrer">Van Hiele Level 0 to 1</a> thought process. For instance, when talking about parallel line angle relationships, once they real... | paul garrett | 63 | <p>Unable to resist continuing my comment-rant: it is perverse to disbelieve a thing that appears to be true. Sure, appearances can be deceiving, but... mostly not so much. It is also perverse for mathematics teachers to create situations in which appearances are deceiving... since, in my opinion, the goal is not to te... |
10,537 | <p>In Finnish middle school (yläkoulu) the students learn to measure distances and angles, draw geometric figures and do certain calculations (area, volume, surface measure, trigonometry). There are also geometric constructions.</p>
<p><strong>As a teacher</strong>, to what extent is it important and useful for me to ... | JTP - Apologise to Monica | 64 | <p>I've gotten used to "drawing not to scale" and perhaps a bit too complacent in accepting this. Unless we are out to play tricks on students, a square should have its sides "close enough" so when it's stated that we have a square, there's no doubt. Don't offer a square image and declare beneath it that the rectangle ... |
10,537 | <p>In Finnish middle school (yläkoulu) the students learn to measure distances and angles, draw geometric figures and do certain calculations (area, volume, surface measure, trigonometry). There are also geometric constructions.</p>
<p><strong>As a teacher</strong>, to what extent is it important and useful for me to ... | Benoît Kloeckner | 187 | <p>I think Jessica B's answer is spot-on (and it relates with me strongly with respect to my research in geometry), but I would like to add something in that direction, which I get from Anne Parreau who was teaching math to future primary schools teachers.</p>
<p><strong>There is no such thing as an exact drawing.</st... |
674,564 | <p>I came up with a problem that's been bugging me:</p>
<p><em>How many ways can you tile an NxM rectangle with L-polyominos?</em></p>
<p>The L shapes can be any size, so long as they aren't lines.</p>
<p>For clarification:<br>
$L(1,m) = 0$,<br>
$L(2,2) = 0$,<br>
$L(2,3) = 2$,<br>
$L(2,4) = 2$,<br>
$L(3,3) = 0$...</... | Robert Israel | 8,508 | <p>It's not an open cover. Take a point of a "sprout" that is not in open set number $n$. These can be chosen to converge (and in any case some
subsequence must converge). The limit is in the snowflake but not in any of
your open sets.</p>
|
3,490,416 | <p>Problem:</p>
<blockquote>
<p>From any fixed point in the axis of a general conic, a line is drawn perpendicular to the tangent at <span class="math-container">$P$</span> (on the conic) meeting <span class="math-container">$SP$</span> in <span class="math-container">$R$</span> where <span class="math-container">$S... | Jean Marie | 305,862 | <p>(I will refer to the excellent drawing and notations in the solution by Aretino, my solution being in the same spirit).</p>
<p>It is a consequence of the following property of an ellipse (see proof below) :</p>
<p>The locus of the (variable) symmetrical point <span class="math-container">$T$</span> of one of the ... |
857,207 | <p>I need help to confirm my answer for the following question</p>
<p>"There is an alphabet of size 40 and this alphabet is used for forming messages in a communication system. If 10 of these alphabets can be used only as the first or last symbols in the message and the rest 30 can be used anywhere, how many messages ... | John Machacek | 155,418 | <p>If the remaining 30 letters can be used anywhere then you are correct. In this case you have 40 letters available for the first and last symbols and only 30 available for the 23 symbols in between. So, the count is $40^2 \cdot 30^{23}$.</p>
<p>What johannesvalks is answering is if the first and last symbols must co... |
3,264,330 | <p>I am really struggling with a contour integration question, which I am revising for an exam. I want to show that the contour integral of </p>
<p><span class="math-container">$$\int_\Gamma e^{t(z+z^{-1})}z^{-2} dz=\sum_{m=0}^\infty b_mt^{2m+1}$$</span> where the <span class="math-container">$b_m$</span> are constant... | eranreches | 208,983 | <p><strong>Hint:</strong></p>
<p><span class="math-container">$$e^{t\left(z+z^{-1}\right)}z^{-2}=z^{-2}\sum_{n=0}^{\infty}\dfrac{t^{n}}{n!}\left(z+z^{-1}\right)^{n}=\\\\=z^{-2}\sum_{n=0}^{\infty}\dfrac{t^{n}}{n!}\sum_{k=0}^{n}\binom{n}{k}z^{-k}z^{n-k}=\sum_{n=0}^{\infty}\dfrac{t^{n}}{n!}\sum_{k=0}^{n}\binom{n}{k}z^{n-... |
3,425,499 | <ul>
<li><p><span class="math-container">$\sqrt{1+x} - \sqrt{1-x}$</span> is a subtractive cancellation when <span class="math-container">$x \approx 0$</span>, and</p>
</li>
<li><p><span class="math-container">$1 - \cos(x)$</span> is a catastrophic cancellation when <span class="math-container">$x \approx \pi$</span>.<... | IDC | 422,221 | <p>I guess it is because the first difference is of the order <span class="math-container">$x$</span>, while the second one is of the order <span class="math-container">$(x-\pi)^2$</span> (from Taylor's expansion).</p>
|
270,683 | <p>How can I make a function, say, <code>R[expression]</code> such that it works on arguments like this <code>Integrate[f[x],{x,0,Infinity}]</code>, applying some operator <code>r[]</code> to the function under integral, and leaving the epression unevaluated if the first level is not an integral from zero to infinity?<... | Edmund | 19,542 | <p>Not at my computer but the following should give slightly more general solution.</p>
<pre><code>ClearAll[rInt]
SetAttributes[rInt, HoldFirst]
rInt[Integrate[f_, x:{_Symbol,0,Infinity}..]] :=Integrate[r[f], x]
</code></pre>
<p>Then the following give the expected result</p>
<pre><code>rInt[Sin[x]]
rInt[Integrate[Sin... |
1,083,301 | <p>I'm reading Knuth/Graham/Patashnik's Concrete Mathematics:</p>
<blockquote>
<p><img src="https://i.imgur.com/YpjYJwd.png" alt="enter image description here"></p>
</blockquote>
<p>I don't understand how he goes from $(r-k){r \choose r-k}$ to $r{r-1 \choose r-k-1}$ using $(5.6)$. The mentioned property has a $k$ i... | d125q | 112,944 | <p>$(5.6)$ states:
$$
\color{green}{k} \binom{\color{red}{r}}{\color{green}{k}} = \color{red}{r} \binom{\color{red}{r} - 1}{\color{green}{k} - 1}
$$
So, in this case:
$$
(\color{green}{r - k}) \binom{\color{red}{r}}{\color{green}{r - k}} = \color{red}{r} \binom{\color{red}{r} - 1}{\color{green}{r - k} - 1}
$$</p>
<hr>... |
15,508 | <p>In probability/statistics, there is a notion of two things being "independent", which would basically mean that any information we can get about one thing has no effect on our (probabilistic) knowledge of the other.</p>
<p>What are the possible notions of "independent" for the natural numbers? Un... | Shiva Kaul | 1,656 | <p>Independence has nothing to do with the uniform distribution. It is extremely common to define an infinite sequence of measures whose joint measure is the product measure. It is also common to define finite-dimensional distributions and then use an extension theorem (Daniell-Kolmogorov, <a href="https://www.stat.cmu... |
3,808,774 | <p>I have this limit I've been trying to solve, and so far I can only solve it using the property of geometric series.</p>
<p>So we get
<span class="math-container">$$\lim_{n \to \infty}\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^n}=\sum_{n=2}^{\infty}\left(\frac{1}{3}\right)^n=\frac{1}{1-\frac{1}{3}}=\frac{3}{2}$$</spa... | Tinotenda Chigasa | 819,719 | <p>We know that <span class="math-container">$S_n=1/9+1/27+...+1/3^n$</span>, so our <span class="math-container">$r=1/3$</span>, now <span class="math-container">$rS_n=1/27+1/81+....+1/3^{n+1}$</span>, so <span class="math-container">$S_n-rS_n=1/9-1/3^{n+1}$</span> which implies <span class="math-container">$S_n=\frac... |
2,330,368 | <p>I have calculate the sum of the following power series: $$\sum_{n=0}^{\infty} (-1)^n(n+1)^2x^n$$</p>
<p>I've determined the convergence radius $R = 1$convergence interval $I_c = (-1,1)$ and the convergence set $A = (-1,1)$ for this power series. Furthermore, I've tired to bring it to a from which can be easily comp... | Donald Splutterwit | 404,247 | <p>We have the following geometric series
\begin{eqnarray*}
\sum_{n=0}^{\infty} (-1)^n x^n = \frac{1}{1+x}
\end{eqnarray*}
Now multiply this by $x$ and differentiate
\begin{eqnarray*}
\sum_{n=0}^{\infty} (-1)^n (n+1) x^n = \frac{(1+x)-x}{(1+x)^2}=\frac{1}{(1+x)^2}
\end{eqnarray*}
& now do exactly the same again to ... |
505,804 | <p>Let $X$ be a metric space and let $G$ be a group of homeomorphisms $X \to X$ acting on $X$. We say $G$'s action is properly discontinuous in case for every $x \in X$ and compact $K \subseteq X$, there are at most finitely many $g \in G$ such that $g(x) \in K$. Equivalently (and this is not hard to show), $G \cdot x$... | Marek | 3,223 | <p>Suppose there is a compact $K$ such that $g(K) \cap K \neq 0$ for infinitely many $g$. Denote by $L_k := g_k(K) \cap K$ where $g_k, k\in \mathbb N$ are among those elements of $G$ for which the inequality holds. Now pick any sequence $y_k \in L_k$. Because $K$ is compact, we have a convergent subsequence $y_{k_n} \... |
1,770,180 | <p>I was solving this limit from the Demidovich's book of exercises:
$$\lim_{n\to\infty} \frac{\sqrt[n]{\vphantom{\Large a}\, n!\,}}{n}$$</p>
<p>and I managed to get it to this state but then I got stuck:</p>
<p>$$e^{\frac{1}{n}(\log(n) + \log(n - 1) + ... + \log(1)) - \log(n)}$$</p>
<p>where $\log x$ is a natural l... | Jack D'Aurizio | 44,121 | <p>Apply the ratio test to the sequence $\{a_n\}_{n\geq 1}$ given by:
$$ a_n = \frac{n!}{n^n}. $$
We have:</p>
<p>$$ \frac{a_{n+1}}{a_n} = \frac{1}{\left(1+\frac{1}{n}\right)^n} $$
hence $\lim_{n\to +\infty}\frac{a_{n+1}}{a_n}=\frac{1}{e}$ implies $\lim_{n\to +\infty}\sqrt[n]{a_n}=\frac{1}{e}$, no integrals needed.</p... |
2,554,060 | <p>The book said: "if $f$ is a bounded linear functional then ker $f$ is a closed subspace of the Hilbert space $H$."</p>
<p>But I do not know why, could anyone clarify this for me please? </p>
| Parcly Taxel | 357,390 | <p>After the last equation in the question, this will require some rearrangements…
$$b^2c^2-2ab^2c+a^2b^2
-4a^2bc+4ab^2c+4a^2c^2-4abc^2=0$$
$$(ab)^2+4(ac)^2+(bc)^2-4(ac)(bc)+2(ab)(bc)-4(ab)(bc)=0$$
$$(ab-2ac+bc)^2=0$$
$$ab-2ac+bc=0$$
$$ab+bc=2ac$$
$$\frac1c+\frac1a=\frac2b$$
and thus $a,b,c$ are in harmonic prog... |
2,014,156 | <p><a href="http://www.wolframalpha.com/input/?i=lim%20(sum%201%2F(a%2Bn)%5E2,%20n%20%3D%201%20to%20infinity),%20a%20to%20infinity" rel="nofollow noreferrer">WolframAlpha</a> says that the sequence
$$
S_n = \sum\limits_{i=1}^{\infty}\frac{1}{(n+i)^2}
$$
tends to $0$. The proof given by the site uses notations unfamilia... | Simply Beautiful Art | 272,831 | <p>For positive monotonically decreasing functions, <a href="https://en.wikipedia.org/wiki/Integral_test_for_convergence#Remark" rel="nofollow noreferrer">the integral test not only applies, but it sets bounds</a>:</p>
<p>$$\int_{N+1}^\infty\frac1{x^2}dx\le\sum_{i=N+1}^\infty\frac1{i^2}\le\frac1{(N+1)^2}+\int_{N+1}^\i... |
2,108,245 | <p>While studying Fourier analysis last semester, I saw an interesting identity:</p>
<p>$$\sum_{n=1}^{\infty}\frac{1}{n^2-\alpha^2}=\frac{1}{2\alpha^2}-\frac{\pi}{2\alpha\tan\pi\alpha}$$
whenever $\alpha \in \mathbb{C}\setminus \mathbb{Z}$, which I learned two proofs using Fourier series and residue calculus.</p>
<p>... | math | 382,933 | <p>According to Wikipedia (<a href="https://en.wikipedia.org/wiki/Basel_problem">https://en.wikipedia.org/wiki/Basel_problem</a>),
Euler was the first to give a representation of the sine function as an infinite product:
$$(*) \hspace{2cm}\sin (\pi \alpha)=\pi \alpha \prod\limits_{n=1}^{\infty}(\frac{n^2-\alpha^2}{n^... |
62,002 | <p>I have a two-part question:</p>
<p>(1) First and foremost: I have been going through the paper by Dijkgraaf and Witten "Group Cohomology and Topological Field Theories." (See <a href="https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-129/issue-2/Topological-gauge-theories-and... | Scott Carter | 36,108 | <p>Takefumi Nosaka has written a paper with Eri Hatakenaka that describes connections between DW and quandle cocycle invariants.</p>
|
3,735,388 | <blockquote>
<p>They ask me to solve
<span class="math-container">$$y' +2y + \int_{0}^{x} y(t)dt = f(x)$$</span> with <span class="math-container">$y(0)=0$</span> and
<span class="math-container">$$
f(x) =
\begin{cases}
0, & x < 5 \\
2, & x \geq 5
\end{cases}
$$</span></p>
</blockquote>
<p>I don't know how ... | gt6989b | 16,192 | <p><strong>HINT</strong></p>
<ol>
<li>Assuming <span class="math-container">$y' = \frac{dy}{dx}$</span>, differentiate once to get
<span class="math-container">$$
y'' + 2y' + y = 0
$$</span>
everywhere except <span class="math-container">$x = 5$</span> where the RHS does not exist. Can you solve that?</li>
<li>Once you... |
2,055,695 | <p>Use mathematical induction to prove each of the following statements.
let <span class="math-container">$$g(n) = 1^3 + 2^3 + 3^3 + ... + n^3$$</span></p>
<p>Show that the function <span class="math-container">$$g(n)= \frac{n^2(n+1)^2}{4}$$</span> for all n in <strong>N</strong></p>
<p>so the base case is just g(1) ... | Hans Hüttel | 289,137 | <p><em>Hint:</em> Think of what you are supposed to prove: That a property $P(n)$ is true for every natural number. </p>
<p>A proof by induction essentially proves that the property $P(n)$ holds for the first natural number and is preserved, as we count upwards.</p>
<p>In this case the property $P(n)$ is </p>
<p>$$P... |
1,339,041 | <p>The Fourier coefficients are defined (in our course) as: $$\hat{f(n)}={1\over 2\pi}\int_{0}^{2\pi}{f(t)e^{-int}dt}$$</p>
<p>I am asked to express g's coefficients as a combination of f's ones, given $g(x)=f(x+c)$. The answer says: $$\hat{g(n)}=e^{2\pi inc}\hat{f(n)}$$ </p>
<p>But I don't understand why. What I did... | Maestro13 | 144,992 | <p>You forget to mention one vital thing: $f$ is periodic with periodicity $2\pi$.
The fourier coefficient is as you mention:
$$f_n=\frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-inx}dx$$
An interesting property of this integral is that one may move the interval of integration: as $f(x+2\pi)=f(x)$ and also $e^{-in(x+2\pi)}=e^{-2n\... |
3,396,816 | <p>Let <span class="math-container">$k\in \mathbb{N}$</span>. Which of <span class="math-container">$[0]_3, [1]_3, [2]_3$</span> is <span class="math-container">$[5^k]_3$</span> equal to? Prove your answer.</p>
<p>Below is my proof so far. I figured out what it equals when <span class="math-container">$k$</span> is ev... | user289143 | 289,143 | <p>See that <span class="math-container">$[5]_3=[-1]_3$</span> and therefore <span class="math-container">$[5^k]_3=[(-1)^k]_3$</span>.</p>
<p>Now it's quite straightforward: we have <span class="math-container">$(-1)^k=1$</span> for <span class="math-container">$k$</span> even and <span class="math-container">$(-1)^k=... |
1,731,227 | <p>Find he local maximum and minimum value and saddle points of the function: $$f(x,y)=x^2-xy+y^2-9x+6y+10$$</p>
<p>The answer is a min of $(-4,1), f(-4,1)=73$</p>
<p>I got a min of $(12/5,-21/5)$
my
$$
f_x=2x-y-9\\
f_y=-x+2y+6
$$
set $f_x = 0 = f_y$ and we get $x=(9+y)/2$ so $y=-21/5$ and $x=12/5$</p>
<p>I found ou... | Robert Israel | 8,508 | <p>You're both wrong (or maybe you made a mistake in copying $f$): the minimum is at $(4,-1)$ where the value is $-11$.</p>
|
31,113 | <p>Zagier has a very short proof (<a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1041893" rel="noreferrer">MR1041893</a>, <a href="https://www.jstor.org/stable/2323918" rel="noreferrer">JSTOR</a>) for the fact that every prime number $p$ of the form $4k+1$ is the sum of two squares.
The proof defines an inv... | Daniel Litt | 6,950 | <p><a href="http://www.math.tugraz.at/~elsholtz/WWW/papers/zagierenglish9thjuly2002.ps">This paper</a> by Christian Elsholtz seems to be exactly what you're looking for. It motivates the Zagier/Liouville/Heath-Brown proof and uses the method to prove some other similar statements. Here is a <a href="http://www.math.t... |
31,113 | <p>Zagier has a very short proof (<a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1041893" rel="noreferrer">MR1041893</a>, <a href="https://www.jstor.org/stable/2323918" rel="noreferrer">JSTOR</a>) for the fact that every prime number $p$ of the form $4k+1$ is the sum of two squares.
The proof defines an inv... | Alexey Ustinov | 5,712 | <p>It is interesting that not only Zagier took this proof from Heath-Brown. Heath-Brown (according to his own words) took this proof from Uspensky. </p>
<p>This trick has different applications, see articles of Bykovskii <a href="http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=dvmg&paperid=247&op... |
2,907,487 | <p>Trying to better understand how an "interpretation" works in logic: <a href="https://en.wikipedia.org/wiki/Interpretation_(logic)" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Interpretation_(logic)</a></p>
<p>For the most part it seems to be the idea that we have some syntactical symbol and we can assig... | Noah Schweber | 28,111 | <p>"Interpretations" in this context is a technical term, a synonym for the notion of <a href="https://en.wikipedia.org/wiki/Structure_(mathematical_logic)" rel="nofollow noreferrer"><strong>models (or structures)</strong></a>. So when we say "every interpretation," we don't mean "everything you could possibly think of... |
275,558 | <p>I recently had a homework problem asking to prove that the direct product of rings (or rings with identity) are still rings (with identity), and it seemed really silly to go through all the steps in order to conclude what was extremely trivial. </p>
<p>My question is, are there instances when this is not the case? ... | ncmathsadist | 4,154 | <p>Such properties are sometimes called " productive". For example, a product of compact topological spaces is compact (Tychonov). Not all properties meet this [stringent] criterion.</p>
|
1,989,764 | <blockquote>
<p>Let $f,g$ be continuous functions on $[a,b]$ and suppose $g(x) \geq 0
$ for all $x \in [a,b]$. Prove there exists $\xi \in [a,b]$ such that </p>
<p>$$ \int\limits_a^b f(x) g(x) \mathrm{d} x = f( \xi ) \int\limits_a^b
g(x) \mathrm{d} x $$</p>
</blockquote>
<p>My attempt:</p>
<p>Mean value theo... | Martin Argerami | 22,857 | <p>What you want to do is write your vector as
$$
\begin{bmatrix}x\\ y\\ 1\end{bmatrix}.
$$
Then you apply the three matrices one after the other (it doesn't matter if you apply them one by one, or if you apply them to the resulting vector; the only important thing is that you respect the order: first goes right). </p... |
155,143 | <p>Let $X$ be a Banach space an let $(e_n)_{n=1}^{\infty}$ be a Schauder basis of $X$. If we denote the sequence of the natural projections associated with $(e_n)_{n=1}^{\infty}$ by $(S_n)_{n=1}^{\infty}$ then by Uniform Boundedness principle we can observe that $$\sup_n||S_n||< \infty. $$</p>
<p>And the number $K=... | Bill Johnson | 2,554 | <p>Your first question is too general. As Michael pointed out, there is always an equivalent norm under which the basis has basis constant one, and sometimes you can choose the equivalent norm to have good geometric properties. As it is nice to have approximation of the identity operators by finite rank projections o... |
3,137,775 | <p>Let <span class="math-container">$Z$</span> be the matrix </p>
<p><span class="math-container">$$Z =
\begin{pmatrix} 1 & 2\\ 3 & 4 \end{pmatrix}$$</span></p>
<p>And the set <span class="math-container">$\text{Commu}(Z) = \{B ∈ M_{2\times2} | ZB = BZ\}$</span>.</p>
<p>What is the set <span class="math-cont... | José Carlos Santos | 446,262 | <p>Note that<span class="math-container">$$\frac{\tan^2\theta}{\sec^3\theta}=\sin^2\theta\cos\theta.$$</span>A primitive of this will be <span class="math-container">$\frac13\sin^3\theta$</span>.</p>
|
3,137,775 | <p>Let <span class="math-container">$Z$</span> be the matrix </p>
<p><span class="math-container">$$Z =
\begin{pmatrix} 1 & 2\\ 3 & 4 \end{pmatrix}$$</span></p>
<p>And the set <span class="math-container">$\text{Commu}(Z) = \{B ∈ M_{2\times2} | ZB = BZ\}$</span>.</p>
<p>What is the set <span class="math-cont... | Lai | 732,917 | <p><span class="math-container">$$
\begin{aligned}
& \int \frac{\sqrt{x^{2}-1}}{x^{4}} d x \\\stackrel{y=\frac{1}{x}}{=} &\int\frac{\sqrt{\frac{1}{y^{2}}-1}}{\frac{1}{y^{4}}}\left(-\frac{1}{y^{2}} d y\right)\\
=& -\int y \sqrt{1-y^{2}}d y \\
=&\frac{\left(1-y^{2}\right)^{\frac{3}{2}}}{3}+C
\\=&\fra... |
4,205,525 | <p>I have to prove by mathematical induction that <span class="math-container">$L = R$</span></p>
<p>Question:</p>
<p><a href="https://i.stack.imgur.com/3UYsv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3UYsv.png" alt="enter image description here" /></a></p>
<p>What I've tried so far:</p>
<p><a ... | Deepak | 151,732 | <p>You seem to be solving it like an equation. What you need to do is simply to <em>show</em> that:</p>
<p><span class="math-container">$\displaystyle \frac{n(5n+3)} 2 + (5(n+1)-1)$</span></p>
<p>equals to:</p>
<p><span class="math-container">$\displaystyle \frac{(n+1)(5(n+1)+3)}2$</span></p>
<p>by algebraically manipu... |
1,345,503 | <p>I have no idea about the answer, but I'm viewing the question this way;</p>
<p>If the smallest score obtained from the any three throws of the die is $1$, then largest among the other two throws must be $3$, and likewise
if the smallest score obtained from any of the three throw of the die is $2$ then a $6$ must a... | Brian M. Scott | 12,042 | <p>One can also solve the problem using the <a href="https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle" rel="nofollow">inclusion-exclusion principle</a> to count the possible outcomes in each of two cases.</p>
<p>There are $3^3=27$ outcomes in which all of the dice are between $1$ and $3$ inclusive. ... |
3,168,214 | <blockquote>
<p>Suppose that <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are related by the equation <span class="math-container">$\displaystyle x=\int_0^y\frac{1}{\sqrt{1+4t^2}}\,dt$</span>. Show that <span class="math-container">$d^2y/dx^2$</span> is proportional to <span cla... | Xin Fu | 544,425 | <p><span class="math-container">$$\frac{dy}{dx}=\left(\frac{dx}{dy}\right)^{-1}=\left(\frac{d}{dy}\int_{0}^y\frac{1}{\sqrt{1+4t^2}}\right)^{-1}=\sqrt{1+4y^2}$$</span>
so
<span class="math-container">$$\frac{d^2y}{dx^2}=\frac{d}{dx}\sqrt{1+4y^2}=\frac{4y\frac{dy}{dx}}{\sqrt{1+4y^2}}=4y$$</span>
the proportionality is <s... |
3,168,214 | <blockquote>
<p>Suppose that <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are related by the equation <span class="math-container">$\displaystyle x=\int_0^y\frac{1}{\sqrt{1+4t^2}}\,dt$</span>. Show that <span class="math-container">$d^2y/dx^2$</span> is proportional to <span cla... | Jean Marie | 305,862 | <p>Here is a solution that gives explicitly the expression of functions we are working on.</p>
<p>You may know that </p>
<p><span class="math-container">$$\text{the antiderivative of} \ \dfrac{1}{\sqrt{1+t^2}} \ \text{is x=sinh}^{-1}(t)+k$$</span></p>
<p>where <span class="math-container">$k$</span> is an arbitrary ... |
4,188,176 | <p>Consider the second order ordinary differential equation with one parameter <span class="math-container">$k$</span>
<span class="math-container">$$(x-3)^7 ((x^2-4 x+3) f''(x)+2 x f'(x))+3456 k (3-2 x)^2 (x-1) x^2 f(x)=0, \quad 0 \leq x \leq 1$$</span>
and a set of boundary conditions
<span class="math-container">$$... | Claude Leibovici | 82,404 | <p>I would have done the same.</p>
<p>If you use a CAS, it is very quickly done. The problem is that there are many solutions. Assuming that <span class="math-container">$k=\frac n {10}$</span> <span class="math-container">$n$</span> being a positive integer, the first solutions generate for <span class="math-container... |
1,751,531 | <p>I have a question concerning the component test for conservative fields. So the component test tells us that the vector field is conservative if the following three conditions are met.
$$
P_y = N_z \\
M_z = P_x \\
N_x = M_y
$$
I am having trouble memorizing, because I can't seem to find why these values are used... | robjohn | 13,854 | <p>Using the <a href="https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm" rel="nofollow noreferrer">Extended Euclidean Algorithm</a> as implemented in <a href="https://math.stackexchange.com/a/68021">this answer</a>, we get
$$
\begin{array}{r}
&&1&x+6&(x-8)/53\\\hline
1&0&1&-x-6&... |
43,043 | <p>I'm working through Carl Bender's <a href="http://youtu.be/LYNOGk3ZjFM">Mathematical Physics lectures</a> on YouTube (which are great fun), and I'd like <em>Mathematica</em>'s help solving terms in the perturbation series. It would be convenient if expressions like</p>
<pre><code>SeriesCoefficient[Sum[a[n] b^n, {n,... | Emilio Pisanty | 1,000 | <p>You could always build a pattern matcher that will recover the coefficients you want:</p>
<pre><code>seriescoefficient[
Sum[coeff_ b_^n_, {n_, min_, max_}], {b_, b0_, m_}] := coeff /. {n -> m}
</code></pre>
<p>will evaluate</p>
<pre><code>seriescoefficient[Sum[a[n] b^n, {n, 0, ∞}], {b, 0, 3}]
</code></pre>
... |
2,164,338 | <p>I am stuck up with this simple problem related to averages. Please help me out in explaining the complete solution to this problem.</p>
<blockquote>
<p>Consider a class of <span class="math-container">$40$</span> students whose average weight is <span class="math-container">$40\ \mathrm{kg}$</span>. <span class="... | Doug M | 317,162 | <p>m+n = 50</p>
<p>The average weight with the new students will be</p>
<p>$AvW = \frac {40\cdot 40 + m\cdot n}{40+m}$</p>
<p>maximize $AvW$ constrained by $m+n = 50
$n = 50-m$</p>
<p>$AvW = \frac {40\cdot 40 + m\cdot (50-m)}{40+m}$</p>
<p>Will be optimized when $\frac {d}{dm} AvW = 0$</p>
|
2,041 | <p>I'd like to process data like the following, from my most recent seven-question exam.</p>
<pre><code> Item 1 2 3 4 5 6 7
PtsPossible 9 13 11 12 12 11 12
Median 8 9 8 7 10 10 12
Mean 7.10 9.10 7.25 6.65 9.27 9.39 10.57... | Roland | 108 | <p>The comment of Mon Kee Poo raises an important point: The actual distributions of the points per item might offer some more insight than just the median and the standard deviation.</p>
<p>For example, if there is a 'hit or miss' question, the points will not be normally distributed, but rather bimodal, i.e. one por... |
980,470 | <p>How to find the continued fraction of $\sqrt{n}$, for an integer $n$?</p>
<p>I saw a site where they explained it, but it required a calculator. Is it possible to do it without a calculator? </p>
| Empy2 | 81,790 | <p>The key is to find reciprocals, using conjugates. For example, the reciprocal of $\sqrt{7}-2$ is $\frac{\sqrt{7}+2}3$, which you can check by multiplying those two together. To start with,<br>
$$\sqrt{7}=2+(\sqrt{7}-2)\\=2+\frac{1}{\frac{\sqrt{7}+2}3}\\=2+\frac{1}{1+\frac{\sqrt{7}-1}3}$$<br>
Carry on like that. ... |
415,743 | <p><strong>Given Problem</strong> is to solve this separable differential equation:</p>
<p>$$y^{\prime}=\frac{y}{4x-x^2}.$$</p>
<p><strong>My approach</strong>: was to build the integral of y':</p>
<p>$$\int y^{\prime} = \int \frac{y}{4x-x^2}dy = \frac{y^2}{2(4x-x^2)}.$$</p>
<p>But now i am stuck in differential eq... | Mercy King | 23,304 | <p>For $y \ne 0$ we have
$$
y'=\frac{y}{4x-x^2} \iff \frac{y'}{y}=\frac{1}{4x-x^2}=\frac{1}{4}\left(\frac{1}{x}+\frac{1}{4-x}\right).
$$
Integrating the two sides of the latter identity we get:
$$
\ln|y/a|=\frac{1}{4}\left(\ln|x|-\ln|4-x|\right)=\ln\sqrt[4]{\left|\frac{x}{4-x}\right|},
$$
where $a$ is a nonzero constan... |
1,071,981 | <p>I have the following integral:
$$\int_V ((\vec\nabla.\vec A)(\vec\nabla.\vec B)+\vec B.\vec\nabla(\vec\nabla.\vec A))dV$$</p>
<p>Using Gauss' theorem I can convert this into a surface integral. However, I need to rewrite the integrandum into something of the form $\vec\nabla.\vec V$ in order to apply Gauss.</p>
<p... | user_of_math | 161,022 | <p>Write everything in index notation. Your integrand is
$$
\frac{\partial A_i}{\partial x_i}\frac{\partial B_j}{\partial x_j} + B_j \hat{e}_j \cdot \left ( \hat{e}_m \frac{\partial}{\partial x_m} \left ( \frac{\partial A_p}{\partial x_p} \right)\right)
$$
which reduces, on simplification, to
$$
\frac{\partial A_i}{\pa... |
1,780,751 | <p>If $$I=\int\sin 2x\, \mathrm{d}x$$
since $$(\sin^2x)'=2\cos x\sin x=\sin 2x$$ then $$I=\sin^2x+C$$</p>
<p>However using substitution
$$I=\int \frac{\sin2x}{-2\sin 2x}\,\mathrm{d}u,~~~~\textrm{s.t}~~~ u=\cos2x$$
giving $$I=-\frac{1}{2}\int\,\mathrm{d}u=-\frac{1}{2}\cos2x+C=\sin^2x-\frac{1}{2}+C$$</p>
<p>I have chec... | lab bhattacharjee | 33,337 | <p>Both are correct as $$C+\sin^2x=C+1-\cos^2x=K-\cos^2x$$
where $K=C+1$</p>
<p>as $$\cos2x=2\cos^2x-1=1-2\sin^2x$$</p>
|
3,549,582 | <blockquote>
<p>Let V and w be finite-dimensional vector spaces with ordered bases <span class="math-container">$\beta={v_1,v_2,...,v_n}$</span> and <span class="math-container">$\gamma={w_1,w_2,...,w_m}$</span>. Let <span class="math-container">$T:V \rightarrow$</span> be linear. Then for each j, <span class="math-c... | Eric Towers | 123,905 | <blockquote>
<p>... The fisherman always fishes for at least 2 hours. If during these 2 hours he catches at least 1 fish, he goes back home, else, he keeps fishing until he catches his first fish and then immediately leaves ... </p>
</blockquote>
<p>He will <em>always</em> fish for at least two hours, regardless of... |
240,894 | <p>I am importing an excel sheet into Mathematica. The result is a rectangular array. The array contains sequences of commas. Whenever a cell is empty in the Excel sheet consecutive commas appear in the imported array. How to get rid of them? So I am looking for a function <span class="math-container">$f$</span> such t... | Armin Vollmer | 54,621 | <p>Remove empty words (<code>""</code>) or <code>Null</code> by replacing by <code>Sequence[]</code></p>
<pre><code>In[1]:= list = {1,2,3,2,1,,,,6,5,4,3,,,,8};
In[2]:= list /. {Null -> Sequence[], "" -> Sequence[]}
Out[2]= {1,2,3,2,1,6,5,4,3,8}
</code></pre>
|
1,771,845 | <p>The problem is to find the number of real solutions of a cubic equation. This exercise is in a book, in the chapter about functions, limits and continuity. This chapter is before the chapter about derivatives therefore I assume that they cannot be used to solve the exercise.</p>
<p>$$f(x) = x^3-9x^2+24x-17 = 0$$</p... | lulu | 252,071 | <p>Easiest, I think, to just look for sign changes. We note that $$f(1)=-1\;\;f(2)=3\;\;f(3)=1\;\;f(4)=-1\;\;f(5)=3$$</p>
<p>It follows that there are real roots in each of the intervals $[1,2],\;[3,4],\;[4,5]$. As a cubic can't have more than three roots, the answer is $\fbox 3$.</p>
|
4,267,359 | <p><strong>The problem:</strong></p>
<p>If <span class="math-container">$f(x)=\begin{cases}\frac{\sin^2(ax)}{x^2}&x\neq0\\ 1&x=0\end{cases}$</span>, show that <span class="math-container">$f(x)$</span> is discontinuous at <span class="math-container">$x=0$</span>. If <span class="math-container">$f(x)$</span> i... | FShrike | 815,585 | <p>We know that:</p>
<p><span class="math-container">$$\lim_{x\to0}\frac{\sin^2ax}{x^2}=\lim_{ax\to0}a^2\cdot\frac{\sin^2ax}{(ax)^2}=a^2$$</span></p>
<p>Which gives that <span class="math-container">$a=\pm1$</span> for <span class="math-container">$f$</span> to be continuous at <span class="math-container">$0$</span>, ... |
1,516,482 | <p>I need to proof that $\sqrt{3} + \sqrt{2}$ is irrational, without using the fact that an irrational number plus a rational number equals irrational. also, i can't use the rational root theorem. that's why i posted a new question..</p>
<p>thanks for help! </p>
| Alex Fish | 149,127 | <p>Assume to get contradiction $\sqrt{3} + \sqrt{2}=\frac{p}{q}$ where $p,q\in \mathbb{N}$. $\sqrt{3} + \sqrt{2}=\frac{p}{q} \Rightarrow (\sqrt{3} + \sqrt{2})^2=(\frac{p}{q})^2 \Rightarrow 3+2\sqrt{6}+2=\frac{p^2}{q^2} \Rightarrow \sqrt{6}=\frac{1}{2}(\frac{p^2}{q^2}-5)$. So now we have $\sqrt{3}+\sqrt{2}\not \in \math... |
173,187 | <p>Does anyone how to solve this matrix equation?
$$ PXQ^T+P^TXQ=A$$
where all matrices are real and square. Can you provide me with some guidelines?
Thanx</p>
| Mahdi | 53,059 | <p>Use <a href="http://en.wikipedia.org/wiki/Vectorization_%28mathematics%29" rel="nofollow">vectorization operator</a> $\mathrm{vec}(\cdot)$ to convert your matrix equation into linear equation form.</p>
<p>Note that
$\mathrm{vec}(PXQ^T)=(Q \otimes P) \mathrm{vec(X)}$. </p>
<p>So it is sufficient to solve linear ... |
2,125,327 | <p>How can we solve these questions using limit comparison test, specifically how can we choose $b_n$ for any series,
(The $n^{th}$ terms are given)</p>
<p><strong>(1)</strong> $\dfrac{1.2.3.....n}{1.3.5....(2n-1)}$</p>
<p><strong>(2)</strong> $\dfrac{(2n-1)}{n!}$</p>
<p><strong>(3)</strong> $\dfrac{(n+1)(1)}{(n)(4^... | Upstart | 312,594 | <p>Both the balls and the urns are distinguishable</p>
<p>$n=1,r=1$ gives $1!$ ways</p>
<p>$n=2,r=2$ gives $2!$ ways</p>
<p>$n=6,r=10$ you have a choice of selecting 6 urns from 10 and then put the balls in them that gives you $10\choose6$.$6!$ </p>
<p>In case of balls being identical you can disting which ball w... |
2,946,759 | <p><span class="math-container">$L =$</span> {<span class="math-container">$(01)^a$$x(10)^b$</span> | <span class="math-container">$a=b, b > 0, x∈$</span>{<span class="math-container">$0,1$</span>}*}</p>
<p>This is a question where according to the key, yes, the language is regular, but no explanation is given. How... | Bjørn Kjos-Hanssen | 61,578 | <p>Since <span class="math-container">$x$</span> can be anything, it can swallow most of the 01s and 10s.
So it's the same as <span class="math-container">$\{01x10:x\in\{0,1\}^*\}$</span>.</p>
|
285,965 | <p>I was told to assume $f(x)$ is a polynomial with degree $d\geq 1$ with integer coefficients and positive leading coefficient.</p>
<p>(i) I need to show that there are infinitely many $x$ such that $f(x)$ isn't prime.</p>
<p>(ii) I also need to show that if $f(x_0) = m$, where $m>0$, then $f(x) \equiv 0 \pmod m$... | André Nicolas | 6,312 | <p>For (i), prove first that there is an $N$ such that $f(x)$ is <strong>increasing</strong> in the interval $(N,\infty)$, and greater than $1$. Now suppose $a\gt N$ and $f(a)=m$. Then $f(a+m)\gt m$, and $f(a+m)$ is divisible by $m$. </p>
<p>Note that we used the fact (ii) that you were asked to prove. For the proof... |
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