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 |
|---|---|---|---|---|
351,325 | <p>It's known that order topology is completely normal, so the lexicographic ordering on the unit square is also completely normal. It's also known that the lexicographic ordering on the unit square is not metrizable. I am interested in whether it is perfectly normal. (A space is <em>perfectly normal</em> if for any tw... | Will Brian | 70,618 | <p>As Nate Eldredge points out in the comments, the book <em>Counterexamples in Topology</em> states that this space is not perfectly normal, but does not provide a proof. Here is the idea for a proof.</p>
<p>A <em>perfectly normal</em> space is a normal space in which every closed set is a <span class="math-container... |
351,325 | <p>It's known that order topology is completely normal, so the lexicographic ordering on the unit square is also completely normal. It's also known that the lexicographic ordering on the unit square is not metrizable. I am interested in whether it is perfectly normal. (A space is <em>perfectly normal</em> if for any tw... | Ramiro de la Vega | 17,836 | <p>It is a known fact that any perfectly normal (countably) compact space <span class="math-container">$X$</span> is ccc. The proof is what you would try: start with an uncountable cellular family <span class="math-container">$\mathcal{U}$</span>, choose a point <span class="math-container">$p_U \in U$</span> for each ... |
351,325 | <p>It's known that order topology is completely normal, so the lexicographic ordering on the unit square is also completely normal. It's also known that the lexicographic ordering on the unit square is not metrizable. I am interested in whether it is perfectly normal. (A space is <em>perfectly normal</em> if for any tw... | Henno Brandsma | 2,060 | <p>For a compact Hausdorff <span class="math-container">$X$</span> it is equivalent that <span class="math-container">$X$</span> is hereditarily Lindelöf or that <span class="math-container">$X$</span> is perfectly normal. Sketch: <span class="math-container">$X$</span> is hereditarily Lindelöf implies that every open ... |
1,506,805 | <p>$\lim _{x\to 0}\left(\frac{\cos\left(x+\frac{\pi }{2}\right)}{x}\right)\:$</p>
| Angelo Mark | 280,637 | <p>$\cos(x+\frac{\pi}{2})=-\sin(x)$</p>
<p>So $\displaystyle\lim_{x \to 0} \frac{-\sin(x)}{x}=-1$</p>
|
1,506,805 | <p>$\lim _{x\to 0}\left(\frac{\cos\left(x+\frac{\pi }{2}\right)}{x}\right)\:$</p>
| Alekos Robotis | 252,284 | <p>Another suggestion which may work more generally (although the above solutions are better in this case) is to apply L'Hôpital's Rule for an indeterminate form $\frac{0}{0}$. That is, in this case, the limit of the derivatives of the numerator and denominator is equal to the limit of the original expression:</p>
<p>... |
2,759,063 | <p>$A=\left\{n+\frac{1}{2n}|n\in\mathbb{N}\right\}$<br/>
Is this set closed?
<br/>
I was reading answer form <a href="https://math.stackexchange.com/questions/125709/example-to-show-the-distance-between-two-closed-sets-can-be-0-even-if-the-two-se">Example to show the distance between two closed sets can be 0 even if t... | user | 293,846 | <p>By <a href="https://en.wikipedia.org/wiki/Law_of_sines" rel="nofollow noreferrer">law of sines</a>:
$$
\frac{\sin CAM}{\sin BAM}=\frac{\sin C}{\sin B}=:\lambda.
$$</p>
<p>Setting $\sin BAM=x$, and solving the resulting equation for cosine of the sum:
$$
\sqrt{1-x^2}\cdot\sqrt{1-\lambda^2x^2}-x\cdot\lambda x=\cos A... |
2,759,063 | <p>$A=\left\{n+\frac{1}{2n}|n\in\mathbb{N}\right\}$<br/>
Is this set closed?
<br/>
I was reading answer form <a href="https://math.stackexchange.com/questions/125709/example-to-show-the-distance-between-two-closed-sets-can-be-0-even-if-the-two-se">Example to show the distance between two closed sets can be 0 even if t... | Blue | 409 | <p>Let $A = 2\alpha$, $B = 2\beta$, $C = 2\gamma$. We can coordinatize, with
$$A = (0,0) \qquad B = c\left(\cos\alpha,-\sin\alpha\right) \qquad C = b\left(\cos\alpha,\sin\alpha\right) \qquad M = \frac12(B+C)$$ </p>
<p>Since bisector $\overline{AD}$ coincides with the $x$-axis, $\angle DAM$ is simply the direction ang... |
1,321,826 | <p>I need to find residue of function $f(z) = \frac{z}{1-\cos(z)}$ at $z=2\pi k$, where $k\in \Bbb Z$.
I know residue at $z=0$ from <a href="https://math.stackexchange.com/questions/168355/residue-of-fz-fracz1-cosz-at-z-0">here</a>.
I got a hint that need to substitute $z=\hat z+2\pi k$, so $\hat z=z-2\pi k$, so I have... | Nitin Uniyal | 246,221 | <p>Hint: As the denominator 1-cosz has a simple pole at z=2kπ,
Res(f;2kπ)= lim┬(z→2kπ)〖z/(d/dz(1-cosz))〗</p>
|
1,321,826 | <p>I need to find residue of function $f(z) = \frac{z}{1-\cos(z)}$ at $z=2\pi k$, where $k\in \Bbb Z$.
I know residue at $z=0$ from <a href="https://math.stackexchange.com/questions/168355/residue-of-fz-fracz1-cosz-at-z-0">here</a>.
I got a hint that need to substitute $z=\hat z+2\pi k$, so $\hat z=z-2\pi k$, so I have... | klackna | 472,041 | <p>no it has pole of order $2$ for $2k\pi$.see image.</p>
<p><a href="https://i.stack.imgur.com/LW1Zc.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/LW1Zc.png" alt="enter image description here"></a></p>
|
3,557,398 | <p>Here are two second-order differential equations.</p>
<p><span class="math-container">$$ y''+9y=\sin(2t) \tag 1 $$</span></p>
<p><span class="math-container">$$ y'' +4y =\sin(2t) \tag 2 $$</span></p>
<p>I am told to use undetermine coefficients method to solve.</p>
<p>For 1), I use <span class="math-container">... | Community | -1 | <p>Yes, if <span class="math-container">$x+y<z$</span> then <span class="math-container">$x<z$</span> and <span class="math-container">$y<z$</span>.
Can you see how to approach the other cases?</p>
|
1,799,559 | <blockquote>
<p>Let $f:[a,b]\to\mathbb{R}$ be a continious function. Show that if $$\int_a^b f(x)g(x)dx=0$$
for all continious functions $g:[a,b]\to\mathbb{R}$ with $g(a)=g(b)=0$, then $f(x)=0$ $\forall x\in[a,b]$</p>
</blockquote>
<p>I have difficulties proving this question. Consider for example $g(x)=0$ $\fora... | mariob6 | 326,028 | <p>The proof starts like this:</p>
<p>Suppose, by contraddiction that </p>
<p>$ \exists x_0 \ \in \ (a,b) \ :\ f(x_0)>0 $ then $ \exists \delta>0 \ :\ f(x)>0 \ \ \forall x \in(x_0-\delta,x_0+\delta) $</p>
<p>Can you reach the contraddiction?</p>
|
3,810,856 | <p>I'm having trouble understanding how this form of the principle (<a href="https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle" rel="nofollow noreferrer">on wiki</a>) results in the form below.</p>
<p>Wiki form:
<a href="https://i.stack.imgur.com/6IXQZ.png" rel="nofollow noreferrer"><img src="https://... | mjw | 655,367 | <p>We want <span class="math-container">$$\frac{\pi}{\sin^2 \frac{\pi}{x}} \gt \frac{x}{\tan \frac{\pi}{x}}$$</span></p>
<p>When the denominators are non-zero, we can rewrite this as:</p>
<p><span class="math-container">$$\frac{\pi}{x} > \sin \frac{\pi}{x} \cos \frac{\pi}{x}$$</span></p>
<p>Well, for all <span class... |
501,993 | <p>I have never encounter "proof" questions before in my career, but this question in textbook troubled me and I have totally no clue where to start.</p>
<p>Prove that the limit
$$\lim_{x\rightarrow0} \frac{1-\sqrt{1-x^2}}{x^2} = \frac{1}{2}$$</p>
| Pedro | 23,350 | <p><strong>Hint</strong> Multiply by the conjugate of the numerator.</p>
|
2,803,956 | <p>Let $\mathfrak{g}$ be a Lie algebra. Then its automorphism group $Aut(\mathfrak{g})$ is a Lie group, and hence we may take its Lie algebra $Lie(Aut(\mathfrak{g}))$.</p>
<p>I'd like to say that this is equal to the Lie algebra of derivations $Der(\mathfrak{g})$. Is this true? Where can I find a reference?</p>
| Dietrich Burde | 83,966 | <p>This is a standard fact from the theory of Lie algebras. There are several references, say over $\Bbb{R}$ and $\Bbb{C}$, e.g., Proposition 1.25 in the book <a href="https://books.google.at/books?id=RXbLBQAAQBAJ&pg=PA35&lpg=PA35&dq=show+that+Der(L)+is+the+Lie+algebra+of+the+automorphism+group+of+L&sou... |
4,632,869 | <p>Suppose <span class="math-container">$\Omega \subset \mathbb{R}$</span> is a bounded domain. Then using
Poincare-Wirtinger, one can prove that for functions <span class="math-container">$f \in W^{1,2}(\Omega)$</span> there exists <span class="math-container">$C>0$</span> such that</p>
<p><span class="math-contain... | LL 3.14 | 731,946 | <p>If <span class="math-container">$\Omega$</span> has regular boundary (say Lipshitz) you can use the fact that since <span class="math-container">$u' \in L^2(\Omega)$</span> then by Sobolev-Morrey inequality, <span class="math-container">$u\in C^{0,\alpha}(\Omega)$</span> for some <span class="math-container">$\alpha... |
101,256 | <p>Can you please help me finding an exact description of the set:</p>
<p>$$ E_{R}=\{\cos{z} | z \in \mathbb{C}, |z|>R\} $$</p>
<p>For any $0<R \in \mathbb{R}$.</p>
<p>My feeling is the $E_R = \mathbb{C}$, for any $R$, but I don't know how to show it, if it's true.</p>
| Fabian | 7,266 | <p><em>Hint:</em>
Make use of <a href="http://en.wikipedia.org/wiki/Picard_theorem" rel="nofollow">Picard theorem</a> to show that it the image is all of $\mathbb{C}$ with at most a single point missing.</p>
|
1,951,964 | <p>I think the answer is <span class="math-container">$\mathbb{Z}_n$</span> where <span class="math-container">$n$</span> is all the factors of <span class="math-container">$26$</span>. Is this right?</p>
| E.H.E | 187,799 | <p>$${\frac {1}{1+x}}=\sum _{n=0}^{\infty }(-1)^nx^{n}$$
let $x\rightarrow x^2$
$${\frac {1}{1+x^2}}=\sum _{n=0}^{\infty }(-1)^nx^{2n}$$
then multiply by $x$
$${\frac {x}{1+x^2}}=\sum _{n=0}^{\infty }(-1)^nx^{2n+1}$$</p>
|
4,023,342 | <p>Blue: <span class="math-container">$x \ln x-x$</span></p>
<p>Brown: <span class="math-container">$\ln \frac{\Gamma(x+1/2)}{\sqrt{\pi}}$</span></p>
<p><a href="https://i.stack.imgur.com/CbVYi.png" rel="noreferrer"><img src="https://i.stack.imgur.com/CbVYi.png" alt="enter image description here" /></a></p>
| Anixx | 2,513 | <p>Well, I post my own answer. The real reason is that (up to some constant terms and factors), the first function is <span class="math-container">$D^{-1}D^{-1}[\frac1x]$</span> and the second function is <span class="math-container">$D^{-1}\Delta^{-1}[\frac1x]$</span>. So, in one case we replace one normal integration... |
3,652,087 | <p>I need to find an example of a sequence that on the one hand is divergent and on the other hand has two subsequences that <span class="math-container">$$\lim_{n\to \infty} \mathit{a}_{3n} = \lim_{n\to \infty} \mathit{a}_{5n}.$$</span></p>
<p>Thanks a lot.</p>
| J. W. Tanner | 615,567 | <p>What about <span class="math-container">$1,2,0,4,0,0,7,8,0,0,11,0,13...$</span>?</p>
|
3,652,087 | <p>I need to find an example of a sequence that on the one hand is divergent and on the other hand has two subsequences that <span class="math-container">$$\lim_{n\to \infty} \mathit{a}_{3n} = \lim_{n\to \infty} \mathit{a}_{5n}.$$</span></p>
<p>Thanks a lot.</p>
| Sahiba Arora | 266,110 | <p>Let <span class="math-container">$$a_n=\begin{cases}1, \text{if } 3\mid n \text{ or }5\mid n \\0, \text{otherwise }\end{cases},$$</span>
then <span class="math-container">$a_{3n}=a_{5n}=1$</span> for all <span class="math-container">$n$</span> and hence <span class="math-container">$$\lim_{n\to \infty} \mathit{a}_{3... |
1,794,025 | <p>Show that <span class="math-container">$Z(G) = \cap_{a \in G} C(a)$</span>.</p>
<p>Let <span class="math-container">$a \in Z(G)$</span>. Then <span class="math-container">$ax=xa$</span> for all <span class="math-container">$x$</span> in <span class="math-container">$G$</span>. In particular we can say that <span cla... | Bernard | 202,857 | <p>Maybe this is the shortest:
<span class="math-container">$$\begin{align}
x\in Z(G)&\iff \forall a\in G,\;ax=xa\\
&\iff\forall a\in G,\;x\in C(a)\\
&\iff x\in\bigcap_{a\in G}C(a).
\end{align}$$</span></p>
|
459,067 | <p>In set theory, numbers are often constructed, e.g. from nestings of sets which eventually contain the empty set. The operations are defined in term of taking unions etc.etc. </p>
<p>The extended real number line hat plus and minus infinity $\infty,-\infty$ in addition to the reals, and they fulfill <a href="https:/... | Hagen von Eitzen | 39,174 | <p>All you need is two objects that are not in $\mathbb R$, so you may let $\infty=\{\mathbb R\}$ and $-\infty=\{\{\mathbb R\}\}$, for example.</p>
|
1,185,846 | <p>Hey all I am looking for help on a problem. I will post it, and than I will add what I have tried and my ideas etc. The question has been up now for a few days, I'm sure someone out there can help! I even put a bounty, I have spent a lot of time on this question!</p>
<p>I am interested in calculating the torsion (... | Frieder | 221,616 | <p>First trace (red)
$x-2 y=6$
has parametrization like this
$(2 t+6,t,0) $
for $-5<t<-1$
Straight-Line is complete in x-y- plane</p>
<p>Second trace (green)
$x+3 z=6$
has parametrization like this
$(6 - 3 t, 0, t) $
for $-1<t<1$
Straight-Line is complete in x-z- plane</p>
<p>Third trace (magenta)
$-2 ... |
2,702,347 | <blockquote>
<p>Prove that $$\left|\dfrac{1}{z^4-4z^2+3}\right|\leq \dfrac{1}{3},\quad\text{ if}\quad |z|=2$$</p>
</blockquote>
<p>I know that I can complete squares on the denominator ($(z^2-2)^2-1$). I know the triangular inequality and some other few things... but I can't work so well with absolute value on $\mat... | user | 505,767 | <p><strong>HINT</strong></p>
<p>Recall that $$\left|\frac1w \right|=|w|^{-1}$$</p>
<p>and then refer to Martin R answer.</p>
|
949,320 | <p>how can I solve this:</p>
<blockquote>
<p>How many different ways can 8 children be divided into two groups of 3
and one group of 2?</p>
</blockquote>
<p>My method was:</p>
<p>8C3*5C3*2C2+8C3*5C2*3C3+8C2*6C3*3C3=560+560+560=1680</p>
<p>But the answer is:</p>
<p>8C3*5C3*2C2=560</p>
<p>I feel close to gettin... | Arthur | 15,500 | <p>It doesn't matter what order you pick out the groups. So first picking $3$ children, then $3$ children then $2$ (the first term in your sum) yields a result that in the end is indistinguishable from any of the two other orders you can do it.</p>
<p>Which group you pick out children for <em>first</em> is irrelevant ... |
1,143,832 | <p>Let $\zeta$ be a 12th primitive root over $\mathbb Q$. Determine all intermediate field of $\mathbb Q(\zeta)/\mathbb Q$.</p>
<p>My problem is that this is a task from an old exam where you were not allowed to use some sheets which you prepared at home.</p>
<p>So I would like to know if there are some possibilities... | MooS | 211,913 | <p>The Galois group is $(\mathbb Z/12\mathbb Z)^* \cong V_4$, hence there are $3$ proper intermediate fields of degree $2$, which are made up of square roots.</p>
<p>Obviously the fourth and the third root of unity give rise to the intermediate fields $\mathbb Q(i)$ and $\mathbb Q(i\sqrt{3})$. So the third one is give... |
1,143,832 | <p>Let $\zeta$ be a 12th primitive root over $\mathbb Q$. Determine all intermediate field of $\mathbb Q(\zeta)/\mathbb Q$.</p>
<p>My problem is that this is a task from an old exam where you were not allowed to use some sheets which you prepared at home.</p>
<p>So I would like to know if there are some possibilities... | Lubin | 17,760 | <p>It’s worth while recognizing what the twelfth roots of unity <em>are</em>. You know that the cube roots of unity are, they’re $1$ and $(-1\pm i\sqrt3)/2$. You also know what the fourth roots of unity are, they’re $\pm1$ and $\pm i$. Combine them in an appropriate way and you’ll get the twelfth roots; but looking at ... |
41,940 | <p>For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides required?</p>
<p>Using the Wolfram Alpha site, this input gave an almost-square:
<code>PolarPlot(0.75 + ArcSin(Sin(2x... | André Nicolas | 6,312 | <p>The following is probably not in the spirit of the game, but what about a <em>parametric equation</em>? If we are willing to use complex numbers to represent points in the plane, we could use
$$z=t\exp(2\pi ik/n) +(1-t)\exp(2\pi i(k+1)/n),\qquad 0 \le t<1,\quad k=0, 1, \dots, n-1$$ </p>
|
41,940 | <p>For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides required?</p>
<p>Using the Wolfram Alpha site, this input gave an almost-square:
<code>PolarPlot(0.75 + ArcSin(Sin(2x... | NJS | 129,614 | <p>How's this (Mathematica code)? It gives the n-gons inscribed in the unit circle, with vertices at the nth roots of unity.</p>
<p>Manipulate[PolarPlot[Cos[Pi/n] Sec[(2/n) ArcTan[Cot[(n t/2)]]], {t, 0, 2 Pi}], {n, 3, 40, 1}]</p>
|
41,940 | <p>For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides required?</p>
<p>Using the Wolfram Alpha site, this input gave an almost-square:
<code>PolarPlot(0.75 + ArcSin(Sin(2x... | JoeB | 243,738 | <p>$\arcsin \left(\sin \left(\left[\sin \left(60\cdot \frac{\pi }{180}\right)\left(\frac{\sqrt{3}}{3}\cdot x+y\right),x\right]\right)\right)+\left(\frac{1}{3}\right)\cdot \arcsin \left(\sin \left(\left[\cos \left(30\cdot \frac{\pi }{180}\right)\left(x-\frac{\sqrt{3}}{3}y\right),y\right]\cdot \sqrt{3}\right)\right)=0$</... |
3,753,205 | <p>In my general topology textbook there is the following exercise:</p>
<blockquote>
<ul>
<li>(i) - Let <span class="math-container">$\mathcal B$</span> be a basis for a topology <span class="math-container">$\tau$</span> on a non-empty set <span class="math-container">$X$</span>. If <span class="math-container">$\math... | freakish | 340,986 | <p>You are on the right track. So you have</p>
<p><span class="math-container">$$\mathcal B= \{(a,b):a,b\in \mathbb R\}$$</span></p>
<p>Now for any <span class="math-container">$x\in\mathbb{R}$</span> add a single open set to <span class="math-container">$\mathcal B$</span>:</p>
<p><span class="math-container">$$\mathc... |
343,993 | <p>Let $I$ be an ideal of $\mathbb{C}[x,y]$ such that its zero set in $\mathbb{C}^2$ has cardinality $n$. Is it true that $\mathbb{C}[x,y]/I$ is an $n$-dimensional $\mathbb{C}$-vector space (and why)?</p>
| Jim | 56,747 | <p>No, the zero set of $I = (x, y^k)$ has cardinality $1$, it's just the point $(0, 0)$. But $\mathbb C[x, y]/(x, y^k)$ has dimension $k$.</p>
|
343,993 | <p>Let $I$ be an ideal of $\mathbb{C}[x,y]$ such that its zero set in $\mathbb{C}^2$ has cardinality $n$. Is it true that $\mathbb{C}[x,y]/I$ is an $n$-dimensional $\mathbb{C}$-vector space (and why)?</p>
| Alex Becker | 8,173 | <p>As Jim points out, this is false in general. The correct statement is that $\mathbb C[x,y]/\sqrt{I}$ is an $n$-dimensional vector space.</p>
<p><strong>Proof:</strong> Write $\sqrt{I} = P_1\cap \cdots \cap P_m$, where $P_i$ are the primes minimal over $I$. Since $V(P_i)\subseteq V(I)$, we see that each $V(P_i)$ is ... |
281,387 | <p>I've been reading Girard et al's 'Proofs and Types', which in Chapter 6 presents a proof of strong normalisation for the simply typed lambda calculus with products and base types. The proof is based on Tait's method of defining a set of 'reducible terms for type $T$' by induction on the type $T$. For example a term ... | gallais | 14,972 | <p>The typical strategy when you're building a model to perform normalisation by evaluation (i.e. the computational part of Tait's method, see [1]) with sum types is to have as a semantics for <code>A + B</code>:</p>
<ul>
<li>either the semantics of <code>A</code></li>
<li>or the semantics of <code>B</code></li>
<li>o... |
281,387 | <p>I've been reading Girard et al's 'Proofs and Types', which in Chapter 6 presents a proof of strong normalisation for the simply typed lambda calculus with products and base types. The proof is based on Tait's method of defining a set of 'reducible terms for type $T$' by induction on the type $T$. For example a term ... | cody | 36,103 | <p>Guillaume's answer is correct, that is you may define the candidate for $A+B$ as the smallest set containing
$$ \mathrm{inl}(a)\ a\in R_A(a)$$
$$ \mathrm{inr}(b)\ b\in R_B(b)$$</p>
<p>and which is a candidate, i.e. closed under anti-reductions and inclusion of neutral terms. But it is possible to define an "elim" c... |
98 | <p>I am looking for a reference (book or article) that poses a problem that seems to be a classic, in that I've heard it posed many times, but that I've never seen written anywhere: that of the possibility of a man in a circular pen with a lion, each with some maximum speed, avoiding capture by that lion. </p>
<p>Refe... | Jan Gorzny | 50 | <p>Since you're asking for a reference, perhaps this will do?</p>
<p>Wolfram Mathworld says the problem was listed by Rado in 1925. The reference is on the problem description page, <a href="http://mathworld.wolfram.com/LionandManProblem.html">here</a>.</p>
|
334,541 | <p>The standard frame for <span class="math-container">$S^3$</span> consists of <span class="math-container">$X_i,X_j,X_k$</span> with <span class="math-container">$X_i(a)=ia, X_j(a)=ja, X_k(a)=ka$</span> where <span class="math-container">$i,j,k$</span> are standard quaternion numbers, <span class="math-container">$a\... | Gael Meigniez | 105,095 | <p>Seifert actually proved that every vector field close enough to X_i has a periodic orbit. (In other words, if you perturbate slightly X_i whose all orbits are compact, then most orbits will of course in general become noncompact, but there will remain at least one compact orbit).
K. Kuperberg built on S^3 a smooth (... |
2,658,368 | <p>I know that one can use Category Theory to formulate polynomial equations by modeling solutions as limits. For example, the sphere is the equalizer of the functions
\begin{equation}
s,t:\mathbb{R}^3\rightarrow\mathbb{R},\qquad s(x,y,z):=x^2+y^2+z^2,~t(x,y,z)=1.
\label{equalizer}
\end{equation}
One could then fin... | Community | -1 | <p>There is this observation of Marvan <a href="https://ncatlab.org/nlab/files/MarvanJetComonadPDE.pdf" rel="nofollow noreferrer">A Note on the Category of PDEs</a> that the jet bundle construction in ordinary differential geometry has the structure of a <a href="https://en.wikipedia.org/wiki/Monad_(category_theory)#Co... |
391,572 | <p>I never really understood what $e$ means and I'm always terrified when I see it in equations. What is it? Can somebody dumb it down for me? I know it's a constant. Is it as simple as that?</p>
| Hagen von Eitzen | 39,174 | <p>On can show that there are functions $f(x)$ with the property
$$\tag1f'(x)=f(x)$$ for all $x$. If one additionally requires $f(0)=1$, the solution is <em>unique</em> and called $\exp(x)$. From this property alone, it follows that the general solution to $f'(x)=f(x)$ is $f(x)=c\exp(x)$ with $c=f(0)$. And also that $... |
1,609,854 | <p>If a polyhedron is made only of pentagons and hexagons, how many pentagons can it contain? With the assumption of three polygons per vertex, one can prove there are 12 pentagons.</p>
<p>Let's not make that assumption, and only use pentagons. </p>
<p>12 pentagons: dodecahedron and <a href="https://en.wikipedia.org... | Vincenzo Zaccaro | 269,380 | <p>Another way is: </p>
<p>$\dfrac {1}{z^2}=\dfrac {\left(1-\dfrac{1}{z}\right)^2}{(z-1)^2}=
\dfrac {(1-z)^2+2(1-z)^3+3(1-z)^4+4(1-z)^5+ \ldots}{(1-z)^2}=1+2(1-z)+3(1-z)^2+4(1-z)^3+ \ldots$ </p>
<p>using the product of convergent power series.</p>
|
1,869,981 | <p>I have a question in a book which says in how many ways can 6 different things be divided between 2 boys and (my understanding of) the explanation goes something along the lines of:</p>
<p>Items: 1 1 1 1 1 1</p>
<p>The first item can be distributed to either boy 1 or boy 2, i.e. 2 choices the second item could be ... | smcc | 354,034 | <p>Your first method correctly counts the number of ways of distributing six different things to the two boys.</p>
<p>Your second method is wrong because it does not take account of the fact that the items are different. </p>
<p>Your second method (if you do it correctly) would give the number of ways of dividing six... |
1,057,172 | <p>How many 3digit numbers can be written with $2,4,4,6,6$ ? </p>
<p>I tried $\frac{5.4.3}{2!.2!} = 15$ but it's wrong. </p>
<p>when I solved the question "how many 3digit numbers can be written with $1,1,2$"
the solution $\frac{3.2.1}{2!}$ was correct but why this way doesn't work for above question?</p>
| Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcomma... |
112,889 | <p>I am new in Wolfram Math and I need a help in making a simple program. Let's suppose we have a list:</p>
<pre><code>list = {{1, 11}, {2, 7}, {4, 2}, {7, 9}, {-2, 3}, {-1, 10}};
</code></pre>
<p>Now, I need to collect the first elements of sublists, but not all of them, only those whose second elements are larger t... | kglr | 125 | <pre><code>Pick[list[[All, 1]], # > 8 & /@ list[[All, 2]]]
</code></pre>
<p>or</p>
<pre><code>Pick[#[[1]], # > 8 & /@ #[[2]]] &@Transpose[list]
</code></pre>
<blockquote>
<p>{1, 7, -1}</p>
</blockquote>
|
427,663 | <p>If $|z|=|w|=1$, and $1+zw \neq 0$, then $ {{z+w} \over {1+zw}} \in \Bbb R $</p>
<p>i found one link that had a similar problem.
<a href="https://math.stackexchange.com/questions/343982/prove-if-z-1-and-w-1-then-1-zw-neq-0-and-z-w-over-1">Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w}... | user1337 | 62,839 | <p>We have $|\zeta|=1 \Leftrightarrow \bar{\zeta}=\frac{1}{\zeta}$.</p>
<p>$|z|=|w|=1 \Leftrightarrow \bar{z}=\frac{1}{z} \text{and } \bar{w}=\frac{1}{w} \Leftrightarrow \frac{\bar{z}+\bar{w}}{1+\bar{z}\bar{w}}=\frac{\frac{1}{z}+\frac{1}{w}}{1+\frac{1}{z} \frac{1}{w}} \Leftrightarrow \overline{ \left(\frac{z+w}{1+zw}\... |
427,663 | <p>If $|z|=|w|=1$, and $1+zw \neq 0$, then $ {{z+w} \over {1+zw}} \in \Bbb R $</p>
<p>i found one link that had a similar problem.
<a href="https://math.stackexchange.com/questions/343982/prove-if-z-1-and-w-1-then-1-zw-neq-0-and-z-w-over-1">Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w}... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>As $z \bar z=|z|^2=1$ and similarly for $w,$</p>
<p>$$\frac{z+w}{1+zw}=\frac{\frac1{\bar z}+\frac1{\bar w}}{1+\frac1{\bar z}\frac1{\bar w}}=\frac{\bar w+\bar z}{1+\bar w\bar z}$$</p>
<p>$$\text{Using }\frac ab=\frac cd=\frac{a+c}{b+d},$$</p>
<p>$$\frac{z+w}{1+zw}=\frac{\bar w+\bar z}{1+\bar w\bar z... |
1,644,715 | <p>Define a locally lipschitz and nonnegative function $f\colon\mathbb{R}^n\to\mathbb{R}$. Let $M\in\mathbb{R}^{n\times n}$ and $\eta>0\in\mathbb{R}$. Consider the function $h\colon\mathbb{R}^n\to\mathbb{R}^n$ defined as </p>
<p>$$h(\mathbf{x})=
\begin{cases}
\frac{1}{\Vert M\mathbf{x}\Vert}M\mathbf{x}, &\text{... | copper.hat | 27,978 | <p>Note that if $x$ is such that $f(x)=0$ or $Mx=0$ then there is a neighbourhood $U$ of $x$ such that $f(y) \|My\| < \eta$ for $y \in U$. Hence $h$ is locally Lipschitz in a neighbourhood of any point such that $f(x)=0$ or $Mx=0$.</p>
<p>Suppose $x$ is such that $f(x) \neq 0$ and $Mx \neq 0$.
Let $\phi(y) = \min({... |
2,835,177 | <p>For example 6,</p>
<p>convert to the base 2 = 110</p>
<p>number of 1's is 2(prime), number of zero's is 1(prime - 1)</p>
<p>or 496 = (111110000)2</p>
<p>5(prime) times 1 and 4 times 0</p>
<p>Is this always correct?</p>
| Graham Kemp | 135,106 | <p>Use the Law of Total Probability: for example $d6, d10$ the results of independen six and ten sided dice.</p>
<p>$$\begin{align}\mathsf P(d6>d10) &= \mathsf P(d10>6)\mathsf P(d6>d10\mid d10>6)+\mathsf P(d10\leq 6)\mathsf P(d6>d10\mid d10\leq 6) \\ &=\tfrac 4{10}\cdot 1+\tfrac 6{10}\cdot\maths... |
54,395 | <p>A generic question: </p>
<p>are there any spectral techniques to estimate the genus of a graph? I am interested in complete balance multipartite graph.</p>
| David E Speyer | 297 | <p>I believe that this is the subject of Jon Kelner's paper <a href="http://math.mit.edu/~kelner/Publications/Docs/LowGenusJournal.pdf" rel="nofollow">Spectral partitioning, eigenvalue bounds, and circle packings for graphs of bounded genus</a>. In particular, he proves a lower bound of the form $O(g/n)$ for $\lambda_2... |
341,946 | <p>Using $n$-th Taylor polynomial for $f_1(x)=\frac{1}{1-x}$ with center in $0$, find $4$-th derivative of $f_2(x)=\frac{1+x+x^2}{1-x+x^2}$ in the point $0$ without calculating it's $1$,$2$ or $3$ derivative. </p>
<p>I'm looking for hints, it's a homework. I've tried using the Taylor expansion with Lagrange rest, but... | Chris Godsil | 16,143 | <p>This an answer to the version of the question given in Mike's comment. Note first that for any polynomials $p$ and $q$ we have $(xp,q)=(p,xq)$. Second, note that $P_n$ is orthogonal to all polynomials of degree less than $n$. </p>
<p>So if $m \le n-2$, then
$$
(xP_n,P_m) = (P_n,xP_m) = 0.
$$</p>
<p>But we know t... |
747,286 | <p>Pardon my ignorance. I don't know enough calculus to understand this. I assume this is a very easy question for this amazing site. I saw this on the The Theory of Riemann Zeta Function Book.</p>
<p>$$\sum_{n=2}^{\infty}\pi (n)\int_{n}^{n+1}\frac{s}{x(x^{s}-1)}dx =s\int_{2}^{\infty}\frac{\pi (x)}{x(x^{s}-1)}dx$$</p>... | ploosu2 | 111,594 | <p>Linearity of integral and $\pi$ is constant $\pi (n)$ on the interval $[n, n+1)$.</p>
<p>EDIT: Also: the set $[2, \infty)$ is broken to intervals $[n, n+1],$ that's not called linearity... (is it additivity?)</p>
|
747,286 | <p>Pardon my ignorance. I don't know enough calculus to understand this. I assume this is a very easy question for this amazing site. I saw this on the The Theory of Riemann Zeta Function Book.</p>
<p>$$\sum_{n=2}^{\infty}\pi (n)\int_{n}^{n+1}\frac{s}{x(x^{s}-1)}dx =s\int_{2}^{\infty}\frac{\pi (x)}{x(x^{s}-1)}dx$$</p>... | Brian | 114,928 | <p>First, remember that if $a<b<c$, then $\int_{a}^{c}f\left(x\right)\;dx=\int_{a}^{b}f\left(x\right)\;dx+\int_{b}^{c}f\left(x\right)\;dx$. Also, since $\pi$ is constant on $\left[n,n+1\right)$, you can push it into the integral as $\pi\left(x\right)$. Then, if you look at the partial sums, at the $n$th term, you... |
1,717,775 | <p>I have defined, for a probability measure $\eta$ we have the fourier transform as $\hat{\eta} = \int e^{itx} \ d\eta(x)$, and for a function $h: \mathbb{R} \to \mathbb{R}$ we have that the fourier transform of $h$ is given by $\hat{h}(t) = \int e^{itx} h(x) \ dx$.</p>
<p>I then am told that the fourier transform of... | Henry | 313,321 | <p>Since you are dealing with non-normal distributions, shouldn't you be using tests for non-normally distributed data? Also, when the sample size is less than 30, you would usually conduct a t-test. In this case, the t-test equivalents for non-normally distributed data are the following:Mann-Whitney test, Mood’s media... |
1,717,775 | <p>I have defined, for a probability measure $\eta$ we have the fourier transform as $\hat{\eta} = \int e^{itx} \ d\eta(x)$, and for a function $h: \mathbb{R} \to \mathbb{R}$ we have that the fourier transform of $h$ is given by $\hat{h}(t) = \int e^{itx} h(x) \ dx$.</p>
<p>I then am told that the fourier transform of... | grand_chat | 215,011 | <p>Try computing confidence intervals for samples from distributions that deviate more from normality. For example, try generating observations from a Poisson distribution with $\lambda=.001$. You'll find that your $z$-theory intervals will not have the advertised coverage.</p>
<p>In general, you'll find that the norm... |
4,553,447 | <p>In order to create a Monte-Carlo Simulation for Mertons Jump Diffusion Model I need to simulate a random variable <span class="math-container">$Y$</span>, which is log-normally distributed. The mean of <span class="math-container">$Y$</span> is supposed to be <span class="math-container">$m$</span> and the variance ... | Narasimham | 95,860 | <p>From trig figure directly</p>
<p><a href="https://i.stack.imgur.com/Yq9Vt.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Yq9Vt.png" alt="enter image description here" /></a></p>
<p><span class="math-container">$$ \text{L is ladder length},~x = L\sin \theta~, \frac{dx}{d\theta} =L\cos \theta=50\co... |
4,553,447 | <p>In order to create a Monte-Carlo Simulation for Mertons Jump Diffusion Model I need to simulate a random variable <span class="math-container">$Y$</span>, which is log-normally distributed. The mean of <span class="math-container">$Y$</span> is supposed to be <span class="math-container">$m$</span> and the variance ... | zeeshan khan | 1,107,512 | <p>@Narasimham did a nice job but I'll give another answer to this</p>
<p><span class="math-container">$$x/50 =tan(θ) $$</span>
<span class="math-container">$$x = 50 [tan(θ)]$$</span>
<span class="math-container">$$dx/dθ = 50 sec^2(θ)$$</span>
<span class="math-container">$$put θ= 45$$</span>
<span class="math-containe... |
4,553,447 | <p>In order to create a Monte-Carlo Simulation for Mertons Jump Diffusion Model I need to simulate a random variable <span class="math-container">$Y$</span>, which is log-normally distributed. The mean of <span class="math-container">$Y$</span> is supposed to be <span class="math-container">$m$</span> and the variance ... | Narasimham | 95,860 | <p>If the base is of <span class="math-container">$50$</span> meters length, then the figure should be scaled up by a magnification factor <span class="math-container">$\sqrt2.$</span></p>
<p>From trig figure directly</p>
<p><a href="https://i.stack.imgur.com/YVq4r.png" rel="nofollow noreferrer"><img src="https://i.sta... |
89,666 | <p>As I am trying to get the solution of the simultaneous ordinary differential equations </p>
<pre><code> b1'[z]-1I*beta1*b1[z]-C1*b2[z]==0,
b2'[z]-1I*beta2*b2[z]+C1*b1[z]==0
</code></pre>
<p>with boundary conditions </p>
<pre><code>b1[1.581825567*10^-6]==0.876212, b2[1.581825567*10^-6]==0.481925
</code... | m_goldberg | 3,066 | <p>We have seen this problem posted by you before. This time you don't seem to understand that you can't have more equations than you have dependent variables. You have three equations </p>
<pre><code>b1'[z] - 1 I*beta1*b1[z] - C1*b2[z] == 0
b2'[z] - 1 I*beta2*b2[z] + C1*b1[z] == 0
(b1[z])^2 + (b2[z])^2 == 1
</code></... |
325,236 | <blockquote>
<p>Is there a bijection between $\mathbb N$ and $\mathbb N^2$?</p>
</blockquote>
<p>If I can show $\mathbb N^2$ is equipotent to $\mathbb N$, I can show that $\mathbb Q$ is countable. Please help. Thanks,</p>
| Dhruv Ranganathan | 42,999 | <p>You're trying to find a way to "count" the things in $\mathbb N^2$ in a manner that gets you to any pair $(a,b)$ in some finite time. Notice that "counting" something is essentially finding this bijection - there is a "first" pair, and a "second pair" and so on.</p>
<p>There are many ways to do this, but most of th... |
1,902,407 | <p>Expand given function $f$ as Taylor series around $c=3$
$$f(x) = \frac{x-3}{(x-1)^2}+\ln{(2x-4)} $$</p>
<p>and find out open interval at what that series is convergent. What is radius of convergence?</p>
<p>This is what i have so far. We know that $\ln{(1+x)} = \sum_{n=1}^{\infty} \frac{x^n}{n}$ and $\frac{1}{1-x}... | Robert Z | 299,698 | <p>Try 3 and 11 by using <a href="https://en.wikipedia.org/wiki/Divisibility_rule" rel="nofollow">divisibility criteria</a>.
Note that $10^{100}-1$ is a sequence of 100 digits all equal to $9$.</p>
|
3,799,421 | <p>While I was finding the potential at a point in the plane of a uniformly charged ring, I got the following integral as the solution.
<span class="math-container">$$\int_{0}^{2\pi} \frac{d\theta}{\sqrt{R^2+x^2-2Rx\cos\theta}},$$</span>
where <span class="math-container">$R$</span> and <span class="math-container">$x$... | Claude Leibovici | 82,404 | <p><span class="math-container">$$\frac{1}{\sqrt{R^2+x^2-2 R x \cos (\theta )}}=\frac{1}{\sqrt{R^2+x^2}}\frac{1}{\sqrt{1-k\cos (\theta )}}$$</span> with <span class="math-container">$k=\frac{2Rx}{R^2+x^2}$</span>.
<span class="math-container">$$\int \frac{d\theta}{\sqrt{1-k\cos (\theta )}}=\frac 2 {\sqrt{{1-k}}} F\left... |
2,202,498 | <p>Test whether the series $\sum_{n=1}^\infty\frac{1+\sin(n)}{2^n}$ is converges or diverges. I used the limit comparison method in order to test its convergence. I chose $a_n = \frac{1+\sin(n)}{2^n}$ and $b_n = \frac{1}{2^n}$.</p>
<p>$$
\lim_{n \rightarrow \infty}\frac{\frac{1+\sin(n)}{2^n}}{\frac{1}{2^n}} = \lim_{n ... | AlexT | 428,860 | <p>You have a map $\phi: \mathbb{Z_7} \rightarrow M_2(\mathbb{Z_7})$ such that $\ f \mapsto f \begin{pmatrix} 0 & 2 \\ 2 & 2 \end{pmatrix}$. Basically you evaluate all possible linear combinations of the matrix $\begin{pmatrix} 0 & 2 \\ 2 & 2 \end{pmatrix}$, which we'll call $A$ for brevity, as f varie... |
1,137,914 | <p>Can Someone please help me prove that the number 1729 is a Pseudoprime?</p>
<p>So a pseudoprime is a composite $n$ such that $n |(2^n − 2)$. And every prime
number also has this property.</p>
| Mikko Korhonen | 17,384 | <p>Suppose that $G$ has a composition series. If you take any series (proper inclusions)</p>
<p>$$1 < N_1 < \cdots < N_t < N_{t+1} = G$$</p>
<p>with each $N_i$ normal in $G$, by Jordan-Hölder the length of this series is at most the composition length of $G$. Hence we can find a minimal normal subgroup $N... |
262,985 | <p>$Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative.</p>
<p>What should $A$ satisfy to guarantee the equation set have only zero solution?</p>
| Dima Pasechnik | 11,100 | <p>If a nonzero $x\geq 0$ is a solution to $Ax=0$ then $\sum_j x_j=a>0$, and $x$ is also a solution to the system
$$A x=0,\quad 0\leq x,\quad e^\top x\leq a,\qquad\qquad\qquad\qquad(*) $$
where $e$ denotes the all-1 vector.
It is a standard application of Farkas lemma in an appropriate form (one might want to use i... |
4,174,481 | <p>I'm going through a proof in which at one point, the authors make the following statement :</p>
<blockquote>
<p>Let <span class="math-container">$(X,d)$</span> be a complete metric space and <span class="math-container">$f:X\to X$</span> and <span class="math-container">$f_m:X\to X$</span> Lipschitz with <span class... | Son Gohan | 865,323 | <p>I have a strong feeling that one needs uniform convergence and not only pointwise convergence to conclude this. Let me give you a partial answer (I'll think about a counterexample) when we assume <span class="math-container">$$\lim_{m\to\infty} \sup_{x \in X} d(f_m(x),f(x)) = 0 $$</span></p>
<p>Indeed note that unde... |
3,789,429 | <blockquote>
<p><strong>Problem</strong>:
Can an <span class="math-container">$f$</span> function be created where:<span class="math-container">$$f\colon\mathbb Q_{+}^{*}\to \mathbb Q_{+}^{*}$$</span>
The function is defined on the set of fully positive rational numbers and is achieved:
<span class="math-container">$\f... | freakish | 340,986 | <p>For <span class="math-container">$x=1$</span> we get <span class="math-container">$f(f(y))=f(f(1))/y$</span>. For <span class="math-container">$y=1$</span> we get <span class="math-container">$f(xf(1))=f(f(x))$</span>. And so if we replace <span class="math-container">$y$</span> with <span class="math-container">$x$... |
137,915 | <blockquote>
<p>Given two continuously differentiable functions $f,\
g:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(x_0) = g(x_0)$ and
$f'(x_0) < g'(x_0)$, there exists $\epsilon > 0$ such that $f(x) <
g(x)$ for $x \in (x_0, x_0 + \epsilon)$.</p>
</blockquote>
<p>The above result is not too difficult... | Siminore | 29,672 | <p>If $f$ and $g$ are differentiable at $x_0$, there exist two functions $\omega_f$ and $\omega_g$ such that $\omega_f$ and $\omega_g$ are continuous at $x_0$ and $$f(x)=f(x_0)+\omega_f(x)(x-x_0), \quad g(x)=g(x_0)+\omega_g (x)(x-x_0)$$ for all $x$ (in some neighborhood of $x_0$). From the assumptions you get $\omega_f... |
2,491,560 | <p>Given a sequence $(s_n)$ in $\mathbb R$ such that
$$\lim \limits_{n \to \infty}( s_{n+1}-s_n)=0,$$
I am asked to prove
$(s_n)$ converges. </p>
<p>I know all Cauchy sequences converge in $\mathbb R^k$. So I want to prove that $(s_n)$ is Cauchy. </p>
<p>I am stuck as to how to show the given sequence is a Cauchy. ... | José Carlos Santos | 446,262 | <p>Another counter-example is: $s_n=\log n$. Then$$\lim_{n\to\infty}\bigl(\log(n+1)-\log(n)\bigr)=\lim_{n\to\infty}\log\left(1+\frac1n\right)=0.$$</p>
|
1,826,435 | <p>I was given a problem:
(Edited) </p>
<blockquote>
<p>Show $$ S=\bigl\{S_p:p\in\mathbb{P}\bigr\}\cup\bigl\{\{1\}\bigr\} $$
where $\mathbb{P}$ is the set of prime numbers, and $S = \{n \in
\mathbb{N}: n \text{ is a multiple of } p\}$ is a subbasis on
$\mathbb{N}$</p>
</blockquote>
<p>The easiest way to do th... | 5xum | 112,884 | <p>It's actually really easy to prove every number has a prime divisor, and the proof is rather nice:</p>
<hr>
<p>Take a number $n\in\mathbb N$. If $n$ is prime, then $n=1\cdot n$ and $n$ is a multiple of a prime. </p>
<p>If $n$ is not prime, then $n$ has a divisor that is not $1$ and is not $n$ (because otherwise i... |
1,379,878 | <blockquote>
<p>Let $M_n(\mathbb{C})$ denote the vector space over $\mathbb{C}$ of all $n\times n$ complex matrices. Prove that if $M$ is a complex $n\times n$ matrix then $C(M)=\{A\in M_n(\mathbb{C}) \mid AM=MA\}$ is a subspace of dimension at least $n$.</p>
</blockquote>
<p>My Try:</p>
<p>I proved that $C(M)$ is ... | Community | -1 | <p>Let <span class="math-container">$K$</span> be a field, <span class="math-container">$A\in M_n(K)$</span> and <span class="math-container">$C(A)$</span> be the commutant of <span class="math-container">$A$</span>. Clearly, if <span class="math-container">$A,B$</span> are similar, then <span class="math-container">$\... |
1,183,643 | <p>Given a integer $h$</p>
<blockquote>
<p>What is $N(h)$ the number of full binary trees of height less than $h$?</p>
</blockquote>
<p><img src="https://i.stack.imgur.com/XcNVi.jpg" alt="enter image description here"></p>
<p>For example $N(0)=1,N(1)=2,N(2)=5, N(3)=21$(As pointed by <a href="https://math.stackexc... | TravisJ | 212,738 | <p>(Partial Answer) First, $N(3)=21$. Second, and probably more useful, you can construct all the trees of height $h$ by adding two children to every non-empty subset of lowest vertices on a tree of height $h-1$. So, if at height $h-1$ you have trees $T_{1},...,T_{n}$ and tree $T_{i}$ has $v_{i}$ vertices on the lowe... |
1,183,643 | <p>Given a integer $h$</p>
<blockquote>
<p>What is $N(h)$ the number of full binary trees of height less than $h$?</p>
</blockquote>
<p><img src="https://i.stack.imgur.com/XcNVi.jpg" alt="enter image description here"></p>
<p>For example $N(0)=1,N(1)=2,N(2)=5, N(3)=21$(As pointed by <a href="https://math.stackexc... | Ross Millikan | 1,827 | <p>You can define $T(h,b)$ as the number of trees of height $h$ with $b$ leaves at the bottom. Your $N(h)$ is then the sum over $b$ of $T(h,b)$ To get a tree of height $h+1$ and $b$ leaves on the bottom, you pick a tree of height $h$ and at least $b/2$ leaves on the bottom, then pick $b/2$ leaves to hang new leaves f... |
1,986,007 | <p>What is the intuitive explanation of the Inverse Function Theorem (and generalized to multiple dimensions)?</p>
| Christian Blatter | 1,303 | <p>If you have $n\geq1$ independent variables $x_k$ and $n$ functions $$f_i:\quad U\to{\mathbb R},\quad (x_1,\ldots, x_n)\mapsto y_i:=f_i(x_1,\ldots, x_n)\qquad(1\leq i\leq n)$$ defined in a neighborhood $U$of ${\bf0}$, whereby ${\bf f}({\bf 0})={\bf 0}$, then under certain "technical assumptions", this setup defines t... |
173,360 | <p>From my lecture notes: "The notation $\mathbb T$ will be used for the additive circle and $S^1$ for the multiplicative circle."</p>
<p>What I understand: As a topological group, $S^1$ has the subspace topology of $\mathbb R^2$ and multiplication is defined as $(e^{ia}, e^{ib}) \mapsto e^{i(a + b)}$. </p>
<p>My gu... | Alex Becker | 8,173 | <p>In my experience the two are used interchangeably, as the two groups are isomorphic, even as topological groups, and this isomorphism (as topological groups) is unique! You are right that "$\bmod 1$" is the same as $\mathbb R/\mathbb Z$, which is best viewed as $[0,1]$ with the endpoints identified. The only differe... |
3,028,385 | <p>How to determine if there is a surjective homomorphism from <span class="math-container">$U(15)$</span> to <span class="math-container">$\mathbb{Z}_4$</span>? <span class="math-container">$U(15)$</span> is the multiplicative group of positive integers less than <span class="math-container">$n$</span> that are coprim... | Mauro ALLEGRANZA | 108,274 | <p>A formula is an <em>expression</em> of the language of set theory built up acoording to the <a href="https://en.wikipedia.org/wiki/First-order_logic#Syntax" rel="nofollow noreferrer">rules of the syntax</a>. </p>
<p>Examples : <span class="math-container">$∃y \ ∀x \ ¬(x ∈ y), ∀x \ ¬(x ∈ \emptyset)$</span>.</p>
<p>... |
3,028,385 | <p>How to determine if there is a surjective homomorphism from <span class="math-container">$U(15)$</span> to <span class="math-container">$\mathbb{Z}_4$</span>? <span class="math-container">$U(15)$</span> is the multiplicative group of positive integers less than <span class="math-container">$n$</span> that are coprim... | Alberto Takase | 146,817 | <p>Put simply in the language of set theory we start with atomic formulas
<span class="math-container">$$(x\in y)$$</span>
or
<span class="math-container">$$(x=y)$$</span>
where <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are variables.</p>
<p>Then we expand the definition of fo... |
1,178,969 | <p>$$\sum_{k=1}^{\infty}\ln\frac{k+5}{k+4}$$</p>
<p>$$
\frac{\ln\frac{k+6}{k+5}}{\ln\frac{k+5}{k+4}}
=\cdots
$$</p>
<p>(ln(k+6/k+5)/ln(k+5/k+4))</p>
<p>ln((k+6/k+5)*(k+4/k+5))</p>
<p>simplify</p>
<p>lim as k approaches infinity of ln(kˆ2 +10k +24/ kˆ2 +10k+25)= ln(1) = 0.</p>
<p>why doesn't the series converge?</... | Claude Leibovici | 82,404 | <p>Concerning the ratio test, if you consider $$a_k=\ln\frac{k+5}{k+4}$$ then $$\frac{a_{k+1}}{a_k}=\frac{\log \left(\frac{k+6}{k+5}\right)}{\log \left(\frac{k+5}{k+4}\right)}=\frac{\log \left(1+\frac{1}{k+5}\right)}{\log \left(1+\frac{1}{k+4}\right)}$$ Now, use that, for small values of $y$, $$\log(1+y)=y-\frac{y^2}{2... |
2,476,267 | <p>Here's a problem presented in my proofs class that I cannot for the life of me figure out. Apparently the answer is P2 according to the answer book but I have no idea how Help is much appreciated, thanks in advance!</p>
<p>Anna wrote down the following 3 statements, denoted P1, P2, P3, on a blank piece of paper:</p... | Alex B. | 492,378 | <p>Suppose that P1 is True. Then P2 and P3 must be false, a contradiction to P1. </p>
<p>If you follow the same reasoning for each statement, you will find your answer.</p>
|
1,160,726 | <p>I am trying to figure out why the equation,</p>
<p>$2x^2\:-5y^2\:=\:1$</p>
<p>has no solutions. I have tried taking the equation mod $2$,$4$, and $5$, but when I do this I am not sure what I am supposed to look for. I would really appreciate an explanation on how to solve this.</p>
<p>Thanks!</p>
| Bill Dubuque | 242 | <p>${\rm mod}\ 5\!:\ 1 \equiv 2x^2\overset{(\ \ )^{\large 2}}\Rightarrow 1\equiv 4x^4\overset{\rm Fermat}\equiv 4\,\Rightarrow\, 5\mid 4-1\,\Rightarrow\!\Leftarrow$</p>
|
18,174 | <p>I am an undergraduate secondary math education major. In <span class="math-container">$2$</span> weeks I have to give a <a href="https://www.mathmammoth.com/lessons/number_talks.php" rel="nofollow noreferrer">Number Talk</a> in my math ed class on the problem "<span class="math-container">$3.9$</span> times <span cl... | Adam | 4,791 | <p>Combining your last two methods: <span class="math-container">$3.9*7.5 = 4*8 - 0.1*8 - 4 * 0.5 + 0.1*0.5$</span>, which can be thought of as computing the area of the big rectangle below, cutting off the two extra strips along the edge, then adding back in a copy of the corner piece since you have removed it twice, ... |
946,738 | <p>I ran into a nice question from one book in Discrete Mathematics. I want to someone lean me how solve such a problem, because I prepare for entrance exam. </p>
<blockquote>
<blockquote>
<p>if the time is "Wednesday 4 afternoon", after $47^{74}$ hours, we are in what hours? and what day? </p>
</blockquote>
<... | paw88789 | 147,810 | <p>Since $47^{74}$ hours is more than a googol ($10^{100}$) years, by all estimates, the sun will be long gone, and we won't have days anymore.</p>
<p>But if you ignore all that, you should try to reduce $47^{74} \pmod{168}$ (the number of hours in a week).</p>
|
155,897 | <p>I would like to exclude non-western characters and words from a text file. I do not know how to insert the text file here, but I suppose you can do without it. All your suggestions will be much appreciated.</p>
<p>Update: I have used:</p>
<pre><code>Alphabet["Russian"]
Select[dict21, Not@StringContainsQ[#, Alterna... | b3m2a1 | 38,205 | <p>Here's a better <code>StringDelete</code> than my first method, courtesy of Alexey Popkov:</p>
<pre><code>text = Import["https://sv.wikipedia.org/wiki/Kategori:Sjötermer"];
pattern =
Except[
Join[Alphabet["Swedish"],
ToUpperCase@Alphabet["Swedish"], {PunctuationCharacter,
WhitespaceCharacter}
]... |
3,700,575 | <p>I have to find Taylor series at <span class="math-container">$a=1$</span> for </p>
<p><span class="math-container">$
f(x)=\begin{cases}
\frac{e^{x}-e}{x-1},\quad &\text{if } x\ne1\\
e,\quad &\text{if } x=1\\
\end{cases}
$</span></p>
<p>I haven't found Taylor series for ... | Kavi Rama Murthy | 142,385 | <p>Using the series expansion of <span class="math-container">$e^{x-1}$</span> we get <span class="math-container">$f(x)=e \frac {e^{x-1}-1}{x-1}=e\sum\limits_{k=1}^{\infty} \frac {(x-1)^{k}} {(x-1)k!} =\sum\limits_{k=1}^{\infty}\frac { e{(x-1)^{k-1}}} {k!}=\sum\limits_{k=0}^{\infty}\frac { e{(x-1)^{k}}} {(k+1)!}$</spa... |
3,700,575 | <p>I have to find Taylor series at <span class="math-container">$a=1$</span> for </p>
<p><span class="math-container">$
f(x)=\begin{cases}
\frac{e^{x}-e}{x-1},\quad &\text{if } x\ne1\\
e,\quad &\text{if } x=1\\
\end{cases}
$</span></p>
<p>I haven't found Taylor series for ... | vin92 | 795,021 | <p>First I compute </p>
<p><span class="math-container">$$f'(x)=\dfrac{e^x-f(x)}{x-1}.$$</span>
And </p>
<p><span class="math-container">$$
f''(x)=\dfrac{e^x-2f'(x)}{x-1}.$$</span></p>
<p>So that I find a pattern</p>
<p><span class="math-container">$$
f'''(x)=\dfrac{[e^x-2f''(x)](x-1)-[e^x-2f'(x)]}{(x-1)^2}=\dfrac{... |
3,381,543 | <p>Let <span class="math-container">$X \sim \text{Poisson}(\lambda)$</span>. If <span class="math-container">$P(X = 1) = 0.1$</span> and <span class="math-container">$P(X = 2) = 0.2$</span>, what is the value of <span class="math-container">$P(X = 3)?$</span></p>
<p>I've tried to expand the given equalities, finding <... | Ongky Denny Wijaya | 556,947 | <p>Given <span class="math-container">$X\sim \text{POI}(\lambda)$</span>, so <span class="math-container">$X$</span> have p.d.f.
<span class="math-container">$$P(X=x)=\dfrac{e^{-\lambda}\lambda^x}{x!}, x=0,1,\ldots.$$</span>
So, we have
<span class="math-container">\begin{eqnarray}
P(X=1)=0.1\iff \dfrac{e^{-\lambda}\la... |
3,068,381 | <p>I calculated, using Mathematica, that for <span class="math-container">$4\leq k \leq 100$</span>,
<span class="math-container">$$ \sum_{j=k}^{2k} \sum_{i=j+1-k}^j (-1)^j 2^{j-i} \binom{2k}{j} S(j,i) s(i,j+1-k) = 0,$$</span>
where <span class="math-container">$s(i,j)$</span> and <span class="math-container">$S(i,j)$<... | Marko Riedel | 44,883 | <p>It looks like my first response interprets the problem statement to
use unsigned Stirling numbers of the first kind. We find for signed
ones,</p>
<p><span class="math-container">$$S_n = (-1)^{n+1} \sum_{j=n}^{2n} \sum_{k=j+1-n}^j
(-1)^k 2^{j-k} {2n\choose j} {j\brace k}
{k\brack j+1-n}.$$</span></p>
<p>With... |
220,170 | <p>My question is rather philosophical : without using advanced tools as Perlman-Thurston's geometrisation, how can we get convinced that the class of closed oriented $3$-manifolds is large and that simple invariants as Betti number are not even close to classify ?</p>
<p>For example i would start with :</p>
<ol>
<li... | Marc Kegel | 84,120 | <p>I would explain it like this:</p>
<p>Start with a framed link in $S^3$ and do <a href="https://en.wikipedia.org/wiki/Dehn_surgery" rel="nofollow noreferrer">Dehn-surgery</a> along it to get a new $3$-manifold. In fact one can get every compact oriented $3$-manifold in this way as mentioned earlier.</p>
<p>To disti... |
2,559,623 | <p>The question is as follows:<p>
Calculate $\lim_{n \to +\infty} \int_{0}^{+\infty} \frac{n \sin(\frac{x}{n})}{1 + x^2} dx$.<p></p>
<p>$\textbf{Some ideas:}$<p>
We can use the fact that $\sin(\frac{x}{n}) \simeq \frac{x}{n} $.But then we find that <p>
$\lim_{n \to +\infty} \int_{0}^{+\infty} \frac{n \sin(\frac{x}{n})... | Deepak | 151,732 | <p>Easiest to solve geometrically with a quick Argand diagram sketch. The first locus is a ray in the first quadrant making an angle with the horizontal real axis of $\frac{\pi}{6}$ with a "hole" at the origin (since the origin is excluded from $C_1$). The second locus $C_2$ is a circle of undetermined radius but cente... |
2,535,111 | <p>Let $n$ be a natural number, and let $1\leq j,k\leq n$ be two randoms.
Show that if $k\mid n$ ($k$ divides $n$) then the probabilty of
$k$ also dividing $j$ is $\frac{1}{k}$, that is $\text{P}\left(k\mid j\right)=\frac{1}{k}$.</p>
<p>One way i was thinking about is to look straight forward for
$$\text{P}\left(k\mi... | ℋolo | 471,959 | <p>By symmetry I can tell that it is on the line $x=y=0$</p>
<p>Now to get $z$ we need to find few things:</p>
<p>the volume of the shape is: $\frac23\pi r^3=\frac23\pi$</p>
<p>Now the integral: we are searching for an integral of $dz$ so we need a function that "have" the change of $z$ in it, let's take the functio... |
1,576,561 | <p>I'd like to show that $$\sum\limits_{n = 1}^\infty {{{{x^{n + 1}}} \over {n(n + 1)}}} $$ absolutely converges for $|x| < 1$</p>
| Community | -1 | <p>$$\sum\limits_{n = 1}^\infty \left|{{{{x^{n + 1}}} \over {n(n + 1)}}}\right|<\sum\limits_{n = 1}^\infty {{{{|x|^{n + 1}}}} }=\lim_{n\to\infty}x^2\frac{1-|x|^n}{1-|x|}.$$</p>
|
3,284,729 | <p>I was wondering how to calculate the following:
I know the average chance of death per year for a certain age.
What is the average chance of death for a 5 year period then?</p>
<p>Is it simply additive?
The chances are listed here (jaar is the dutch word for Year)</p>
<p><a href="https://i.stack.imgur.com/09wt9.pn... | heropup | 118,193 | <p>What you have is an excerpt of a life table, which is used in the actuarial sciences. The entries you have are ages <span class="math-container">$x$</span>, and the corresponding probabilities <span class="math-container">$q_x$</span> of death within one year for a life aged <span class="math-container">$x$</span>.... |
214,705 | <p>If $a(x) + b(x) = x^6-1$ and $\gcd(a(x),b(x))=x+1$ then find a pair of polynomials of $a(x)$,$b(x)$.</p>
<p>Prove or disprove, if there exists more than 1 more distinct values of the polynomials.</p>
| Brian M. Scott | 12,042 | <p>Since $\gcd\{a(x),b(x)\}=x+1$, there are polynomials $p(x)$ and $q(x)$ such that $a(x)=(x+1)p(x)$ and $b(x)=(x+1)q(x)$. </p>
<p>$$x^6-1=(x+1)\left(x^5-x^4+x^3-x^2+x-1\right)\;,$$</p>
<p>so </p>
<p>$$p(x)+q(x)=x^5-x^4+x^3-x^2+x-1\;.\tag{1}$$</p>
<p>You can easily find several pairs of relatively prime polynomials... |
1,914,536 | <p><strong>Problem</strong>: Solve $3^x \equiv 2 \pmod{29}$ using Shank's Baby-Step Giant-Step method.</p>
<p>I choose $k=6$ and calculated $3^i \pmod {29}$ for $i=1,2,...,6$.
$$3^1 \equiv 3 \pmod {29}$$
$$3^2 \equiv 9 \pmod {29}$$
$$3^3 \equiv 27 \pmod {29}$$
$$3^4 \equiv 23 \pmod {29}$$
$$3^5 \equiv 11 \pmod {29}$$
... | Darío A. Gutiérrez | 353,218 | <p>$$3^x \equiv 2 (\text{ mod } 29)$$
$\varphi(29) = 28$<br>
$m = \lceil \sqrt{28} \rceil = 6$</p>
<p>$\begin{array}{c|ccccc}
j& 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
3^j & 1 & 3& 9 & 27 & 23 & \color{#f00}{11}
\end{array}$
<br><br>
$3^{-6}= 3^{28 -6 } = 3^{22} = 22$</p>
<... |
2,842,481 | <p>If the minimum and the maximum values of $y = (x^2-3x+c)/(x^2+3x+c)$ are $7$ and $1/7$ , then the value of $c$ is?</p>
<p>I cross multiplied the equation and tried to find it's discriminant, but I don't think it gets me anywhere. A little hint would be appreciated!</p>
| dxiv | 291,201 | <p>Hint: suppose $\,c, x \gt 0\,$, then $\,u = x + \dfrac{c}{x} \ge 2 \sqrt{c}\,$ by AM-GM with equality iff $\,x = \sqrt{c}\,$, and:</p>
<p>$$
f(x)=\frac{x^2-3x+c}{x^2+3x+c} \color{red}{\cdot \frac{\;\;\cfrac{1}{x}\;\;}{\cfrac{1}{x}}}=\dfrac{x+ \cfrac{c}{x}-3}{x+ \cfrac{c}{x}+3} = \dfrac{u-3}{u+3}=1 - \cfrac{6... |
40,618 | <p>Let $f_{=}$ be a function from $\mathbb{R}^{2}$ be defined as follows:
(1) if $x = y$ then $f_{=}(x,y) = 1$;
(2) $f_{x,y} = 0$ otherwise.</p>
<p>I would like to have a proof for / a reference to a textbook proof of the following theorem (if it indeed is a theorem):</p>
<p>$f_{=}$ is uncomputable even if one restr... | slimton | 9,355 | <p>Suppose that $f_=$ is computable when restricted to computable real numbers, which means that there exists a Turing machine that, given as input the encoding of two Turing machines $M_1$ and $M_2$ that compute the fractional digits of two computable real numbers $r_1$ and $r_2$ in $[0,1]$, produces $1$ if $r_1 = r_2... |
40,618 | <p>Let $f_{=}$ be a function from $\mathbb{R}^{2}$ be defined as follows:
(1) if $x = y$ then $f_{=}(x,y) = 1$;
(2) $f_{x,y} = 0$ otherwise.</p>
<p>I would like to have a proof for / a reference to a textbook proof of the following theorem (if it indeed is a theorem):</p>
<p>$f_{=}$ is uncomputable even if one restr... | Carl Mummert | 5,442 | <p>The main difficulty in finding a reference for this is that it's so well known :). The fact that equality of reals is only (negatively) semidecidable is a basic and important result in both computable analysis and constructive analysis. </p>
<p>The underlying phenomenon here <em>is</em> about continuity. As Gerald... |
1,456,262 | <p>Show that the equation to a circle with center at $z_0$ and radius $r$ can be written as $$|z-z_0| = r$$ or as $$z\bar{z} - z\bar{z_0} - \bar{z}z_0 + |z_0|^2 = r^2$$</p>
<p>I let $z_0 = (x_0 + iy_0) = (x_0,y_0)$. Now I have $$(x-x_0)^2 + (y-y_0)^2 = r^2$$</p>
<p>I know $x=\frac{1}{2}(z+\bar{z})$ and $y = \frac{1}{... | Urgje | 95,681 | <p>Subtract the second equation from the first. Then
\begin{equation*}
x^{\prime }(t)-y^{\prime }(t)=x(t)-y(t)+4t-e^{t}
\end{equation*}
from which $u(t)=x(t)-y(t)$ can be found. Next
\begin{equation*}
x^{\prime }(t)=2x(t)-y(t)+4t=x(t)+u(t)+4t
\end{equation*}
from which $x(t)$ can be obtained.</p>
|
1,729,220 | <p>Let</p>
<p>$$f(x)=\frac{1}{x^2+3x+2}$$</p>
<p>I must find $$\lim_{n\to\infty}\sum_{i=0}^n \frac{(-1)^i}{i!}f^{(i)}(1)$$</p>
<p>How should I proceed?</p>
| Mark Viola | 218,419 | <p>Let $f(x)=\frac{1}{x^2+3x+2}$. Then, using partial fraction expansion, we find that</p>
<p>$$f(x)=\frac{1}{x+1}-\frac{1}{x+2}$$</p>
<p>The $i$'th derivative of $f(x)$ is given by </p>
<p>$$f^{(i)}(x)=(-1)^i\, i!\left(\frac{1}{(x+1)^{i+1}}-\frac{1}{(x+2)^{i+1}}\right)$$</p>
<p>Therefore, we find that</p>
<p>$$\... |
1,797,245 | <p>How does one show that $I_n = \int\limits_0^1 x^n e^x dx$ is decreasing?</p>
<p>The best I came up with is this: $I_{n-1} - I_n= \int\limits_0^1 e^xx^{n-1}(1-x)dx$, but how do we go from here?</p>
<p>I'd appreciate some hints.</p>
| Jack D'Aurizio | 44,121 | <p>You may prove much more than monotonicity: since $e^x$ is bounded between $1$ and $e$ on $[0,1]$,</p>
<p>$$ \int_{0}^{1} x^n(1-x)\,dx\leq \int_{0}^{1}e^x x^n(1-x)\,dx \leq e\int_{0}^{1} x^n(1-x)\,dx \tag{1}$$
hence:</p>
<blockquote>
<p>$$ \frac{1}{(n+1)(n+2)}\leq \left(I_n-I_{n+1}\right)\leq\frac{e}{(n+1)(n+2)}.... |
3,528,946 | <p><a href="https://i.stack.imgur.com/JG89d.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/JG89d.png" alt="enter image description here"></a></p>
<p>This is the example from Friedberg's Linear Algebra book. It is giving an example about Null space (denoted by <span class="math-container">$N(T)$</sp... | egreg | 62,967 | <p>If <span class="math-container">$\lambda=0$</span> the statement may be true or false, depending on the matrix <span class="math-container">$A$</span>.</p>
<p>The easiest example is for <span class="math-container">$A=0$</span> a null matrix: the only eigenvalue of both <span class="math-container">$A^T\!A$</span> ... |
1,628,386 | <p>I had following limit of two variables as a problem on my calculus test. How does one show whether the limit below exists or does not exist? I think it does not exist but I was not able to show that rigorously. There was a hint reminding that $\lim_{t\to 0}\sin t / t=1$.</p>
<p>$$\lim_{(x,y)\to (0,0)} \frac{3x^2\si... | محمد حسین فصیحی هرندی | 417,896 | <p>$$\lim_{(x,y)\to(0,0)}\frac{3}{2}(\frac{x^2sin^2(y)}{x^4+siny^4})$$
If Dividing by $y^4$
$$\lim_{(x,y)\to(0,0)}\frac{3}{2}(\frac{\frac{x^2sin(y)sin(y)}{y^2yy}}{\frac{x^4}{y^4}+\frac {sin(y^4)}{y^4}})$$
Know:$\lim_{t\to0}\frac{sint}{t}=1$</p>
<p>$(y^4=t)$,$(y=t)$</p>
<p>$$\lim_{(x,y)\to(0,0)}\frac{3}{2}(\frac{\fr... |
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