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 |
|---|---|---|---|---|
783 | <p>In a category I have two objects $a$ and $b$ and a morphism $m$ from $a$ to $b$ and one $n$ from $b$ to $a$. Is this always an isomorphism? Why is it emphasized that this has to be true, too: $m \circ n = \mathrm{id}_b$ and $n \circ m = \mathrm{id}_a$?</p>
<p>I am looking for an example in which the id-part is not ... | Eric O. Korman | 9 | <p>If you have no restrictions on m and n, then clearly they cannot be isomorphisms in general. For instance, take any two groups G and H and let m: G --> H, n: H --> G be the zero homomorphisms.</p>
<p>Even if you say that n and m are monomorphisms, then it is still not true in general that they are isomorphisms. I... |
61,406 | <p>Can I say this is a bad performance from the new V10 function <code>SubsetQ</code>?</p>
<p>Here are some tests comparing it to <code>Complement[l2, l1] === {}</code></p>
<pre><code>count1[data_,list_]:=Module[{r},
r=SubsetQ[#,list]&/@data;
Counts[r]
]
count2[data_,list_]:=Module[{r},
r=Complement[li... | Taliesin Beynon | 7,140 | <p><code>SubsetQ</code> <em>is</em> implemented in top-level using <code>Complement[a, b] === {}</code>. It has some overhead because it has to treat associations specially, plus it has to go through the requisite error-handling rigmarole. But has the same time complexity in the length of the first argument:</p>
<p><i... |
2,942,244 | <p>I need to find unmeasurable dense subset of circle. I think, that I found the unmeasurable set, but I can't show that it is dense. Here is my construction. Let's take <span class="math-container">$\alpha\in\mathbb{R}\setminus\mathbb{Q}$</span>, consider the irrational rotation of the circle on angle <span class="mat... | R.C.Cowsik | 293,582 | <p>Take any unmeasurable subset <span class="math-container">$X$</span> and add a countable dense subset <span class="math-container">$Y$</span>, that is take <span class="math-container">$X \cup Y$</span>. </p>
|
51,732 | <p>A <em>Perron number</em> is a real algebraic integer $\lambda$ that is larger than the absolute value of any of its Galois conjugates. The Perron-Frobenius theorem says that any
non-negative integer matrix $M$ such that some power of $M$ is strictly positive has a
unique positive eigenvector whose eigenvalue is a Pe... | Community | -1 | <p>(This is more of an extended comment than an answer.)</p>
<p>You speculate whether imposing artificial bounds on the coefficients imposes a bias on the pictures your are producing. There is reason to think that this is possible. Edelman and Kostlan have some nice results on "random" polynomials, where a possible c... |
4,102,443 | <p>Let <span class="math-container">$a,b \in [0,\infty)$</span> with <span class="math-container">$a \leq b$</span>.</p>
<p><span class="math-container">$D(x,y)=\mid \frac{x}{2+x}-\frac{y}{2+y} \mid$</span>.</p>
<p>There exist constants <span class="math-container">$c_{1}, c_{2} \in [0,\infty)$</span> such that <span c... | paul blart math cop | 571,438 | <p>A group action on an object <span class="math-container">$X$</span> can be thought of as a group homomorphism <span class="math-container">$G \longrightarrow Aut(X)$</span>. This is a special case of a monoid action, which would be a monoid homomorphism <span class="math-container">$M \longrightarrow End(X)$</span>,... |
556,150 | <blockquote>
<p>Prove that a metric space is totally bounded if and only if every sequence has a Cauchy subsequence. </p>
</blockquote>
<p>I think I proved the Cauchy subsequence part:</p>
<p>Let $a_{0},a_{1}, a_{2}, a_{3}, a_{4},...\in X$ be a sequence.</p>
<p>For each $k$, let $F \subseteq X$ be a finite $\frac1... | Moss | 23,468 | <p>You have proved that totally bounded implies every sequence has Cauchy subsequence, so I will prove the other implication. This is a proof using the contrapositive, that is, not totally bounded implies that there is a sequence with no Cauchy subsequence.</p>
<p>Suppose that $X$ is not totally bounded. Then there is... |
664,347 | <p>For each of the following values of ($a,b$), find the largest number that is not of the form $ax+by$ with $x\geq 0$ and $y \geq 0$.</p>
<p>$(i) (a,b) = (3,7)$</p>
<p>$(ii) (a,b) = (5,7)$</p>
<p>$(iii) (a,b) = (4,11)$</p>
<p>I know how to compute numbers of these forms (clear), but have no idea how to generate on... | Tom-Tom | 116,182 | <p>To help you, let us have a look on how it works with the first example.
With $x\geq0$ and $y\geq0$, what are the smallest numbers $3x+7y$ that you can generate ? </p>
<ul>
<li>Clearly 0</li>
<li>The next smallest one is 3, using $x=1$ and $y=0$</li>
<li>you can't generate 4 or 5 so the next one is $6=3\times2+7\tim... |
479,851 | <p>I toss a coin a many times, each time noting down the result of the toss. If at any time I have tossed more heads than tails I stop.
I.e. if I get heads on the first toss I stop.
Or if toss T-T-H-H-T-H-H I will stop.
If I decide to only toss the coin at most 2n+1 times, what is the probability that I will get more ... | user84413 | 84,413 | <p>As in some previous answers, assume that you continue tossing the coin even if you get more heads than tails at some point, so you have a total of $2n+1$ tosses and $2^{2n+1}$ possible outcomes of the tosses.</p>
<p>To count the number of sequences of tosses in which we never get more heads than tails, we can use A... |
22,415 | <p>Very often here when a post does not show any work gets down voted to hell, <a href="https://math.stackexchange.com/questions/1611839/how-to-solve-problem-2014-iran-tst">this question</a> is getting upvoted +5 without having shown any work.</p>
<p>I have seen others people's post get downvoted severely with more wo... | Jyrki Lahtonen | 11,619 | <p>I have other reasons not to upvote that question, but I would not downvote or vote to close it either.</p>
<p>The official reason for closing many a question is "off-topic -> missing context". Some users mistakenly equate "missing context" with "missing work/effort shown".</p>
<p>The linked question gives context ... |
22,415 | <p>Very often here when a post does not show any work gets down voted to hell, <a href="https://math.stackexchange.com/questions/1611839/how-to-solve-problem-2014-iran-tst">this question</a> is getting upvoted +5 without having shown any work.</p>
<p>I have seen others people's post get downvoted severely with more wo... | zyx | 14,120 | <p>Whatever its advantages or disadvantages in other categories, the "work and effort" question closure movement never made any sense for postings in the (contest-math) tag. Where there are downvotes or close votes on tagged contest questions it is mainly the result of believers in the work-effort-context philosophy v... |
1,252,857 | <p>Let $f\colon \mathbb R \to \mathbb R$ be defined by $f(x)= 5x^3+3$. Is it onto?</p>
<p>According to me, if $y=5x^3+3$, then $x = \sqrt[3]{(y-3)/5}$ is not an element of $\mathbb R$ for all $y \in (-\infty,3)$ so all numbers in the codomain $(-\infty,3)$ wont have pre-images.</p>
<p>But many say $5x^3+7$ as an odd ... | Bat Dejean | 196,331 | <p>$x=\sqrt[3]{\frac{y-3}5}$ is an inverse as you claimed; keep in mind that odd radicals ($\sqrt[3]{}$, $\sqrt[5]{}$, etc.) are defined for all real numbers, whereas even ones are not (unless we introduce extensions such as $\mathbb{C}$.) Therefore all numbers in the codomain $\mathbb{R}$ have preimages given by the r... |
2,751,300 | <blockquote>
<p>Figure out with distinct real numbers the system of equations.</p>
<p><span class="math-container">$$\frac{xy}{x-y} = \frac{1}{30}$$</span>
<span class="math-container">$$\frac{x^2y^2}{x^2+y^2} = \frac{1}{2018}$$</span></p>
</blockquote>
<p>I multiplied x-y both side on the first equation and square on ... | Dr. Sonnhard Graubner | 175,066 | <p>From the first equation we get:
$$30xy=x-y$$ or $$y(30x+1)=x$$ so $$y=\frac{x}{30x+1}$$ substituting this in the second equation we get
$${\frac {1}{2018}}\,{\frac { \left( 43\,x+1 \right) \left( 13\,x-1
\right) }{450\,{x}^{2}+30\,x+1}}
=0$$ Can you finish?</p>
|
2,751,300 | <blockquote>
<p>Figure out with distinct real numbers the system of equations.</p>
<p><span class="math-container">$$\frac{xy}{x-y} = \frac{1}{30}$$</span>
<span class="math-container">$$\frac{x^2y^2}{x^2+y^2} = \frac{1}{2018}$$</span></p>
</blockquote>
<p>I multiplied x-y both side on the first equation and square on ... | Cesareo | 397,348 | <p>$$
\left\{
\begin{array}{rcl}
\frac{xy}{x-y} & = & \frac{1}{30}\\
\frac{x^2y^2}{x^2+y^2} & = & \frac{1}{2018}
\end{array}
\right.
\Rightarrow
\left\{
\begin{array}{rcl}
900 x^2y^2 & = & x^2+y^2-2 xy\\
2018x^2y^2 & = & x^2+y^2
\end{array}
\right.
\Rightarrow x y = \frac{1}{559}
$$</p>... |
18,136 | <p>I introduced the hypercube (to undergraduate students in the U.S.) in the
context of generalizations of the Platonic solids, explained its
structure, showed it rotating.
I mentioned <a href="https://matheducators.stackexchange.com/a/1824/511">Alicia Stott</a>, who discovered the <span class="math-container">$6$</spa... | Owen Reynolds | 13,766 | <p>That reminds me of my first college programming course where they drew a square picture of a 2D array, a cube for a 3D array, and then said 4D arrays were very hard to understand. But I'd already made 4D arrays that were fine, since they weren't representing points in 4d space. I'd played a computer dungeon game whe... |
4,038,685 | <p>Suppose that <span class="math-container">$u(x)$</span> is continuous and satisfies the integral equation</p>
<p><span class="math-container">\begin{equation}\label{1.4.1}
u(x) = \int_0^x \sin(u(t))u(t)^p dt
\end{equation}</span></p>
<p>on the interval <span class="math-container">$0 \leq x \leq 1$</span... | doobdood | 800,483 | <p>WARNING: As pointed out by Matthew in the comments, this isn't as sound as I wanted it to be and has a few holes - namely continuity for <span class="math-container">$p<1$</span> and the interval for which uniqueness is guaranteed. Check out his answer, if he posts one!</p>
<p>I'm going to write the differential ... |
1,328,082 | <p>I've been trying to prove the following inequality, but until now I've had problems coming up with a solution:</p>
<p>$$
2^{mn} \ge m^n
$$</p>
<p>$m$ and $n$ can assume any natural number.</p>
<p>I wasn't able to find any counterexample that would invalidate this inequality, so I am assuming that this statement i... | gt6989b | 16,192 | <p><strong>Hint</strong> Suffices to show $2^m \ge m$ for all natural $m$. How about using Mathematical induction?</p>
|
1,328,082 | <p>I've been trying to prove the following inequality, but until now I've had problems coming up with a solution:</p>
<p>$$
2^{mn} \ge m^n
$$</p>
<p>$m$ and $n$ can assume any natural number.</p>
<p>I wasn't able to find any counterexample that would invalidate this inequality, so I am assuming that this statement i... | Guest | 102,415 | <p>This is true for $m,n\in\mathbb{Z}_{\geq1}$.</p>
<p>Since $2^{mn}=(2^m)^n$ and $x^n$ is an increasing function on $x>0$, all that has to be shown is $2^m\geq m$ when $m\geq1$. This can be done via induction.</p>
|
2,076,831 | <blockquote>
<p>Is the following set path-connected?</p>
<p><span class="math-container">$A=\{(x,y):y=x\sin \frac{1}{x},x>0\}$</span></p>
</blockquote>
<p>I am unable to understand that should I prove it or disprove it .</p>
<p>Will someone please give me some hints.</p>
| Fabio Somenzi | 123,852 | <p>The product of two integers is the best-known candidate for one-way function. It's fast to compute the product of two integers. However, it is not known whether integer factorization can be solved in polynomial time (even though primality testing is in P).</p>
<p>In complexity theory it is common to call <em>easy... |
2,076,831 | <blockquote>
<p>Is the following set path-connected?</p>
<p><span class="math-container">$A=\{(x,y):y=x\sin \frac{1}{x},x>0\}$</span></p>
</blockquote>
<p>I am unable to understand that should I prove it or disprove it .</p>
<p>Will someone please give me some hints.</p>
| ameed | 119,300 | <p>"Easy", roughly speaking, means a computer can do it efficiently, and "hard" means a computer can't do it efficiently. Usually the threshold for "efficiently" is that the problem is solvable in polynomial time; <em>i.e.</em> there exists some algorithm that, given an input of size $n$, solves the problem in $O(n^p)$... |
14,746 | <p>Is there a case to be made that the topic of line integrals should only involve vector fields?</p>
<p>My colleagues and our textbook take the position that line integrals should only be taught from a vector field perspective (specifically for calculating "work"). [In fact, our textbook <em>defines</em> a line integ... | guest | 10,126 | <p>Caveat: I'm just a random Internet poster, not an instructor, so take this with a grain of salt. And get some advice from blooded veterans.</p>
<p>A. It's too much. 1 and 2 are plenty for 12 minutes.</p>
<p>B. Write out the entire 50 minute lecture, practice it at once, and write up a synthesis of the last 35... |
14,746 | <p>Is there a case to be made that the topic of line integrals should only involve vector fields?</p>
<p>My colleagues and our textbook take the position that line integrals should only be taught from a vector field perspective (specifically for calculating "work"). [In fact, our textbook <em>defines</em> a line integ... | WeCanLearnAnything | 7,151 | <p>I'd take an entirely different approach, though the faculty that is assessing you may not view it favorably.</p>
<p>Mathematicians tend to view the logic of math as intrinsically beautiful.</p>
<p>The vast majority of people do not. The vast majority considers math to be hoop jumping and weirdly arbitrary rules th... |
2,585,828 | <p>I saw in an article that for every real number, there exists a Cauchy sequence of rational numbers that converges to that real number. This was stated without proof, so I'm guessing it is a well-known theorem in analysis, but I have never seen this proof. So could someone give a proof (preferably with a source) for ... | Mohammad Riazi-Kermani | 514,496 | <p>Note that the set of real numbers is a complete set that is every Cauchy sequence of real numbers is convergent also every convergent sequence is a Cauchy sequence. Therefore once you have a rational sequence converging to a real number the same sequence is a rational Cauchy sequence converging to the same real nu... |
22,269 | <p>How to verify, which version of MathJax is used here? It seems that v. 2.5 is not implemented yet. It has very useful features, as <code>\delimiterfactor</code>, which is needed here: <a href="https://tex.stackexchange.com/questions/284809/getting-large-braces-in-mathjax">https://tex.stackexchange.com/questions/284... | Community | -1 | <ol>
<li>Right-click any formula, like this one: $\int_0^1 e^x\,dx$</li>
<li>Choose "About MathJax"</li>
<li>You will see the version. Currently MathJax v2.6.0-beta.2</li>
</ol>
|
3,138,450 | <p>Let <span class="math-container">$A \subseteq \mathbb{R}$</span> be bounded above and let <span class="math-container">$c \in \mathbb
{R}$</span>. Define the set <span class="math-container">$c + A = \{c + a : a \in A\} $</span></p>
<p>Now since <span class="math-container">$a \leq \sup A , \forall a\in A$</span>. ... | Theo Bendit | 248,286 | <p>You have shown that <span class="math-container">$c + \sup A$</span> is an upper bound to <span class="math-container">$c + A$</span>, and hence <span class="math-container">$\sup(c + A) \le c + \sup A$</span>. Replacing <span class="math-container">$A$</span> with <span class="math-container">$c + A$</span> and <sp... |
391,509 | <p>We have $$\dfrac{1+2+3+...+ \space n}{n^2}$$</p>
<p>What is the limit of this function as $n \rightarrow \infty$?</p>
<p>My idea:</p>
<p>$$\dfrac{1+2+3+...+ \space n}{n^2} = \dfrac{1}{n^2} + \dfrac{2}{n^2} + ... + \dfrac{n}{n^2} = 0$$</p>
<p>Is this correct?</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Such application of limit (esp. to $\infty$) to individual summand is applicable only when the number of summand is finite.</p>
<p>$$\dfrac{1+2+3+...+ \space n}{n^2} =\frac{\frac{n(n+1)}2}{n^2}=\frac12\cdot\left(1+\frac1n\right)$$</p>
|
4,311,731 | <p>I was thinking about solutions of equation <span class="math-container">$x^2=i$</span> . First thought coming to my mind was <span class="math-container">$x=\pm \sqrt{i}$</span> . ( I know it's wrong ) .
Then I thought if we solve this equation like real problem then
<span class="math-container">$|x|=\sqrt i$</span>... | Community | -1 | <blockquote>
<p>But what if this distance is not real so let consider complex numbers <span class="math-container">$z$</span> whose <span class="math-container">$|z|=i$</span>.</p>
</blockquote>
<p>No, the "distance (from a complex number to another)" is a nonnegative real number by definition. It does not ma... |
3,042,308 | <p>Suppose a matrix <span class="math-container">$A$</span> has eigenvalues 0, 3, 7 and eigenvectors <span class="math-container">$\mathbf{u, v, w,}$</span> respectively. Find the least square minimum length solution for <span class="math-container">$A\mathbf{x} = \mathbf{u+v+w}$</span>.</p>
<p>This was on our enginee... | Hermione | 593,163 | <p>You can consider the function <span class="math-container">$f$</span> defined as
<span class="math-container">$$ f (x, y) = 1 \quad \text{if } x=0 \text{ for all } y \in \mathbb{R}$$</span>
<span class="math-container">$$ f (x, y) = 1 \quad \text{if } x>0 \text{ and } y \geq 0$$</span>
<span class="math-container... |
2,699,536 | <blockquote>
<p>Given a continuous function $f:[0,1]\to\mathbb R$, prove that $$\forall t>0, \frac{1}{t}\cdot\ln\left(\int_{0}^{1}e^{-tf(x)}dx\right)\leq-\min f(x).$$</p>
</blockquote>
<p>I have no idea where to begin. Thought I could use the FTOC to come up with some form of antiderivative but I don't know if th... | Community | -1 | <p>A good first approach might be exploring what happens for constant functions $f$, or for something simplie like $f(x) = ax$.</p>
<hr>
<p>Note that</p>
<p>$$\int_0^1 e^{-t f(x)} dx \le \int_0^1 e^{-t \min f} dx = e^{-t \min f}$$</p>
<p>so that the left hand side of your inequality is</p>
<p>$$\frac 1 t \ln \left... |
3,048,381 | <p>We found the orthonormal basis for the eigen spaces.</p>
<p>We got <span class="math-container">$C$</span> to be the matrix</p>
<pre><code>[ 1/squareroot(2) 1/squareroot(6) 1/squareroot(3)
-1/squareroot(2) 1/squareroot(6) 1/squareroot(3)
0 -2/squareroot(6) ... | nicomezi | 316,579 | <p>The matrix <span class="math-container">$C$</span> is orthogonal. Hence <span class="math-container">$C^{-1}=C^T$</span>.</p>
|
324,557 | <p>Isbell gave, in <a href="https://eudml.org/doc/213746" rel="noreferrer"><em>Two set-theoretic theorems in categories</em> (1964)</a>, a necessary criterion for categories to be concretisable (i.e. to admit some faithful functor into sets). Freyd, in <a href="https://www.sciencedirect.com/science/article/pii/00224049... | Tim Campion | 2,362 | <p>I did this once with the category of schemes in response to <a href="https://mathoverflow.net/questions/2015/can-the-category-of-schemes-be-concretized/2050">this question</a>, with help from Laurent Moret-Bailly. But then Zhen Lin Low pointed out there's an obvious concretizing functor. Maybe it wasn't so obvious u... |
324,557 | <p>Isbell gave, in <a href="https://eudml.org/doc/213746" rel="noreferrer"><em>Two set-theoretic theorems in categories</em> (1964)</a>, a necessary criterion for categories to be concretisable (i.e. to admit some faithful functor into sets). Freyd, in <a href="https://www.sciencedirect.com/science/article/pii/00224049... | Martti Karvonen | 136,562 | <p>An inverse category can be defined as a category where every <span class="math-container">$f$</span> admits a unique regular inverse, i.e. a map <span class="math-container">$g$</span> such that <span class="math-container">$fgf=f$</span> and <span class="math-container">$gfg=g$</span>. In [1], Kastl proves that any... |
4,401,028 | <p>Let <span class="math-container">$p$</span> be an odd prime and <span class="math-container">$ω=e^{2\pi i /p}$</span>. Determine if <span class="math-container">$1-ω$</span> is prime in <span class="math-container">$\mathbb{Z}[ω]=\mathbb{Z}+\mathbb{Z}ω+\mathbb{Z}ω^2+...+\mathbb{Z}ω^{p-2}$</span></p>
<p><strong>My at... | Zaz | 3,157 | <p>Well it's not true in general that <span class="math-container">$\sin(x) = 0$</span>. You should have been taught that <span class="math-container">$\sin(0) = 0$</span>.</p>
<p><span class="math-container">$\sin(x)$</span> is the value of the sine function at <span class="math-container">$x$</span> - it is an unknow... |
1,796,008 | <p>I am using DFT with windows. The way I understand how a window makes the DFT "look" better, is that multiplication in time domain is convolution in frequency domain. Therefore a window with following FT (Hann window), will suppress the side lobes found in a signal FT (second picture) : </p>
<p><a href="https://i.st... | Chappers | 221,811 |
$
\newcommand{\sech}{\mathop{\rm sech}\nolimits}
\newcommand{\csch}{\mathop{\rm csch}\nolimits}
$
<p>Your calculation of the curvature of the tractrix is fine. The problem comes from the other part: the plane in which the circle $(\cos{u}\sech{v},\sin{u}\sech{v},v-\tanh{v})$ (with constant $v$) lies is <... |
1,796,008 | <p>I am using DFT with windows. The way I understand how a window makes the DFT "look" better, is that multiplication in time domain is convolution in frequency domain. Therefore a window with following FT (Hann window), will suppress the side lobes found in a signal FT (second picture) : </p>
<p><a href="https://i.st... | Narasimham | 95,860 | <p>Gauss curvature is product of principal <em>curvatures of the meridian and its perpendicular line</em>. This is not curvature of radius of pseudosphere in cylindrical coordinate system as you incorrectly found.</p>
<p><span class="math-container">$v$</span> is the angle of rotation of a point on asymptotic line (ze... |
2,430,529 | <p>How should I define a vector, that has equal angles to vectors $\vec{i}, \vec{i} + \vec{j}$ and $\vec{i} + \vec{j} + \vec{k}$?</p>
<p>After looking at the problem in a graphical way, I tried taking average from $\vec{i}$ and $\vec{i} + \vec{j} + \vec{k}$, modifying the $\vec{i}$ vector to be of same length than the... | achille hui | 59,379 | <p>In general, if you have $3$ vectors $\vec{v}_1, \vec{v}_2, \vec{v}_3$ and you want
to find a vector $\vec{v}$ making equal angles to them. You first construct the unit vectors for those vectors $\hat{v}_1, \hat{v}_2, \hat{v}_3$ and then solve for the equation:</p>
<p>$$\hat{v}_1 \cdot \vec{v} = \hat{v}_2 \cdot \vec... |
4,022,269 | <p>I'm refreshing some Probability Theory (it's been a while) and came accross the following example:</p>
<p>Let <span class="math-container">$X$</span> be a random variable with distributuion function <span class="math-container">$$F_X(x) = \left\{\begin{array}{l r}0 & \text{if }x<0\\\frac{1}{2}+\frac{x}{2}&... | John L | 852,522 | <p>A random variable is continuous if the cdf is a continuous function. This is not continuous at x=0.
A random variable is discrete if it can take only a countable number of values. This can take any value in (0.5,1) which is uncountably infinite.</p>
<p>So it is neither discrete or continuous.</p>
<p>To find <span cl... |
4,022,269 | <p>I'm refreshing some Probability Theory (it's been a while) and came accross the following example:</p>
<p>Let <span class="math-container">$X$</span> be a random variable with distributuion function <span class="math-container">$$F_X(x) = \left\{\begin{array}{l r}0 & \text{if }x<0\\\frac{1}{2}+\frac{x}{2}&... | HallsofIvy | 879,895 | <p>The <strong>distribution</strong> function, F, is not continuous because the limit, at = 0, "from the left" is 0 while the limit "from the right" is <span class="math-container">$\frac{1}{2}+ \frac{0}{2}= \frac{1}{2}$</span>.</p>
<p>But the statement that "P(x= 0)= 1/2" is simply WRONG.... |
1,591,197 | <p>Let $ A \trianglelefteq G $ and $ B \trianglelefteq A $ a Sylow normal subgroup of $ A $. My textbook says then that $ B \trianglelefteq G $.</p>
<p>I don’t understand why that is.</p>
| Lauren | 293,855 | <p>let $x\in G$ we'll show that $xBx^{-1}=B$, we have :</p>
<p>$B$ subgrouf of $A$ means $xBx^{-1}\subset xAx^{-1}=A$ ( because $A \trianglelefteq G$, so $xBx^{-1}\subset A$, which means that $B$ and $xBx^{-1}$ are two $p$-subgroups of sylow of $A$. So $B=xBx^{-1}$ because $B \trianglelefteq A$.</p>
<p>Finally $B \tr... |
1,107,932 | <p>Is the map $f:S_n \to A_{n+2}$ given by </p>
<p>$$f(s)= \begin{cases}
s & s\ \text{is even}\\
s \circ (n+1,\ n+2) & s\ \text{is odd}
\end{cases}$$ </p>
<p>an injective homomorphism? I can show that if it is a homomorphism then it is injective but having difficulty in showing that $f$ is a homomorphism. Pl... | Timbuc | 118,527 | <p>As disjoint cycles commute and $\;(n+1\;n+2)^2=1\;$ , for any two <strong>cycles</strong> $\;\sigma,\pi\in S_n\;$ we get</p>
<p>$$f(\sigma\pi):=\begin{cases}\sigma\pi&,\text{both cycles have same parity}\\{}\\\sigma\pi(n+1\;n+2)=\sigma(n+1\;n+2)\pi=f(\sigma)f(\pi)&,\text{otherwise}\end{cases}$$</p>
<p>wher... |
416,407 | <blockquote>
<p>What examples are there of habitual but unnecessary uses of the axiom of
choice, in any area of mathematics except topology?</p>
</blockquote>
<p>I'm interested in standard proofs that use the axiom of choice, but where
choice can be eliminated via some judicious and maybe not quite obvious
rephrasing. ... | Michael Hardy | 6,316 | <p>Sometimes people prove the <a href="https://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" rel="noreferrer">Schröder–Bernstein theorem</a> by saying it follows easily from the well-ordering theorem, which is equivalent to the axiom of choice. But it can be proved without the axiom of choice. The theo... |
416,407 | <blockquote>
<p>What examples are there of habitual but unnecessary uses of the axiom of
choice, in any area of mathematics except topology?</p>
</blockquote>
<p>I'm interested in standard proofs that use the axiom of choice, but where
choice can be eliminated via some judicious and maybe not quite obvious
rephrasing. ... | Willie Wong | 3,948 | <p>One of my papers, <a href="https://arxiv.org/abs/1310.1318v3" rel="nofollow noreferrer">A comment on the construction of the maximal globally hyperbolic Cauchy development</a>, did this for the existence of the maximal globally hyperbolic Cauchy development for the initial value problem in general relativity.</p>
<p... |
416,407 | <blockquote>
<p>What examples are there of habitual but unnecessary uses of the axiom of
choice, in any area of mathematics except topology?</p>
</blockquote>
<p>I'm interested in standard proofs that use the axiom of choice, but where
choice can be eliminated via some judicious and maybe not quite obvious
rephrasing. ... | Sam Sanders | 33,505 | <p>My favourite example is from Reverse Mathematics, namely <em>Pincherle's theorem</em> stating that</p>
<p><em>a locally bounded function on Cantor space is bounded there</em>.</p>
<p>The obvious proof proceeds by contradiction and uses AC:</p>
<ol>
<li><p>Suppose <span class="math-container">$F:2^\mathbb{N}\rightarr... |
2,645,406 | <p>I am reading Rudin's real and complex analysis. On page 14 we see $\underset{n} {\text{sup }} f_n$, and on page 15 we see max$\{f,g\}$. These two are obviously different. I searched some answers and found an example: </p>
<p>sup $\{f,g\} = f(x)$ wehn $f(x)\geq g(x)$ and $g(x)$ when $f(x) < g(x)$ </p>
<p>But the... | LucasSilva | 154,194 | <p><strong>Supremum and Maximum</strong></p>
<p>Let $A$ be a set of real numbers. </p>
<p>If $x$ is a real number such that $a \leq x$ for all $a \in A$, then $x$ is called an <strong>upper bound</strong> for $A$. Example: $1$ is an upper bound for the interval $(0,1)$. So is $2$. </p>
<p>If $x$ is an upper bound fo... |
248,325 | <p>is there some connection between a curve in the algebraic geometry sense, e.g.</p>
<blockquote>
<p>Separated scheme of finite type over spec($k$)</p>
</blockquote>
<p>for a field $k$</p>
<p>and a curve in the sense of a smooth map from an interval in $\mathbb R$ to $\mathbb R^n$?</p>
| Simon Rose | 1,703 | <p>Well, first of all, a separated scheme of finite type over $Spec(k)$ is not necessarily a curve. A <em>one dimensional</em> separated scheme of finite type etc. etc. may be a curve, but this is also not quite a curve in the sense that you describe.</p>
<p>A curve in the algebro-geometric sense is a one dimensional ... |
2,414,721 | <p>For all perfect numbers $N$, $\sigma (N) = 2N$, where $\sigma$ is the divisor sigma function.</p>
<p>Let $s$ be a perfect number of the form $3^m 5^n 7^k$, where $m,n,k \geq 1$ are integers.</p>
<p>Then $\sigma (s)= \sigma (3^m 5^n 7^k)$</p>
<p>$ =\sigma (3^m) \sigma (5^n) \sigma (7^k)$ since $3, 5,$ and $7$ are ... | hardmath | 3,111 | <p>Note that the Question stipulates the exponents considered are $m,n,k\ge 1$. As @lulu points out, the cases where one of the exponents is zero can (if desired) be ruled out by <a href="https://math.stackexchange.com/questions/1148588/prove-that-pj-qi-cannot-be-a-perfect-number-for-p-q-odd-distinct-primes">this prev... |
1,394,141 | <blockquote>
<p>Compute the tangent space $T_pM$ of the unit matrix $p=I$ when $$(i)\,M=SO(n)\\ (ii)\,M=GL(n)\\ (iii)\,M=SL(n).$$</p>
</blockquote>
<p><strong>My attempt:</strong>
I think I have computed the tangent space in the case that $M=SL(n)$. We can write $SL(n)=\det^{-1}(1)$ and I've proved in an earlier exe... | ASCII Advocate | 260,903 | <p>These are foundational calculations in what used to be called the theory of "continuous groups", now called Lie groups. The group structures are in fact smooth in today's terminology, not only continuous. The beginning material of any text on <strong>Lie groups and Lie algebras</strong> will have what you are looki... |
464,586 | <blockquote>
<p>Find a basis for $U=\{A\in\mathbb{M}_{22}\mid A^T=-A\}$.</p>
</blockquote>
<p>$\mathbb{M}_{22}$ denotes the set of all $2 \times 2$ matrices. This question appeared on an examination I wrote yesterday. Does a basis even exist? I can't think of any matrices in which $A^T=-A$ except for the zero matrix... | Branimir Ćaćić | 49,610 | <p>Well, let
$$
A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} \in \mathbb{M}_{22}.
$$
Then
$$
A + A^T = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} + \begin{pmatrix} A_{11} & A_{21} \\ A_{12} & A_{22} \end{pmatrix} = \begin{pmatrix} 2 A_{11} & A_... |
3,841,535 | <p>I'm having a lot of trouble solving this question via the differentiating with respect to a parameter method. I can get the correct result for the integral containing sine, but I'm totally lost when it comes to evaluating the integral containing cosine. Here's the problem statement:</p>
<p>Given:</p>
<p><span class=... | Abhijeet Vats | 426,261 | <p><strong>Hint:</strong></p>
<p>Have you considered differentiating:</p>
<p><span class="math-container">$$\int_{0}^{\infty} e^{-ax} \sin(kx) \ dx = \frac{k}{a^2+k^2}$$</span></p>
<p>with respect to <span class="math-container">$k$</span> instead? :-)</p>
|
674,621 | <p>I am trying to figure out what the three possibilities of $z$ are such that </p>
<p>$$
z^3=i
$$</p>
<p>but I am stuck on how to proceed. I tried algebraically but ran into rather tedious polynomials. Could you solve this geometrically? Any help would be greatly appreciated.</p>
| colormegone | 71,645 | <p>I believe your "polynomial" approach would also have worked, if this is what you meant :</p>
<p>[In this, we are supposing that we knew nothing of the "Euler Identity", DeMoivre's Theorem, or roots of unity, all of which provide quite efficient devices]</p>
<p>If we (probably safely) assume that the solution(s) ar... |
530,605 | <blockquote>
<p>Let <span class="math-container">$A$</span> be open in <span class="math-container">$\mathbb{R}^m$</span>; let <span class="math-container">$g:A\rightarrow\mathbb{R}^n$</span>. If <span class="math-container">$S\subseteq A$</span>, we say that <span class="math-container">$S$</span> satisfies the <stron... | Community | -1 | <p>Intuitively, a counterexample must be a function which is very steep without having a jump or other sort of discontinuity. Consider, for example, $$g(x) = x^{1/3}$$ at $0$. Then</p>
<p>$$\frac{|g(x) - g(0)|}{|x - 0|} = x^{-2/3}$$</p>
<p>This cannot be bounded in a neighborhood of $0$.</p>
|
1,601,427 | <blockquote>
<p>Let $a,b,c$ be three nonnegative real numbers. Prove that $$a^2+b^2+c^2+3\sqrt[3]{a^2b^2c^2} \geq 2(ab+bc+ca).$$</p>
</blockquote>
<p>It seems that the inequality $a^2+b^2+c^2 \geq ab+bc+ca$ will be of use here. If I use that then I will get $a^2+b^2+c^2+3\sqrt[3]{a^2b^2c^2} \geq ab+bc+ca+3\sqrt[3]{a... | chenbai | 59,487 | <p>$x^3=a^2,y^3=b^2,z^3=c^2 \implies x^3+y^3+z^2 +3xyz \ge 2(\sqrt{(xy)^3}+\sqrt{(yz)^3}+\sqrt{(xz)^3})$</p>
<p>we have $x^3+y^3+z^3 +3xyz \ge xy(x+y)+yz(y+z)+xz(x+z)$</p>
<p>$xy(x+y)\ge 2xy\sqrt{xy}=2\sqrt{(xy)^3} \implies xy(x+y)+yz(y+z)+xz(x+z)\ge 2(\sqrt{(xy)^3}+\sqrt{(yz)^3}+\sqrt{(xz)^3})$</p>
|
3,828,205 | <blockquote>
<p>Find all integers <span class="math-container">$x$</span> and <span class="math-container">$y$</span> that satisfy <span class="math-container">$$x^4-12x²+x^2y^2+30 < 0$$</span></p>
</blockquote>
<p>Letting <span class="math-container">$a = x^2$</span> and <span class="math-container">$b = y^2$</span... | lab bhattacharjee | 33,337 | <p>Clearly, <span class="math-container">$x\ne0$</span>,</p>
<p><span class="math-container">$$\dfrac{x^4-12x^2+30}{x^2}<-y^2$$</span></p>
<p><span class="math-container">$$\iff x^2+\dfrac{30}{x^2}<12-y^2\le12$$</span></p>
<p>Now <span class="math-container">$x^2+\dfrac{30}{x^2}<12\iff0>x^4-12x^2+30=(x^2-6... |
1,165,828 | <p>I run into a problem when I'm trying to prove how $\tan^2x+1 = \sec^2x$, and $1+\cot^2x=\csc^2x$</p>
<p>I understand that $\sin^2x+\cos^2x = 1$. (To my understanding 1 is the Hypotenuse, please correct me if I'm wrong). If referring to a Pythagorean triangle, let's say a triangle where $a=3$, $b=4$, and $c=5$, or $... | Felipe Faria | 219,178 | <p>Your issue is your understanding of the relationship between your basic trigonometry values and the sides of a triangle. You must keep in mind that the values of trigonometric functions are utilising the sides of your triangles in the following manner: $\sin(x) = \frac{opp}{hyp}$, $\cos(x) = \frac{adj}{hyp}$, and $\... |
1,756,567 | <p>Let A be a 3*3 matrix and $A^{2014}=0$. Must $A^3$ be the zero matrix? I can work out that I-A is invertible, but I don't know how to proceed further.</p>
| Harald Hanche-Olsen | 23,290 | <p><strong>Hint:</strong> Note that $\operatorname{rank}(A^{n+1})\le\operatorname{rank}(A^n)$. Then show that if $\operatorname{rank}(A^{n+1})=\operatorname{rank}(A^n)$, then all powers $A^k$ with $k\ge n$ have the same rank.</p>
|
1,756,567 | <p>Let A be a 3*3 matrix and $A^{2014}=0$. Must $A^3$ be the zero matrix? I can work out that I-A is invertible, but I don't know how to proceed further.</p>
| Ben Grossmann | 81,360 | <p><strong>Hint:</strong> Let $R(A)$ denote the range (AKA image or column space) of $A$. If $A^k=0$ for some $k$ (and if $A^2 \neq 0$), we <em>must</em> have
$$
R(A^3) \subsetneq R(A^2) \subsetneq R(A) \subsetneq \Bbb R^3
$$</p>
|
198,049 | <p>Let $S$ be a smooth projective surface over $\mathbb{C}$. (I guess this can be more general—higher dimension, other ground fields, non-projective, maybe even singular?—and I'dd like to hear that.) Let $s \in S$ be a point. Let $\beta \colon X \to S$ be the blowup of $s \in S$. Suppose that $H^{i}(S, T_{S})$ is known... | BlaCa | 14,514 | <p>Let $S$ be a surface and $Z=\{p_1,...,p_n\}\subset S$ be a reduced subscheme of dimension zero. Let $\epsilon:\widetilde{S}\rightarrow S$ be the blow-up of $S$ at $Z$. Consider the exact sequence
$$0\mapsto \epsilon^{*}\Omega_{S}\rightarrow \Omega_{\widetilde{S}}\rightarrow i_{*}\Omega_{E/Z}\mapsto 0$$
where $i:E\ho... |
238,809 | <p><a href="https://i.stack.imgur.com/W5ILn.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/W5ILn.jpg" alt="enter image description here" /></a>
How to construct a tree like <a href="https://i.stack.imgur.com/XKYl9.png" rel="nofollow noreferrer">this</a>? I was looking <code>CompleteKaryTree</code> i... | Ian Ford | 76,874 | <p>In 12.3 or later, you can use NestTree:</p>
<pre><code>choice[{_, list_List}]:=
MapIndexed[{e, pos} |-> {e, Delete[list, pos]}, list]
permutationTree[list_List] :=
NestTree[choice, {Null, list}, Infinity, First]
permutationTree[n_Integer] :=
permutationTree[Range[n]]
</code></pre>
<p><a href="https://i.sta... |
1,565,406 | <p>Which interpolation method should I use for complicated "smooth" curves such as </p>
<p>$\frac{sin(x)}{x}$ for $x>0$.</p>
| Rorimac | 296,921 | <p>There are many different interpolation methods and depending on the data you want to interpolate some might work better than others. If you know that the dataset comes from a trigonometric function, say $\frac{sin(x)}{x}$, then using <a href="https://en.wikipedia.org/wiki/Trigonometric_interpolation" rel="nofollow">... |
1,360,658 | <p>prove: $$\frac{1-\cos 2\theta}{1-\cos\theta}=2\cos\theta-2$$</p>
<p>I'm thinking that there will be something to square in this? Because I notice that the $LHS$ looks like the half-angle identity....</p>
<p>Edit: I am so sorry guys, my grave mistake, the expression should have been equal to 2 instead like,</p>
<p... | Mary Star | 80,708 | <p><strong>Hint:</strong> </p>
<p>Use the property:
$$\cos 2 \theta=2 \cos^2 \theta-1$$ </p>
|
2,619,344 | <p>I just struck with a doubt today</p>
<blockquote>
<p>Why do most of the standard inequalities require the variables to be positive.</p>
</blockquote>
<p>For example if we want to find minimum value of a certain expression say <span class="math-container">$a+b+c$</span> the very first thought that comes in our mind ... | carmichael561 | 314,708 | <p>Since $f$ is continuous on $[0,1]$ there is some constant $C$ such that $|f(x)|\leq C$ for all $x\in[0,1]$, hence
$$\Big|\int_0^1f(x)e^{-nx}\;dx\Big|\leq C\int_{0}^1e^{-nx}\;dx=C\frac{1-e^{-n}}{n}\leq \frac{C}{n}$$
Therefore the sum converges absolutely.</p>
|
1,036,907 | <p>For <span class="math-container">$n$</span> in the natural numbers let</p>
<p><span class="math-container">$$a_n = \int_{1}^{n} \frac{\cos(x)}{x^2} dx$$</span></p>
<p>Prove, for <span class="math-container">$m ≥ n ≥ 1$</span> that <span class="math-container">$|a_m - a_n| ≤ \frac{1}{n}$</span> and deduce <span class... | DeepSea | 101,504 | <p>$|a_m-a_n| = \left|\displaystyle \int_{n}^m \dfrac{\cos x}{x^2}dx\right| \leq \displaystyle \int_{n}^m \dfrac{|\cos x|}{x^2}dx \leq \displaystyle \int_{n}^m \dfrac{1}{x^2}dx = \dfrac{1}{n} - \dfrac{1}{m} < \dfrac{1}{n}$. Thus $\{a_n\}$ is a Cauchy sequence, hence converges.</p>
|
223,718 | <p>Suppose I have two lists to which I apply the <code>Line</code> command for example</p>
<pre><code>A = {{-4,0},{4,0}}
B = {{0,4},{0,-4}}
</code></pre>
<p>and I take <code>Line[A]</code> and <code>Line[B]</code>. Is there a way to get Mathematica to tell me the intersection points of the line? Of course this is a v... | flinty | 72,682 | <p>Join each of your collections of lines using a <code>RegionUnion</code> into a single region, then intersect both regions with <code>RegionIntersection</code> like so:</p>
<pre><code>redLines = {
Line[{{-2.`, 4.`}, {-1.5`, 2.25`}, {-1.`, 1.`}, {-0.5`, 0.25`}, {0.`, 0.`}, {0.5`, 0.25`}, {1.`, 1.`}, {1.5`, 2.25`}, ... |
2,262,001 | <p>Prove this inequality where $a$, $b$ and $c$ are sides of a triangle and $S$ its Area.
$$\frac{ab + bc + ca}{4S}\ge \operatorname{ctg} \frac{\pi}{6}$$</p>
| Jack D'Aurizio | 44,121 | <p>By the sine theorem, the given inequality is equivalent to
$$ \frac{1}{\sin A}+\frac{1}{\sin B}+\frac{1}{\sin C} \geq 2\sqrt{3} \tag{1}$$
and since $\frac{1}{\sin(x)}$ is a convex function on the interval $(0,\pi)$, $(1)$ is a straightforward consequence of <a href="https://en.wikipedia.org/wiki/Jensen%27s_inequalit... |
2,598 | <p>Should Wolfram Alpha Notebook questions be considered on-topic?</p>
<p>Here's an example: <a href="https://mathematica.stackexchange.com/questions/240780/calculating-double-integral-bounded-by-domain-in-wolfram-alpha-notebook">Calculating double integral bounded by domain in Wolfram Alpha Notebook</a></p>
<p>Here ar... | Szabolcs | 12 | <p><strong>No, Wolfram|Alpha questions should be explicitly off-topic, regardless of whether W|A is being used through the web interface or Wolfram|Alpha Notebook</strong>.</p>
<p>Vote here if you agree (and post the opposite answer for voting, with appropriate arguments, if you disagree).</p>
<p>Arguments:</p>
<ol>
<l... |
1,139,789 | <p>Is $f(z)=z^n$ holomorphic?</p>
<p>I have tested a number of other functions using the Cauchy Riemann equations $u_x=v_y$, $v_x=-u_y$. However in the case of $f(z)=z^n$ I cannot think of a way to find the functions $u(x,y)$ and $v(x,y)$ without using a binomial expansion of $(x+iy)^n$. </p>
<p>Any help or pointers ... | egreg | 62,967 | <p>Consider the function $f(x,y)=(x+iy)^n$. Then the real and imaginary parts of this function are
$$
u(x,y)=\frac{1}{2}((x+iy)^n+(x-iy)^n)\\
v(x,y)=\frac{1}{2i}((x+iy)^n-(x-iy)^n)$$
so
$$
\frac{\partial u}{\partial x}=\frac{n}{2}((x+iy)^{n-1}+(x-iy)^{n-1})
$$
whereas
$$
\frac{\partial v}{\partial y}=
\frac{n}{2i}(i(x+... |
1,139,789 | <p>Is $f(z)=z^n$ holomorphic?</p>
<p>I have tested a number of other functions using the Cauchy Riemann equations $u_x=v_y$, $v_x=-u_y$. However in the case of $f(z)=z^n$ I cannot think of a way to find the functions $u(x,y)$ and $v(x,y)$ without using a binomial expansion of $(x+iy)^n$. </p>
<p>Any help or pointers ... | Mårten W | 58,780 | <p>HINT: Show that if $f=u+iv$ and $g=\hat{u}+i\hat{v}$ satisfy the Cauchy-Riemann equations, then so does $fg$. Use this together with the fact that $z^n$ is a finite product of holomorphic functions (since $z$ is holomorphic).</p>
|
2,783,224 | <p>$$\lim_{x \rightarrow 0}\frac{x^2 \sin(Kx)}{x - \sin x} =1$$</p>
<p>Here K is a constant whose value I want to find,</p>
<p>I got it by writing the series expansion of $\sin(\theta)$, but couldn't by L'hospital rule or standard limits,</p>
<p>This is what I tried:</p>
<p>$$\lim_{x \rightarrow 0}\frac{x^2 \sin(Kx... | Jiabin Du | 442,428 | <p>$$\lim_{x \rightarrow 0}\frac{x^2 \sin(Kx)}{x - \sin x} =\lim_{x\to 0}\frac{Kx^3}{x-\sin x}=\lim_{x\to 0}\frac{3Kx^2}{1-\cos x}=\lim_{x\to 0}\frac{6Kx}{\sin x}=6K$$</p>
|
3,703,011 | <p>So, I am an absolute beginner in mathematics; only being knowledgeable in some basic ideas of the subject. My interest in math started only recently, while reading about set theory and cardinality (particularly the concept of higher infinities) in some other forums. Can you guys recommend me any farily accessible bo... | Mark S. | 26,369 | <p>If you are an "absolute beginner", then I would recommend starting by working through <a href="https://www.people.vcu.edu/~rhammack/BookOfProof/" rel="nofollow noreferrer">Book of Proof</a> by Richard Hammack, which builds up basic naive set theory, tours through a variety of good foundations for any math subject, a... |
3,703,011 | <p>So, I am an absolute beginner in mathematics; only being knowledgeable in some basic ideas of the subject. My interest in math started only recently, while reading about set theory and cardinality (particularly the concept of higher infinities) in some other forums. Can you guys recommend me any farily accessible bo... | Anonymous | 794,282 | <p>For a real beginner in mathematics who is particularly interested in set theory and cardinalities, I might recommend <em>Stories about Sets</em> by Vilenkin, which is aimed at a high school audience.</p>
<p>I don't recommend studying an axiomatic presentation of set theory until you have significant experience with... |
4,492,250 | <p>Say I have a function <span class="math-container">$f(x) = x^2 + 2$</span></p>
<p>This function never touches the x-axis, but it could be easily transformed to touch it by cancelling the constant as in <span class="math-container">$g(x) = (x^2 + 2) - 2$</span></p>
<p>Is there any way to generalize this, so that I ca... | Pearson | 992,512 | <p>Simply letting <span class="math-container">$g(x)=f(x)-f(x_0)$</span> for any <span class="math-container">$x_0\in\mathbb R$</span> such that <span class="math-container">$f$</span> is well-defined at <span class="math-container">$x_0$</span> works</p>
|
253,208 | <p>Here's a problem I'm working on: </p>
<p>Find the matrix of T with respect to the standard bases $P_{3}$ and $\mathbb{R}^{3}$: </p>
<p>$T(p(x)) = \left( \begin{array}{cc} p'(0) \\ p(0) \\ p(0) - p'(0)\end{array} \right)$</p>
<p>So I'll list the steps that I've been taking and hopefully someone will be able to tel... | lab bhattacharjee | 33,337 | <p>2.</p>
<p>So, $103x=444+999a$ for some integer $a$</p>
<p>or, $103x=111(4+9a)$ or $\frac{103x}{111}=4+9a$ an integer. </p>
<p>$\implies 111\mid x$</p>
<p>Let $x=111y\implies 103y=4+9a$
$\implies 103y\equiv4\pmod 9$
$\implies 4y\equiv4\pmod 9$
$\implies y\equiv1\pmod 9$ as $(4,9)=1,y=9b+1$ for some integer $b$</p... |
263,718 | <p>I have 2d data in the form <code>{x,y,f(x,y)}</code> which is randomly stored. the random <code>{x,y}</code> for specific shape (square for simplification here) can be created as</p>
<pre><code>Regn = {{-\[Pi], -\[Pi]}, {-\[Pi], \[Pi]}, {\[Pi], \[Pi]}, {\[Pi], -\
\[Pi]}, {-\[Pi], -\[Pi]}};
xylist=Select[RegionMembe... | Shin Kim | 85,037 | <p>If you want to produce the derivatives directly using only the raw data,</p>
<pre><code>(*w.r.t y*)
datxy = Sort[Sort[datxy, #1[[1]] < #2[[1]] &],
Which[ #1[[1]] == #2[[1]] && #1[[2]] < #2[[2]], True,
#1[[1]] == #2[[1]] && #1[[2]] > #2[[2]], False ] &];
Dy = {};
Do[If[... |
97,329 | <p>This is inspired by <a href="https://mathoverflow.net/questions/97307/polynomials-all-of-whose-roots-are-rational">this </a> question. Let $f(x)=a_nx^n+...+a_0$ be a polynomial with rational coefficients. The sandard procedure of finding a rational root $p/q$ involves checking all $p$ that divide $a_0$ and all $q$ t... | Michael Stoll | 21,146 | <p>Didn't Lenstra, Lenstra and Lovász in their famous LLL paper prove that factorization of polynomials over $\mathbb Q$ can be done in polynomial time? You get a rational root if and only if there is a factor of degree 1, and the polynomial has only rational roots if and only if all factors have degree 1.</p>
<p>Lens... |
3,471,585 | <blockquote>
<p>All roots of polynomial <span class="math-container">$x^3+ax^2+17x+3b$</span>, <span class="math-container">$a,b\in \Bbb Z$</span> are integers. Prove that this polynomial doesn't have same roots.</p>
</blockquote>
<p>My plan was to find all three solutions and then compare them.</p>
<p>I don't know... | nonuser | 463,553 | <p>Say it has, so <span class="math-container">$x_1=x_2 = m$</span> and <span class="math-container">$x_3=k$</span>, then we have by (second) Vieta formula <span class="math-container">$$m^2+2mk =17$$</span> and (third) <span class="math-container">$$m^2k =-3b$$</span></p>
<p>So <span class="math-container">$m$</span>... |
3,471,585 | <blockquote>
<p>All roots of polynomial <span class="math-container">$x^3+ax^2+17x+3b$</span>, <span class="math-container">$a,b\in \Bbb Z$</span> are integers. Prove that this polynomial doesn't have same roots.</p>
</blockquote>
<p>My plan was to find all three solutions and then compare them.</p>
<p>I don't know... | Community | -1 | <p><strong>Hint:</strong> There is a theorem that <span class="math-container">$f$</span> and <span class="math-container">$f'$</span> are relatively prime iff the polynomial <span class="math-container">$f$</span> has no repeated roots in a splitting field. . </p>
|
3,471,585 | <blockquote>
<p>All roots of polynomial <span class="math-container">$x^3+ax^2+17x+3b$</span>, <span class="math-container">$a,b\in \Bbb Z$</span> are integers. Prove that this polynomial doesn't have same roots.</p>
</blockquote>
<p>My plan was to find all three solutions and then compare them.</p>
<p>I don't know... | Maverick | 171,392 | <p>Let <span class="math-container">$f(x)=x^3+ax^2+17x+3b$</span></p>
<p><span class="math-container">$f'(x)=3x^2+2ax+17$</span></p>
<p>Putting <span class="math-container">$f'(x)=0$</span> we have <span class="math-container">$$x=\frac{-a\pm\sqrt{a^2-51}}{3}$$</span></p>
<p>If <span class="math-container">$f(x)=0$<... |
3,471,585 | <blockquote>
<p>All roots of polynomial <span class="math-container">$x^3+ax^2+17x+3b$</span>, <span class="math-container">$a,b\in \Bbb Z$</span> are integers. Prove that this polynomial doesn't have same roots.</p>
</blockquote>
<p>My plan was to find all three solutions and then compare them.</p>
<p>I don't know... | lhf | 589 | <p>A polynomial has a multiple root iff its discriminant is zero.</p>
<p>The discriminant of <span class="math-container">$x^3+ax^2+17x+3b$</span> is <span class="math-container">$-12 a^3 b + 289 a^2 + 918 a b - 243 b^2 - 19652$</span>.</p>
<p>Mod <span class="math-container">$3$</span>, this discriminant reduces to ... |
4,337,320 | <p>Let <span class="math-container">$G$</span> be a group and <span class="math-container">$F:G^n \to G$</span> with the following property: If <span class="math-container">$x_1,…,x_n,h \in G$</span>, then <span class="math-container">$F(hx_1,…,hx_n)=hF(x_1,…,x_n)$</span>. Is there a name for this type of function prop... | Milten | 620,957 | <p>I would like to generalise Thomas Andrews's answer a bit more. Let <span class="math-container">$G$</span> act on two sets <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>, and consider functions <span class="math-container">$F:X\to Y$</span> such that <span class="math-container">... |
1,315,922 | <blockquote>
<p>show that
$$\sum_{k=2}^{n}\left(\dfrac{2}{k}+\dfrac{H_{k}-\frac{2}{k}}{2^{k-1}}\right)\le 1+2\ln{n}$$where $ n\ge 2,H_{k}=1+\dfrac{1}{2}+\cdots+\dfrac{1}{k}$</p>
</blockquote>
<p>Maybe this $\ln{k}<H_{k}<1+\ln{k}$?</p>
| Macavity | 58,320 | <p>WLOG, let $a_n$ be positive. Now there are only $n-1$ coefficients left! hence the maximum number of sign changes possible is also exactly $n-1$, hence by Descartes rule of signs, that is the maximum possible number of positive roots.</p>
|
108,890 | <p>Consider $V_{(n-1, 1)}$, the $n-1$ dimensional irreducible representation of $S_n$, i.e. the "standard" or "defining" representation. Is there a nice formula for how the $k$-th tensor power of $V_{(n-1, 1)}$ decomposes into irreps?</p>
| Richard Stanley | 2,807 | <p>For convenience consider the representation $Y=V_n\oplus V_{n-1,1}$ instead of $V_{n-1,1}$. Then the multiplicity of the representation of $S_n$ indexed by the partition $\lambda$ of $n$ in the $k$th tensor power of $Y$ equals the scalar product of the symmetric function $s_1^k$ (where $s_1=x_1+x_2+\cdots$ denotes a... |
108,890 | <p>Consider $V_{(n-1, 1)}$, the $n-1$ dimensional irreducible representation of $S_n$, i.e. the "standard" or "defining" representation. Is there a nice formula for how the $k$-th tensor power of $V_{(n-1, 1)}$ decomposes into irreps?</p>
| Abdelmalek Abdesselam | 7,410 | <p>The problem has been solved in the reference indicated in the comment
by Vasu Vineet, namely:
<a href="http://www.mat.univie.ac.at/~slc/wpapers/s54goupchau.html" rel="noreferrer">"Combinatorial Operators for Kronecker Powers of Representations of Sn"</a> by
Alain Goupil and Cedric Chauve. However, one cannot say tha... |
2,933,577 | <p>Eliminate the parameters to find a Cartesian equation of the curve and sketch the curve.</p>
<p><span class="math-container">$x = e^t – 1, y = e^{2t}$</span>.</p>
<p>My attempt:</p>
<p><span class="math-container">$x = e^{t} - 1$</span></p>
<p><span class="math-container">$x + 1 = e^{t}$</span></p>
<p><span cla... | José Carlos Santos | 446,262 | <p>As you were told in the comments, there is an error in your computations and the answer is <span class="math-container">$y=e^{2\ln(x+1)}=\left(e^{\ln(x+1)}\right)^2=(x+1)^2$</span>. But there is an even shorter path to that conclusion: since <span class="math-container">$x+1=e^t$</span>, then <span class="math-conta... |
4,380,559 | <p><span class="math-container">$P$</span> is a <span class="math-container">$\mathbb{R}^{2\times2}$</span> positive semidefinite matrix satisfying <span class="math-container">$P=P^2$</span>, <span class="math-container">$P \neq 0$</span>, and <span class="math-container">$P \neq I$</span>. Show that
<span class="math... | M. Winter | 415,941 | <p>As a positive semi-definite matrix, <span class="math-container">$P$</span> has positive eigenvalues, and the matrix equation <span class="math-container">$P=P^2$</span> implies <span class="math-container">$\lambda=\lambda^2$</span> for each eigenvalue <span class="math-container">$\lambda$</span>. That is <span cl... |
95,616 | <p>I have tried to adapt <a href="https://mathematica.stackexchange.com/a/16290/29734">this answer</a> to my problem of calculating some bosonic commutation relations, but there are still some issues.</p>
<p>The way I'm implementing the commutator is straightforward:</p>
<pre><code>commutator[x_, y_] :=x**y-y**x
</co... | evanb | 7,936 | <p>You get get far by just implementing the linearity rules for the commutator (and that 0 and 1 "escape" the matrix product. The first two aren't strictly needed, but I include them for completeness:</p>
<pre><code>commutator[Plus[a_,A__],B_]:=commutator[a,B]+commutator[Plus[A],B]
commutator[A_,Plus[b_,B__]]:=commut... |
3,155,241 | <p>Let <span class="math-container">$\mathbb{A}^n$</span> denote the set of n-tuples of elements from field <span class="math-container">$k$</span> and <span class="math-container">$I(X)$</span> the ideal of polynomials in <span class="math-container">$k[x_1,...,x_n]$</span> that vanish every point in <span class="math... | Maxime Ramzi | 408,637 | <p>Let me just write up a proof without going through a contradiction (it's almost literally the same proof but it doesn't use contradiction so is somewhat cleaner and clearer)</p>
<p>It proceeds by induction, like dcolazin's proof . The proof for <span class="math-container">$n=1$</span> is the same. </p>
<p>Now let... |
2,243,900 | <p>I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is:</p>
<blockquote>
<p>What exactly is calculus? </p>
</blockquote>
<p>I know who invented it, the Leibniz controversy, etc., but I'm not exactly sure what it is. I think I ... | Ben Grossmann | 81,360 | <p>In a nutshell: calculus is about derivatives and integrals. A <em>derivative</em> generalizes the idea of slope to graphs that are not lines. For instance, you might look at the graph of $y = x^2$ and notice that it gets steeper as $x$ increases, but how can we make this observation precise? You might ask what th... |
2,243,900 | <p>I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is:</p>
<blockquote>
<p>What exactly is calculus? </p>
</blockquote>
<p>I know who invented it, the Leibniz controversy, etc., but I'm not exactly sure what it is. I think I ... | Mauro ALLEGRANZA | 108,274 | <p>Maybe useful:</p>
<ul>
<li>George Exner, <a href="https://books.google.it/books?id=IjP0BwAAQBAJ&printsec=frontcover" rel="nofollow noreferrer">Inside Calculus</a>, Springer (2000), <strong>Introduction</strong> - <em>Propaganda: For Students</em>, page vii:</li>
</ul>
<blockquote>
<p>Calculus is really two t... |
2,243,900 | <p>I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is:</p>
<blockquote>
<p>What exactly is calculus? </p>
</blockquote>
<p>I know who invented it, the Leibniz controversy, etc., but I'm not exactly sure what it is. I think I ... | Kyle Strand | 52,057 | <p>This is not a "standard" definition, and I am going to gloss over many details, but I think it provides some insight.</p>
<p>I consider the <strong>fundamental insight</strong> of calculus as a mathematical discovery (i.e., the basis for the branches of mathematics that you will eventually study) to be the idea tha... |
2,243,900 | <p>I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is:</p>
<blockquote>
<p>What exactly is calculus? </p>
</blockquote>
<p>I know who invented it, the Leibniz controversy, etc., but I'm not exactly sure what it is. I think I ... | Tom Au | 12,506 | <p>I consider calculus to be the study of "infintessimals."</p>
<p>As in, what happens to a secant line if you take smaller and smaller distances from the starting point. (It approaches the tangent line, which is its "derivative.")</p>
<p>Or what happens to your calculation of the area under a curve as you approximat... |
31,782 | <p>This question is edited following the comment of Joseph. He pointed out that the main object of the first version of this question is the cut locus.</p>
<p>Recall that the cut locus of a set <span class="math-container">$S$</span> in a geodesic space <span class="math-container">$X$</span>
is the closure of the set... | Joseph O'Rourke | 6,094 | <p>Let me continue the comments above here so I can include a figure.
Here are examples of the <em>medial axis</em> of two different convex polygons
(from my own work):
<hr>
<img src="https://i.stack.imgur.com/v7kba.jpg" alt="medaxis">
<hr>
The term <em><a href="https://en.wikipedia.org/wiki/Medial... |
162,147 | <p>I have</p>
<pre><code>J = Table[{x10, y10, x10*y10}, {x10, 0, 1, 0.5}, {y10, 0, 1, 0.5}]
L = Table[{x10, y10, 2.0*x10*y10}, {x10, 0, 1, 0.5}, {y10, 0, 1, 0.5}]
</code></pre>
<p>I want the third elements of J and L to be added and the first and second elements are as they are (as they are the same in both cases) fo... | Picaud Vincent | 42,847 | <p>Something like: </p>
<pre><code>MapThread[{#1[[1]], #1[[2]], #1[[3]] + #2[[3]]} &, {J, L}, 2]
</code></pre>
<p>do the job. </p>
|
162,147 | <p>I have</p>
<pre><code>J = Table[{x10, y10, x10*y10}, {x10, 0, 1, 0.5}, {y10, 0, 1, 0.5}]
L = Table[{x10, y10, 2.0*x10*y10}, {x10, 0, 1, 0.5}, {y10, 0, 1, 0.5}]
</code></pre>
<p>I want the third elements of J and L to be added and the first and second elements are as they are (as they are the same in both cases) fo... | mrz | 19,892 | <pre><code>j = Table[{x10, y10, x10*y10}, {x10, 0, 1, 0.5}, {y10, 0, 1, 0.5}];
l = Table[{x10, y10, 2.0*x10*y10}, {x10, 0, 1, 0.5}, {y10, 0, 1, 0.5}];
Partition[
Transpose[{Partition[Flatten[j], 3], Partition[Flatten[l], 3]}] /.
{{x1_, y1_, r1_}, {x2_, y2_, r2_}} :> {x1, y1, r1 + r2}
, 3]
{{{0., 0., 0.}, {0., ... |
177,091 | <p>What would $\int\limits_{-\infty}^\infty e^{ikx}dx$ be equal to where $i$ refers to imaginary unit? What steps should I go over to solve this integral? </p>
<p>I saw this in the Fourier transform, and am unsure how to solve this.</p>
| BlueTrin | 36,894 | <p>Isn't it </p>
<p>$$
-{i\over k} e^{i k x} + C
$$</p>
<p>Just start by deriving </p>
<p>$$
e^{i k x}
$$</p>
<p>And workout what constants are needed to adjust the result or alternatively use the formula by noticing that you are integrating</p>
<p>$$
e^{c x}
$$</p>
<p>with:</p>
<p>$$
c=ik
$$</p>
|
3,986,785 | <p>On page 153 of <em>Linear Algebra Done Right</em> the second edition, it says:</p>
<blockquote>
<p>Define a linear map <span class="math-container">$S_1: \text{range}(\sqrt{T^*T} ) \to \text{range}(T)$</span> by:</p>
</blockquote>
<blockquote>
<p><strong>7.43:</strong> <span class="math-container">$S_1 (\sqrt{T^* ... | Community | -1 | <p>I'll describe well-definedness in terms of a simpler function. Let's say that I wanted to describe a function on non-zero rational numbers by <span class="math-container">$$f(p/q)=q/p$$</span> In this case, it falls on me to show that the value of <span class="math-container">$f$</span> at a certain rational numbe... |
3,986,785 | <p>On page 153 of <em>Linear Algebra Done Right</em> the second edition, it says:</p>
<blockquote>
<p>Define a linear map <span class="math-container">$S_1: \text{range}(\sqrt{T^*T} ) \to \text{range}(T)$</span> by:</p>
</blockquote>
<blockquote>
<p><strong>7.43:</strong> <span class="math-container">$S_1 (\sqrt{T^* ... | Hagen von Eitzen | 39,174 | <p>Ultimately, well-defined means simply: defined, however in a special kind of defining functions.
Namely, we want to define a map <span class="math-container">$f\colon A\to B$</span> in terms of given maps <span class="math-container">$ g\colon C\to A$</span> and <span class="math-container">$h\colon C\to B$</span>. ... |
3,319,903 | <p>What is exactly generating function of ordered partitions and how can I get number of ordered partitions from that?</p>
<p>Example:</p>
<p><span class="math-container">$$ 4 = 1+1+1+1 \\ = 2+2 \\ = 1+1+2 \\ = 1+2+1 \\ = 2+1+1 \\ = 1+3 \\ = 3+1 \\ = 4 $$</span>
so we have <span class="math-container">$8$</span> par... | mathsdiscussion.com | 694,428 | <p>Let us say we want to partition <span class="math-container">$n$</span>.</p>
<p>There are <span class="math-container">$\underbrace{1.1.1. \ldots .1}_{n ~ \text{times}}$</span> </p>
<p>Between these <span class="math-container">$n$</span> times <span class="math-container">$1$</span> there exist <span class="mat... |
18,090 | <p>I am wondering if anyone could prove the following equivalent definition of recurrent/persistent state for Markov chains:</p>
<p>1) $P(X_n=i,n\ge1|X_0=i)=1$<p>
2) Let $T_{ij}=min\{n: X_n=j|X_0=i\}$, a state $i$ is recurrent if for $\forall j$ such that $P(T_{ij}<\infty)>0$, one has $P(T_{ij}<\infty)=1$</p>... | leonbloy | 312 | <p>I think the first property is wrong. That would be a strictly periodic state. Perhaps you missed something?</p>
|
1,361,517 | <p>$K_1 K_2 \dotsb K_{11}$ is a regular $11$-gon inscribed in a circle, which has a radius of $2$. Let $L$ be a point, where the distance from $L$ to the circle's center is $3$. Find
$LK_1^2 + LK_2^2 + \dots + LK_{11}^2$.</p>
<p>Any suggestions as to how to solve this problem? I'm unsure what method to use. </p>
| Michael Hardy | 11,667 | <p>Let $O$ be the center of the circle. One would like to say $LO^2 + OK_j^2 = LK_j^2\quad$ --- a Pythagorean theorem of sorts, except that $LO$ is not at a right angle with $OK_j$. So
\begin{align}
LO^2 + OK_1^2 & \ne LK_1^2 \\
LO^2 + OK_2^2 & \ne LK_2^2 \\
LO^2 + OK_3^2 & \ne LK_3^2 \\
& \,\,\,\vdot... |
133,109 | <p>How can I integrate,</p>
<p>$$
\int_n^{+\infty} x \exp\{-ax^2+bx+c\}dx
$$</p>
<p>and what's the result w.r.t the Gaussian function's p.d.f $p(x)$ and c.d.f $\phi(x)$?</p>
<p>Thanks!</p>
| David Mitra | 18,986 | <p>Completing the square in the quadratic:
$$\eqalign{
\int_n^\infty x \exp(-ax^2+bx+c)\,dx
&= \int_n^\infty\kern-5pt x \exp( -a(x-{\textstyle{b\over 2a})^2 +c+{b^2\over 4a} } )\,dx\cr
&= \alpha \int_n^\infty\kern-5pt x \exp( \textstyle {-(x-{\textstyle{b\over 2a})^2 }\over 1/a } )\,dx\cr
&= \alpha \in... |
3,912,722 | <blockquote>
<p>Circle of radius <span class="math-container">$r$</span> touches the parabola <span class="math-container">$y^2+12x=0$</span> at its vertex. Centre of circle lies left of the vertex and circle lies entirely within the parabola. What is the largest possible value of <span class="math-container">$r$</span... | Math Lover | 801,574 | <p><span class="math-container">$(x+r)^2+y^2=r^2$</span> (check the equation)</p>
<p><span class="math-container">$y^2 + 12x = 0$</span></p>
<p>Equating the two,</p>
<p><span class="math-container">$x^2 -12x + 2xr = 0$</span></p>
<p><span class="math-container">$x^2 + (2r-12)x = 0$</span></p>
<p>So two solutions in <sp... |
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