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 |
|---|---|---|---|---|
156,285 | <p>I have been working on this exercise for a while now. It's in B.L. van der Waerden's <em>Algebra (Volume I)</em>, page $19$. The exercise is as follows:</p>
<blockquote>
<p>The order of the symmetric group $S_n$ is $n!=\prod_{1}^{n}\nu$. (Mathematical induction on $n$.)</p>
</blockquote>
<p>I don't comprehend ho... | 000 | 22,144 | <p>Here's my answer using induction (it's a similar proof, but seems more concise and understandable):</p>
<p>Our base is true.</p>
<p>Assume $S_n$ has $n!$ permutations.</p>
<p>Define all the original $n!$ permutations as permutations where $(n+1)$ is sent to itself. Thus, by definition, all other permutations ("th... |
7,761 | <p>Our undergraduate university department is looking to spruce up our rooms and hallways a bit and has been thinking about finding mathematical posters to put in various spots; hoping possibly to entice students to take more math classes. We've had decent success in finding "How is Math Used in the Real World"-type po... | Joseph O'Rourke | 511 | <p>One potential source is the AMS blog <a href="http://blogs.ams.org/visualinsight/" rel="noreferrer">Visual Insights</a>
run by John Baez. Each image comes with a clear mathematical story.
Most would not fit on one page, but you could make a poster by printing out
several pages and pasting them on a poster (or doing ... |
2,629,744 | <p>I have done the sum by first plotting the graph of the function in the Left Hand Side of the equation and then plotted the line $y=k$. For the equation to have $4$ solutions, both these two curves must intersect at $4$ different points, and from the two graphs, I could see that for the above to occur, the value of $... | Michael Rozenberg | 190,319 | <p>Now, let $a_n=b_n+1$ for some sequence $b$.</p>
<p>Thus, $$b_{n+1}=(b_n+1)\frac{n-1}{n+1}+\frac{2}{n+1}$$ or
$$b_{n+1}+1=\frac{n-1}{n+1}b_n+\frac{n-1}{n+1}+\frac{2}{n+1}$$ or
$$b_{n+1}=\frac{n-1}{n+1}b_n$$ and use your work.</p>
|
4,375,994 | <blockquote>
<p>Question:</p>
<p>Show that, <span class="math-container">$$\pi =3\arccos(\frac{5}{\sqrt{28}}) +
3\arctan(\frac{\sqrt{3}}{2}) ~~~~~~ (*)$$</span></p>
</blockquote>
<p><em>My proof method for this question has received mixed responses. Some people say it's fine, others say that it is a verification, inst... | lhf | 589 | <p>Using complex numbers:
<span class="math-container">$$
\arctan(\frac{\sqrt{3}}{5})+\arctan(\frac{\sqrt{3}}{2})
= \arg((5+\sqrt{3}i)(2+\sqrt{3}i))
= \arg(7+7\sqrt{3}i)
= \arctan(\sqrt{3})
$$</span></p>
|
246,862 | <p>I have stumbled upon this problem which keeps me from finishing a proof:</p>
<p>$(\sum_{n} {|X_n|})^a \leq \sum_{n} {|X_n|}^a$,
where $n \in \mathbb{N}$ and $ 0 \leq a \leq 1 $</p>
<p>I have no idea how to prove this. It is something like the Cauchy-Schwarz inequality which applies in case $0 \leq a \leq 1$?</p>
... | loved.by.Jesus | 272,774 | <p>This is kind of the complementary of the <a href="https://math.stackexchange.com/questions/2735722/has-it-been-proven-that-the-sum-of-powers-is-greater-than-the-power-of-the-sum/2735741">question here</a>.</p>
<p>So, I adapt the answer of <em>Saulspatz</em> to your case.</p>
<p>I will adapt the notation, instead of ... |
3,137,599 | <p>I actually have a doubt about the solution of this question given in my book. It uses the equations tan 2A = - tan C (from A=B, A+B+C = 180 degrees) and 2 tan A + tan C = 100, thereby formulating the cubic equation <span class="math-container">$x^3 - 50x^2 + 50=0$</span>. The discriminant is <span class="math-contai... | Michael Rozenberg | 190,319 | <p>Because <span class="math-container">$$2\alpha+\gamma=180^{\circ}$$</span> or
<span class="math-container">$$\alpha=90^{\circ}-\frac{\gamma}{2}<90^{\circ},$$</span> which says that <span class="math-container">$\alpha$</span> is an acute angle and we need to take <span class="math-container">$x>0$</span> only.... |
521,589 | <p>In a rectangle $ABCD$, the coordinates of $A$ and $B$ are $(1,2)$ and $(3,6)$ respectively and some diameter of the circumscribing circle of $ABCD$ has equation $2x-y+4=0$. Then the area of the rectangle is:</p>
<p>My work: I found the equations of $AD$ and $BC$ of the rectangle. Taking the points $C$ and $D$ as $(... | dibyendu | 95,293 | <p>HINT: The slope of the line $AB$ (i.e $2$) is equal to that of the diameter, that is they are parallel.</p>
<p>$\therefore$ The diameter must go through the mid-ponts of $BC$ and $AD$.</p>
<p>$\therefore$ Perpendicular distance between the diameter and $AB$ is $\frac{1}{2}BC$.</p>
|
46,631 | <p>I'm writing a program to play a game of <a href="http://en.wikipedia.org/wiki/Pente" rel="noreferrer">Pente</a>, and I'm struggling with the following question:</p>
<blockquote>
<p>What's the best way to detect patterns on a two-dimensional board?</p>
</blockquote>
<p>For example, in Pente a pair of neighboring ... | Mr.Wizard | 121 | <p>One function comes to mind that already implements matching of multidimensonal rules: <a href="http://reference.wolfram.com/mathematica/ref/CellularAutomaton.html" rel="nofollow noreferrer"><code>CellularAutomaton</code></a>. Allow me to represent your board data like this:</p>
<pre><code>board = SparseArray[
a ... |
3,521,534 | <p>I tried solving a calculus problem and I got the right result, but I don't understand the solution provided at the end of the exercise. Even though I got the same answer, I would like to understand what's happening in the given solution aswell.</p>
<blockquote>
<p>Consider the function: <span class="math-containe... | DanielWainfleet | 254,665 | <p><span class="math-container">$f(x)$</span> is continuous on <span class="math-container">$(-\infty,0]$</span> and on <span class="math-container">$(0,\infty).$</span> So if <span class="math-container">$F'(x)=f(x)$</span> for all <span class="math-container">$x,$</span> then by the Fundamental Theorem of Calculus th... |
394,085 | <p>How is it possible to establish proof for the following statement?</p>
<p>$$n = \frac{1}{2}(5x+4),\;2<x,\;\text{isPrime}(n)\;\Rightarrow\;n=10k+7$$</p>
<p>Where $n,x,k$ are $\text{integers}$.</p>
<hr>
<p>To be more verbose:</p>
<p>I conjecture that;</p>
<p>If $\frac{1}{2}(5x+4),\;2<x$ is a prime number, ... | Ross Millikan | 1,827 | <p>$x$ must be even or$n$ is not a natural. So let $x=2y$ and your conjecture is that if $5y+2$ is prime, it is$10k+7$. As $5$ divides into $10$, $5y+2 \equiv 2,7 \pmod {10}$. Any number $2\pmod {10}$ is even and (if $\gt 2)$ not prime.</p>
|
227,562 | <p>Let $K\subset \mathbb{R}^n$ be a compact convex set of full dimension. Assume that $0\in \partial K$. </p>
<p><strong>Question 1.</strong> Is it true that there exists $\varepsilon_0>0$ such that for any $0<\varepsilon <\varepsilon_0$ the intersection $K\cap \varepsilon S^{n-1}$ is contractible? Here $\var... | Mikhail Katz | 28,128 | <p>Given a convex set $K$ in Euclidean space and a point $O\in\partial K$, there exists an $\epsilon>0$ such that for all $r<\epsilon$ the intersection $S(O,r)\cap K$ is connected.</p>
<p>Thus, let $O\in \partial K$. Call $P\in \partial K$ a <em>critical point</em> if $\langle O-P, X-P\rangle \geq 0$ for all $X\... |
606,356 | <p>I would appreciate if somebody could help me with the following problem</p>
<p>Q: Quadratic Equation $x^4+ax^3+bx^2+ax+1=0$ have four real roots
$x=\frac{1}{\alpha^3},\frac{1}{\alpha},\alpha,\alpha^3(\alpha>0)$ and $2a+b=14$.</p>
<p>Find $a,b=?(a,b\in\mathbb{R})$</p>
| DonAntonio | 31,254 | <p>Hint:</p>
<p>$$0=x^4+ax^3+bx^2+ax+1=(x-\alpha)(x-\alpha^3)\left(x-\frac1\alpha\right)\left(x-\frac1{\alpha^3}\right)$$</p>
<p>Now compare coefficients in both sides (Viete's Formulas), for example:</p>
<p>$$-a=\alpha+\alpha^3+\frac1\alpha+\frac1{\alpha^3}\;,\;\;b=\alpha^4+\frac1{\alpha^4}+\alpha^2+\frac1{\alpha^2... |
881,159 | <p>I'm very lost on the following problem and will appreciate your help very much.</p>
<p>How large should $n$ be to guarantee that the Simpson's Rule approximation on the $\int_0^1 19e^{x^2} \, dx$ is accurate to within $0.0001$?</p>
| RRL | 148,510 | <p>A bound on the error in using Simpson's rule with $n$ subintervals to approximate the integral of $f(x)$ over $[a,b]$ is</p>
<p>$$E \leq \frac{(b-a)^5}{180n^4}\max_{a \leq x \leq b}|f^{iv}(x)|.$$</p>
<p>Differentiating $f(x) = 19e^{x^2}$ four times we have for $x \in [0,1]$</p>
<p>$$f^{iv}(x) = 19e^{x^2}(12+48x^2... |
881,159 | <p>I'm very lost on the following problem and will appreciate your help very much.</p>
<p>How large should $n$ be to guarantee that the Simpson's Rule approximation on the $\int_0^1 19e^{x^2} \, dx$ is accurate to within $0.0001$?</p>
| Ala Pawelek | 481,449 | <p>A bound on the error in using Simpson's rule with nn subintervals to approximate the integral of $f(x)$ over $[a,b]$ is</p>
<p>Differentiating $f(x)=19e^{x^2}$ four times we have for $x\in [0,1]$</p>
<p>$$f(x)=38e^{x^2}(8+24x^2+8x^4)$$ </p>
<p>plugging $1$ into $x$
we get
$$38e^{x^2}(8+24x^2+8x^4) = 1444e$$</p... |
1,181,631 | <p>Let $f : \mathbb R \to \mathbb R$ continuous. Prove that graph $G = \{(x, f(x)) \mid x \in \mathbb R\}$ is closed.</p>
<p>I'm a little confused on how to prove $G$ is closed. I get the general strategy is to show that every arbitrary convergent sequence in $G$ converges to a point in $G$.</p>
<p>Here is what I tri... | Olórin | 187,521 | <p>Almost, but when learning topology, better try to understand what is purely topological and what proper to metric spaces. Here, what you want to prove is not proper at all to metric spaces, nor to the fact that $\mathbf{R}$'s addition $(x,y)\mapsto x+y$ is continuous, but is <em>proper to continuous maps with Hausdo... |
76,753 | <p>I am having a hard time getting to factor this binomial: I have tried other methods but they do not seem to work... ah well.
$$4m^2-\frac{9}{25}.$$</p>
<p>Thanks.</p>
| Ross Millikan | 1,827 | <p>If you mean $4m^2-\frac{9}{25}$, note that the second term is $(\frac{3}{5})^2$ and you have a difference of squares.</p>
|
1,132,003 | <blockquote>
<p><strong>Problem</strong> Find the value of $$\frac{1}{\sqrt 1 + \sqrt 3} + \frac 1 {\sqrt 3 + \sqrt 5} + \dots + \frac 1 {\sqrt {1087} + \sqrt{1089}}$$</p>
</blockquote>
<p>I cant figure out how to solve this problem. I cant use summation.</p>
| Workaholic | 201,168 | <p><strong>Hint:</strong> $$\dfrac1{\sqrt{n}+\sqrt{n+2}}=\dfrac{\sqrt{n}-\sqrt{n+2}}{(\sqrt{n}+\sqrt{n+2})(\sqrt{n}-\sqrt{n+2})}=\dfrac{\sqrt{n}-\sqrt{n+2}}{n-n-2}=\ldots$$</p>
|
1,356,545 | <p>Given a fair 6-sided die, how can we simulate a biased coin with P(H)= 1/$\pi$ and P(T) = 1 - 1/$\pi$ ?</p>
| Barry Cipra | 86,747 | <p>A fair die can obviously simulate a fair coin, so it suffices to show that a fair coin can simulate a biased one. There's a simple way to do so.</p>
<p>Write $p$, the desired (biased) probability of getting Heads, in binary, with $.111\ldots$ if the probability is $1$. Now toss the fair coin until the result of t... |
1,887,536 | <p>Howdy just a simple question,</p>
<p>I know when A is diagonalizable, the eigenvalues of F(A) are just simply $F(\lambda_i)$ where $\lambda_i \exists \sigma (A)$</p>
<p>I'm interested in the case of when A is not diagonalizable. I look at A as a Jordan form, but I cannot seem to show that when $A$ is not diagonali... | egreg | 62,967 | <p>Assuming the coefficients are real (otherwise the problem would be underdetermined), you know that also $1-4i$ is a root and, from Viète's formulas, that
$$
\begin{cases}
-\dfrac{b}{a}=2+1+4i+1-4i=4 \\[6px]
\dfrac{c}{a}=2(1+4i)+2(1-4i)+(1+4i)(1-4i)=21 \\[6px]
-\dfrac{d}{a}=2(1+4i)(1-4i)=34
\end{cases}
$$
Hence $b=-4... |
365,631 | <p>Suppose we want to prove that among some collection of things, at least one
of them has some desirable property. Sometimes the easiest strategy is to
equip the collection of all things with a measure, then show that the set
of things with the desired property has positive measure. Examples of this strategy
appear in... | Ian Agol | 1,345 | <p><a href="https://en.wikipedia.org/wiki/Sard%27s_theorem" rel="noreferrer">Sard's theorem</a> implies that the measure of the set of critical points of a smooth function <span class="math-container">$f:M_1\to M_2$</span> between smooth manifolds has measure zero. Hence the preimage <span class="math-container">$f^{-1... |
365,631 | <p>Suppose we want to prove that among some collection of things, at least one
of them has some desirable property. Sometimes the easiest strategy is to
equip the collection of all things with a measure, then show that the set
of things with the desired property has positive measure. Examples of this strategy
appear in... | Ian Agol | 1,345 | <p><a href="https://en.wikipedia.org/wiki/Jeremy_Kahn" rel="noreferrer">Kahn</a> and <a href="https://en.wikipedia.org/wiki/Vladimir_Markovic" rel="noreferrer">Markovic</a> showed the <a href="https://annals.math.princeton.edu/2012/175-3/p04" rel="noreferrer">existence of immersed essential surfaces in closed hyperboli... |
438,336 | <p>This a two part question:</p>
<p>$1$: If three cards are selected at random without replacement. What is the probability that all three are Kings? In a deck of $52$ cards.</p>
<p>$2$: Can you please explain to me in lay man terms what is the difference between with and without replacement.</p>
<p>Thanks guys!</p>... | ZZ7474 | 210,085 | <p>4/52*3/51*2/50 =(24/13260)or(1/5525)same as (0.0018)remember when there's an "or" put
a cross over the letter "O" & remember to add rather than mutliply.When there's a plus sign
between remember to multiply.</p>
|
3,068,031 | <blockquote>
<p>Let <span class="math-container">$G$</span> be a group and <span class="math-container">$H$</span> be a subgroup of <span class="math-container">$G$</span>. Let also <span class="math-container">$a,~b\in G$</span> such that <span class="math-container">$ab\in H$</span>.</p>
<p>True or false? <span cla... | John Douma | 69,810 | <p>Consider <span class="math-container">$S_3$</span>.</p>
<p>Let <span class="math-container">$a=(1 2 3)$</span> and <span class="math-container">$b=(2 3)$</span>. Then <span class="math-container">$ab=(1 2)$</span> and <span class="math-container">$a^2b^2=(1 3 2)$</span> </p>
<p>Let <span class="math-container">$H=... |
1,572,351 | <p>Solve the differential equation;</p>
<p>$(xdx+ydy)=x(xdy-ydx)$</p>
<p>L.H.S. can be written as $\frac{d(x^2+y^2)}{2}$ but what should be done for R.H.S.?</p>
| achille hui | 59,379 | <p>In polar coordinate $(r,\theta)$, $xdx + ydy = rdr$ and $xdy- ydx = r^2d\theta$.
The equation at hand becomes
$$rdr = r^3\cos\theta d\theta
\iff \frac{1}{r^2} dr = \cos\theta d\theta
\iff d\left(\frac{1}{r} + \sin\theta\right) = 0\\
\iff \frac{1+y}{r} = K
\iff (1+y)^2 = K^2(x^2+y^2)
$$
for some constant $K$.</p>
|
1,572,351 | <p>Solve the differential equation;</p>
<p>$(xdx+ydy)=x(xdy-ydx)$</p>
<p>L.H.S. can be written as $\frac{d(x^2+y^2)}{2}$ but what should be done for R.H.S.?</p>
| azc | 166,613 | <p>I think that there is actually a general solution to a general kind of ODE. So I give here its general form and its solution. After that, the solution to the ODE in the thread above is also given.
Consider the following ODE
\begin{gather*}
A(x,y)(xd x+yd y)=B(x,y)(xd y-yd x)\tag{1}
\end{gather*}
where $A$ and $B... |
3,110,508 | <p>I read that implication like a=>b can be proof using the following steps :
1) suppose a true.
2) Then deduce b from a.
3) Then you can conclude that a=>b is true.</p>
<p>Actually my real problem is to understand why step 1 and 2 are sufficient to prove that a=>b is true. I mean, how can you prove the truth table of... | Dave | 334,366 | <p>In this case of <span class="math-container">$n=3$</span> your result follows from this other theorem by considering <span class="math-container">$x^3+y^3=x^3-(-y)^3$</span>. The pattern with odd <span class="math-container">$n$</span> is that <span class="math-container">$-y^n=(-y)^n$</span>.</p>
|
3,110,508 | <p>I read that implication like a=>b can be proof using the following steps :
1) suppose a true.
2) Then deduce b from a.
3) Then you can conclude that a=>b is true.</p>
<p>Actually my real problem is to understand why step 1 and 2 are sufficient to prove that a=>b is true. I mean, how can you prove the truth table of... | nonuser | 463,553 | <p><span class="math-container">$$x^3+y^3 = x^3+\color{red}{x^2y-x^2y}+y^3 $$</span>
<span class="math-container">$$ = x^2(x+y)-y(x^2-y^2)$$</span>
<span class="math-container">$$ = x^2(x+y)-y(x-y)(x+y)$$</span>
<span class="math-container">$$ = (x+y)(x^2-y(x-y))$$</span>
<span class="math-container">$$ = (x+y)(x^2-xy+... |
3,405,914 | <blockquote>
<p>It's known that <span class="math-container">$\lim_{n \to \infty} \left(1 + \frac{x}{n} \right)^n = e^x$</span>.</p>
<p>Using the above statement, prove <span class="math-container">$\lim_{n \to \infty} \left(\frac{3n-2}{3n+1}\right)^{2n} = \frac{1}{e^2}$</span>.</p>
</blockquote>
<h2>My attempt... | Community | -1 | <p>Let <span class="math-container">$m:=3n+1$</span>. We have</p>
<p><span class="math-container">$$\left(\frac{3n-2}{3n+1}\right)^{2n}=\left(1-\frac3m\right)^{2(m-1)/3}=\left(\left(1-\frac3m\right)^m\right)^{2/3}\left(1-\frac3m\right)^{-2/3}.$$</span></p>
<p>Hence the limit is <span class="math-container">$e^{-3\cdo... |
2,933,375 | <p>I have a set of vectors, <span class="math-container">$M_1$</span> which is defined as the following:
<span class="math-container">$$M_1:=[\begin{pmatrix}1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix}0 \\ 1 \\ 1 \end{pmatrix}]$$</span>
I have to show that <span class="math-container">$M_1$</span> isn't a generating set ... | Scientifica | 164,983 | <p>And in Enderton's <em>Elements of Set Theory</em>, he presents the axioms of ZFC in first-order logic as follows:<a href="https://i.stack.imgur.com/ZPefC.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ZPefC.png" alt="enter image description here"></a></p>
<p>There's neither ':' nor ','. Also, if... |
1,196,424 | <p>So I'm reviewing my notes and I just realized that I can't think of how to show that a particular integer mod group is abelian. I know how to do it with symmetric but not with integers themselves.</p>
<p>For example, lets say I was asked to show $\mathbb{Z_5}$ is abelian.</p>
<p>I know for symmetric groups, lets s... | Qudit | 210,368 | <p>Note that symmetric groups are <em>not</em> Abelian unless $n < 3$. See my answer <a href="https://math.stackexchange.com/questions/1186094/for-what-values-n-is-the-group-s-n-cyclic/1186099#1186099">here</a> for a proof.</p>
<p>As for how to see that $\Bbb{Z}_n$ is Abelian, note that the group $\Bbb{Z}$ is Abel... |
1,196,424 | <p>So I'm reviewing my notes and I just realized that I can't think of how to show that a particular integer mod group is abelian. I know how to do it with symmetric but not with integers themselves.</p>
<p>For example, lets say I was asked to show $\mathbb{Z_5}$ is abelian.</p>
<p>I know for symmetric groups, lets s... | Asinomás | 33,907 | <p>The symmetric group $S_n$ is not abelian for $n\geq 3$ since $(12)(123)=(23)$ and $(123)(12)=(13)$.</p>
<p>On the other hand any cyclic group is abelian. This is because in reality every element $m$ can be thought of as $\underbrace{1+1+1\dots+1}_{\text{m times}}$.</p>
<p>And so</p>
<p>$m+n=(\underbrace{1+1+1\dot... |
834,678 | <p>An object $X$ is a <em>generator</em> of a category $\mathcal{C}$ if the functor $Hom_{\mathcal{C}}(X,\_) : \mathcal{C} \rightarrow Set$ is faithful. </p>
<p>I encountered the notion in the context of Morita-equivalence of rings, but I don't understand what its use is.
Why is $X$ called a "generator"? What does it ... | Martin Brandenburg | 1,650 | <p>Generators don't generate the category. This explains why "generator" is an unfortunate terminology. A better one would be "separator". I've seen this suggestion in many places. This makes sense, because every two non-equal morphisms $f,g : A \to B$ may be separated by a morphism $i : X \to A$, i.e. $fi \neq gi$.</p... |
2,050,760 | <p>The question:</p>
<blockquote>
<p>Find a recurrence for the number of n length ternary strings that contain "00", "11", or "22".</p>
</blockquote>
<p>My answer:</p>
<p>$3(a_{n-2}) + 3(a_{n-1} - 1)$</p>
<p>Proof:</p>
<p>Cases:</p>
<p>______________00 (a_(n-2))</p>
<p>______________11 (a_(n-2))</p>
<p>__... | Jean Marie | 305,862 | <p>I would like to add something to the answer given by @Robert Israel.</p>
<p>The set $U$ of matrices :</p>
<p>$$ U_c=\begin{pmatrix}1 & c \\ 0 & 1\end{pmatrix}$$</p>
<p>is a subgroup of $GL(2,\mathbb{K})$ (for multiplication, of course).</p>
<p>Mapping:</p>
<p>$$f: U \rightarrow \mathbb{K}, \ \ U_c \map... |
3,207,767 | <p>What is the general solution of differential equation <span class="math-container">$y\frac{d^{2}y}{dx^2} - (\frac{dy}{dx})^2 = y^2 log(y)$</span>.</p>
<p>The answer to this DE is <span class="math-container">$log(y) = c_1 e^x + c_2 e^{-x}$</span></p>
<p>I don't know the method to solve differential equation with d... | user120123 | 120,123 | <p>With thanks to Michael for pointing this beautiful topic.
For proving
<span class="math-container">$$(x^2+3)(y^2+3)(z^2+3)(t^2+3)\geq16(x+y+z+t)^2,$$</span>
I have used the following result discovered by me:
<span class="math-container">$$(x^2+3)(y^2+3)(z^2+3)(t^2+3)\geq16(x+y+z+t)^2+$$</span>
<span class="math-cont... |
791,535 | <blockquote>
<p>Let $f,g:\mathbb [a,b] \to \mathbb [a,b]$ be monotonically increasing functions
such that $f\circ g=g\circ f$</p>
<p>Prove that $f$ and $g$ have a common fixed point.</p>
</blockquote>
<p>I found this problem in a problem set, it's quite similar to this <a href="https://math.stackexchange.com... | Gabriel Romon | 66,096 | <p>Let $A=\{x \in [a,b]/ x \leq f(x) \; \text{and} \; x \leq g(x) \}$</p>
<ul>
<li><p>$a\in A$</p></li>
<li><p>let $u=\sup A$</p></li>
<li><p>Let us prove that $f(u)$ and $g(u)$ are upper bounds for $A$</p></li>
</ul>
<p>Indeed let $x\in A$.</p>
<p>Then $x\leq u$. Hence $f(x) \leq f(u)$, thus $x\leq f(x) \leq f(u)$ ... |
393,712 | <p>I studied elementary probability theory. For that, density functions were enough. What is a practical necessity to develop measure theory? What is a problem that cannot be solved using elementary density functions?</p>
| not all wrong | 37,268 | <p>Simple answer: Tossing a coin.</p>
<p>Longer answer: You know that you treat discrete events like the above with probability mass functions or similar, but continuous things with probability density functions. Imagine you had $X$ which is randomly uniform on $[0,1]$ half the time and $5$ the rest of the time. Perfe... |
1,684,095 | <p>How can I evaluate the following series.</p>
<p>$$\sum_{k=1}^{\infty}\frac{1}{(k+1)(k-1)!}.$$</p>
| Hanul Jeon | 53,976 | <p>Consider the class $C = \{(\alpha, n) : \alpha\in\mathrm{On}\text{ and } n=0, 1\}$ and define an ordering $\prec$ over $C$ as lexicographical order. </p>
<p>Our $C$ is well-ordered: let $B\subset C$ is a subclass. If the intersection of $B$ and the initial segment $S=\{x\in C : x\prec (0,1)\}$ is not empty, then $B... |
3,747,453 | <p>Isn't it wrong to write the following with only the percent sign? Instead of <span class="math-container">$100 \%$</span>?</p>
<blockquote>
<p>The change in height as a percentage is
<span class="math-container">$$
\frac{a - b}{a} \% \tag 1
$$</span>
where <span class="math-container">$a$</span> is the initial heigh... | dan_fulea | 550,003 | <p>Consider the composition of rings <span class="math-container">$\Bbb Z\to \Bbb Z[X]\to \Bbb Z$</span>, the first morphism is the inclusion <span class="math-container">$f:\Bbb Z\to \Bbb Z[X]$</span>, mapping <span class="math-container">$n$</span> into the constant polynomial <span class="math-container">$n=n\cdot X... |
3,011,758 | <p>So I was going through a sum for </p>
<p>Prove <span class="math-container">$ex \leq e^x$</span> , <span class="math-container">$\forall x \in \mathbb{R} $</span></p>
<p>I took <span class="math-container">$g(x) = e^x - ex$</span></p>
<p>Then <span class="math-container">$g'(x)= e^x - e$</span></p>
<p>I understo... | user | 505,767 | <p>We have that</p>
<p><span class="math-container">$$g(x)=e^x-ex \implies g'(x)=e^x-e=0\quad x=1$$</span></p>
<p>moreover <span class="math-container">$g''(x)=e^x > 0$</span>, therefore <span class="math-container">$x=1$</span> is a point of minimum for <span class="math-container">$g(x)$</span>.</p>
|
498,056 | <p>If $A$ is a subset (not proper subset) of $B$, does that mean that $B$ is a subset (not proper subset) of $A$ and that $A=B$?</p>
| J. W. Perry | 93,144 | <p>Your parentheses give me cause for concern as they introduce a degree of possible ambiguity.</p>
<p>The following is true:</p>
<p>$$(A \subseteq B) \wedge (A \not \subset B) \Rightarrow A=B.$$</p>
<p>Read this as "If $A$ is a subset of $B$ and $A$ is not a proper subset of $B$, then $A=B$".</p>
<p>However, the f... |
498,056 | <p>If $A$ is a subset (not proper subset) of $B$, does that mean that $B$ is a subset (not proper subset) of $A$ and that $A=B$?</p>
| ILikeMath | 86,744 | <p>Yes. We have $A \subseteq B$ but $A \not\subset B$. Of the second statement $A \not\subseteq B$ or $A= B$ (this is the negation of $A \subset B$ equivalent to $A \subseteq B$ and $A \ne B$), therefore $A=B$. It follows that $B\subseteq A$ too.</p>
|
3,984,930 | <p>I am studying maths purely out of interest and have come across this question in my text book:</p>
<p>A rectangular piece of paper ABCD is folded about the line joining points P on AB and Q on AD so that the new position of A is on CD. If AB = a and AD = b, where <span class="math-container">$a \ge\frac{2b}{\sqrt3}$... | Math Lover | 801,574 | <p>If <span class="math-container">$\angle AQP = \theta$</span>, <span class="math-container">$\angle APQ = 90^0 - \theta$</span>.</p>
<p>Draw a perp from <span class="math-container">$A_1$</span> to line segment <span class="math-container">$AP$</span> and say it is point <span class="math-container">$A_2$</span> on l... |
4,246,048 | <p>As I understand it, Cantor defined two sets as having the same cardinality iff their members can be paired 1-to-1. He applied this to infinite sets, so ostensibly the integers (Z) and the even integers (E) have the same cardinality because we can pair each element of Z with exactly one element of E.</p>
<p>For infi... | md2perpe | 168,433 | <p>There are many definitions where you kind of need to know the answer before you use the definition. I write "kind of" because <strong>you don't need to have a proof for it, but you need to have a guess</strong>. Then you write a proof to show that the guess is actually correct.</p>
<p>The definition of equ... |
42,787 | <p>I am using <code>ListPlot</code> to display from 5 to 12 lines of busy data. The individual time series in my data are not easy to distinguish visually, as may be evident below, because the colors are not sufficiently different.</p>
<p><img src="https://i.stack.imgur.com/PiMMh.png" alt="enter image description here... | mfvonh | 5,059 | <p>As mentioned by others, you should read up on <code>ColorDataFunctions</code>. For example, you could evenly space colors across a continuous color scheme, for an arbitrary number of lines, with:</p>
<pre><code>d = Table[i*Range[0, 10], {i, 1, 5, 0.5}];
ListPlot[d, Frame -> True, Joined -> True, PlotRange -&g... |
227,797 | <p>I have this function and I want to see where it is zero.
<span class="math-container">$$\frac{1}{16} \left(\sinh (\pi x) \left(64 \left(x^2-4\right) \cosh \left(\frac{2 \pi x}{3}\right) \cos (y)+\left(x^2+4\right)^2+256 x \sinh \left(\frac{2 \pi x}{3}\right) \sin (y)\right)+\left(x^2-12\right)^2 \sinh \left(\frac... | C. E. | 731 | <p>Not a complete answer but one direction to go in:</p>
<pre><code>cp = ContourPlot[f[x, y] == 0, {x, 3.465728, 3.465729}, {y, 1.046786, 1.046795}, PlotPoints -> 500]
{l1, l2} = Cases[Normal[cp], _Line, Infinity];
{{x1, y1}} = MinimalBy[First@l1, RegionDistance[l2]];
{{x2, y2}} = MinimalBy[First@l2, RegionDistance... |
394,580 | <p>Let <span class="math-container">$S$</span> be a smooth compact closed surface embedded in <span class="math-container">$\mathbb{R}^3$</span> of genus <span class="math-container">$g$</span>.
Starting from a point <span class="math-container">$p$</span>, define a random walk as taking discrete steps
in a uniformly r... | Carlo Beenakker | 11,260 | <p>This random walk is known in the literature as the "geodesic random walk". For a manifold with positive curvature, theorems 1 and 4 of <A HREF="https://arxiv.org/abs/1609.02901" rel="noreferrer">arXiv:1609.02901</A> prove that the uniform measure on the manifold is the unique stationary distribution of the... |
394,580 | <p>Let <span class="math-container">$S$</span> be a smooth compact closed surface embedded in <span class="math-container">$\mathbb{R}^3$</span> of genus <span class="math-container">$g$</span>.
Starting from a point <span class="math-container">$p$</span>, define a random walk as taking discrete steps
in a uniformly r... | R W | 8,588 | <p>This problem was first considered and solved by Sunada, see his 1983 paper <a href="http://www.numdam.org/item/?id=CM_1983__48_1_129_0" rel="noreferrer">Mean-value theorems and ergodicity of certain geodesic random walks</a>. Alas, the authors of the quoted arxiv paper were not aware of this. Any assumptions on curv... |
473,508 | <p>Let $p > 2$ be a prime. Can someone prove that for for any $t \leq p-2$ the following identity holds</p>
<blockquote>
<p>$\displaystyle \sum_{x \in \mathbb{F}_p} x^t = 0$</p>
</blockquote>
| azimut | 61,691 | <p>Because of $t < p-1$, there exists an $a\in\mathbb F_p^\times$ with $a^t\neq 1$. Now
$$
\sum_{x\in\mathbb F_p} x^t
= \sum_{x\in\mathbb F_p} (ax)^t
= a^t \sum_{x\in\mathbb F_p} x^t.
$$
So
$$\underbrace{(1 - a^t)}_{\neq 0}\sum_{x\in\mathbb F_p} x^t = 0$$
and therefore
$$\sum_{x\in\mathbb F_p} x^t = 0.$$</p>
|
474,587 | <p>Does $\|Tv\|\leq\|v\|$ (for all $v \in V$) leads to $T$ is normal?</p>
<p>If not, when I add the additional information that every e.e of $T$ is of the absolute value 1, can I prove $T$ is unitary? </p>
<p>Thanks!</p>
| user8268 | 8,268 | <p>For the second question the answer is yes. I'll suppose that $T$ is diagonalizable (i.e. no Jordan blocks). If $T$ is not unitary then there are eigenvectors $u,v$ with eigenvalues $a,b$, $a\neq b$ ($|a|=|b|=1$) such that $(u,v)\neq 0$. If $c$ is a complecx number then $\|u+cv\|^2=\|u\|^2+|c|^2\|v\|^2+2Re\, c(v,u)$ ... |
4,353,203 | <p>While looking for an explanation to the first inequality, I spied another similar inequality. So, I will ask about both of them here.</p>
<p><span class="math-container">$a$</span>, <span class="math-container">$b$</span>, and <span class="math-container">$c$</span> are positive numbers.
<span class="math-container"... | md5 | 301,549 | <p>Here it is not too hard to explicitly compute the distribution of <span class="math-container">$s_{2n}$</span>. By counting the number of <span class="math-container">$x_k$</span> that match <span class="math-container">$a_k$</span>, we see that for all <span class="math-container">$x\in\{0,\ldots,n\}$</span>,
<span... |
318,351 | <p>As we know, the Ky Fan norm is convex, and so is the Ky Fan k-norm. My question is, does this imply that the difference between them is a non-convex function, since it results from "difference between two convex" functions ?</p>
| Muphrid | 45,296 | <p>Differential forms are a way to talk about fields that aren't vector or scalar fields--or at least, to talk about those and things beyond them. In traditional vector calculus, you only talk about scalar fields and vector fields. But using differential forms, you can talk about something like $\omega = 2y \, dx \we... |
12,717 | <p>In the familiar case of (smooth projective) curves over an algebraically closed fields, (closed) points correspond to DVR's.</p>
<p>What if we have a non-singular projective curve over a non-algebraically closed field? The closed points will certainly induce DVR's, but would all DVR's come from closed points? Is th... | Pete L. Clark | 1,149 | <blockquote>
<p>What if we have a non-singular projective curve over a non-algebraically closed field? The closed points will certainly induce DVR's, but would all DVR's come from closed points?</p>
</blockquote>
<p>Yes: let <span class="math-container">$L/k$</span> be a function field in one variable, so it can be giv... |
1,442,240 | <p>I have a little question, that run threw my thoughts, when i saw this exercise: $$\lim _{x\to \infty }\left(\frac{\int _{sin\left(x\right)}^xe^{\sqrt{t^2+1}}dt}{e^{\sqrt{x^2-1}}}\:\right)$$</p>
<p>Of course I want to implement here Lophital's rule, but without showing a calculation, is there intuitive and logical e... | Crostul | 160,300 | <p>You can see that
$$\int_{\sin x}^x e^{\sqrt{t^2-1}} dt = \int_0^x e^{\sqrt{t^2-1}} dt + \int_{\sin x}^0 e^{\sqrt{t^2-1}} dt$$
and clearly the first summand diverges to $\infty$, while the second summand is bounded becuase
$$\left| \int_{\sin x}^0 e^{\sqrt{t^2-1}} dt \right| \le \int_{-1}^1 e^{\sqrt{t^2-1}} dt = con... |
316,055 | <p>I have no idea of how to solve the following: </p>
<p>$$\displaystyle \lim_{x\rightarrow 0}\frac{e^x-1}{3x}$$</p>
<p>I know about the notable special limit $$\displaystyle \lim_{x\rightarrow 0}\frac{e^x-1}{x}=1$$, and I know that I have to do some algebraic manipulation and change what I have above to the notable ... | Jeel Shah | 38,296 | <p>Note, that the limit that you have provided is indeterminate.
$$\begin{align}
\displaystyle \lim_{x\rightarrow 0}\frac{e^x-1}{3x}
&=\frac{e^0-1}{3(0)}
=\frac{0}{0}
\end{align}$$</p>
<p>Therefore, you can use L'Hopital's Rule which states: </p>
<p>$$\displaystyle \lim_{x \rightarrow 0}\frac{f(x)}{g(x)} = \disp... |
674,259 | <p>I am trying to understand why the derivative of $f(x)=x^\frac{1}{2}$ is $\frac{1}{2\sqrt{x}}$ using the limit theorem. I know $f'(x) = \frac{1}{2\sqrt{x}}$, but what I want to understand is how to manipulate the following limit so that it gives this result as h tends to zero:</p>
<p>$$f'(x)=\lim_{h\to 0} \frac{(x+h... | Dan | 79,007 | <p>Hint: Simplify
$$
\lim_{h\to0}\frac{\sqrt{x+h}-\sqrt{x}}{h} = \lim_{h\to0}\frac{\sqrt{x+h}-\sqrt{x}}{h}\cdot\frac{\sqrt{x+h} + \sqrt{x}}{\sqrt{x+h} + \sqrt{x}}
$$</p>
|
347,385 | <p>Assume $f(x) \in C^1([0,1])$,and $\int_0^{\frac{1}{2}}f(x)\text{d}x=0$,show that:
$$\left(\int_0^1f(x)\text{d}x\right)^2 \leq \frac{1}{12}\int_0^1[f'(x)]^2\text{d}x$$</p>
<p>and how to find the smallest constant $C$ which satisfies
$$\left(\int_0^1f(x)\text{d}x\right)^2 \leq C\int_0^1[f'(x)]^2\text{d}x$$</p>
| math110 | 58,742 | <p>let <span class="math-container">$$\displaystyle\int_{0}^{\frac{1}{2}}f(x)=0\Longrightarrow \int_{0}^{\frac{1}{2}}xf'(x)dx=\dfrac{1}{2}f(\dfrac{1}{2})$$</span></p>
<p>so
<span class="math-container">\begin{align}
&(\int_{0}^{1}f(x)dx)^2=\left[\int_{\frac{1}{2}}^{1}(f(x)-f(\dfrac{1}{2}))dx+\dfrac{1}{2}f(\dfrac{1... |
147,994 | <p><strong>Preamble</strong></p>
<p>Consider a <a href="http://reference.wolfram.com/language/ref/SetterBar.html" rel="nofollow noreferrer"><code>SetterBar</code></a>:</p>
<pre><code>SetterBar[1, StringRepeat["q", #] & /@ Range@5]
</code></pre>
<blockquote>
<p><a href="https://i.stack.imgur.com/b4eqb.png" rel=... | Michael E2 | 4,999 | <p>I usually use <code>Pane</code> to solve such alignment problems. String padding in a variable-width font does not produce reliable results, and <code>Pane</code> can be used to get around that. (Fortunately or unfortunately, a vertical <code>SetterBar</code> automatically pads out the buttons to be the same sizes... |
1,335,483 | <p>Given a relation $R \subseteq A \times A$ with $n$ tuples, I am trying to prove that its transitive closure $R^+$ has at the most $n^2$ elements.</p>
<p>My initial idea was to use the following definition of the transitive closure to identify an argument why the statement to be proven must be true:</p>
<p>$$R^+ = ... | ajotatxe | 132,456 | <p>For each ordered pair of different tuples $\{(a,b),(b,c)\}$ we have to add at most one tuple, namely $(a,c)$.</p>
<p>Since there are at most $n(n-1)=n^2-n$ of such pairs, then $|R^+|\le n+n^2-n$</p>
|
706,980 | <p>If I know that $f(z)$ is differentiable at $z_0$, $z_0 = x_0 + iy_0$.
How do I prove that $g(z) = \overline{f(\overline{z})}$ is differentiable at $\overline z_0$?</p>
| copper.hat | 27,978 | <p>First note that $z \to z_0$ <strong>iff</strong> $\bar{z} \to \bar{z_0}$. In particular, the map $z \mapsto \bar{z}$ is continuous. </p>
<p>Then note that $\lim_{z \to z_0} { f(\bar{z})-f(\bar{z_0}) \over \bar{z} - \bar{z_0} } = f'(\bar{z_0})$.</p>
<p>Finally, ${ g(z) -g(z_0) \over z - z_0} = \overline{\left( { f(... |
3,291,549 | <p>How does one prove that the exponential and logarithmic functions are inverse using the definitions:</p>
<p><span class="math-container">$$e^x= \sum_{i=0}^{\infty} \frac{x^i}{i!}$$</span>
and
<span class="math-container">$$\log(x)=\int_{1}^{x}\frac{1}{t}dt$$</span></p>
<p>My naive approach (sort of ignoring issue... | Julien D | 688,390 | <p><span class="math-container">$\log(e^x)=\int_1^{e^x}\frac{1}{t}dt=\int_0^x\frac{1}{e^u}e^u du=x$</span> and since it's easy to prove that <span class="math-container">$e^x$</span> is bijective then <span class="math-container">$\log$</span> is its inverse.</p>
|
1,182,953 | <p>Does anyone know the provenance of or the answer to
the following integral</p>
<p>$$\int_0^\infty\ \frac{\ln|\cos(x)|}{x^2} dx $$</p>
<p>Thanks.</p>
| Jack D'Aurizio | 44,121 | <p>Lucian's answer is just fine (as always), but from
$$ \sum_{n\in\mathbb{Z}}\frac{1}{(x+n\pi)^2}=\frac{1}{\sin^2 x}\tag{1}$$
for any $x\in(-\pi,\pi)$ it also follows that:
$$ I = \frac{1}{2}\int_{0}^{+\infty}\frac{\log\cos^2 x}{x^2}\,dx = \frac{1}{2}\int_{-\pi/2}^{\pi/2}\frac{\log\cos x}{\sin^2 x}\,dx=-\frac{1}{2}\in... |
201,458 | <p>Consider $W\subseteq V$, a subspace over a field $\mathbb{F}$ and $T:V\rightarrow V$ a linear transformation with the stipulation that $T(W)\subseteq W$. Then we have the induced linear transformation $\overline{T}:V/W \rightarrow V/W$ such that $\overline{T}=T(v)+W$.</p>
<p>I'm supposed to show that this induced ... | Michael Joyce | 17,673 | <p>Hint: Given a linear isomorphism $T : V \rightarrow V'$ of finite-dimensional vector spaces, can you prove that the restricted linear transformation $T|_W : W \rightarrow T(W)$ is an isomorphism?</p>
<p>If $V' = V$ and $T(W) \subseteq W$ as in your problem, can you prove that $T(W) = W$? In that case, you'll have... |
201,458 | <p>Consider $W\subseteq V$, a subspace over a field $\mathbb{F}$ and $T:V\rightarrow V$ a linear transformation with the stipulation that $T(W)\subseteq W$. Then we have the induced linear transformation $\overline{T}:V/W \rightarrow V/W$ such that $\overline{T}=T(v)+W$.</p>
<p>I'm supposed to show that this induced ... | M Turgeon | 19,379 | <p>When you restrict to $W$, you still get a linear transformation -- this much is clear. Since $T$ is injective, its restriction to $W$ is still injective. Therefore, the dimension of the image $T(W)$ is equal to the dimension of $W$. Since $T(W)\subseteq W$, it readily follows that we have an equality. Since $T|_W$ i... |
544,464 | <p>Show that any subset of $\{1, 2, 3, ..., 200\}$ having more than $100$ members must contain at least one pair of integers which add to $201$.</p>
<p>I think it is doable using the Pigeonhole Principle.</p>
| 1233dfv | 102,540 | <p>Consider the subset {$1,2,3,...,100$}. No two integers from this set sum to $201$. However, the very next integer you include in this set gives you the smallest subset with more than $100$ integers in it where two of the integers sum to $201$. Any other subset of the desired size will always have two integers whose ... |
311,677 | <p>The problem from the book. </p>
<blockquote>
<p>$\dfrac{\mathrm{d}y}{\mathrm{d}x} = 6 -y$ </p>
</blockquote>
<p>I understand the solution till this part. </p>
<p>$\ln \vert 6 - y \vert = x + C$ </p>
<p>The solution in the book is $6 - Ce^{-x}$ </p>
<p>My issue this that this solution, from the book, doesn't s... | Iuli | 33,954 | <p>$$\dot{y}(x)=6-y(x)$$
$$\frac{\dot{y}(x)}{6-y(x)}=1$$
$$\int{\frac{\dot{y}(x)}{6-y(x)}}dx=\int{1}dx$$
$$-\ln{|6-y(x)|}=x+c$$
$$y(x)=6-e^{-x-c}.$$</p>
|
71,636 | <p>For a self-map $\varphi:X\longrightarrow X$ of a space $X$, many important notions of entropy are defined through a limit of the form $$\lim_{n\rightarrow\infty}\frac{1}{n}\log a_n,$$ where in each case $a_n$ represents some appropriate quantity (see, for example, <a href="https://mathoverflow.net/questions/69218/if... | Joe Silverman | 11,926 | <p>Let $\phi:\mathbb{P}^N\to\mathbb{P}^N$ be a rational map. The <em>algebraic entropy</em> of $\phi$ is the quantity
$$h_{alg}(\phi) = \limsup_{n\to\infty} \frac{1}{n}\log \deg(\phi^n).$$</p>
<p>Suppose now that $\phi$ is defined over $\overline{\mathbb{Q}}$.
Since you're using $h$ for entropy, I will let $w:\mathbb... |
1,122,926 | <p>Question: The product of monotone sequences is monotone, T or F?</p>
<p>Uncompleted Solution: There are four cases from considering each of two monotone sequences, increasing or decreasing.</p>
<p>CASE I: Suppose we have two monotonically decreasing sequences, say ${\{a_n}\}$ and ${\{b_n}\}$. Then, $a_{n+1}\leq a_... | Mnifldz | 210,719 | <p>In general the answer is no. Take $a_n = \left ( \frac{5}{4} \right )^n$ and $b_n = \frac{1}{n}$. We then have </p>
<p>\begin{eqnarray*}
a_1b_1 & = & \frac{5}{4} \;\; = \;\; 1.25 \\
a_2b_2 & = & \frac{25}{32} \;\; \approx\;\; 0.781 \\
a_3b_3 & = & \frac{125}{192} \;\; \approx \;\; 0.651 \\... |
19,495 | <p>I was told that one of the most efficient tools (e.g. in terms of computations relevant to physics, but also in terms of guessing heuristically mathematical facts) that physicists use is the so called "Feynman path integral", which, as far as I understand, means "integrating" a functional (action... | Allen Knutson | 391 | <p>Here's a relatively recent paper by Jonathan Weitsman: <a href="http://arxiv.org/abs/math/0509104" rel="nofollow">http://arxiv.org/abs/math/0509104</a></p>
<p>He has more recent papers, but I'm not entirely sure that they're following the program he meant to initiate with this paper.</p>
|
19,495 | <p>I was told that one of the most efficient tools (e.g. in terms of computations relevant to physics, but also in terms of guessing heuristically mathematical facts) that physicists use is the so called "Feynman path integral", which, as far as I understand, means "integrating" a functional (action... | Tim van Beek | 1,478 | <p>First, there are several rigorous definitions of integration in infinite dimensional spaces, like the Bochner integral in Banach spaces (see Wikipedia), or see the book by Parthasarathy: "Probability measures on metric spaces" (this includes the Gaussian probability measures used by constructive QFT already mentione... |
1,874,634 | <blockquote>
<p>Corollary (of Schur's Lemma): Every irreducible complex representation of a finite abelian group G is one-dimensional.</p>
</blockquote>
<p>My question is now, why has the group to be abelian? As far as I know, we want the representation $\rho(g)$ to be a $Hom_G(V,V)$, where $V$ is the representation... | H. H. Rugh | 355,946 | <p>Any finite group is isomorphic to a direct product of its irreducible representations, acting on a direct sum of vector spaces. If all irreducible representations are one-dimensional then this faithful representation consists of diagonal matrices which commute. Whence the group is abelian.</p>
|
1,741,765 | <p>Problem description: Show that well-ordering is not a first-order notion. Suppose that $\Gamma$ axiomatizes the class of well-orderings. Add countably many constants $c_i$ and show that $\Gamma \cup \{c_{i+1} < c_i \mid i \in \mathcal{N}\}$ has a model. </p>
<p>So, I don’t quite get how to go about this. <a href... | rschwieb | 29,335 | <p>Let's forget the hashes and just write with juxtaposition.</p>
<p>From $aba=b$ we get $aa=abab=bb$. Call the value that everything squares to "$e$". It is an identity: $ae=aaa=ea=a$ for all $a$, using the hypothesis. So it is at least a monoid.</p>
<p>Further, $aa=aea=e$, so every element is its own inverse.</p>... |
70,946 | <p>I'm an REU student who has just recently been thrown into a dynamical system problem without basically any background in the subject. My project advisor has told me that I should represent regions of my dynamical system by letters and look at the sequence of letters formed by the trajectory of a point under the iter... | Sam Nead | 1,650 | <p>I am not an expert in this topic. I would guess that "minimality" is required to make anything work. For example, take any system with any labelling (also called a partition, or Markov partition if it satisfies various properties). Take any point $x_0$. Let $x_i$ be the $i$-th iterate of $x_0$. Attach a blob $B... |
2,771,240 | <p>Let $\mathbb F$ be a field and $\mathbb K $ be an extension field of $\mathbb F$ such that $\mathbb K$ is algebraically closed. </p>
<p>Let $\mathbb L$ be the field of all elements of $\mathbb K$ which are algebraic over $\mathbb F$. Then $\mathbb L_{|\mathbb F}$ is an algebraic extension. </p>
<p>My question is... | angryavian | 43,949 | <p>Let $m_n = \lfloor n/T \rfloor$.</p>
<p>\begin{align}
\frac{1}{n} \int_0^n f(x) \, dx
&= \frac{1}{n} \int_0^{m_n T} f(x) \, dx + \frac{1}{n} \int_{m_n T}^n f(x) \, dx
\\
&= \frac{m_n T}{n} \cdot \left(\frac{1}{T} \int_0^T f(x) \, dx\right)
+ \frac{1}{n} \int_{m_n T}^n f(x) \, dx.
\end{align}</p>
<p>The sec... |
2,821,323 | <blockquote>
<p>How to show that a rational polynomial is irreducible in $\mathbb{Q}[a,b,c]$? For example, I try to show this polynomial $$p(a,b,c)=a(a+c)(a+b)+b(b+c)(b+a)+c(c+a)(c+b)-4(a+b)(a+c)(b+c)(*)$$ is irreducible, where $a,b,c\in \mathbb{Q}$.</p>
</blockquote>
<p>The related problem is <a href="https://math.... | Batominovski | 72,152 | <p>Suppose contrary that $p(a,b,c)$ is reducible over $\mathbb{Q}$. You can write $p(a,b,c)$ as
$$a^3+b^3+c^3-3(b+c)a^2-3(c+a)b^2-3(a+b)c^2-5abc\,.$$
It suffices to regard $p(a,b,c)$ as a polynomial over $\mathbb{F}_3$ (why?). Over $\mathbb{F}_3$,
$$p(a,b,c)=a^3+b^3+c^3+abc=a^3+(bc)a+(b+c)^3\,.$$
Since $p(a,b,c)$ is... |
121,865 | <p>Can someone please help me clarify the notations/definitions below:</p>
<p>Does $\{0,1\}^n$ mean a $n$-length vector consisting of $0$s and/or $1$s?</p>
<p>Does $[0,1]^n$ ($(0,1)^n$) mean a $n$-length vector consisting of any number between $0$ and $1$ inclusive (exclusive)?</p>
<p>As a related question, is there... | Alex Becker | 8,173 | <p>The notation $\{0,1\}^n$ refers to the <em>space</em> of all $n$-length vectors consisting of $0$s and $1$s, while the notation $[0,1]^n$ ($(0,1)^n$) refers to the space of all $n$-length vectors consisting of real numbers between $0$ and $1$ inclusive (exclusive).</p>
<p>Edit: I often find wikipedia's <a href="htt... |
3,208,412 | <p>I have to prove the following:</p>
<p><span class="math-container">$$ \sqrt{x_1} + \sqrt{x_2} +...+\sqrt{x_n} \ge \sqrt{x_1 + x_2 + ... + x_n}$$</span></p>
<p>For every <span class="math-container">$n \ge 2$</span> and <span class="math-container">$x_1, x_2, ..., x_n \in \Bbb N$</span></p>
<p>Here's my attempt:<... | Martin Pekár | 658,253 | <p>I'm will just show it my way, since I find it simpler.</p>
<p>First, we will start off with the basis step where <span class="math-container">$n = 1$</span>.</p>
<h2>Basis step</h2>
<p><span class="math-container">$\sqrt{x_1} \geq \sqrt{x_1}$</span>, which is of course true.</p>
<h2>Inductive step</h2>
<p>We let the... |
1,392,661 | <p>For a National Board Exam Review: </p>
<blockquote>
<p>Find the equation of the perpendicular bisector of the line joining
(4,0) and (-6, -3)</p>
</blockquote>
<p>Answer is 20x + 6y + 29 = 0</p>
<p>I dont know where I went wrong. This is supposed to be very easy:</p>
<p>Find slope between two points:</p>
<p... | Harish Chandra Rajpoot | 210,295 | <p>Notice, the mid=point of the line joining $(4, 0)$ & $(-6, -3)$ is given as $$\left(\frac{4+(-6)}{2}, \frac{0+(-3)}{2}\right)\equiv \left(-1, -\frac{3}{2}\right)$$ The slope of the perpendicular bisector
$$=\frac{-1}{\text{slope of line joining}\ (4, 0)\ \text{&}\ (-6, -3)}$$
$$=\frac{-1}{\frac{-3-0}{-6-4... |
2,961,686 | <p>Consider a matrix <span class="math-container">$A$</span> which we subject to a small perturbation <span class="math-container">$\partial A$</span>. If <span class="math-container">$\partial A$</span> is small, then we have <span class="math-container">$(A + \partial A)^{-1} \approx A^{-1} - A^{-1} \partial A A^{... | user1551 | 1,551 | <p>The usual argument is that, if you perturb <span class="math-container">$A$</span> by a small <span class="math-container">$X$</span> and get <span class="math-container">$(A+X)^{-1}=A^{-1}+Y+O(\|X\|^2)$</span>, where <span class="math-container">$Y$</span> is the first-order (i.e. linear) change in <span class="mat... |
1,356,900 | <p>For section 1 on Fields, there is a question 2c:</p>
<p>2.</p>
<p>a) Is the set of all positive integers a field?</p>
<p>b) What about the set of all integers?</p>
<p>c) Can the answers to both these question be changed by re-defining addition or multiplication (or both)?</p>
<p>My answer initially to 2c was th... | jeo15 | 252,647 | <p>I just want to answer one of your questions. If we have a map, say g, such that $g:\mathbb{F}\times \mathbb{F}\rightarrow \mathbb{F}$ this means that g is a function with domain in $\mathbb{F} \times \mathbb{F}$ with its codomain in $\mathbb{F}$. Here, $\mathbb{F}$ means the field.</p>
<p>In other words, if $\mathb... |
661,026 | <p>prove or disprove this
$$\sum_{k=0}^{n}\binom{n}{k}^3\approx\dfrac{2}{\pi\sqrt{3}n}\cdot 8^n,n\to\infty?$$</p>
<p>this problem is from when Find this limit
$$\lim_{n\to\infty}\dfrac{\displaystyle\sum_{k=0}^{n}\binom{n}{k}^3}{\displaystyle\sum_{k=0}^{n+1}\binom{n+1}{k}^3}=\dfrac{1}{8}?$$</p>
<p>first,follow I c... | André Nicolas | 6,312 | <p>We can do it by looking at the expression for different ranges of $x$. It is clearly positive if $x\le 0$. </p>
<p>If $0 < x \le 4$, then $x^3+x\lt 100$, so our expression is positive even without the aid of $\frac{1}{2}x^4$. If $x\gt 4$, then $\frac{1}{2}x^4\gt 2x^3$, so $\frac{1}{2}x^4-x^3-x\gt 0$.</p>
|
3,043,780 | <p><a href="https://i.stack.imgur.com/h1M7D.png" rel="nofollow noreferrer">the image shows right-angled triangles in semi-circle</a></p>
<p>In Definite Integration, we know that area can be found by adding up the total area of each small divided parts.</p>
<p>So, base on the Definite Integration, we may say the area ... | Shubham Johri | 551,962 | <p>If the point is in the <span class="math-container">$3^{rd}$</span> quadrant, the angle it makes with the <span class="math-container">$x$</span> axis in the anti-clockwise direction is <span class="math-container">$180+\arctan(\Big|\frac yx\Big|)$</span></p>
|
787,358 | <p>Consider the equation: ay'' +by'+cy=0</p>
<p>If the roots of the corresponding characteristic equation are real, show that a solution to the differential equation either is everywhere zero or else can take on the value zero at most once.</p>
<p>hmm I have no idea how to do this one, I think it might have to do som... | user314551 | 314,551 | <ol>
<li>Consider the linear 2nd order ODE
ay′′ + by′ + cy = 0. (1) </li>
</ol>
<p>Since the equation is linear, with constant coefficients, and no terms not involving y (i.e. the
right hand side is 0), a logical solution to try is</p>
<p>y(t) = ce^(rt), (2)</p>
<p>where c and r are unknown constants.</p>
<p>(a) Wh... |
1,115,222 | <blockquote>
<p>Suppose <span class="math-container">$f$</span> is a continuous, strictly increasing function defined on a closed interval <span class="math-container">$[a,b]$</span> such that <span class="math-container">$f^{-1}$</span> is the inverse function of <span class="math-container">$f$</span>. Prove that,
... | Julián Aguirre | 4,791 | <p>Let $\{x_0,x_1,\dots,x_N\}$ be a partition of $[a,b]$. Then $\{f(x_0),f(x_1),\dots,f(x_N)\}$ is a partition of $[f(a),f(b)]$. The following equality holds:
$$
\sum_{i=0}^{N-1}f(x_i)(x_{i+1}-x_i)+\sum_{i=0}^{N-1}x_i(f(x_{i+1})-f(x_i))+\sum_{i=0}^{N-1}(x_{i+1}-x_i)(f(x_{i+1})-f(x_i))=b\,f(b)-a\,f(a).
$$
The first two ... |
2,324,850 | <p>How to find the shortest distance from line to parabola?</p>
<p>parabola: $$2x^2-4xy+2y^2-x-y=0$$and the line is: $$9x-7y+16=0$$
Already tried use this formula for distance:
$$\frac{|ax_{0}+by_{0}+c|}{\sqrt{a^2+b^2}}$$</p>
| Michael Rozenberg | 190,319 | <p>We need to find the minimum of $$\frac{|9x-7y+16|}{\sqrt{9^2+7^2}},$$
where $x+y=2(x-y)^2$ or
$$\min\frac{|(9x-7y)(x+y)+32(x-y)^2|}{2\sqrt{130}(x-y)^2}$$ or
$$\min\frac{|41x^2-62xy+25y^2|}{2\sqrt{130}(x-y)^2}$$ or
$$\min\frac{41x^2-62xy+25y^2}{2\sqrt{130}(x-y)^2},$$</p>
<p>which is $\frac{8}{\sqrt{130}}$ because
$$... |
2,324,850 | <p>How to find the shortest distance from line to parabola?</p>
<p>parabola: $$2x^2-4xy+2y^2-x-y=0$$and the line is: $$9x-7y+16=0$$
Already tried use this formula for distance:
$$\frac{|ax_{0}+by_{0}+c|}{\sqrt{a^2+b^2}}$$</p>
| hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>The parametric equations are</p>
<p>for parabola
$$2 (x-y)^2=x+y $$
$$x-y=t $$
$$x+y=2t^2$$
thus</p>
<blockquote>
<p>$$x=t^2+t/2 \;,\;y=t^2-t/2$$</p>
</blockquote>
<p>the distance from a point of parabola to the line is </p>
<p>$$D=\frac {|9 (t^2+t/2)-7 (t^2-t/2)+16|}{\sqrt{81+... |
217,291 | <p>I am trying to recreate the following image in latex (pgfplots), but in order to do so I need to figure out the mathematical expressions for the functions</p>
<p><img src="https://i.stack.imgur.com/jYGNP.png" alt="wavepacket"></p>
<p>So far I am sure that the gray line is $\sin x$, and that
the redline is some ver... | Yong Hao Ng | 31,788 | <p>Disclaimer: This is a long post and I am not originally a Mathematician. I just wanted to offer my viewpoint as someone who is not just in the process of learning it, but also had prior experience in its application. </p>
<p><strong>What you might be able to appreciate from an introductory course</strong><br>
Pers... |
217,291 | <p>I am trying to recreate the following image in latex (pgfplots), but in order to do so I need to figure out the mathematical expressions for the functions</p>
<p><img src="https://i.stack.imgur.com/jYGNP.png" alt="wavepacket"></p>
<p>So far I am sure that the gray line is $\sin x$, and that
the redline is some ver... | kjetil b halvorsen | 32,967 | <p>I just found the following book:
<a href="http://rads.stackoverflow.com/amzn/click/0521457181" rel="nofollow">http://www.amazon.com/Fourier-Analysis-Applications-Mathematical-Society/dp/0521457181/ref=sr_1_1?s=books&ie=UTF8&qid=1350785755&sr=1-1&keywords=audrey+terras</a></p>
<p>which are written ju... |
453,295 | <p>I wanna show that the non-zero elements of $\mathbb Z_p$ ($p$ prime) form a group of order $p-1$ under multiplication, i.e., the elements of this group are $\{\overline1,\ldots,\overline{p-1}\}$. I'm trying to prove that every element is invertible in the following manner:</p>
<blockquote>
<p><strong>Proof (a)</s... | DonAntonio | 31,254 | <p>Don't talk about rings if you don't want to, but you still must talk about the <em>additive abelian group</em> $\,\Bbb Z_p\,$ and you can work with it all along, remarking</p>
<p>$$\forall\,k\in\Bbb Z\;,\;\;kp=\overline 0=0\pmod p$$</p>
<p>Then you can work comfortably with Bezout's lemma as you did, including $\... |
1,511,078 | <p><strong>Show that the product of two upper (lower) triangular matrices is again upper (lower) triangular.</strong></p>
<p>I have problems in formulating proofs - although I am not 100% sure if this text requires one, as it uses the verb "show" instead of "prove". However, I have found on the internet the proof belo... | Hagen von Eitzen | 39,174 | <p>$n\times n$ matrices are (or at least readily can be viewed) as linear maps from $n$-dimensional space to itself.</p>
<p>An upper triangular matrix is one where each vector from the standard basis is mapped to a linear combination of itself and the <em>preceeding</em> base vectors alone. In other words: The subspac... |
292,831 | <p>Usually the question whether the <a href="https://en.wikipedia.org/wiki/Diamond_principle" rel="noreferrer">diamond principle</a> $\diamondsuit(\kappa)$ holds for some large cardinal $\kappa$ only concerns large cardinal notions of very low consistency (among the weakly compacts). Partly since it <em>does</em> hold ... | Not Mike | 8,843 | <p>Assuming that by "inaccessible Jónsson" you meant a regular limit cardinal of uncountable cofinality which is Jónsson; then using the arguments of [1] (Theorem 15 p.115), we have</p>
<blockquote>
<p>if $\mathbb{P}$ is c.c.c. and $\kappa$ is Jónsson then for any $V$-generic $G\subset \mathbb{P}$, $V[G]\vDash$ "$\k... |
4,259,561 | <p>I am looking for the derivation of the closed form along any given diagonal <span class="math-container">$a$</span> of Pascal's triangle,<br />
<span class="math-container">$$\sum_{k=a}^n {k\choose a}\frac{1}{2^k}=?$$</span>
Numbered observations follow. As for the limit proposed in the title given by:</p>
<p><stron... | MXXZ | 966,405 | <p><strong>Edit 1:</strong> I misinterpreted the question to mean that (at least) <span class="math-container">$2$</span> sit next to each other, so the following solution is incorrect. I'll leave it if anyone is interested in that case.</p>
<p><strong>Edit 2:</strong> Even in that situation it wouldn't work (see the c... |
4,076,006 | <p>I would like to know the number of valuation rings of <span class="math-container">$\Bbb Q_p((T))$</span>.
I know <span class="math-container">$\Bbb Q_p$</span> has <span class="math-container">$2$</span> valuation rings, that is,<span class="math-container">$\Bbb Q_p$</span> and <span class="math-container">$\Bbb ... | reuns | 276,986 | <p>There is only one (non-trivial) <strong>discrete</strong> valuation on <span class="math-container">$\Bbb{Q}_p((T))$</span>.</p>
<p>For all <span class="math-container">$f\in 1+p\Bbb{Z}_p+T \Bbb{Q}_p[[T]]$</span> the binomial series gives that <span class="math-container">$f^{1/n}\in \Bbb{Q}_p((T))$</span> whenever ... |
763,199 | <p>I am trying to understand more about the Bidualspace (or double dual space). The whole idea is that $V$ and $V^{**}$ are canonically isomorphic to one another, <s>which means that they are isomorphic without the choice of a basis</s>, which means there exists an isomorphism between them which <strong>does not</stron... | M Turgeon | 19,379 | <p>The first thing you mention is that, if you fix a basis for $V$, you get a non-canonical isomorphism $V\cong V^*$. Similarly, fixing a basis for $V^*$, you get a non-canonical isomorphism $V^*\cong (V^*)^*=V^{**}$. The "magic" is that, when you compose these two isomorphisms, you get an isomorphism $V\cong V^{**}$ w... |
763,199 | <p>I am trying to understand more about the Bidualspace (or double dual space). The whole idea is that $V$ and $V^{**}$ are canonically isomorphic to one another, <s>which means that they are isomorphic without the choice of a basis</s>, which means there exists an isomorphism between them which <strong>does not</stron... | Community | -1 | <p>The canonical map $\theta$ from a vector space to its double-dual has a particularly clean pointwise forumula:</p>
<p>$$\theta(v)(\omega) = \omega(v)$$</p>
<p>where $v$ is a vector and $\omega$ is a covector (an element of $V^*$).</p>
<p>Since $\omega$ ranges over all covectors, this gives a pointwise definition ... |
1,408,036 | <p>Five points are drawn on the surface of an orange. Prove that it is possible to cut the orange in half in such a way that at least four of the points are on the same hemisphere. (Any points lying along the cut count as being on both hemispheres.)</p>
| davidlowryduda | 9,754 | <p>Through any two points, there is a great circle that passes through those two points. Such a cut will split the other 3 pigeons — <em>oh, I mean points</em> — among 2 halves. </p>
<p>[You can now handle additional points being on the great circle on your own, I believe.]</p>
|
146,813 | <p>Is sigma-additivity (countable additivity) of Lebesgue measure (say on measurable subsets of the real line) deducible from the Zermelo-Fraenkel set theory (without the axiom of choice)?</p>
<p>Note 1. Follow-up question: Jech's 1973 book on the axiom of choice seems to be cited as the source for the Feferman-Levy m... | Asaf Karagila | 7,206 | <p>No, you can't have that. It is consistent that the real numbers are a countable union of countable sets, in which case you immediately have that there is no nontrivial measure which is countably additive on the real numbers.</p>
<p>There are other models, however, in which $\aleph_1$ is singular, the countable unio... |
21,262 | <p><strong>Bug introduced in 9.0 and fixed in 11.1</strong></p>
<hr>
<p><code>NDSolve</code> in Mathematica 9.0.0 (MacOS) is behaving strangely with a piecewise right hand side. The following code (a simplified version of my real problem):</p>
<pre><code>sol = NDSolve[{x'[t] ==
Piecewise[{{2, 0 <= Mod[t... | Albert Retey | 169 | <p>This is not really an answer (the answer is of course that this is a bug), but it is too long for a comment and probably gives a hint where the problem is and how one can avoid it in other cases. The workaround is to define the piecewise function as an external definition only for numeric arguments. It looks like ot... |
21,262 | <p><strong>Bug introduced in 9.0 and fixed in 11.1</strong></p>
<hr>
<p><code>NDSolve</code> in Mathematica 9.0.0 (MacOS) is behaving strangely with a piecewise right hand side. The following code (a simplified version of my real problem):</p>
<pre><code>sol = NDSolve[{x'[t] ==
Piecewise[{{2, 0 <= Mod[t... | xzczd | 1,871 | <p>Another possible fix to the bug is to use <code>Simplify`PWToUnitStep</code> to expand the <code>Piecewise</code> into a combination of <code>UnitStep</code>:</p>
<pre><code>Table[NDSolveValue[{x'[t] ==
Simplify`PWToUnitStep@
Piecewise[{{2, 0 <= Mod[t, 1] < 0.5}, {-1, 0.5 <= Mod[t, 1] < 1}... |
206,723 | <p>Can any one explain why the probability that an integer is divisible by a prime $p$ (or any integer) is $1/p$?</p>
| Steven Stadnicki | 785 | <p>As I said in a comment, the notion of 'probability' over the set of all integers (or equivalently, the natural numbers) is fraught with some peril. A better statement of the question is that the <em>natural density</em> of the numbers divisible by $p$ is $\frac{1}{p}$. Natural density captures what people think of... |
948,329 | <p>I have come across this trig identity and I want to understand how it was derived. I have never seen it before, nor have I seen it in any of the online resources including the many trig identity cheat sheets that can be found on the internet.</p>
<p>$A\cdot\sin(\theta) + B\cdot\cos(\theta) = C\cdot\sin(\theta + \Ph... | beep-boop | 127,192 | <p>We can write
$$A\sin(\theta)+B\cos(\theta)$$ in the form $C \sin(\theta+\phi)$ for some $\phi$ and $C$.</p>
<p>i.e. $$A\sin(\theta)+B\cos(\theta) \equiv C\sin(\theta+\phi).$$</p>
<p>Let's expand the RHS using the addition identity for sine.</p>
<p>$$A\sin(\theta)+B\cos(\theta) \equiv C\underbrace{[\sin(\theta)\c... |
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