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 |
|---|---|---|---|---|
4,615,947 | <p>Let <span class="math-container">$a,b\in\Bbb{N}^*$</span> such that <span class="math-container">$\gcd(a,b)=1$</span>. How to show that <span class="math-container">$\gcd(ab,a^2+b^2)=1$</span>?</p>
| Claude Leibovici | 82,404 | <p><em>Using algebra</em></p>
<p>Consider the more general case of
<span class="math-container">$$I=\int_{-\infty}^{+\infty} \frac{\sin(t)}{(t-a)(t-b)} \, dt$$</span> where <span class="math-container">$(a,b)$</span> are complex numbers.</p>
<p>Using partial fraction decomposition
<span class="math-container">$$I=\frac... |
1,038,713 | <p>Suppose I am given a circle $C$ in $\Bbb C^*$ and two points $w_1,w_2$. Given another circle $C'$ and points $z_1,z_2$, what is the procedure to find a Möbius transformation that sends $C\to C'$, $w_i\to z_i,i=1,2$? Here $z_1\in C\not\ni z_2$; $w_1\in C'\not\ni w_2$. For example, take $|z|=2$, $w_1=-2,w_2=0$. Then,... | Christian Blatter | 1,303 | <p>Each instance of your problem involves a lot of data, whereby you haven't even indicated in which format the two circles are given. It follows that there is no simple "one formula suits it all" solution. Given an instance of the problem one can reduce it to the special case $z_1=w_1=0$, $z_2=w_2=\infty$. The Moebiu... |
1,382,374 | <p>This is an exercise from Rudin's <em>Principles of Mathematical Analysis</em>, Chapter $6$.</p>
<blockquote>
<p>Suppose $f$ is a real, continuously differentiable function on $[a, b]$, $f(a) = f(b) = 0$, and
$$\int_a^b f^2(x)\, dx = 1.$$
Prove that
$$\int_a^b xf(x)f'(x)\,dx = -\frac{1}{2}$$
and that
$$... | karmalu | 258,239 | <p>The differential equalities gives $f(x)=e^{\lambda \frac{x^2}{2}+c}$ which is never equal to 0 or f constant which can't satisfy your requests.</p>
|
3,736,580 | <p>Show that for <span class="math-container">$n>3$</span>, there is always a <span class="math-container">$2$</span>-regular graph on <span class="math-container">$n$</span> vertices. For what values of <span class="math-container">$n>4$</span> will there be a 3-regular graph on n vertices?</p>
<p>I think this q... | Rhys Hughes | 487,658 | <p>Such a sequence clearly exists, for example I could say:</p>
<p><span class="math-container">$$\frac{1}{7}(4+3+3+3+3+3+3)=\frac{22}{7}\approx\pi$$</span></p>
<p>and, continuing such a process towards an infinite number of terms, there exists a configuration that can get as close to <span class="math-container">$\pi$... |
2,057,813 | <p>How can I prove that for $s \in \mathbb{C}$, with real part of $s$ being equal to 1,
\begin{equation}
\sum_{n=1}^{\infty}\frac{1}{n^{s}}
\end{equation}
diverges?</p>
<p>Thanks a lot!</p>
| vidyarthi | 349,094 | <p>Note that the Riemann Zeta function diverges only for <span class="math-container">$s=1$</span>. For <span class="math-container">$s\neq 1$</span>, however, the function is convergent.As for divergence at <span class="math-container">$s=1$</span>, there are many proofs extant, one of which is the <a href="https://en... |
927,188 | <p>This question has been on my mind for a very long time, and I thought I'd finally ask it here. </p>
<p>When I was 6, my dad pulled me out of school. The classes were too easy; the professors, too dull. My father had been man of philosophy his entire life (almost got a PhD in it) and regretted not having a more q... | user4951 | 22,644 | <p>This would be my personal answer. I have medals in international physics olympiad. What do I do now?</p>
<p>I am a businessman.</p>
<p>Yes. I love Math. I love money and women more.</p>
<p>No I will not be a great mathematician or physicist. Why would I?</p>
<p>I still love Math. I read calculus, accounting. But... |
927,188 | <p>This question has been on my mind for a very long time, and I thought I'd finally ask it here. </p>
<p>When I was 6, my dad pulled me out of school. The classes were too easy; the professors, too dull. My father had been man of philosophy his entire life (almost got a PhD in it) and regretted not having a more q... | Masacroso | 173,262 | <p>Do whatever you want to do... you dont need to be "great" or compete with anyone... LOL, it is obvious you are from USA. Just be happy all that you can as you want to do.</p>
<p>Unfortunately nobody can help you very much on this kind of personal questions. You cant know if your decisions will be fine or not, sorry... |
1,216,392 | <blockquote>
<p>$P,Q$ are polynomials with real coefficients and for every real $x$ satisfy $P(P(P(x)))=Q(Q(Q(x)))$. Prove that $P=Q$.</p>
</blockquote>
<p>I see only that these polynomials are same degree</p>
| Hagen von Eitzen | 39,174 | <p>Since one easily messes things up when starting from highest coefficients downward (as I did in my first revision of this answer), we shall consider the function $f(z)=\frac1{P(z^{-1})}$, which allows us nicer acccess to the high coefficients of $P$.
As a matter of notation, write $$f^{\circ k}:=\underbrace{f\circ\l... |
2,227,280 | <p>For every positive number there exists a corresponding negative number. Would that imply that the number of positive numbers is "equal" to the number of negative numbers? (Are they incomparable because they both approach infinity?)</p>
| P Vanchinathan | 28,915 | <p>If there is SOME function that gives a bijection between two sets, then these two sets are considered equally big. (Even if there is some other function between those two sets that is not onto/not 1-1). For example the set of multiples of 10 among positive integers being a proper subset of all positive integers seem... |
69,378 | <p>Updated Question : How to show that in TH we never reach a state where there are no paths to the solution? ( without reversing moves, as if reversing is allowed this becomes trivial )</p>
<p>Edit : Thanks to <strong>Stéphane Gimenez</strong> for pointing out the distinction between “A deadlock would never occur” an... | Henry | 6,460 | <p>There cannot be deadlock in the Towers of Hanoi, as you almost always have three moves: you can move the smallest disk to one of the other two pegs, and unless all the disks are on the same peg you can always move another disk.</p>
<p>There are many ways of proving that any Towers of Hanoi position is solvable. On... |
2,409,268 | <p><strong>Confirm that the identity $1+z+...+z^n=(1-z^{n+1})/(1-z)$ holds for every non-negaive integer $n$ and every complex number $z$, save for $z=1$</strong></p>
<p>I have tried to prove this by induction but I am not sure that I am doing things right, for $ n = 1 $ we have $ (1-z ^ 2) / (1-z) = (1-z) (1+ z) / (1... | hamam_Abdallah | 369,188 | <p>By induction,</p>
<p>Assume that
$$1+z+z^2+...+z^n=\frac {1-z^{n+1}}{1-z} $$</p>
<p>then
$$1+z+...+z^n+z^{n+1}=$$
$$\frac {1-z^{n+1}}{1-z}+z^{n+1}= $$
$$\frac {1-z^{n+1}+z^{n+1}-z^{n+2}}{1-z} $$</p>
<p>Done.</p>
|
2,621 | <p>Let $A$ be a commutative Banach algebra with unit.
It is well known that if the Gelfand transform $\hat{x}$ of $x\in A$ is non-zero, then $x$ is invertible in $A$ (the so called Wiener Lemma in the case when $A$ is the Banach algebra of absolutely convergent Fourier series).</p>
<p>As a converse of the above, let ... | John Channing | 1,032 | <p>Monte Carlo methods are used extensively in <a href="http://en.wikipedia.org/wiki/Monte_Carlo_methods_in_finance" rel="nofollow">financial mathematics</a> for the pricing of complex or "exotic" financial derivatives. With equity options, for example, the value of the stocks in the option contract is simulated using... |
1,344,284 | <p>if $Z=X+iy$ then determine the locus of the equation $\left | 2Z-1 \right | = \left | Z-2 \right |$.I can tell that it a circle equation and it is $x^2 + y^2 = 1$.There are a lot of equation in my book such as $\left | Z-8 \right | +\left | Z+8 \right |=20$,$\left | Z-2 \right | = \left | Z-3i \right |$,$\left | 2Z... | Doppelschwert | 251,434 | <p>Part 1 is covered by
<a href="https://math.stackexchange.com/questions/1344573/a-theorem-of-symmetric-positive-definite-matrix">A theorem of symmetric positive definite matrix.</a></p>
<p>Part 2 should follow if you consider $A+\epsilon I$ and let $\epsilon$ go toward 0.</p>
|
776,627 | <p>How many integer factors of 0 are there, and what are they?</p>
<p>I'm just curious, but what counts as a factor of 0? My guess is that there are an infinite number of factors of 0, but is there a proof?</p>
| user138710 | 138,710 | <p>Take any non-zero k, it can be written 0=k.0</p>
|
776,627 | <p>How many integer factors of 0 are there, and what are they?</p>
<p>I'm just curious, but what counts as a factor of 0? My guess is that there are an infinite number of factors of 0, but is there a proof?</p>
| David | 119,775 | <p>The statement</p>
<blockquote>
<p>$a$ is a factor of $b$</p>
</blockquote>
<p>means</p>
<blockquote>
<p>$b=ka$ for some integer $k$.</p>
</blockquote>
<p>Take $b=0$: then no matter what $a$ is, the equation $0=ka$ is always true for some value of $k$, namely, $k=0$. So every integer is a factor of $0$.
<hr>... |
776,627 | <p>How many integer factors of 0 are there, and what are they?</p>
<p>I'm just curious, but what counts as a factor of 0? My guess is that there are an infinite number of factors of 0, but is there a proof?</p>
| Marc van Leeuwen | 18,880 | <p>This question is about terminology, and one can only give an answer if you provide the precise definition of "factor" you are using. Curiously the term does not appear to often get a formal definition, contrary to "divisor" which the term more conventionally associated to the divisibility relation. One may then for ... |
27,455 | <p>Let $(f_n)_{n \geq 1}$ be disjointly supported sequence of functions in $L^\infty(0,1)$. Is the space $\overline{\mathrm{span}(f_n)}$ (the closure of linear span) complemented in $L^\infty(0,1)$? By complemented we mean that $L^\infty(0,1) = \overline{\mathrm{span}(f_n)} \oplus X$, where $X$ is a subspace of $L^\inf... | Plop | 2,660 | <p>$X$ can be taken to be $\left\{ f \in L^{\infty} | \forall n,\ \int_{[0,1]} f f_n =0 \right\}$</p>
|
2,894,126 | <blockquote>
<p>$$\int \sin^{-1}\sqrt{ \frac{x}{a+x}} dx$$</p>
</blockquote>
<p>We can substitute it as $x=a\tan^2 (\theta)$ . Then:</p>
<p>$$2a\int \theta \tan (\theta)\sec^2 (\theta) d\theta$$</p>
<p>Using integration by parts will be enough here. But I wanted to know if this particular problem can be solved by... | Dr. Sonnhard Graubner | 175,066 | <p>With Integration by parts we get</p>
<p>$$x\arcsin(\sqrt{\frac{x}{a+x}})-\int\frac{1}{2}\sqrt{\frac{ax}{(a+x)^2}}dx$$
now we get by substituting</p>
<p>$$u=\sqrt{x},du=\frac{1}{2\sqrt{x}}dx$$ we get</p>
<p>$$x\arcsin(\sqrt{\frac{x}{a+x}})-\int\frac{\sqrt{a}u^2}{a+u^2}du$$
Write the last Integrand in the form</p>
... |
764,632 | <p>The question is this :</p>
<p>$$\lim_{x\to-\infty} {\sqrt{x^2+x}+\cos x\over x+\sin x}$$</p>
<p>The solution is $-1$ and this seems to be only obtained from the change variable strategy, such as $t=-x$.</p>
<p>However, I have no idea why this isn't just solved by simply eliminating $x$ in numerator and denominato... | evil999man | 102,285 | <p>Note that the trigonometric terms are negligible as $x \to-\infty $. Hence,</p>
<p>$$\lim_{x\to-\infty}\frac{\sqrt{x^2+x}}{x}$$</p>
<p>You cannot take the x in the square root. But, your problem is that x is negative. So, you must : </p>
<p>$$\lim_{x\to-\infty}\frac{\sqrt{x^2+x}}{-(-x)}$$</p>
<p>Now, $-x$ is pos... |
1,873,596 | <p>Near the end of <a href="http://www.maa.org/sites/default/files/pdf/upload_library/2/Rice-2013.pdf" rel="nofollow noreferrer">this MAA piece about elliptic curves</a>, the author explains why the complex domain of the cosine function is a sphere: since it's periodic, its domain can be taken as a cylinder, wrapping u... | Richard Anthony Baum | 356,531 | <p>Trigonometric functions are circular functions. Complex cosine is:
cos z = (e^(iz) + e^(-iz))/2 where z = x + iy where x and y are real numbers and z is a complex number. Suppose we restrict z so modulus z = 1 = modulus (x + iy). Hence, the image of z is the unit circle in the complex plane. What if we ... |
652,025 | <p>Assume $A$, $B$, and $C$ are three independent predicates. Maybe $A$ stands for "my age is 20," and $B$ "stands for tomorrow is a good day."</p>
<p>So is it true that $(A \lor B) \land C \iff (A \lor C) \land (B \lor C)$?</p>
| hmmmm | 18,301 | <p>$ \begin{array}{|c|c|c|c|} A & B & C & (A \bigvee B) \bigwedge C \\ T & T & T & T\\
T & T & F & F \\
T & F & T &T \\ T & F & F & F \\ F & T & T & T \\ F & T & F & F \\ F & F & T & F \\ F & F & F & F \end{array... |
1,516,925 | <p>Let $x,y,z$ be 3 non-zero integers defined as followed: </p>
<p>$$(x+y)(x^2-xy+y^2)=z^3$$</p>
<p>Let assume that $(x+y)$ and $(x^2-xy+y^2)$ are coprime
and set $x+y=r^3$ and $x^2-xy+y^3=s^3$</p>
<p>Can one write that $z=rs$ where $r,s$ are 2 integers?
I am not seeing why not but I want to be sure.</p>
| PM 2Ring | 207,316 | <p>Yes.</p>
<p>If $z^3 = r^3s^3$ we can take cube roots of both sides to get $z = rs$. It's valid to do this because $n$ is the only solution to $({n^3})^{1/3}$ for real $n$. If we go to complex numbers we have to be more careful because cube roots have 3 solutions.</p>
|
4,066,942 | <p>This is a problem from Kenneth A Ross 2nd Edition Elementary Analysis:</p>
<p>Show that the infinite series,<span class="math-container">$$\sum_{n=1}^{\infty} \frac{(-1)^n}{n+x^2}$$</span> converges uniformly for all <span class="math-container">$x$</span>, and by termwise differentiation, compute <span class="math-... | Thomas | 89,516 | <p>Same solution probably written differently and in a more intricate way but it came to me like that before reading MartinR solution:</p>
<p>Suppose wlog <span class="math-container">$x<y$</span> and per absurdum that <span class="math-container">$|f(y)-f(x)|>(b-a)/2$</span>. In such a case we would have:</p>
<p... |
4,050,831 | <p>Suppose 40% of all seniors have a computer at home and a sample of 64 is taken. What is the probability that more than 30 of those in the sample have a computer at home?"</p>
<p>My attempt:</p>
<p>n=64</p>
<p>0.4x64=25.6</p>
<p>p=?</p>
<p>x=??</p>
<p>A>30=??</p>
<p>Don't have an idea of what equation would b... | user2661923 | 464,411 | <p>Given an event <span class="math-container">$E$</span> with probability <span class="math-container">$p$</span> of occurring, where <span class="math-container">$0 < p < 1$</span> <br>
and <span class="math-container">$q = (1-p)$</span>, and given <span class="math-container">$n$</span> independent trials of t... |
101,191 | <p>A few years ago I <a href="http://math.sfsu.edu/federico/Articles/arrangem.pdf">computed</a> the Tutte polynomials of the matroids given by the classical Coxeter groups, and found that their generating functions are all simple variations of the series $\sum_n \frac{x^n y^{n^2}}{n!}$.
I've wondered if there is a mor... | Gjergji Zaimi | 2,384 | <p>Here is an attempt at a "soft" answer inspired by a paper that appeared after this question was asked here. <a href="http://arxiv.org/abs/1305.6621" rel="nofollow">The arithmetic Tutte polynomials of the classical root systems</a> by Ardila, Castillo, and Henley. In particular this will not contain anything the OP d... |
1,204,566 | <p>I tried asking this on StackOverflow and it was quickly closed for being too broad, so I come here to get the mathematical part nailed down, and then I can do the rest with no help, most likely.</p>
<p>From <a href="http://www.afjarvis.staff.shef.ac.uk/sudoku/sudgroup.html" rel="nofollow">this web page</a>, I learn... | Community | -1 | <p>What about brute forcing ?</p>
<p>You can explore the tree of all possible grids of 81 digits, enforcing the placement constraints. Without constraints, there are 9^81 grids and generating all and checking the constraints is out of question. But checking the constraints as you go filling the grid might not be so un... |
69,137 | <p>Is there any reference for gluing in the context of Morse homology on Hilbert manifolds?</p>
<p>Gluing is pretty standard in Morse homology for finite-dimensional manifolds. Unfortunately, in the infinite-dimensional case the sources I know avoid gluing. For proving that the Morse boundary operator squares to zero ... | Daniel Moskovich | 2,051 | <p>There's quite a bit of literature on gluing theory for instantons and monopoles, which take the place of flow lines of Morse functions in instanton Floer homology and in Seiberg-Witten Floer homology correspondingly. It is indeed quite a bit harder than the finite dimensional case, and yes, it's an active research t... |
1,561,563 | <p>Two circles $\Gamma_1,\Gamma_2$ have centers $O_1,O_2$. Let $\Gamma_1\cap\Gamma_2=A,B$, with $A\neq B$. An arbitrary line through $B$ intersects $\Gamma_1$ at $C$ and $\Gamma_2$ at $D$. The tangents to $\Gamma_1$ at $C$ and to $\Gamma_2$ at $D$ intersect at $M$. Let $N=AM\cap CD$. Let $l$ be a line through $N$ paral... | Jan Eerland | 226,665 | <p>BIG HINT:</p>
<p>$$\int\frac{1}{x+\sqrt{1-x^2}}\space\text{d}x=$$</p>
<hr>
<p>Substitute $x=\sin(u)$ and $\text{d}x=\cos(u)\space\text{d}u$.</p>
<p>Then $\sqrt{1-x^2}=\sqrt{1-\sin^2(u)}=\cos(u)$ and $u=\arcsin(x)$:</p>
<hr>
<p>$$\int\frac{\cos(u)}{\sin(u)+\cos(u)}\space\text{d}u=$$
$$\int\frac{\sec^3(u)}{\sec^... |
1,499,949 | <p>Prove that for all event $A,B$</p>
<p>$P(A\cap B)+P(A\cap \bar B)=P(A)$</p>
<p><strong>My attempt:</strong></p>
<p>Formula: $\color{blue}{P(A\cap B)=P(A)+P(B)-P(A\cup B)}$</p>
<p>$=\overbrace {P(A)+P(B)-P(A\cup B)}^{=P(A\cap B)}+\overbrace {P(A)+P(\bar B)-P(A\cup \bar B}^{=P(A\cap \bar B)})$</p>
<p>$=2P(A)+\un... | GEdgar | 442 | <p>There is no difference. In the same way as I write $e^x$ when $x$ is simple enough, and $\exp(x)$ otherwise, I also write $\sqrt{x}$ when $x$ is simple enough, and $(x)^{1/2}$ otherwise.</p>
<p>For example, I write
$$
\left(\frac{1+\frac{9}{x^2}}{\sin \frac{\pi}{9}+2}\right)^{1/2}
\qquad\text{and not the ugly}\qqu... |
380,452 | <p>A relation R is defined on ordered pairs of integers as follows :</p>
<p>$(x,y) R(u,v)$ if $x<u$ and $y>v.$ </p>
<p>Then R is </p>
<ol>
<li><p>Neither a Partial Order nor an Equivalence relation</p></li>
<li><p>A Partial Order but not a Total Order</p></li>
<li><p>A Total Order </p></li>
<li><p>An Equivalen... | Martin Brandenburg | 1,650 | <p>Let $M$ be an $R$-module, $f \in R$ and let $N$ be the colimit of $M \xrightarrow{f} M \xrightarrow{f} \dotsc$. Directed colimits are easy to construct: Elements come from elements of the individual modules, and are identified if they get sent to the same element by some transition map. So in our case, if $i_n : M \... |
3,715,824 | <p>I proved that <span class="math-container">$$\lim_{n\to\infty}\left(1+\frac{x^2}{n^2}\right)^{\frac{n}{2}}=1$$</span>
using L'Hospital's rule. But is there a way to prove it without L'Hospital's rule? I tried splitting it as
<span class="math-container">$$\lim_{n\to\infty}n^{-n}(n^2+x^2)^{\frac{n}{2}},$$</span>
but ... | Mark Viola | 218,419 | <p><strong>METHODOLOGY <span class="math-container">$1$</span>: Direct Application of Bernoulli's Inequality</strong></p>
<p>Note that for <span class="math-container">$n>|x|$</span></p>
<p><span class="math-container">$$1\le \left(1+\frac{x^2}{n^2}\right)^{n/2}\le \frac1{\left(1-\frac{x^2}{n^2}\right)^{n/2}}\le ... |
3,715,824 | <p>I proved that <span class="math-container">$$\lim_{n\to\infty}\left(1+\frac{x^2}{n^2}\right)^{\frac{n}{2}}=1$$</span>
using L'Hospital's rule. But is there a way to prove it without L'Hospital's rule? I tried splitting it as
<span class="math-container">$$\lim_{n\to\infty}n^{-n}(n^2+x^2)^{\frac{n}{2}},$$</span>
but ... | Alex | 38,873 | <p>For the upper bound using Bernoulli inequality note that it applies for exponents <span class="math-container">$t: t \leq 0 \cup t \geq 1$</span>, so for <span class="math-container">$\frac{n}{2} < 0$</span>:
<span class="math-container">$$
\bigg(1+\frac{x^2}{n^2} \bigg)^\frac{n}{2}= \frac{1}{\bigg(1+\frac{x^2}{n... |
3,715,824 | <p>I proved that <span class="math-container">$$\lim_{n\to\infty}\left(1+\frac{x^2}{n^2}\right)^{\frac{n}{2}}=1$$</span>
using L'Hospital's rule. But is there a way to prove it without L'Hospital's rule? I tried splitting it as
<span class="math-container">$$\lim_{n\to\infty}n^{-n}(n^2+x^2)^{\frac{n}{2}},$$</span>
but ... | Paramanand Singh | 72,031 | <p>The <a href="https://math.stackexchange.com/a/1451245/72031">lemma of Thomas Andrews</a> can be used here:</p>
<blockquote>
<p><strong>Lemma</strong>: If <span class="math-container">$n(a_n-1)\to 0$</span> then <span class="math-container">$a_n^n\to 1$</span>.</p>
</blockquote>
<p>Now use this with <span class="... |
2,655,018 | <p>I have a quick question regarding a little issue.</p>
<p>So I'm given a problem that says "$\tan \left(\frac{9\pi}{8}\right)$" and I'm supposed to find the exact value using half angle identities. I know what these identities are $\sin, \cos, \tan$. So, I use the tangent half-angle identity and plug-in $\theta = \f... | Steven Alexis Gregory | 75,410 | <p>Because the period of tangent is $\pi$,
$\tan \dfrac{9 \pi}{8} = \tan \dfrac{\pi}{8}$</p>
<p>You could just look this up, but its pretty easy to derive.</p>
<p>$$ \tan \frac x2
= \frac{\sin \frac x2}{\cos \frac x2}
= \frac{2 \sin \frac x2 \ \cos \frac x2}{1 + 2\cos^2 \frac x2 - 1}
= \frac{\sin x}{1 + \cos... |
3,371,638 | <p>Measure space <span class="math-container">$(X, \mathcal{A}, ν)$</span> has <span class="math-container">$ν(X) = 1$</span>. Let <span class="math-container">$A_n \in \mathcal{A} $</span> and denote </p>
<p><span class="math-container">$B := \{x : x ∈ A_n$</span> for infinitly many n }.</p>
<p>I want to prove that... | Paul Frost | 349,785 | <p>We shall a give a proof which is (hopefully) intuitive but can be made precise if desired.</p>
<p>The "unit circle with its interior" is the set <span class="math-container">$D = \{(x,y) \in \mathbb{R}^2 ; x^2+y^2 \leq 1 \}$</span>. Its boundary is the unit circle <span class="math-container">$C = \{(x,y) \in \math... |
1,085,491 | <p>Prove that the following number is an integer:
$$\left( \dfrac{76}{\dfrac{1}{\sqrt[\large{3}]{77}-\sqrt[\large{3}]{75}}-\sqrt[\large{3}]{5775}}+\dfrac{1}{\dfrac{76}{\sqrt[\large{3}]{77}+\sqrt[\large{3}]{75}}+\sqrt[\large{3}]{5775}}\right)^{\large{3}}$$</p>
<p>How can I prove it?</p>
| Dr. Sonnhard Graubner | 175,066 | <p>we have $$ \left( 76\, \left( \left( \sqrt [3]{77}-\sqrt [3]{75} \right) ^{-1}-
\sqrt [3]{5775} \right) ^{-1}+ \left( 76\, \left( \sqrt [3]{77}+\sqrt
[3]{75} \right) ^{-1}+\sqrt [3]{5775} \right) ^{-1} \right) ^{3}
$$
after expanding we obtain
$$438976\, \left( \left( \sqrt [3]{77}-\sqrt [3]{75} \right) ^{-1}-
\s... |
1,085,491 | <p>Prove that the following number is an integer:
$$\left( \dfrac{76}{\dfrac{1}{\sqrt[\large{3}]{77}-\sqrt[\large{3}]{75}}-\sqrt[\large{3}]{5775}}+\dfrac{1}{\dfrac{76}{\sqrt[\large{3}]{77}+\sqrt[\large{3}]{75}}+\sqrt[\large{3}]{5775}}\right)^{\large{3}}$$</p>
<p>How can I prove it?</p>
| Bernard | 202,857 | <p>The conjugate expression of $\sqrt[3]{a} \pm\sqrt[3]{b}$ is $\sqrt[3]{a^2} \mp\sqrt[3]{ab}+\sqrt[3]{b^2} $. You can use that to rationalise the denominators. The expression inside the parentheses is $\sqrt[3]{77}$ so that finally you get $77$.</p>
|
110,373 | <p>Are there classes of infinite groups that admit Sylow subgroups and where the Sylow theorems are valid?</p>
<p>More precisely, I'm looking for classes of groups <span class="math-container">$\mathcal{C}$</span> with the following properties:</p>
<ul>
<li><span class="math-container">$\mathcal{C}$</span> includes th... | Anton Klyachko | 24,165 | <p>You may also read Chapter 13 of Kurosh's book <a href="https://books.google.ru/books/about/Theory_of_Groups.html?id=B3lHYIuqPuQC&redir_esc=y" rel="nofollow noreferrer">Theory of groups, volume 2</a>.
For instance, it contains a proof of Baer's theorem (<a href="https://mathoverflow.net/a/110375">cited</a> by @Ig... |
656,185 | <blockquote>
<p>let sequence $\{G_{n}\}$ such
$$G_{1}=1,G_{3}=3,G_{2n}=G_{n}$$
$$G_{4n+1}=2G_{2n+1}-G_{n},G_{4n+3}=3G_{2n+1}-2G_{n}$$</p>
</blockquote>
<p>If such $G_{n}=n$, then we said $n$ is 'good'.
How many 'good' numbers $n$, such that $n<2^{100}?$</p>
<p><strong>My try:</strong></p>
<p>since
$$\begin{... | Barry Cipra | 86,747 | <p>This is sequence <a href="http://oeis.org/A030101" rel="nofollow">A030101</a> in the OEIS. That is, $G(n)$ is the number obtained by reversing the digits of $n$ when written base $2$, e.g. $G(25)=G(11001_2)=10011_2=19$.</p>
<p>This is easy to check: If $n=d_0+2d_1+\cdots+2^rd_r$, then</p>
<p>$$
\begin{align}
G(n... |
1,575,397 | <p>I need help calculating
$$\lim_{n\to\infty}\left(\frac{1}{n^{2}}+\frac{2}{n^{2}}+...+\frac{n}{n^{2}}\right) = ?$$</p>
| Jan Eerland | 226,665 | <p>HINT:</p>
<p>$$\lim_{n\to\infty}\left(\frac{1}{n^{2}}+\frac{2}{n^{2}}+...+\frac{n}{n^{2}}\right)=\lim_{n\to\infty}\sum_{m=1}^{n}\frac{m}{n^2}=\lim_{n\to\infty}\frac{n+1}{2n}=\lim_{n\to\infty}\frac{1+\frac{1}{n}}{2}=\frac{1+0}{2}=\frac{1}{2}$$</p>
|
1,526,474 | <p>Find the natural number $k <117$ such that $2^{117}\equiv k \pmod {117}$.</p>
<p>I know $117$ is the product of $3$ and $37$.</p>
<p>$2^{117}\equiv 2 \pmod 3$
$2^{117}\equiv 31 \pmod {37}$.
But $2^{117}\equiv 44 \pmod {117}$.</p>
<p>I can't seem to understand how to get $44$. Can anyone help me understand?</p... | Peter | 82,961 | <p>You can use the chinese remainder theorem </p>
<p>$$2^{117}\equiv \ 2^3 = 8 \ (\ mod\ 9)$$</p>
<p>$$2^{117}\equiv \ 2^9 \equiv 5\ (\ mod\ 13\ )$$</p>
<p>Take it from here.</p>
|
1,991,238 | <p>How can I integrate this? $\int_{0}^{1}\frac{\ln(x)}{x+1} dx $</p>
<p>I've seen <a href="https://math.stackexchange.com/questions/108248/prove-int-01-frac-ln-x-x-1-d-x-sum-1-infty-frac1n2">this</a> but I failed to apply it on my problem.</p>
<p>Could you give some hint?</p>
<p>EDIT : From hint of @H.H.Rugh, I'v... | user361424 | 361,424 | <p>The Fibonacci numbers increase as $\phi^n$ (where $\phi$ is the golden mean $\frac{1+\sqrt{5}}{2}$), and harmonic numbers increase as $\log n$ (i.e., the natural log). Therefore, the difference between the harmonic numbers for successive Fibonacci numbers will approach $\log\phi \approx 0.481211825...$</p>
<p>To e... |
4,292,618 | <p>I have the following function <span class="math-container">$$\frac{1}{1+2x}-\frac{1-x}{1+x} $$</span>
How to find equivalent way to compute it but when <span class="math-container">$x$</span> is much smaller than 1? I assume the problem here is with <span class="math-container">$1+x$</span> since it probably would b... | PierreCarre | 639,238 | <p>I think the purpose of the exercise is simply to provide an alternative, but equivalent, expression for <span class="math-container">$f$</span>. For this reason, I would rule out power series approximations (they do the job quite well but strictly speaking, they are not equivalent to the original expression). If you... |
1,071,040 | <p>I found <a href="https://math.stackexchange.com/questions/549065/how-exactly-do-you-measure-circumference-or-diameter">How exactly do you measure circumference or diameter?</a> but it was more related to how people measured circumference and diameter in old days.</p>
<p><strong>BUT</strong> now we have a formula, b... | slinshady | 194,678 | <p>The correct answer to the question what the circumference of a circle with diameter $d$ would be $\pi \cdot d $. Of course this is not a satisyfing answer. But since this ridiculous number $\pi$ cannot even be described by the root of a polynomial with coefficients in $\mathbb Q$ we can only approximate $\pi$. This ... |
3,091,353 | <p>There are 2 definitions of <strong><em>Connected Space</em></strong> in my lecture notes, I understand the first one but not the second. The first one is:</p>
<blockquote>
<p>A topological space <span class="math-container">$(X,\mathcal{T})$</span> is connected if there does not exist
<span class="math-conta... | postmortes | 65,078 | <p>The second definition is equivalent to the first: suppose there is a set <span class="math-container">$U$</span> which is neither <span class="math-container">$X$</span> nor <span class="math-container">$\emptyset$</span> and is both open and closed. Then <span class="math-container">$U^c$</span>, the complement of... |
3,091,353 | <p>There are 2 definitions of <strong><em>Connected Space</em></strong> in my lecture notes, I understand the first one but not the second. The first one is:</p>
<blockquote>
<p>A topological space <span class="math-container">$(X,\mathcal{T})$</span> is connected if there does not exist
<span class="math-conta... | Mark | 470,733 | <p>First of all the definitions are equivalent, you already got a few answers about that. I'll try to add some intuition. If <span class="math-container">$X$</span> is a topological space and <span class="math-container">$A\subseteq X$</span> then you can split the space into three parts: the interior of <span class="m... |
1,482,104 | <p>Let $X$ have a uniform distribution with p.d.f. $f(x) = 1$, $x$ is in $(0, 1)$, zero elsewhere.
Find the p.d.f. of $Y = -2 \ln X$.</p>
<p>I don't think this is a very difficult question, I just don't really understand what it is asking or where to start. Any help would be very much appreciated. Thank you! </p>
<p>... | alecsphys | 164,662 | <p>you are asked to find the probability distribution of the random variable $Y$ that is related to the random variable $X$ by the relation $Y=-2\ln X$, being $X$ uniform in the interval $[0,1]$. You can solve it considering a change of variable applied to the cumulative function $F(x)$:
$$F(x)=\int_0^x f(x)dx = \int_\... |
749,714 | <p>Does anyone know how to show this preferable <strong>without</strong> using modular</p>
<p>For any prime $p>3$ show that 3 divides $2p^2+1$ </p>
| mathse | 136,490 | <p>Since $p>3$ it holds that $p\equiv 1,2\pmod{3}$. Then $p^2\equiv 1\pmod{3}$ and then $2p^2\equiv 2\pmod{3}$. Adding $1$ yields the result.</p>
|
4,941 | <p>I was reviewing my class notes and found the following:</p>
<p>"The name 'torsion' comes from topology and refers to spaces that are twisted, ex. Möbius band"</p>
<p>In our notes we used the following definition for torsion element and torsion module:
An element m of an R-module M is called a torsion element if $r... | Matt E | 221 | <p>When you compute the homology groups of "twisted" spaces (which are abelian groups), you (sometimes) find that they contain non-zero torsion elements; furthermore, the presence of these particular elements in the homology is due to the twisting (in that, when you compute the homology groups, you see that is the twis... |
2,672,908 | <p>Hey I was given this question in my discrete math class, and I'm unsure of what I should do!</p>
<blockquote>
<p>Prove that if $x$ is coprime with $6$ and $x$ is coprime with $8$, then $x$ is coprime with 24.</p>
</blockquote>
<p>I think I have to use the GCD theorem or co-primality theorem but I don't think wha... | Donald Splutterwit | 404,247 | <p>By Bezout we have
\begin{eqnarray*}
Ax+6B=1 \\
Cx+8D=1.
\end{eqnarray*}
Multiply these equations
\begin{eqnarray*}
x(ACx+6BC+8AD)+24\times 2BD =1.
\end{eqnarray*}</p>
|
4,043,787 | <p>I have <span class="math-container">$$f_n(x)=\begin{cases}
\frac{1}{n} & |x|\leq n, \\
0 & |x|>n .
\end{cases}$$</span></p>
<p>Why cannot be dominated by an integrable function <span class="math-container">$g$</span> by the Dominated Convergence Theorem? I am also wondering what exactly it... | saulspatz | 235,128 | <p><span class="math-container">$f_n\to 0$</span> on <span class="math-container">$\mathbb{R}$</span>, so if the <span class="math-container">$f_n$</span> were dominated by an integrable function, DCT would give <span class="math-container">$$\lim_{n\to\infty}\int_{\mathbb{R}}f_n(x)\,\mathrm{d}x=0$$</span> whereas it's... |
2,763,735 | <p>Is it true that $$\mathbb{Z/4Z\subseteq Z/2Z}$$
Why precisely? Or the reverse $$\mathbb{Z/2Z \subseteq Z/4Z}$$ holds? I'm a beginner. How do I justify the true inclusion?
How do I visualize $$\mathbb{Z/2Z \subseteq Z/4Z}$$
Thank you very much.</p>
| lhf | 589 | <p>$\mathbb{Z}/4\mathbb{Z}\subseteq \mathbb{Z}/2\mathbb{Z}$ cannot be true because $\mathbb{Z}/4\mathbb{Z}$ has $4$ elements but $\mathbb{Z}/2\mathbb{Z}$ has only $2$ elements.</p>
<p>$\mathbb{Z}/2\mathbb{Z}\subseteq \mathbb{Z}/4\mathbb{Z}$ is not true because a class mod $2$ is not a class mod $4$.</p>
<p>Neverthele... |
1,262,036 | <p>In complex analysis, this seems to be a really helpful way to avoid having to expand out Laurent series. I am unclear, however, when it is appropriate to use this property.</p>
<p>In specific, I'm worried I CAN'T use this method on the following:</p>
<p>$$\frac{e^z}{z^3 \sin(z)}$$ at the origin. This looks really ... | Demosthene | 163,662 | <p>Using the <a href="http://en.wikipedia.org/wiki/Residue_%28complex_analysis%29#Limit_formula_for_higher_order_poles" rel="nofollow">limit formula for higher order poles</a>, and the fact that $f(z)=\dfrac{e^z}{z^3\sin z}$ admits an order $4$ pole at $z_0=0$, we get:
$$\mathrm{Res}(f,0)=\dfrac{1}{3!}\lim_{z\to 0}\dfr... |
2,934,906 | <p>This is a shot in the dark and a pretty tall order, but I am wondering if anybody could give a good explanation of the spectral theorem for the reals to high schoolers who have only seen computational calculus of one variable and have <strong>not</strong> taken a linear algebra course before? An overview of the proo... | Rushabh Mehta | 537,349 | <p>I don't really know why you'd want to show this theorem to your students, but find below a decent motivation of why the theorem can be helpful in a simple context, as well as an example of how it's used. I've also linked to a paper that presents a rather simple and elegant proof of the theorem as well, if you would ... |
2,934,906 | <p>This is a shot in the dark and a pretty tall order, but I am wondering if anybody could give a good explanation of the spectral theorem for the reals to high schoolers who have only seen computational calculus of one variable and have <strong>not</strong> taken a linear algebra course before? An overview of the proo... | sasquires | 99,630 | <p>Note that before students can understand the spectral theorem at all, they have to have a solid understanding of what eigenvalues and eigenvectors even are, so I'll start there. </p>
<p>A colleague (who is a smart guy but never took a linear algebra course) recently asked me to explain eigendecomposition to him. ... |
1,891,831 | <p>In linear algebra we have vectors:$$
\mathbf{A}=(x,y,z)=x\mathbf{\hat e}_x+y\mathbf{\hat e}_y+z\mathbf{\hat e}_z$$
We have vector algebra, i.e. <a href="https://en.wikipedia.org/wiki/Euclidean_vector#Basic_properties" rel="nofollow">vector addition, dot product, lines, planes, etc</a>. A vector have a magnitude and ... | Emilio Novati | 187,568 | <p>Multivariable calculus is essentially the study of functions between vector spaces. A function $f: \mathbb{R}^m \to \mathbb{R}^n$ is a function of $m$ variables that represents a field of $n-$dimensional vectors.</p>
|
2,668,447 | <p>Let <span class="math-container">$F$</span> be a subfield of a field <span class="math-container">$K$</span> and let <span class="math-container">$n$</span> be a positive integer. Show that a nonempty linearly-independent subset <span class="math-container">$D$</span> of <span class="math-container">$F^n$</span> rem... | Mariano Suárez-Álvarez | 274 | <p>Hint: express the linear independence as the non-vanishing of some determinant.</p>
|
3,050,497 | <p>The operator is given by
<span class="math-container">$$A=\begin{pmatrix}
1 & 0 & 0\\
1 & 1 & 0\\
0 & 0 & 4
\end{pmatrix}$$</span>
I have to write down the operator <span class="math-container">$$B=\tan(\frac{\pi} {4}A)$$</span>
I calculate <span class="math-container">$$\mathcal{R} (z) =\fra... | TonyK | 1,508 | <p>Hint:</p>
<p><span class="math-container">$$a+b+c+ab+ac+bc+abc=(1+a)(1+b)(1+c)-1$$</span></p>
<p>Therefore <span class="math-container">$(1+a)(1+b)(1+c)=?$</span></p>
|
785,188 | <p>I found a very simple algorithm that draws values from a Poisson distribution from <a href="http://www.akira.ruc.dk/~keld/research/javasimulation/javasimulation-1.1/docs/report.pdf" rel="nofollow">this project.</a></p>
<p>The algorithm's code in Java is:</p>
<pre><code>public final int poisson(double a) {
... | Lightspark | 239,618 | <p>Suppose that we have $k$ independent and identically distributed exponential random variables $X_1, \dotsc, X_k$ with parameter $\lambda$. If we define a counting process $\{N(t)\}_{t \geq 0}$ such that $S_k := X_1 + \dotsb + X_k$ is the occurring time of the $k$-th event, then this is a Poisson process with rate $\... |
628,682 | <p>As both a programmer and a math student, I am trying to come up with a fool-proof way to handle errors from subtractive cancellation caused by trying to evaluate $x-y$, where x,y are extended (long double) precision floating-point numbers. (Obviously, if x is very close to y, this causes problem.) I found two equiva... | Ross Millikan | 1,827 | <p>Once you have $x$ and $y$ and want to subtract them, you won't do better. What is useful is (if $x \approx y$, which is where the problem is) to find some number $a$ that is close to $x,y$ and you can subtract from them analytically. Now you are calculating $(x-a)-(y-a)$. For example, suppose $x=1+10^{-4}$ and $y... |
3,392,749 | <p>I'm trying to draw a dfa for this description</p>
<p>The set of strings over {a, b, c} that do not contain the substring aa,</p>
<p>current issue i'm facing is how many states to start with, any help how to approach this problem?</p>
| Andreas Blass | 48,510 | <p>It seems to me that you can do this with three states, whose "meanings" are:</p>
<p>(1) I haven't seen two consecutive <span class="math-container">$a$</span>'s and either I'm just starting (haven't seen anything yet) or the last symbol I saw was not <span class="math-container">$a$</span>.</p>
<p>(2) I haven't se... |
977,446 | <p>Prove that $A\cap B = \emptyset$ iff $A\subset B^C$. I figured I could start by letting $x$ be an element of the universe and that $x$ is an element of $A$ and not an element of $B$. </p>
| Community | -1 | <p>$\textbf{Claim: } A\cap B=\emptyset$ iff $A\subset B^c$.</p>
<p>$Proof:\; A\cap B=\emptyset \Leftrightarrow (a\in A\Rightarrow a\notin B)\Leftrightarrow (a\in A\Rightarrow a\in B^c)\Leftrightarrow A\subset B^c$</p>
|
3,489,347 | <p><strong>Is there a simple way to characterize the functions in <span class="math-container">$C^\infty((0,1])\cap L^2((0,1])$</span>?</strong></p>
<p>That is, given a function <span class="math-container">$f(t)\in C^\infty((0,1])$</span>, is there a necessary/sufficient condition I can check to see if it's square in... | Community | -1 | <p>First divide by <span class="math-container">$2N$</span> on both sides, </p>
<p><span class="math-container">$5\log N> N$</span> (since <span class="math-container">$N>0$</span> then the inequality stays the same)</p>
<p>Then by raising to the <span class="math-container">$e$</span> power on both sides (the ... |
253,152 | <p>So I was given $f(x)$ continuous and positive on $[0,\infty)$, and need to show that $g(x)$ increasing on $(0,\infty)$</p>
<p>And $g(x)={\int_0^xtf(t)dt\over \int_0^xf(t)dt} $</p>
<p>So my approach is I want to show that $g'(x)>0$, so I used FTC and quotient rule to take the derivative of $g'(x)$, but then I go... | martini | 15,379 | <p>We have
\begin{align*}
g'(x) &= \frac{xf(x)\cdot \int_0^x f(t)\,dt - \int_0^x tf(t)\,dt \cdot f(x)}{(\int_0^x f(t)\, dt)^2}
\end{align*}
Now the denominator is positive, we look at the numerator
\begin{align*}
xf(x)\int_0^x f(t)\,dt - \int_0^x tf(t)\, dt \cdot f(x)
&= \int_0^x xf(x)f(t)\, dt - \int_0^... |
265,537 | <p>I have a set of inequalities</p>
<pre><code>Cos[a]Cos[b]>=Cos[t-a]Cos[b]&&Cos[a]Cos[b]>=Cos[t/2]&&Cos[a]Cos[b]>=Sin[t/2]&&a<=t<=Pi
</code></pre>
<p>How to solve this to get a range of values for <code>a,b,t</code>?</p>
| Ulrich Neumann | 53,677 | <p><code>RegionPlot3D</code>shows the region of possible solutions</p>
<pre><code>RegionPlot3D[Cos[a] Cos[b] >= Cos[t - a] Cos[b] && Cos[a] Cos[b] >= Cos[t/2] &&Cos[a] Cos[b] >= Sin[t/2] && a <= t <= Pi, {a, -Pi/2, Pi}, {t, 0, Pi}, {b, -1, 1}, AxesLabel -> {a, t, b}]
</code></p... |
3,409,598 | <p>Given three equation</p>
<p><span class="math-container">$$\log{(2xy)} = (\log{(x)})(\log{(y)})$$</span>
<span class="math-container">$$\log{(yz)} = (\log{(y)})(\log{(z)})$$</span>
<span class="math-container">$$\log{(2zx)} = (\log{(z)})(\log{(x)})$$</span></p>
<p>Find the real solution of (x, y, z)</p>
<p>What s... | azif00 | 680,927 | <p>Once you have convinced yourself that
<span class="math-container">$$\frac{\partial f(x)}{\partial x_i}=\frac{\partial \|x\|}{\partial x_i}=\frac{x_i}{\|x\|}$$</span>
Then, recall that, the <strong>Hessian</strong> matrix of <span class="math-container">$f$</span> is the <span class="math-container">$n\times n$</sp... |
1,365,489 | <p>What is the value of the following expression?</p>
<p>$$\sqrt[3]{\ 17\sqrt{5}+38} - \sqrt[3]{17\sqrt{5}-38}$$</p>
| Leucippus | 148,155 | <p>Start by noticing that $(2 + \sqrt{5})^{3} = 38 + 17 \sqrt{5}$ and $(2 - \sqrt{5})^{3} = 38 - 17 \sqrt{5}$. Now,
\begin{align}
\sqrt[3]{\ 17\sqrt{5}+38} - \sqrt[3]{17\sqrt{5}-38} &= \sqrt[3]{(2 + \sqrt{5})^{3}} - \sqrt[3]{(\sqrt{5} - 2)^{3}} \\
&= (2 + \sqrt{5}) - (-2 + \sqrt{5}) \\
&= 4.
\end{align}</p... |
3,511,445 | <p>Well it is the problem from jmo odisha. It is of 5 marks
I tried a lot of ways but I can't get the answer. Only elementary mathematics is allowed.</p>
| Robert Israel | 8,508 | <p>An obvious solution is <span class="math-container">$x=y=z=0$</span>, but it's not the only one: you could have
<span class="math-container">$x=48/5$</span>, <span class="math-container">$y = 48/7$</span>, <span class="math-container">$z=48$</span>. </p>
<p>Take <span class="math-container">$(z-6)$</span> times eq... |
1,288,584 | <p>If <span class="math-container">$f(x)=f(x_0)+f'(x_0)(x-x_0)+\ldots+\frac{f^{(n-1)}(x_0)}{(n-1)!}(x-x_0)^{n-1}+\frac{f^{(n)}(\xi(x))}{n!}(x-x_0)^n,$</span> prove that <span class="math-container">$x \rightarrow f^{(n)}(\xi(x)) $</span> is continuous on <span class="math-container">$[x_0-\beta, x_0+\beta]$</span>, if... | Aleksandar | 240,930 | <p>Take the function, $f(x)$ the radius of convergence is the distance between the number you are expanding at in this case $x_0$ and the nearest singularity. If the function has no singularities the disc of convergence is infinite. </p>
<p>Let $\alpha$ be the nearest singularity. The radius of the disc of convergence... |
446,456 | <p>Educators and Professors: when you teach first year calculus students that infinity isn't a number, how would you logically present to them $-\infty < x < +\infty$, where $x$ is a real number?</p>
| Alexey | 48,558 | <p>To make sense out of $-\infty < 1000 < \infty$, for example, you do not need $\infty$ and $-\infty$ to be "numbers," it is enough to extend the definition of "$<$" so that the truth value of $x < y$ was defined for all $x,y\in\mathbb R\sqcup\{\pm\infty\}$. This is easy to do, in an obvious way.</p>
|
446,456 | <p>Educators and Professors: when you teach first year calculus students that infinity isn't a number, how would you logically present to them $-\infty < x < +\infty$, where $x$ is a real number?</p>
| Daniel McLaury | 3,296 | <p>I'd make a couple points:</p>
<p>The word "number," on its own, doesn't really mean anything, aside from "something that somehow resembles something else we're already calling 'numbers.'" Lots of people have called lots of different things "numbers." As a few examples, the modern notion of an ideal in a ring gets... |
1,866,931 | <p>I would like to see a proof to this fact.</p>
<blockquote>
<p>If $A$ is an invertible matrix and $B \in \mathcal{L}(\mathbb{R}^n,\mathbb{R}^n)$, that is an bounded linear opertor in $\mathbb{R}^n$. Then, if there holds
$$
\|B-A\| \|A^{-1}\| <1,
$$
we have that B is invertible.</p>
</blockquote>
<p>Moreov... | egreg | 62,967 | <p>The relation is transitive when</p>
<blockquote>
<p>for all $x,y,z\in A$, if $x\mathrel{R}y$ and $y\mathrel{R}z$, then $x\mathrel{R}z$.</p>
</blockquote>
<p>Thus you have to start with $x,y,z\in A$, such that $x\mathrel{R}y$ and $y\mathrel{R}z$, and prove that $x\mathrel{R}z$.</p>
<p>Now, $x\mathrel{R}y$ and $y... |
889,155 | <blockquote>
<p>There are $2n-1$ slots/boxes in all and two objects say A and B; total number of A's are $n$ and total number of B's are $n-1$. (All A's are identical and all B's are identical.) In how many ways can we arrange A's and B's in $2n-1$ slots.</p>
</blockquote>
<p>My approach: there are $2n-1$ boxes in... | Rebecca J. Stones | 91,818 | <p>Your answer appears correct to me (assuming all $2n-1$ slots need to be filled), but could be simplified:</p>
<p>We choose the positions of the A's, which can be done in $\binom{2n-1}{n}$ ways. The remaining positions all contain B's.</p>
<p>(While it's technically correct, it seems pointless to account for the $... |
889,155 | <blockquote>
<p>There are $2n-1$ slots/boxes in all and two objects say A and B; total number of A's are $n$ and total number of B's are $n-1$. (All A's are identical and all B's are identical.) In how many ways can we arrange A's and B's in $2n-1$ slots.</p>
</blockquote>
<p>My approach: there are $2n-1$ boxes in... | AnonymousMaths | 167,278 | <p>Since A's are identical, and B's are identical, it follows that you do not need to permute the A's among themselves, and B's among themselves, and just need to determine the places in which $n$ A's and hence, $n-1$ B's can be placed.
First, select the $n$ places from $2n-1$ places for the A's in $C(2n-1,n)$ ways. Au... |
990,512 | <p>Suppose that the probability of $x=0$ is $p$, and the probability of $x=1$ is $1-p=q$. Consider the random sequence $X=\{X_i\}_{i=1}^{\infty}$. We map this sequence by $C$ to a point in the interval $[0,1]$ as below:</p>
<p>$1)$ we look at the first random variable. If it is $0$, then we update the interval to $I_1... | Indrajit | 147,241 | <p>Intuitively, we observe that probability of $C(X)$ being inside an interval (for example $[p,1)$) is proportional to the length of the interval.</p>
<p>Notice that $C(X)$ is not well defined for $p=0,1$. So let us assume $0<p<1$. Define new random variables $Y_{i}=p(1-X_{i})+qX_{i}$. Then $Y_{i}=p$ with proba... |
11,609 | <p>What's the code for multiple alignment in MathJAX? An analogous question for Latex is at <a href="https://tex.stackexchange.com/questions/43464/multiple-alignment-in-equations">https://tex.stackexchange.com/questions/43464/multiple-alignment-in-equations</a>, but it doesn't appear to function here. Thank you.</p>
<... | Willie Wong | 1,543 | <p>Another option is to <em>abuse</em> the <code>alignat</code> environment. </p>
<pre><code>$$\newcommand{\myeqa}{=}
\begin{alignat}{5}
A & \myeqa & B \\
& & B & \myeqa & C \\
& & & ... |
373,578 | <p><a href="http://en.wikipedia.org/wiki/Quiver_%28mathematics%29" rel="noreferrer">Quivers</a> are directed graphs where loops and multi-arrows are allowed. And we can talk about representations of quivers by assigning each vertex a vector space and each arrow a homomorphism. Moreover, Gabriel gives <a href="http://en... | Alistair Savage | 74,366 | <p>I would say that one of the reasons (and probably the main motivation for the work of Gabriel that you mention) is that the representation theory of quivers is intimately related to the representation theory of finite-dimensional associative algebras. If $A$ is a finite-dimensional algebra over some field $k$, then... |
885,627 | <p>John invites 12 people to a dinner party, half of which are men. Exactly one man and one woman are bringing desserts. If one person from this group is selected at random,what is the probability that it is a woman, or a man who is not bringing a dessert?</p>
| Roy Sheehan | 79,095 | <p>I could be wrong, but I would have said that 10 people didn't bring deserts so we are looking at a 1/10 probability? </p>
|
885,627 | <p>John invites 12 people to a dinner party, half of which are men. Exactly one man and one woman are bringing desserts. If one person from this group is selected at random,what is the probability that it is a woman, or a man who is not bringing a dessert?</p>
| drhab | 75,923 | <p>There are $6+5=11$ persons of the $12$ that fall in one of the classes: 1) women 2) men that do not bring a desert. So the probability is $\frac{11}{12}$.</p>
<p>(If the first class is meant to be: women that do not bring a desert, then there are $5+5=10$ persons and the probability is $\frac{10}{12}$)</p>
|
4,350,699 | <blockquote>
<blockquote>
<p><span class="math-container">$r:$</span>All prime numbers are either even or odd, Is it a true statement?</p>
</blockquote>
</blockquote>
<p>I was studying Mathematical Logic then i came across above question.
Since here connecting word is "OR"
so if i separate two statement then ... | User5678 | 632,875 | <p>You erroneously distributed the “all”. The correct interpretation is:</p>
<p>For every prime <span class="math-container">$p$</span>: <span class="math-container">$p$</span> is even or <span class="math-container">$p$</span> is odd</p>
<p>Since every integer is either even or odd, we have:</p>
<p>(<span class="math-... |
3,835,514 | <p>For some c > <span class="math-container">$0$</span>, the cumulative distribution function of a continuous random variable X is given by:</p>
<p><span class="math-container">$$
F_X(x) = \begin{cases} 0 & \text{if } x \le0 \\ cx(x+1) & \text{if } 0 \lt x <1 \\ 1 & \text{if } x \ge 1\end{cases}
$$</s... | Siong Thye Goh | 306,553 | <p>You have chosen an example that is not easy to factorize.</p>
<p>By using the quadratic formula which states that the roots of <span class="math-container">$ax^2+bx+c$</span> is <span class="math-container">$\frac{-b \pm \sqrt{b^2-4ac}}{2a}$</span></p>
<p>The roots are</p>
<p><span class="math-container">$$x_1 = \fr... |
2,087,084 | <p>Fix an arbitrary abelian category $\mathscr{A}$, and let $$0\to A\xrightarrow{f}B\xrightarrow{g}C\to 0$$ be a short exact sequence in the category of chains $\mathscr{A}_\bullet$, where $A$, $B$, and $C$ have chain maps $\varphi^A_n:A_n\to A_{n-1}$, $\varphi^B_n:B_n\to B_{n-1}$, $\varphi^C_n:C_n\to C_{n-1}$ respecti... | Alex Mathers | 227,652 | <p>Are you able to use tools such as the long exact homology sequence, or the snake lemma, as Pedro has suggested? If not, this can be done directly. Write out what this short exact of sequences actually looks like:</p>
<p>$$
\newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!}
\newc... |
224,968 | <p>Does the above Diophantine equation have other integer solutions besides $(x,y)=(1,2)$ and $(x, y) = (0, -1)$?</p>
| Igor Rivin | 11,142 | <p>Yes. Here is another one: $x=0, y=-1.$</p>
|
224,968 | <p>Does the above Diophantine equation have other integer solutions besides $(x,y)=(1,2)$ and $(x, y) = (0, -1)$?</p>
| so-called friend Don | 16,510 | <p>Theorem 6.4.30 in Cohen's <em>Number Theory: Volume I</em> asserts: For each nonzero integer $d$, there is at most one pair of integers $(X,Y)$ with $Y\ne 0$ and $X^3+dY^3=1$. Apply this with $X=-y$, $Y=x$, and $d=9$, to see that that there are no more solutions. Theorem 6.4.30 is attributed to Skolem.</p>
|
2,069,573 | <p>I'm trying to solve the following inequality $\dfrac{(\log_2 (8x) \times \log_{x/8} 2)}{\log_{x/2} 16} \leq 0.25$ </p>
<p>Wolfram alpha gives the answer $(0, 0.5], [1,8)$ but surely $x \not= 2$ since log to base $1$ is undefined. But is the fact that it basically shrinks the fraction down to $0$ sufficient to satis... | Community | -1 | <p>You can otherwise write the inequality as: $$\frac{\log_2 8x \times \log_{\frac{x}{8}}2 \times \log \frac{x}{2}}{\log 16} \leq 0.25$$ $$\Rightarrow \frac{\log \frac{x}{2} \times \log 8x}{\log 16 \times \log \frac{x}{8}} \leq 0.25$$ What we just did is to use the logarithmic identity: $\log_{a}b = \frac{\log b}{\log... |
2,069,573 | <p>I'm trying to solve the following inequality $\dfrac{(\log_2 (8x) \times \log_{x/8} 2)}{\log_{x/2} 16} \leq 0.25$ </p>
<p>Wolfram alpha gives the answer $(0, 0.5], [1,8)$ but surely $x \not= 2$ since log to base $1$ is undefined. But is the fact that it basically shrinks the fraction down to $0$ sufficient to satis... | Dr. Sonnhard Graubner | 175,066 | <p>we write your inequality in the form
$$\frac{\frac{\ln(8x)}{\ln(x/8)}}{\frac{\ln(16)}{\ln(x/2)}}\le \frac{1}{4}$$
simplifying this we get
$$\frac{(3\ln(2)+\ln(x))(\ln(x)-\ln(2))}{4\ln(2)(\ln(x)-3\ln(2))}\le \frac{1}{4}$$
simplifying again we get $$\frac{\ln(x)^2+2\ln(2)\ln(x)-3\ln(2)^2}{(\ln(x)-3\ln(2))\ln(2)}\le 1... |
1,448,363 | <p>I have gotten to the next stage where you write it as $\frac{1}{\left(\frac 34\right)}$ to the power of $3$, now I am stuck</p>
<p>I've got it now, thanks everyone.</p>
| Dr. Sonnhard Graubner | 175,066 | <p>it is $(\frac{4}{3})^3=\frac{4^3}{3^3}=\frac{64}{27}$</p>
|
1,242,760 | <p>Now proving by induction is fairly simple. However, this is a multiple choice problem whose answers don't make any sense to me. The actual problem goes as follows:</p>
<p><em>To prove by induction that $n^2 - 7n - 2$ is divisible by $2$ is true for all positive integers $n$, we assume $k^2 - 7k - 2$ is divisible b... | Adhvaitha | 228,265 | <p><strong>HINT</strong>: For the inductive step, we assume that $2$ divides $n^2-7n-2$, i.e., $n^2-7n-2 = 2M$. We now have
\begin{align}
(n+1)^2 - 7(n+1) - 2 & = n^2 + 2n + 1 - 7n - 7 +2 = n^2 - 7n -2 + 2(n-3)\\
& = 2(M+n-3)
\end{align}</p>
|
1,003,379 | <p>I've been working problems all day so maybe I'm just confusing myself but in order to do this. I have to the take the integral along each contour $C_1-C_4$. My issue is how to convert to parametric functions in order to this so that I can integrate</p>
<p><img src="https://i.stack.imgur.com/HWRoM.jpg" alt="enter im... | user2345215 | 131,872 | <p>No because $\dfrac1z$ is not defined when $z=0$. You need a holomorphic function on the whole square for this to hold. This integral should be the same as for the circle, namely $2\pi i$.</p>
|
2,553,175 | <p>How can I verify that
$$1-2\sin^2x=2\cos^2x-1$$
Is true for all $x$?</p>
<p>It can be proved through a couple of messy steps using the fact that $\sin^2x+\cos^2x=1$, solving for one of the trigonemtric functions and then substituting, but the way I did it gets very messy very quickly and you end up with a bunch of ... | Parcly Taxel | 357,390 | <p>$$1=\cos^2x+\sin^2x$$
$$2=2\cos^2x+2\sin^2x$$
$$2-2\sin^2x=2\cos^2x$$
$$1-2\sin^2x=2\cos^2x-1$$</p>
|
2,553,175 | <p>How can I verify that
$$1-2\sin^2x=2\cos^2x-1$$
Is true for all $x$?</p>
<p>It can be proved through a couple of messy steps using the fact that $\sin^2x+\cos^2x=1$, solving for one of the trigonemtric functions and then substituting, but the way I did it gets very messy very quickly and you end up with a bunch of ... | Bernard | 202,857 | <p>Both are equal to $\cos 2x=\cos^2x-\sin^2x$ by the duplication formula!</p>
<p>If you don't want to use this formula, just rewrite the equality as
$$1+1=2\sin^2x+2\cos^2x $$
and it boils down to <em>Pythagoras' identity</em>.</p>
|
1,350,085 | <p>Consider $f(t)$, continuous on $[0,1]$, and $\alpha > 1$, and:</p>
<p>$$\int_0^1 \frac{f(t)}{t^{\alpha + 1}} \ dt$$</p>
<p>How can we tell this integral diverges? Basically since $f$ is continuous it reaches it's minimum at $[0,1]$ so we could make a comparison with $\int_0^1 \frac{f(x_{min})}{t^{\alpha+1}}\ dt... | Rory Daulton | 161,807 | <p>If $A$ has $18$ elements, and $B\subseteq A$ has fewer than $9$ elements, then $B^c\subseteq A$ has more than nine elements. Also, if $B$ has more than $9$ elements then $B^c$ has fewer than $9$ elements. That means the number of subsets of $A$ with fewer than $9$ elements equals the number of subsets of $A$ with mo... |
35,987 | <p><em>cross post in <a href="https://stackoverflow.com/questions/3513660/multivariate-bisection-method">StackOverflow</a></em></p>
<p>I need an algorithm to perform a 2D bisection method for solving a $2$x$2$ non-linear problem. Example: two equations $f(x,y)=0$ and $g(x,y)=0$ which I want to solve simultaneously. I ... | Per Vognsen | 2,036 | <p>A remarkable generalization of bisection to multiple dimensions is the subgradient method from convex optimization theory. If $f$ and $g$ are convex then $h = f^2 + g^2$ is also convex, and a simultaneous zero of $f$ and $g$ is a minimum of $h$.</p>
<p>Unfortunately, the subgradient method has more theoretical than... |
851,712 | <p>Build quadratic extension of field that contains $5$ elements. And solve $x^2+x+2=0$ in this field.</p>
<p>As I understand we need to build $\mathbb{F}_{5^{2}}$.</p>
<p>Field $\mathbb{F}_5$ contains $\{0,1,2,3,4\}.$</p>
<p>And as I understand field $\mathbb{F}_{5^{2}}$ contains $\{0,1,2,3,4\}$ and polynomials $\{... | André Nicolas | 6,312 | <p>The easiest description of the quadratic extension is that it has the same $25$ elements as yours, with addition defined in the obvious way, and $x^2$ being defined as $-x-2$, that is, $4x+3$. Then $x$ is a solution of your quadratic equation.</p>
<p>Once we have defined $x^2$, the product $ax+b)(cx+d)$ can be de... |
851,712 | <p>Build quadratic extension of field that contains $5$ elements. And solve $x^2+x+2=0$ in this field.</p>
<p>As I understand we need to build $\mathbb{F}_{5^{2}}$.</p>
<p>Field $\mathbb{F}_5$ contains $\{0,1,2,3,4\}.$</p>
<p>And as I understand field $\mathbb{F}_{5^{2}}$ contains $\{0,1,2,3,4\}$ and polynomials $\{... | Jyrki Lahtonen | 11,619 | <p>An alternative to the trick in André's answer (+1) would be to use the usual quadratic formula. That works over any field where you can <em>complete the square</em>, which is the case whenever you can divide by two. This in turn is always possible, when $2\neq0$, i.e. when the characteristic is not equal to two.</p>... |
1,074,177 | <p>Suppose a problem
$$\min_{x \in \mathbb{R}^{n}} f(x)$$</p>
<p>subject to $x \in \Omega$ which is a closed and convex set. If $\nabla f(x)$ is Lipschitz continuous in $\Omega$, then prove that</p>
<p>$$e(x) = x - P_{\Omega}(x- \nabla f(x))$$</p>
<p>is also Lipschitz continuous in $\Omega$.</p>
<p>Thanks in advanc... | copper.hat | 27,978 | <p>You need to show that $P_\Omega$ is Lipschitz.</p>
<p>In general, $\hat{x}$ minimises a convex function $g$ over a convex set $C$
<strong>iff</strong> $\langle \nabla g(\hat{x}), c-x \rangle \ge 0$ for all $c \in C$.</p>
<p>Taking $g(c) = {1 \over 2} \|c -x \|^2$, we see that $\hat{c}$ minimises $g$ over $\Omega$ ... |
1,497 | <p>If (C,tensor,1) is a symmetric monoidal category and f:A-->B is a morphism of PROPs (or monoidal cats = colored PROPs), one gets a forgetful functor f^*:B-Alg(C)-->A-Alg(C) (where B-Alg(C)=tensor-preserving functors from B to C) defined by precomposing with f.</p>
<p>Does anyone conditions on A,B,C under which this... | Aleks Kissinger | 800 | <p>Paul-André Melliès has quite an interesting paper on this topic:</p>
<p><a href="http://hal.archives-ouvertes.fr/docs/00/33/93/31/PDF/free-models.pdf" rel="nofollow">http://hal.archives-ouvertes.fr/docs/00/33/93/31/PDF/free-models.pdf</a></p>
<p>...but phrased in the more general terms of T-algebras of a pseudomon... |
1,241,864 | <p>I would like to know if there is formula to calculate sum of series of square roots $\sqrt{1} + \sqrt{2}+\dotsb+ \sqrt{n}$ like the one for the series $1 + 2 +\ldots+ n = \frac{n(n+1)}{2}$.</p>
<p>Thanks in advance.</p>
| Alex | 38,873 | <p>For an easier solution notice that $f(x) = \sqrt{x}$ is a monotone increasing function, hence for every $[k ,k+1]$
$$
\int_{k-1}^{k} \sqrt{x} dx<\sqrt{k}<\int_{k}^{k+1} \sqrt{x} dx
$$
Now sum over k, you'll get a sharp approximation</p>
|
2,055,803 | <p>Simplify $\left(\dfrac{4}{5} - \dfrac{3}{5}i\right)^{\!75}$</p>
<p>I've searched around on the internet and haven't found a very straightforward answer for this particular problem. I believe this problem has something to do with Euler's Formula, but I'm not sure how to use it in this case.</p>
<p>EDIT: We are not ... | Vidyanshu Mishra | 363,566 | <p>As well mentioned by @dxiv in the comment, you can easily see that $x$ cannot be zero (otherwise in the expression on the left we will do something which is not permitted to do so [which I shall let you find]). So, you can multiply the equation by $x$.</p>
<p>On multiplying whole equation by $x$, you get $1+2x^{2}=... |
2,529,387 | <p><a href="https://i.stack.imgur.com/3M7iS.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3M7iS.jpg" alt="enter image description here"></a></p>
<p>The fourth question is how can I do?</p>
<blockquote>
<p>Q: Given that $a, b \in \mathbb{Z}$ and $a$ is even, show that if $a^2b$ is not divisible ... | Paolo Intuito | 395,372 | <p>If $a$ is even, then there exists $n$ such that $a=2n$. Hence, $a^2b= 4n^2b$.</p>
<hr>
<p>Can you take it from here, knowing that $8=2^3$?</p>
|
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