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 |
|---|---|---|---|---|
258,332 | <blockquote>
<p>Prove that if $a^x=b^y=(ab)^{xy}$, then $x+y=1$.</p>
</blockquote>
<p>How do I use logarithms to approach this problem?</p>
| Shaun Ault | 13,074 | <p>Using logarithms:</p>
<p>Since $a^x = b^y$,
$$
\log a^x = \log b^y \quad \Rightarrow \quad x \log a = y \log b
\quad \Rightarrow \quad \log a = \frac{y}{x} \log b
$$
Then, since $b^y = (ab)^{xy}$,
$$
\log b^y = \log (ab)^{xy}
\quad \Rightarrow \quad
y \log b = xy \log (ab) = xy \left( \log a + \log b\righ... |
77,744 | <p>Hopefully a simple one for you (or at least seemingly)!</p>
<p>I import a .txt file, from which i make a ListLinePlot. I simply want to read in the name of the file, store it in a variable so I can use it to tag my plots later.</p>
<pre><code> Data = Import["C:\\Users\\Name\\Folder\\test2.txt", "CSV"];
ListLine... | Gordon Coale | 19,027 | <p>Given you already have the filename as a text string you could specify the file name and path separately when you first run, and create the full path & filename using FileNameJoin. FileNameJoin is OS agnostic so puts in forward or backslashes depending on whether you are running Windows, OSX or Linux.</p>
<p>A... |
4,618,433 | <p>Just a heads up: "<span class="math-container">$a$</span>" and "<span class="math-container">$α$</span>" are different</p>
<p>Let <span class="math-container">$a,b \in \Bbb R$</span> and suppose <span class="math-container">$a^2 − 4b \neq 0$</span>. Let <span class="math-container">$\alpha$</span... | Essaidi | 708,306 | <ol>
<li> If <span class="math-container">$a^2 - 4 b > 0$</span> then the roots are :
<span class="math-container">$$\alpha = \dfrac{-a + \sqrt{a^2 - 4 b}}{2} \text{ and } \beta = \dfrac{-a - \sqrt{a^2 - 4 b}}{2}$$</span>
so :
<span class="math-container">$$\alpha - \beta = \sqrt{a^2 - 4 b}$$</span>
Let <span class... |
313,437 | <p>I have to find out the convergence of the next integral:
$$\int^{\pi/2}_0{\frac{\ln(\sin(x))}{\sqrt{x}}}dx$$
Any help? Thanks</p>
| Jonathan | 37,832 | <p>If you integrate by parts, then you find
$$\int_0^{\pi/2}dx\,\frac{\ln\sin x}{\sqrt{x}}=2\sqrt{x}\ln\sin x\Bigg|_0^{\pi/2}+2\int_0^{\pi/2}dx\,\sqrt{x}\frac{\cos x}{\sin x}.$$
It is fairly clear at this point that the integral converges because $\sqrt{x}/\sin x\sim x^{-1/2}$ at $x=0$.</p>
|
1,617,890 | <blockquote>
<p>Question: Solve $\sin(3x)=\cos(2x)$ for $0≤x≤2\pi$.</p>
</blockquote>
<p>My knowledge on the subject; I know the general identities, compound angle formulas and double angle formulas so I can only apply those.</p>
<p>With that in mind</p>
<p>\begin{align}
\cos(2x)=&~ \sin(3x)\\
\cos(2x)=&~... | Anurag A | 68,092 | <p>Use $\sin 3x=3 \sin x - 4 \sin^3x$ and $\cos 2x=1-2\sin^2x$. To get
$$3 \sin x - 4 \sin^3x=1-2\sin^2x.$$
Now call $\sin x=t$. Thus we have
$$4t^3-2t^2-3t+1=0.$$
Observe that $t=1$ is definitely a solution, so we have
$$(t-1)(4t^2+2t-1)=0.$$
The quadratic factor will be zero for
$$t=\frac{-1\pm \sqrt{5}}{4}$$
I hop... |
1,580,586 | <p>Question goes as follows:
Consider the points on a line; $A(1,3,-1)$ and $B(-1,4,-2)$. Find the point $Q$ on $L$ closest to the point $P(1,1,0)$.</p>
<p>My thinking:
Closest distance from $a$ to $b$ is always a straight line, $90$ degree angle.
Therefore:
$$
Q⋅P=0
$$</p>
<p>$$
L=
\left(\begin{array}{cc}
1\\
3\\
-1... | Martin Argerami | 22,857 | <p>If you pay attention to your graph, you will see that what you need is $(-2,1,-1)\cdot(P-Q)=0$. That is,
\begin{align}
0&=-2(1-(1-2t))+(1-(3+t))-(-(-1-t))\\
&=-6t-3,\\
\end{align}
so $t=-1/2$, and your point in the line is
$$
\left(\begin{array}{cc}
1-2(-1/2)\\
3+(-1/2)\\
-1-(-1/2)\\
\end{array}\right)
=
\... |
142,105 | <p>In trying to deduce the lower bound of the ramsey number R(4,4) I am following my book's hint and considering the graph with vertex set $\mathbb{Z}_{17}$ in which $\{i,j\}$ is colored red if and only if $i-j\equiv\pm2^i,i=0,1,2,3$; the set of non-zero quadratic (mod 17) and blue otherwise. This graph shows that $R(4... | Gerry Myerson | 8,269 | <p>By symmetry, you just have to show that $0$ is not in any monochromatic $K_4$. $0$ is adjacent to $1,2,4,8,9,13,15,16$. Suppose $1$ is involved. $1$ is adjacent to $2,9,16$. No two of these are adjacent, so that rules out a red $K_4$ involving $0$ and $1$. Now you have to do the same kind of analysis for $0$ and $2$... |
308,856 | <p>A set $E\subseteq \mathbb{R}^d$ is said to be Jordan measurable if its inner measure $m_{*}(E)$ and outer measure $m^{*}(E)$ are equal.However, Lebesgue mesure theory is developed with only outer measure. </p>
<p>A function is Riemann integrable iff its upper integral and lower integral are equal.However, in Lebesg... | Pietro Majer | 6,101 | <p>My two cents. Outer measures are sub-additive on countable coverings: $$A\subset \cup _{j\in\mathbb{N}} A_j\quad \Rightarrow \quad \mu( A)\le \sum_{{j\in\mathbb{N}}}\mu(A_j)$$
which is somehow a nicer and more practical property than the analogous dual property of super-additivity for inner measures (even in the ca... |
2,705,794 | <p>I ran across this problem on a practice Putnam worksheet. Completely stumped.</p>
<p>Is $$\large \frac{m^{6} + 3m^{4} + 12m^{3} + 8m^{2}}{24}$$ an integer for all $m \in \mathbb{N}$?</p>
<p>I suspect it is an integer for any $m$. It checks out for small cases.</p>
<p>Any hints for proving the general case?</p>
| lhf | 589 | <p>You can just compute $f(m)$ at $7$ consecutive integers. If these values are all integers, then $f(m)$ is always an integer because of <a href="http://en.wikipedia.org/wiki/Newton_series#Newton.27s_series" rel="nofollow noreferrer">Newton's interpolation formula</a>. Try $m=-3,\dots,3$.</p>
|
1,875,351 | <p>I saw this problem in a book (not homework),</p>
<p>Assuming $L(n) = F(n)$ for$ n = 1,2,\cdots, k$
where $L(n)$ is Lucas Number and $F(n)$ is Fibonacci number.</p>
<p>$$\qquad L(k+1) = L(k) + L(k-1) \qquad \tag{by definiton}$$</p>
<p>$$ \qquad\qquad= F(k) + F(k-1) \tag{assumption}$$</p>
<p>$$ \ =F(k+1)... | Seewoo Lee | 350,772 | <p>We have to prove $F(1)=L(1)$ and $F(2)=L(2)$, which is false. That is the reason why the proof is incorrect. </p>
|
1,875,351 | <p>I saw this problem in a book (not homework),</p>
<p>Assuming $L(n) = F(n)$ for$ n = 1,2,\cdots, k$
where $L(n)$ is Lucas Number and $F(n)$ is Fibonacci number.</p>
<p>$$\qquad L(k+1) = L(k) + L(k-1) \qquad \tag{by definiton}$$</p>
<p>$$ \qquad\qquad= F(k) + F(k-1) \tag{assumption}$$</p>
<p>$$ \ =F(k+1)... | Kitter Catter | 166,001 | <p>This looks corny after someone already pointed this out, but you have to establish the assumption is true for some set of points. Now you can't just show $L(1)=F(1)$ because you need $L(k)$ and $L(k-1)$ to build on. Thus you would have to show it to be true for two consecutive values of $L$</p>
|
3,135,386 | <p>Our teacher tells us to convert it this way <span class="math-container">$ 3^x = e^{\ln 3^x}= e^{x\cdot\ln 3}$</span> and then use the rule <span class="math-container">$e^u\cdot u'$</span> but I can't understand where <span class="math-container">$\ln$</span> comes from and how <span class="math-container">$\ln 3^x... | Michael Rozenberg | 190,319 | <p>Because by the definition of <span class="math-container">$\ln$</span> we have <span class="math-container">$\ln3^x=x\ln3$</span> and from here: <span class="math-container">$$\left(3^x\right)'=\left(e^{x\ln3}\right)'=e^{x\ln3}\cdot\ln3=3^x\ln3.$$</span>
Actually, <span class="math-container">$\log_ab$</span> it's ... |
3,016,386 | <p>Hi I am struggling with this exercise, which may be perceived as simple. so I was trying to write tangents as follows:</p>
<p><span class="math-container">$$\tan(z)=-i\frac{e^{iz}-e^{-iz}}{e^{iz}+e^{-iz}}$$</span> and then <span class="math-container">$$z=a+bi$$</span>, which led me to <span class="math-container">... | egreg | 62,967 | <p>The equation you have to solve is
<span class="math-container">$$
-i\frac{e^{iz}-e^{-iz}}{e^{iz}+e^{-iz}}=w
$$</span>
that can be rewritten as
<span class="math-container">$$
i-ie^{2iz}=we^{2iz}+w
$$</span>
hence
<span class="math-container">$$
e^{2iz}=\frac{i+w}{i-w}
$$</span>
The denominator vanishes for <span cla... |
2,258,557 | <p>Why the equation of an arbitrary straight line in complex plane is $zz_o + \bar z \bar z_0 = D$ where D $\in R$</p>
<p>I understand that a vertical straight line can be defined by the equation $z+\bar z= D$ because suppose $z =x+yi$ then $\bar z = x-yi$ Thus, $z+\bar z = x+yi+x-yi=2x$ which is an arbitrary vertic... | Nosrati | 108,128 | <p>You know that a vertical straight line can be defined as $z+\bar z= D$, so if you rotate it's points with angle $\theta$ you get $(e^{i\theta}z)+ \overline{(e^{i\theta}z)}= D$ or $e^{i\theta}z + e^{-i\theta}\overline{z}= D$ and with arbitrary real $r\neq0$,
$$re^{i\theta}z + re^{-i\theta}\overline{z}= rD$$
gives us
... |
59,495 | <p>Suppose $K$ is an $n$-dimensional $C^2$ convex body in $\mathbb{R}^{n+1}$. We choose two distinct directions $z_0, z_1\in\mathbb{S^{n}}$. If $P_1$ and $P_2$ are the corresponding hyperplanes($z_0\perp P_1$ and $z_1\perp P_2$) and $K'$ is the projection of $K$ on $P_1\cap P_2$, what is the $Vol(K')$?
We know the supp... | Liviu Nicolaescu | 20,302 | <p>You should definitely check <a href="http://www.nd.edu/~lnicolae/val-simple.pdf" rel="nofollow">these notes</a> generated by three bright undergraduates for an REU project that I supervised a few years ago. I promise you, it will be worth your time.</p>
|
4,450,169 | <p>Here is a (seemingly) simple problem in group theory. Given a non-elementary finite nilpotent group <span class="math-container">$N$</span>, show there exist <span class="math-container">$p \neq q$</span> primes such that <span class="math-container">$N$</span> has a quotient <span class="math-container">$\Bbb Z_{pq... | spin | 12,623 | <p>A finite group is nilpotent if and only if it is a direct product of its Sylow subgroups.</p>
<p>So say you have a nilpotent group <span class="math-container">$G$</span>, then <span class="math-container">$$G = P_1 \times P_2 \times \cdots \times P_t$$</span> with <span class="math-container">$P_i$</span> a <span ... |
2,255,617 | <p>I am trying to learn how to do proofs by contradiction. The proof is,</p>
<p>"Prove by Contradiction that there are no positive real roots of $x^6 + 2x^3 +4x + 5$"</p>
<p>I understand that now I am attempting to prove that there is a positive real root of this equation, so I am able to contradict myself within the... | Hayden | 27,496 | <p>This will depend somewhat on your definition of a ring and a ring homomorphism, as there are a few standard ways of defining these notions. </p>
<p>Firstly, for rings, there is sometimes a distinction between a <em>ring</em> and a <em>unital ring</em>, which additionally has a multiplicative identity $1$ (whereas a... |
1,285,443 | <blockquote>
<p>Let us denote solution to the equation</p>
<p>$$(x+a)^{x+a}=x^{x+2a}$$</p>
<p>with $X_a$.</p>
<p>($a$ is a non-zero real number)</p>
<p>Prove that:</p>
<p>$$\lim_ {a \to 0} X_a = e$$</p>
</blockquote>
<p>This is something that I noticed while making numerical experiments for ... | mathlove | 78,967 | <p>This might not be rigorous, but note that one has
$$(x+a)^x=x^{x+a}\Rightarrow x\ln(x+a)=(x+a)\ln x\Rightarrow \frac{\ln(x+a)}{x+a}=\frac{\ln x}{x}.$$</p>
<p>Then, let $f(x)=\frac{\ln x}{x}$. One has $f'(x)=0\iff x=e$, and considering the graph of $y=f(x)$ should give you the answer.</p>
|
1,285,443 | <blockquote>
<p>Let us denote solution to the equation</p>
<p>$$(x+a)^{x+a}=x^{x+2a}$$</p>
<p>with $X_a$.</p>
<p>($a$ is a non-zero real number)</p>
<p>Prove that:</p>
<p>$$\lim_ {a \to 0} X_a = e$$</p>
</blockquote>
<p>This is something that I noticed while making numerical experiments for ... | Claude Leibovici | 82,404 | <p>If $a$ is small, a first order Taylor expansion built at $a=0$ gives $$(x+a)^{x+a}-x^{x+2a}=-a\, x^x\, \big(\log (x)-1\big)+O\left(a^2\right)$$ Do you think that this is sufficient to prove that $$\lim_ {a \to 0} X_a = e$$</p>
<p>Another way is starting from zoli's answer $$\frac{(x+a)\log(x+a)-x\log(x)}{a}=2\log(x... |
3,433,492 | <p>I know that a function can admitted multiple series representation (according to Eugene Catalan), but I wonder if there is a proof for the fact that each analytic function has only one unique Taylor series representation. I know that Taylor series are defined by derivatives of increasing order. A function has one an... | Calvin Khor | 80,734 | <p>Well its possible for e.g. <span class="math-container">$f(x) = \sum a_n x^n = \sum b_n (x-1)^n$</span> simultaneously, but that probably isn't what you meant. Instead lets just consider the behavior at one point, say expanding around <span class="math-container">$x=0$</span>.</p>
<p>Let's fix notation-</p>
<blockqu... |
120,687 | <p>Consider the following code</p>
<pre><code>styles = {Red, Blue, {Red, Dashed}, {Blue, Dashed}}
pt1 = Plot[{x^2, 2 x^2, 1/x^2, 2/x^2}, {x, 0, 3}, Frame -> True,
PlotStyle -> styles, PlotLegends -> {"1", "2", "1", "2"}]
</code></pre>
<p>I would like the two red lines to carry the same label "1" and the two... | Anton Antonov | 34,008 | <p>This answer shows how to define <a href="https://mathematica.stackexchange.com/q/118324/34008">a new <code>NIntegrate</code> rule</a> that evaluates <code>f</code> in the list of two integrands <code>{f[x],g[f[x]]+h[x]}</code> only once per sampling point. The answer can be also easily modified into an answer of <a... |
12,927 | <p>The problem:</p>
<p><strong><em>Three poles standing at the points $A$, $B$ and $C$ subtend angles $\alpha$, $\beta$ and $\gamma$ respectively, at the circumcenter of $\Delta ABC$.If the heights of these poles are in arithmetic progression; then show that $\cot \alpha$, $\cot \beta$ and $\cot \gamma $ are in harmon... | Aryabhata | 1,102 | <p>My guess is the poles are of (different) heights $h_A$, $h_B$, $h_C$ and the angle is from the foot of pole to the circumcentre of $\triangle ABC$ to the top of the pole.</p>
|
761,823 | <blockquote>
<p>Suppose that $G$ is a finite abelian group that does not contain a subgroup isomorphic to $\mathbb Z_p\oplus\mathbb Z_p$ for any prime $p$. Prove that $G$ is cyclic.</p>
</blockquote>
<p><strong>Attempt</strong>: If $G$ is a finite abelian group, then let $H$ be any subgroup of $G$</p>
<p>It's given... | egreg | 62,967 | <p>Hint: decompose $G$ into the direct sum of its primary components; then examine each primary component and deduce it's cyclic because it has a minimum nonzero subgroup (that is, the intersection of its nonzero subgroups is nonzero).</p>
|
295,545 | <p>The following figure depicts the paths from home to work. SAM never travels through the park when going to work.</p>
<p><img src="https://i.stack.imgur.com/IANqM.png" alt="enter image description here"></p>
| anegligibleperson | 17,248 | <p>You know the plane passes through the origin, and you need two orthogonal vectors that span this plane. Do you see it now?</p>
|
7,981 | <p>I've read so much about it but none of it makes a lot of sense. Also, what's so unsolvable about it?</p>
| Roupam Ghosh | 2,320 | <p>A direct translation of RH (Riemann Hypothesis) would be very baffling in layman's terms. But, there are many problems that are equivalent to RH and hence, defining them would be actually indirectly stating RH. Some of the equivalent forms of RH are much easier to understand than RH itself. I give what I think is th... |
3,615,117 | <p>I want to find the intersection of the sphere <span class="math-container">$x^2+y^2+z^2 = 1$</span> and the plane <span class="math-container">$x+y+z=0$</span>. </p>
<p><span class="math-container">$z=-(x+y)$</span> that gives <span class="math-container">$x^2+y^2+xy= \frac 12$</span></p>
<p>How do I represent thi... | lab bhattacharjee | 33,337 | <p><span class="math-container">$$\dfrac12=\left(x+\dfrac y2\right)^2+y^2\left(1-\dfrac14\right)$$</span></p>
<p>Let <span class="math-container">$x+\dfrac y2=X, y=Y$</span></p>
<p>Alternately, use <a href="https://en.wikipedia.org/wiki/Rotation_of_axes#Rotation_of_conic_sections" rel="nofollow noreferrer">this</a> t... |
1,407,641 | <p>If $T$ is a linear transformation and is said to be one to one or onto- this only makes sense when we specify what domain and range is right?
$T: V \rightarrow V$ may not be onto or one to one
but $T: V \rightarrow Im(T)$ is certainly onto and may or may not be one to one.
Is this right?</p>
| Ben Grossmann | 81,360 | <p>Yes, you are correct. We can "make" a linear transformation onto by restricting the codomain to the image of the transformation.</p>
|
2,040,041 | <p>I was able to think that the numerator will always be positive and will overpower the denominator as well. But couldn't proceed from there.</p>
| mathcounterexamples.net | 187,663 | <p>Using the Mean-value forms of the remainder of <a href="https://en.m.wikipedia.org/wiki/Taylor%27s_theorem" rel="nofollow noreferrer">Taylor's theorem</a> you get the existence of $c \in (0,x)$ such that
$$e^x=1+x+\frac{x^2}{2}e^c$$ Hence
$$e^x = 1+x+\frac{x^2}{2}e^c <1+x+\frac{x^2}{2} e^x$$ and the desired resu... |
3,086,758 | <p>I know that if <span class="math-container">$\mathbb{E}[X]=\mathbb{E}[X|Y] , \mathbb{E}[Y]=\mathbb{E}[Y|X]$</span>, <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> can be dependent, for example a ‘uniform’ distribution in a unit circle.
Now we add the variance, if
<span class="mat... | Dasherman | 177,453 | <p>Consider <span class="math-container">$X\sim N(0,1)$</span> and <span class="math-container">$Y\sim N(0,2)$</span> if <span class="math-container">$X>0$</span> and <span class="math-container">$Y\sim t_4$</span> otherwise. Then <span class="math-container">$X, Y$</span> are dependent, but your conditions hold (th... |
2,644,910 | <p>Ali Baba is trying to enter a cave. At the entrance, there is a drum with four openings, in each of which there is a pot with a herring inside. The herring may be lying with its tail up or down. Ali Baba can put his hands into any two
openings, feel the herrings, and put any one or both of them either tail up or tai... | fred goodman | 124,085 | <p>This should be a comment, but I had trouble with formatting in the comments. </p>
<p>Evidently, the same proof will work for $R/I \to R/J$ where $R$ is a PID. The more general problem was considered
<a href="https://mathoverflow.net/questions/31495/when-does-a-ring-surjection-imply-a-surjection-of-the-group-of-un... |
2,274,736 | <p>I am finding particular subgroups of $Q_{12}$ and had a couple of questions about it.</p>
<p>$Q_{12}=\langle a,b:a^6=1,b^2=a^3,ba=a^{-1}b\rangle$</p>
<p>Firstly here is part of a solution I came across: </p>
<p>The first step is to establish the orders of the elements. So $1$ has order 1, $a^3$ has order 2, $a^2$... | Michael Hardy | 11,667 | <p>The notation $p\in(a,b)$ means $a<p<b$, and $p\in[a,b]$ means $a\le p \le b$, and $p\in(a,b]$ means $a<p\le b$ and $p\in[a,b)$ means $a\le p<b.$ The notation $p\in\{a,b\},$ on the other hand, means either $p=a$ or $p=b.$</p>
<p>You have
$$
L(p) ={n \choose x_1,x_2,x_3} (1 - 2p)^{x_1}\cdot p^{x_2+x_3}.
$... |
247,553 | <p>Let $f(x)=\binom{x}{2}+\binom{x}{4}+\cdots+\binom{x}{2u}$, where $u\in\mathbb{Z}^+$ and $\binom{x}{l}=\frac{x(x-1)\dots(x-l+1)}{l!}$ for all $l\in\mathbb{Z}^+$.
Then can we prove $f(x)$ is a convex function on $[0,+\infty)$?</p>
<p>Updates:</p>
<p>1) It was pointed out by @user44191 that, observing $\binom{x}{i}=... | Pietro Majer | 6,101 | <p><em>(Here is a proof of the convexity of $\sum_{k=0}^n\binom{x}{k}$ on $[0,\infty)$
for any large $n$; with a bit more care the argument should work for all positive even integer $n$ on $[-1,\infty)$, which is the original problem)</em></p>
<p>We may consider the problem of showing the convexity of $S_n(x):=\sum... |
247,553 | <p>Let $f(x)=\binom{x}{2}+\binom{x}{4}+\cdots+\binom{x}{2u}$, where $u\in\mathbb{Z}^+$ and $\binom{x}{l}=\frac{x(x-1)\dots(x-l+1)}{l!}$ for all $l\in\mathbb{Z}^+$.
Then can we prove $f(x)$ is a convex function on $[0,+\infty)$?</p>
<p>Updates:</p>
<p>1) It was pointed out by @user44191 that, observing $\binom{x}{i}=... | esg | 48,831 | <p>Here's an alternative proof based on probabilistic arguments (showing different aspects). Let
$$f_n(x):=\sum_{j=0}^n { x \choose j}=[t^n]\,\frac{(1+t)^x}{1-t}\;\;,$$
and let $^\prime$ denote derivative with resp. to $x$.</p>
<p>We have to show that for even $n=2k$ the second derivative $f_{2k}^{\prime\prime}(x)... |
2,137,332 | <p>On the MathWorld page: </p>
<p><a href="http://mathworld.wolfram.com/FermatPseudoprime.html" rel="nofollow noreferrer">http://mathworld.wolfram.com/FermatPseudoprime.html</a></p>
<p>in the first table, I expect to see $561$ on every line, but it is not on the line for base $3$.</p>
<p>When you click on the link ... | Laray | 396,534 | <p>To get the mapping of the lines, you first to describe the lines by using parameters. </p>
<p>Theline between $-1$ and $1$ ist desribed by $\lambda \in [-1, 1]$ and can easily be mapped onto the positive real numbers upto $1$
After that, you need to transform $\lambda \cdot 1 + (1-\lambda)\cdot i$ for $\lambda \in ... |
1,284,938 | <p>I was revising for one of my end of year maths exams, then I came across this example on how to find lines of tangents to ellipses outside the curve. Personally, I'd use differentiation and slopes to find such lines, but the lecturer does something simpler and more elegant.</p>
<p>The question is: "Find the equatio... | JJacquelin | 108,514 | <p>The solution of the PDE :
$${\partial z \over \partial x}+(2e^x-y){\partial z \over \partial y}=0.$$
according to the boundary condion $z(0,y)=y \:$ is :
$$z(x,y)=e^x y-e^{2x}+1$$
as shown below :</p>
<p><img src="https://i.stack.imgur.com/QRLtT.jpg" alt="enter image description here"></p>
<p>One can check the abo... |
23,911 | <p>I am teaching a course on Riemann Surfaces next term, and would <strong>like a list of facts illustrating the difference between the theory of real (differentiable) manifolds and the theory non-singular varieties</strong> (over, say, $\mathbb{C}$). I am looking for examples that would be meaningful to 2nd year US g... | Dan Zaffran | 2,109 | <p>Here is a list biased towards what is remarkable in the complex case. (To the potential peeved real manifold: I love you too.) By "complex" I mean holomorphic manifolds and holomorphic maps; by "real" I mean $\mathcal{C}^{\infty}$ manifolds and $\mathcal{C}^{\infty}$ maps. </p>
<ul>
<li><p>Consider a map $f$ betwee... |
23,911 | <p>I am teaching a course on Riemann Surfaces next term, and would <strong>like a list of facts illustrating the difference between the theory of real (differentiable) manifolds and the theory non-singular varieties</strong> (over, say, $\mathbb{C}$). I am looking for examples that would be meaningful to 2nd year US g... | Community | -1 | <p>Some embedding statements.</p>
<p>A compact complex subvariety of ${\mathbb{C}}^n$ is a point. However, every compact real manifold of dimension $n$ can be realized as a submanifold of some ${\mathbb{R}}^{2n}$.</p>
<p>There are compact complex manifolds that cannot be embedded into complex projective space. An exa... |
3,943,199 | <p>How to put <span class="math-container">$(-\sqrt{3}-i)^{\frac{5}{7}}$</span> into polar form and find all roots.</p>
<p>What I tried:</p>
<p><span class="math-container">$$w = -\sqrt{3}-i$$</span>
<span class="math-container">$$\arg(w)=\arctan(\frac{1}{\sqrt 3})-\pi = \frac{\pi}{6} - \pi = \frac{-5\pi}{6}$$</span>
<... | Z Ahmed | 671,540 | <p><span class="math-container">$$e^{i\phi}=\cos \phi +i \sin \phi,~~ e^{-i\phi}=\cos \phi -i\sin \phi$$</span>
add these two to get <span class="math-container">$$\sin \phi= \frac{e^{i\phi}-e^{-i\phi}}{2i}$$</span></p>
|
1,424,273 | <p>Let $(a_n)$ be a convergent sequence of positive real numbers. Why is the limit nonnegative?</p>
<p>My try: For all $\epsilon >0$ there is a $N\in \mathbb{N}$ such that $|a_n-L|<\epsilon$ for all $n\ge N$. And we know $0< a_n$ for all $n\in \mathbb{N}$, particularly $0<a_n$ for all $n\ge N$. Maybe by c... | user118494 | 118,494 | <p>See, you have $$L\lt 0\lt a_{n} \ ,\ \ for\ \ all\ \ n\in N\ $$ Now,take $\epsilon={{|L|}\over {2}}$. Look at the $\epsilon$-nbd of $L$. This has no positive number at all in it, let alone the elements of the sequence $\{a_{n}\}$. So this contradicts the fact that $L$ is the limit poin... |
4,080,776 | <p>I am doing an individual study of an abstract algebra for number theory course online. I just started, so I hope my question just note come off as too trivial. The lecture notes state that the ring of <span class="math-container">$p$</span>-adic integers does not have a ring endomorphism.</p>
<h3>Questions:</h3>
<p>... | roxas3582 | 596,046 | <p>For your reference, there is a more general perspective from the theory of "Witt vectors". Throughout, we let <span class="math-container">$p$</span> denote a fixed prime and <span class="math-container">$\mathbb{F}_p$</span> the finite field with <span class="math-container">$p$</span> elements.</p>
<p>Th... |
351,870 | <p>Let <span class="math-container">$(X_n)$</span> be a sequence of <span class="math-container">$\mathbb{R}^d$</span>-valued random variables converging in distribution to some limiting random variable <span class="math-container">$X$</span> whose CDF is absolutely continuous with respect to the Lebesgue measure.</p>
... | Mateusz Kwaśnicki | 108,637 | <p>What is essential here is that the distribution of <span class="math-container">$X$</span> assigns little mass to sets which are essentially <span class="math-container">$(d-1)$</span>-dimensional.</p>
<hr>
<p>The standard approach to problems of this kind is to estimate
<span class="math-container">$$ \operatorna... |
351,870 | <p>Let <span class="math-container">$(X_n)$</span> be a sequence of <span class="math-container">$\mathbb{R}^d$</span>-valued random variables converging in distribution to some limiting random variable <span class="math-container">$X$</span> whose CDF is absolutely continuous with respect to the Lebesgue measure.</p>
... | Iosif Pinelis | 36,721 | <p><span class="math-container">$\newcommand{\R}{\mathbb{R}}
\newcommand{\ep}{\varepsilon}
\newcommand{\p}{\partial}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}$</span>
This is to try to provide a simplification and detalization of the answer by Mateusz Kwaśnicki. </p>
<p>Suppose that the distribution of <span c... |
2,330,196 | <p>The question asks me to draw a Hasse diagram for the given set of rules.
$$ (\{n\in \mathbb N: n\mid 100\ \lor\ n = 75 \}, {}\mid{} ) $$</p>
<p>My approach is to write down the set satisfying for $n\mid 100$, but I dont get what's with "or" $n =75.$</p>
<p>Could someone help me figure out what that means? is it s... | Community | -1 | <p><strong>Hint</strong> : show that if $p \sim q$ (as divisor) then $X \cong \Bbb P^1$. Show then that Riemann Roch implies that there is $p,q \in X$ with $p \sim q$. </p>
|
2,897,785 | <blockquote>
<p>Fix a $2\times 2$ real matrix $A$. Let $V$ be the set of all $2\times 2$ real matrices $X$ such that $AX=XA$. Show that $V$ is a vector space of dimension of at least 2.</p>
</blockquote>
<p>I'm struggling to see a good way to approach this problem. There's the brute force style method of algebraical... | Arnaud Mortier | 480,423 | <p>Hint:</p>
<ul>
<li><p>Showing that it's a vector space should be easy enough without actually considering the entries of the matrices.</p></li>
<li><p>To see that the dimension is at least $2$, consider two cases:</p>
<ul>
<li>What if $A$ is a multiple of the identity matrix?</li>
<li>What if $A$ isn't a multiple ... |
12,204 | <p>A tag named <a href="https://math.stackexchange.com/questions/tagged/tricks" class="post-tag" title="show questions tagged 'tricks'" rel="tag">tricks</a> has recently been created in <a href="https://math.stackexchange.com/questions/616672/2-tricks-to-prove-every-group-with-an-identity-and-xx-identity-is-abe... | Grigory M | 152 | <p>Certainly not. Let's remove it.</p>
<p>It hard to imagine someone adding this tag to favorite or ignored tag. It's not useful for search.</p>
<p>It's a meta-tag. It has unclear and potentially very vast area of usage. What questions exactly should be tagged (trick)? Who should tag it — e.g. if an answer is 'trick'... |
274,908 | <p>I would like to plot a molecule in 3D and use different colors for the same atom type in the molecule. For example, by using:</p>
<pre><code>MoleculePlot3D[Molecule["NC(=O)C[C@H](C(=O)O)N"], ColorRules -> {"C" -> Black}]
</code></pre>
<p>all C atoms become Black. But how can I make, for exa... | Jason B. | 9,490 | <p>This is underdocumented but the <code>"ColorRules"</code> option can take both atom indices and patterns.</p>
<pre><code>MoleculePlot3D[Molecule["NC(=O)C[C@H](C(=O)O)N"],
ColorRules -> {2 -> Green, "C" -> Black, "O" -> Orange, 8 -> Pink}]
</code></pre>
<p><a h... |
2,292,520 | <p>I know that the logical negation of $$\neg(a \rightarrow b)= a \wedge \neg b $$ I am not clear what that means in the following simple setting:</p>
<p>So its clear that $$x\geq 2 \to x^2\geq 4.$$ Now I can write the logical negation of $a\to b$ as $a \wedge \neg b$, but what does that intuitively mean? </p>
<p>Sup... | Graham Kemp | 135,106 | <p>Our statement is $\neg (a\to b)$</p>
<p>This reads: "It is false that $a$ (materially)implies $b$". </p>
<p>Recall that a <em>material implication</em> is falsified only when the antecedant is false and the consequent is true.</p>
<p>So our statement must be infering that "$a$ is true and $b$ is false."</p>
<p>W... |
2,710,703 | <p>Given any non abelian group, how can I prove that every proper subgroup may be abelian? I know the definition of "abelian," but I don't know the difference between a group and a subgroup, nor do I understand how the two interconnect.</p>
| Dietrich Burde | 83,966 | <p>There are many non-abelian groups all of whose proper subgroups are abelian. Studying such groups of low order, we immediately find examples, such as $S_3$ or $Q_8$, the quaternion group. Because we know all subgroups explicitly for these groups, it is easy to prove that they are abelian.</p>
<p>One might ask what ... |
1,804,042 | <p><strong>Edit:</strong> Here is the original problem; it is possible that my recurrence for the stationary distribution $\pi$ is incorrect.</p>
<blockquote>
<p>Consider a single server queue where customers arrive according to a Poisson process with intensity $\lambda$ and request i.i.d. $\mathsf{Exp}(\mu)$ servic... | Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 ... |
462,921 | <p>I simply don't get the following question answered:</p>
<p>How can i proof the equality $\lim_{a\to 0}\sup_{z\in\mathbb{Z}}2-2\cos(2\pi a z)=0$?</p>
<p>Or is it even false?</p>
<p>Thanks in advance!</p>
| R.T. | 89,230 | <p>I think this is wrong. </p>
<p>If $a$ is rational, then $\cos(2\pi az)$ will take the value $-1$ at some $z$. Hence, for $a\in \mathbb{Q}$, you have
$$\sup_z(2-2\cos(2\pi az)=4.$$</p>
<p>Since you can go arbitrarily close to zero with rational numbers, the limit cannot be zero...</p>
|
1,977,306 | <p>This is from a math competition so it must not be something really long
If a parabola touches the lines $y=x$ and $y=-x$ at $A(3,3) $ and $b(1,-1)$ respectively, then </p>
<p>(A) equation of axis of parabola is $2x+y=0$ </p>
<p>(B)slope of tangent at vertex is $1/2$</p>
<p>(C) Focus is $(6/5,-3/5)$</p>
<p>(D) ... | Narasimham | 95,860 | <p>Conditions (A),(B),(C),(D) are not needed to determine the tilted parabola ( $xy$ term non-zero). Because out of five constants needed to determine a conic, one can be reduced as zero determinant for parabola.You have given two points and two slopes which are quite sufficient.</p>
<p>$$ (x, y, y^{ \prime} ) = (3,3... |
730,198 | <blockquote>
<p>Show $f(x)=\sqrt{x^4+1} - \sqrt{x^4+x^2} \rightarrow -1/2$ for $x \rightarrow \infty$, $x \in \mathbb R$.</p>
</blockquote>
<p>I've tried $$\frac {(\sqrt{x^4+1} - \sqrt{x^4+x^2})(\sqrt{x^4+1} + \sqrt{x^4+x^2})}{\sqrt{x^4+1} + \sqrt{x^4+x^2} } = \frac {1-x^2} {\sqrt{x^4+1} + \sqrt{x^4+x^2}}$$</p>
<p>... | 5xum | 112,884 | <p>Yes, is enough to verify the last expression. This is because of $f$ and $g$ both have limits as $x$ approaches $\infty$, then $\lim_{x\to\infty}f(x)+g(x) = \lim_{x\to\infty}f(x) + \lim_{x\to\infty} g(x)$.</p>
<p>When calculating the last limit, you can simply divide both sides of the fraction by $x^2$ to get</p>
... |
1,853,464 | <p>I am using the Lorentz Force Equation and the electric-cross-magnetic field velocity equation] to solve for the $E$ and $B$ fields given the known path of a particle moving in 3D. </p>
<p>So with that I have the following equations where a and v are known:
<a href="https://i.stack.imgur.com/oL7fg.gif" rel="nofollow... | Robert Israel | 8,508 | <p>Consider the case where the particle is moving in a straight line with constant velocity. Then the magnetic field in the direction of that velocity has no effect.</p>
|
620,045 | <p>What is the mean and variance of Squared Gaussian: $Y=X^2$ where: $X\sim\mathcal{N}(0,\sigma^2)$?</p>
<p>It is interesting to note that Gaussian R.V here is zero-mean and non-central Chi-square Distribution doesn't work.</p>
<p>Thanks.</p>
| Cm7F7Bb | 23,249 | <p>We can avoid using the fact that $X^2\sim\sigma^2\chi_1^2$, where $\chi_1^2$ is the chi-squared distribution with $1$ degree of freedom, and calculate the expected value and the variance just using the definition. We have that
$$
\operatorname E X^2=\operatorname{Var}X=\sigma^2
$$
since $\operatorname EX=0$ (see <a ... |
1,102,638 | <p>Let $n\in \mathbb{N}$. Can someone help me prove this by induction:</p>
<p>$$\sum _{i=0}^{n}{i} =\frac { n\left( n+1 \right) }{ 2 } .$$</p>
| abel | 9,252 | <p>$$\sqrt{1 + x + x^2} = 1 + \dfrac{1}{2}(x+x^2) + \cdots = 1 + \dfrac{1}{2}x + \cdots$$ so the
$$\lim \limits_{x \to 0}{\frac{\sqrt{1 + x + x^2} - 1}{x}} =
\lim \limits_{x \to 0}\dfrac{1 + 1/2 x + \cdots - 1}{x} = \dfrac{1}{2}$$</p>
|
3,857,494 | <p>I have the following sequence given recursively by:</p>
<p><span class="math-container">$$A_n - 2A_{n-1} - 4A_{n-2} = 0$$</span></p>
<p>Where:</p>
<p><span class="math-container">$$A_0 = 1, A_1 = 3, A_2 = 10, A_3 = 32, etc.$$</span></p>
<p>To find the generating function, I have done the following:</p>
<p><span clas... | Joshua P. Swanson | 86,777 | <p>To answer your literal question, you can have software expand your rational function to make sure the first terms are correct, <a href="https://www.wolframalpha.com/input/?i=Series%5B%281%2Bx%29%2F%281-2x-4x%5E2%29%2C%20%7Bx%2C0%2C50%7D%5D" rel="nofollow noreferrer">like this</a>.</p>
|
3,065,818 | <blockquote>
<p>If <span class="math-container">$$z=\dfrac{\sqrt{3}-i}{2}$$</span> then <span class="math-container">$$(z^{95}+i^{67})^{94}=z^n$$</span> then, <span class="math-container">$\text{find the smallest positive integral value of}$</span> <span class="math-container">$n$</span> <span class="math-container">... | tarit goswami | 579,780 | <p>Here is an alternate method. </p>
<p>Note that set of primes dividing <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are same. Take any prime <span class="math-container">$p$</span> diving <span class="math-container">$x$</span>(and
hence <span class="math-container">$y$</span>... |
1,309,728 | <p>I know what a 3x10 looks like, but I cannot seem to find a distinguishable pattern to extend it to a 3x14.</p>
<p>The 3x10 pattern I'm using looks like the one at the top right of figure 6 of <a href="http://faculty.olin.edu/~sadams/DM/ktpaper.pdf" rel="nofollow">this paper</a>.</p>
<p>Any help would be greatly ap... | Bill Dubuque | 242 | <p>${\rm mod}\ x\!-\!a,\,y\!-\!b\!:\,\ x\equiv a,\,y\equiv b\,\Rightarrow\, xy-1\equiv ab-1\equiv0 $</p>
|
436,172 | <p>let $a,b,c,d$ are real numbers,show that
$$2\sqrt{a^2+c^2}+\sqrt{a^2+c^2+3(b^2+d^2)-2\sqrt{3}(ab+cd)}+\sqrt{a^2+c^2+3(b^2+d^2)+2\sqrt{3}(ab+cd)}\ge6\sqrt{|ad-bc|}$$</p>
<p>This problem is creat by China's famous mathematician hua luogeng,<a href="http://en.wikipedia.org/wiki/Hua_Luogeng" rel="nofollow">http://en.... | S.B. | 35,778 | <p>Let $x=[a,c]^T$ and $y=\sqrt{3}[b,d]^T$ be two vector in the plane and denote the cross product by $\times$. The inequality is equivalent to $$2\Vert x\Vert+\Vert x+y\Vert+\Vert x-y\Vert\geq 2\sqrt{3\Vert x\times y\Vert}\iff\\4\Vert x\Vert^2+(\Vert x+y\Vert+\Vert x-y\Vert)^2+4\Vert x\Vert(\Vert x+y\Vert+\Vert x-y\Ve... |
436,172 | <p>let $a,b,c,d$ are real numbers,show that
$$2\sqrt{a^2+c^2}+\sqrt{a^2+c^2+3(b^2+d^2)-2\sqrt{3}(ab+cd)}+\sqrt{a^2+c^2+3(b^2+d^2)+2\sqrt{3}(ab+cd)}\ge6\sqrt{|ad-bc|}$$</p>
<p>This problem is creat by China's famous mathematician hua luogeng,<a href="http://en.wikipedia.org/wiki/Hua_Luogeng" rel="nofollow">http://en.... | zyx | 14,120 | <p>It is the isoperimetric inequality for a triangle with vertices $0$, $x+y$, and $2x$, where $x=(a,c)$ and $y = \sqrt{3} (b,d)$. </p>
<p>If $P$ is the perimeter of the triangle and $A$ its area, Hua's inequality says $$|2x| + |x-y| + |x+y| \geq 6\sqrt{\frac{A}{\sqrt{3}}} .$$ The sum on the left is $P$ and the squa... |
1,517,698 | <p>What notation should I use for the set of the form</p>
<p>$$\{a_1,a_2,a_3,a_4\}$$</p>
<p>where $a_i \in \{0,1\}$ for $i = 1,2,3,4$?</p>
<p>It's an output from an indicator function that is evaluated over some "domain" $\{b_1,b_2,b_3,b_4\}$. I.e. it produces 0 or 1 for each $b_i$. So the result (I think) should be... | vadim123 | 73,324 | <p>A set is an unordered collection of elements. $\{1,1,0,1\}$ is no different from $\{0,1\}$. </p>
<p>It appears that OP is looking for an ordered collection of four elements. The natural notation for this is a <em>function</em>. We define $$f:\{b_1,b_2,b_3,b_4\}\to \{0,1\}.$$</p>
<p>We can also use a shorthand ... |
1,517,698 | <p>What notation should I use for the set of the form</p>
<p>$$\{a_1,a_2,a_3,a_4\}$$</p>
<p>where $a_i \in \{0,1\}$ for $i = 1,2,3,4$?</p>
<p>It's an output from an indicator function that is evaluated over some "domain" $\{b_1,b_2,b_3,b_4\}$. I.e. it produces 0 or 1 for each $b_i$. So the result (I think) should be... | Emanuele Paolini | 59,304 | <p>What you are speaking about are not sets. Sets are collections of elements without any order. There are only three sets composed by only zeros and ones: $\{\}, \{0\}, \{1\}, \{0,1\}$.</p>
<p>What you are speaking about are $n$-uples. Just use $()$ instead of $\{\}$ e.g.:
$$
(a_1,a_2,a_3,a_4) = (0,1,1,0).
$$</p>
|
1,917,313 | <p>I am to find a combinatorial argument for the following identity:</p>
<p>$$\sum_k \binom {2r} {2k-1}\binom{k-1}{s-1} = 2^{2r-2s+1}\binom{2r-s}{s-1}$$</p>
<p>For the right hand side, I was think that would just be number of ways to choose at least $s-1$ elements out of a $[2r-s]$ set. However, for the left hand sid... | Brian M. Scott | 12,042 | <p>HINT: We have <span class="math-container">$2r-s$</span> white balls numbered <span class="math-container">$1$</span> through <span class="math-container">$2r-s$</span>. We pick <span class="math-container">$s-1$</span> of them and paint those balls red, and we stick gold stars on any subset of the remaining white b... |
3,554,188 | <blockquote>
<p>I was given <span class="math-container">$$|ax - 11| = 4x - 10$$</span> has a positive integral solution and <span class="math-container">$a$</span> is a positive integer.</p>
<p>I was asked what was <span class="math-container">$x, a$</span></p>
</blockquote>
<p><span class="math-container">$$ax > 1... | Will Jagy | 10,400 | <p>safer to split into cases and confirm with the original item.</p>
<p>We can have the thing in the absolute value either positive or negative,</p>
<p>(I) positive
<span class="math-container">$$ ax-11 = 4x-10 $$</span>
<span class="math-container">$$ (a-4)x = 1 $$</span>
Since we demand positive integers, we get <s... |
4,326,073 | <p><a href="https://i.stack.imgur.com/RTyOy.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RTyOy.jpg" alt="enter image description here" /></a>
I came across questions in the free module section of my abstract algebra text. In the text, the notation <span class="math-container">$End_{R}(V)$</span> d... | QC_QAOA | 364,346 | <p>Let <span class="math-container">$\epsilon>0$</span> be given. By definition, there exists <span class="math-container">$N\in\mathbb{N}$</span> such that <span class="math-container">$n\geq N$</span> implies <span class="math-container">$|a_n-a|<\epsilon$</span>. This then implies <span class="math-container">... |
4,326,073 | <p><a href="https://i.stack.imgur.com/RTyOy.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RTyOy.jpg" alt="enter image description here" /></a>
I came across questions in the free module section of my abstract algebra text. In the text, the notation <span class="math-container">$End_{R}(V)$</span> d... | TravorLZH | 748,964 | <p>We rely on the following lemma:</p>
<p><strong>Lemma (Stolz-Césaro):</strong> Suppose <span class="math-container">$y_n\ge0$</span>, <span class="math-container">$x_n=o(y_n)$</span>, and <span class="math-container">$\sum_{k\le n}y_k\to+\infty$</span> as <span class="math-container">$n\to+\infty$</span>. Then</p>
<p... |
2,596,213 | <p>I'm having huge troubles with problems like this. I know the following:</p>
<p>$$\frac{\sin{x}}{x}=1-\frac{x^2}{3!}+\frac{x^4}{5!}-\frac{x^6}{7!}+O(x^7)$$</p>
<p>and </p>
<p>$$\ln{(1+t)}=t-\frac{t^2}{2}+\frac{t^3}{3}+O(t^4)$$</p>
<p>So</p>
<p>$$\ln{\left(1+\left(-\frac{x^2}{3!}+\frac{x^4}{5!}-\frac{x^6}{7!}+O(x... | Dylan | 135,643 | <p>If you want a series expansion up to $x^6$, then</p>
<p>$$ \left( -\frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + O(x^8)\right)^2 =
\frac{x^4}{(3!)^2} - 2\frac{x^6}{3!5!} + O(x^8) $$</p>
<p>$$ \left( -\frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + O(x^8)\right)^3 = -\frac{x^6}{(3!)^3} + O(x^8) $$</p>
<p... |
109,298 | <p>I'm an applied model theorist, and open image theorems are important in the mathematical structures I study (they limit the number of types of elements being realised, and therefore keep things model theoretically nice e.g. stable). </p>
<p>So I have some idea as to why these open image theorems should hold from a ... | Kevin Ventullo | 5,513 | <p>Well, this isn't explicitly diophantine, but here goes:</p>
<p>If $f$ is a level one weight $k$ eigenform with rational coefficients, the image of the attached Galois representation </p>
<p>$\rho_f:G_{\mathbb{Q}}\rightarrow GL_2(\hat{\mathbb{Z}})$ </p>
<p>is open in the subgroup $G$ defined by demanding </p>
<p>... |
1,019,078 | <p>Let $\alpha_1=[ 2,1,3,0] $
$\alpha_2=[ 1,1,1,-1] $, $\alpha_3=[ 2,-1,5,4] $, $\alpha_4=[ 1,2,0,-3] $, $\alpha_5=[ 3,1,6,1] $
be vectors from $\mathbb{R}^4$ . From vectors system ($\alpha_1,\alpha_2, \alpha_3, \alpha_4, \alpha_5 $) choose basis of vector space $V=lin(\alpha_1,\alpha_2, \alpha_3, \alpha_4, \alpha_5)\s... | mfl | 148,513 | <p>Since</p>
<p>$$\mathrm{rank}\begin{pmatrix}2 & 1 & 2 & 1 & 3\cr 1 & 1 & −1 & 2 & 1\cr 3 & 1 & 5 & 0 & 6\cr 0 & −1 & 4 & −3 & 1\end{pmatrix}=3$$</p>
<p>the vector subspace $\mathrm{span}\{\alpha_1,\cdots,\alpha_5\}$ has dimension $3.$ So, you need to f... |
3,729,851 | <p>So I have the following question here.</p>
<blockquote>
<p>Suppose that <span class="math-container">$y_1$</span> solves <span class="math-container">$2y''+y'+3x^2y=0$</span> and <span class="math-container">$y_2$</span> solves <span class="math-container">$2y''+y'+3x^2y=e^x$</span>. Which of the following is a solu... | Lutz Lehmann | 115,115 | <p>This is just linear algebra. On the left side you have a linear (differential, but that is not so important here) operator, call it <span class="math-container">$L$</span>. Then you are given <span class="math-container">$L(y_1)=0$</span> and <span class="math-container">$L(y_2)=f$</span>. Now you want to solve <spa... |
3,729,851 | <p>So I have the following question here.</p>
<blockquote>
<p>Suppose that <span class="math-container">$y_1$</span> solves <span class="math-container">$2y''+y'+3x^2y=0$</span> and <span class="math-container">$y_2$</span> solves <span class="math-container">$2y''+y'+3x^2y=e^x$</span>. Which of the following is a solu... | Community | -1 | <p>By linearity, it suffices to substitute the RHS:</p>
<ul>
<li><p>a) <span class="math-container">$3y_1-2y_2\to-2e^x,$</span></p>
</li>
<li><p>b) <span class="math-container">$y_1+2y_2\to2e^x,$</span></p>
</li>
<li><p>c) <span class="math-container">$2y_1-y_2\to-e^x,$</span></p>
</li>
<li><p>d) <span class="math-cont... |
1,113,760 | <blockquote>
<p>$\frac{4}{3} e^{3x} + 2 e^{2x} - 8 e^x$</p>
</blockquote>
<p>I have some confusion especially because of the e </p>
<p>how can I approach the solution?</p>
<p>The solution of the x-intercept is 0.838</p>
<p>Many thanks</p>
| bjd2385 | 167,604 | <p>\begin{align}
0&=\frac{4}{3}e^{3x}+2e^{2x}-8e^x\tag{1} \\[1em]
& = \frac{\frac{4}{3}e^{3x}+2e^{2x}-8e^{x}}{2e^x}\tag{2} \\[1em]
& = \frac{2}{3}e^{2x}+e^x-4\tag{3} \\[1em]
\end{align}
Now let $\xi=e^x,\therefore e^{2x}=\left(e^x\right)^2=\xi^2.$ This gives us
\begin{align}
0&=\frac{2}{3}\xi^2 +\xi-4\t... |
57,195 | <p>Let $f$ be a morphism of schemes $f: (X,\mathcal{O}_X)\to (Y,\mathcal{O}_Y)$, and $\mathcal{F},\mathcal{G}$ be sheaves of $\mathcal{O}_Y$-modules. I am trying to prove (I do NOT claim this to be true):</p>
<p>$f^{\ast}\mathcal{F}\otimes_{\mathcal{O}_X}f^{\ast}\mathcal{G}\cong f^{\ast}(\mathcal{F}\otimes_{\mathcal{O... | Martin Brandenburg | 1,650 | <p><em>Alternative proof, using only adjunctions.</em></p>
<p>First, notice that there is an isomorphism in $\mathsf{Mod}(Y)$</p>
<p>$$f_* \underline{\hom}_X(f^* G,H) = \underline{\hom}_Y(G,f^* H)$$</p>
<p>for $G \in \mathsf{Mod}(Y)$ and $H \in \mathsf{Mod}(X)$. In fact, on an open subset $V \subseteq Y$, we have</p... |
3,379,837 | <p>I know that the two semigroups <span class="math-container">$(\{0,1,2,\dots \},\times)$</span> and <span class="math-container">$(\{0,1,2,\dots \},+)$</span> are not isomorphic because if we want to map identity elements together then it can be see that we can't have injective function between them,but what can we ... | ΑΘΩ | 623,462 | <p>As you have noted, it is easy to see that <span class="math-container">$(\mathbb{N}, +)$</span> and <span class="math-container">$(\mathbb{N}, \cdot)$</span> are not isomorphic as semigroups, since the latter possesses an algebraic feature that the first one does not, namely an <strong>absorptive element</strong> (a... |
234,340 | <p>Suppose that I have two real-valued matrices $\bf{A}$ and $\bf{B}$. Both matrices are exactly the same size. I multiply both matrices together in a point-by-point fashion similar to the Matlab <code>A .* B</code> operation.</p>
<p>Under what conditions can I approximately separate $\bf{A}$ and $\bf{B}$ using Prin... | lerije | 48,917 | <p>If the entries are non-negative then you could use <a href="http://en.wikipedia.org/wiki/Non-negative_matrix_factorization" rel="nofollow">NMF</a> (non-negative matrix factorization). Or let $\textbf{C} = \textbf{A} \textbf{B}$. Then you could use singular value decomposition on $\textbf{C}$. </p>
|
617,927 | <p>Find the taylor expansion of $\sin(x+1)\sin(x+2)$ at $x_0=-1$, up to order $5$.</p>
<p><strong>Taylor Series</strong></p>
<p>$$f(x)=f(a)+(x-a)f'(a)+\frac{(x-a)^2}{2!}f''(a)+...+\frac{(x-a)^r}{r!}f^{(r)}(a)+...$$</p>
<p>I've got my first term...</p>
<p>$f(a) = \sin(-1+1)\sin(-1+2)=\sin(0)\sin(1)=0$</p>
<p>Now, I... | mathlove | 78,967 | <p>You did nothing wrong. However, building the Taylor series for each term is better and easier. </p>
<p>Also, using the decimal is not good. Keep using $\sin(1).$</p>
<p>Also, notice that actually $\sin(1)\not =0.8414709848$. This is only an approximate value.</p>
|
2,117,225 | <p>I need to find an expression that satisfies the qualifying conditions for a quintic polynomial.</p>
<p>$f(0)=3$ and $f(-2)=f(\frac{1}{2})=f(1)=0$.</p>
<p>With this information, I found that the zeros are $2, -\frac{1}{2},$ and $-1$.</p>
<p>By plugging $0$ into $f(x)$, I found that $F=3$ using the form $ax^5+bx^4... | Robert Israel | 8,508 | <p>Plugging in $x=-2$, $1/2$ and $1$ into $f(x)$ will give you three linear equations in the four remaining unknown parameters.</p>
|
2,117,225 | <p>I need to find an expression that satisfies the qualifying conditions for a quintic polynomial.</p>
<p>$f(0)=3$ and $f(-2)=f(\frac{1}{2})=f(1)=0$.</p>
<p>With this information, I found that the zeros are $2, -\frac{1}{2},$ and $-1$.</p>
<p>By plugging $0$ into $f(x)$, I found that $F=3$ using the form $ax^5+bx^4... | WW1 | 88,679 | <p>One set of possibilities is ...
$$ f(x) = k \left [ (x+2)^n(x-\frac12)^p(x-1)^q \right ]$$
where $n,p,q$ are positive integers that sum to $5$</p>
<p>Choose any three you like and then use $f(0)=3$ to calculate $k$</p>
|
19,098 | <p>There are already <a href="https://math.meta.stackexchange.com/questions/4277/people-who-ask-homework-questions-and-then-remove-them">a</a> <a href="https://meta.stackexchange.com/questions/155933/preventing-misuse-of-question-self-deletion">lot</a> <a href="https://math.meta.stackexchange.com/questions/8528/why-do-... | apnorton | 23,353 | <p>Self-deletion after receiving an answer is an abuse of the site. (We're meant to be a repository of question and answer pairs--if someone deletes their question immediately, they're working against the purpose of the site.)</p>
<p>Therefore, you shouldn't let the issue rest; if the person deletes again, keep flagg... |
19,098 | <p>There are already <a href="https://math.meta.stackexchange.com/questions/4277/people-who-ask-homework-questions-and-then-remove-them">a</a> <a href="https://meta.stackexchange.com/questions/155933/preventing-misuse-of-question-self-deletion">lot</a> <a href="https://math.meta.stackexchange.com/questions/8528/why-do-... | Scott Morrison | 28 | <p>Over at MathOverflow, we (the moderators, sometimes asking for help) have semi-regularly gone on an undeleting spree. We use judgement; if there's some genuine reason for deleting (even embarrassment) we leave it deleted, but if it's a worthwhile question and we can't see any reason it's fair game.</p>
|
3,817,340 | <p>I'm trying to draw the bio-hazard symbol for <a href="https://codegolf.stackexchange.com/questions/191294/draw-the-biohazard-symbol">a codegolf challenge</a> in Java, for which I've been given the following picture (later referred to as unit diagram):</p>
<p><a href="https://i.stack.imgur.com/fIsNl.png" rel="nofollo... | LCFactorization | 148,887 | <p>Here there is an answer: <a href="https://www.reddit.com/r/geogebra/comments/on54iw/how_to_create_such_a_biohazard_symbol_in_geogebra/" rel="nofollow noreferrer">https://www.reddit.com/r/geogebra/comments/on54iw/how_to_create_such_a_biohazard_symbol_in_geogebra/</a></p>
<p>If you know how to use geogebra, the soluti... |
1,675 | <p>This is a follow-up to <a href="https://mathoverflow.net/questions/1039/explicit-direct-summands-in-the-decomposition-theorem">this post</a> on the Decomposition Theorem. Hopefully, this will also invite some discussion about the theorem and perverse sheaves in general.</p>
<p>My question is how does one use the De... | Mike Skirvin | 916 | <p>There's a recent paper by Mark Andrea de Cataldo and Luca Migliorini (<a href="http://arxiv.org/abs/0712.0349" rel="nofollow">http://arxiv.org/abs/0712.0349</a>) which gives an excellent introduction to the decomposition theorem. In particular, they discuss semi-small maps in the context of Springer theory and Hilb... |
4,236,878 | <p>Given a symmetric matrix <span class="math-container">$S$</span> and positive definite matrix <span class="math-container">$B$</span>, with <span class="math-container">$S,B \in \mathbb{R}^{n \times n}$</span> can one prove that</p>
<p><span class="math-container">\begin{align*}
\text{tr}((S-B)B) \le -\mu(S) \text{t... | FelipeCruzV10 | 957,011 | <p>If <span class="math-container">$\mu(S)>0$</span> it doesn't necessarily hold.</p>
<p>Taking <span class="math-container">$S=\begin{pmatrix}100 & 50 \\ 50 & 100\end{pmatrix}$</span> and <span class="math-container">$B=\begin{pmatrix}0.1 & 0 \\ 0 & 0.1\end{pmatrix}$</span>, we have that <span class... |
361,201 | <p>Let $\left\{ x_\alpha : \alpha \in \mathscr{A}\right\} \subset (0, + \infty ) $ be a set of positive real numbers such that for every countable subcollection $ \left\{ x_{\alpha_n} \right\} $ of distinct points it holds $ x_{\alpha_n} \rightarrow 0 $. Then $ \mathscr{A} $ is a countable set. \</p>
<p>I think that t... | Asaf Karagila | 622 | <p><strong>Hint:</strong> Let $X$ denote this collection. Prove that for every $n\in\Bbb N$, $X\cap\left(\frac1n,+\infty\right)$ must be finite.</p>
|
361,201 | <p>Let $\left\{ x_\alpha : \alpha \in \mathscr{A}\right\} \subset (0, + \infty ) $ be a set of positive real numbers such that for every countable subcollection $ \left\{ x_{\alpha_n} \right\} $ of distinct points it holds $ x_{\alpha_n} \rightarrow 0 $. Then $ \mathscr{A} $ is a countable set. \</p>
<p>I think that t... | Hagen von Eitzen | 39,174 | <p>For $\epsilon>0$ the set $A_\epsilon:=\{x_\alpha\mid x_\alpha>\epsilon\}$ must be finite as otherwise we'd find a countable subcollection with limit $\ge\epsilon$. Therefore
$$\{x_\alpha\mid \alpha\in\mathscr A\}=\bigcup_{n\in\mathbb N}A_{\frac1n}$$
is the countable union of finite sets, hence countable.</p>
|
2,107,685 | <p><a href="https://i.stack.imgur.com/GtU6e.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/GtU6e.png" alt="laaa"></a></p>
<p>I have to represent the function on the left as a power series, and this is the solution to it but I don't know how to calculate this for example when n=1?</p>
| NiU | 163,915 | <p>In the second case where $\Omega=[\tfrac{1}{4}, \tfrac{1}{3}) \cup ([\tfrac{1}{3}, \tfrac{1}{2}) \cap \mathbb{Q})$ there is no definite answer <em>even if we assume the existence of a cyclic vector</em>.</p>
<p>First, consider $A$ to be multiplication by $f(x)=x$ on $L^2[0,1]$ with the usual Lebesgue measure. A cyc... |
871,581 | <p>I am trying to prove the identity below to help with the simplification of another function that I'm investigating as it doesn't appear to be a standard trig identity.</p>
<p>$$
\tan\left(x\right) + \tan\left( y \right) = \frac{{\sin\left( {x + y} \right)}}{{\cos\left( x \right)\cos\left( y \right)}}
$$</p>
<p>Any... | Blue | 409 | <p>For fun, here's a picture-proof:</p>
<p><img src="https://i.stack.imgur.com/vKC0Ym.png" alt="enter image description here"></p>
<p>$$\begin{align}
2\;|\triangle OAB| = \qquad\qquad |\overline{OR}|\;|\overline{AB}| \;&=\; |\overline{OA}|\;|\overline{OB}|\;\sin\angle AOB \\[6pt]
1\cdot (\;\tan\alpha + \tan\beta\... |
636,467 | <p>What is it that makes something a paradox? It seems to me that paradoxes are just, in many cases, misunderstandings about the properties some object can have and so misunderstandings about definitions. Is there something I might be missing? How is this kind of thought handled in logic?</p>
| Peter Smith | 35,151 | <p>Two thoughts to add to Carl Mummert's fine answer. He writes that a serious use of the work of "paradox" </p>
<blockquote>
<p>refers to a result that shows that a particular naive intuition is not sound. </p>
</blockquote>
<p>Perhaps it is slightly better, as indeed his own examples show, to say that the interes... |
636,467 | <p>What is it that makes something a paradox? It seems to me that paradoxes are just, in many cases, misunderstandings about the properties some object can have and so misunderstandings about definitions. Is there something I might be missing? How is this kind of thought handled in logic?</p>
| MJD | 25,554 | <p>You may enjoy W.V.O. Quine's essay "The ways of paradox", which tries to answer many of the same questions. Quine suggests early on:</p>
<blockquote>
<p>May we say in general, then, that a paradox is just any conclusion that at first sounds absurd but that has an argument to sustain it? In the end I think this a... |
514 | <p>I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.</p>
<p>I'm sure that everyone here is familiar with it; it describes an operation on a natural number – <span class="math-container">$n/2$</span> if it is even, <span class="math-container">$3n+1$</spa... | Doug Chatham | 273 | <p>Further counterexamples can be found here: <a href="https://mathoverflow.net/questions/15444/the-phenomena-of-eventual-counterexamples">https://mathoverflow.net/questions/15444/the-phenomena-of-eventual-counterexamples</a></p>
|
514 | <p>I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.</p>
<p>I'm sure that everyone here is familiar with it; it describes an operation on a natural number – <span class="math-container">$n/2$</span> if it is even, <span class="math-container">$3n+1$</spa... | Mr Pie | 477,343 | <p>Let’s take the number $12$. This number is not prime. It is a composite number, equal to $2^2\times 3$. Also, $121$ is not prime either. It is equal to $11^2$. And $1211$ is not prime as well. It is equal to $7\times 173$. Now you might notice a pattern here.</p>
<blockquote>
<p>Let $$12\,\|\, \underbrace{1\,\|\,... |
443,578 | <blockquote>
<p>Is the limit
$$
e^{-x}\sum_{n=0}^N \frac{(-1)^n}{n!}x^n\to e^{-2x} \quad \text{as } \ N\to\infty \tag1
$$
uniform on $[0,+\infty)$? </p>
</blockquote>
<p>Numerically this appears to be true: see the difference of two sides in (1) for $N=10$ and $N=100$ plotted below. But the convergence is ve... | Pedro | 23,350 | <p>Credits should go to <a href="https://math.stackexchange.com/users/46120/landscape">Landscape</a>.</p>
<p>Define $$r_n(x)=\sum_{k=n+1}^\infty (-1)^k\frac{x^k}{k!}$$</p>
<p>Note that by Taylor's theorem with <a href="https://en.wikipedia.org/wiki/Taylor's_theorem#Explicit_formulae_for_the_remainder" rel="nofoll... |
2,621,932 | <p><strong>Question:</strong> In a chess match, there are 16 contestants. Every player has to be each other player (like a round-robin). The player with the most wins/points wins the tournament.</p>
<p>a) How many games must be played until there is a victor? </p>
<p>b) If every player has to team up with each other ... | Rohan Shinde | 463,895 | <p>For the first part for a better clarity begin with some smaller cases. </p>
<p>Let's consider there are 3 people playing the chess. The what will be the number of matches until a Victor is decided. Through simple counting we get this as 3( which is simply $\binom {3}{2}$ )</p>
<p>Let's consider 4 people to get ... |
1,055,091 | <p>I've been asked to estimate a y coordinate by using differentials. This normally isn't overly difficult, however, I'm not sure what to do in a case like this when y cannot be separated and used as a function of x. Can anyone point me in the right direction? I suspect I'll have to use implicit differentiation but I c... | Matt Samuel | 187,867 | <p>Use the Leibniz formula, namely $d(fg)=gdf+fdg$. We have
$$d(2x^3+2y^3-9xy)=6x^2dx+6y^2dy-d(9xy)=6x^2dx+6y^2dy-9ydx-9xdy=(6x^2-9y)dx+(6y^2-9x)dy$$
so since this is equal to $0$ we have
$$(6x^2-9y)dx=(9x-6y^2)dy$$
hence
$$dy=\frac{6x^2-9y}{9x-6y^2}dx$$</p>
<p>Using the standard routine in assuming $dy\approx \Delta ... |
103,776 | <p>I am curious as to why Wolfram|Alpha is graphing a logarithm the way that it is. I was always taught that a graph of a basic logarithm function $\log{x}$ should look like this:</p>
<p><img src="https://i.stack.imgur.com/3SRqI.png" alt="enter image description here"></p>
<p>However, Wolfram|Alpha is graphing it lik... | Tib | 23,349 | <p>As well as being an $\mathbb{R^+} \to \mathbb{R}$ function, the logarithm can also be extended to a multi-valued complex function. Wolfram Alpha interprets the logarithm as the complex logarithm, then restricts it to real line again for graphing. See <a href="http://enwp.org/wiki/Complex_logarithm" rel="nofollow">ht... |
340,855 | <p>Say if there is a matrix A:</p>
<p>$$\begin{bmatrix} 1 & 2 & 0 & 2 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$</p>
<p><strong>What the column space of A?</strong> : I am confused whether to exclude NON-pivot columns.</p>
<p><strong>What is the dimension of column space?</... | Jim | 56,747 | <p>The column space is not a list of vectors so it's not clear what you mean when you ask if you should exclude non-pivot columns. The column space is the linear span of the columns. Each column (including the non-pivot columns) is contained in this space.</p>
<p>What you may be confusing yourself with is the column... |
897,043 | <p>I'm having issues getting my head around cartesian products and their cardinalities.</p>
<p>$A = \{0, 1, \{2, 3, 4\}\}$<br>
$B = \{1,5\}$<br>
$D = B \times N$ (where $N$ is the set of natural numbers)</p>
<p><strong>The first problem:</strong> What is the cardinality of:</p>
<p>(a) $A \times B$ (cartesian produ... | Fargle | 157,905 | <p>For (a) and (b), you were right, but more specifically, the cardinality of $A \times D$ is $\aleph_0$, or countable infinity. (The same cardinality as $\Bbb N$.</p>
<p>For part 2 (a), you were wrong, however. $D$ does not contain $\Bbb N$, because $1 \neq (b,n)$ for any $b \in B, n \in \Bbb N$. Put more simply, $A ... |
125,610 | <p>I have question about sets. I need to prove that: $$X \cap (Y - Z) = (X \cap Y) - (X \cap Z)$$</p>
<p>Now, I tried to prove that from both sides of the equation but had no luck.</p>
<p>For example, I tried to do something like this: $$X \cap (Y - Z) = X \cap (Y \cap Z')$$ but now I don't know how to continue.</p>
... | Arturo Magidin | 742 | <p>You can do it two ways: with manipulations using the properties of unions, intersections, and complements, or through double inclusion.</p>
<ol>
<li><p>To prove it by double inclusion, we must show that $X\cap(Y-Z)\subseteq (X\cap Y)-(X\cap Z)$, and that $(X\cap Y)-(X\cap Z)\subseteq X\cap(Y-Z)$.</p>
<p>I'll show ... |
1,848,222 | <p>Very simple and quick question. Usually distribution notation is such that you give the name of the distribution, then its mean, and finally the variance, for example for normal distribution:</p>
<p>$$N(0,1)$$</p>
<p>The 0 means that the distribution has mean zero, and the 1 tells that the variance is one. However... | Em. | 290,196 | <p>I don't know where you are getting this
"Usually distribution notation is such that you give the name of the distribution, then its mean, and finally the variance"
from.</p>
<p>As for the uniform distribution, the best way to imagine the uniform distribution is to know where it <em>starts</em> and where it <em>ends... |
3,953,153 | <p>I need help to prove this:
If <span class="math-container">$\gcd(a,b)=1$</span> then <span class="math-container">$\gcd(a+b, a^2-ab+b^2)$</span> is equal to <span class="math-container">$1$</span> or <span class="math-container">$3$</span>.
I have done this:</p>
<p>Let <span class="math-container">$d$</span> be the ... | fleablood | 280,126 | <p>Well try to factor <span class="math-container">$a^2 -ab +b^2$</span> into terms of <span class="math-container">$a+b$</span> or <span class="math-container">$a,b$</span>.</p>
<p><span class="math-container">$a^2 -ab + b^2 = a^2 + 2ab + b^2 - 3ab= (a+b)^2 -3ab$</span> so</p>
<p><span class="math-container">$\gcd(a+b... |
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