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 |
|---|---|---|---|---|
1,264,195 | <p>So there is a way of extending the set $N$ of natural numbers with 0, equipped with ordinary multiplication, to its Grothendieck group, the group of integers with respect to addition.</p>
<p>This group, $Z$, is the <em>best</em> extension of $N$ in the sense that ...? (question)</p>
<p>So, here are my questions:</... | jgon | 90,543 | <p>Since you're interested in polynomial curves in the plane, I thought I'd write up the following.</p>
<p>Suppose your curve $C$ is the zero set in the plane $\mathbb{R}^2$ or $\mathbb{C}$ of a polynomial function $f$ in $\mathbb{R}[X,Y]$ (i.e. the set of points $x\in \mathbb{R}^2$ such that $f(x)=0$), then for any l... |
863,561 | <p>A Lambertian surface reflects or emits radiation proportional to the cosine of the angle subtended between the exiting angle and the normal to that surface. The integral of surface of the hemisphere which describes the exiting radiance is supposed to be equal to π. Is there a way I can prove that the surface of the ... | Ron Gordon | 53,268 | <p>Note that <strong>intensity</strong> is proportional to $|\cos{\theta}|$, as this is a positive quantity. The integrated quantity you seek (total intensity divided by radiance) is then</p>
<p>$$\pi \int_0^{\pi} d\theta \, \sin{\theta} |\cos{\theta}| = 2 \pi \int_0^{\pi/2} d\theta \, \sin{\theta} \cos{\theta} = \p... |
3,996,090 | <p>I believe I have found the recurrence relation to be
<span class="math-container">$$B\left(n\right)=B\left(n-1\right)+2^{n-1}-1$$</span>
with Initial Condition B(0)=0 (I am a bit unsure about the initial condition though but I think it is correct)</p>
<p>Now I am trying to solve B(n) using iteration, this is what I ... | Brian M. Scott | 12,042 | <p>You’re on the right track:</p>
<p><span class="math-container">$$\begin{align*}
B(n)&=B(n-1)+2^{n-1}-1\\
&=B(n-2)+2^{n-2}+2^{n-1}-2\\
&=B(n-3)+2^{n-3}+2^{n-2}+2^{n-1}-3\\
&\;\;\vdots\\
&=B(n-k)-k+\sum_{i=1}^k2^{n-i}\\
&\;\;\vdots\\
&=B(0)-n+\sum_{i=1}^n2^{n-i}\\
&=-n+\sum_{i=0}^{n-1}2... |
115,821 | <p>Here are some sample data:</p>
<pre><code>data = {{1}, {50., 53, 52, 52}, {100., 105, 104, 104}, {150., 157, 156,
156}, {200., 209, 208, 208}, {250., 261, 260, 260}, {300., 313, 312,
313}, {2}, {50., 53, 52, 51}, {100., 106, 105, 102}, {150., 158,
157, 153}, {200., 211, 210, 204}, {250., 265, 264, 256}, {300., 31... | Karsten 7. | 18,476 | <p>The easiest workaround is to use</p>
<pre><code>Table[Labeled[
ListVectorPlot[data[[i]], PlotRange -> All,
VectorColorFunction ->
Function[{x, y, vx, vy, n},
ColorData["Rainbow"][Rescale[n, {min, max}]]],
VectorColorFunctionScaling -> False, ImageSize -> 300],
BarLegend[{"Rainbow"... |
61,509 | <p>I'm trying to read Elias Stein's "Singular Integrals" book, and in the beginning of the second chapter, he states two results classifying bounded linear operators that commute (on $L^1$ and $L^2$ respectively).</p>
<p>The first one reads:</p>
<p>Let $T: L^1(\mathbb{R}^n) \to L^1(\mathbb{R}^n)$ be a bounded linear ... | robjohn | 13,854 | <p>For the first one:</p>
<p>Let $\phi_\epsilon(x)=\phi(x/\epsilon)/\epsilon^n$ where
$$
\phi(x)=\left\{\begin{array}{cl}c\;e^{|x|^2/(|x|^2-1)}&\text{if }|x|<1\\0&\text{if }|x|\ge1\end{array}\right.
$$
and $c$ is chosen so that $\int_{\mathbb{R}^n}\phi(x)\;\mathrm{d}x=1$.</p>
<p>$\|T\phi_\epsilon\|_{L^1}... |
39,387 | <p>It is usual to use second, third and fourth moments of a distribution to describe certain properties. Do partial moments or moments higher than fourth describe any useful properites of a distribtution.</p>
| Spencer | 4,281 | <p>The typical example I remember being given at school is that of <a href="http://en.wikipedia.org/wiki/Kurtosis" rel="nofollow">kurtosis</a>, which uses fourth moment and is easy to understand. </p>
<p>Wiki says: "Higher kurtosis means more of the variance is the result of infrequent extreme deviations, as opposed t... |
1,669,096 | <p>How do I show that <span class="math-container">$\ell^{ \infty}$</span> is a normed linear space,
where <span class="math-container">$\ell^{ \infty}$</span> is define as <span class="math-container">$$\|\{a_n\}_{n=1}^{\infty}\|_{\ell^\infty}=\sup_{1 \leq k \leq \infty} |a_k|?$$</span>
There are three properties that... | Mel | 566,704 | <p>Well begin by noticing that for <span class="math-container">$a,b\in \mathbb{R}$</span> and <span class="math-container">$(x_n),(y_n)\in \ell^\infty$</span> you have that <span class="math-container">$(ax_n+by_n)$</span> is bounded, therefore it belongs in <span class="math-container">$\ell^\infty$</span> as well.</... |
2,363,840 | <blockquote>
<p>As the heading states, find all $p(x)$ for $p(x+1)=p(x)+3x(x+1)+1$ for all real $x$.</p>
</blockquote>
<p>I have no idea how to approach this. Any solution or guide how to solve these kinds of questions would be appreciated! </p>
| Ennar | 122,131 | <p>Let $g(x) = p'''(x)$. Deriving the condition $p(x+1) = p(x) + 3x(x+1)+1$ three times, we get $$p'''(x+1) = p'''(x)\implies g(x+1) = g(x).$$</p>
<p>Now, $g$ is still polynomial and the last line tells us that if $\alpha$ is a root of $g$, so is $\alpha \pm 1$ and hence, by induction, so is $\alpha + n$ for any integ... |
1,303,485 | <p>Evaluate the integral $$\int_{C}\frac{z^2}{z^2+9}dz$$
where C is the circle $|z|=4$</p>
<p>I know that if f is analytic in simply connected domain $D$, $C$ a simple closed positively oriented contour that lies in D and $z_o$ lies interior to $C$, then
$$\int_{C}\frac{f(z)}{z-z_o}dz=2\pi i f(z_o)$$</p>
<p>But for t... | Ron Gordon | 53,268 | <p>You can rewrite</p>
<p>$$\frac{z^2}{z^2+9} = 1 - \frac{9}{z^2+9} = 1 - \frac{9}{(z+i 3)(z-i 3)} = 1+\frac{i 3/2}{z-i 3} - \frac{i 3/2}{z+i 3}$$</p>
<p>Now apply Cauchy's integral formula.</p>
|
3,084,479 | <p><span class="math-container">$h\in \mathbb{R}$</span>, because we have defined the Trigonometric Functions only on <span class="math-container">$\mathbb{R}$</span> so far.</p>
<p>I have a look at <span class="math-container">$e^{ih}=\sum_{k=0}^{\infty}\frac{(ih)^k}{k!}=1+ih-\frac{h^2}{2}+....$</span> </p>
<p><stro... | Paramanand Singh | 72,031 | <p>Here is a simple proof which I first read in Hardy's <em>A Course of Pure Mathematics</em>.</p>
<p>To reiterate the question for clarity we have <span class="math-container">$h\in\mathbb{R} $</span> and the symbol <span class="math-container">$e^{ih} $</span> is defined by the series <span class="math-container">$\... |
2,809,090 | <p>The stochastic vector $(X,Y)$ has a continuous distribution with pdf:
$$f(x,y) =
\begin{cases}
xe^{-x(y+1)} & \text{if $x,y>0$} \\[2ex]
0 & \text{otherwise}
\end{cases}$$<br>
Define $Z:=XY$.<br>
I would like to know what exactly $XY$ is. It seems to me that the function $f(x,y)$ has but one output, so wh... | egreg | 62,967 | <p>Your expression is full of red herrings. Try first removing the denominators (you can multiply by $1/6$ at the end) and writing $t$ for $x/2$. You get:
$$
\cos t\cos 3t -3\cos^2t+3\sin^2t + \sin t\sin 3t
$$
You could be tempted to expand $\cos3t$ and $\sin3t$, which is of course possible, but it would be longer than... |
1,382,587 | <p>In $\mathbb R^3$, let $C$ be the circle in the $xy$-plane with radius $2$ and the origin as the center, i.e., </p>
<p>$$C= \Big\{ \big(x,y,z\big) \in \mathbb R^3 \mid x^2+y^2=4, \ z=0\Big\}.$$</p>
<p>Let $\Omega$ consist of all points $(x,y,z) \in \mathbb R^3$ whose distance to $C$ is at most $1$. Compute
$$\int... | zzchan | 252,453 | <p>Since you only want hints...</p>
<p>For each point on the circle of radius $2$ consider a solid sphere of radius $1$. Now consider the union of all these spheres. What do you get?</p>
|
1,392,257 | <p><strong>The definition of a conjugate element</strong> </p>
<p>We say that $x$ is conjugate to $y$ in $G$ if $y = g^{-1}xg $ for some $g \in G$</p>
<p>Now for the group $G=Q_8$ , we have the group presentation $$Q_8 = \big<a,b: a^4 =1,b^2 = a^2, b^{-1}ab = a^{-1} \big>$$</p>
<p>Now the elements of $Q_8$ a... | Geoff Robinson | 13,147 | <p>In a finite group $G$ of odd order $2n+1$, only the identity is conjugate to its inverse. For if $gxg^{-1} = x^{-1}$, then it is easy to see that $g^{2}xg^{-2} =x$, so that $g^{2}$ commutes with $x$. But $g = (g^{2})^{n+1}$ is a power of $g^{2}$, so that $g$ commutes with $x$. Hence $x = x^{-1}$, so that $x^{2} = 1_... |
306,788 | <p>I understand determinants but I cannot understand the following question, can someone explain it to me ?</p>
<p>Suppose that a $4 x 4$ matrix with rows $v_1,v_2,v_3$ and $v_4$ has determinant det A = -1. Find the following determinants: </p>
<p>$$det \begin{bmatrix}v_1\\6v_2\\v_3\\v_4 \end{bmatrix}=$$
$$det \beg... | Clayton | 43,239 | <p>Hint: For the first one, use the property that $$\det(A\cdot B)=\det(A)\cdot\det(B).$$See if you can change the identity matrix just right to get the 'new' matrix, where the second row is multiplied by $6$.</p>
<p>For the second one, use a similar idea, but this time the rows of the identity matrix get jumbled up a... |
3,430,008 | <p>Currently stuck on the last part of 15.2.8(e) of this problem:</p>
<p><a href="https://i.stack.imgur.com/UvGJK.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/UvGJK.png" alt="enter image description here"></a></p>
<p>I don't know how to apply Fubini's theorem since one index relies on the other.... | Matematleta | 138,929 | <p>I think Rudin's approach is very clear. Following him then, without loss of generality <span class="math-container">$a=0,$</span> so <span class="math-container">$(b-s,b+s)\subseteq (-r,r).$</span> Then, by the binomial theorem,</p>
<p><span class="math-container">$\sum^{\infty}_{n=0}c_nx^n=\sum^{\infty}_{n=0}c_n((... |
4,435,088 | <p>In an <span class="math-container">$8×8$</span> table one of the square is colored black and all the others are white . Prove that one cannot make all the boxes white by recoloring the rows and columns . "Recoloring" is the operation of changing the color of all boxes in a row or in a column .</p>
<p>This ... | JMP | 210,189 | <p>The only moves that affect the corner cells are the top/bottom rows and left/right columns.</p>
<p>Neither of these alter the total parity of the corner cells, and as the initial parity is <span class="math-container">$1$</span>, parity <span class="math-container">$0$</span> is unachievable.</p>
|
3,742,456 | <blockquote>
<p>If <span class="math-container">$ABC$</span> is a right angled triangle at <span class="math-container">$C$</span> then prove that <span class="math-container">$a^n+b^n<c^n$</span> for all <span class="math-container">$n>2$</span></p>
</blockquote>
<p>This is an olympiad book problem. I know that ... | atul ganju | 788,520 | <p>Well, we know <span class="math-container">$a^2+b^2 = c^2$</span> and we know <span class="math-container">$a\lt c$</span> and <span class="math-container">$b\lt c$</span> so we can write this:</p>
<p><span class="math-container">$$a^n+b^n = a^2a^{n-2}+b^2b^{n-2} \leq a^2(\max\{a, b\})^{n-2}+b^2(\max\{a, b\})^{n-2} ... |
118,074 | <p>Just based on some reading, I know that every Möbius transformation is a bijection from the Riemann sphere to itself. </p>
<p>I'm curious about the converse. For any holomorphic bijection on the sphere, why is it necessarily a Möbius transformation? Is there a proof or reference of why this converse is true? Thanks... | M.C. | 752,637 | <p>Here is another proof using results from the theory of Riemann surfaces.</p>
<p>Any holomorphic automorphism <span class="math-container">$F\colon\mathbb{P}^1\rightarrow \mathbb{P}^1$</span> of the Riemann sphere <span class="math-container">$\mathbb{P}^1$</span> can be viewed as a meromorphic function <span class="... |
2,732,562 | <p>I'm trying to figure out how to find the number of ternary strings of length $n$ that have 3 or more consecutive 2's.
So far I've been able to establish that there is $n(2^{n-1})$ with a single 2.
And I think (but am not certain) that this can be extrapolated to the number of strings with a single group of 2's of le... | Christian Blatter | 1,303 | <p>Denote by $c_n$ the number of ternary words containing no $222$. Then $c_1=3$, $c_2=9$, $c_3=26$, and we have the recursion
$$c_n=2c_{n-1}+2c_{n-2}+2c_{n-3}\qquad(n\geq4)\ .\tag{1}$$
<em>Proof.</em> An admissible word $w$ can begin with $Z$, $2Z$, or $22Z$ with $Z\in\{0,1\}$, followed by an admissible word $w'$.</p>... |
2,011,003 | <p>I stumbled upon this logic question in a math class recently. </p>
<p>My teacher told us that a statement that is not tested/is empty is true. For example, that if I stated that: "if the team A wins the game, I am gonna buy you a coke", and then team B goes on and wins the game, the statement would be true, indepen... | PMar | 388,671 | <p>The short answer is, this is true because we have adopted that resolution by convention, because, for all its faults, doing so comes closest to resembling the intuitive notion of 'if-then' while having a definite logical value in all mathematical situations. To see what I am getting at, consider the following 'trut... |
1,095,621 | <p>I am looking for a way to integrate $$\int \sqrt{x^2-4}\ dx $$ using trigonometric substitutions. </p>
<p>All my attempts so far lead to complicated solutions that were uncomputable.</p>
| Git Gud | 55,235 | <p>I'll focus on $\displaystyle \int \sqrt{x^2-1}\,\mathrm dx$ for simplicity. This is how I think about this sort of problem when I don't remember the appropriate substitution. </p>
<p>If you suspect that it's useful to use a trigonometric substitution, since the fundamental property $$\forall \theta \in\mathbb R\lef... |
1,095,621 | <p>I am looking for a way to integrate $$\int \sqrt{x^2-4}\ dx $$ using trigonometric substitutions. </p>
<p>All my attempts so far lead to complicated solutions that were uncomputable.</p>
| Dylan | 135,643 | <p>Using the substitution $x = \sec \theta$, we end up with</p>
<p>$$ 4\int \sec \theta \tan^2 \theta \,d\theta $$</p>
<p>This can be done using integration by parts</p>
<p>$$ \begin{align} \int \sec \theta \tan^2 \theta \,d\theta &= \int \tan \theta \,d(\sec \theta)
\\&= \tan \theta \sec \theta - \int \sec... |
3,054,898 | <h3>Problem</h3>
<p>Evaluate <span class="math-container">$$\int_0^{2\pi}(t-\sin t)(1-\cos t)^2{\rm d}t.$$</span></p>
<h3>Comment</h3>
<p>It's very complicated to compute the integral applying normal method. I obtain the result resorting to the skillful formula</p>
<blockquote>
<p><span class="math-container">$$\int_0^... | mechanodroid | 144,766 | <p>Notice that your integral is in fact an area integral of the function <span class="math-container">$(x,y) \mapsto x$</span> over one arch of a cycloid given by
<span class="math-container">$$\begin{cases} x = t - \sin t \\ y = 1-\cos t\end{cases}$$</span>
Indeed, if we assume that the cycloid is explicitly given by ... |
3,054,898 | <h3>Problem</h3>
<p>Evaluate <span class="math-container">$$\int_0^{2\pi}(t-\sin t)(1-\cos t)^2{\rm d}t.$$</span></p>
<h3>Comment</h3>
<p>It's very complicated to compute the integral applying normal method. I obtain the result resorting to the skillful formula</p>
<blockquote>
<p><span class="math-container">$$\int_0^... | Nosrati | 108,128 | <p>With substitution <span class="math-container">$t=\pi+x$</span> we see that
<span class="math-container">$$I=\int_0^{2\pi}(t-\sin t)(1-\cos t)^2{\rm d}t=\int_{-\pi}^{\pi}(\pi+x+\sin x)(1+\cos x)^2{\rm d}x.$$</span>
the part <span class="math-container">$(x+\sin x)(1+\cos x)^2$</span> is an odd function so it's integ... |
2,130,807 | <p>How to Prove $G$ is connected, if $G$ is an acyclic graph on $n \ge 1$ vertices containing exactly $n − 1$ edges?</p>
| Smylic | 100,361 | <p>Induction on $n$ is a good option. Hint:</p>
<blockquote class="spoiler">
<p> If $|G| \ge 2$ and $G$ is acyclic and $G$ has at least one edge then $G$ has a vertex of degree 1.</p>
</blockquote>
|
3,267,499 | <p>Let <span class="math-container">$k$</span> be a field.
<span class="math-container">$k[x,y]$</span> is a UFD by the following known argument taken from <a href="https://en.wikipedia.org/wiki/Unique_factorization_domain" rel="nofollow noreferrer">wikipedia</a>:
"If <span class="math-container">$R$</span> is a UFD, t... | Con | 682,304 | <p>1) Localization of a UFD is again a UFD (at least if you are not inverting <span class="math-container">$0$</span>). Maybe you can try to prove that. </p>
<p>2) Is there any condition on <span class="math-container">$k$</span>?</p>
|
4,650,979 | <blockquote>
<p>Solve the following recurrence equation: <span class="math-container">$T(n) = T(n-2)+n^2$</span>, having <span class="math-container">$T(0)=1$</span>, <span class="math-container">$T(1)=5$</span>.</p>
</blockquote>
<p>I need to solve this equation but when I get to the particular solution with <span cla... | Alexander Burstein | 499,816 | <p>Since <span class="math-container">$n^2=2\binom{n}{2}+\binom{n}{1}$</span>, it follows from the hockey-stick identity that
<span class="math-container">$$
\begin{split}
\sum_{k=0}^{n}{k^2}&=2\binom{n+1}{3}+\binom{n+1}{2}=\frac{(n+1)n(n-1)}{3}+\frac{(n+1)n}{2}\\
&=\frac{(n+1)n}{6}(2(n-1)+3)=\frac{n(n+1)(2n+1)... |
1,664,081 | <p>Solve the equation $2^x - 3^{x-1}=-(x+2)^2$</p>
<p>How I got this question? I created this question so I know the answer. The answer is 5. But I have no idea how to solve it. Take note that I cannot do logarithm, guess and check and modulus. Does anybody know how to solve this? I have no idea how to start.</p>
<p>... | SS_C4 | 242,290 | <p>The only solution is $x=5$.</p>
<p>You can see this by graphing the LHS and RHS separately and seeing that there is only one point of intersection.
<a href="https://i.stack.imgur.com/1JJWv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/1JJWv.png" alt="Graph"></a></p>
|
1,664,081 | <p>Solve the equation $2^x - 3^{x-1}=-(x+2)^2$</p>
<p>How I got this question? I created this question so I know the answer. The answer is 5. But I have no idea how to solve it. Take note that I cannot do logarithm, guess and check and modulus. Does anybody know how to solve this? I have no idea how to start.</p>
<p>... | Piquito | 219,998 | <p>HINT.-You have $$3^{x-1}-(x+2)^2=2^x$$ Searching for an eventual rational solution one has $$(3^{\frac{x-1}{2}}+x+2)(3^{\frac{x-1}{2}}-x-2)=2^x$$
We do now $$3^{\frac{x-1}{2}}+x+2=2^{x-h}$$ $$3^{\frac{x-1}{2}}-x-2=2^h$$ Hence
$$2x+4=2^{x-h}-2^h\iff x+2=2^{x-h-1}-2^{h-1}\qquad (*)$$ Trying with $h=1$ we find out the... |
222,555 | <p>I would like to find a simple equivalent of:</p>
<p>$$ u_{n}=\frac{1}{n!}\int_0^1 (\arcsin x)^n \mathrm dx $$</p>
<p>We have:</p>
<p>$$ 0\leq u_{n}\leq \frac{1}{n!}\left(\frac{\pi}{2}\right)^n \rightarrow0$$</p>
<p>So $$ u_{n} \rightarrow 0$$</p>
<p>Clearly:</p>
<p>$$ u_{n} \sim \frac{1}{n!} \int_{\sin(1)}^1 (... | WimC | 25,313 | <p>Substitute $x = \sin(t)$ to get</p>
<p>$$
u_n = \frac{1}{n!}\int_0^{\pi/2} t^n \cos(t) dt.
$$</p>
<p>The following equalities hold for all integers $m \geq 0$ (which can be checked by partial integration):</p>
<p>$$
\frac{1}{n!}\int_0^{\pi/2} t^n \left(\tfrac{\pi}{2} - t \right)^m dt = \frac{m!}{(n + m + 1)!}\lef... |
2,698,121 | <p>Came across a question about CDF and PDF in my homework:</p>
<p><a href="https://i.stack.imgur.com/bYqvr.jpg" rel="nofollow noreferrer">Click here for the question</a></p>
<p><a href="https://i.stack.imgur.com/bYqvr.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/bYqvr.jpg" alt="enter image desc... | Dave | 334,366 | <p>I'll get you going on the CDF. You're correct that the derivative of the CDF is the PDF. By definition, the CDF (denote it $F(x)$) is $F(x):=P(X<x)=\int_{-\infty}^xf(x)~dx$, where $f(x)$ is the PDF. The PDF has a piecewise definition, and the support is cut up into sections. To compute the CDF we need to examine ... |
1,089,635 | <blockquote>
<p>Is it possible to display a result in Mathcad as a function of $\pi$? </p>
</blockquote>
<p>I'm studying physics and I need to show exact results at the exams. I know I can set Mathcad to give me the result in decimals or in fractions, but none of them are good enough. </p>
<p><strong>Example: calcu... | Antonio Velazquez Bustamante | 250,223 | <p>If you want to see results as an expression of something else, then put that at the end of the result (in this case, after 6.283). In that way you can express results in different units or if you want to see it in PIs, then inset a pi symbol (p and alt+g) at the end. If the result is not shown properly, then highlig... |
1,299,266 | <p>How many zeros are there in the number $50!$?</p>
<p>My attempt:</p>
<p>The zeros in every number come from the 10s that make up the number. The 10s are, in turn, made up of 2s and 5s.</p>
<p>So: $\frac{50}{5*2} = 5$ zeros?</p>
| David Quinn | 187,299 | <p>In answer to your specific question there are not only 12 zeros at the end of the number but an additional 7 zeros occurring amongst the other non-zero digits</p>
|
1,379,513 | <p>A hot dog stand has 12 different toppings available. How many different kinds of hot dogs can be made, assuming the order of the toppings does not make a difference. I believe the correct answer is 882050, with the maximum varieties per number of toppings selected being 665280 when there six toppings. I am also n... | user176717 | 257,988 | <p>It's not necessarily a yes or no question on each topping.</p>
<p>For each topping, I would make 4 different 'states'</p>
<ul>
<li>none</li>
<li>regular</li>
<li>extra</li>
<li>easy</li>
</ul>
<p>$4^{12}=16,777,216$ individual hot dogs, which would definitely not fit in the stand. You could refine it more to have... |
3,191,402 | <p>I have tried to answer by taking change the variable <span class="math-container">$\theta$</span> to <span class="math-container">$\theta/2$</span>, so the integration is now over unit circle, then I have taken <span class="math-container">$z=e^{i\theta}$</span>. Now I tried to use residue formula for integration, b... | Peter Foreman | 631,494 | <p>This method solves the integral when <span class="math-container">$n\in\mathbb{N}$</span> and <span class="math-container">$a\in\mathbb{C}$</span>.</p>
<p>By using binomial expansion the integral becomes
<span class="math-container">$$\int_0^\pi\sum_{k=0}^n\left( \binom{n}{k} a^{n-k} \cos^k{(\theta)} \right)d\theta... |
4,315,283 | <p>Let <span class="math-container">$X$</span> be a continuous random variable, having pdf <span class="math-container">$f(x)$</span> and let be <span class="math-container">$Y=f(X)$</span>.
I have already proved these two following results:</p>
<ol>
<li>If <span class="math-container">$f(x)$</span> is strictly increas... | Siong Thye Goh | 306,553 | <p><span class="math-container">$Y\in [0,1]$</span>,</p>
<p><span class="math-container">$$Y=\begin{cases} X, & X \in [0,1] \\ \frac1X, & X \in [1, e^\frac12]\end{cases}$$</span></p>
<p>If <span class="math-container">$y \in [0, e^{-\frac12}],$</span></p>
<p><span class="math-container">\begin{align}
P(Y \le y)... |
2,725,317 | <blockquote>
<p>For which $p,q\in \mathbb R$ is the following system stable?$$\frac{\mathrm dx}{\mathrm dt} = \begin{bmatrix} p & -q \\ q & p \end{bmatrix}x(t)$$ </p>
</blockquote>
<p>If I'm correct about this, isn't it just when the eigenvalues are $< 1?$ Or is there something more fancy to it? Any and ... | Lutz Lehmann | 115,115 | <p>No, the condition you think of is for discrete-time dynamical systems. </p>
<p>Here in the continuous-time case you need that the eigenvalues $p\pm i\,q$ have non-positive real part. You need an extra test for the purely imaginary case, as that can not in general be decided from the eigenvalues alone.</p>
|
1,156,907 | <p>I don't know anything about measure theory, I'm studying real analysis and this showed up in the book I'm reading as a way to characterize integrable functions. The author defined that a subset $X \subset \mathbb{R}$ has measure zero if for each $\epsilon > 0$ we can find infinitely countable open intervals $I_n$... | user2566092 | 87,313 | <p>Equalities and inequalities with series of series are preserved under exchanging order of summation indices and also taking inner limits before outer limits, vs. summing all the terms in any order you want, so long as the series are either all positive or all negative terms. Thus you don't have to worry about the do... |
4,036,049 | <p>In order for <span class="math-container">$n^5 - n$</span> divisible by <span class="math-container">$5$</span>, <span class="math-container">$n^5 - n = 5 x + 0$</span> (for some <span class="math-container">$x$</span>, <span class="math-container">$x$</span> is a natural number)
I simplified the <span class="math-c... | Paul Frost | 349,785 | <p>You have to solve the cubic equation <span class="math-container">$-3t^3+45t^2+600t = 3750$</span> which is equivalent to
<span class="math-container">$$t^3 -15t^2 -200t +1250 = 0. \tag{1}$$</span></p>
<p>In my opinion the best strategy is to verify first whether it has a rational root. Here all coefficients are int... |
1,793,854 | <p>I am messed up on solving this question. What should I do first in order to get the answer ?</p>
<p><a href="https://i.stack.imgur.com/hE4rG.png" rel="nofollow noreferrer">This is the trigonometric function</a></p>
<p>$$ \lim \limits_{x \rightarrow 0} \frac{(a+x)\sec(a+x) - a \sec(a)}{x} $$</p>
| user5713492 | 316,404 | <p>Method 1: This is the definition of
$$\frac d{da}(a\sec a)=\sec a+a\sec a\tan a$$
Method 2: Change those secants to cosines
$$
\begin{align}
\lim_{x\rightarrow0}\frac{(a+x)\cos a-a\cos(a+x)}{x\cos a\cos(a+x)}&=\lim_{x\rightarrow0}\frac{a\cos a+x\cos a-a\cos a\cos x+a\sin a\sin x}{x\cos a\cos(a+x)}\\
&=\lim_{... |
1,738,968 | <blockquote>
<p>Let $V$ be a vector space and let $T \in \operatorname{End}(V)$. If $\operatorname{rank}(T)$ and $\operatorname{null}(T)$ are finite, prove that $\dim(V)$ is finite.</p>
</blockquote>
<p>I cannot use the Rank-Nullity Theorem as it only applies to finite
dimensional vector space and I don't know wheth... | mathlove | 78,967 | <p>We have two lines :
$$L_1 : a_1x+b_1y+c_1=0,\quad L_2 : a_2x+b_2y+c_2=0$$</p>
<p>and the angle bisectors :
$$L_{\pm} : \frac{a_1x+b_1y+c_1}{\sqrt{a_1^2+b_1^2}}=\pm\frac{a_2x+b_2y+c_2}{\sqrt{a_2^2+b_2^2}}$$</p>
<p>If we let $\theta$ be the (smaller) angle between $L_+$ and $L_1$, then we have
$$\cos\theta=\frac{\le... |
751,053 | <p>T : Rn → Rm is a linear transformation where n,m>= 2</p>
<p>Let V be a subspace of Rn
and let W ={T(v ) | v ∈ V}
. Prove completely that W
is a subspace of Rm. </p>
<p>For this question how do I show that the subspace is non empty, holds under scaler addition and multiplication! I have never proved subspaces with... | Vincent Boelens | 94,696 | <p>$W$ is nonempty since $0=T(0)\in W$. Now, suppose $x,y\in W$. Then there are $u,v\in V$ such that $Tu=x$ and $Tv=y$. Use this to show $x+y\in W$. A similar argument can be used to prove $ax\in W$ for all scalars $a$.
For b), use the fact that :$$0=\sum_{i=1}^n\lambda_iT(a_i)=T\left(\sum_{i=1}^n\lambda_ia_i\right) \i... |
3,903,016 | <p>Why is
<span class="math-container">$$
\sum_{n=1}^{\infty} \left( \frac{1}{(10n-9)^2} + \frac{1}{(10n-1)^2} \right) = \frac{\pi^2}{50} \frac{1}{1- \cos \frac{\pi}{5}}
$$</span>
?</p>
<p>Each form of
<span class="math-container">$$
\sum_{n=1}^{\infty} \frac{1}{(10n-9)^2}
$$</span>
and
<span class="math-container">$$
... | xpaul | 66,420 | <p>Using
<span class="math-container">$$\sum_{n\in\mathbb{Z}}\frac{1}{(n-a)^2+b^2}=\frac{\pi\sinh2\pi b}{b\left(\cosh2\pi b-\cos2\pi a\right)}$$</span>
from <a href="https://math.stackexchange.com/questions/386667/closed-form-for-sum-n-infty-infty-frac1n-a2b2">this</a>, one has
<span class="math-container">\begin{eqnar... |
3,631,587 | <p>Assume that our samples are high dimensional points (i.e., d is large) and we use
PCA to reduce it to k = 10 dimensions. After this step, we found that all the 10 new dimensions have
continuous values (e.g., in other words, each feature in the transformed dimension is not from discrete
domain, but rather, continuous... | Aderinsola Joshua | 395,530 | <p><span class="math-container">$circumference^2 = 4×\pi×area$</span></p>
|
1,760,242 | <p>Can anybody tell me where can I find some REAL problems (i.e. form real life) that can be solved using a 3x3 system of linear equations? Or, can anybody give me an example? A solution could be a circuit in electrical engineering, but this is not very interesting, and it doesn't seem to be so real.
Thank you!</p>
| Robert Israel | 8,508 | <p>A factory employs three classifications of worker, each type having a different wage rate: \$10/hour for $A$'s, \$15/hour for $B$'s, \$20/hour for $C$'s. Each $A$ worker works $20$ hours per week, while $B$'s and $C$'s each work $40$ hours per week. Unfortunately, some will have to be laid off. Management has dec... |
1,760,242 | <p>Can anybody tell me where can I find some REAL problems (i.e. form real life) that can be solved using a 3x3 system of linear equations? Or, can anybody give me an example? A solution could be a circuit in electrical engineering, but this is not very interesting, and it doesn't seem to be so real.
Thank you!</p>
| Doug M | 317,162 | <p>One which I deal with in on a somewhat regular basis. Treasury notes have 3 primary risk factors (called shift, twist and butterfly). These are in fact all dimension of interest rate risk. However, once you have determined the factor loading for any given security, it behaves as a linear factor.</p>
<p>You and y... |
88,156 | <p>I understand that <code>Round</code> give the nearest even integer for cases where the number is between two integers, i.e. <code>Round[2.5] = 2</code> and <code>Round[3.5] = 4</code> (see this <a href="https://mathematica.stackexchange.com/questions/29122/why-do-numberform-and-round-apparently-use-different-tie-bre... | Sjoerd C. de Vries | 57 | <p>Welcome in the amazing world of machine precision arithmetic!</p>
<p>If you examine the binary representation of both numbers you see the following:</p>
<pre><code>RealDigits[1.2, 2]
(* {{1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1,
1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0... |
60,152 | <p>What is the motivation behind topology?</p>
<p>For instance, in real analysis, we are interested in rigorously studying about limits so that we can use them appropriately. Similarly, in number theory, we are interested in patterns and structure possessed by algebraic integers and algebraic prime numbers.</p>
<p>So... | Michael Hardy | 11,667 | <p>This will answer only the question in the second paragraph.</p>
<p>Imagine the half-open interval $[0,1)$ with the usual open sets. In particular, an open neighborhood of $0$ contains $0$ itself and all positive numbers sufficiently close to $0$.</p>
<p>Now alter the definition of "open set", so that every open n... |
60,152 | <p>What is the motivation behind topology?</p>
<p>For instance, in real analysis, we are interested in rigorously studying about limits so that we can use them appropriately. Similarly, in number theory, we are interested in patterns and structure possessed by algebraic integers and algebraic prime numbers.</p>
<p>So... | Tilo Wiklund | 13,504 | <p>This may not be quite the answer you are looking for but I think a "birds-eye" view of topology is somewhat hard. One example which I found somewhat enlightening though, was when I first thought about the topology of Baire space.</p>
<p>The topology is defined on the set of infinite sequences over the natural numbe... |
2,567,607 | <p>$$\arctan 2x +\arctan 3x = \left(\frac{\pi}{4}\right)$$
$$\arctan \left(\frac{2x+3x}{1-2x*3x}\right)=\frac {\pi}{4}$$
$$\frac {5x}{1-6x^2}=\tan \frac{\pi}{4}=1$$
$$6x^2 + 5x -1 = 0$$
$$(6x-1)(x+1)=0$$
$$x=-1, \frac{1}{6}$$</p>
<p>The answer however rejects the solution $x=-1$ saying that it makes the L.H.S of the e... | Michael Rozenberg | 190,319 | <p>Because
$$\arctan(-2)+\arctan(-3)=-135^{\circ}<0$$
By the way, $\tan(-135^{\circ})=1.$</p>
|
398,010 | <p>1.17 THEOREM</p>
<p>Let $f: X \to [0, \infty)$ be measurable. There exist simple measurable functions $s_n$ on $X$ such that</p>
<p>(a) $0 \le s_1 \le s_2 \le \dots \le f.$</p>
<p>(b) $s_n(x) \to f(x)$ as $n \to \infty \forall x$</p>
<p>PROOF Put $\delta_n = 2^{-n}$. To each positive integer $n$ and each real nu... | Sargera | 30,988 | <p>The construction partitions a subset of the range of $f$, i.e. the values of $f$ that fall in the interval $[0,n]$. The definition of $\phi_{n}$ ensures that the interval goes to $[0,\infty)$ (i.e. covers the entire range of $f$) while the requirement $\phi$ be equal to $k_{n}(t)\delta_{n}$ ensures that the range i... |
398,010 | <p>1.17 THEOREM</p>
<p>Let $f: X \to [0, \infty)$ be measurable. There exist simple measurable functions $s_n$ on $X$ such that</p>
<p>(a) $0 \le s_1 \le s_2 \le \dots \le f.$</p>
<p>(b) $s_n(x) \to f(x)$ as $n \to \infty \forall x$</p>
<p>PROOF Put $\delta_n = 2^{-n}$. To each positive integer $n$ and each real nu... | AlexM | 72,471 | <p>What would we lose? Well, we need to keep our eyes on the prize: approximating $f$ by simple functions. The $k_n(t)\delta_n$ part is what is actually doing it. </p>
<p>Imagine filling up a glass with some ice in the bottom, ripe for filling. As you pour water in, ignoring displacement there will be a level it would... |
1,307,269 | <p>In my textbook, before introducing the epsilon delta definition, they gave a working definition of what a limit is. The definition sounded something like this "$\lim \limits_{x \to a}f(x) = L$, if when $x$ gets closer to $a$, $f(x)$ gets closer to $L$" </p>
<hr>
<p>But is that always the case with limits? What if... | Ittay Weiss | 30,953 | <p>You are correct. You present a nuance of the 'working definition', a special case where "getting closer" is misleading. You should interpret "getting closer" as "getting as close as you like". This is a more accurate 'working definition' in any case. Then the constant function scenario works just fine. You can get a... |
457,557 | <p>Use a triple integral to find the volume of the solid: The solid enclosed by the cylinder $$x^2+y^2=9$$ and the planes $$y+z=5$$ and $$z=1$$<br>
This is how I started solving the problem, but the way I was solving it lead me to 0, which is incorrect. $$\int_{-3}^3\int_{-\sqrt{9-y^2}}^{\sqrt{9-y^2}}\int_{1}^{5-y}dzdx... | Christian Blatter | 1,303 | <p>This was an exercice on triple integrals; but at the end we are inclined to check our result against geometric reasoning. Such reasoning tells us that the given solid is half a cylinder of radius $3$ and height $2\cdot(5-1)=8$. Therefore its volume is $36\pi$.</p>
|
915,631 | <blockquote>
<p>Let <span class="math-container">$V$</span> be the vector space of <span class="math-container">$n\times n$</span> matrices over <span class="math-container">$K$</span> under addition and let the linear operator <span class="math-container">$f$</span> be given by <span class="math-container">$f(A)=A^{T}... | Hamou | 165,000 | <p>$f(A)=\lambda A$, hence $f(f(A))=\lambda f(A)=\lambda^2 A$, but $f(f(A))=A$, then
$A=\lambda^2 A$, so $\lambda^2=1$.</p>
|
1,740,458 | <p>I saw somewhere on Math Stack that there was a way of finding integrals in the form $$\int \frac{dx}{a+b \cos x}$$ <strong>without</strong> using Weierstrass substitution, which is the usual technique. </p>
<p>When $a,b=1$ we can just multiply the numerator and denominator by $1-\cos x$ and that solves the problem ... | omegadot | 128,913 | <p>Another way to get to the same point as C. Dubussy got to is the following:
\begin{align*}
\frac{1}{a + b \cos x} &= \frac{1}{a \left (\cos^2 \frac{x}{2} + \sin^2 \frac{x}{2} \right ) + b \left (\cos^2 \frac{x}{2} - \sin^2 \frac{x}{2} \right )}\\
&= \frac{1}{(a - b) \sin^2 \frac{x}{2} + (a + b) \cos^2 \frac{... |
3,046,979 | <p>Recently I've been trying to tackle the integral <span class="math-container">$\int_0^{\frac{\pi}{2}} \ln(\sin(\theta))d\theta$</span> using the Beta function
<span class="math-container">$$\frac{B(\frac{x}{2},\frac{1}{2})}{2}=\int_0^{\frac{\pi}{2}} \sin^{x-1}(\theta)d\theta=\frac{\sqrt{\pi}}{2}\left(\Gamma\left(\fr... | J.G. | 56,861 | <p>You forgot the <span class="math-container">$\Gamma(x/2)$</span> factor.</p>
|
3,046,979 | <p>Recently I've been trying to tackle the integral <span class="math-container">$\int_0^{\frac{\pi}{2}} \ln(\sin(\theta))d\theta$</span> using the Beta function
<span class="math-container">$$\frac{B(\frac{x}{2},\frac{1}{2})}{2}=\int_0^{\frac{\pi}{2}} \sin^{x-1}(\theta)d\theta=\frac{\sqrt{\pi}}{2}\left(\Gamma\left(\fr... | Ininterrompue | 622,553 | <p>Another way that <em>only</em> requires knowledge of the beta function (and expansion of the gamma function) is given as follows. Consider the integral</p>
<p><span class="math-container">$$\begin{aligned}\int_{0}^{\pi/2}\sin^{\epsilon}x\,\mathrm{d}x &= \int_{0}^{\pi/2}e^{\epsilon\ln\sin x}\,\mathrm{d}x = \sum_... |
2,650,182 | <ol>
<li>Between every two distinct real numbers, there is a rational number </li>
</ol>
<p>Answer: There is no rational numbers between two non-distinct real numbers.</p>
<ol start="2">
<li>For all natural numbers $n ∈ N, \sqrt n$ is either a natural number or an
irrational number</li>
</ol>
<p>Answer: For all natu... | Ove Ahlman | 222,450 | <p>The general rule is that "For all" statements should be changed to "There exists" and "There exists" should be changed to "For all". Then apply this inward in the statement. It is often easier to rewrite statements in the syntactic logical way to get a better formal grip. (i.e. using $\exists x, \forall y $ etc.)</p... |
3,179,505 | <p>Help me please , I am not able to solve this problem.I have tried in many ways to figure out such as Ration test , Integral test , Comparison test , Limit Comparison Test , Root Test but i can't find the way out . This is my first question and i'm not good at English. If there is something wrong or you are not comfo... | Mark | 470,733 | <p>If you let <span class="math-container">$d$</span> be <strong>any</strong> metric on <span class="math-container">$X$</span> then of course the answer is no. Let's say we take <span class="math-container">$\mathbb{R}$</span> with the usual metric and <span class="math-container">$\mathbb{R^2}$</span> with the discre... |
1,915,782 | <p>I'm attempting to teach myself some vector calculus before starting university next month in hope of getting my head around some of the concepts as I can foresee this being a weak topic for me.</p>
<p>I have been 'learning' from some online lecture notes related to my course. The notes talk about line integrals but... | Bernard | 202,857 | <p>You have to make a substitution with the parameter. I'll denote paths under the generic name $\gamma$.</p>
<p>For the first integral, the parametrisation of the straight line is clearly $\;\begin{cases}x=t\\y=t\\z=t\end{cases}\enspace(0\le t\le 1)$, so $\;\mathrm d\mkern1mux=\mathrm d\mkern1mut=\mathrm d\mkern1muy... |
3,970,959 | <blockquote>
<p><span class="math-container">$S = \frac{1}{1001} + \frac{1}{1002}+ \frac{1}{1003}+ \dots+\frac{1}{3001}$</span>.</p>
</blockquote>
<blockquote>
<p>Prove that <span class="math-container">$\dfrac{29}{27}<S<\dfrac{7}{6}$</span>.<br></p>
</blockquote>
<p>My Attempt:<br>
<span class="math-container... | WA Don | 542,712 | <p>Dividing the sum into groups as you have done will obtain the limits if you take enough buckets (around 10).</p>
<p>A similar approach is to add <span class="math-container">$1/1000$</span> and then pair the terms, with two terms left over added to the end
<span class="math-container">\begin{align}
S&=\frac{1}{1... |
254,030 | <p>I am following a course in basic algebra, and we have covered rings & groups in class, but I am having trouble visualising them. Are there applications of group &/or ring theory that can be more easily visualized than the abstract object? For instance, are there objects, or properties of objects, that behave... | 000 | 22,144 | <p>This post hinges on whether or not you consider a polygon an abstract object. That is a highly debatable philosophical claim. As a result, I'm submitting this answer on good faith.</p>
<p>The most basic thing that comes to mind for me is the group of rotations of a polygon about its center point $P_k$. For instance... |
1,128,623 | <p>Question: Let S be the set of sequences of $0$s and $1$s. For $x = (x_1, x_2, x_3, ...)$ and $y = (y_1, y_2, y_3, ...)$. Define</p>
<p>$d(x,y)=\sum_{i=1}^\infty \dfrac{|x_i - y_i|}{2^i}$ </p>
<p>Proof the infinite sum in the definition of $d(x,y)$ converges for all $x$ and $y$. </p>
<p>Incomplete answer: Since th... | Community | -1 | <p>The sequence $a_n = \sum_{k=1}^n \frac{|x_k-y_k|}{2^k}$ is monotonic and bounded from above, thus converges.</p>
<p>$$a_n \le \sum_{k=1}^n \frac{1}{2^k} \le \sum_{k=1}^\infty \frac{1}{2^k} = 1$$</p>
|
3,680,658 | <p>In how many ways are we able to arrange <span class="math-container">$k$</span> identical non-overlapping dominoes on a circle of <span class="math-container">$2n$</span> labelled vertices?</p>
<hr>
<p>The problem can be reduced to the number of ways to choose <span class="math-container">$k$</span> non-consecutiv... | Will Orrick | 3,736 | <p>Let's use <span class="math-container">$d_k$</span> to denote the number non-overlapping arrangements of <span class="math-container">$k$</span> dominoes on a circle of <span class="math-container">$2n$</span> vertices.</p>
<p>Your calculation contains an error. If vertex <span class="math-container">$1$</span> is ... |
3,680,658 | <p>In how many ways are we able to arrange <span class="math-container">$k$</span> identical non-overlapping dominoes on a circle of <span class="math-container">$2n$</span> labelled vertices?</p>
<hr>
<p>The problem can be reduced to the number of ways to choose <span class="math-container">$k$</span> non-consecutiv... | Brian M. Scott | 12,042 | <p>You appear to be off a bit: in your first case <span class="math-container">$3$</span> vertices are unavailable, not <span class="math-container">$2$</span>.</p>
<p>I’ve numbered the vertices from <span class="math-container">$1$</span> through <span class="math-container">$2n$</span>. For my first case I put a dom... |
4,508,796 | <p>How to find the integral
<span class="math-container">$$\int_0^1 x\sqrt{\frac{1-x}{1+x}}dx$$</span></p>
<p>I tried by substituting <span class="math-container">$x=\cos a$</span>. But it's leading to a form <span class="math-container">$\sin2a\cdot\tan a/2$</span> which I can't integrate further.</p>
| Ucchash Hore | 878,812 | <p>Well I tried this problem yesterday so here is my solution .</p>
<p>Let <span class="math-container">$x= \cos (2a)$</span>; hence ,<span class="math-container">$\mathrm dx=2\sin(2a) \mathrm da$</span>
putting this value you will get<br />
<span class="math-container">$x\cdot \sqrt{\frac{1-x}{1+x}}\\=\cos (2a)\cdot \... |
615,375 | <p>I would appreciate if somebody could help me with the following problem</p>
<p>Q: Let $f:[0,1]\longrightarrow \Bbb R$ be a continuously function such that
$$m\leq f(x)\leq M, m+M=1 (m:\text{minimum of} f(x), M:\text{maximum of} f(x) )$$
Prove that for every $x \in[0,1]$, there exists $c\in[0,1]$ such that $f(c)=1-... | copper.hat | 27,978 | <p>We have $f([0,1]) = [m,M]$, and $m+M=1$. Hence if we let $\phi(x) = 1-f(x)$, we have $\phi([0,1]) = [1-M,1-m] = [m,M]$.</p>
<p>Hence $f([0,1]) = \phi([0,1])$.</p>
<p>It should be straightforward to finish from here.</p>
|
773,324 | <p>I know Dijkstra's algorithm to find the shortest way between 2 nodes, but is there a way to find the shortest path between 3 nodes among $n$ nodes? Here are the details:</p>
<p>I have $n$ nodes, some of which are connected directly and some of which are connected indirectly, and I need to find the shortest path bet... | Lily Chung | 32,765 | <p>For the case of a start node S and two target nodes X and Y, one could use the following algorithm:</p>
<p>Use Dijkstra's shortest-path algorithm to find the shortest path from S to X and the shortest path from S to Y. If path from S to X is shorter, use Dijkstra's shortest-path algorithm to find the shortest path ... |
2,304,448 | <blockquote>
<p>Does any group homomorphism $\Bbb Z \to \Bbb Z/n$ have kernel isomorphic to $\Bbb Z$?</p>
</blockquote>
<p>Here $n$ is any natural number.</p>
| Tomasz Kania | 17,929 | <p>I am confused because what is required here is almost the version of the open mapping theorem that I was taught as an undergraduate:</p>
<p><em>Suppose that $X$ and $Y$ are Banach spaces and $T\colon X\to Y$ is a bounded linear operator. If the range of $T$ is not meagre in $Y$, then $T$ is surjective.</em></p>
<p... |
2,580,822 | <p>first I will describe my problem (1), then solutions I found (2) and then pose the question (3).</p>
<p>1) I am a student making simple program for statistical analysis of data for biological laboratory. The situation is as follows: They have a protein and measure some property three times (to eliminate measurement... | Archis Welankar | 275,884 | <p>A short hint. Since the number is low and same batch of proteins is used to do two different tasks you can choose t-test $t=\frac {d\bar-\mu }{\frac {s}{\sqrt {n}}} $ where $d\bar$ difference between results of two tests and $\mu,s$ are the mean and standard deviations of the $d\bar $. n is the total number of samp... |
2,580,822 | <p>first I will describe my problem (1), then solutions I found (2) and then pose the question (3).</p>
<p>1) I am a student making simple program for statistical analysis of data for biological laboratory. The situation is as follows: They have a protein and measure some property three times (to eliminate measurement... | BruceET | 221,800 | <p>This semi-answer used to be a Comment, but too abbreviated to make sense:</p>
<p>Everything "they say" is not necessarily helpful. I think a two-sample t test is the right thing to use. The two groups in any two-sample test have <em>something</em> in common. Details are sparse, but the argument that you don't have ... |
4,059,426 | <ul>
<li>In a longer derivation I ran into the following quantity:
<span class="math-container">$$
\nabla\left[\nabla\cdot\left(%
{\bf r}_{0}\,{\rm e}^{{\rm i}{\bf k} \cdot {\bf r}}\,\right)
\right]
$$</span>
( i.e., the gradient of the divergence ) where <span class="math-container">${\bf k}$</span> is a vector of con... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
3,445,768 | <p>I am trying to solve a nonlinear differential equation of the first order that comes from a geometric problem ; <span class="math-container">$$x(2x-1)y'^2-(2x-1)(2y-1)y'+y(2y-1)=0.$$</span></p>
<p>edit1 <strong><em>I am looking for human methods to solve the equation</em></strong> </p>
<p>edit2 the geometric pr... | Francois Rideau | 728,345 | <p>This problem is about <a href="http://mathworld.wolfram.com/IsotomicTransversal.html" rel="nofollow noreferrer">isotomic transversals</a>: </p>
<p>In general, each point of the plane is on a pair of isotomic transversals with respect to some given triangle <span class="math-container">$ABC$</span>.
So is defined a ... |
409,626 | <p>I'm studying elementary group theory, and just seeing the ways in which groups break apart into simpler groups, specifically, a group can be broken up as the sort of product of any of its normal subgroups with the quotient group of that subgroup. So I wondered how you could do the inverse of that operation:</p>
<ol... | Zev Chonoles | 264 | <p>You're correct that $A\times B$ <em>satisfies</em> (<a href="http://www.jmilne.org/math/words.html" rel="nofollow">see here</a>) the stated property, but in general it will not be the only such group. </p>
<p>The simplest example is with $A=B=\mathbb{Z}/2\mathbb{Z}$, in which case $\mathbb{Z}/4\mathbb{Z}$ also has ... |
409,626 | <p>I'm studying elementary group theory, and just seeing the ways in which groups break apart into simpler groups, specifically, a group can be broken up as the sort of product of any of its normal subgroups with the quotient group of that subgroup. So I wondered how you could do the inverse of that operation:</p>
<ol... | Dan Rust | 29,059 | <p>Your idea about the direct product working is true. The general idea you are looking for however is that of a <a href="http://en.wikipedia.org/wiki/Semidirect_product" rel="nofollow">semidirect product</a> which refutes your claim of uniqueness in the general setting.</p>
<p>It's interesting to ask <em>when</em> th... |
3,245,945 | <p>Suppose <span class="math-container">$A$</span> is a linearly ordered set without maximum or minimum and every closed interval is a finite set. I want to show <span class="math-container">$A$</span> is isomorphic to the set of integers with the usual order.</p>
<p>I know that if <span class="math-container">$A$</sp... | Community | -1 | <p>Consider <span class="math-container">$x_0\in A$</span> and the function <span class="math-container">\begin{align}f:A&\to \Bbb Z\\ f(x)&=\lvert[x_0,x)\rvert-\lvert (x,x_0]\rvert\end{align}</span></p>
<p>This function is injective (which is sufficient for the specific passage you need). It's also surjective... |
972,530 | <blockquote>
<p>If $S_1 = \sqrt{2}$, and</p>
<p>$S_{n+1} = \sqrt{2 + \sqrt{S_n}}$ (n = 1,2,3....),</p>
<p>prove that $\{S_n\}$ converges, and that $S_n < 2$ for all $n \in \Bbb{N}$</p>
<p>This is one the questions from Principles of Mathematical Analysis by Rudin. I am not sure how to proceed with t... | Kevin Sheng | 150,297 | <p>Well if we prove that $S_n<2$ for $n=1,2,3,\ldots$ then the first part comes easily as $S_n$ is a sequence in $\mathbb{R}$ which is complete. Since $S_n$ is a monotone increasing sequence with an upper bound (namely 2), it converges. We can prove $S_n<2$ for $n \ge 1$ via induction as follows:</p>
<p>$\textbf... |
381,067 | <p>I am asked this question:</p>
<blockquote>
<p>Prove that $x^5 - 5x + 1$ has no double roots in $\mathbb C[x]$. </p>
</blockquote>
<p>Now here's what I said:</p>
<blockquote>
<p>$p(x) \in \mathbb C[x]$ has no double roots if and only if $gcd(p,p') = 1$. Now we need to prove that. So:
$p(x) = x^5 - 5x + 1$ a... | Ma Ming | 16,340 | <p>You are almost finished. $\gcd(-4x+1, \frac{5}{4} x^3-5)=1$</p>
|
449,413 | <p>I'm trying to construct a norm on the space $\mathcal{D}(\Omega) := \{ f \in C^\infty(\Omega) | f $ is compactly supported on $ \Omega \}$ where $\Omega$ is an open subset of $\mathbb{R}$. I want this norm to include, somehow, the $L^\infty$-norms of <strong>all</strong> the derivatives of the smooth function to whi... | AD - Stop Putin - | 1,154 | <p>If $\Omega$ is open then there is no norm on $C^\infty(\Omega)$, it is a classical Montel space. </p>
<p>In addition, for compact $K\subset\Omega$ we may put $\sup_{x\in K}|f^{(k)}(x)|$ is finite, but $\sup_K\sup_{x\in K}|f^{(k)}(x)|$ may be infinite.</p>
|
2,919,561 | <p>What is the following limit ? How do I solve it ?
$$\lim \limits_{x\to0}\frac{6\sin x}{x-3\tan x}$$</p>
| user | 505,767 | <p><strong>HINT</strong></p>
<p>We don't need l'Hopital indeed we have</p>
<p>$$\lim \limits_{x\to0}\frac{6\sin x}{x-3\tan x}=\lim \limits_{x\to0}\frac{6\frac{\sin x}x}{1-3\frac{\tan x}x}$$</p>
<p>then refer to standard limits.</p>
|
439,745 | <blockquote>
<p>Prove:$|x-1|+|x-2|+|x-3|+\cdots+|x-n|\geq n-1$</p>
</blockquote>
<p>example1: $|x-1|+|x-2|\geq 1$</p>
<p>my solution：(substitution)</p>
<p>$x-1=t,x-2=t-1,|t|+|t-1|\geq 1,|t-1|\geq 1-|t|,$</p>
<p>square,</p>
<p>$t^2-2t+1\geq 1-2|t|+t^2,\text{Since} -t\leq -|t|,$</p>
<p>so proved.</p>
<p><em>ques... | Thomas Andrews | 7,933 | <p>Actually, $|x-n| + |x-1| = |n-x| + |x-1| \geq |n-1|=n-1$. Don't need the other terms.</p>
<p>Your first proof is correct. You have to take some care moving from $U^2\geq V^2$ to $U\geq V$, but in this case, $U=|t-1|$ is non-negative.</p>
|
3,484,136 | <blockquote>
<p>Show that <span class="math-container">$n^2+n$</span> is even for all <span class="math-container">$n\in\mathbb{N}$</span> by contradiction.</p>
</blockquote>
<p>My attempt: assume that <span class="math-container">$n^2+n$</span> is odd, then <span class="math-container">$n^2+n=2k+1$</span> for all <... | ViHdzP | 718,671 | <p>Other users have already pointed out your big mistake: you can’t say that <span class="math-container">$n^2+n=2k+1$</span> <strong>for all <span class="math-container">$\mathbf k$</span></strong>, since being odd only means that this is true for a <strong>specific <span class="math-container">$\mathbf k$</span></str... |
1,130,855 | <p>I've searched this website and while there are a few questions similar to mine, I couldn't find what I was looking for/a specific method for what I want to do.</p>
<p>I want to understand how one would prove that the remainder of $5^{336}$ by $23$ is $8$, or in other words, $$5^{336}\equiv 8 \pmod{23}.$$</p>
<p>Ca... | AlexR | 86,940 | <p>$23$ is prime, so $\phi(23) = 22$. You can thus <em>always</em> take the exponent modulo $22$ (by <a href="http://en.wikipedia.org/wiki/Euler%27s_theorem" rel="nofollow">Euler's theorem</a>, or in this special case <a href="http://en.wikipedia.org/wiki/Fermat%27s_little_theorem" rel="nofollow">Fermat's little theore... |
1,250,132 | <p>Below is part of a solution to a critical points question. I'm just not sure how the equation on the left becomes the equation on the right. Could someone please show me the steps in-between? Thanks.</p>
<blockquote>
<p>$$\frac{-1}{x^2}+2x=0 \implies 2x^3-1=0$$</p>
</blockquote>
| Jeel Shah | 38,296 | <p>The solution has multiplied both left and right side by $x^2$</p>
<p>Observe:
\begin{align}
&-\frac{1}{x^2} * x^2 + 2x(x^2) = 0 * x^2 \\
&\implies -\frac{x^2}{x^2} + 2x^3 = 0 \\
&\implies -1 + 2x^3 = 0
\end{align}</p>
|
1,014,476 | <p>I pick 6 cards from a set of 13 (ace-king). If ace = 1 and jack,queen,king = 10 what is the probability of the sum of the cards being a multiple of 6? </p>
<p><strong>Tried so far:</strong>
I split the numbers into sets with values:
6n, 6n+1, 6n+2, 6n+3
like so:</p>
<p>{6}{1,7}{2,8}{3,9}{4,10,j,q,k}{5}</p>
<p>and... | David K | 139,123 | <p>It will help to explicitly list the values of cards in each combination, as other answers have done, <em>then</em> count the number of ways you can make each combination.
If doing this by hand rather than by computer, you were wise to group the cards by
their remainders after division by $6$; you can use just the re... |
1,221,914 | <blockquote>
<p>Let <span class="math-container">$P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 ,\ n\ge 1$</span> have <span class="math-container">$n$</span> roots <span class="math-container">$x_1,x_2,\ldots,x_n \le -1$</span> and <span class="math-container">$a_0^2+a_1a_n=a_n^2+a_0a_{n-1}$</span>.
Find all such <... | Karanko | 227,830 | <p>Let $s_i$ denote the <a href="https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial#Definition" rel="nofollow">elementary symmetric polynomials</a> in $x_1,x_2,...,x_n$.</p>
<p>Then $a_0/a_n=(-1)^ns_n$, $a_1/a_n=(-1)^{n-1}s_{n-1}=(-1)^{n-1}s_n\left(\sum_{k=1}^{n}\frac{1}{x_k}\right)$, and $a_{n-1}/a_n=-s_1$... |
4,572,336 | <p>This is the
<a href="https://i.stack.imgur.com/BRjsf.jpg" rel="nofollow noreferrer">question</a></p>
<p>I have managed to find the formula for part a) which is n(n+1) for all natural numbers n, but I'm not sure how to prove it directly. Can I just say that the summation of 2i is the same as the sums of the summation... | Scene | 723,326 | <p>We have
<span class="math-container">$$
2+4+\dots+2n =2(1+2+\dots+n) =2\left (\frac{n(n+1)}{2}\right)=n(n+1)
$$</span></p>
|
4,572,336 | <p>This is the
<a href="https://i.stack.imgur.com/BRjsf.jpg" rel="nofollow noreferrer">question</a></p>
<p>I have managed to find the formula for part a) which is n(n+1) for all natural numbers n, but I'm not sure how to prove it directly. Can I just say that the summation of 2i is the same as the sums of the summation... | Dean Menezes | 452,340 | <p><span class="math-container">$$\begin{align} 2 + 4 + \dots + 2n &= (1 + \dots + n) + (1 + \dots + n) \cr & = (1 + 2 + \dots + n) + (n + (n-1) + \dots + 1) \cr & = (n+1) + \dots + (n+1) = n(n+1)\end{align}$$</span></p>
|
4,572,336 | <p>This is the
<a href="https://i.stack.imgur.com/BRjsf.jpg" rel="nofollow noreferrer">question</a></p>
<p>I have managed to find the formula for part a) which is n(n+1) for all natural numbers n, but I'm not sure how to prove it directly. Can I just say that the summation of 2i is the same as the sums of the summation... | Nayjer | 1,117,302 | <p>Yes, that should be enough. You can also say:</p>
<p><span class="math-container">$\sum_{i=1}^{n}2i = 2 \cdot \sum_{i=1}^{n}i = 2 \cdot \frac{n(n+1)}{2}$</span>.</p>
|
4,572,336 | <p>This is the
<a href="https://i.stack.imgur.com/BRjsf.jpg" rel="nofollow noreferrer">question</a></p>
<p>I have managed to find the formula for part a) which is n(n+1) for all natural numbers n, but I'm not sure how to prove it directly. Can I just say that the summation of 2i is the same as the sums of the summation... | user | 293,846 | <p>Hint for even <span class="math-container">$n$</span>:
<span class="math-container">$$\left.\begin{align}
&2+2n&=2(n+1)\\
&4+(2n-2)&=2(n+1)\\
&6+(2n-4)&=2(n+1)\\
&\dots\\
&n+(n+2)&=2(n+1)
\end{align}\right \}\frac n2\text{ rows }$$</span></p>
|
14,735 | <p>This question is somewhat related to <a href="https://mathematica.stackexchange.com/questions/4576/changing-the-order-of-elements-in-a-chart-legend">this</a> one.</p>
<p>Let's say this is the <code>BarChart</code> i want to make:</p>
<pre><code>BarChart[Range[5], ChartStyle -> "Rainbow", BarOrigin -> Left,
... | J. M.'s persistent exhaustion | 50 | <p>This might be what you were expecting:</p>
<pre><code>In[]:=
{{-1/4, 1/4 + 1/Sqrt[2], -1/2 + 1/(2 Sqrt[2])},
{1/4 - 1/Sqrt[2], -1/4, -1/2 - 1/(2 Sqrt[2])},
{-1/2 - 1/(2 Sqrt[2]), -1/2 + 1/(2 Sqrt[2]), 1/2}} // Eigenvalues // ExpToTrig
Out[]=
{-1/2 + (I Sqrt[3])/2, -1/2 - (I Sqrt[3])/2, 1}
</code></pre>
<p... |
1,068,719 | <p>The problem is to prove that the quintic
$$x^5+10x^4+15x^3+15x^2-10x+1$$
is irreducible in the rationals. </p>
<p>I don't have much knowledge in group theory, and certainly not in Galois theory, and I'm pretty sure this problem can be solved without those tools. </p>
<p>I know about Eisenstein's criterion, but it ... | Will Jagy | 10,400 | <p>Major simplification: result of Gauss that one may factor a polynomial with integer coefficients and get the same answer as with rational numbers, <a href="http://en.wikipedia.org/wiki/Gauss%27s_lemma_%28polynomial%29" rel="nofollow">http://en.wikipedia.org/wiki/Gauss%27s_lemma_%28polynomial%29</a> </p>
<p>This is ... |
135,911 | <p>Liouville's theorem gives such a proof for antiderivatives of functions like <span class="math-container">$e^x/x$</span> or <span class="math-container">$e^{x^2}$</span>, and differential Galois theory extends that to Bessel functions, say. But what tools exist for implicit functions like Lambert's W?</p>
| Igor Khavkine | 2,622 | <p>It seems that the non-linear differential equation satisfied by the Lambert <span class="math-container">$W$</span> function is simple enough for this question to have already been answered. This paper proves that the Lambert <span class="math-container">$W$</span> is non-elementary by appealing to a result of Rosen... |
164,309 | <p>I came across this recurrence function:</p>
<blockquote>
<p>$$F(n) = a \times F(n-1) + b$$</p>
</blockquote>
<p>where $F(0) =1$. We have to solve for $F(n) \pmod {m}$</p>
<p>But for very large $n$, solving it with computer is also taking time. Is there anyway to simplify this. I think the values will be repeate... | pritam | 33,736 | <p>$F(n)=a.F(n-1)+b=a(a.F(n-2)+b)=a^2F(n-2)+ab=a^2(a.F((n-3)+b)+ab=a^3F(n-3)+a^2b+ab=\ldots=a^nF(0)+a^{n-1}b+\cdots +ab$</p>
<p>Now if $a\mod m\ne 1$ $$F(n)\mod m=(a\mod m)^n+(b\mod m)\frac{1-(a\mod m)^n}{1-a\mod m}$$</p>
<p>If $a\mod m=1$ then $a^i\mod m=1$ for all $i$, hence $F(n)\mod m=1+(n-1)(b\mod m)$</p>
|
164,309 | <p>I came across this recurrence function:</p>
<blockquote>
<p>$$F(n) = a \times F(n-1) + b$$</p>
</blockquote>
<p>where $F(0) =1$. We have to solve for $F(n) \pmod {m}$</p>
<p>But for very large $n$, solving it with computer is also taking time. Is there anyway to simplify this. I think the values will be repeate... | robjohn | 13,854 | <p>If $m$ is small enough, compute when the sequence
$$
1,1+a,1+a+a^2,1+a+a^2+a^3,\dots\tag{1}
$$
becomes $0\bmod{m}$. If the $k^{\text{th}}$ term of $(1)$ is $0\bmod{m}$, then
$$
\frac{a^k-1}{a-1}\equiv0\pmod{m}\quad\quad\text{and}\quad\quad a^k\equiv1\pmod{m}\tag{2}
$$
Reduce $n'\equiv n\pmod{k}$ and you have $F(n)\e... |
194,664 | <p>How to generate a list of fixpoint free permutations of n elements in mathematica?</p>
| Carl Woll | 45,431 | <p>You can use <a href="http://reference.wolfram.com/language/ref/Solve" rel="noreferrer"><code>Solve</code></a> to find the roots. Here is a function that finds the roots between <span class="math-container">$-\pi$</span> and <span class="math-container">$\pi$</span>:</p>
<pre><code>f[x_] := a /. Solve[Sin[a]/a==x &a... |
2,061,363 | <p>I have the complex power series $ \sum_{k=1}^{\infty}(\frac{z^4}{4} - \frac{\pi}{7})^k$. </p>
<p>Through algebraic manipulation I obtain $ \sum_{k=1}^{\infty}(\frac{1}{4})^k(z^4 - \frac{4}{7}\pi)^k$. I now argue that this is a power series around $\frac{4}{7}\pi$ with radius of convergence R = 4, using the euler ro... | zhw. | 228,045 | <p>The series in this problem is not a power series, so the concept of "radius of convergence" doesn't really apply. Because of basic results for geometric series, the series convergeges iff $|z^4/4 - \pi/7|<1.$ The set of such $z$ is not a disc, so what does "radius" mean? This set actually looks kind of complicat... |
3,547,384 | <p>I saw this equation<span class="math-container">$$S(q)=\int_a^bL(t,q(t),\dot q(t))dt$$</span>
in <a href="https://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation" rel="nofollow noreferrer">wikipedia</a>.</p>
<p>So I would think that <span class="math-container">$f(x,y)$</span> must be equal to <span class="ma... | Quiver | 564,698 | <p>If we write <span class="math-container">$f(x,y,t)$</span> we mean an <em>explicit</em> dependence of time. This makes the assertion you make </p>
<blockquote>
<p>Because it is always correct that <span class="math-container">$$\frac{df}{dt} = \frac{\partial f}{\partial t} $$</span></p>
</blockquote>
<p><strong>... |
1,241,695 | <p>I was reading A First Course in Probability by Sheldon Ross. I think I quite understood the below problem but I still feel fuzzy.</p>
<blockquote>
<p><strong>Problem</strong>: In answering on a multiple choice test, a student either know the answer or guesses. Let p be the probability that the students knows the ans... | elDin0 | 195,164 | <p>The way I work through these is by thinking "the probability of <strong>B GIVEN A</strong> is equal to the probability of <strong>A AND B</strong> divided by the probability of <strong>just A</strong>. </p>
<p>So in this case, the probability that the student knew the answer given he answered it right is the probab... |
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