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 |
|---|---|---|---|---|
16,532 | <p>I want to export a plot into an eps file. Below I present a sample plot code in order to demonstrate the two issues that appear. </p>
<p>(1). <code>LabelStyle</code>: When I use "Times" or "Helvetica" style all seem to work fine. However, when I choose "Traditional" or "Tahoma" the style of the labels in the export... | cormullion | 61 | <p>Your plot, exported to EPS, looks OK on my system (<em>Mathematica</em> 9, Mac OS X):</p>
<p>The version using Helvetica: </p>
<p><img src="https://i.stack.imgur.com/b07jt.png" alt="Helvetica"></p>
<p>A version using Georgia:</p>
<p><img src="https://i.stack.imgur.com/HX06l.png" alt="Georgia"></p>
<p>(I used Ge... |
16,532 | <p>I want to export a plot into an eps file. Below I present a sample plot code in order to demonstrate the two issues that appear. </p>
<p>(1). <code>LabelStyle</code>: When I use "Times" or "Helvetica" style all seem to work fine. However, when I choose "Traditional" or "Tahoma" the style of the labels in the export... | Jens | 245 | <p>There was another issue in the comment to @cormullion's answer, related to a missing sliver of the exported graphic on the side. If you see that kind of problem, it can help to set the <code>PlotRegion</code> to be slightly less than 100% of the available width or height determined by <code>ImageSize</code>. </p>
<... |
424,209 | <p>I am a Computer Science student. While going through some random maths topics I came across Chaos Theory. I wanted to know if there are any applications of it in CS.
I tried searching on the internet about this but ended up only with <a href="https://security.stackexchange.com/questions/31000/does-chaos-theory-have-... | Fixed Point | 30,261 | <p>Cryptography is another application of chaos to computer science. There are a few chaotic encryption schemes around.</p>
|
1,937,762 | <p>Let's say I have a ratio of polynomials as follows</p>
<p>$P(x)=\frac{a_0x^n+a_1x^{n-2}+a_2x^{n-4}+...}{b_0x^n+b_1x^{n-2}+b_2x^{n-4}+...}$.</p>
<p>The polynomials are finite. Is there a procedure to convert it into a polynomial</p>
<p>$P(x) = A_0 + A_1 f(x) + A_2g(x) + ...$</p>
<p>where $f(x)$ and $g(x)$ are som... | Dosetsu | 139,331 | <p>This is not possible. Think of $P(x)=1/x$. This does not have a Taylor expansion, so even your second questions, the series expansion is not possible.</p>
|
13,166 | <p>I taught IT in an engineering school during three years in <a href="https://en.wikipedia.org/wiki/Problem-based_learning" rel="nofollow noreferrer">problem based learning</a> (PBL) only. Now I teach maths to pupils between 10 and 15 years old who have a lot of educational difficulties.</p>
<p>I'm thinking to use PB... | Photon | 3,542 | <p>I cannot provide any research results supporting my claim, but intuitively, I'd assume that it can work out if the level of the exercises is low enough to provide success experience. </p>
<p>For example, I had a pupil aged about 15 who had really big problems in maths (and also other subjects because she attended a... |
3,858,517 | <p>Is it possible to count exactly the number of binary strings of length <span class="math-container">$n$</span> that contain no two adjacent blocks of 1s of the same length? More precisely, if we represent the string as <span class="math-container">$0^{x_1}1^{y_1}0^{x_2}1^{y_2}\cdots 0^{x_{k-1}}1^{y_{k-1}}0^{x_k}$</s... | leonbloy | 312 | <p>An aproximation for large <span class="math-container">$n$</span></p>
<p>The runs of <span class="math-container">$0$</span>s and <span class="math-container">$1$</span>s can be approximated by iid geometric random variables (with <span class="math-container">$p=1/2$</span>, mean <span class="math-container">$2$</sp... |
3,783,878 | <p>Hey everyone can anyone help me in simplifying the following boolean expression with explanation?</p>
<p><span class="math-container">\begin{equation}[((p\land q)\implies r)\implies((q\land r')\implies r')]\land[(p \land q)\implies(q\iff p)]\end{equation}</span></p>
| Hussain-Alqatari | 609,371 | <p>Based on the following table, the given statement is always true for any <span class="math-container">$p,q,r$</span>:</p>
<p><a href="https://i.stack.imgur.com/LGTS7.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/LGTS7.png" alt="enter image description here" /></a></p>
|
3,197,046 | <p>I'm interested in <span class="math-container">${\bf integer}$</span> solutions of </p>
<p><span class="math-container">$$abcd+1=(ecd-c-d)(fab-a-b)$$</span></p>
<p>subject to <span class="math-container">${\bf a,b,c,d \geq 2}$</span>, and <span class="math-container">${\bf e,f \geq 1}$</span>. </p>
<p><span class... | Jeff | 562,680 | <p>There are finitely many such solutions.</p>
<p><span class="math-container">${\bf Proof.}$</span></p>
<p>It follows directly from the following (not too hard to verify) statements:</p>
<p>(1) <span class="math-container">$e$</span> or <span class="math-container">$f$</span> is <span class="math-container">$1$</sp... |
2,637,812 | <p>Here is dice game question about probability.</p>
<p>Play a game with $2$ die. What is the probability of getting a sum greater than $7$?</p>
<p>I know how the probability for this one is easy, $\cfrac{1+2+3+4+5}{36}=\cfrac 5{12}$.</p>
<p>I don't know how to solve the follow-up question:</p>
<p>Play a game with ... | user284331 | 284,331 | <p>\begin{align*}
\lim_{x\rightarrow 0}\dfrac{x^{2}-\log(1+x^{2})}{x^{2}\sin^{2}x}&=\lim_{x\rightarrow 0}\dfrac{x^{2}-x^{2}+\dfrac{1}{2}x^{4}-\dfrac{1}{3}x^{6}\cdots}{x^{2}\sin^{2}x}\\
&=\lim_{x\rightarrow 0}\dfrac{\dfrac{1}{2}x^{2}-\dfrac{1}{3}x^{4}\cdots}{\sin^{2}x}\\
&=\lim_{x\rightarrow 0}\dfrac{\dfrac{... |
2,637,812 | <p>Here is dice game question about probability.</p>
<p>Play a game with $2$ die. What is the probability of getting a sum greater than $7$?</p>
<p>I know how the probability for this one is easy, $\cfrac{1+2+3+4+5}{36}=\cfrac 5{12}$.</p>
<p>I don't know how to solve the follow-up question:</p>
<p>Play a game with ... | Barry Cipra | 86,747 | <p>Note that</p>
<p>$${x^2-\log(1+x^2)\over x^2\sin^2x}={x^2\over\sin^2x}\cdot{x^2-\log(1+x^2)\over x^4}={x^2\over\sin^2x}\cdot{u-\log(1+u)\over u^2}$$</p>
<p>where $u=x^2\to0^+$ as $x\to0$. If we take ${x\over\sin x}\to1$ for granted, then L'Hopital takes care of the rest:</p>
<p>$$\lim_{u\to0^+}{u-\log(1+u)\over u... |
396,085 | <p>The length of three medians of a triangle are $9$,$12$ and $15$cm.The area (in sq. cm) of the triangle is</p>
<p>a) $48$</p>
<p>b) $144$</p>
<p>c) $24$</p>
<p>d) $72$</p>
<p>I don't want whole solution just give me the hint how can I solve it.Thanks.</p>
| robjohn | 13,854 | <p><strong>Area of a Triangle from the Medians</strong></p>
<p>A triangle is divided in to $6$ equal areas by its medians:</p>
<p>$\hspace{2cm}$<img src="https://i.stack.imgur.com/c63v0.png" alt="enter image description here"></p>
<p>In the case where the two blue triangles share a common side of the triangle, it is... |
396,085 | <p>The length of three medians of a triangle are $9$,$12$ and $15$cm.The area (in sq. cm) of the triangle is</p>
<p>a) $48$</p>
<p>b) $144$</p>
<p>c) $24$</p>
<p>d) $72$</p>
<p>I don't want whole solution just give me the hint how can I solve it.Thanks.</p>
| vaasie | 225,294 | <p>The area of a triangle made by the medians taken as sides is 75% of the triangle of which the medians are given. Now you can find the area by heron formula and the area thus you get will be 75% of the area of the triangle of which the medians are given. </p>
|
2,275,785 | <p>I asked a similar question last night asking for an explanation of the statement, however I was unable to find how to prove such a statement, so I have a proof, however I think it is wrong, so I'm just asking for it to be checked and if it is, for it to be corrected, thanks! </p>
<p><strong>Question</strong></p>
<... | DeepSea | 101,504 | <p><strong>hint</strong>: You have $ \lfloor x \rfloor \le x \le \lfloor x \rfloor + 1 < \lfloor x \rfloor + 2$</p>
|
2,966,010 | <p>how to show that <span class="math-container">$$\sum_{n=1}^{\infty}(-1)^{n}\dfrac{3n-1}{n^2 + n} = \log\left({32}\right) - 4$$</span>? Can I use the Alternating Series test and how? </p>
| Darío A. Gutiérrez | 353,218 | <p><span class="math-container">$$S_1= \sum_{n=1}^{\infty}(-1)^{n}\dfrac{3n-1}{n^2 + n}$$</span>
<span class="math-container">$$\dfrac{3n-1}{n^2 + n} = \dfrac{3-\frac{1}{n}}{n + 1} = \left(\frac{4}{n+1} - \frac{1}{n}\right)$$</span>
So
<span class="math-container">\begin{align}
S_1 &= \sum_{n=1}^{\infty}(-1)^{n}\le... |
207,243 | <p>I am using FindFit function in order to fit my data and get two parameters: c and m.</p>
<p>The function that I am using has the following form:</p>
<pre><code>function = (m/x*(x/c)^m)*Exp[-1*(x/c)^m];
</code></pre>
<p>The answer should be c = 64.68 and m = 2.47, but I am constantly getting the error message Over... | mikado | 36,788 | <p>You might find the following works. I'll create some points</p>
<pre><code>pts = Table[(5 + t) {Cos[t], Sin[t]}, {t, 0, 5}] // N
(* {{5., 0.}, {3.24181, 5.04883}, {-2.91303, 6.36508}, {-7.91994,
1.12896}, {-5.88279, -6.81122}, {2.83662, -9.58924}} *)
</code></pre>
<p>and extract the cells of the triangulation<... |
397,040 | <p>What is the domain for $$\dfrac{1}{x}\leq\dfrac{1}{2}$$</p>
<p>according to the rules of taking the reciprocals, $A\leq B \Leftrightarrow \dfrac{1}{A}\geq \dfrac{1}{B}$, then the domain should be simply $$x\geq2$$</p>
<p>however negative numbers less than $-2$ also satisfy the original inequality. When am I missin... | Zev Chonoles | 264 | <p>The equivalence
$$A\leq B\iff \frac{1}{A}\geq\frac{1}{B}$$
only holds for numbers $A$ and $B$ that <strong>have the same sign</strong> (i.e., are both positive or both negative). Remember, when $c>0$, we have
$$A\leq B \iff c A\leq c B$$
and when $c<0$, we have
$$A\leq B\iff cA\geq cB.$$
To go from $A\leq B$ t... |
4,056,073 | <p>I need help with this task, if anyone had a similar problem it would help me !</p>
<p>The task is: Determine the type of interruption at the point x = 0 for the function</p>
<p><span class="math-container">$$f(x)=2^{-\frac{1}{x^{2}}}$$</span></p>
<p>I did:</p>
<p><span class="math-container">$$L=\lim_{x\to 0^{-}} 2^... | Adam Latosiński | 653,715 | <p>The ordering of the roots doesn't give you any new information about the polynomial. You have a map
<span class="math-container">$$ {\mathbb C}^n \ni (r_1,r_2,\dots r_n) \mapsto (c_0,c_1\dots c_{n-1}) \in \mathbb{C}^n$$</span>
This map is surjective, but not injective. That kind of things may happen because <span cl... |
1,627,050 | <p>Let $f:[a,b]\rightarrow R$ be differentiable at $c\in [a,b]$. Show that for every $\epsilon >0$, there is a $\delta(\epsilon) >0$ s.t if $0<|x-y|<\delta(\epsilon)$ and $a\leq x \leq c\leq y \leq b$, then\
$$ |{\frac{f(x)-f(y)}{x-y}-f'(c)}|<\epsilon$$
I can only think of using triangle inequality, but ... | David C. Ullrich | 248,223 | <p>Hint: If $x=c<y$ or $x<c=y$ this is immediately immediate from the definition. So assume $x<c<y$. Then $$\frac{f(y)-f(x)}{y-x}-f'(c)
=\frac{y-c}{y-x}\left(\frac{f(y)-f(c)}{y-c}-f'(c)\right)+\frac{c-x}{y-x}\left(\frac{f(c)-f(x)}{c-x}-f'(c)\right).$$</p>
<p>Now you can use the triangle inequality there; <... |
1,677,035 | <p>I'm new to this website so I apologize in advance if what I'm going to ask isn't meant to be posted here.</p>
<p>A bit of background though: I haven't been to school in 6 years and the last level I've graduated was Grade 7 due to financial problems, as well as my mom frequently being in and out of the hospital. I a... | H Huang | 604,218 | <p><a href="http://tutorial.math.lamar.edu" rel="nofollow noreferrer">http://tutorial.math.lamar.edu</a> is really good for calculus. I’ve never used the non-calc parts, but from a cursory look, they look of the same quality. Don’t use it exclusively though, since it is designed for a single college course, it has to s... |
1,643,201 | <p>The spectrum-functor
$$
\operatorname{Spec}: \mathbf{cRng}^{op}\to \mathbf{Set}
$$
sends a (commutative unital) ring $R$ to the set $\operatorname{Spec}(R)=\{\mathfrak{p}\mid \mathfrak{p} \mbox{ is a prime ideal of R}\}$ and a morpshim $f:S\to R$ to the map $\operatorname{Spec}(R)\to \operatorname{Spec}(S)$ with $\m... | Win Vineeth | 311,216 | <p>You are absolutely right, and that's the shortest way of doing it as far as I know.</p>
|
3,916,092 | <blockquote>
<p>A ball rotates at a rate <strong><span class="math-container">$r$</span></strong> rotations per second and simultaneously revolves around a stationary point <strong><span class="math-container">$O$</span></strong> at a rate <strong><span class="math-container">$R$</span></strong> revolutions per second... | Math Lover | 801,574 | <p><a href="https://i.stack.imgur.com/OlccJ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/OlccJ.png" alt="enter image description here" /></a></p>
<p>As rotation and revolution are both in the same direction (say, counterclockwise), the point will come in the line connecting centers after one compl... |
3,357,841 | <p>In the diagram (which is not drawn to scale) the small triangles each have the area shown. Find the area of the shaded quadrilateral.</p>
<p><a href="https://i.stack.imgur.com/DK8sn.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DK8sn.png" alt="enter image description here"></a></p>
| Jean Marie | 305,862 | <p>There is a simple solution obtained by splitting the area to be found into two areas <span class="math-container">$x$</span> and <span class="math-container">$y$</span> by diagonal <span class="math-container">$CM$</span> (see figure).</p>
<p><a href="https://i.stack.imgur.com/BtR4J.jpg" rel="nofollow noreferrer"><... |
192,125 | <p>Solve: $$\sqrt{x-4} + 10 = \sqrt{x+4}$$
Little help here? >.<</p>
| Robert Israel | 8,508 | <p>There are no real solutions, nor any complex solutions if you use the principal branch of the square root. Squaring both sides and simplifying gives you $20 \sqrt{x-4} = -92$.</p>
<p>EDIT: More generally, for any $a, b \ge 0$, $\sqrt{a + b} \le \sqrt{a} + \sqrt{b}$. Since
$(x+4) - (x-4) = 8$, the most $\sqrt{x+4}... |
1,323,943 | <p>Identify the parent function </p>
<p>$g(x)=\sqrt {-x+2 }$</p>
<p>1-Identify the parent function</p>
<p>2-Identify the transformations being applied, in appropriate order</p>
<p>3-sketch a graph of the transformed function</p>
<p>my work for first one </p>
<p>from parent function <img src="https://i.stack.imgur... | wythagoras | 236,048 | <ul>
<li><p>Translation $(-2,0)$ gives $\sqrt{x+2}$</p></li>
<li><p>Multiplication through the y-axis by -1 gives $\sqrt{-x+2}$.</p></li>
</ul>
<p>Graph: The start point is $(2,0)$. Then draw a square root graph to the left. </p>
|
1,323,943 | <p>Identify the parent function </p>
<p>$g(x)=\sqrt {-x+2 }$</p>
<p>1-Identify the parent function</p>
<p>2-Identify the transformations being applied, in appropriate order</p>
<p>3-sketch a graph of the transformed function</p>
<p>my work for first one </p>
<p>from parent function <img src="https://i.stack.imgur... | John Joy | 140,156 | <p>Sometimes its just easier thinking about the equation inside out looking at $g^{-1}$. Consider
$$\begin{array}{lll}
g(x)=f(a(x-b))\\
f^{-1}(g(x))=f^{-1}(f(a(x-b)))\\
f^{-1}(g(x))=a(x-b)\\
\frac{1}{a}[f^{-1}(g(x))] + b=x\\
\end{array}$$
Which looks somewhat cryptic, but lets look at your particular example.
$$\begin{... |
4,434,832 | <p>I have taken this question from molodovian national MO 2008
The question is as follows</p>
<p>The sequence <span class="math-container">$(a_p)_p\ge 0$</span> is defined as <span class="math-container">$$a_p=\sum_{i=0}^p (-1)^i\frac{\binom{p}{i}}{(i+2)(i+4)}$$</span></p>
<p>Now let's find the limit</p>
<p><span clas... | Stefan Lafon | 582,769 | <p>Let <span class="math-container">$$\begin{split}
S_{k,p} &= \sum_{i=0}^p\binom{p}{i}\frac{(-1)^i}{i+k}\\
&= \sum_{i=0}^p\binom{p}{i}(-1)^i\int_0^1 x^{i+k-1}dx\\
&= \int_0^1 \sum_{i=0}^p\binom{p}{i}(-1)^ix^{i+k-1}dx\\
&= \int_0^1 x^{k-1}(1-x)^pdx
\end{split}$$</span>
Then since <span class="math-conta... |
351,846 | <p>The following problem was on a math competition that I participated in at my school about a month ago: </p>
<blockquote>
<p>Prove that the equation $\cos(\sin x)=\sin(\cos x)$ has no real solutions.</p>
</blockquote>
<p>I will outline my proof below. I think it has some holes. My approach to the problem was to... | Micah | 30,836 | <p>A possibly-shorter way of getting there would be to write
$$\cos (\sin x) - \sin(\cos x)=\cos (\sin x) - \cos(\pi/2-\cos x)$$
and then use a sum-to-product identity to turn this last expression into:
\begin{eqnarray}
&&−2 \sin \left(\frac{\sin x + \pi/2 - \cos x}{2}\right)\sin\left(\frac{\sin x - \pi/2 + \co... |
210,658 | <p>$$\sum_{i=1}^n t^{i-1}$$
I am stuck with the proof of this equality. </p>
| Mikasa | 8,581 | <p>Use induction on $n\in \mathbb N, n\ge1$ with $P(n):=\sum_{i=1}^n t^{i-1}=\frac{1-t^n}{1-t}$:
For $n=1$ you have: $$P(1): 1=\frac{1-t}{1-t}$$
which is correct. Suppose that $P(n)$ is true, now check it for $n+1$. You have $$P(n+1): \sum_{i=1}^{n+1} t^{i-1}=\sum_{i=1}^n t^{i-1}+t^{n+1}=\frac{1-t^n}{1-t}+t^{n+1}=\fr... |
210,658 | <p>$$\sum_{i=1}^n t^{i-1}$$
I am stuck with the proof of this equality. </p>
| Per Erik Manne | 33,572 | <p>Hint: Write out the summation
$$\sum_{i=1}^n t^{i-1}=1+t+t^2+\cdots+t^{n-1}$$
Now multiply by $1-t$, and observe which terms will cancel and which will survive.</p>
|
1,387,184 | <p>Can someone show how to compute the residue of this function:
$$\frac{z}{e^z - 1}$$</p>
<p>I think can represent the Taylor series of $e^z$ as
$$e^z = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots$$
Then, we have
$$\frac{z}{e^z - 1} = \frac{z}{(1 +z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots) -1}$$
$$ = 1 + 1... | Pedro | 23,350 | <p>Matrices are Morita equivalent to their underlying ring, and Morita equivalent rings have isomorphic Hochschild cohomology. A field as an algebra over itself has trivial cohomology for nonzero degrees.</p>
|
1,386,683 | <p>I posted early but got a very tough response.</p>
<p>Point $A = 2 + 0i$ and point $B = 2 + i2\sqrt{3}$ find the point $C$ $60$ degrees ($\pm$) such that Triangle $ABC$ is equilateral. </p>
<p>Okay, so I'll begin by converting into polar form:</p>
<p>$A = 2e^{2\pi i}$ and $B = 4e^{\frac{\pi}{3}i}$</p>
<p>$\overli... | Empty | 174,970 | <p><strong>From your calculation :</strong></p>
<p>$$=\lim_{\overset{x\to0}{y\to0}}\frac{(x^2-2xyi-y^2)(x-iy)}{x^2+y^2}$$</p>
<p>$$=\lim_{(x,y)\to (0,0)}\frac{x^3-3xy^2}{x^2+y^2}-i\lim_{(x,y)\to (0,0)}\frac{3x^2y-y^3}{x^2+y^2}$$</p>
<p>From here, show that both the limits are <strong>zero</strong> by changing polar ... |
4,495,044 | <p>Edit: There is an answer at the bottom by me explaining what is going on in this post.</p>
<p>Define a function <span class="math-container">$f : R \to R$</span> by <span class="math-container">$f(x) = 1$</span> if <span class="math-container">$x = 0$</span> and <span class="math-container">$f(x) = 0$</span> if <spa... | AlvinL | 229,673 | <blockquote>
<p>Proof: Suppose that <span class="math-container">$\lim_{x\to 0; x \in R}f(x)=L$</span>. Then for every <span class="math-container">$\varepsilon > 0$</span> there exists a <span class="math-container">$\delta > 0$</span> such that for all those <span class="math-container">$x \in R$</span> for whi... |
2,879,883 | <p>Suppose that $f$ and $g$ are differentiable functions on $(a,b)$ and suppose that $g'(x)=f'(x)$ for all $x \in (a,b)$. Prove that there is some $c \in \mathbb{R}$ such that $g(x) = f(x)+c$.</p>
<p>So far, I started with this:</p>
<p>Let $h'(x)=f'(x)-g'(x)=0$, then MVT implies $\exists$ c $\in \mathbb{R}$ such that... | Robert Lewis | 67,071 | <p>If $F(x)$ is a differentiable function on $(a, b)$ with</p>
<p>$F'(x) = 0, \; \forall x \in (a, b), \tag 1$</p>
<p>then </p>
<p>$\exists c \in \Bbb R, \; F(x) = c; \tag 2$</p>
<p>for, picking any $x_1, x_2 \in (a, b)$ with $x_1 < x_2$, we have, by the fundamental theorem of calculus,</p>
<p>$F(x_2) - F(x_1) ... |
2,601,088 | <p>I'm new to the group theory and want to get familar with the theorems in it, so I choose a number $52$
to try making some obseveration on all group that has this rank. Below are my thoughts. I don't know if there is any better way to think of these (i.e., an experienced group theorist would think), and I still have... | BallBoy | 512,865 | <p>Are you familiar with the Sylow theorems? They are the most commonly used tool to answer questions about groups of a given order. (For example, they will immediately tell you that a subgroup of order $4$ must exist.) Look up the Sylow theorems, and if you search "groups of order $x$" for many different values of $x$... |
2,371,108 | <p>Cubic equations of the form $ax^3+bx^2+cx+d$ can be solved in various ways. Some are easy to easy to factor in a pair, for some the roots can be found out by trial-and-error, some are one-of-a-kind, some can be reduced to a quadratic equation. A compilation of all possible ways to solve cubic equations would be very... | Mathlover | 22,430 | <p>$ax^3+bx^2+cx+d=0$</p>
<ol>
<li>Divide to a to get $x^3$ term 1</li>
</ol>
<p>$$x^3+\frac{b}{a}x^2+\frac{c}{a} x+\frac{d}{a}=0$$</p>
<p>$$x^3+b_1 x^2+c_1 x+d_1=0$$</p>
<ol start="2">
<li>Eliminate $x^2$ term via using $x=z-\frac{b_1}{3}$ transform
Then you will get </li>
</ol>
<p>$$(z-\frac{b_1}{3})^3+b_1(z-\fr... |
1,040,932 | <p>I have a system of congruence equations</p>
<p>$$
\begin{cases}
x \equiv 17 \pmod{15} \\
x \equiv 14 \pmod{33}
\end{cases}
$$</p>
<p>I need to investigate the system and see if they've got any solutions.</p>
<p>I know that I should use the Chinese remainder theorem "in a reverse order" so I think I should split e... | user194150 | 194,150 | <p>$
\left\{ \begin{array}{l}
x \equiv 17\left[ {15} \right] \\
x \equiv 14\left[ {33} \right] \\
\end{array} \right. \Rightarrow x \equiv 47\left[ {495} \right]
$</p>
|
1,462,379 | <p>I have been given the task to compute $\int_{-1}^1 \sqrt{1-x^2} dx$ by means of calculus. We got the hint to substitute $x=\sin u$, but that only seems to make things more complicated:</p>
<p>$$\int_{-1}^1 \sqrt{1-x^2}dx = \int_{\arcsin-1}^{\arcsin1} \sqrt{1-\sin^2u} \frac{d\arcsin u}{du} du = \int_{\arcsin-1}^{\ar... | Jack D'Aurizio | 44,121 | <p>$$\begin{align*} \int_{-1}^{1}\sqrt{1-x^2}\,dx &= 2\int_{0}^{1}\sqrt{1-x^2}\,dx \\&= 2\int_{0}^{\pi/2}\cos^2\theta\,d\theta\\&=\int_{0}^{\pi/2}(\cos(2\theta)+1)\,d\theta\\&=\int_{0}^{\pi/2}1\,d\theta=\color{red}{\frac{\pi}{2}}\end{align*} $$
as expected. We exploited the parity of the function $\sqrt... |
3,438,653 | <p>I have this thing written on my notes: let <span class="math-container">${x}, {y}\in\mathbb R^n$</span> be two distinct points, then the set <span class="math-container">$$\{ \lambda x + (1-\lambda){y}\;\lvert\; \lambda \in [0,1] \}$$</span> contains all the points on the line segment that connects <span class="math... | user | 505,767 | <p>Let consider</p>
<p><span class="math-container">$$P(\lambda)=\lambda x + (1-\lambda){y}$$</span></p>
<p>then </p>
<p><span class="math-container">$$P(0)=y, \quad P(1)=x$$</span></p>
<p>and </p>
<p><span class="math-container">$$P(\lambda)-P(0)=\lambda (x-y)$$</span></p>
<p>therefore for <span class="math-cont... |
336,834 | <p>It is a well known theorem, that every signed measure can be split into its positive and negative parts (Hahn-Jordan-Decomposition). My question is, if something similar is possible for functionals on Sobolev spaces.</p>
<p>To be precise, let $\Omega \subset \mathbb{R}^n$ be some open domain and $\mu \in H^{-1}(\Om... | user67888 | 67,888 | <p>No, such a decomposition is not possible. <a href="http://en.wikipedia.org/wiki/Distribution_%28mathematics%29#Functions_as_distributions">The reason is that</a> </p>
<blockquote>
<p>every distribution which is non-negative on non-negative functions is of the form $\varphi\mapsto \int \varphi \, d\mu$ for some (p... |
355,740 | <p>Today in class we learned that for exponential functions $f(x) = b^x$ and their derivatives $f'(x)$, the ratio is always constant for any $x$. For example for $f(x) = 2^x$ and its derivative $f'(x) = 2^x \cdot \ln 2$</p>
<p>$$\begin{array}{c | c | c | c}
x & f(x) & f'(x) & \frac{f'(x)}{f(x)}\\ \hline
-1... | Zev Chonoles | 264 | <p>Given a ring $R$, if
$$\underbrace{1_R+\cdots+1_R}_{n\text{ times}}=0_R,$$
then for any $a\in R$, we have (by the distributive property) that
$$\underbrace{a+\cdots+a}_{n\text{ times}}=a\cdot\underbrace{\left(1_R+\cdots+1_R\right)}_{n\text{ times}}=a\cdot 0_R=0_R,$$
so whatever the number is for the multiplicative ... |
20,314 | <p>Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.</p>
| Rasmus | 1,291 | <p>Gel'fand, I. M. and Shilov, G. E.: <em>Generalized Functions</em></p>
|
20,314 | <p>Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.</p>
| Piero D'Ancona | 7,294 | <p>What do you need distributions for? Your request is strange, PDEs are the fundamental application, the origin, and the main source of examples for distribution theory, so no surprise all the books on distributions after a while steer to PDEs.</p>
<p>Thus maybe my advice is misguided since I do not understand your n... |
20,314 | <p>Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.</p>
| Bon Clarke | 11,157 | <p>Two very readable, wide ranging and well motivated accounts are "Generalised Functions and Partial Differential Equations" by Georgi E. Shilov, published by Gordon and Breach 1968, and "Advanced Mathematical Analysis" by Richard Beals, published by Springer 1973 (International student edition). Both are unfortunatel... |
4,291,864 | <p>I have the following equation:</p>
<p><span class="math-container">$y=\frac{3x}{x^{2}+1}$</span></p>
<p>and I want to obtain x in terms of y, so far what I have done is the following:</p>
<p><span class="math-container">$3x=y(x^{2}+1)$</span></p>
<p><span class="math-container">$3x=x^{2}y+y$</span></p>
<p><span clas... | Henno Brandsma | 4,280 | <p>For every open set <span class="math-container">$O$</span> in a topology <span class="math-container">$(X, \mathcal T)$</span>, we form its complement. These sets together are called the closed sets of<span class="math-container">$X$</span>. So <span class="math-container">$\emptyset = X^\complement$</span> and <spa... |
662,403 | <p>I'm working on a homework assignment concerning convex optimization and I came across a problem involving the convexity of the function and the convexity of the domain of the function.</p>
<p>Consider the function $f : [0,1]^3 \in R$ with the following form
$$
f(x,y,z) = xlnx + ylnz + zlnz + \alpha ( x + y + z - 1)... | Nick Alger | 3,060 | <ol>
<li><p>This is a really interesting function as it is the Lagrangian for <a href="http://en.wikipedia.org/wiki/Principle_of_maximum_entropy" rel="nofollow">constrained maximum information entropy</a>, of, for example, a probability distribution on a set with 3 points.</p></li>
<li><p>One can see that $f$ is convex... |
1,125,842 | <p>In $\sf ZFC$ we have the axiom of infinity and thus can define the natural numbers $$\mathbb N \equiv \bigcap\{X:\emptyset\in X\land \forall n(n\in X\implies n\cup\{n\}\in X)\}.$$ From this it's not particularly hard (exercises 1.6 and 1.7 in Jech - <em>Set Theory</em>) to prove that, firstly, every $n\in\mathbb N$... | hmakholm left over Monica | 14,366 | <p>All the sets we define and use in the development of ordinary mathematics are automatically well-founded. Essentially, any set you can prove <em>exists</em> can also be proven to be well-founded without using the axiom of foundation.</p>
<p>The role of the axiom of foundation is not to tell us anything <em>new</em>... |
753,881 | <p>I want to know some typical forms of system of equations generating from practical problems in engineering/economics/physics,etc.</p>
<p>Some examples or research articles would be good.</p>
<p>Specifically, I am looking for some examples of nonlinear system of equations generated from practical problems.</p>
<p>... | ml0105 | 135,298 | <p>So you have four odd numbers. They must occupy slots $2, 4, 6, 8$. So you permute $\{1, 1, 1, 3\}$. There are $\dfrac{4!}{3!}$ ways to do this by the multinomial distribution.</p>
<p>Now we permute $\{2, 2, 4, 4\}$ in slots $1, 3, 5, 7$. There are $\dfrac{4!}{2! * 2!}$ ways to do this by the multinomial distributio... |
1,762,268 | <p>Let $X$ be a Hausdorff space and let $f:X\to \mathbb{R}$. If grapph of $f$ is compact we have to show that $f$ is continuous. </p>
<p>Since every closed subset of a Hausdorff space is closed, therefore grapph of $f$ is closed. WE know that if $f:X\to Y$ and $Y$ is compact, then graph of $f$ is clsed implies $f$ is... | Forever Mozart | 21,137 | <p>Proof without nets:</p>
<p>Let $G$ be the graph of $f$. </p>
<p>Suppose $G$ is compact. </p>
<p>Let $A$ be closed in $\mathbb R$. We show $f^{-1} [A]$ is closed.</p>
<p>Note that $f^{-1}[A]=\pi_X [G\cap (X\times A)]$. </p>
<p>Since $G\cap (X\times A)$ is compact and the projection $\pi _X$ is continuous, $f^{-... |
422,799 | <p>Maschke's theorem says that every <em>finite-dimensional</em> representation of a finite group is completely reducible. Is there a simple example of an infinite-dimensional representation of a finite group which is not completely reducible?</p>
<p>EDIT: As mentioned in the answers, there is actually no finite-dime... | Keenan Kidwell | 628 | <p>For a ring $R$, not necessarily commutative, but with a multiplicative identity, a non-zero $R$-module is a direct sum of simple modules if and only if it is a sum of simple modules, if and only if every submodule has a complement. A representation of a finite group $G$ over $\mathbf{C}$ is the same thing as a $\mat... |
422,799 | <p>Maschke's theorem says that every <em>finite-dimensional</em> representation of a finite group is completely reducible. Is there a simple example of an infinite-dimensional representation of a finite group which is not completely reducible?</p>
<p>EDIT: As mentioned in the answers, there is actually no finite-dime... | rschwieb | 29,335 | <p>I was not able to find any reference for Maschke's theorem talking <em>only</em> about the finite dimensional representations.</p>
<p>The "modern" statement of Maschke's theorem (or at lest, the one ring theorists like) is this:</p>
<blockquote>
<p>For any commutative ring $R$ and finite group $G$, the group rin... |
873,992 | <p>I am having problem with the onto part of this problem.</p>
<p>$\mathbb{N}\rightarrow \mathbb{E}$</p>
<p>My function or pattern is </p>
<p>$x \rightarrow f(x)=2x$ </p>
<p>Which take my natural to even.</p>
<p><strong>One to One</strong></p>
<p>$f(x)=f(y)$</p>
<p>$2x=2y$</p>
<p>$x=y$</p>
<p><strong>Onto</str... | amWhy | 9,003 | <p>You can approach this literally, more or less. We are being told that there does not exist an $x$ such that $x$ is both naive ($N(x)$) and bad ($B(x)$).</p>
<p>It makes sense to let "all people" to be the domain (or universe) in which $x$ resides. </p>
<p>$$\begin{align} \lnot \exists x(N(x) \land B(x)) & \eq... |
3,077,084 | <blockquote>
<p>If <span class="math-container">$a,b,c>0.$</span> Then minimum value of</p>
<p><span class="math-container">$(8a^2+b^2+c^2)\cdot (a^{-1}+b^{-1}+c^{-1})^2$</span></p>
</blockquote>
<p>Try: Arithmetic geometric inequality</p>
<p><span class="math-container">$8a^2+b^2+c^2\geq 3\cdot 2\sqrt{2}(abc)^{1/3}... | Macavity | 58,320 | <p><strong>Hint:</strong> Another way is to consider <a href="https://en.wikipedia.org/wiki/H%C3%B6lder%27s_inequality#Counting_measure" rel="nofollow noreferrer">Hölder's Inequality</a></p>
<p><span class="math-container">$$(8a^2+b^2+c^2)(a^{-1}+b^{-1}+c^{-1})^2 \geqslant (2+1+1)^3$$</span></p>
<p>Equality is possib... |
44,562 | <p>The question is motivated from the definition of $C^r(\Omega)$ I learned from S.S.Chern's <em>Lectures on Differential Geometry</em>:</p>
<p>Suppose $f$ is a real-valued function defined on an open set $\Omega\subset{\bf R}^m$. If all the $k$-th order partial derivatives of $f$ exist and are continuous for $k\leq r... | Asaf Karagila | 622 | <p>Suppose $A\subseteq\mathbb R$ can be well-ordered by the usual $<$, and fix an enumeration of the rationals, i.e. $\mathbb Q = \langle q_n\mid n\in\omega\rangle$.</p>
<p>For $a\in A$ denote $S(a)$ the successor of $a$ in $A$, if $b\in A$ is a maximal element of $A$ then $S(b) = b+1$.</p>
<p>For every $a\in A$ s... |
44,562 | <p>The question is motivated from the definition of $C^r(\Omega)$ I learned from S.S.Chern's <em>Lectures on Differential Geometry</em>:</p>
<p>Suppose $f$ is a real-valued function defined on an open set $\Omega\subset{\bf R}^m$. If all the $k$-th order partial derivatives of $f$ exist and are continuous for $k\leq r... | André Nicolas | 6,312 | <p>The following idea uses some set-theoretic machinery, has the advantage of coming from a simple geometric visualization. </p>
<p>Suppose to the contrary that there is an uncountable set $A$ of reals which is well-ordered under the natural order.</p>
<p>Then $A$ is order isomorphic to an uncountable ordinal. It fo... |
3,652,518 | <p>I am no mathematician but have studied mathematics some 20 years ago. So I know basics of number theory but have lost the skills to solve problems. </p>
<p>I was wondering if the equation <span class="math-container">$3ax^2 + (3a^2+6ac)x-c^3=0$</span> in which <span class="math-container">$a$</span> and <span cla... | Jingeon An-Lacroix | 471,868 | <p>Let <span class="math-container">$$\alpha:=x+\cdots+x^n.$$</span> Then <span class="math-container">$$\alpha x=x^2+\cdots+x^{n+1}.$$</span> Therefore, <span class="math-container">$$\alpha(x-1)=x^{n+1}-x=x(x^n-1).$$</span> If <span class="math-container">$x\neq 1$</span>, divide both sides with <span class="math-con... |
2,706,872 | <p>I'm working on my latest linear algebra assignment and one question is as follows: </p>
<p>In $\mathbb R^3$ let <em>R</em> be the reflection over the null space of the matrix </p>
<p><em>A</em> = [4 4 5]</p>
<p>Find the matrix which represents <em>R</em> using standard coordinates. </p>
<p>I am familiar with the... | RCT | 424,406 | <p>I could imagine that there's some ugly formula for this, but I don't know it, and it's probably more instructive to solve the problem from basic principles anyway, so let's try that! My first thought when reading this problem is to start by changing to a basis in which it's obvious what the reflection does.</p>
<p>... |
804,483 | <p>The following integrals look like they might have a closed form, but Mathematica could not find one. Can they be calculated, perhaps by differentiating under the integral sign?</p>
<p>$$I_1 = \int_{-\infty }^{\infty } \frac{\sin (x)}{x \cosh (x)} \, dx$$
$$I_2 = \int_{-\infty }^{\infty } \frac{\sin ^2(x)}{x \sinh (... | Urgje | 95,681 | <p>I do not know of a closed form but it seems that the integral can be
converted into an infinite series. Let
$$
I_{1}(k)=\int_{-\infty }^{+\infty }dx\frac{\sin kx}{x\cosh x}
$$
so $I_{1}=I_{1}(1).$ Now (note that $I_{1}(0)$ vanishes)
\begin{eqnarray*}
\partial _{k}I_{1}(k) &=&\int_{-\infty }^{+\infty }dx\frac... |
804,483 | <p>The following integrals look like they might have a closed form, but Mathematica could not find one. Can they be calculated, perhaps by differentiating under the integral sign?</p>
<p>$$I_1 = \int_{-\infty }^{\infty } \frac{\sin (x)}{x \cosh (x)} \, dx$$
$$I_2 = \int_{-\infty }^{\infty } \frac{\sin ^2(x)}{x \sinh (... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}
\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{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\... |
685,681 | <p>I want to prove that $\dim V/(X \cap Y)$ in finite, if $V$ be a vector space and $X$, $Y$ two sub spaces of $V$ such that $\dim V/Y$ and $\dim V/X$ are finite.</p>
| ajd | 90,897 | <p>A much easier argument - consider the map $V\to (V/X)\times (V/Y)$ given by the product of the two projection maps. The kernel of this map is $X\cap Y$. Therefore, we have an injection $V/(X\cap Y)\to (V/X)\times (V/Y)$. Therefore, $\dim V/(X\cap Y)\le \dim [(V/X)\times (V/Y)]=\dim(V/X)\dim(V/Y)<\infty$.</p>
|
878,939 | <p>I have found the eigen vaues, I also know that you can find the eigenvectors through a Gausian Jordan.
-- x1, gauss jordan gives me rows(1 -1/3 ,, 0 0 ), so [a, b] = [1,3]
For vector x2, GJ gives (1 -2/5 ,, 0 0 ), I would assume [a,b] = [2,5], but why did they choose to go with [-2,-5]. I don't get it?</p>
<p... | mfl | 148,513 | <p>Every even integer $n$ can be written as $n=2m,$ for some integer $m.$ Now, there are two possibilities: $m$ is even ($m=2q$) or $m$ is odd ($m=2q+1$) for some integer $q.$ In the first case we have</p>
<p>$$n=2m=4q$$</p>
<p>and in the second case</p>
<p>$$n=2m=2(2q+1)=4q+2.$$</p>
<p>Another way to see why we ca... |
508,791 | <p>I have an integer list that is <code>n</code> long and each value can be ranging from <code>1 .. n</code>.</p>
<p>I need a formula that tells me how many of all possible lists for a given n, that have one or more consecutive sequences of a length of exactly 2 of the same number and no other consecutive sequences th... | miracle173 | 11,206 | <p>How many sequences of length $q$ of numbers from $\{1,...,p\}$ are there such that consecutive elements are always different?
For the first element of such a sequence we can selcet one of the $p$ different values of $\{1,...,p\}$. For the following $q-1$ positions we can select
always all values from $\{1,...,p\... |
2,073,410 | <p>If $$ a-(a \bmod x)<b$$ how do I prove that $$c-(c\bmod x)<b \;\forall c<a?$$ </p>
| user64066 | 64,066 | <p>Write $a=k_a+p_a x$ and $c=k_c+p_cx$ where $k_a, k_c\geq 0$. Then we have $p_a x <b$, and since $c<a$ implies $p_c x<p_a x$. Thus, the result follows. </p>
|
3,191,233 | <p>Why is <span class="math-container">$ce^λ=1$</span> equal to <span class="math-container">$c=e^{-λ}$</span>?</p>
| Kwin van der Veen | 76,466 | <p>The answer by Michael Stachowsky is incomplete. It is indeed correct that the time evolution of a continuous time LTI state space model</p>
<p><span class="math-container">$$
\left\{
\begin{align}
\dot{x}(t) &= A\,x(t) + B\,u(t), \\
y(t) &= C\,x(t),
\end{align}
\right. \tag{1}
$$</span></p>
<p>can be expre... |
4,408,772 | <p>I'm read about a Lienard System in Perko books, but I don't understand how this applies
<a href="https://i.stack.imgur.com/PFVeR.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PFVeR.png" alt="enter image description here" /></a>
<a href="https://i.stack.imgur.com/vgttf.png" rel="nofollow noreferr... | Lutz Lehmann | 115,115 | <p>I too can not see the value of the cited theorem without further context.</p>
<p>Let's fixate what is obviously true:</p>
<ul>
<li><p>As long as <span class="math-container">$x>0$</span> we have <span class="math-container">$\dot y=-g(x)<0$</span>, so <span class="math-container">$y$</span> is at least non-inc... |
3,800,521 | <p>Let <span class="math-container">$x=\tan y$</span>, then
<span class="math-container">$$
\begin{align*}\sin^{-1} (\sin 2y )+\tan^{-1} \tan 2y
&=4y\\
&=4\tan^{-1} (-10)\\\end{align*}$$</span></p>
<p>Given answer is <span class="math-container">$0$</span></p>
<p>What’s wrong here?</p>
| Alexey Burdin | 233,398 | <p><span class="math-container">$$\arcsin(\sin(x))=\left\{
\begin{array}{ll}
x-2\pi n&\hbox{for }-\frac{\pi}{2}+2\pi n\le x\le \frac{\pi}{2}+2\pi n\\
\pi-x-2\pi n&\hbox{for }\frac{\pi}{2}+2\pi n\le x\le \frac{3\pi}{2}+2\pi n\\
\end{array}\right.,$$</span>
<span class="math-container">$$\arctan(\tan(x))=x-\left\... |
3,800,521 | <p>Let <span class="math-container">$x=\tan y$</span>, then
<span class="math-container">$$
\begin{align*}\sin^{-1} (\sin 2y )+\tan^{-1} \tan 2y
&=4y\\
&=4\tan^{-1} (-10)\\\end{align*}$$</span></p>
<p>Given answer is <span class="math-container">$0$</span></p>
<p>What’s wrong here?</p>
| lab bhattacharjee | 33,337 | <p>Let <span class="math-container">$\tan^{-1}\dfrac{2x}{1-x^2}=u\implies-\dfrac\pi2<u<\dfrac\pi2$</span></p>
<p><span class="math-container">$\tan u=\dfrac{2x}{1-x^2}$</span></p>
<p><span class="math-container">$\implies\sec u+\sqrt{1+\left(\dfrac{2x}{1-x^2}\right)^2}=\dfrac{1+x^2}{|1-x^2|}$</span></p>
<p><span ... |
17,134 | <p>On a very regular basis we see new users that are not accustomed with the use of MathJaX on MSE. Sometimes even some users that aren't that new to the site. Most of us, when this happens, kindly bring to this users attention that there is a <a href="http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tut... | daOnlyBG | 173,397 | <p>The motivation behind your idea is good; after all, the lack of MathJax editing can stir up confusion for both the asker and the helper. We all know MathJax's benefits.
However, I'm not quite sure that adding a "tutorial" of some sort would really help out- it's easy to skip through the tutorial steps, especially wh... |
17,134 | <p>On a very regular basis we see new users that are not accustomed with the use of MathJaX on MSE. Sometimes even some users that aren't that new to the site. Most of us, when this happens, kindly bring to this users attention that there is a <a href="http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tut... | ELC | 189,635 | <p>I just joined today, and it took me a few minutes to locate a tutorial for syntax, but it wasn't too difficult. </p>
<p>I think it would be useful to have this tutorial available in a more obvious place or perhaps sent along with the registration email. </p>
|
896,940 | <p>i tried 9 D + (-10 D)</p>
<p>9= 0000 1001</p>
<p>10= 0000 1010</p>
<p>Reverse 10 = 1111 0101 and add 1 become 1111 0110</p>
<p>after that add up 9 D + (-10 D) == 0000 1001 + 1111 0110 but the answer is equal to 1111 1111 whch is 255 in decimal but the answer should be -1 right? anything goes wrong?</p>
<p>Thank... | Adriano | 76,987 | <p>In two's complement, we deal with signed numbers by checking the leftmost bit. If this leftmost bit is a $0$, then it's positive, so we proceed like we do with unsigned numbers. Otherwise, if the leftmost bit is a $1$, then it's negative, so we have to do the "reverse the bits then add one" trick that you did when y... |
6,887 | <p>Let x_1, x_2, ... be iid draws from a laplace distribution with scale parameter b. Is there a relatively nice closed form for x_1+x_2+...x_n? I've seen a derivation floating around for when b=1, but I couldn't figure out a generalisation. </p>
| Andrew M Ross | 2,128 | <p>One way to generate a Laplace random variable is to generate two IID (independent and identically distributed) exponential random variables and then subtract them:
x_i = y_i - z_i
with y_i and z_i ~ exponential(parameter=b), and of course everything independent.
Then the sum of the x_i is simply (sum y_i) - (sum z_i... |
329,600 | <p>Let $U,V,W$ be vector spaces over $F$ and $S: U \to V$, and $T: V \to W$ linear maps.</p>
<p>(a) Show that if $S$ and $T$ are isomorphisms, then $T\circ S$ is an isomorphism, too.</p>
<p>(b) Show that if $U$ is isomorphic to $V$ and $V$ is isomorphic to $W$, then $U$ is isomorphic to $W$.</p>
| azimut | 61,691 | <p><strong>Hints:</strong></p>
<ul>
<li><p>For (a): Check that the conditions on an isomorphism hold for $T\circ S$.</p></li>
<li><p>For (b): Apply (a).</p></li>
</ul>
|
2,646,363 | <p>Let $A_1, A_2, \ldots , A_{63}$ be the 63 nonempty subsets of $\{ 1,2,3,4,5,6 \}$. For each of these sets $A_i$, let $\pi(A_i)$ denote the product of all the elements in $A_i$. Then what is the value of $\pi(A_1)+\pi(A_2)+\cdots+\pi(A_{63})$?</p>
<p>Here is the solution </p>
<p>For size 1: sum of the elements, whi... | Jose L. Arregui | 892,303 | <p>The answer is 5039 just because, if you add 1 (let 1 be the product of the elements of the empty set) you must get <span class="math-container">$7!$</span>.</p>
<p>This is a general property: if <span class="math-container">$\pi(A)$</span> is the product of the elements of <span class="math-container">$A$</span> (wi... |
1,282,489 | <p>I have a simple problem that I need to solve. Given a height (in blue), and an angle (eg: 60-degrees), I need to determine the length of the line in red, based on where the green line ends. The green line comes from the top of the blue line and is always 90-degrees.</p>
<p>The height of the blue line is variable.... | Hagen von Eitzen | 39,174 | <p>Reflect at the green line to obtain a triangle, which must be equilateral.</p>
|
3,101,098 | <p>From 11, 12 in the book Logic in Computer Science by M. Ryan and M. Huth:</p>
<p>**</p>
<blockquote>
<p>"What we are saying is: let’s make the assumption of ¬q. To do this,
we open a box and put ¬q at the top. Then we continue applying other
rules as normal, for example to obtain ¬p. But this still depends o... | Graham Kemp | 135,106 | <blockquote>
<p>How can an assumption only have scope inside the box, but once you finish what you want to prove it is no more part of the assumption box and is accessible universally in the proof? </p>
</blockquote>
<p>Assumptions are raised and discharged.</p>
<p>When you raise an assumption, you open a box (rath... |
2,286,540 | <p>$(x,y)=(x,\sqrt{x})$
$d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$
$=\sqrt{x^4-5x^2+9}$</p>
<p>$g(x)=x^4-5x^2+9$ , $g'(x)=4x^3-10x=0$ $: x=0, x=+-\sqrt{10}/2$</p>
<p>The question is: Should I put those x-values in g(x) or the orginal graph,$
y=\sqrt{x}$. To me, it's logical to put it into g(x), but in an example it was re... | Jack D'Aurizio | 44,121 | <p>I feel an issue shared by many students: to apply algorithms without really understanding what they are doing. What is the question? The question is <em>locate a point such that$\ldots$</em>. So the answer is a point, i.e. a couple of coordinates. $f(x)$ is defined only on $[0,+\infty]$, so the answer is given by $(... |
2,286,540 | <p>$(x,y)=(x,\sqrt{x})$
$d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$
$=\sqrt{x^4-5x^2+9}$</p>
<p>$g(x)=x^4-5x^2+9$ , $g'(x)=4x^3-10x=0$ $: x=0, x=+-\sqrt{10}/2$</p>
<p>The question is: Should I put those x-values in g(x) or the orginal graph,$
y=\sqrt{x}$. To me, it's logical to put it into g(x), but in an example it was re... | Mercy King | 23,304 | <p>You want to minimize the function
$$
f(x)=\|(x,\sqrt{x})-(3,0)\|^2=\|(x-3,\sqrt{x})\|^2=(x-3)^2+(\sqrt{x})^2=x^2-5x+9
$$
The function $f$ reaches its minimum value at
$$
h=-\dfrac{-5}{2(1)}=\dfrac{5}{2}
$$
and the minimum value is
$$
k=f(h)=\dfrac{11}{4}
$$
Hence the point of $y=\sqrt{x}$ closest to the point $(3,0... |
573,964 | <blockquote>
<p>Let set $S$ be the set of all functions $f:\mathbb{Z_+} \rightarrow \mathbb{Z_+}$. Define a realtion $R$ on $S$ by $(f,g)\in R$ iff there is a constant $M$ such that $\forall n (\frac{1}{M} < \frac{f(n)}{g(n)}<M). $ Prove that $R$ is an equivalence relation and that there are infinitely mane equ... | sheriff | 49,030 | <p>But $k$ must live in a field, and $\mathbb{Z}$ isn't.</p>
|
2,404,176 | <p>From the days I started to learn Maths, I've have been taught that </p>
<blockquote>
<p>Adding Odd times Odd numbers the Answer always would be Odd; e.g.,
<span class="math-container">$$3 + 5 + 1 = 9$$</span></p>
</blockquote>
<p>OK, but look at this question </p>
<p><a href="https://i.stack.imgur.com/TmYsJ.... | Robert Soupe | 149,436 | <p>Do you <em>have</em> to fill in <em>all</em> three boxes? I would just put 15 in <em>two</em> of the boxes, e.g., $$\fbox{ } + \fbox{15} + \fbox{15} = 30,$$ and hope people understand the first box as being an implicit 0.</p>
|
2,854,671 | <p>Consider a $n\times n$ Hankel Matrix</p>
<p>$$
H = \begin{bmatrix}
x_{1} & x_{2} & \dots & x_{n} \\
x_{2} & x_{3} & \dots & x_{n+1} \\
\vdots \\
x_{n} & x_{n+1} & \dots & x_{2n}
\end{bmatrix}
$$
, where all $x_i \in \mathbb{Z}_p = \{ 0,\dots,p-1 \}$, where $p$ is ... | littleO | 40,119 | <p>This is not a full answer, but it's too long for a comment. The Hankel matrix
$$
H = \begin{bmatrix} x_1 & x_2 & x_3 & x_4 \\
x_2 & x_3 & x_4 & x_5 \\
x_3 & x_4 & x_5 & x_6 \\
x_4 & x_5 & x_6 & x_7 \\
\end{bmatrix}
$$
can be enlarged to an anti-circulant matrix
$$
\ti... |
2,631,342 | <p>$$\lim_{x\rightarrow 14}\frac{\sqrt{x-5}-3}{x-14}$$</p>
<p>How do I evaluate the limit when I put x = 14 and I got 0/0?</p>
| Michael Rozenberg | 190,319 | <p>$$\frac{\sqrt{x-5}-3}{x-14}=\frac{x-14}{(x-14)(\sqrt{x-5}+3)}\rightarrow\frac{1}{6}.$$</p>
|
106,131 | <blockquote>
<p>Define $f:\mathbb{R}\rightarrow\mathbb{R}$ by<br>
$\
f(x) =
\begin{cases}
1/q
& \text{if } x =p/q \space(\mathrm{lowest}\space \mathrm{terms},\space\mathrm{nonzero})\\
0 & \text{if } x = 0\space\mathrm{or}\space x\not\in\mathbb{Q}
\end{cases}
$<br>
Show that f is continuo... | David Mitra | 18,986 | <p>Hints:</p>
<p>Keep in mind that to show $f$ is continuous at $p=0$ or at $p$ irrational, that $f(p)=0$. So to show continuity at $p$, you need to show that $|f(z)|$ can be made as small as desired by taking $z$ sufficiently close to $p$.</p>
<p>Towards this end, use (and prove) the fact that one can select an op... |
154,722 | <p>Let $A = \pmatrix{1 & 0 \\ \alpha & 1} $ and $ B = \pmatrix{1 & 1 \\ 0 & 1}$, where $\alpha \in \mathbb{C}$ is a complex parameter.</p>
<p>Now consider the family of representations $r_{\alpha}$ of the free group on two generators $F_2 = \langle a,b\rangle$ in $\mathrm{SL}(2, \mathbb{C})$ setting ... | Alexandre Eremenko | 25,510 | <p>Yes, the answer is complicated, it is related to holomorphic dynamics,
and the question was much studied, see for example:
MR0869581
Lyubich, M. Yu.; Suvorov, V. V.
Free subgroups of SL2(C) with two parabolic generators.</p>
|
154,722 | <p>Let $A = \pmatrix{1 & 0 \\ \alpha & 1} $ and $ B = \pmatrix{1 & 1 \\ 0 & 1}$, where $\alpha \in \mathbb{C}$ is a complex parameter.</p>
<p>Now consider the family of representations $r_{\alpha}$ of the free group on two generators $F_2 = \langle a,b\rangle$ in $\mathrm{SL}(2, \mathbb{C})$ setting ... | Ian Agol | 1,345 | <p>The free discrete subgroups consist of the closure of the Riley slice of Schottky space (notice that one may assume $Re(\alpha)\geq 0, Im(\alpha)\geq 0$, since $\alpha \mapsto -\alpha, \overline{\alpha}$ preserves discreteness).
Here's a picture of the first quadrant of the Riley slice:
<img src="https://i.stack.im... |
2,721,718 | <blockquote>
<p>$AD, BE$ and $CF$ are three concurrent lines in $\triangle ABC$, meeting the opposite sides in $D, E$ and $F$ respectively. Show that the joins of the midpoints of $BC, CA$ and $AB$ to the midpoints of $AD, BE$ and $CF$ are concurrent.</p>
</blockquote>
<p>Let $D', E'$ and $F'$ be the midpoints of $B... | Tsemo Aristide | 280,301 | <p>Let $f:X\times Y-A\times B\rightarrow\{0,1\}$ be a continuous function. There exists $x\in X-A, y\in Y-B$, suppose $f(x,y)=0$. We have $f(x\times Y)=f(X\times y)=0$ since $x\times Y$ and $X\times y\subset :X\times Y-A\times B$. Let $(u,v)\in :X\times Y-A\times B$. You have $u$ is not in $X$ or $v$ is not in $Y$. If... |
2,721,718 | <blockquote>
<p>$AD, BE$ and $CF$ are three concurrent lines in $\triangle ABC$, meeting the opposite sides in $D, E$ and $F$ respectively. Show that the joins of the midpoints of $BC, CA$ and $AB$ to the midpoints of $AD, BE$ and $CF$ are concurrent.</p>
</blockquote>
<p>Let $D', E'$ and $F'$ be the midpoints of $B... | Henno Brandsma | 4,280 | <p>Pick $p \in X \setminus A$, $q \in Y \setminus B$, by properness of inclusions.</p>
<p>For $x \in X$ define $V_x = \{x\} \times Y$ the vertical stalk at $X$, homeomorphic to $Y$, so connected. </p>
<p>Likewise for $y \in Y$, $H_y = X \times \{y\}$, the horizontal "stalk" at $y$, homeomorphic to $X$ and hence conne... |
4,545,364 | <blockquote>
<p>Solve the quartic polynomial :
<span class="math-container">$$x^4+x^3-2x+1=0$$</span>
where <span class="math-container">$x\in\Bbb C$</span>.</p>
<p>Algebraic, trigonometric and all possible methods are allowed.</p>
</blockquote>
<hr />
<p>I am aware that, there exist a general quartic formula. (Ferrari... | Parcly Taxel | 357,390 | <p>Substitute <span class="math-container">$x=y+1$</span>, then <span class="math-container">$z=y+1/y$</span> to get
<span class="math-container">$$y^4+5y^3+9y^2+5y+1=0$$</span>
<span class="math-container">$$y^2((y+1/y)^2+5(y+1/y)+7)=0$$</span>
<span class="math-container">$$z^2+5z+7=0$$</span>
Then <span class="math-... |
2,179,253 | <p>$$n{n-1 \choose 2}={n \choose 2}{(n-2)}$$
Give a conceptual
explanation of why this formula is true.</p>
| Joffan | 206,402 | <p>$$\begin{align}
n{n-1 \choose 2}&=\binom n1 \binom{n-1}{2} \tag {choose 1 then 2}\\[1ex]
&= \binom n2 \binom{n-2}{1} \tag {choose 2 then 1}\\[1ex]
&={n \choose 2}{(n-2)}
\end{align}$$</p>
|
2,068,986 | <p>Consider the function</p>
<p>$$K(u) = \frac 1 {\sqrt {2\pi}} \left( \Bbb e ^{-\frac 1 2 \left( \frac {u-5.3} h \right)^2 } + \Bbb e ^{-\frac 1 2 \left( \frac {u-1.6} h \right)^2 } + \Bbb e ^{-\frac 1 2 \left( \frac {u-2.1} h \right)^2 } + \Bbb e ^{-\frac 1 2 \left( \frac {u-1.7} h \right)^2 } + \Bbb e ^{-\frac 1 2 ... | John Hughes | 114,036 | <p>Write $\ln 2$ as a binary number with bits $b_1, b_2, \ldots$. </p>
<p>As others have suggested in briefly-present answers, taking logs is the secret: you want
$$
\lim_{n\to+\infty}\frac{a_n}{2^n}=\ln(2)
$$</p>
<p>and hence
$$a_n=2^n\ln(2).$$
but this choice of $a_n$ is not an integer. On the other hand, the
f... |
3,164,280 | <p>By accident, I find this summation when I pursue the particular value of <span class="math-container">$-\operatorname{Li_2}(\tfrac1{2})$</span>, which equals to integral <span class="math-container">$\int_{0}^{1} {\frac{\ln(1-x)}{1+x} \mathrm{d}x}$</span>.</p>
<p>Notice this observation</p>
<p><span class="math-co... | Jack D'Aurizio | 44,121 | <p>Well, ignoring the dilogarithm reflection formula, we still have
<span class="math-container">$$ \sum_{n=1}^{N}\frac{\log(2)}{n}=\log(2)H_N,\qquad \sum_{n=1}^{N}\frac{H_n}{n}\stackrel{\text{sym}}{=}\frac{H_n^2+H_n^{(2)}}{2} $$</span>
and
<span class="math-container">$$ \sum_{n=1}^{N}\frac{H_{2n}}{n}\stackrel{\text{... |
3,164,280 | <p>By accident, I find this summation when I pursue the particular value of <span class="math-container">$-\operatorname{Li_2}(\tfrac1{2})$</span>, which equals to integral <span class="math-container">$\int_{0}^{1} {\frac{\ln(1-x)}{1+x} \mathrm{d}x}$</span>.</p>
<p>Notice this observation</p>
<p><span class="math-co... | omegadot | 128,913 | <p>Here is an approach that avoids knowing the value of <span class="math-container">$\operatorname{Li}_2 (\frac{1}{2})$</span>. </p>
<p>Let
<span class="math-container">$$S = \sum_{n = 1}^\infty \frac{1}{n} \left (H_{2n} - H_n - \ln 2 \right ).$$</span>
Observing that
<span class="math-container">$$\int_0^1 \frac{x^{... |
1,112,081 | <p>Does $\int_0^\infty e^{-x}\sqrt{x}dx$ converge? Thanks in advance.</p>
| dalastboss | 194,935 | <p>Hint: Note that $\sqrt xe^{-x}$is monotonically decreasing for large enough $x$ and as such the integral converges only if the infinite sum of the same converges. Then apply ratio test. </p>
|
1,890,040 | <p>This is related to <a href="https://math.stackexchange.com/questions/1888881/expanding-a-potential-function-via-the-generating-function-for-legendre-polynomi">this previous question of mine</a> where (with lots of help) I show that $$\sum_{l=0}^\infty \frac{R^l}{a^{l+1}}P_l(\cos\theta)=\frac{1}{\sqrt{a^2-2aR\cos\the... | Toutatis | 360,924 | <p>I found 432 in the book
"Les nombres remarquables" by Francois Le Lionnais (Herman, 1983)
(which is a list of interesting numbers), but there is no reference or proof. Among these 432 parts 192 are bounded.</p>
|
4,324,493 | <p>According to Rick Miranda's Algebraic curves and Riemann surfaces, a hyperelliptic curve is defined as the Riemann surface obtained by gluing two algebraic curves, <span class="math-container">$y^2=h(x)$</span> and <span class="math-container">$w^2 = k(z)$</span> (where <span class="math-container">$h$</span> has di... | KReiser | 21,412 | <p>This question has been <a href="https://mathoverflow.net/questions/14024/degree-2-branched-map-from-the-torus-to-the-sphere">addressed on MathOverflow</a>. Here's the accepted answer there, from user <a href="https://mathoverflow.net/users/3384/stankewicz">stankewicz</a>:</p>
<blockquote>
<p>One example: lay your <s... |
1,818,764 | <p>I think everything I have done is kosher, but unless I am missing an identity it is a different answer than the online quiz and wolfram alpha give.</p>
<p>I tried to use the trig substitution
$$ x=2\sin(\theta)\Rightarrow dx=2\cos(\theta)$$</p>
<p>Which yields
$$\int\frac{x^2}{\sqrt{4-x^2}}dx=\int\frac{4\sin^2(\t... | SchrodingersCat | 278,967 | <p>There are $3$ mistakes that you have made:</p>
<ol>
<li>Missed the $d\theta$ in the first line. </li>
</ol>
<blockquote>
<p>$$ x=2\sin(\theta)\Rightarrow dx=2\cos(\theta)$$</p>
</blockquote>
<ol start="2">
<li>Then you did the integration in the following step as:</li>
</ol>
<blockquote>
<p>$$=2\int (1-\cos(... |
32,137 | <p>I have an equation that I evaluate at some point (let's say $x=1$) that have terms of the form</p>
<pre><code>f[1,y] D[g[1,y],{y,2}]
</code></pre>
<p>Is there an easy way to replace [1,y] by [x,y] with a simple replacement rule? The thing is that </p>
<pre><code>g[1,y] /. g[1,y] -> g[x,y]
</code></pre>
<p>wil... | István Zachar | 89 | <p>A more general approach that replaces argument lists <code>[1, y]</code> <em>anywhere</em> (note, that this might result in unwanted replacements compared to Szabolcs's solution):</p>
<pre><code>(f[1, y] D[g[1, y], {y, 2}]) /. h_[1, y] :> h[x, y]
(* ==> f[x, y]*Derivative[0, 2][g][x, y] *)
</code></pre>... |
1,323,845 | <p>For a nonnegative integer $n$, a composition of $n$ means a partition in which the order of the parts matters.</p>
<p>Consider the generating function
$$C(x) = \sum_{n=0}^{\infty} c_nx^n,$$
where $c_n$ is the number of distinct compositions of $n$ (note that $c_0=1$ by convention).</p>
<p>What is the value of $C\l... | Tim Raczkowski | 192,581 | <p>I don't think this is true. Take two points in an open ball. Since there is a path between the points and the image of the path is uncountable, the open ball is uncountable.</p>
|
1,323,845 | <p>For a nonnegative integer $n$, a composition of $n$ means a partition in which the order of the parts matters.</p>
<p>Consider the generating function
$$C(x) = \sum_{n=0}^{\infty} c_nx^n,$$
where $c_n$ is the number of distinct compositions of $n$ (note that $c_0=1$ by convention).</p>
<p>What is the value of $C\l... | Matematleta | 138,929 | <p>At least, if the open ball $B$ is connected, this is false because then if $B$ is countable, it has only has one point. To see this, suppose there are two distinct points $a,b\in B$. Then $d$ restricted to $B$ is a metric on $B$. </p>
<p>Consider S= $\left \{ d(a,x) :x\in B\right \}$. If $S$ contains every real num... |
4,128,046 | <p>I am working through a pure maths book as a hobby. This question puzzles me.</p>
<p>The line y=mx intersects the curve <span class="math-container">$y=x^2-1$</span> at the points A and B. Find the equation of the locus of the mid point of AB as m varies.</p>
<p>I have said at intersection:</p>
<p><span class="math-c... | Vishu | 751,311 | <p>The <span class="math-container">$x$</span>-coordinate of the midpoint is the the sum of the other two <em>divided by <span class="math-container">$2$</span></em>, i.e. <span class="math-container">$$ x=\frac m2$$</span> And <span class="math-container">$$y=mx =\frac{m^2}{2} $$</span> and <span class="math-container... |
4,128,046 | <p>I am working through a pure maths book as a hobby. This question puzzles me.</p>
<p>The line y=mx intersects the curve <span class="math-container">$y=x^2-1$</span> at the points A and B. Find the equation of the locus of the mid point of AB as m varies.</p>
<p>I have said at intersection:</p>
<p><span class="math-c... | Parcly Taxel | 357,390 | <p>You made a small mistake near the end: the midpoint's <span class="math-container">$x$</span>-coordinate is not the <em>sum</em> of those of the intersection points, but half that sum, which is still <span class="math-container">$m/2$</span>. Then the corresponding <span class="math-container">$y$</span>-coordinate ... |
422,941 | <p>How can we expand the following by the binomial expansion, upto the term including $x^3$? That'll be 4 terms.</p>
<p>This the expression to be expanded: $\sqrt{2+x\over1-x}$</p>
<p>I understand how to do the numerator and denominator individually. Now this is what I'm doing - having expanded the denominator (u... | André Nicolas | 6,312 | <p>You have tow alternatives, neither completely pleasant.</p>
<p>1) Let our function be $f(x)$, Compute $f(0)$, $f'(0)$, $f''(0)$, and $f'''(0)$. Then the answer is
$$f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f'''(0)}{3!}x^3.$$</p>
<p>2) Or else (easier) compute the expansion of $(2+x)^{1/2}$ and $(1-x)^{-1/2}$ up to ... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.