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 |
|---|---|---|---|---|
1,405,809 | <p>So we have to find $\lim\limits_{x\to 0} \frac{a^x-1}{x}$ without using any series expansions or the L'Hopital's rule.<br>
I did it using both but I have no idea how to do it. I tried many substitutions but nothing worked. Please point me in the right direction.</p>
| ybtang21c | 285,657 | <p>Firt prove $\lim\limits_{x\to 0}\frac{e^x-1}x=1$. In fact, let $y=e^x-1$, that is $x=\ln(1+y)$, we have
$$\lim\limits_{x\to 0}\frac{e^x-1}x=\lim\limits_{y\to 0}\frac{y}{\ln(1+y)}
=\lim\limits_{y\to 0}\frac1{\frac1y\ln(1+y)}=\frac1{\ln\left[\lim\limits_{y\to 0}
(1+y)^{1/y}\right]}=\frac1{\ln e}=1.$$
Then
$$\lim\limit... |
2,263,520 | <p>I'm trying to understand how to simplify summations. My text says that: $$\sum_{i=1}^{\frac{n}{2}} \sum_{j=i}^{n-i} \sum_{k=1}^{j} 1 = \frac{n^3}{8}$$</p>
<p>But does not explain how to get to the right-hand side.</p>
<p>I think the above nested summation evaluates to $\sum_{i=1}^{n/2} \left[\sum_{j=i}^{n-i} j = i... | Dirk | 379,594 | <p>Hint:
$$\sum_{j=i}^{n-i} j = \sum_{j=1}^{n-i} j - \sum_{j=1}^{i-1} j.$$
The right hand side now should look rather familiar (Gauss...).</p>
<p>There is also a rule for
$$\sum_{i=1}^n i^2,$$
if you look this up and prove it or if you already know it, that should give the final result.</p>
|
2,140,965 | <p>Arithmetic progression
:$a_n=\{a,a+d,a+2d,...,a+nd,...\}$</p>
<p>Geometric progression
:$b_n=\{b,bq,bq^2,...,bq^n,...\}$</p>
<p>if : $a_r,a_s,a_t,$ be a Geometric progression($b_n$)</p>
<p>then : What is q?</p>
<p>my way :
let :$r>s>t$</p>
$$ a_{r} =a +(r-1)d $$
$$ a_{s} =a +(s-1)d $$
$$ a_{t} =a +(t-1... | G Cab | 317,234 | <p>Recall what is the application of the dot product between vectors in physics:
$$
\mathbf{v} \cdot \mathbf{w} = \text{scalar} = \left| {proj_\mathbf{w} \mathbf{v}} \right| \cdot \left| \mathbf{w} \right|
$$
then
$$
\left| {proj_\mathbf{w} \mathbf{v}} \right| = \frac{{\mathbf{v} \cdot \mathbf{w}}}
{{\left| \mathbf{w} ... |
411,549 | <p>Here is an modular equation</p>
<p>$$5x \equiv 6 \bmod 4$$</p>
<p>And I can solve it, $x = 2$.</p>
<p>But what if each side of the above equation times <strong>8</strong>, which looks like this</p>
<p>$$40x \equiv 48 \bmod 4$$</p>
<p>Apparently now, $x = 0$. Why is that? Am I not solving the modular equation in... | amWhy | 9,003 | <p>Note that in multiplying each side of the original congruence by $8$, you multiplied the congruence by a <em>multiple of the modulus</em>, $4$, hence, since $4\mid 8$, both sides are thereby divisible by $4$, i.e., each side of the congruence is then a multiple of the modulus, and so congruent, by definition, to $0$... |
4,358,080 | <p>I'm making a pump and I need to make a circular (from the front perspective) hole in a side of a pipe. I can't use a drill and I have to print out a shape that I will stick onto it and cut and file away. Will this circle projected off center onto a cylinder be an ellipse, or is it not an exact ellipse and I have to ... | sirous | 346,566 | <p><a href="https://i.stack.imgur.com/qB7xV.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/qB7xV.jpg" alt="enter image description here" /></a></p>
<p><a href="https://i.stack.imgur.com/7gys0.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7gys0.jpg" alt="enter image description h... |
198,105 | <p>I have runtimes for requests on a webserver. Sometimes events occur that cause the runtimes to skyrocket (we've all seen the occasionaly slow web page before). Sometimes, they plummet, due to terminated connections and other events. I am trying to come up with a consistent method to throw away spurious events so tha... | Henry | 6,460 | <p>Your data is not close to being normally distributed even if you exclude outliers so the whole procedure is rather dubious. </p>
<p>That being said, if you follow the <a href="http://en.wikipedia.org/wiki/Chauvenet%27s_criterion">procedure set out in Wikipedia</a>, then you would reject the two most extreme points... |
1,950,466 | <p>Let $x_1, x_2, ...,x_n \in \mathbb{H}$, $\mathbb{H}$ is the Hilbert space. $x_j \neq 0$ for every $j$. If $x_i \perp x_j$ for $i \neq j$ then show that $x_j$ 's are linearly independent. </p>
<p>I think I need to use the fact that $<\alpha x + \beta y , z> = \alpha<x,z> + \beta<y,z>$ but how?</p... | Community | -1 | <p>Always try a simpler case first when you are lost. Suppose $n=2$ and write $ax_1+bx_2=0$. Try to do the inner products with $x_1$ and $x_2$ on both sides. Then you see $a=b=0$. Can you generalize the argument?</p>
|
789,802 | <p>How do I solve $\displaystyle \lim_{x \to 0} \sqrt{x^2 + x^3} \sin \frac{\pi}{x}$ using squeeze Theorem?</p>
<p>My book only teaches me the simplest use of the Theorem. I have no idea what should I do with a function as complex as this...</p>
<p>I know I have to start with:</p>
<p>$$ -1 \le \sin \frac{\pi}{x} \le... | Adrian Keister | 30,813 | <p>Then you could say
$$-\sqrt{x^{2}+x^{3}}\le \sqrt{x^{2}+x^{3}} \, \sin\left(\frac{\pi}{x}\right)\le \sqrt{x^{2}+x^{3}}.$$
The outermost functions approach the desired limit, and so by the Squeeze Theorem, you get the desired result.</p>
|
2,721,372 | <p>A sequence is defined by $a_1=2$ and $a_n=3a_{n-1}+1 $ .Find the sum $a_1+a_2+\cdots+a_n$</p>
<p>how to find sum $a_1=2,a_2=7,\ldots$</p>
<p>Also i found the value of $a_n=\frac{5}{6}\cdot3^n-\frac{1}{2}$</p>
| fleablood | 280,126 | <p>1)</p>
<p>$A\subset B$ if all elements of $A$ are elements of $B$. As $\emptyset$ has no elements, then all of them are in $B$.</p>
<p>That's vacuously true.</p>
<p>So $\emptyset \subset B$.</p>
<p>2)</p>
<p>$A\subset B$ if any elements not in $B$ are not in $A$ either. As any element that is not in $B$ is no... |
2,317,406 | <p>How many two-digit positive integers $N$ have the property that the sum of $N$ and the number obtained by reversing the order of the digits of $N$ is a perfect square?
Any help would be much appreciated.</p>
| M. Winter | 415,941 | <p>I wrote a program and it gave me this:</p>
<p>$0 + 0 = 0$ (if you consider $00$ as a two digit number)</p>
<p>$92 + 29 = 121$</p>
<p>$83 + 38 = 121$</p>
<p>$74 + 47 = 121$</p>
<p>$65 + 56= 121$</p>
|
198,298 | <p>Is there a tool for editing the spelling dictionary? Now that <em>Mathematica</em> 12 does spellchecking on the fly, I've been using it. And of course, as I was wondering how I might delete a word accidentally added to the dictionary, I accidentally added a misspelled word to the dictionary. I'm sure I can hunt it ... | ciao | 11,467 | <p>For removing a misspelled word from the correct word list, go to Edit->Preferences, select the Advanced tab, select Open Option Inspector, lookup/search for SpellingDictionaries, click on the edit button for the CorrectWords entry, highlight the incorrect entry and select remove.</p>
<p><a href="https://i.stack.img... |
4,473,264 | <p>I have part of a circle described by three two dimensional vectors.</p>
<ul>
<li>start point <code>s1</code></li>
<li>center point <code>c1</code></li>
<li>end point <code>e</code></li>
</ul>
<p>I move the start point <code>s1</code> by <code>m1</code>, which is a <strong>known</strong> two dimensional vector. My qu... | prets | 12,880 | <p>The <strong>indefinite</strong> integral <span class="math-container">$\displaystyle \int f(x) \, d x$</span> represents the collection of all functions whose derivatives with respect to <span class="math-container">$x$</span> equals <span class="math-container">$f(x)$</span>---they're antiderivatives.
This is in co... |
24,361 | <p>Let $X$ be a topological space and let $\mathcal{F}$ and $\mathcal{G}$ be two sheaves over $X$.</p>
<p>Of course, if one has a morphism $f : \mathcal{F} \to \mathcal{G}$ such that for all $x\in X$, $f_x : \mathcal{F}_x \to \mathcal{G}_x$ is an isomorphism, then it is known that $f$ itself is an isomorphism.</p>
<p... | Harry Gindi | 1,353 | <p>A conceptual explanation (since the other answers have given counterexamples) for why this is untrue is as follows: A sheaf consists of local data and global data specifying how those local data fit together. Even if all of the local data of two sheaves are isomorphic, there is no reason to believe that those isomo... |
1,870,305 | <p>I am trying to calculate the Jacobian of a function that has quaternions and 3D points in it. I refer to quaternions as <span class="math-container">$q$</span> and 3D points as <span class="math-container">$p$</span>
<span class="math-container">$$h_1(q)=A C(q)p $$</span>
<span class="math-container">$$h_2(q)=q_1\ot... | Yonatan Simson | 351,327 | <p>I needed to solve an easier problem then what I posted.</p>
<p>If $\mathbf{a}=(0,a_x,a_y,a_z)^T$ and the differential is by
$\bar{\mathbf{a}}=(a_x,a_y,a_z)^T$ the answer simply becomes:</p>
<p>$$ \frac{\partial(\mathbf{q} \otimes \mathbf{a} \otimes \mathbf{q\ast})}{\partial \bar{\mathbf{a}}} =
\frac{\partial(\ma... |
1,217,771 | <p>Let $x,y \in R$.
If $0 \leq y < x$ for all $x > 0$, then $y=0$.</p>
<p>Proof by contradiction: </p>
<p>Assume the opposite that is; "If $0 \leq y < x$ for all $x > 0$, then $y\neq0$".
Subtract $x$ from each part of the inequality to get,
$0-x \leq y-x < 0$
Th... | Marco Cantarini | 171,547 | <p>We have $$\frac{1}{2}=\lim_{x\rightarrow\infty}\frac{\sqrt{x}}{2\sqrt{x}\left(\sqrt{\left(1+\frac{1}{\sqrt{x}}\right)}\right)}\leq\lim_{x\rightarrow\infty}\frac{\sqrt{x}}{\sqrt{x\left(1+\frac{1}{\sqrt{x}}\right)}+\sqrt{x}}\leq\lim_{x\rightarrow\infty}\frac{1+\sqrt{x}}{\sqrt{x+\sqrt{x}}+\sqrt{x-1}}\leq\lim_{x\rightar... |
138,866 | <p>I have data in a csv file. The first row has labels, and the first column, too.</p>
<pre><code>Datos = Import["C:\\Users\\jodom\\Desktop\\Data.csv"]
</code></pre>
<p>Tha data in the csv file is that:</p>
<pre><code>{{"No", "Vol", "Vel"}, {1, 500, 45}, {2, 700, 67}, {3, 350, 87}, {4,
123, 23}, {5, 587, 45}, {6, 4... | GenericAccountName | 38,159 | <p>As of 11.2, I would suggest doing either:</p>
<p><code>Import["test.csv", "Dataset", "HeaderLines" -> 1]</code></p>
<p>or</p>
<p><code>Import["test.csv", "Dataset", "HeaderLines" -> {1, 1}]</code></p>
<p><a href="https://i.stack.imgur.com/FkENo.png" rel="noreferrer"><img src="https://i.stack.imgur.com/FkEN... |
150,084 | <p>I need to say whether or not $f_n(x)=n\left(\sqrt{x+\frac{1}{n}}-\sqrt{x}\right)$ is uniformly convergent on $(0,\infty)$.</p>
<p>I've found that the function is locally convergent to $f(x)=\frac{1}{2\sqrt{x}}$ and was trying to find $\sup{|f_n(x)-f(x)|}$.</p>
<p>I got the derivative $f_n'(x)= \frac{2nx\left(x-\sq... | Mark McClure | 21,361 | <p>In fact, $f_n'(x)<0$ for all $x>0$. Thus, each $f_n$ is continuous, positive, and decreasing on $[0,\infty)$. It follows that
$$\sup\{f(x):x>0\} = f(0) = \sqrt{n}.$$</p>
<p>As $f$ is unbounded, you can't have uniform convergence.</p>
|
4,216,243 | <p>I have to solve the following problem which seems difficult:</p>
<p>Find
<span class="math-container">$$ \iint_S \nabla \times F\ dS $$</span>
where <span class="math-container">$S$</span> is given by</p>
<p><span class="math-container">$$r(t,s)=\left( 9+(\cos t)(\sin s)\left(2+\frac{\sin (5s)}{2}\right), \ \ \ 9+(\... | rebo79 | 814,741 | <p>As far as I can see the surface <span class="math-container">$S$</span> encloses a solid volume (I plotted it in mathematica)
<a href="https://i.stack.imgur.com/X3vLc.jpg" rel="noreferrer"><img src="https://i.stack.imgur.com/X3vLc.jpg" alt="enter image description here" /></a>
Then if <span class="math-container">$V... |
509,856 | <p>Let $K_n$ be a complete $n$ graph with a color set $c$ with $c=\{\text{Red}, \text{Blue}\}$. Every edge of the complete $n$ graph is colored either $\text{Red}$ or $\text{Blue}$. Since $R(3, 3)=6$, the $K_6$ graph must contain at least one monochromatic $K_3$ graph. How can I prove that this graph must contain anoth... | StatisticalMechanic | 514,930 | <p>There is another counting argument that one can come up with. A possible quantity of interest is a $monochromatic$ $angle$. An angle is monochromatic if both its arms are of the same colour.</p>
<p>Using pigeonhole principle, we can see that, at each vertex of $K_6$, there are at least three edges of the same colou... |
4,096,516 | <p>Calculate <span class="math-container">$\operatorname{tg}( \alpha), $</span> if <span class="math-container">$\frac{\pi}{2} < \alpha<\pi$</span> and <span class="math-container">$\sin( \alpha)= \frac{2\sqrt{29}}{29}$</span>. Please provide a hint.<br />
I know that <span class="math-container">$\operatorname{t... | José Carlos Santos | 446,262 | <p>Since <span class="math-container">$\frac\pi2<\alpha<\pi$</span>,<span class="math-container">$$\cos(\alpha)=-\sqrt{1-\sin^2(\alpha)}=-\frac5{\sqrt{29}},$$</span>and therefore<span class="math-container">$$\tan(\alpha)=\frac{\sin(\alpha)}{\cos(\alpha)}=-\frac25.$$</span></p>
|
2,460,195 | <p>I had the following question:</p>
<p>Three actors are to be chosen out of five — Jack, Steve, Elad, Suzy, and Ali. What is the probability that Jack and Steve would be chosen, but Suzy would be left out?</p>
<p>The answer given was:
Total Number of actors = $5$;
Since Jack and Steve need to be in the selection and... | Robert Z | 299,698 | <p>More generally, the equation $y+z=2n-2x$ has $2n-2x+1$ non negative solutions for all $x=0,1,\dots, n$. Therefore the number of non negative solutions is
$\sum_{x=0}^n(2n+1-2x)$, that is the sum of all odd numbers in the interval $[1,2n+1]$ which is equal to $(n+1)^2$. </p>
|
677,708 | <p>Haven't done this for a long time, just want to know if this is the right method for a really simple example. Say we have two (obviously equal) sets $$A= \Big\{\begin{bmatrix}a & b\\c & d\end{bmatrix} : a,b,c,d \in \Bbb R, a+b=c \Big\}$$ $$B= \Big\{\begin{bmatrix}e & f\\g & h\end{bmatrix} : e,f,g,h ... | ricardio | 124,869 | <p>Yes that works. In general, you take something from $A$, and show it has whatever property necessary to belong to $B$. </p>
|
4,116,732 | <p>If we have a topological category and the underlying category forgetting the topological structure, are the nerves same. They should be, is what my guess is from the definition of nerve of a category. Then, I have some facts which leads to contradiction (which should not be, thus I am missing something). The facts a... | Zhen Lin | 5,191 | <p>I am sure that Segal does not mean them to be the same.
In the cited paper, notice that Segal deliberately mentions "semi-simplicial spaces" (in modern terminology: "simplicial spaces") and their realisations.
From that one infers that the nerve of a topological category is a simplicial space.
By... |
383,735 | <p>Given this (very) tricky determinant, how can we calculate it easily?</p>
<p>$$\begin{pmatrix} \alpha + \beta & \alpha \beta & 0 & ... & ... & 0 \\ 1 & \alpha + \beta & \alpha \beta & 0 & ... & 0 \\ 0 & 1 & \alpha + \beta & \alpha \beta & ... & ... \\ ...... | Riccardo | 74,013 | <p>You can consider factorizing the matrix using L-U factorization.
Let A be your matrix, $L$ the matrix $$\begin{pmatrix}
p_1 & 0 & 0 & ... & ... & 0 \\
1 & p_2 & 0 & 0 & ... & 0 \\
0 & 1 & p_3 & 0 & ... & ... \\
... & ... & ... & ... & .... |
3,938,951 | <p>The problem:</p>
<p>Given a group <span class="math-container">$G$</span> or order <span class="math-container">$n$</span>, and a Cayley embedding <span class="math-container">$\phi \ :\ G\to S_{n}$</span>. Prove that some <span class="math-container">$g\in G$</span> is of order <span class="math-container">$m$</spa... | Community | -1 | <p>Let's consider the action by left multiplication of <span class="math-container">$\langle g\rangle$</span> on <span class="math-container">$G$</span>. For <span class="math-container">$x\in G$</span>, the stabilizer is given by:</p>
<p><span class="math-container">\begin{alignat}{1}
\operatorname{Stab}(x) &= \{g... |
183,039 | <p>I'm pretty weak in the field of mathematics, but a strong programmer. I am looking for a mathematical solution that, given two points on a line will give me a curve between them, including those two points within the curve itself.</p>
<p>For instance, if I have a set of points { (0, 3) (1,10) } I'd like a mathemati... | Matt Calhoun | 1,417 | <p>You probably want to check out <a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve" rel="nofollow">Bézier Curves</a>. You can find a live demo <a href="http://jsdraw2d.jsfiction.com/demo/curvesbezier.htm" rel="nofollow">here</a>, just be sure to pick 4 points and then click draw bezier. I know Bézier curves ar... |
2,965,865 | <blockquote>
<p>Finding the minimum value of <span class="math-container">$\displaystyle \frac{x^2 +y^2}{y}.$</span> where <span class="math-container">$x,y$</span> are
real
numbers satisfying <span class="math-container">$7x^2 + 3xy + 3y^2 = 1$</span></p>
</blockquote>
<p>Try: Equation <span class="math-conta... | Aleksas Domarkas | 562,074 | <p>Lagrange function is
<span class="math-container">$$L=k\, \left( 3 {{y}^{2}}+3 x y+7 {{x}^{2}}-1\right) +\frac{{{y}^{2}}+{{x}^{2}}}{y}$$</span>
Solve system:
<span class="math-container">$$L'_x=0,\quad L'_y=0,\quad L'_k=0.$$</span></p>
<p>We get two solutions
<span class="math-container">$$k=1/4,y=-2/5,x=-1/5$$</sp... |
1,953,628 | <p>0 choices for the 1st person.
17 choices for the 2nd person (must exclude 1st and his/her two neighbours) </p>
<p>For 2 of these choices of 2nd person, there is one shared neighbour, so 15 remaining choices. (e.g. if they are numbered 1 to 20 in a circle, 1st person is #1, 2nd is #3, then people 20,1,2,3,4 are exc... | Brian M. Scott | 12,042 | <p>There are $20$ ways to choose a block of $3$ consecutive people. There are also $20$ ways to choose a pair of adjacent people, and for each of those pairs there are $16$ ways to choose a third person who is not adjacent to either of them. Those are the only sets of $3$ people that we don’t want. There are altogether... |
2,485,109 | <blockquote>
<p>Proof by induction that $1^2 + 2^2 + 3^2 +.......+ n^2 = \frac{1}{6}\cdot n\cdot (1+n)\cdot (1+2n)$</p>
</blockquote>
<p>I tried showing that </p>
<p>$1^2 + 2^2 + 3^2 +.......+ (k+1)^2 = \frac{1}{6}\cdot (k+1)\cdot (1+(k+1))\cdot (1+2(k+1))$</p>
<p>By using the left side:</p>
<p>$1^2 + 2^2 + 3^2 +... | Bruce Ikenaga | 62,033 | <p>Since $5 \cdot 3 + 7 \cdot (-2) = 1$, it follows that $5 \cdot 3702 + 7 \cdot (-2468) = 1234$. Thus, $x = 3702$, $y = -2468$ is a particular solution to the equation. The theory of linear Diophantine equations gives the general solution $x = 3702 + 7 t$ and $y = -2468 - 5 t$.</p>
<p>Since we want positive solutions... |
1,730,445 | <p>I am checking whether the limit is true or not. $z$ is complex number
\begin{equation}
\lim_{z \rightarrow 0} z\sin(\frac{1}{z})=0
\end{equation}
I found Laurent series of $z\sin(\frac{1}{z})$ which is $\sum_{n=0}^{\infty} (\frac{1}{z})^{2n}\frac{1}{(2n+1)!}(-1)^n$ </p>
<p>the series is not defined $z =0$.</p>
... | Arthur | 15,500 | <p>It might be more intuitive to do the substitution $z=1/w$ to get
$$
\lim_{|w|\to\infty}\frac{\sin w}{w}
$$
which clearly goes to $0$ along the real axis. But along the imaginary axis, the sine function grows exponentially, which outweighs the division by $w$ to give an infinite limit. Therefore the limit is not defi... |
3,709,331 | <p>How can I integrate
<span class="math-container">$$
\int \frac{x\,dx}{(a-bx^2)^2}
$$</span> I've tried to use partial fraction decomposition, but I'm getting six equations for four variables, and they don't give uniform answers.</p>
| Harish Chandra Rajpoot | 210,295 | <p><span class="math-container">$$\int \frac{x\,dx}{(a-bx^2)^2}=-\frac{1}{2b}\int \frac{(-2bx)\,dx}{(a-bx^2)^2}=-\frac1{2b}\int \frac{d(a-bx^2)}{(a-bx^2)^2}=\frac{1}{2b(a-bx^2)}+c$$</span> </p>
|
1,193,472 | <p>Let $ X \subseteq \mathbb{R}^n$ be compact and $ f:X \to \mathbb{R} $ be a function such that for all $ t \in \mathbb{R} $, the set $ f^{-1} [t, \infty) $ is closed. Which one of the following statement is correct and why the rest are incorrect? </p>
<p>1) There exists a $ x_0 \in X $ such that $$ f(x_0)= \sup_{x\... | Alex Zorn | 73,104 | <p>There is a nice theorem that will help you here: Suppose $X$ is compact, and let $V_0 \supset V_1 \supset V_2 \supset \cdots$ be a decreasing collection of closed subsets such that $\cap_i V_i = \emptyset$. Then there exists some $k$ such that $V_k = \emptyset$. Now lets prove some stuff.</p>
<p><strong>I: $f$ is b... |
806,779 | <p>Prove, without expanding, that
\begin{vmatrix}
1 &a &a^2-bc \\
1 &b &b^2-ca \\
1 &c &c^2-ab
\end{vmatrix} vanishes.</p>
<p>Any hints ?</p>
| Robert Israel | 8,508 | <p>e.g. if your matrix is $A$, consider $(b-c,c-a,a-b) A$</p>
|
2,294,991 | <p>I am trying to find the limit as $x\to 8$ of the following function. What follows is the function and then the work I've done on it. </p>
<p>$$ \lim_{x\to 8}\frac{\frac{1}{\sqrt{x +1}} - \frac{1}{3}} {x-8}$$</p>
<hr>
<p>\begin{align}\frac{\frac{1}{\sqrt{x +1}} - \frac{1}{3}} {x-8} &= \frac{\frac{1}{\sqrt{x +1... | Zain Patel | 161,779 | <p>We have $$\begin{align}\frac{\frac{1}{x+1} - \frac{1}{9}}{(x-8)((x+1)^{-1/2} + 1/3)} &= \frac{\frac{8-x}{9(x+1)}}{(x-8)((x+1)^{-1/2} + 1/3)} \\&= -\frac{1}{9(x+1)((x+1)^{-1/2} + 1/3))} \\ &\longrightarrow-\frac{1}{9(9)(2/3)} = \color{blue}{-\frac{1}{54}}.\end{align}$$</p>
<p>In particular you forgot to ... |
1,620,175 | <p>Given metric space $M = (\mathbb{R}^2, d)$ where $d = \operatorname{max}\{|x_1 - y_1|, |x_2 - y_2|\}$, how can one measure distance from some arbitrary point $X$ to the line $y = 3$, let's say?</p>
<p>How it can be done, by use of the identity
$$\operatorname{max}\{x, y\} = \frac{1}{2}(x + y+ |x - y|)\text{?}$$</p>... | levap | 32,262 | <p>Denote your point by $(x_0,y_0)$ and the line $y = 3$ by $l$. You have</p>
<p>$$ d((x_0,y_0),l) = \inf_{x \in \mathbb{R}} d((x_0,y_0), (x,3)) = \inf_{x \in \mathbb{R}} \max \{ |x_0 - x|, |y_0 - 3| \}. $$</p>
<p>It is clear that $\max \{ |x_0 - x|, |y_0 - 3| \} \geq |y_0 - 3|$ and so $d((x_0,y_0),l) \geq |y_0 - 3|$... |
603,290 | <p>this might be a stupid question, but is any $C^2$ function $f:\mathbb{R}\to\mathbb{C}$ of period $f(x+L)=f(x)$ automatically analytic (and in particular, infinitely often differentiable)?</p>
<p>I learned that for a $L$-periodic $C^k$ function with Fourier coefficients $f_n$, they converge to zero like $|f_n| |n|^k... | user1337 | 62,839 | <p>Consider $f(x)=\left| \sin \frac{x}{L} \right|^5$.</p>
<p>The first derivative is, by the chain rule $$f'(x)=5 \left| \sin \frac{x}{L} \right|^4 \left[ \text{sgn} \sin \frac{x}{L} \right] \left( \cos \frac{x}{L} \right) \frac{1}{L}$$ for $x \neq 0$. The derivative at $x=0$ can be computed from the definition $$f'(0... |
2,388,613 | <p>1) a right cone has a surface area 12 m${^2}$ and radius 1.3 m</p>
<p><strong>here is my answers:</strong></p>
<p>1) $s$ = 1.28</p>
<p><a href="https://i.stack.imgur.com/a8ovb.jpg" rel="nofollow noreferrer">(photo of how i got my answer)</a></p>
<p><strong>textbook answer:</strong> 1.64 m</p>
<p>how did they ge... | K Split X | 381,431 | <p>The formula to calculate the surface area of a cone is:</p>
<p>$$SA=\pi r(r+\sqrt{h^2+r^2})$$</p>
<p>We know the $SA$ and $r$. Now solve for $h$.</p>
<p>$$12=\pi(1.3)(1.3+\sqrt{h^2+(1.3)^2})$$</p>
<p>$$\frac{12}{1.3\pi}-1.3=\sqrt{h^2+(1.3)^2}\text{ moved stuff we already know to left side}$$</p>
<p>$$(\frac{12}... |
87,271 | <p>For a category $C$, let $C-Set$ denote the category of set-valued functors $\delta\colon C\to Set$. Given categories $C$ and $D$, and a functor $F\colon C\to D$, composition with $F$ yields a functor that I'll denote by $$\Delta_F\colon D-Set\longrightarrow C-Set.$$ The functor $\Delta_F$ has both a left adjoint, wh... | Ricardo Andrade | 21,095 | <p>I will try to give an answer to your first question.</p>
<p>The functor $\Delta_F$ verifies automatically nearly all the conditions of the monadicity theorem:</p>
<ul>
<li>it is a right adjoint;</li>
<li>it is a left adjoint with cocomplete domain, and thus coequalizers exist in the source and are preserved by $\D... |
31,992 | <p>I am attempting to generate partially transparent images for PNG <code>Export</code>, but seem to run to the following issue. If I <code>Rasterize</code> a simple piece of <code>Graphics</code> with <code>Background -> None</code> (transparent) it looks worse:</p>
<pre><code>rings = Image[
Rasterize[
Gr... | halirutan | 187 | <p>One easy idea is to set the alpha channel directly but whether this satisfy your needs really depends on what exactly you want to have. Btw, it is worth that you use <a href="http://reference.wolfram.com/mathematica/ref/ColorSeparate.html" rel="nofollow noreferrer"><code>ColorSeparate</code></a> to see for yourself ... |
31,992 | <p>I am attempting to generate partially transparent images for PNG <code>Export</code>, but seem to run to the following issue. If I <code>Rasterize</code> a simple piece of <code>Graphics</code> with <code>Background -> None</code> (transparent) it looks worse:</p>
<pre><code>rings = Image[
Rasterize[
Gr... | ybeltukov | 4,678 | <p>Another easy idea: use difference between white and black backgrounds to set alpha channel</p>
<pre><code>SetAlphaChannel[#2, ColorNegate@ImageSubtract[##]] & @@ (
Rasterize[
Graphics[{Black, Disk[], White, Disk[{1, 1}/64, 1 - Sqrt[2]/64]}],
Background -> #, RasterSize -> 400,
ImageSize ->... |
3,503,518 | <blockquote>
<p>Determine if <span class="math-container">$ \sum_{n=2}^{\infty} \frac{(\sin{n})\sum\limits_{k=1}^{n} \frac{1}{k}}{(\log n)^2}$</span> is convergent or divergent.</p>
</blockquote>
<hr>
<p>[My attempt]</p>
<p>It seems like Dirichlet test, so I tried to show that <span class="math-container">$a_n := ... | Eugene Sirkiza | 469,093 | <p><span class="math-container">$$ \frac{\sum_{k=1}^{n} \frac{1}{k}}{(\log{n})^2} \ \ \lor \ \ \frac{\sum_{k=1}^{n+1} \frac{1}{k}}{(\log{(n+1)})^2}$$</span></p>
<p><span class="math-container">$$ \left(\sum_{k=1}^{n} \frac{1}{k}\right) \log^2(n+1) - \left(\sum_{k=1}^{n+1} \frac{1}{k}\right) \log^2(n) \ \ \lor \ \ 0... |
381,093 | <p>I would like to prove Chebyshev's sum inequality, which states that:</p>
<p>If <span class="math-container">$a_1\geq a_2\geq \cdots \geq a_n$</span> and <span class="math-container">$b_1\geq b_2\geq \cdots \geq b_n$</span>, then<br />
<span class="math-container">$$
\frac{1}{n}\sum_{k=1}^n a_kb_k\geq \left(\frac{1}{... | Fedor Petrov | 4,312 | <p>I do not know whether this counts, but you may do the following.</p>
<p>At first, you may suppose that <span class="math-container">$a_n>0$</span> (otherwise replace all <span class="math-container">$a_i$</span> to <span class="math-container">$a_i-a_n+1$</span>). Consider the probabilistic distribution <span cla... |
706,514 | <p>I know that the fundamental group of homeomorphic spaces are isomorphic. Is the converse true? I mean, can we say the two spaces with isomorphic fundamental groups are homeomorphic? </p>
| Ayman Hourieh | 4,583 | <p>The closest one can get to your statement is probably <a href="http://en.wikipedia.org/wiki/Whitehead_theorem" rel="nofollow">Whitehead's theorem</a>:</p>
<blockquote>
<p>If a map $f$ between two connected CW complexes $X$, $Y$ induces isomorphisms on all homotopy groups, then $f$ is a homotopy equivalence betwee... |
3,921,255 | <p>If a function <span class="math-container">$f:M\to\mathbb{R}$</span> is continuous at point <span class="math-container">$x_0$</span> we know that for an arbitrary <span class="math-container">$\epsilon>0$</span> there exists a <span class="math-container">$\delta>0$</span> such that for all <span class="math-... | user3482749 | 226,174 | <p>For manifolds with boundary: no, you can't take boundary points to non-boundary points.</p>
<p>For manifolds without boundary: yes. For 1-manifolds, this is easy. For <span class="math-container">$n > 1$</span>, take a neighbourhood of a (shortest) path between your points. After removing problems (by taking a su... |
3,678,908 | <p>Let</p>
<p><span class="math-container">$$
V = \{ax^3+bx^2+cx+d|a,b,c,d \in Z_7\}
$$</span></p>
<p>Let the subspace <span class="math-container">$U$</span> Be a subspace of <span class="math-container">$V$</span>:</p>
<p><span class="math-container">$$
U = \{p(x) \in V | p(3)=p(5)=0\}
$$</span></p>
<p>The answer... | Community | -1 | <p>The LHS is a piecewise linear, continuous function. If you evaluate it at the "corner points", you will know in which interval(s) there are solutions.</p>
<p><span class="math-container">$$\begin{array}{}x&-\infty&-\frac32&1&\infty\\\hline f(x)&\infty&-\frac52&5&\infty\end{array}.$$<... |
796,787 | <p>Given a Lipschitz function $g$ (i.e. $|g(x) - g(y)| \leq L |x - y|, \forall x, y \in dom(g)$), and an function $f$ integrable on $[a, b]$, how do we prove $g \circ f$ is integrable on $[a, b]$, preferably using Darboux integrals/sums?</p>
<p>Let us assume $g, f$ have appropriate domains and codomains.</p>
| copper.hat | 27,978 | <p>Note that if $M=\sup_{x \in [a,b]} f(x), m = \inf_{x \in [a,b] } f(x)$, then
\begin{eqnarray}
\sup_{x \in [a,b]} g(x) -\inf_{x \in [a,b] } g(x) &=& \sup_{x,y\in [a,b]}
g(x)-g(y) \\
&=& \sup_{x,y\in [a,b]} |g(x)-g(y)| \\
&\le & L \sup_{x,y\in [a,b]} |f(x)-f(y)| \\
&\le & L \sup_{x,y\i... |
154,955 | <blockquote>
<p>Let $M\in M_n(F)$ and define $\phi:M_n(F)\to M_n(F)$ by $\phi(X)=AX$
for all $X\in M_n(F)$. Prove that $\det(\phi)=\det(A)^n$.</p>
</blockquote>
<p>I can prove it by considering the matrix representation with respect to the usual basis, which turns out to be a block diagonal form consisting of $n$ ... | Qiaochu Yuan | 232 | <p>Think of $M_n(F)$ as $V \otimes V^{\ast}$ where $V = F^n$ and $V^{\ast}$ is the <a href="http://en.wikipedia.org/wiki/Dual_space" rel="nofollow">dual space</a>. The multiplication map $M_n(F) \times M_n(F) \to M_n(F)$ is a map
$$(V \otimes V^{\ast}) \otimes (V \otimes V^{\ast}) \to V \otimes V^{\ast}$$</p>
<p>which... |
2,936,236 | <p>I have two 2D disks, <span class="math-container">$(C_1, r_1)$</span> (blue) and <span class="math-container">$(C_2, r_2)$</span> (red), where the blue disk somehow overlaps the red disk (the figure below shows one example). I’m interested in finding the positive distance along the <span class="math-container">$y$<... | davewy | 595,478 | <p>Just to codify @amd 's answer, the formula appears to just require combining the Pythagorean theorem with the <span class="math-container">$y$</span>-offset of the two circles:</p>
<p><span class="math-container">$dy = \sqrt{(r_2-r_1)^2 - (x_2-x_1)^2} - (y_1-y_2)$</span></p>
<p>where <span class="math-container">$... |
85,228 | <p>I wonder if the periodic paths of a lightray trapped between two nonconcentric circles,
each perfectly reflecting, are known.
The behavior of such rays seems chaotically complicated. For example, left below the highlighted
initial ray has slope <span class="math-container">$\frac{1}{12}$</span>, while that to the r... | Ian Agol | 1,345 | <p>I don't know how to answer your question, but I'll make a reformulation of the question. </p>
<p>Consider the unit tangent bundle to the outer circle, and consider
the subset consisting of vectors pointing into the circle. This is
an annulus, parameterized by $(\theta,\varphi)\in A=[0,2\pi]\times [0,\pi]/\{ (0,\va... |
3,335,257 | <p>This is the full question:</p>
<blockquote>
<p>show that if <span class="math-container">$p$</span> is an odd prime then the number of ordered pair solutions of the congruence<span class="math-container">$x^2-y^2 \equiv a \pmod p,$</span> is <span class="math-container">$p-1$</span> unless <span class="math-conta... | Mike | 544,150 | <p>ETA: For whatever reason I thought the question here was to prove that the statement in the OP's thread in the shaded yellow box, which is what I did below.</p>
<p>You already answered your own question for the case where <span class="math-container">$a$</span> is zero so we now consider the case where <span class=... |
2,279,660 | <p>On $C^1([0,1],\mathbb{R})$ I have two norms
\begin{align*}
N_1(f)&=|f(0)|+\|f'\|_{\infty}\\
N_2(f)&=\|f\|_{\infty}+\|f'\|_{\infty}.
\end{align*}</p>
<p>We have always that $N_1(f)\leq N_2(f)$ but what about the other inequality please?</p>
<p>And please how to see If $N_1$ and $N_2$ are equivalent to $\le... | copper.hat | 27,978 | <p>Note that $f(x) = f(0) + \int_0^x f'(t) dt$ hence
$\|f\|_\infty \le |f(0)| + \|f'\|_\infty$.</p>
<p>Hence $N_2(f) = \|f\|_\infty + \|f'\|_\infty \le |f(0)| + 2\|f'\|_\infty \le 2 N_1(f)$.</p>
<p>For the second part, take $f_n(t) = \sin (2 \pi nt)$. Clearly $\|f_n\|_\infty = 1$, but
$N_2(f_n) = 1+2\pi n$.</p>
|
3,817,092 | <p>How to show that we may replace the property of a norm <span class="math-container">$$||x||=0 \iff x=0$$</span> by <span class="math-container">$$||x||=0 \implies x = 0$$</span> without altering the concept
of a norm.</p>
| Qiaochu Yuan | 232 | <p>Some points.</p>
<ol>
<li><p>By Poincare duality the Euler characteristic of a closed odd-dimensional manifold vanishes so we restrict our attention to even-dimensional manifolds. By the classification of surfaces <span class="math-container">$S^2$</span> is the only example in dimension <span class="math-container"... |
3,294,446 | <p>I'm currently working on a definite integral and am hoping to find alternative methods to evaluate. Here I will to address the integral:
<span class="math-container">\begin{equation}
I_n = \int_0^\frac{\pi}{2}\ln^n\left(\tan(x)\right)\:dx
\end{equation}</span>
Where <span class="math-container">$n \in \mathbb{N}$</s... | Martin Gales | 2,323 | <p>Slightly different way - use well known results</p>
<p><span class="math-container">$$\int_0^{\frac{\pi}{2}}\tan^ay\;dy=\frac{\pi}{2}\frac{1}{\sin \frac{\pi}2(a+1) } $$</span></p>
<p>( this integral is considered in this site probably many times.)
<span class="math-container">$$\frac{1}{\sin x}=\frac{1}{x}+\sum _{n... |
1,141,412 | <p>Prove that the ring of integers of $\mathbb Q (\sqrt{-5})$ does not have unique factorisation.</p>
<p>Since $-5\equiv 3\pmod 4$, I know that the ring of integers of $\mathbb Q (\sqrt{-5})$ is $\mathbb Z [\sqrt{-5}]$. </p>
<p>I assume the way to prove that this does not have a unique factorisation is to give two di... | Bernard | 202,857 | <p>We can write two decompositions of $6$:
$$6=2\cdot 3=(1+\sqrt{-5})(1-\sqrt{-5})$$
All factors are irreducible, since their norms can't be decomposed as the product of two norms $\neq 1$ and the factors on the rhs are not associated with the factors on the lhs, since their norms are different.</p>
<p>This proves $\m... |
1,424,561 | <p>I know that $\sum_0^\infty \frac{\lambda^x} {x!} = e^\lambda$, but I'm having a really difficult time dealing with the extra $x$.</p>
| Michael Hardy | 11,667 | <p>\begin{align}
\sum_{x=0}^\infty \frac{x\lambda^x}{x!} & = \sum_{x=1}^\infty \frac{x\lambda^x}{x!} & & \text{(since the first term is $0$)} \\[10pt]
& = \sum_{x=1}^\infty \frac{\lambda^x}{(x-1)!} & & \text{(since $\dfrac x {x!} = \dfrac 1 {(x-1)!}$)} \\[10pt]
& = \lambda\sum_{x=1}^\infty ... |
17,259 | <p>I am using <em>Mathematica</em> to go through the examples and exercises on the book <em>Modern Control Systems</em>, 6th edition by Dorf. On page 605, there is a table (Table 8.5) with the Bode plot for several transfer functions. In what follows there is a piece of code that attempts to build the very same table... | Nasser | 70 | <p>To address the above comment by Ed</p>
<blockquote>
<p>However if we change the example. Matlab - <code>bode(tf(10*[10 1],[1 0 0
0]))</code> - phase is negative (-270 to -180). Mathematica - See plots above
- phase from +90 to +180. If instead of <code>10(10s+1)/sˆ3</code>, one uses <code>10(10s+1)/(s+0.0001)... |
17,259 | <p>I am using <em>Mathematica</em> to go through the examples and exercises on the book <em>Modern Control Systems</em>, 6th edition by Dorf. On page 605, there is a table (Table 8.5) with the Bode plot for several transfer functions. In what follows there is a piece of code that attempts to build the very same table... | Suba Thomas | 5,998 | <p>This is fixed for v10. In v10 there is also a <code>PhaseRange</code> option for <code>BodePlot</code> and <code>NicholsPlot</code> that can be used to override the default range (if you need to wrap it between $-\pi$ and $\pi$, etc).</p>
<pre><code>With[{τ1 = 20, τ2 = 2, τ3 = 0.4, τ4 = 0.05, τa = 10, τb = 1, k = 1... |
2,339,964 | <p>I wondered whether it is possible to define a sequence without relying on sets or natural numbers and tried with this definition.</p>
<ol>
<li>A symbol which is not a comma is a sequence. </li>
<li>If $S$ is sequence and $s$ a symbol which is not a comma:
<ul>
<li>$S, s$ is a sequence, where every symbol occur... | DanielV | 97,045 | <p>You seem to be attempting a recursive definition of finite sequences. A recursive definition needs 3 things: construction, distinguishing, and exclusion.</p>
<p>For construction:</p>
<blockquote>
<p>A symbol which is not a comma is a sequence.</p>
</blockquote>
<p>It is very very bad style not to distinguish ... |
4,095,361 | <p>Given <span class="math-container">$f : [0, 1] \rightarrow \mathbb{R}$</span> be a bounded function which is continuous on <span class="math-container">$[0,1]$</span> except at <span class="math-container">$1/2$</span>. Let <span class="math-container">$\alpha(x) = x^2$</span>. No further description of function <s... | Rahul Madhavan | 439,353 | <p>Another way to solve this is by ysing the heaviside step function <span class="math-container">$H(x)$</span> and the dirac delta function <span class="math-container">$\delta(x)$</span>:</p>
<p><span class="math-container">\begin{align*}
F(s)=
\begin{cases}
0, & s<0\\
s, & 0\leq s<5/6\\
\frac{5}{6} + \... |
820,490 | <p>I am studying pre-calculus mathematics at the moment, and I need help in verifying if $\sin (\theta)$ and $\cos (\theta)$ are functions? I want to demonstrate that for any angle $\theta$ that there is only one associated value of $\sin (\theta)$ and $\cos (\theta)$. How do I go about showing this?</p>
| Amir Ayupov | 155,158 | <p>It follows from the definition of $\sin(\theta)$ as a ratio of length of $\theta$'s opposite side to hypotenuse (<a href="http://en.wikipedia.org/wiki/Sine" rel="nofollow">Wikipedia</a>).</p>
<p>As long as hypotenuse length is not equal to zero (otherwise the triangle degenerates into line or point), the division i... |
549,254 | <blockquote>
<p>$$\frac{1}{3}=.33\bar{3}$$ </p>
</blockquote>
<p>is a rational number, but the $3$ keeps on repeating indefinitely (infinitely?). How is this a ratio if it shows this continuous pattern instead of being a finite ratio? </p>
<p>I understand that $\pi$ is irrational because it extends infinitely <em>... | Khosrotash | 104,171 | <p>$$\begin{align}
0.3333333333333\ldots &= 0.3 +0.03 +0.003 +0.0003+ \ldots\\
&=\frac{3}{10} + \frac{3}{100} + \frac{3}{1000}+ \frac{3}{10000} +\ldots
\end{align}$$</p>
<p>If you know the sum of a geometric sequence, that is:
$$a+aq+aq^2+aq^3+\ldots = \frac{a}{1-q} \quad\text{ if $|q| < 1$}$$</p>
<p>you c... |
549,254 | <blockquote>
<p>$$\frac{1}{3}=.33\bar{3}$$ </p>
</blockquote>
<p>is a rational number, but the $3$ keeps on repeating indefinitely (infinitely?). How is this a ratio if it shows this continuous pattern instead of being a finite ratio? </p>
<p>I understand that $\pi$ is irrational because it extends infinitely <em>... | Michael Hoppe | 93,935 | <p>Eye catcher: $0.123\overline{5678}=\dfrac{12345678-123}{9999000}$.</p>
|
925,558 | <p>How do I prove that the inductive sequence $y_{n+1}= \dfrac {2y_n + 3}{4}$ is bounded? $y(1)=1$</p>
<p><strong>Attempt:</strong> Let us assume that the given sequence is unbounded. : </p>
<p>Then, $y_{n+1} \rightarrow \infty$ either for some finite $n$ or when $n \rightarrow \infty$</p>
<blockquote>
<p>CASE $... | André Nicolas | 6,312 | <p>It is clear that $y_n\gt 0$ for all $n$. Let us prove by induction that $y_n\lt 4$. This is certainly true if $n=1$. And if $y_n\lt 4$, then $\frac{2y_n+3}{4}\lt 4$. </p>
<p>There are smaller upper bounds than $4$, but to prove boundedness we need not find the least upper bound.</p>
|
530,920 | <p>I want to know if there is a way to find for example $\ln(2)$, without using the calculator ?</p>
<p>Thanks </p>
| Mats Granvik | 8,530 | <p>$$\log (x)=\sum _{n=1}^{\infty } \frac{\left(\frac{x-1}{x}\right)^n}{n}$$
when $x>1$</p>
|
63,064 | <p>If one defines on a $\mathbb{R},\mathbb{C}$-vector space a norm this gives rise to a metric. Why are particularly mappings that satisfy the norm axioms so important that in every book for beginners on linear algebra/functional analysis norms are studied ? </p>
<p>Aren't there also other functions that always give r... | Jadmrial | 10,957 | <p>The standard notion of a metric is weaker than that of a norm- see the discrete metric, for example. By adding the multiplicative condition, we introduce extra structure on our space that makes it nicer and more intuitive. </p>
<p>Indeed, as Asaf Karagila says, the really nice thing about the standard norms on $\ma... |
3,693,814 | <p>For <span class="math-container">$x,y,z>0.$</span> Prove<span class="math-container">$:$</span> <span class="math-container">$$P={x}^{4}y+{x}^{4}z+3\,{x}^{3}{y}^{2}-11\,{x}^{3}yz+3\,{x}^{3}{z}^{2}+3
\,{x}^{2}{y}^{3}+3\,{x}^{2}{y}^{2}z+3\,{x}^{2}y{z}^{2}+3\,{x}^{2}{z}^{
3}+x{y}^{4}-11\,x{y}^{3}z+3\,x{y}^{2}{z}^{2}... | Michael Rozenberg | 190,319 | <p>There is also the following SOS:
<span class="math-container">$$\sum_{cyc}(x^4y+x^4z+3x^3y^2+x^3z^2-11x^3yz+3x^2y^2z)=$$</span>
<span class="math-container">$$=\sum_{cyc}(x^4y+x^4z+3x^3y^2+3x^3z^2-18x^3yz+10x^2y^2z)+\sum_{cyc}7x^3yz-7x^2y^2z)=$$</span>
<span class="math-container">$$=\sum_{cyc}z(x-y)^2(x+y-3z)^2+7.5... |
2,277,117 | <p>For $m = 2$, the fraction is $\frac{3}{4}$. for $m=3$, the fraction is$\frac{8}{15}$. I was wondering why numerators and demoninators of $\frac{1}{m} + \frac{1}{m+2}$ show primitive pythagorean triples a and b.</p>
| helloworld112358 | 300,021 | <p>Notice that $$\frac{1}{m}+\frac{1}{m+2}=\frac{2m+2}{m^2+2m}.$$ We can check that $$(m^2+2m)^2+(2m+2)^2=m^4+4m^3+4m^2+4m^2+8m+4=m^4+4m^3+8m^2+8m+4=(m^2+2m+2)^2.$$ Note that this is always a primitive triple if you put the fraction in lowest form, but $2m+2$ and $m^2+2m$ are not coprime in general (and thus the triple... |
2,277,117 | <p>For $m = 2$, the fraction is $\frac{3}{4}$. for $m=3$, the fraction is$\frac{8}{15}$. I was wondering why numerators and demoninators of $\frac{1}{m} + \frac{1}{m+2}$ show primitive pythagorean triples a and b.</p>
| Jay Zha | 379,853 | <p>$$\frac{1}{m} + \frac{1}{m+2} = \frac{2m+2}{m^2 + 2m}$$</p>
<p>Now notice that</p>
<p>$$(2m+2)^2 + (m^2 + 2m)^2=8m^2+4m+4+m^4+4m^3=(m^2+2m+2)^2$$</p>
|
1,545,019 | <p>I'm sure this is probably an extremely simple problem but I'm stuck figuring this out.<br>
For example: </p>
<p>$(\frac{1}{5})^{x} + (\frac{7}{10})^{x} = 1$</p>
<p>What would be the steps to solve for x?</p>
| Brevan Ellefsen | 269,764 | <p>This is a hard problem. There is no general closed form for $a^x + b^x = c$, and the best way is probably for you to use something like Newton's method to get an approximation. Here is a <a href="https://math.stackexchange.com/questions/1371722/solve-axbx-c-for-x">link</a> to an answer outlining this</p>
|
3,769,222 | <p>Evaluate: <span class="math-container">$\lim_{n\to \infty } \int_0^n (1-\frac{x}{n})^n \cos(\frac{x}{n})dx$</span></p>
<p>For fixed <span class="math-container">$n$</span>, I first rewrite my integral as
<span class="math-container">$$\int_0^n \left(1-\frac{x}{n}\right)^n \cos\left(\frac{x}{n}\right)dx = \int_0^\inf... | juancodmw | 517,676 | <p>You can see that <span class="math-container">$\int_{0}^n (1-\frac{x}{n})^n\cos{(\frac{x}{n})}dx = \int_{0}^{1}n(1-t)^{n}\cos{(t)}dt$</span> and you can integrate by parts</p>
|
1,125 | <p>I know volume preserving diffeomorphisms of a $sphere^2$ make a grou'p sdiff($S_2$). I would to know if it is a Lie group, which I assume if it is that makes interpolation easier (like with rotations). </p>
<p>So that is one question, is it a Lie group?</p>
<p>Also is the group path connected? If so, how can I int... | Jason DeVito | 331 | <p>First, as others have pointed out, the group of volume preserving diffeomorphisms will be infinite dimensional.</p>
<p>For the second question, there is a beautiful technique known as Moser's trick which answers it. Moser's trick, in fancy language, says that if <span class="math-container">$(M,w)$</span> and <spa... |
271,554 | <p>Is $\chi_{p(\displaystyle\lim_{n\to\infty} f_n)}=\displaystyle\lim_{n\to\infty}\chi_{p(f_n)}$, assuming $\lim f_n$ exists? Here $\chi_{p(f)}$ is $1$ for the set where the proposition $p$ on the function $f$ is true and $0$ for otherwise. Here the functions are $[0,1]\to\mathbb{R}$ and the limits are almost everywher... | Did | 6,179 | <p>No. Try $p=$ "being bounded" and $f_n(x)=\min\{n,1/x\}$.</p>
|
74,290 | <p>I have this function $f(x)$ which is continuous and differentiable on $\mathbb R$. Is the following true - without the assumption that $f$ being absolutely continuous!</p>
<p>$$ \int_{a}^{b} f'(x) dx=f(b)-f(a)$$</p>
<p>Edit: $f$ is infinitely differentiable on $\mathbb R$. I think this may change everything!</... | kahen | 1,269 | <p>No, the derivative need not even be integrable! See Wikipedia's article on <a href="http://en.wikipedia.org/wiki/Volterra%27s_function" rel="nofollow">Volterra's function</a>.</p>
<p>PS: If $f$ is differentiable, then $f$ is continuous.</p>
|
1,443,166 | <blockquote>
<p>If the proposition ¬p→v is true, then the truth value of the
proposition ¬p∨(p→q), where ¬ is negation, ∨ is inclusive OR and → is
implication, is</p>
<ol>
<li>True</li>
<li>False</li>
<li>Multiple Values</li>
<li>Cannot be determined</li>
</ol>
</blockquote>
<hr>
<p><strong>I try... | Mauro ALLEGRANZA | 108,274 | <p>The formula $¬p→v$ is True in two cases :</p>
<blockquote>
<p>either when $p$ is True (i.e. $\lnot p$ False) or when $v$ is True.</p>
</blockquote>
<p>Thus, the assumption does not force a truth-value for $p$.</p>
<p>Consider now $¬p∨(p→q)$; we have three cases :</p>
<p><em>(i)</em> $q$ True; in this case $p→q... |
76,457 | <p>I have an ellipse centered at $(h,k)$, with semi-major axis $r_x$, semi-minor axis $r_y$, both aligned with the Cartesian plane.</p>
<p>How do I determine if a point $(x,y)$ is within the area bounded by the ellipse? </p>
| Srivatsan | 13,425 | <p>The region (disk) bounded by the ellipse is given by the equation:
$$
\frac{(x-h)^2}{r_x^2} + \frac{(y-k)^2}{r_y^2} \leq 1. \tag{1}
$$
So given a test point $(x,y)$, plug it in $(1)$. If the inequality is satisfied, then it is inside the ellipse; otherwise it is outside the ellipse. Moreover, the point is on the b... |
76,457 | <p>I have an ellipse centered at $(h,k)$, with semi-major axis $r_x$, semi-minor axis $r_y$, both aligned with the Cartesian plane.</p>
<p>How do I determine if a point $(x,y)$ is within the area bounded by the ellipse? </p>
| Victor Engel | 49,770 | <p>I have another solution. In summary, transform everything so you can test whether a point is within a circle centered at (0,0). Since the ellipse is oriented orthogonally to the Cartesian plane, we can simply scale one of the dimensions by the quotient of the two axes.</p>
<p>First subtract (h,k) from both points.<... |
211,123 | <p>I am writing some tests and would like to introduce large chunks of random alphabetic text (just characters a-z) to the input. The way I am generating the text now is like this:</p>
<pre><code>RandomString[length_Integer] :=
StringJoin[
Table[
FromLetterNumber[RandomInteger[{1, 26}]],
... | MarcoB | 27,951 | <p>The following will give you a long string of text using common words separated by a space; it is very fast after the first execution (which loads some indices):</p>
<pre><code>StringRiffle@RandomWord["CommonWords", 100]
</code></pre>
<blockquote>
<p>"dexterous calibration ethical nocturnal misfortune ruining com... |
898,082 | <p>I have the answer for this, but my teacher hadn't taught the whole "when cosine is an even, the value of $-\arccos (-0.7)$ is a solution too." </p>
<p>Please: </p>
<p>-tell me when a $\cos$/$\sin$ function is even/odd </p>
<p>-what happens if its odd? </p>
<p>-how to use the "$±\arccos(-0.7) + 2kπ$" ( don't unde... | drhab | 75,923 | <p>I restrict to a solution of the title of your question.</p>
<p>In general $\cos x=\cos\alpha$ if and only if $\frac{x-\alpha}{2\pi}$
is an integer or $\frac{x+\alpha}{2\pi}$ is an integer. </p>
<p>Another way
to express that is: if and only if $x=\pm\alpha+2k\pi$ where $k$
stands for an integer. Take notice of sig... |
1,715,324 | <p>I am wondering what is the difference between algebraic sets and algebraic varieties in complex projective space.</p>
<p>It seems that both are zero sets of polynomials, so what is the difference?</p>
| silvascientist | 108,405 | <p>According to <a href="https://en.wikipedia.org/wiki/Algebraic_variety" rel="nofollow">Wikipedia</a>, some sources require that a variety be irreducible, which means that it is not the union of two smaller sets that are closed in the <a href="https://en.wikipedia.org/wiki/Zariski_topology" rel="nofollow">Zariski topo... |
102,966 | <p>Let $T^*$ denote upper triangular matrices (of the appropriate size) with positive diagonal entries and $\mathrm{UT}$ upper triangular matrices with all diagonal entries equal to 1.</p>
<blockquote>
<p>Does every (abstract group) embedding $\varphi:\mathrm{UT}(n,\mathbb{R})\to\mathrm{UT}(m,\mathbb{R})$ extend to ... | YCor | 14,094 | <p>Here's a counterexample ($m=n=3$) which is a continuous homomorphism and actually probably also works for your second question with $\mathbf{Q}$.</p>
<p>In short: most automorphisms of $UT(3)$ do not extend to $T^*(3)$.</p>
<p>Since I deal with continuous automorphisms, it boils down to a Lie algebra problem. An a... |
440,744 | <p>The vector space dimension of the cohomology group of the <span class="math-container">$2$</span>-plane Grassmannian <span class="math-container">$\mathrm{Gr}_{2,n}$</span> is given by the number of tuples <span class="math-container">$(\lambda_1,\lambda_2)$</span> satisfying
<span class="math-container">$$
n - 2 \g... | Vladimir Dotsenko | 1,306 | <p>I accidentally (looking for something else) came across another paper where a very elegant explanation is given:</p>
<p><a href="https://www.jstor.org/stable/24903230" rel="nofollow noreferrer">Dan Laksov, Anders Thorup: Schubert Calculus on Grassmannians and Exterior Powers</p>
<p>Indiana University Mathematics Jou... |
1,110,122 | <p>Prove $$\large\int_{-\pi}^{\pi}\sin (\sin x) \,dx =0$$ without using the fact that $\sin(x)$ is odd.</p>
<p>Computing this in wolfram says that it is uncomputable, which leads me to believe that the only way to find this would be methods for solving definite integrals. I am wondering if it is possible with any othe... | Robert Israel | 8,508 | <p>$$\int_{-\pi}^\pi \sin(\sin(x)) \; dx = \sum_{n=0}^\infty \dfrac{(-1)^n}{(2n+1)!}\int_{-\pi}^\pi \sin^{2n+1}(x)\; dx$$ (the interchange of sum and integral justified by uniform absolute convergence)
so it suffices to show that $\int_{-\pi}^\pi \sin^{2n+1}\; dx = 0$ for nonnegative integers $n$.
Now using the substit... |
1,110,122 | <p>Prove $$\large\int_{-\pi}^{\pi}\sin (\sin x) \,dx =0$$ without using the fact that $\sin(x)$ is odd.</p>
<p>Computing this in wolfram says that it is uncomputable, which leads me to believe that the only way to find this would be methods for solving definite integrals. I am wondering if it is possible with any othe... | mickep | 97,236 | <p>I think this question is a bit funny (and in principal, I think that any calculation leading to the desired conclusion uses that sine is odd in one or another way). </p>
<p>Are we allowed to use $\sin(t\pm\pi)=-\sin(t)$ and that $\sin t=\frac{\exp(it)-\exp(-it)}{2i}$ (I agree that it follows from the second identit... |
1,616,460 | <blockquote>
<p>If $p(x)$ be a quadratic equation with real coefficient satisfying $$x^2-2x+2\leq p(x)\leq 2x^2-4x+3\;\forall x\in \mathbb{R}$$ and $p(11) = 181.$ Then $p(16)= $ </p>
</blockquote>
<p>$\bf{My\; Try::}$ We can Write Inequality as $$(x-1)^2+1\leq p(x)\leq 2(x-1)^2+1$$</p>
<p>Here from above inequalit... | Future | 299,525 | <p>Put the maps into the diagram. Now take anything in $\ker \iota$, say $n$. By the diagram in Thm 8, it gets sent to $0$ in $S \otimes N$ which, since $\Phi$ is a homomorphism, then gets mapped to $0$ in $L$. Thus, by commutativity of the diagram, so then does $\varphi (n) = 0$ meaning that $n \in \ker \varphi$. </p>... |
690,822 | <p>Let <span class="math-container">$G$</span> be a graph of order <span class="math-container">$n$</span> and let <span class="math-container">$k$</span> be an integer with <span class="math-container">$1\leq k\leq n-1$</span>. Prove that if <span class="math-container">$\delta(G)\geq (n+k-2)/2$</span>, then <span cla... | Calvin Lin | 54,563 | <p><strong>Hint:</strong> Apply Dirac's Theorem.</p>
<p>Let $K\subset G$ be a set of $k-1$ vertices. Consider $G-K$, which has $n-k+1 $ vertices.</p>
<blockquote class="spoiler">
<p> The minimum degree is $ \frac{ n+k-2}{2} - (k-1) = \frac{n-k}{2}.$</p>
</blockquote>
<p>Consider $G-K + v$, where we add a special v... |
2,367,788 | <p>Need to find the total numbers out of all 6 digit numbers where a digit is repeated exactly 4 times in the number. </p>
<p>Eg. 111122, 111123 is valid
But 111121 is not valid.</p>
| Tony | 258,406 | <p>If you mean any digit combinations as "numbers":</p>
<p>$10\times1\times1\times1\times1\times9\times9+...+10\times9\times9\times1\times1\times1\times1 = 10\times9^2\times \binom64=10\times9^2\times 15=12150$</p>
<p>If the qualities of numbers are meant to be satisfied (leading zeros excluded), simply multiply by $... |
2,012,080 | <p>Consider the PDE for $v(x,y)$:</p>
<blockquote>
<p>$v_{xx} − v_{xy} − 6v_{yy} − 5v_{x} − 10v_{y} + 25 = 0$</p>
</blockquote>
<p>Taking $\xi = y-2x$ and $\eta=y+3x$ that I found from $b^2-4ac$ I have transformed the equation to the form</p>
<p>$v_{\eta\xi} + v_{\eta}=1$</p>
<p>I have been posed the question:</p... | Mahmoud Hassan | 282,900 | <p>we have :
$$v_{\eta\xi} + v_{\eta}=1$$
Now let $u=v_{\eta} $
$$ u=v_{\eta} \Rightarrow u_{\xi}=v_{\eta\xi}$$
$$\Rightarrow u_{\xi}+u = 1$$
$$\frac{-d_{\xi}}{1}=\frac{-d_{\eta}}{0}=\frac{du}{u-1}$$
which implies :
$$\eta=a \ , \ u-1=be^-{\xi}\Rightarrow u=1+f(\eta)e^-{\xi}$$
Finally :
$$v_{\eta}=1+f(\xi)e^-{\xi}\Righ... |
3,902,836 | <p>There're similar questions already, but I somewhat struggled to apply their reasoning to this particular statement.</p>
<p>I wanted to ask if my proof is correct (and clarify several things which I've seemingly figured out when writing it here). Here's the proof:</p>
<p><span class="math-container">$$\mbox{(1) }\lim... | WA Don | 542,712 | <p>The proof you give becomes a bit unwieldy and it does not make clear that <span class="math-container">$N$</span> must be negative.</p>
<p>I think it can be shortened as follows:</p>
<hr />
<p>Assume <span class="math-container">$\lim_{x\to0^-} f(1/x)$</span> exists and equals <span class="math-container">$\ell$</sp... |
251,799 | <p>I have a PDE in the form of
$$
\frac{\partial u}{\partial t} + \frac{\partial u}{\partial x} + u = \delta(x-1),
$$
with initial condition $u(x,0)=100$. I'm trying to solve it numerically, but I have no idea on which method should I use. Most of the examples that I refer to has no delta function in the PDE.</p>
<p>... | Pragabhava | 19,532 | <p>You can restate the problem by integrating the equation in $x \in (1-\epsilon,1+\epsilon)$ and taking $\epsilon \to 0$.</p>
<p>$$
\int_{1-\epsilon}^{1+\epsilon} \left(\frac{\partial u}{\partial t} + \frac{\partial u}{\partial x} + u\right) dx = \int_{1-\epsilon}^{1+\epsilon} \delta(1-x)dx = 1.
$$</p>
<p>Now we hav... |
2,636,188 | <p>How would I go about proving:</p>
<blockquote>
<p>For every non-prime $n \in \mathbb N$ there exists $m \in \mathbb Z / n\mathbb Z \setminus \{0\}$ with $m^{n-1} \not\equiv 1 \mod n$.</p>
</blockquote>
<p>I already proved the other way around (straight forward using Euler-Fermat), but am stuck at this point.</p>... | Dave | 334,366 | <p>I think it makes sense that we need to use the fact that $n$ is composite somehow. So what is the one thing we know about composite numbers: they have nontrivial diviors! So choose $m$ to be a divisor of $n$ different from $1$. Now is it possible for $ma\equiv 1\pmod n$ for any $a\in\Bbb Z/n\Bbb Z$?</p>
|
711,922 | <p>I know there are well-ordered sets that are not countable.</p>
<p>Suppose you are given an uncountable, well-ordered set $S$.</p>
<p>Isn't it possible to provide a bijection $f:\mathbb{N} \rightarrow S$ as following?</p>
<p>$S$ is well-ordered, so it has the smallest element, say $s_1$. $S \setminus$ {$s_1$} is a... | user126154 | 126,154 | <p>Even if you made a mistake about surjectivity, as pointed out by 5xum and alex, you did a good thing: you proved that </p>
<p>given any two well-ordered sets, one is always the intial segment of the other! (which is, of course, true. The proof needs transfinite induction in general, but works exactly as your attemp... |
3,279,503 | <p>I'm relatively new in tensor space theory, and while reading some materials i've came across authors describing a inner product as a <span class="math-container">$(0,2)$</span> tensor. I'm not sure why it is, but i think if i write a map <span class="math-container">$f$</span> as <span class="math-container">$$f: P ... | Lutz Lehmann | 115,115 | <p>The linear map <span class="math-container">$T:V\to W$</span> belongs to the tensor space <span class="math-container">$W\otimes V^*$</span>. As that is not a product of <span class="math-container">$V$</span> and <span class="math-container">$V^*$</span>, it is not really appropriate to call it an <span class="math... |
626,837 | <p>If $a+b+c+d = 30$ and $a,b,c,d$ lie between $0$ and $9$. How to find number of solutions of this equation.</p>
| robjohn | 13,854 | <p><strong>Hint:</strong> Consider the coefficient of $x^{30}$ in
$$
\left(\frac{x^{10}-1}{x-1}\right)^4
$$
Since
$$
(1-x)^{-4}=\sum_{n=0}^\infty\binom{n+3}{3}x^n
$$
and
$$
\left(1-x^{10}\right)^4=\sum_{k=0}^4(-1)^k\binom{4}{k}x^{10k}
$$
we get the coefficient of $x^n$ to be
$$
\sum_{k=0}^4(-1)^k\binom{4}{k}\binom{n-10... |
3,170,742 | <p>I have a system of recurrence relations in the following form:</p>
<p><span class="math-container">$$
\begin{pmatrix}
f(n+1)\\
g(n+1)\\
\end{pmatrix}
= \textbf{A}
\begin{pmatrix}
f(n)\\
g(n)\\
\end{pmatrix} +\vec{b}
$$</span></p>
<p>which hold for all <span class="math-container">$n \in \{ 0,1,2,...,N\}.$</span> ... | Dr. Sonnhard Graubner | 175,066 | <p>Use that <span class="math-container">$$\sin(x-y)=\sin(x)\cos(y)-\sin(y)\cos(x)$$</span></p>
|
3,663,374 | <p>I asked a question about if we have in general that if <span class="math-container">$G=G_1\times \cdots \times G_n$</span> (where <span class="math-container">$G_i$</span> are characteristic in <span class="math-container">$G$</span> for <span class="math-container">$i=1,\cdots ,n$</span>), then
<span class="math-co... | Community | -1 | <p>It's true if <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are normal in <span class="math-container">$G$</span> and <span class="math-container">$H$</span> respectively. Then <span class="math-container">$A×B\triangleleft G×H$</span>. Define <span class="math-container">$i:(G... |
1,437,961 | <p>Are comparison notations such as <, >, ≤, ≥, =, ≠ valid for sets?</p>
<p>I'm interested in stating size (number of elements) relations between sets.</p>
| Siminore | 29,672 | <p>Equality of course is. I have never seen the notation $A<B$ to say that $A$ has less elements than $B$. Everybody uses a more descriptive notation as $\#A < \# B$, for instance.</p>
|
1,437,961 | <p>Are comparison notations such as <, >, ≤, ≥, =, ≠ valid for sets?</p>
<p>I'm interested in stating size (number of elements) relations between sets.</p>
| Ove Ahlman | 222,450 | <p>$=$ and $\neq$ are defined and valid (see the axiom of extensionality).</p>
<p>$\leq, \geq, <$ and $>$ are more tricky. We may compare the number of elements in two sets, called the cardinality, and write that $|A|> |B|$ meaning that the cardinality of $A$ is greater than B, however this notation is never ... |
1,437,961 | <p>Are comparison notations such as <, >, ≤, ≥, =, ≠ valid for sets?</p>
<p>I'm interested in stating size (number of elements) relations between sets.</p>
| Gabor Bakos | 270,432 | <p>You can use the $=$ sign, which has the definition: both sets have the same members.</p>
<p>The other comparison notations are not used unless you are comparing the sizes (cardinality, the number of elements) of the sets:</p>
<p>A={1,2,3,4}, B={1,5}, here $≥$ can be used (|A|≥|B|)</p>
<hr>
<p>Some comparison n... |
289,757 | <p>I am writing this, as I am a currently an intern at an aircraft manufactur. I am studying a mixture of engineering and applied math. During the semester I focussed on numerical courses and my applied field is CFD. Even though every mathematician would say I have not heard a lot of math, for myself I would say that I... | Mike Spivey | 2,370 | <p>I've spent the past twelve years as a professor. However, for five months last spring I spent part of a sabbatical working in the long-term forecasting group of an investment firm. I used <em>a lot</em> of math in those five months. (Admittedly, it was mostly a research-type position, and I gravitated toward the ... |
2,466,022 | <p>In Real and Complex Analysis, 3rd Edition, Walter Rudin advances the following:</p>
<p><a href="https://i.stack.imgur.com/XEVwK.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XEVwK.png" alt="enter image description here"></a>
<a href="https://i.stack.imgur.com/dTwbg.png" rel="nofollow noreferrer... | MathematicianByMistake | 237,785 | <p>If there was a $z\in \mathbb{C}$ such that $e^z=0$ we would have </p>
<p>$e^z\cdot e^{-z}=0\cdot e^{-z} \Rightarrow\\1=0$</p>
<p>which simply is not true.</p>
<p><em>EDIT</em> </p>
<p>Another way to look at it is by using the form $e^z=e^{x+iy}=e^x(\cos y+i\sin y)$</p>
<p>with $x,y \in \mathbb{R}$</p>
<p>Thus ... |
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