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 |
|---|---|---|---|---|
366,311 | <blockquote>
<p>Show that the sequence $\displaystyle (x_n)=\left( \sum_{i=1}^n\frac 1 i\right)$ diverge by epsilon delta definition.</p>
</blockquote>
<p>I'm not familiar with proving divergent sequence. Do anyone have any des? Thank you.</p>
| Warren Moore | 63,412 | <p>Are you familiar with the comparison test?</p>
<blockquote>
<p>Let $\sum x_n$ and $\sum y_n$ be series with $x_n\ge y_n\ge 0$ for sufficiently large $n$. Then if $\sum y_n$ diverges, $\sum x_n$ diverges.</p>
</blockquote>
<p>So you just need to compare it with a divergent series from below. </p>
|
34,215 | <p>How do professional mathematicians learn new things? How do they expand their comfort zone? By talking to colleagues? </p>
| Peter Shor | 2,294 | <p>Try to solve a famous open problem in the new field. Even though you'll almost certainly fail, you'll learn a lot of new mathematics on the way.</p>
|
188,087 | <p>Is there a function that can extract a list of variables in an expression?
For example, assume we have an expression</p>
<pre><code>x^2+y^3+z
</code></pre>
<p>This expression has variables x, y and z. The result should be</p>
<pre><code>{x, y, z}
</code></pre>
<p>. Is there a way to get this?</p>
| user49047 | 49,047 | <p>Listing the expression might be an alternative for this simple example</p>
<pre><code>List @@ (x^8 + y^3 + z) /. a_Symbol^n_ -> a
</code></pre>
|
1,858,529 | <p>I know, there are some threads dealing with this sum but I want to solve it with the integral test for convergence(<a href="https://en.wikipedia.org/wiki/Integral_test_for_convergence" rel="nofollow">more</a>)</p>
<blockquote>
<p>$$\sum\limits_{n=3}^{\infty} \frac{1}{n\log(n)\log(\log(n))}$$</p>
</blockquote>
<p... | Jack D'Aurizio | 44,121 | <p>Just apply <a href="https://en.wikipedia.org/wiki/Cauchy_condensation_test" rel="nofollow noreferrer">Cauchy's condensation test</a> twice to get that the series is divergent.</p>
<p>As an alternative, the substitution <span class="math-container">$x=\exp(\exp t)$</span> gives:
<span class="math-container">$$ \int_{... |
1,319,476 | <p>This is a question related to another posted question:</p>
<p>The answer to the following question "Find all solutions to: $e^{ix}=i$" is as follows: </p>
<p>"Euler's formula: $e^{ix}=\cos(x)+i\sin(x)$,</p>
<p>so: $ \cos x+i\sin x=0+1⋅i$</p>
<p>compare real and imaginary parts
$\sin(x)=1$
and
$\cos(x)=0$</p>
<p... | Community | -1 | <p>You said it, you compare real and imaginary parts. $\cos(x)=0$ and $\sin(x)=1$. When is this true?</p>
|
1,272,499 | <p>Definite integral of $$\int_0^{2\pi} \frac{1}{2+\cos x}$$ without using improper integral, I want to solve this without having to use $-\infty$ and $\infty$ on the integrals limits. Is that possible?</p>
<p>The only way I can think of solving that is by using Weierstrass. $u = \tan \frac{x}{2}$, don't you have to m... | Mark Viola | 218,419 | <p>Another way of evaluating this integral is to use contour integration. </p>
<p>Let $z=e^{i\theta}$ so that $d\theta =dz/(iz)$, $\cos \theta =\frac12 (z+z^{-1})$ where $0\le \theta \le 2\pi$, and let $C$ be the unit circle $|z|=1$. </p>
<p>Then, </p>
<p>$$\begin{align}
I&=\int_0^{2\pi}\frac{1}{2+\cos \theta}... |
207,418 | <p>The TAs in my department are stuck in assisting an undergraduate with the following problem:</p>
<p>$$\sum^{2k}_{i=0} C^{4k}_{2i}(-1)^{i}=2^{2k}(-1)^{k}.$$</p>
<p>We tried to solve this via induction (obviously failed), via various combinatorial identities, via generating functions, etc. Aside from the fact that n... | Qiaochu Yuan | 232 | <p>I'll rewrite the identity as</p>
<p>$$\sum_{k=0}^{2n} {4n \choose 2k} (-1)^k = 2^{2n} (-1)^n$$</p>
<p>because I'm about to use $i$ for something else. Write this as</p>
<p>$$\sum_{k=0}^{4n} {4n \choose k} a_k = 4^n (-1)^n$$</p>
<p>where $a_k$ is the sequence of period $4$ which is equal to $0$ when $k$ is odd an... |
207,418 | <p>The TAs in my department are stuck in assisting an undergraduate with the following problem:</p>
<p>$$\sum^{2k}_{i=0} C^{4k}_{2i}(-1)^{i}=2^{2k}(-1)^{k}.$$</p>
<p>We tried to solve this via induction (obviously failed), via various combinatorial identities, via generating functions, etc. Aside from the fact that n... | Brian M. Scott | 12,042 | <p>It can be proved by induction on $k$:</p>
<p>$$\begin{align*}
\sum^{2k}_{i=0}\binom{4k}{2i}(-1)^{i}&=\sum_{i=0}^{2k}\left(\sum_{j=0}^4\binom4j\binom{4k-4}{2i-j}\right)(-1)^i\\
&=\sum_{j=0}^4\binom4j\sum_{i=0}^{2k}\binom{4k-4}{2i-j}(-1)^i\\
&=\sum_{i=0}^{2k}\binom{4k-4}{2i}(-1)^i+4\sum_{i=0}^{2k}\binom{4... |
207,418 | <p>The TAs in my department are stuck in assisting an undergraduate with the following problem:</p>
<p>$$\sum^{2k}_{i=0} C^{4k}_{2i}(-1)^{i}=2^{2k}(-1)^{k}.$$</p>
<p>We tried to solve this via induction (obviously failed), via various combinatorial identities, via generating functions, etc. Aside from the fact that n... | Mike Spivey | 2,370 | <p>Here's a short proof.</p>
<p>$$\sum_{k=0}^{4n} \binom{4n}{k} i^k = (1+i)^{4n} = \left(\sqrt{2} e^{i \pi /4}\right)^{4n} = 4^n e^{i n \pi} = 4^n \left(\cos (n \pi) + i\sin (n \pi)\right) = (-4)^n.$$</p>
<p>Equating real and imaginary parts yields both the OP's identity</p>
<p>$$\sum_{k=0}^{2n} \binom{4n}{2k} (-1)^... |
23,020 | <p>I am in the process of learning about Mapping class groups. At this point it seems like most of what I've read involves very low dimensional (surfaces and 3-manifolds) applications.</p>
<p>I was wondering if they were studied (or arise naturally) in higher dimensional settings?</p>
<p>In particular, any references... | Zoltan Zimboras | 7,317 | <p>As you mentioned, people have been studying the mapping class groups mostly in connection with (two-dimensional) surfaces, and also some work have been done for 3-manifolds - here the most notable result is Kojima's theorem which states that every finite group is the mapping class group of a compact hyperbolic 3-ma... |
3,242,553 | <p>I got two sequences of stochastic process <span class="math-container">$(X_n(t))_{t \in [0,1]}$</span> and <span class="math-container">$(Y_{n}(t))_{t \in [0,1]}$</span>, defined on a probability space <span class="math-container">$(\Omega, \mathcal{F},P)$</span>, and know that their distance in the sup-norm on <spa... | gourab das | 582,064 | <p>One way of interpreting the convergence of a sequence Xn to X is to say that the ''distance'' between X and Xn is getting smaller and smaller. For example, if we define the distance between Xn and X as P(|Xn−X|≥ϵ), we have convergence in probability. One way to define the distance between Xn and X is</p>
<p>E(|Xn−X... |
2,624,837 | <p>What is the integral of $$\int a^{x-1}dx?$$</p>
<p>is it $$\frac{a^{x-1}}{\log(a)} + c?$$</p>
<p>How can we derive the proper integral? Also can you please tell me the definite integral with limits, say b to c?</p>
| Peter Szilas | 408,605 | <p>Let $a>0$, $a,x$ real:</p>
<p>$a^{x-1} =\exp(\ln(a^{x-1}))= $</p>
<p>$\exp((x-1)\ln(a))= \exp(c(x-1))$, </p>
<p>where $c :=\ln(a).$</p>
<p>Use substitution: </p>
<p>$y=c (x-1)$ to integrate $\exp(y).$</p>
|
2,476,181 | <p>Where Ω = {1,2,...,p}, all Ω are equally likely and p is prime how would I show that if A and B are independent events then at least one of A and B is either ∅ or Ω?</p>
| XRBtoTheMOON | 478,087 | <p>You've got 13 letters. Choose 9 of them to make your consonants. Then you've got 21 consonants for the first, 20 for second, etc. Then 5 vowels to put in each of the other spots. So $\binom{13}{9}\cdot 21^{\underline{9}} \cdot 5^4$</p>
|
2,197,790 | <h3>Question</h3>
<blockquote>
<p>A sequence $\{a_n\}$ of real numbers is said to be a Cauchy sequence of for
each $\epsilon$ > 0 there exists a number $N > 0$ such that m, $n > N$ implies
that $|a_n − a_m| <\epsilon$.</p>
<p>Prove that every convergent sequence is a Cauchy sequence</p>
</blockquot... | Jonathan Barkey | 414,649 | <p>A way to do this is to notice that since the sequence $(x_n)$ is convergent, then a tail of that sequence, let's call it $(x_m)$ is also convergent. That is, for all $\epsilon_1\gt0$, there exists $N\in\mathbb{N}$ such that for all $n\ge N$, $|x_n-x|\lt\epsilon_1$. Similarly, for all $\epsilon_2\gt0$, there exists $... |
1,957,453 | <p>Can you please help me with this question?</p>
<p>Question: Find the radius of curvature, and the equation of the osculating circle, for the following curve for <span class="math-container">$t\geq0$</span>.</p>
<p><span class="math-container">$r(t) = \sin(\sqrt{e^t+1}) \hat{i} - \cos(\sqrt{e^t+1}) \hat{j} + 0 \hat... | June Mar Fajardo | 210,883 | <p>the curvature $\kappa(t)$ is $$\dfrac{||\vec{r}'(t)\times\vec{r}''(t)||}{||\vec{r}'(t)||^3}$$ so the radius $\rho$ is $1/\kappa.$ For the equation of osculating circle, note that the vector, with terminal point equal to the center of the osculating circle and initial point equal to the point $\vec{r}(t)$, is paralle... |
4,638,170 | <p>I'm trying to write a proof to show that a tree structure of finite nodes terminate.</p>
<p>Suppose we can say that either a node is a parent of another node (<span class="math-container">$Pqp$</span>: <span class="math-container">$q$</span> is the parent of <span class="math-container">$p$</span>), or it is a termi... | Paul Frost | 349,785 | <p>If we know that <span class="math-container">$U$</span> is both open and closed, we also know that <span class="math-container">$V = X \setminus U$</span> is open. You have <span class="math-container">$a \in U$</span>, thus <span class="math-container">$U \ne \emptyset$</span>. Since <span class="math-container">$U... |
3,853,723 | <p>While brushing up on my old discrete mathematics skills I stumbled upon this problem that I can't solve.</p>
<p>In <span class="math-container">$\mathbb{R^2}$</span> the middle point of two coordinates is <span class="math-container">$(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$</span>. Show that given five points in <sp... | Giovanni Barbarani | 444,892 | <p>EDIT: as pointed out in the comments the approch is wrong but I still think it is a bit valuable so I'm not deleting it for now.</p>
<p>I'll try. <span class="math-container">$B$</span> is a <span class="math-container">$k$</span>-algebra hence there exists a ring homorphism <span class="math-container">$T:k\rightar... |
1,497,495 | <p>How would I solve this linear system equation?</p>
<p>$$\begin{cases}
2w+x-y=4\\
3z-x=6\\
-2y-x+9z+4w=7
\end{cases}$$</p>
<p>First I arranged them so I could make the metric and then I was stuck and I don't realy know how to continue, please help me.</p>
| Jon | 273,053 | <p>Try multiplying the first equation by $2$, and the second one by $3$:
\begin{cases}
4w+2x-2y=8\\
9z-3x=18\\
-2y-x+9z+4w=7
\end{cases}
Then,
\begin{cases}
4w-2y=8-2x\\
9z=18+3x\\
4w-2y+9z=7+x
\end{cases}</p>
<p>And the rest is by positioning the elements from the first two equations in the third one</p>
|
1,497,495 | <p>How would I solve this linear system equation?</p>
<p>$$\begin{cases}
2w+x-y=4\\
3z-x=6\\
-2y-x+9z+4w=7
\end{cases}$$</p>
<p>First I arranged them so I could make the metric and then I was stuck and I don't realy know how to continue, please help me.</p>
| egreg | 62,967 | <p>Write your system into normal form:
\begin{cases}
x-y+2w=4\\
x-3z=-6\\
x+2y-9z-4w=-7
\end{cases}</p>
<p>Now, depending on the tools you have available, there are several possibilities. The most efficient is Gaussian elimination: the matrix of the system is
\begin{align}
\left[\begin{array}{cccc|c}
1 & -1 &... |
1,497,495 | <p>How would I solve this linear system equation?</p>
<p>$$\begin{cases}
2w+x-y=4\\
3z-x=6\\
-2y-x+9z+4w=7
\end{cases}$$</p>
<p>First I arranged them so I could make the metric and then I was stuck and I don't realy know how to continue, please help me.</p>
| k170 | 161,538 | <p>$$2w+x-y=4$$
$$3z-x=6$$
$$-2y-x+9z+4w=7$$
Can be rewritten to
$$2w+x-y+0z=4$$
$$0w-x+0y+3z=6$$
$$4w-x-2y+9z=7$$
$$0w+0x+0y+0z=0$$
So now we have
$$\left[\begin{array}{cccc|c}
2 & 1 & -1 & 0 & 4 \\
0 & -1 & 0 & 3 & 6 \\
4 & -1 & -2 & 9 & 7 \\
0 & 0 & 0 & 0 &... |
3,099,815 | <p>I need some help on how to approach this problem. I can't seem to find any examples that help me understand this, so if anyone has an approach example to post I would be very grateful:</p>
<p>"Consider a relation <span class="math-container">$R$</span> defined on the set of integers. Determine for the following if ... | user247327 | 247,327 | <p>You need to think about the basic definitions. Any relation, x~ y, is "reflexive" if and only if x~ x is true for any x. It is "symmetric" if and only if x~ y implies y~ x. It is "transitive" if and only if x~ y and y~ z implies x~ z.</p>
<p>Reflexive: is it true that x~ x, that is, x= 2x, for any real number x?<... |
3,099,815 | <p>I need some help on how to approach this problem. I can't seem to find any examples that help me understand this, so if anyone has an approach example to post I would be very grateful:</p>
<p>"Consider a relation <span class="math-container">$R$</span> defined on the set of integers. Determine for the following if ... | Sujit Bhattacharyya | 524,692 | <p>I assume you have a clear concept about the definition of Relations.</p>
<p>As given, <span class="math-container">$R=\{(x,y) : x=2y\}$</span> i.e. in the set of integers the relation <span class="math-container">$R$</span> is defined as,</p>
<p>for <span class="math-container">$x,y\in\mathbb{Z}$</span>, <span cla... |
42,301 | <p>everyone! I am sorry, but I am an abcolute novice of Mathematica (to be more precise this is my first day of using it) and even after surfing the web and all documents I am not able to solve the following system: </p>
<pre><code>Solve[{y*(((y*x)/(beta*b))^(1/(beta - 1)) - v) - c*alpha ==
0, ((x/alpha))*(((y*x)... | MathBiolGuy | 12,187 | <p>Thanks everyone for the reply. Didn't expect such an overwhelming response. I did a quick check on the speed of each of the solutions by making a random list of 2x10^7 elements and compared the timing (given in bold) using the 4 solutions given by Yi Wang, halirutan and sakra: </p>
<pre><code>a = RandomInteger[1000... |
212,949 | <p>A simple question:</p>
<p>I have this equation:</p>
<pre><code>eq1=Derivative[0, 1][T1][x, t] - Derivative[1, 0][T0][x, t]^2 -
T0[x, t]*Derivative[2, 0][T0][x, t] - Derivative[2, 0][T1][x, t] == 0;
</code></pre>
<p>I want only to select terms that contain T0 or its derivatives only, that is:</p>
<pre><code>-De... | kcr | 49,048 | <p>Not the most elegant solution, but you can use the <code>Collect</code>command in the following manner</p>
<pre><code>eq1 = Derivative[0, 1][T1][x, t] -
Derivative[1, 0][T0][x, t]^2 -
T0[x, t]*Derivative[2, 0][T0][x, t] -
Derivative[2, 0][T1][x, t];
(Coefficient[#1, {T0[x, t], Deriv... |
4,380,475 | <p>I'm trying to differentiate <span class="math-container">$x\sqrt{4-x^2}$</span> using the definition of derivative.</p>
<p>So it would be something like</p>
<p><span class="math-container">$$\underset{h\to 0}{\text{lim}}\frac{(h+x) \sqrt{4-\left(h^2+2 h x+x^2\right)}-x \sqrt{4-x^2}}{h}$$</span></p>
<p>I was trying t... | 2'5 9'2 | 11,123 | <p>I think your algebra could look more like:</p>
<p><span class="math-container">$$\begin{align}
&\frac{(x+h)\sqrt{4-(x+h)^2}-x\sqrt{4-x^2}}{h}\cdot\frac{(x+h)\sqrt{4-(x+h)^2}+x\sqrt{4-x^2}}{(x+h)\sqrt{4-(x+h)^2}+x\sqrt{4-x^2}}\\
&=\frac{(x+h)^2(4-(x+h)^2)-x^2(4-x^2)}{h\left((x+h)\sqrt{4-(x+h)^2}+x\sqrt{4-x^2}... |
3,966 | <p>This type of answer is what I'm looking for:</p>
<pre><code>In[58]:= ArcTan @ 1
Out[58]= π/4
</code></pre>
<p>This is what mathematica gives me:</p>
<pre><code>In[59]:= ArcTan@2
Out[59]= ArcTan[2]
</code></pre>
<p>Is it possible to express <code>ArcTan</code> in terms of $\pi$? I understand some fractions woul... | F'x | 700 | <p>Short answer: <strong>no</strong>, <code>ArcTan[2]</code> is not fraction of $\pi$. But this is more of a mathematics question than pertaining to <em>Mathematica</em>.</p>
<p>If you want to “check” that the result is not expressable as a fraction of $\pi$, you can check for the continued fraction reprentation of $\... |
14,458 | <p>I want to be able to plot several numerical solutions of an ODE, plus its analytical solution in one plot, in order to see how the numerical solutions converge towards the analytical one with respect to the number of steps. The method I'm using is Euler's method for the equation
$ y'(t) = 1-t +4y(t), y(0)=1$</p>
<p... | Sascha | 4,597 | <p>Instead of implementing an explicit Euler method on your own, you could as well use the built-in option of <code>NDSolve</code>: (the value following <code>StartingStepSize</code> is your actual step size for the whole method since <code>"ExplicitEuler"</code> has no adaptive step size):</p>
<pre><code>AnalyticalSo... |
469,485 | <p>Here's the simple question:</p>
<p>Devon has a piece of poster board 45 cm by 20 cm.
His teacher challenges him to cut the board into parts, then rearrange</p>
<p>the parts to form a square.
a) What is the side length of the</p>
<p>square?
b) What are the fewest cuts</p>
<p>Devon could have made? Explain.</p>
<... | Ross Millikan | 1,827 | <p>Part b doesn't have a nice algorithmic solution that I know of. Clearly you can cut the board int $5 \times 5$ squares and rearrange them to make a $30 \times 30$ square. That is a lot of cuts. Many times the answer is a cut (not a single straight line) that is a stairstep, then you move the stairs one notch. It... |
469,485 | <p>Here's the simple question:</p>
<p>Devon has a piece of poster board 45 cm by 20 cm.
His teacher challenges him to cut the board into parts, then rearrange</p>
<p>the parts to form a square.
a) What is the side length of the</p>
<p>square?
b) What are the fewest cuts</p>
<p>Devon could have made? Explain.</p>
<... | André Nicolas | 6,312 | <p>We describe a nice way to do it, unfortunately in words. It really needs a picture. </p>
<p>Put down your cardboard rectangle, one corner at the origin, the long side along the positive $x$-axis. So the corners of your cardboard rectangle are at $(0,0)$, $(0,45)$, $(45,20)$, and $(0,20)$.</p>
<p>Draw a $30\times 3... |
96,289 | <p>In 1995 (if I'm not mistaken) Taylor and Wiles proved that all semistable elliptic curves over $\mathbb{Q}$ are modular. This result was extended to all elliptic curves in 2001 by Breuil, Conrad, Diamond, and Taylor.</p>
<p>I'm asking this as a matter of interest. Are there any other fields over which elliptic curv... | David Loeffler | 2,481 | <p>Yes, this is a <em>very</em> active area -- one of the major themes of current research in number theory. </p>
<p>Much of the recent work has focussed on proving something slightly weaker, but easier to get at, than modularity. An elliptic curve $E$ over a number field $K$ is said to be <em>potentially modular</em>... |
2,485,261 | <blockquote>
<p>$\displaystyle \sum_{k=0}^n k {n \choose k} p^k (1-p)^{n-k}$ with $0<p<1$</p>
</blockquote>
<p>I know of one way to evaluate it (from statistics) but I was wondering if there are any other ways. </p>
<p>This is the way I know:</p>
<p>Let </p>
<p>$$M(t)=\displaystyle \sum_{k=0}^n e^{kt} {n \c... | epi163sqrt | 132,007 | <p>A slightly different variation of an answer already given which might also be convenient.</p>
<blockquote>
<p>We obtain
\begin{align*}
\color{blue}{\sum_{k=0}^nk\binom{n}{k}p^k(1-p)^{n-k}}
&=np(1-p)^{n-1}\sum_{k=1}^n\binom{n-1}{k-1}\left(\frac{p}{1-p}\right)^{k-1}\tag{1}\\
&=np(1-p)^{n-1}\sum_{k=0}^{n-1... |
4,284,803 | <p>I am solving a question and I can't get over this step: proving <span class="math-container">$$\sin \frac{1}{k} > \frac{1}{k} - \frac{1}{k^2}$$</span>
where <span class="math-container">$k$</span> is a positive integer.</p>
<p>I tried using induction, but I failed. One of my friends managed to prove it using deri... | Albus Dumbledore | 769,226 | <p>well if you don't want to use calculus ...</p>
<p>from <a href="https://math.stackexchange.com/questions/98998/why-x-tanx-while-0x-frac-pi2">this</a> geometric proof by David Mitra we have <span class="math-container">$$\sin t\le t\le \tan t$$</span> Now <span class="math-container">$$\tan t/2\ge t/2 $$</span> <spa... |
207,185 | <p>How would I go about proving this without a truth table?</p>
<p>$[(p \lor q) \implies r ] \implies [ \neg r \implies (\neg p \land \neg q)]$</p>
| RicardoCruz | 36,340 | <p>Let's consider only two different settings in our proof, as shown in figure 1 and 2. You can complete the proof for other settings.</p>
<p>(1) <strong>Case</strong> ($\angle ERM$ and $\angle MRF$ are adjacent angles).</p>
<p>See figure 1.</p>
<p><img src="https://i.stack.imgur.com/Z2J22.png" alt="MiqueSimi"></p>
... |
1,451,301 | <p>We're given a function $P_n(x)$ for $-1\leq x\leq1$ as follows :</p>
<p>$$P_n(x) = \int \limits_0^\pi \dfrac{1}{\pi}(x+i\sqrt{1-x^2} \cos\theta)^n \, d\theta$$</p>
<p>for $n=(0,1,2,3,\ldots)$, we need to prove that $|P_n(x)| \leq 1$.</p>
<p>I tried the following :</p>
<p>Let $z=x+i\sqrt{1-x^2}\cos\theta$ </p>
<... | Community | -1 | <p>why not do it like:
for the above to be true we assume the condition to be true and then equating both we will get $x^2\le 1$ which is always true for the given case. Hence, by true logic.</p>
|
1,451,301 | <p>We're given a function $P_n(x)$ for $-1\leq x\leq1$ as follows :</p>
<p>$$P_n(x) = \int \limits_0^\pi \dfrac{1}{\pi}(x+i\sqrt{1-x^2} \cos\theta)^n \, d\theta$$</p>
<p>for $n=(0,1,2,3,\ldots)$, we need to prove that $|P_n(x)| \leq 1$.</p>
<p>I tried the following :</p>
<p>Let $z=x+i\sqrt{1-x^2}\cos\theta$ </p>
<... | marty cohen | 13,079 | <p>\begin{align}
x^2\sin^2 t+\cos^2 t
&=x^2(1-\cos^2 t+\cos^2 t\\
&=(1-x^2)\cos^2t+x^2\\
\text{so}\\
|x^2\sin^2t+\cos^2t|
&\le|(1-x^2)\cos^2t+x^2|\\
&\le|(1-x^2)\cos^2t|+|x^2|\\
&\le|1-x^2|+|x^2|\\
&\le(1-x^2)+x^2\\
&=1
\end{align}</p>
|
24,305 | <p>I have several functions, let's assume they are:</p>
<pre><code>func1[x_]=x;
func2[x_]=3*x-5;
func3[x_]=0.1*x^2;
</code></pre>
<p>and a lot more like these.</p>
<p>For each and every one of these I want to do the following</p>
<pre><code>xvalues = Range[0, 500, 2.5];
points1 = Map[func1, xvalues];
Do[If points1[... | rcollyer | 52 | <p>I would simplify your code a bit, merging everything into the <code>Map</code> statement, and move everything into a function, as follows:</p>
<pre><code>process[func_, xvals_] :=
Block[{points},
points = Map[ With[{val = func@#}, UnitStep[val] val]&, xvals];
Transpose[{xvals, points}]
]
</code></pre>
<p... |
24,305 | <p>I have several functions, let's assume they are:</p>
<pre><code>func1[x_]=x;
func2[x_]=3*x-5;
func3[x_]=0.1*x^2;
</code></pre>
<p>and a lot more like these.</p>
<p>For each and every one of these I want to do the following</p>
<pre><code>xvalues = Range[0, 500, 2.5];
points1 = Map[func1, xvalues];
Do[If points1[... | Dr. belisarius | 193 | <pre><code>f = {# &, 3*# - 5 &, 0.1*#^2 &};
xvalues = Range[0, 500, 2.5];
t1 = Through[f[xvalues]] /. x_ /; x < 0 -> 0;
ListPlot[t1, DataRange -> {0, 500}]
</code></pre>
<p><img src="https://i.stack.imgur.com/kh38S.png" alt="enter image description here"></p>
|
4,570,329 | <p>In the textbook that I am working through, it is left as an exercise to prove the following claim</p>
<blockquote>
<p>Consider two linear maps <span class="math-container">$P$</span> and <span class="math-container">$Q$</span> from <span class="math-container">$\mathbb{R}^n$</span> to <span class="math-container">$\... | belkacem abderrahmane | 660,639 | <p>I will only assume that <span class="math-container">$m$</span> columns of <span class="math-container">$P$</span> are linearly independent, i.e <span class="math-container">$P$</span> is surjective (Since the columns of <span class="math-container">$P$</span> spans its image), So there's some linear map <span class... |
2,352,821 | <p>So, in order to obtain the required answer, I tried to apply some Taylor expansions, which led me to nowhere actually.
After a while I tried to use the summation theorem </p>
<p>$\sum_{n=-\infty}^{+\infty}{f\left(n\right)}=-\sum_{i=1}^{m}{Res_{z=z_i}{\pi\cot\left(\pi z\right)f\left(z\right)}}$ at $f\left(z\right)$... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\... |
4,291,880 | <p>I don't understand how you would take the conjugate of a quadratic equation and how it would be useful to solve this question.</p>
<p>I would normally show it by saying if <span class="math-container">$b$</span> is real, then it is equal to <span class="math-container">$\alpha$</span> times <span class="math-contain... | Deepak | 151,732 | <p>You have that <span class="math-container">$z^2 + az + b = 0$</span>. The given proposition only holds for real <span class="math-container">$a$</span> and real <span class="math-container">$b$</span>. This needs to be stated.</p>
<p>You need to either show or assume the following properties of complex conjugation:<... |
3,867,834 | <p>I gotta find the value of <span class="math-container">$x+y$</span> in the following image</p>
<p><a href="https://i.stack.imgur.com/j9RPH.png" rel="noreferrer"><img src="https://i.stack.imgur.com/j9RPH.png" alt="enter image description here" /></a></p>
<p>I have no info about if a point is the middle point of a len... | nmasanta | 623,924 | <p>This is nothing but the <em><strong>Trapezoidal Rule with Error Formula</strong></em>, where <span class="math-container">$~-\frac{(b-a)^{3}}{12} f^{\prime \prime}(\eta)~$</span> is the error term <span class="math-container">$(a < η < b)$</span>.</p>
<p><strong>Derivation:</strong> Consider the integral <span... |
1,862,108 | <p>This question is related to maths, so I post here. Actually it's a computer science question and I am facing this type of question while learning Design and Analysis of Algorithms, but we all know that computer science has complete relation with maths. </p>
<p>Arrange the following functions in increasing order of ... | Eduardo Sebastian | 451,521 | <p>Take the limit to the infinity of quotient of each combination and if the result is <span class="math-container">$0$</span> or more generally a constant, what it means? and if the result is <span class="math-container">$\infty$</span>?</p>
<p>That is the way to check what function grows faster that other when <span ... |
2,791,914 | <ul>
<li>$\displaystyle \int_0^\infty \frac{\arctan\frac{x^3}{1+x^2}}{x^2} \, dx$ </li>
</ul>
<hr>
<p>So i know that this one converge from 1 to infinity (by Dirichlet rule), but i'm not sure about the 0 to 1 segment. I kind of think that it converge as well, but can't prove it myself. Any suggestions?</p>
| Andrei | 331,661 | <p><strong>HINT:</strong> The function is positive in that interval. Find the maximum (not at 0). It is a finite value. The integral from $0$ to $1$ must be less than the maximum multiplied by $1-0$</p>
|
1,579,371 | <p>I'm studying for my number theory test tomorrow, and these are the last questions in my study guide. I think I understand Fermat's factorization, however, I can't tell how my professor wants us to answer these questions. One of them is going to be on the exam.</p>
<ol>
<li><p>Set <span class="math-container">$n= 874... | Tito Piezas III | 4,781 | <p>(<em>Too long for a comment</em>.)</p>
<p>You may have missed the small even powers $k=2,4,6$
$$\begin{align}
&32\,\Re\operatorname{Li}_3\left(i\,\phi^2\right)\stackrel{\color{blue}?}=\operatorname{Li}_3\left(\phi^{-8}\right)-4\operatorname{Li}_3\left(\phi^{-4}\right)+20\operatorname{Li}_3\left(\phi^{-2}\right)... |
413,778 | <p>Let <span class="math-container">$G$</span> be a finite abelian group, <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> be two non-empty subsets of <span class="math-container">$G$</span> of equal size. Suppose that for each irreducible character <span class="math-container">$\chi$... | Geoff Robinson | 14,450 | <p>More generally, for any finite group <span class="math-container">$G$</span>, Abelian or not, it is true that <span class="math-container">$Z(\mathbb{C}G)$</span>, the center of the complex group algebra <span class="math-container">$\mathbb{C}G$</span>, has a <span class="math-container">$\mathbb{C}$</span>-basis o... |
2,543,169 | <p>The question is pretty self explanatory, but I’ve encountered situations where, for the length of some vector $\vec{a}$, to denote the length (or magnitude, which ever you prefer) as either $\| \vec{a}\|= \sqrt{a_1^2+a_2^2+\ldots+a_n^2}$ or $|\vec{a}|= \sqrt{a_1^2+a_2^2+\ldots+a_n^2}$ and I was wondering which notat... | GEdgar | 442 | <p>Definitions are not judged "correct" or "incorrect". Both of your notations are used in various contexts in mathematics. There is a problem only if the writer and the reader do not understand each other.</p>
|
919,572 | <p>Do you know any nice way of expressing </p>
<p>$$\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$$
?</p>
<p>Some simple manipulations involving the integrals lead to an expression that also uses<br>
the hypergeometric series. Is there any way of getting a form that doesn't use the HG function?</p>
| Jack D'Aurizio | 44,121 | <p>Yes:
$$\begin{eqnarray*}\color{red}{\sum_{k=1}^{n+1}\frac{H_k}{n+2-k}}=\sum_{k=1}^{n+1}\sum_{r+s=k}\frac{1}{rs}=\color{red}{H_{n+2}^2-H_{n+2}^{(2)}.}\end{eqnarray*}$$
For the proof, see <a href="https://math.stackexchange.com/questions/627936/how-find-this-sum-i-n-sum-k-0n-frach-k1h-n-k1k2">this other</a> question. ... |
1,307,069 | <p>Let's look at $f(x)=\cos(x)$ defined on the interval $[0,\pi]$.</p>
<p>We know that for any function $g$ defined on $[0,\pi]$ we have:</p>
<p>$g(x)=\sum_{k=1}^{\infty}B_k\sin(kx)$ where $B_k=\frac{2}{\pi}\int_{0}^{\pi}g(x)\sin(kx)dx$. And $f$ is no different. So in our case:</p>
<p>$B_k=\frac{2}{\pi}\int_{0}^{\pi... | Hans Lundmark | 1,242 | <p>Your formula after "it can be shown that" is clearly not valid for $k=1$,
so you simply have to compute $B_1$ using some different method. (For example, $\int \cos x \sin x \, dx= \frac12 \sin^2 x + C$.)</p>
|
4,118 | <p>I've recently dipped my toes into the world of number theory; and I've bought a book that to me is quite unconventional: R. P. Burn, <em>A Pathway into Number Theory</em>. I've yet to put the book through its paces, but it seems agreeable enough to me. The book is unique in that it poses a sequence of questions to y... | Jon Bannon | 354 | <p>Check out <a href="http://jiblm.org">http://jiblm.org</a>. There are lots of scripts here, some better than others. </p>
<p>A nice book in this style is <a href="http://www.maa.org/publications/ebooks/distilling-ideas">"Distilling Ideas" by Brain Katz and Michael Starbird</a>.</p>
<p>I also recommend the following... |
4,338,190 | <p>Its required to prove that <span class="math-container">$|x^{1/n} -1| \lt \epsilon$</span> for <span class="math-container">$\epsilon \gt 0$</span> and <span class="math-container">$n \ge N$</span> where <span class="math-container">$N \in \mathbb N$</span>.<br>
Let <span class="math-container">$x^{1/n} -1 = h$</spa... | user2661923 | 464,411 | <blockquote>
<p>Let <span class="math-container">$x^{(1/n)} - 1 = h$</span> for <span class="math-container">$h>0.~~$</span> So <span class="math-container">$x=(1+h)^{(1/n)}$</span>.</p>
</blockquote>
<p>No, this is wrong. The intermediate conclusion is that <br>
<span class="math-container">$x = (1 + h)^n$</span>.... |
2,390,077 | <p>is it possible to find a matrix $B$ which fulfills:</p>
<p>$BAA^TB^T=I$, where $I$ is identity matrix and $A$ strictly lower triangular?</p>
<p>Thank you very much in advance!</p>
| Dietrich Burde | 83,966 | <p>No, this is not possible. It is useful to write it down explicitly for $n=2$. Then either $A=0$, which has no solution, or we may assume that
$$
A=\begin{pmatrix} 0 & 1 \cr 0 & 0 \end{pmatrix},\quad
B=\begin{pmatrix} a & c \cr b & d \end{pmatrix}.
$$
Then the condition $BAA^ TB^ T=I$ is equivalent t... |
707,317 | <p>Let $g: R\rightarrow R$ be a twice differentiable function satisfying $g(0)=1, g'(0)=0$ and $ g''(x)-g(x)=0$, for all $x$ in R</p>
<p>Fix $x$ in R. Show that there exists $M>0$ such that for all natural number n and all θ from 0 to 1 $$ |g^{(n)}(θx)|\leq M$$</p>
<p>Also, find the coefficients of the Taylor expa... | Community | -1 | <p><strong>First Part:</strong></p>
<p>Since <span class="math-container">$g$</span> is differentiable on <span class="math-container">$\mathbb{R}$</span>, it is also continuous on <span class="math-container">$\mathbb{R}$</span>. Note that every continuous function must attain a maximum and a minimum on a closed inte... |
773,880 | <p>What approach would be ideal in finding the integral $\int4^{-x}dx$?</p>
| Ellya | 135,305 | <p>Firstly, the most Important thing here is that $(4^{-x})'=(-\ln 4)4^{-x}$ </p>
<p>So we rewrite our integral as follows:</p>
<p>$\int 4^{-x}\,dx=\frac{-1}{\ln 4}\int -\ln 4\cdot 4^{-x}\,dx=\frac{-1}{\ln 4}\int (4^{-x})'\,dx=-\frac{1}{\ln 4}4^{-x}+C$.</p>
|
248,733 | <p>Assume the following matrix
$$
C_p^{(a,b)}:=\left(
\begin{array}{cccccc}
a &a &0 &\cdots &\cdots &0 \\
0 &0 &a &\ddots &\ddots &\vdots \\
\vdots &\ddots &\ddots &\ddots &\ddots &\vdots \\
\vdots &\ddots &\ddots &\ddots &\ddots &0 \\
0 &a... | Meysam Ghahramani | 84,430 | <p>Let $\left(
\begin{array}{c}
{g_1^{(a,b)}}(1) \\
\\
{g_2^{(a,b)}}(1) \\
\\
\vdots \\
\vdots \\
\\
{g_p^{(a,b)}}(1) \\
\end{array}
\right)$
be the first column of $C_p^{(a,b)}$. From equation ${(C_p^{(a,b)})}^n=C_p^{(a,b)}\cdot {(C_p^{(a,b)})}^{n-1}$ we conclude:</p>
<p>$$\left\{
\begin{array}{c}
g_1^{(a,b)}(... |
385,537 | <p>How would you go about proving the following?</p>
<p>$${1- \cos A \over \sin A } + { \sin A \over 1- \cos A} = 2 \operatorname{cosec} A $$</p>
<p>This is what I've done so far:</p>
<p>$$LHS = {1+\cos^2 A -2\cos A + 1 - \cos^2A \over \sin A(1-\cos A)}$$</p>
<p>....no idea how to proceed .... X_X</p>
| amWhy | 9,003 | <p>You did everything thus far correctly, I just pick up with where you left off in the second line:</p>
<p>$$\begin{align}(1 - \cos A)^2 + \sin^2 A \over \sin A(1 - \cos A)
& = \dfrac{1 - 2 \cos A + \cos^2 A + \sin^2 A}{\sin A(1 - \cos A)} \\ \\
& = {1 \color{blue}{\bf + \cos^2 A} -2\cos A + 1 \color{blue}{\... |
1,958,491 | <p>Let $t^k$ act as the $k$-th derivative operator on the set of polynomials. So</p>
<p>$$t^k(x^n)=t^k x^n=(n)_kx^{n-k}$$</p>
<p>where $(n)_k=n(n-1)(n-2)...(n-k+1)$ is the falling factorial. Then with a formal power series, $f(t)=\sum_{k\ge 0}a_k\frac{t^k}{k!}$, the linear operator $f(t)$ acts as such that</p>
<p>... | Han de Bruijn | 96,057 | <p>Introductionary reference:
<UL><LI>
<A HREF="http://www.alternatievewiskunde.nl/jaar2015/examples.htm" rel="nofollow">Operator Calculus</A>
</LI></UL>
So why not replace $\,t\,$ by the differential operator $\,d/dx\,$ everywhere it occurs? Next reference:</p>
<p><UL><LI>
<A HREF="http://math.stackexchange.com/quest... |
1,484,838 | <p>Ok, so I just learned trig identities and I come across this problem that had it's answer to it, and I have no idea how they got to that answer.</p>
<p>Here is the problem:</p>
<p>$$
\frac{-\sec\theta}{1-\cos\theta}=\frac{-1-\sec\theta}{\sin^2\theta}
$$</p>
<p>Now the problem calls for the left side to be adjust... | Narasimham | 95,860 | <p>Multiply numerator and denominator by $ (1+ \cos \theta) $ and simplify.</p>
|
2,327,273 | <p>If a tree has 5 vertices of degree 2, 3 vertices of degree 3, 4 vertices of degree 4, then how many leaves are there in that tree? </p>
<p>I know the tree has at least 12 vertices and so it must have at least 11 edges. Also the number of leaves must be odd but I could not proceed further. </p>
| Donald Splutterwit | 404,247 | <p><a href="https://i.stack.imgur.com/PEdCG.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PEdCG.png" alt="enter image description here"></a></p>
<p>Note that the number of leaves will be independent of how the verticies of valency $2,3$ & $4$ are joined. We reckon $\color{red}{13}$.</p>
|
181,702 | <p>I am working on getting the hang of proofs by induction, and I was hoping the community could give me feedback on how to format a proof of this nature:</p>
<p>Let $x > -1$ and $n$ be a positive integer. Prove Bernoulli's inequality:
$$ (1+x)^n \ge 1+nx$$</p>
<p><strong>Proof</strong>: </p>
<p>Base Case: For $n... | timur | 2,473 | <p>This looks fine to me. Just a small note on formatting of the inequalities: I would combine the third and fourth inequalities as
$$
(1+x)^{k+1} \geq 1+(k+1)x+kx^2>1+(k+1)x,
$$
so there is no need of the fifth line. Or even
$$
(1+x)^{k+1} = (1+x)(1+x)^{k} \geq (1+x)(1+kx)=1+(k+1)x+kx^2>1+(k+1)x.
$$</p>
|
189,650 | <p>let $S=\{s_1, s_2, s_3 \}$, if $s_1$ can be represented as a linear combination of $s_2$ and $s_3$, $s_2$ can be represented as a linear combination of $s_1$ and $s_3$ but $s_3$ can not be represented as a linear combination of $s_1$ or $s_2$ or $s_1$ and $s_2$, can we call $S$ a linearly dependent set? </p>
| Godot | 38,875 | <p>You said that </p>
<p>$$s_1=as_2+bs_3.$$</p>
<p>Therefore</p>
<p>$$s_1-as_2-bs_3=0$$</p>
<p>and your set is not linearly independent.</p>
|
116,394 | <p>After importing a sound file, how can I add an echo to it?</p>
<pre><code> sound = Import["test.wav", "SampleRate"]
</code></pre>
<p>It needs to be apply after time specified by user. This is as far as I have got:</p>
<pre><code> addEcho[sound_, time_] :=
Module[{tmp = sound, channels, samples, duration},
... | demm | 30,122 | <p>The idea is to delay the sound (sometimes also reduce its amplitude) and add it to your original signal. Assuming that </p>
<p><code>Import["yourWavfile.wav", "Elements"]</code>
yields:
<code>{AudioChannels,AudioEncoding,Data,SampledSoundList,SampleRate,Sound}</code></p>
<p>the duration (in seconds) of your sound... |
3,298,412 | <blockquote>
<p>For an n-dimensional vector space <span class="math-container">$V$</span> and an ordered basis <span class="math-container">$B$</span> of <span class="math-container">$V$</span>
, the mapping <span class="math-container">$\Phi : \mathbb{R}^n → V , \Phi(e_i) = b_i, i = 1,...,n$</span>
is linear , w... | ancient mathematician | 414,424 | <p>As stated this is false unless all the <span class="math-container">$b_i=0$</span> . Let <span class="math-container">$\Phi$</span> be as stated on the <span class="math-container">$e_i$</span> and let <span class="math-container">$\Phi(x)=0$</span> otherwise. </p>
<p>Then <span class="math-container">$\Phi(2e_i)=0... |
3,362,000 | <p>From listing the first few terms, I suspect that the sequence is increasing, so I wanted to use mathematical induction to verify my suspicion.</p>
<p>I have assumed that <span class="math-container">$a_k<a_{k+1}$</span>, I don't see how I can obtain <span class="math-container">$a_{k+1}<a_{k+2}$</span> becaus... | orlp | 5,558 | <p>Hint, prove the following theorems in order:</p>
<blockquote>
<p><strong>Theorem 1.</strong> <span class="math-container">$a_n$</span> is positive.</p>
<p><strong>Theorem 2.</strong> <span class="math-container">$a_n > a_{n-1}$</span></p>
</blockquote>
|
185,112 | <p>Does there exist $m,n\ge1$, an $m \times n$ matrix $A$, and a vector $x \in \mathbb{R}^n$ such that:</p>
<ul>
<li>The entries of $A$ are $\in \{0, 1\}$.</li>
<li>For all pairs of columns $u, v$ of $A$ the entries of $u - v$ are never either all non-negative or all non-positive (i.e. there is a positive entry and a ... | Per Alexandersson | 1,056 | <p>Not if $n$ is too large compared to $m$:
For a fixed number of rows, there is only a finite set of possible columns, $2^m$.
Thus, if $n > 2^n$, some columns are identical. This contradicts property 2.</p>
<p>Using a finer reasoning about the second constraint, it should be easy to strengthen this observation.</... |
2,780,597 | <p>The <strong>definition</strong> of a <em>convex set</em> is geometrically intuitive. But the definition of <em>convex function</em> doesn't seem so intuitive: $S \subset \mathbb{R}^n$ is convex if given $x,y\in S$ the line segment joining $x,y$ is in $S$. </p>
<p>Let $f$ be a real valued function from an open inter... | Masacroso | 173,262 | <p>In the <a href="https://en.wikipedia.org/wiki/Convex_function" rel="nofollow noreferrer">article of wikipedia about convex functions</a> the relation between the two concepts is stated in this way:</p>
<blockquote>
<p>A function $f:\Bbb R^n\to\Bbb R$ is convex if its <a href="https://en.wikipedia.org/wiki/Epigrap... |
3,953,681 | <p>I have a basic question but I have failed in solving it. I have the equation of a cylinder which is <span class="math-container">$y^2 + z^2 = r^2$</span> (centered in the x-axis). The parametric equation (dependent on <span class="math-container">$L$</span> and <span class="math-container">$s$</span>) is <span class... | Travis Willse | 155,629 | <p><strong>Hint</strong> Your method seems to work, provided anyway that <span class="math-container">$a, b > 0$</span>: The substitution <span class="math-container">$x = \sqrt{\frac{a}{b}} \sin \theta$</span>, <span class="math-container">$dx = \sqrt{\frac{a}{b}} \cos \theta \,d\theta$</span> transforms the integr... |
4,204,133 | <p>Trying to solve <a href="https://math.stackexchange.com/questions/4201328/how-can-i-arrange-a-group-of-people-at-tables-and-switch-them-around-so-that-no">this problem</a> led me to consider the following generalization.</p>
<p>Let <span class="math-container">$g$</span> and <span class="math-container">$p$</span> b... | Mark Saving | 798,694 | <blockquote>
<p>Then there is <span class="math-container">$\xi_1 \in U$</span> with <span class="math-container">$\xi_1 = z$</span>.</p>
</blockquote>
<p>No. <span class="math-container">$z \in f(U) \subseteq \mathbb{R}^2$</span>, so <span class="math-container">$z$</span> is an ordered pair, not a single number <span... |
1,511,246 | <blockquote>
<p>What is the value of $0.7\overline{54}$ +$0.69\overline2$?</p>
<p>(a) $\frac{1813}{900}$ (b) $\frac{1783}{910}$ (c) $\frac{14323}{9900} (d) \frac{13243}{9900}$</p>
</blockquote>
<p>I get</p>
<p>@edit</p>
<p>$$754-7/990 + 692-69/900$$=$747$/$990$ + $623$/$900$=$1$/$90$($747$/$11$ + $623$/$10$)<... | SchrodingersCat | 278,967 | <p>The question is not $0.754+0.692=?$ <br>
The question asks $0.7 \overline{54}+0.69\overline2=?$</p>
<p>So the answer will be $0.7 \overline{54}+0.69\overline2=\frac{754-7}{990}+\frac{692-69}{900}=\frac{747}{990}+\frac{623}{900}=1.44\overline{67}=\frac{14323}{9900}$</p>
|
1,511,246 | <blockquote>
<p>What is the value of $0.7\overline{54}$ +$0.69\overline2$?</p>
<p>(a) $\frac{1813}{900}$ (b) $\frac{1783}{910}$ (c) $\frac{14323}{9900} (d) \frac{13243}{9900}$</p>
</blockquote>
<p>I get</p>
<p>@edit</p>
<p>$$754-7/990 + 692-69/900$$=$747$/$990$ + $623$/$900$=$1$/$90$($747$/$11$ + $623$/$10$)<... | fleablood | 280,126 | <p>(754 - 7)/990 + (692 - 69)/900 = 747/990 + 623/900 = 7470/9900 + 6853/9900 = 14323/9900</p>
|
4,416,063 | <p>How to solve <span class="math-container">$\int\frac{\ln(x \ln(x))}{x} dx$</span>?</p>
<p>My work:<br />
Let <span class="math-container">$t = \ln(x) \implies x= e^t ; dt = \dfrac{dx}{x}$</span></p>
<p>So above integral changes to,
<span class="math-container">$$\int t ( e^t t) dt$$</span>
<span class="math-contain... | Lai | 732,917 | <p><span class="math-container">$$
\begin{aligned}
I&= \int \ln (x \ln x) d(\ln x) \\
=& \ln x \cdot \ln (x \ln x)-\int \ln x \cdot \frac{1}{x \ln x}\left(x \cdot \frac{1}{x}+\ln x\right) d x \\
=& \ln x \cdot\ln (x \ln x)-\int \frac{1+\ln x}{x} d x \\
=& \ln x \cdot\ln (x \ln x)-\int(1+\ln x) d(\ln x) ... |
181,499 | <p>In many of the classes that I teach, I require students to learn the basics of Mathematica which we use throughout the semester to do computations and to submit homeworks (in notebook form). Some students really like this and some... not so much. </p>
<p>Since I teach in an engineering department, almost everyone a... | sunt05 | 260 | <p>For those with experience in python, WRI has already provided <a href="http://www.wolfram.com/language/fast-introduction-for-programmers/en/" rel="noreferrer">a nice introductory tutorial</a> along with many analogies.</p>
<p>However, for the class intended for image processing as mentioned by the OP, pure python i... |
213,872 | <p>I'm learning probability theory and I see the half-open intervals $(a,b]$ appear many times. One of theorems about Borel $\sigma$-algebra is that</p>
<blockquote>
<p>The Borel $\sigma$-algebra of ${\mathbb R}$ is generated by inervals of the form $(-\infty,a]$, where $a\in{\mathbb Q}$. </p>
</blockquote>
<p>Also... | shaman | 823,495 | <p>I would say that the answer is best presented in the reverse order of your question:</p>
<p>The (cumulative) distribution function is defined the way it is, because that is a natural way to think about accumulating probabilities as you observe more events in the sample space. So in a sense, the half-open intervals a... |
2,648,626 | <p>Is the set $(e_n)_{n>0}$ a (vector space) basis for the sequence Hilbert space $l^2$? It is a Hilbert space basis anyway.</p>
<p>I would say no, because the sequence $\left(\frac{1}{n}\right)_{n>0}$ is in $l^2$ but it can't be written as a finite linear combination of $e_i$'s.</p>
<p>Is that right?</p>
| José Carlos Santos | 446,262 | <p>Yes, it is entirely correct. Actually, it can be proved that a Hilbert space is infinite-dimensional if and only if no Hilbert basis is a basis in the Linear Algebra sense.</p>
|
112,432 | <p>a) Is true the following statement. Let $h$ be analytic in the unit disk such that $$|h(z)|\le \frac{|z|^2}{1-|z|^2},$$ then $$|h'(z)|\le \frac{2}{(1-|z|^2)^2}.$$
a') Is true the following statement. Let $h$ be analytic in the unit disk such that $$|h(z)|\le \frac{|z|^2}{1-|z|^2},$$ then the inequality $$|h'(z)|\le... | Alexandre Eremenko | 25,510 | <p>Let $|z|=r$, apply the Cauchy estimate to the disc $|\zeta-z|<(1-r)/2$.
We obtain
$$|f'(z)|\leq \frac{2}{1-r}\frac{(1+r)^2}{(1-r)(3+r)}.$$
Maximizing the factor $(1+r)^2/(3+r)$ by Calculus, we obtain that is it at most $1$.</p>
<p>This gives
$$|f'(z)|\leq\frac{2}{(1-|z|)^2}$$
which is worse than conjectured onl... |
3,041,656 | <p>I need some help in a proof:
Prove that for any integer <span class="math-container">$n>6$</span> can be written as a sum of two co-prime integers <span class="math-container">$a,b$</span> s.t. <span class="math-container">$\gcd(a,b)=1$</span>.</p>
<p>I tried to go around with "Dirichlet's theorem on arithmetic ... | templatetypedef | 8,955 | <p>Here's another route you can take to solve this problem. For any <span class="math-container">$n \ge 7$</span>, you want to show that there is a number <span class="math-container">$a$</span> where</p>
<ol>
<li><span class="math-container">$gcd(a, n - a) = 1$</span>,</li>
<li><span class="math-container">$1 < a ... |
2,516,942 | <p>Trying to find all solutions on (-infinity,+infinity) for :
$y''+4y = 0$</p>
<p>I know that the discriminant of the characteristic equation is -16 so the roots are complex. so $k=0.5 \cdot \sqrt{-16} = 2i$</p>
<p>$f_1(x) = e^{(2ix)} = \cos(2x) + i\sin(2x)$</p>
<p>$f_2(x) = e^{(-2ix)} = \cos(2x) - i\sin(2x)$</p>... | fleablood | 280,126 | <p>Let's see:</p>
<p>$(2n!) = \prod\limits_{k=1..2n;n \text{even}}k*\prod\limits_{k=1...2; n\text{ odd}}k=\prod\limits_{k=1..n}2k\prod\limits_{k=1...2n; n\text{ odd}}k=2^n*n!*\prod\limits_{k=1...2n; n\text{ odd}}k$</p>
<p>So $\frac {(2n)!}{(n!)^2}\cdot \frac 1{2^{2n}} = \frac {\prod\limits_{k=1...2n; n\text{ odd}}k}{... |
642,631 | <p>What is $[\mathbb{Q}(i,\sqrt{2},\sqrt{3}):\mathbb{Q}]$?</p>
<p>On the one hand, we have $[\mathbb{Q}(i,\sqrt{2},\sqrt{3}):\mathbb{Q}(i,\sqrt{2})]\cdot[\mathbb{Q}(i,\sqrt{2}):\mathbb{Q}(i)]\cdot[\mathbb{Q}(i):\mathbb{Q}]=2^3=8.$</p>
<p>On the other hand, the minimum polynomial in $\mathbb{Q}[x]$ containing $i,\sqrt... | initial_D | 105,162 | <p>8 sounds right, and your reasoning is right as well. $\mathbb{Q}[i,\sqrt{2},\sqrt{3}]$ is the splitting field of the polynomial you gave. The only thing you might be missing is the degree of the splitting field is not necessarily the degree of the polynomial itself.</p>
|
3,910,053 | <p>The function
<span class="math-container">$u(x,t) = \frac{2}{\sqrt{\pi}}$$\int_{0}^\frac{x}{\sqrt{t}} e^{-s^2}ds$</span>
satisfies the partial differential equation</p>
<p><span class="math-container">$$\frac{\partial u}{\partial t} = K\frac{\partial^2u}{\partial x^2}$$</span></p>
<p>where <span class="math-containe... | Nevzat Eren Akkaya | 611,386 | <p>Let <span class="math-container">$c_1,...,c_n$</span> be eigenvalues of <span class="math-container">$A$</span> then</p>
<p><span class="math-container">$t=c_i$</span> <span class="math-container">$\Rightarrow$</span> <span class="math-container">$\det(A-tI)=0$</span></p>
<p><span class="math-container">$\det((A+2I... |
3,130,939 | <p>Suppose the following function with pi notation, with the pi denoting the iterated product, multiplying from <span class="math-container">$i = 0$</span> to <span class="math-container">$i = n$</span>:</p>
<p><span class="math-container">$$\prod_{i=0}^n \ln(y_i^{x - 1})$$</span></p>
<p>That is, the natural logarith... | Paras Khosla | 478,779 | <p>Define <span class="math-container">$\alpha=\prod_{i=0}^n \ln(y_i)$</span>. Note that <span class="math-container">$\alpha$</span> is simply a constant. Use logarithm properties, use the power rule for derivatives and you're done.</p>
<p><span class="math-container">$$f(x)=\prod_{i=0}^{n}\ln{y_i}^{x-1}=\prod_{i=0}^... |
634,127 | <p>How to prove this (true or not)?</p>
<blockquote>
<p>$f(a,b) = f(a,c)$ must hold if $b = c$</p>
</blockquote>
<p><b>Note:</b> <i><b>f(a,b)</b> is a function with <b>a</b> & <b>b</b></i> parameters</p>
<p>thanks</p>
| Doug Spoonwood | 11,300 | <p>If we assume that $\forall$x(x=x), then yes.</p>
<p>Suppose that b=c.</p>
<p>f(a,b)=f(a,b) since we can derive it from substitution and that $\forall$x(x=x).</p>
<p>Now since b=c we can replace just the second "b" by "c" and obtain</p>
<p>f(a,b)=f(a,c).</p>
|
203,456 | <p>Please help me proof $\log_b a\cdot\log_c b\cdot\log_a c=1$, where $a,b,c$ positive number different for 1.</p>
| DonAntonio | 31,254 | <p>Change all to the natural logarithm $\log\,$:</p>
<p>$$\log_ba\cdot\log_cb\cdot\log_ac=\frac{\log a}{\log b}\frac{\log b}{\log c}\frac{\log c}{\log a}$$</p>
<p>and voila.</p>
|
627,444 | <p>I guess that I'm quite familiar with the basic "everyday algebraic structures" such as groups, rings, modules and algebras and Lie algebras. Of course, I also heard of magmas, semi-groups and monoids, but they seem to be way to general notions as to admit a really interesting theory.</p>
<p>Thus, I'm wondering whet... | Steven Gubkin | 34,287 | <p>Groupoids seem somewhat seem somewhat more exotic than groups, but are actually a lot more natural in many ways. For example the fundamental groupoid is really more "fundamental" than the fundamental group in many ways. </p>
<p><a href="http://en.wikipedia.org/wiki/Operad_theory" rel="nofollow">Operads</a> are al... |
3,576,008 | <p><strong>Question:</strong></p>
<p>In acute <span class="math-container">$\Delta ABC$</span>, let <span class="math-container">$D$</span> be the foot of the altitude from <span class="math-container">$A$</span> to <span class="math-container">$BC$</span>, and let <span class="math-container">$\overline{AD}$</span>
i... | Matt | 263,495 | <p>It does indeed appear to indicate that there is a typo in the question as posed. The circle with the diameter <span class="math-container">$AE$</span> does not have to intersect <span class="math-container">$AB$</span> and <span class="math-container">$AC$</span> at all, and so in particular the point <span class="m... |
1,380,508 | <p>Is there such a proof that states that the Runge Phenomena will always occur when interpolating with higher order polynomials or is this just observed empirically?</p>
| Community | -1 | <p>The Runge Phenomenon does not always occur. You can interpolate, say, $f(x)= e^x$ using equally spaced nodes on any interval $[a,b]$, and the interpolating polynomials will converge to $f$ uniformly. (The same holds for any function whose Taylor series has infinite radius of convergence.)</p>
<p>When the Runge p... |
14,552 | <p>What are good examples of proofs by induction that are relatively low on algebra? Examples might include simple results about graphs.</p>
<p>My aim is to help students get a sense of the logical form of an induction proof (in particular proving a statement of the form 'if $P(k)$ then $P(k+1)$'), independent of the ... | Aeryk | 401 | <p>How about: A tree with $n\ge 1$ vertices has $n-1$ edges.</p>
|
14,552 | <p>What are good examples of proofs by induction that are relatively low on algebra? Examples might include simple results about graphs.</p>
<p>My aim is to help students get a sense of the logical form of an induction proof (in particular proving a statement of the form 'if $P(k)$ then $P(k+1)$'), independent of the ... | ncr | 1,537 | <p>I think tiling problems are good for this kind of thing. See, for example, <a href="https://www.math.hmc.edu/funfacts/ffiles/20002.4.shtml" rel="nofollow noreferrer">this</a>. There they describe how to prove the statement "if you have a $2^n\times 2^n$ chessboard with one square missing, then you can tile it with... |
292,651 | <blockquote>
<p>Does an integer $9<n<100$ exist such that the last 2 digits of $n^2$ is $n$? If yes, how to find them? If no, prove it.</p>
</blockquote>
<p>This problem puzzled me for a day, but I'm not making much progress. Please help. Thanks.</p>
| robjohn | 13,854 | <p>We are solving $n(n-1)=n^2-n\equiv0\pmod{100}$. Since $\gcd(n,n-1)=1$, one of $n$ or $n-1$ must be a multiple of $4$ while the other must be a multiple of $25$.This leads to the equations
$$
\begin{align}
4x-25y=+1\tag{1}\\
4x-25y=-1\tag{2}
\end{align}
$$
For $(1)$, $n=4x$ and $n-1=25y$. For $(2)$, $n=25y$ and $n-1=... |
86,762 | <p>The other day, my teacher was talking infinite-dimensional vector spaces and complications that arise when trying to find a basis for those. He mentioned that it's been proven that some (or all, do not quite remember) infinite-dimensional vector spaces have a basis (the result uses an Axiom of Choice, if I remember ... | David Wheeler | 23,285 | <p>The "hard case" is essentially equivalent to this one:</p>
<p>Find a basis for the real numbers <span class="math-container">$\mathbb{R}$</span> over the field of the rational numbers <span class="math-container">$\mathbb{Q}$</span>.</p>
<p>The reals are obviously an extension field of the rationals, so th... |
611,529 | <p>$$i^3=iii=\sqrt{-1}\sqrt{-1}\sqrt{-1}=\sqrt{(-1)(-1)(-1)}=\sqrt{-1}=i
$$</p>
<p>Please take a look at the equation above. What am I doing wrong to understand $i^3 = i$, not $-i$?</p>
| Brian | 114,928 | <p>We cannot say that $\sqrt{a}\sqrt{b}=\sqrt{ab}$ for negative $a$ and $b$. If this were true, then $1=\sqrt{1}=\sqrt{\left(-1\right)\cdot\left(-1\right)} = \sqrt{-1}\sqrt{-1}=i\cdot i=-1$. Since this is false, we have to say that $\sqrt{a}\sqrt{b}\neq\sqrt{ab}$ in general when we extend it to accept negative numbers.... |
611,529 | <p>$$i^3=iii=\sqrt{-1}\sqrt{-1}\sqrt{-1}=\sqrt{(-1)(-1)(-1)}=\sqrt{-1}=i
$$</p>
<p>Please take a look at the equation above. What am I doing wrong to understand $i^3 = i$, not $-i$?</p>
| Abramo | 3,416 | <p>Multiplying by $i$ in the complex plane amounts to a counterclockwise turn of $90$ degrees. Hence it is geometrically clear that $i^3 = -i$, as the following drawing shows</p>
<p><img src="https://i.stack.imgur.com/W8UJo.png" alt="enter image description here"></p>
<p>Regarding your equations, as remarked already,... |
455,230 | <p>I found this proposition and don't see exactly as to why it is true and even more so, why the converse is false:</p>
<p>Proposition 1. The equivalence between the proposition $z \in D$ and the proposition $(\exists x \in D)x = z$ is provable from the definitory equations of the existential quantifier and of the equ... | Jeremy Rickard | 88,262 | <p>For an Artinian example, let $A$ be the algebra of upper triangular $2\times 2$ matrices over a field $K$, let $J$ be the left ideal consisting of all matrices with bottom row zero, and let $S$ be the simple left $A$-module $A/J$. Then it's easy to check that $Ja\neq0$ for every non-zero element $a$ of $A$, but $Js=... |
3,715,484 | <p>As the title saying , the question is how to find the radius <span class="math-container">$R$</span> of convergence of <span class="math-container">$\sum_{n=1}^{\infty}\frac{\sin n}{n} x^n$</span>. My method is as the following:</p>
<p>When <span class="math-container">$x=1$</span>, it is well known that the ser... | José Carlos Santos | 446,262 | <p>By definition, the partial derivative at <span class="math-container">$(0,0)$</span> with respect to the direction <span class="math-container">$v$</span> is the limit<span class="math-container">$$\lim_{h\to0}\frac{f(hv)-f(0,0)}h.$$</span>So, compute this limit.</p>
|
4,203,079 | <p>I’m trying to grasp the idea behind quotient spaces and reading <a href="https://en.m.wikipedia.org/wiki/Quotient_space_(topology)" rel="nofollow noreferrer">this</a> wikipedia article. In the section ”Examples” they have the unit square <span class="math-container">$S^2$</span> homeomorphism example, which I tought... | Moe Sarah | 787,944 | <p>Let A be the subset consisting of boundary points. Then define an equivalence relation on <span class="math-container">$I^2$</span> by declaring <span class="math-container">$x\sim y$</span> if and only if <span class="math-container">$x=y$</span> or <span class="math-container">$x,y \in A$</span> then the resulting... |
451,131 | <blockquote>
<p><strong>Problem</strong>: If <span class="math-container">$\int f(x) \sin{x} \cos{x}\,\mathrm dx = \frac {1}{2(b^2 - a^2)} \log f(x) +c $</span>. Find <span class="math-container">$f(x)$</span></p>
<p><strong>Solution</strong>: <span class="math-container">$\int f(x) \sin{x} \cos{x}\,\mathrm dx = \f... | Community | -1 | <p>Yes and to complete: we have</p>
<p>$$\int f'(x)(f(x))^{-2}\;dx=C\int\sin(2x)\;dx$$
where $C=b^2-a^2$ so
$$-\frac{1}{f(x)}=-\frac{C}{2}\cos(2x)+C'$$
and you can take $f(x)$ from it.</p>
|
2,856,373 | <blockquote>
<p>If <span class="math-container">$z_{1},z_{2}$</span> are two complex numbers and <span class="math-container">$c>0.$</span> Then prove that</p>
<p><span class="math-container">$\displaystyle |z_{1}+z_{2}|^2\leq (1+c)|z_{1}|^2+\bigg(1+\frac{1}{c}\bigg)|z_{2}|^2$</span></p>
</blockquote>
<p>Try: put <s... | Robert Z | 299,698 | <p>You are on the right track. Following your approach the inequality becomes
$$x^2_{1}+x^2_{2}+2x_{1}x_{2}+y^2_{1}+y^2_{2}+2y_{1}y_{2}\leq (1+c)(x_1^2+y_1^2)+\bigg(1+\frac{1}{c}\bigg)(x_2^2+y_2^2)$$
that is
$$2x_{1}x_{2}+2y_{1}y_{2}\leq c(x_1^2+y_1^2)+\frac{1}{c}(x_2^2+y_2^2).$$
Is it true that $2uv\leq cu^2+\frac{v^2... |
2,548,353 | <p><strong>Find the number of $4\times4$ matrices such that $|a_{ij}| = 1 \forall i,j\in[1,4]$ , and sum of every row and column is zero.</strong></p>
<p>I tried 'counting' the number of matrices that satisfy the above conditions, that is, elements are $1$ or $-1$ and sum of every row and column is zero.</p>
<p>In th... | Isaac Browne | 429,987 | <p>This is quite difficult to generalize to a $n \times n$ matrix. I would suggest generalizing to having two $1$'s in each row and column in an $n \times n$ matrix, as is done is problem $8$ of combinatorics in <a href="https://www.hmmt.co/archive/february/problems/2012/" rel="nofollow noreferrer">2012 February HMMT</... |
2,548,353 | <p><strong>Find the number of $4\times4$ matrices such that $|a_{ij}| = 1 \forall i,j\in[1,4]$ , and sum of every row and column is zero.</strong></p>
<p>I tried 'counting' the number of matrices that satisfy the above conditions, that is, elements are $1$ or $-1$ and sum of every row and column is zero.</p>
<p>In th... | Isaac Browne | 429,987 | <p>And here is a tour of the HMMT solution for the generalized version, for posterity and for understanding.</p>
<p>The problem statement is now as follows: "Find the amount of $n \times n$ matrices do there exist for which all the columns and rows contain exactly two $1$'s and the rest of numbers are $-1$'s"</p>
<p>... |
443,736 | <p>Find the point on the parabola $3x^2+4x-8$ that is closest to the point $(-2,-3)$.</p>
<p>My plan for this problem was to use the distance formula and then that the derivative to get my answer. I'm having a little trouble along the way.</p>
<p>$$ d = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}.$$</p>
| pre-kidney | 34,662 | <p>Suppose the closest point is at $p=(x_0,y_0)$, and set $q=(-2,-3)$. Then the tangent to the parabola at $p$ is perpendicular to $\ell$, the line through $p,q$.</p>
<p>Since $y'=6x+4$, the slope of the tangent is $6x_0+4$, so the slope of $\ell$ is $-\frac{1}{6x_0+4}$. Since it passes through $q$, we see the equatio... |
4,380,992 | <p>I'm trying to do the following exercise:</p>
<p><em>Find a non-homogeneous recurrence relation for the sequence whose general term is</em></p>
<p><span class="math-container">$$a_n = \frac{1}{2}3^n - \frac{2}{5} 7^n$$</span></p>
<p>From this expression we can obtain the roots of the characteristic polynomial <span c... | dxiv | 291,201 | <p>Write the relations for two consecutive terms:</p>
<p><span class="math-container">$$
\begin{align}
\begin{cases}
a_n &=\, \dfrac{1}{2}\,3^n - \dfrac{2}{5}\, 7^n
\\ a_{n+1} &=\, \dfrac{1}{2}\,3^{n+1} - \dfrac{2}{5}\, 7^{n+1} \,=\, \dfrac{3}{2}\,3^n-\dfrac{14}{5}\,7^n
\end{cases}
\end{align}
$$</span></p>
<p>... |
1,621,347 | <p>Is there a closed-form expression for the following definite integral?
\begin{equation}
\mathcal{I} = \int_{\delta_1}^{\delta_2}(1+Ax)^{-L}x^{L}\exp\left(-Bx\right)dx,
\end{equation}
where $A$, $B$, $\delta_1$, and $\delta_2$ are positive constant. $L$ is a positive integer.</p>
<p>I am facing problem due to finite... | Pierpaolo Vivo | 302,446 | <p>Use the identity
$$
\frac{1}{(1+A x)^{L}}=\frac{1}{\Gamma(L)}\int_0^\infty dy\ y^{L-1}e^{-y}e^{-y A x}
$$
and swapping the two integrals (using $\partial_B^{k} e^{-B x}=(-1)^k x^k e^{-B x}$) you get
$$
\mathcal{I}=\frac{(-1)^L}{\Gamma(L)}\partial_B^{(L)}\int_0^\infty dy\ y^{L-1}e^{-y}\underbrace{\int_{\delta_1}^{\de... |
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