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 |
|---|---|---|---|---|
84,735 | <p>Unlike using</p>
<pre><code>Unprotect[In,Out];
Clear[In,Out];
Protect[In,Out];
</code></pre>
<p>to clean the occupied memory of the whole notebook, I want to clean the memory dynamically and accurately. That is, I want to clean out the specific memory I assign whenever I need.</p>
<p>For example, I wrote a loop a... | sakra | 68 | <p>Apply the <code>Clear</code> function to a symbol to clear values and definitions for that symbol, e.g.:</p>
<pre><code>Clear[x]
</code></pre>
|
4,120,122 | <p>If <span class="math-container">$f(x)=\frac{1}{\sqrt{x} (1+\ln{x})}$</span> for <span class="math-container">$x \in (1, \infty)$</span>.</p>
<p>I need to show that <span class="math-container">$ f \in L^p(1, \infty))$</span> if and only if <span class="math-container">$p \geq 2$</span></p>
<p>I suppose that <span cl... | Ian | 83,396 | <p>For <span class="math-container">$p<2$</span>, you can compare to an appropriate <span class="math-container">$x^q$</span> where <span class="math-container">$q<-1$</span>.</p>
<p>For <span class="math-container">$p>2$</span>, you can compare to an appropriate <span class="math-container">$x^q$</span> where... |
1,880,150 | <p>I work in a warehouse where we take components and put them together to make a finished good product.</p>
<p>We have these values for each component:</p>
<ol>
<li><p>Quantity Required</p></li>
<li><p>Quantity Used</p></li>
<li><p>Variance (Quantity Used- Quantity Required)</p></li>
<li><p>Overage ((Variance/Quanti... | Reg Charney | 879,789 | <p>As mentioned in other answers to this question, it depends on what the 0 value represents.</p>
<p>If a sample is missing, then there is no way to impute its values, so the zero sample can be excluded from mean and average calculations.</p>
<p>However, if you have a situation where you started with 10 items and used ... |
258,622 | <p>How do you swap the values and keys of an Association in a manner similar to taking the transpose of a two-dimensional matrix or nested list?
I am trying to solve the Wolfram Challenge <a href="https://challenges.wolfram.com/challenge/capital-cities-near-a-latitude" rel="nofollow noreferrer">Capital Cities Near a La... | Michael E2 | 4,999 | <p>I suppose <code>AssociationThread[Values@assoc, Keys@assoc]</code>, but it's not invertible if the values are not distinct.</p>
<p>Also <code>Association@KeyValueMap[#2 -> #1 &, assoc]</code> with the same caveat.</p>
|
1,349,002 | <blockquote>
<p>Let <span class="math-container">$g(z)= z^4+iz^3 +1$</span>. How many zeros does <span class="math-container">$g$</span> have in <span class="math-container">$\{z\in \Bbb{C}: \text{Re }(z), \text{Im }(z)>0\}$</span>?</p>
</blockquote>
<p>I tried comparing the number of zeros of <span class="math-c... | Eclipse Sun | 119,490 | <p>Put $f(z)=z^4+1$ and $g(z)=iz^3$. </p>
<p>Step 1: On $|z|=2$, $|g(z)|=8<15=|z^4|-1\le|f(z)|$. What does this imply? </p>
<p>Step 2: Consider $\gamma_1=\{z=x+iy:0\le x\le2,y=0\}$. Is it true that $|g(z)|<|f(z)|$ on $\gamma_1$?<br>
(Hint: $|g(z)|=x^3$ and $|f(z)|=x^4+1$, what is the minimum of $x^4+1-x^3$ on... |
1,349,002 | <blockquote>
<p>Let <span class="math-container">$g(z)= z^4+iz^3 +1$</span>. How many zeros does <span class="math-container">$g$</span> have in <span class="math-container">$\{z\in \Bbb{C}: \text{Re }(z), \text{Im }(z)>0\}$</span>?</p>
</blockquote>
<p>I tried comparing the number of zeros of <span class="math-c... | Servaes | 30,382 | <p>Although the question is tagged <code>(complex-analysis)</code>, here's another way to look at the problem:</p>
<p>Setting $f(z)=z^4+iz^3+1$, we might as well look for the roots of the real polynomial
$$g(z):=f(iz)=z^4+z^3+1,$$
in the quadrant $Q:=\{z\in\Bbb{C}:\operatorname{Re}(z)>0>\operatorname{Im}(z)\}$. ... |
481,527 | <p>The following question came up at a conference and a solution took a while to find.</p>
<blockquote>
<p><strong>Puzzle.</strong> Find a way of cutting a pizza into finitely many congruent pieces such that at least one piece of pizza has no crust on it.</p>
</blockquote>
<p>We can make this more concrete,</p>
<blockq... | Anthony | 24,971 | <p>If this violates the parameters in a clear way, consider this a teaching opportunity. Would this count:</p>
<p><img src="https://i.stack.imgur.com/tbek7.png" alt="enter image description here"></p>
|
1,010,820 | <p>I tried the following :</p>
<p>\begin{align}\sec\theta + \tan\theta&=4\\
\frac1{\cos\theta} + \frac{\sin\theta}{\cos\theta}&=4\\
\frac{1+\sin\theta}{\cos\theta}&=4\\
\frac{1+\sin\theta}4&=\cos\theta\end{align}</p>
<p>now don't know how to evaluate further ?</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>As $\sec^2\theta-\tan^2\theta=1$</p>
<p>$$\sec\theta+\tan\theta=4\iff\sec\theta-\tan\theta=\frac14$$</p>
<p>Can you find $\sec\theta$ and then $\cos\theta$?</p>
<hr>
<p>Alternatively if $\sec A+\tan A=x\iff x-\sec A=\tan A$</p>
<p>Squaring we get, $$(x-\sec A)^2=\tan^2A\iff x^2-2x\sec A+\sec^2A=\s... |
814,899 | <p>Suppose that $\{X_n\}$ is an independent sequence and $E[X_n]=0$. If $\sum \operatorname{Var}[X_n] < \infty$, then $\sum X_n$ converges with probability $1$. Is independence necessary condition here ? I am thinking of a counterexample. The intuition behind the other assumptions is clear. </p>
| Ben Grossmann | 81,360 | <p>One method:</p>
<p>Suppose $A$ is connected (otherwise, the statement isn't generally true).</p>
<p>If $f$ is continuous but not monotonic, there exists $a,b,c \in A$ with $[a,c] \subset A$ and $f(a) = f(c) < f(b)$ or $f(a) = f(c) > f(b)$ (verify that this must be the case). Show that
$f((a,c))$
cannot be ... |
299,576 | <p>Suppose $X = \displaystyle\bigsqcup_{i \in I} X_i$ is the disjoint union of infinitely many continua. The components of the Stone-Cech remainder $X^*$ can be described as follows. Treat $I$ as a discrete topological space and consider the continuous map $F:X_i \to I$ that sends each $X_i$ to $i \in I$. The Stone Cec... | KP Hart | 5,903 | <p><a href="https://mathoverflow.net/questions/93719/0-dimensional-locally-compact-space/93762#93762">The answer to this question</a> contains two locally compact zero-dimensional spaces whose Cech-Stone compactification is not zero-dimensional.That may put a limit on what can be said about components of the remainder ... |
2,106,983 | <p><strong>Question</strong></p>
<p>how to evaluate $\tan x-\cot x=2.$</p>
<p>Given that it lies between on $\left[\frac{-\pi} 2,\frac \pi 2 \right]$.</p>
<p><em>My Steps so far</em></p>
<p>I converted cot into tan to devolve into $\frac{\tan^2 x-1}{\tan x}=2$.</p>
<p>Then I multiply $\tan{x}$ on both sides and th... | Community | -1 | <p>We then can solve the quadratic equation thus getting $$\tan x =\frac {2 \pm \sqrt{4-4 (1)(-1)}}{2} = \frac {2\pm 2\sqrt {2}}{2} = 1\pm \sqrt {2} $$ Thus $$x =\arctan ( 1\pm \sqrt {2}) $$ Hope it helps. </p>
|
1,508,753 | <p>Show that for $x,y\in\mathbb{R}$ with $x,y\geq 0$, the arithmetic mean-quadratic mean inequality
$$\frac{x+y}{2}\leq \sqrt{\frac{x^2+y^2}{2}}$$
holds.</p>
<p>After my calculations I'll get: </p>
<p>$$-x^2+2xy-y^2$$
which can't be $\leq 0$.</p>
| Mark Viola | 218,419 | <p>We first choose $\delta'= 1/2$. Then, for $0<|x-2|<\delta'$, we have $\frac{1}{x-1}<2$. Therefore, for all $\epsilon >0$, we have</p>
<p>$$\begin{align}
\left|\frac{1}{1-x}+1\right|&=\left|\frac{x-2}{1-x}\right|\\\\
&<2|x-2|\\\\
&<\epsilon
\end{align}$$</p>
<p>whenever $|x-2|<\de... |
70,539 | <p>I'm looking for resources discussing mathematical notation, the theory, the philosophy, the distinct advantages of various notations. Stuff about notation for computer algebra systems is interesting too.</p>
<p>Anyone have resources they recommend?</p>
<p>I've already read Whitehead's An Introduction to Mathemati... | Jesko Hüttenhain | 9,947 | <p>I think you might like the book <em>"Concrete Mathematics"</em> by Donald Knuth. </p>
|
70,539 | <p>I'm looking for resources discussing mathematical notation, the theory, the philosophy, the distinct advantages of various notations. Stuff about notation for computer algebra systems is interesting too.</p>
<p>Anyone have resources they recommend?</p>
<p>I've already read Whitehead's An Introduction to Mathemati... | Chandan Singh Dalawat | 2,821 | <p>MR1163629 (93f:05001)
Knuth, Donald E.
Two notes on notation.
Amer. Math. Monthly 99 (1992), no. 5, 403–422.</p>
<p>Two notational schemas, with all their advantages, disadvantages and even pitfalls, are discussed. The first goes back to K. E. Iverson and creates a kind of characteristic function by enclosing a (ma... |
1,227,610 | <p>I need to calculate derivative of the following function with respect to the matrix X:</p>
<p>$f(X)=||diag(X^TX)||_2^2$</p>
<p>where $diag()$ returns diagonal elements of a matrix into a vector. How can I calculate $\frac {\partial f(X)} {\partial X}$? Please help me.</p>
<p>Thanks in advance!</p>
| greg | 228,596 | <p>In terms of the Hadamard ($\circ$) and Frobenius ($:$) products, the function can be written as
$$ \eqalign {
f &= \|I\circ(X^TX)\|^2_F \cr
&= I\circ(X^TX):I\circ(X^TX) \cr
&= I\circ(X^TX)\circ I:(X^TX) \cr
&= I\circ I\circ(X^TX):(X^TX) \cr
&= I\circ(X^TX):(X^TX) \cr
&= I:(X^T... |
1,129,567 | <p>Using the substitution $x=\cosh (t)$ or otherwise, find
$$\int\frac{x^3}{\sqrt{x^2-1}}dx$$
The correct answer is apparently
$$\frac{1}{3}\sqrt{x^2-1}(x^2+2)$$
I seem to have gone very wrong somewhere; my answer is way off, can someone explain how to get this answer to me.</p>
<p>Thanks.</p>
<p>My working:
$$\int... | Dylan | 135,643 | <p>Trig substitutions are not necessary. You can substitute $u = x^2 - 1$ and get</p>
<p>$$\int \frac{x^3}{\sqrt{x^2-1}}\,dx = \int \frac{(x^2 - 1 + 1)}{\sqrt{x^2-1}} x\,dx = \frac{1}{2}\int \frac{u+1}{\sqrt{u}} \,du = \frac{1}{2} \int (u^{1/2} + u^{-1/2} )\,du $$</p>
<p>The key word is "or otherwise"</p>
|
4,400,243 | <p>How do we change the equation <span class="math-container">$x^{2} + y^{2} + xy - x - y = 0$</span> to the standard form of ellipse?</p>
<p>Since there have a term <span class="math-container">$xy$</span> , I don't know how to use completing square method.</p>
| Zaragosa | 691,503 | <p>As you can see this equation corresponds to an ellipse rotated an angle <span class="math-container">$\theta$</span>.
<a href="https://i.stack.imgur.com/3nzs0.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3nzs0.png" alt="enter image description here" /></a></p>
<p>Let <span class="math-container... |
269,232 | <p>I'm trying to get my plot ticks in decimal form. The automatic output is</p>
<pre><code>Plot[x, {x, 0, 10^-5}]
</code></pre>
<p><a href="https://i.stack.imgur.com/XbAyw.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XbAyw.png" alt="plot with tick in scientific form" /></a></p>
<p>but I would like... | MarcoB | 27,951 | <p>It seems that the approach from the linked answer is sound but a small modification of the replacement pattern is needed, perhaps because the tick specification format may have changed since that answer:</p>
<pre><code>longticks2 = Show[#, AbsoluteOptions[#, Ticks] /. {pos_, lbl:Except[""], len_} :> {po... |
269,232 | <p>I'm trying to get my plot ticks in decimal form. The automatic output is</p>
<pre><code>Plot[x, {x, 0, 10^-5}]
</code></pre>
<p><a href="https://i.stack.imgur.com/XbAyw.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XbAyw.png" alt="plot with tick in scientific form" /></a></p>
<p>but I would like... | rhermans | 10,397 | <p>You can create the strings in any way you want them.</p>
<pre><code>myticks= Function[
{number},
{
number,
ToString[
AccountingForm[
N[number],
{12, 7},
NumberSigns -> {"-", "+"},
NumberPaddin... |
1,606,709 | <p>I am studing Kähler differentials and I tried to understand the geometric motivation behind these settings. What I do not understand is the role which plays the diagonal in all these theory. The cotangent sheaf is later defined in terms of the diagonal map. Why is this geometrically interesting? I tried to write a s... | Martin R | 42,969 | <p>Denote the $n$-th factor by $x_n$. The sequence $(x_n)$ is increasing, and
$$
x_2 = \sqrt{\frac{1}{2}+\sqrt\frac{1}{2}} > \sqrt{\frac{1}{2}+\frac{1}{2}} = 1
$$
Therefore
$$
x_1 x_2 \cdots x_n \ge x_1 x_2^{n-1} \to \infty \, .
$$</p>
|
2,164,333 | <p>Let $T$ be linear transformation from $V$ to $W$. I know how to prove the result that nullity$(T) = 0$ if and only if $T$ is an injective linear transformation. But I still don't intuitively understand why the kernel only containing the zero vector means that $T$ is injective, and vice versa. In contrast, the relati... | Nicolas FRANCOIS | 288,125 | <p>It comes from the fact that $T(x)=T(y)\iff T(x-y)=0$, so that the kernel contains all the information of the injectivity of $T$ : if it is $\{0\}$, then $T(x)=T(y)$ only has solution $x=y$.</p>
<p>In fact, if $T$ is non injective, the dimension of the set of solutions of equation $T(x)=T(y)$ is independent of $y$ :... |
395,850 | <p>Is Dirac delta a function? What is its contribution to analysis? </p>
<p>What I know about it:
It is infinite at 0 and 0 everywhere else. Its integration is 1 and I know how does it come.</p>
| Sara Tancredi | 76,098 | <blockquote>
<p>$ \delta(x) = \begin{cases} +\infty, & x = 0 \\ 0, & x \ne 0
> \end{cases}$</p>
<p>and which is also constrained to satisfy the identity $
> \int_{-\infty}^\infty \delta(x) \, dx = 1$</p>
</blockquote>
<p>For example, the objects f(x) = δ(x) and g(x) = 0 are equal everywhere except... |
4,316,800 | <p>I have the following two-dimensional SDE:
<span class="math-container">$dX_1=(-\mu X_1 - X_2)dt +\sigma dW_1$</span> and
<span class="math-container">$dX_2=(-\mu X_2 + X_1)dt +\sigma dW_2$</span></p>
<p>I then have to show that <span class="math-container">$E(X_1^2 + X_2^2) = \frac{\sigma^2}{\mu}$</span>.</p>
<p>I k... | Lutz Lehmann | 115,115 | <p>If you apply Ito to <span class="math-container">$Y=X_1^2+X_2^2$</span>, then
<span class="math-container">$$\begin{align}
dY &= 2(X_1dX_1+X_2dX_2)+(dX_1)^2+(dX_2)^2
\\
&= -2\mu(X_1^2+X_2^2)\,dt+2\sigma(X_1dW_1+X_2dW_2)+2\sigma^2\,dt,
\end{align}$$</span>
using the informal notation for the increment of the ... |
3,757,038 | <h2>The problem</h2>
<p>So recently in school, we should do a task somewhat like this (roughly translated):</p>
<blockquote>
<p><em>Assign a system of linear equations to each drawing</em></p>
</blockquote>
<p>Then, there were some systems of three linear equations (SLEs) where each equation was describing a plane in t... | paulinho | 474,578 | <p>If you write your systems of equations as a matrix as follows:
<span class="math-container">$$A \vec{x} = \begin{bmatrix} 1 & -3 & 2 \\ 1 & 3 & -2 \\ 0 & -6 & 4 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} -2 \\ 5 \\ 3\end{bmatrix} = \vec{b}$$</span>
then he... |
370,192 | <p>I have a basic question about the Heegaard diagrams involved in providing a framework
for calculation of Floer-Homology of three-manifolds.</p>
<p>Typically such diagrams look like Figure 1 and Figure 2 <a href="http://www.map.mpim-bonn.mpg.de/Poincar%C3%A9%27s_homology_sphere" rel="nofollow noreferrer">here</a> or ... | Sam Nead | 1,650 | <p>Chapter four of "Knots, Links, Braids and 3-Manifolds" by Prasolov and Sossinsky gives a highly readable (and nicely illustrated) introduction to three-manifolds via Heegaard splittings. Another, more classical, reference is chapter two of "Three-manifolds" by Hempel. Note that Hempel calls han... |
4,509,098 | <p><span class="math-container">$△ABC$</span> - triangle with <span class="math-container">$45°, 105°$</span> and <span class="math-container">$30°.$</span></p>
<p>Perimeter of triangle is <span class="math-container">$\sqrt6 + 2\sqrt3 - \sqrt2.$</span></p>
<p>Find the longest side?</p>
<p>Do you have any ideas? Seems ... | Lion Heart | 809,481 | <p><a href="https://i.stack.imgur.com/2ORZW.png" rel="noreferrer"><img src="https://i.stack.imgur.com/2ORZW.png" alt="enter image description here" /></a>
Draw <span class="math-container">$AD\perp BC$</span></p>
<p>Let <span class="math-container">$AD=x$</span> then <span class="math-container">$AC=2x , DC=x\sqrt3 , B... |
147,551 | <p>My lecturer says as follows;
$C_0 = \{(a,b) : -\infty \le a\le b< \infty\}$</p>
<p>$$C_\mathrm{open} = \{ A \in \mathbb{R} : A\text{ open} \}$$ </p>
<p>He goes on to show that $\sigma(C_0) = \sigma(C_{\mathrm{open}})$; </p>
<p>Clearly, $\sigma(C_0)$ is in $\sigma(C_{\mathrm{open}})$</p>
<p>So now I need to s... | Brian M. Scott | 12,042 | <p>There is in fact a gap in the first proof. First, he didn’t bother to prove Step 3. For that he should be comparing the tables</p>
<p>$$\begin{array}{c|cc}
&1&ab\\ \hline
1&1&ab\\
a^{-1}&a^{-1}&a^{-1}(ab)
\end{array}$$</p>
<p>and</p>
<p>$$\begin{array}{c|cc}
&a^{-1}&b\\ \hline
a&am... |
3,814,502 | <p>I came across the following question:</p>
<blockquote>
<p>Show that <span class="math-container">$(a, b:a^3 = 1, b^2= 1, ba=a^2b)$</span> gives a group of order <span class="math-container">$6$</span>. Show that it is non abelian. Is it the only non abelian group of order <span class="math-container">$6$</span> up t... | Matthew Graham | 378,351 | <p>Check this answer out here: <a href="https://math.stackexchange.com/q/1511511">There are 2 groups of order 6 (up to isomorphism)</a></p>
<p>I think this is one of the answers you’re looking for.</p>
<p>It seems like an exercise by your professor to learn some groups of order <span class="math-container">$6$</span>. ... |
314,329 | <p>When I first learned about factorials in grade school I quickly became interested in the idea and did a lot of playing with them. I noticed, though, that as the factorials got higher and higher they gained more and more trailing zeros.</p>
<pre><code>5! has 1 trailing zero and = 120
10! has ... | Sean Ballentine | 62,751 | <p>Every time you pass a multiple of 10 (or something 5 mod 10) you will accumulate another 0</p>
<p>For example 10! has two trailing zeros, one from multiplying by 10 and the other from multiplying by 5 and 2.</p>
<p>So it makes sense that as you get much higher the number of accumulated zeroes should increase.</p>
|
2,571,746 | <p>$$|x+4| -4 =x $$</p>
<p>I've two questions about this equation. </p>
<ul>
<li><p>Why do we need to build an inequality?</p></li>
<li><p>If we build an inequality, in what cases do we need to analyse?</p></li>
</ul>
<p>Also I'm trying to find the negative values that $x$ can take.</p>
| ArsenBerk | 505,611 | <p>Building an inequality for cases $x < -4$ and $x \ge -4$ is simply in order to make things easier. Because,</p>
<p><strong>Case 1 $(x < -4)$:</strong> In this case, $|x+4| = -x-4$ so we have $-x-4-4 = x \implies x = -4$ but $-4 \not\lt -4$ so this solution is not valid for this case.</p>
<p><strong>Case 2 $(... |
3,915,565 | <p>I am thinking about this question and I don't really know how to tackle it.</p>
<p>The problem is:</p>
<blockquote>
<p>The class has <span class="math-container">$30$</span> students. If you want to call 5 students, each one is independent of the others and randomly with equal probability, what is the probability th... | Jean-Claude Arbaut | 43,608 | <p>Visual proof. <span class="math-container">${}{}{}{}{}{}{}{}{}{}{}$</span></p>
<p><a href="https://i.stack.imgur.com/muQyr.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/muQyr.png" alt="enter image description here" /></a></p>
<p>Of course it doesn't depend on the base triangle, here is the same ... |
3,915,565 | <p>I am thinking about this question and I don't really know how to tackle it.</p>
<p>The problem is:</p>
<blockquote>
<p>The class has <span class="math-container">$30$</span> students. If you want to call 5 students, each one is independent of the others and randomly with equal probability, what is the probability th... | player3236 | 435,724 | <p>Another one, based on the fact that centered hexagonal numbers can be expressed as <span class="math-container">$6T_n+1$</span>:
<a href="https://i.stack.imgur.com/xLiWS.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xLiWS.png" alt="enter image description here" /></a></p>
|
4,354,663 | <p>I'm trying to prove that if <span class="math-container">$x$</span> is a multiple of <span class="math-container">$3$</span>, then <span class="math-container">$x + 1$</span> is not a multiple of <span class="math-container">$3$</span>. This is a rather obvious fact, but I don't think I understand the solution.</p>
... | Mike | 544,150 | <p>What is done by the proof is the following:</p>
<ol>
<li><p>First, the set of integers is closed under addition and subtraction. So as <span class="math-container">$k$</span> and <span class="math-container">$t$</span> are both integers, then so must <span class="math-container">$k-t$</span> be an integer.</p>
</li>... |
1,903,762 | <blockquote>
<p>Largest number that leaves same remainder while dividing 5958, 5430 and 5814 ?</p>
</blockquote>
<hr>
<p>$$5958 \equiv 5430 \equiv 5814 \pmod x$$
$$3\times 17\times 19 \equiv 5\times 181\equiv 3\times 331\pmod x$$
$$969 \equiv 905\equiv 993\pmod x$$</p>
<p>After a bit of playing with the calculato... | Kaligule | 182,303 | <p>The only "small" solutions (below 10000000) seem to be ${1,2,3,4,6,8,12,16,24,48}$.
Of course, you can add $lcm({5958, 5430, 5814}) = 1741612770 $ to a solution and get another, bigger one.</p>
<p>The set above was found by letting do the computer what a computer can do best: Computing stuff. Here is the Haskell C... |
2,174,979 | <p>I am trying to prove that if a function $f:\mathbb{R} \to \mathbb{R}$ is bounded and the second derivative $f''$ is bounded, then the first derivative $f'$ is also bounded. My hint is to use Taylor's theorem.</p>
<p>I know for any $x$ and $y$, $f(y) = f(x) + f'(x)(y-x) + \frac{1}{2} f''(c)(y-x)^2,$ is the Taylor a... | RRL | 148,510 | <p>Hint:</p>
<p>Your inequality is true for <em>every</em> $y \in \mathbb{R}$ and </p>
<p>$$\min_{0 < z < \infty} \left(\frac{2C}{z} + \frac{Dz}{2}\right)= 2 \sqrt{CD}$$</p>
|
346,855 | <p>Let's start with the text of the assignment:</p>
<p><em>Let $V$ be a (finite-dimensional) vector space, and let $L<V$ be a subspace of $V$. Prove that $[a] = a + L$ holds for every vector $a\in{V}$.</em></p>
<p>(where $[S]$ in general is the set of all possible linear combinations of vectors from $S$, a.k.a. th... | Santiago Canez | 1,914 | <p>Given your comment confirming that $[S]$ is defined to be the span of $S$ (which is the common name given to the set of linear combinations of vectors in $S$), then you are correct: the assignment's claim is incorrect, as it should be since there is no relation between $[a]$ and the subspace $L$.</p>
|
2,960,054 | <p>We are told to proof the Minkowski inequality specialized to sequences:</p>
<p><span class="math-container">$$
\left(\sum_{i=1}^n|a_i+b_i|^p\right)^{1/p}\leq
\left(\sum_{i=1}^n|a_i|^p\right)^{1/p}
+\left(\sum_{i=1}^n|b_i|^p\right)^{1/p}
$$</span></p>
<p>when <span class="math-container">$p\geq1$</span> </p>
<p>We... | JDivision | 554,339 | <p>For all <span class="math-container">$1\leq i\leq p$</span> ,<span class="math-container">$|a_i+b_i|\leq|a_i|+|b_i|$</span>. Then we have that <span class="math-container">$$|a_i+b_i|^p \leq \left(|a_i|+|b_i|\right)^p.$$</span> Hence, <span class="math-container">$\displaystyle\sum_{i=1}^{p} |a_i+b_i|^p\leq\sum_{i=1... |
72,209 | <p>Let $\{U_k\}$ be a sequence of independent random variables, with each variable being uniformly distributed over the interval $[0,2]$, and let $X_n = U_1 U_2\cdots U_n$ for $n \geq 1$.</p>
<p>(a) Determine in which of the senses (a.s., m.s., p., d.) the sequence $\{X_n\}$ converges as $n\to\infty$, and identify the... | Community | -1 | <p>In a comment, I noted that $X_n$ is a martingale. Since $X_n\geq 0$, a martingale convergence theorem says that $X_n$ converges almost surely. This implies that $X_n$ also converges in distribution and in probability. That $X_n$ does <em>not</em> converge in mean square follows from $\mathbb{E}(X_n^2)=(4/3)^n$. </p... |
1,645,341 | <p>I know that there exists an elementary matrix that switches the rows in another matrix when they are multiplied, but how do you prove that this elementary matrix actually does this job?</p>
| Gregory Grant | 217,398 | <p>The short answer is no. The function is not continuous at $x=0$ because $\lim_{x\to0} f(x)$ does not exist. </p>
<p>The easiest way to see that is to notice that it oscillates between $-1$ and $+1$ infinitely often in every interval $(0,\epsilon)$ so it cannot be approaching any fixed value.</p>
|
1,645,341 | <p>I know that there exists an elementary matrix that switches the rows in another matrix when they are multiplied, but how do you prove that this elementary matrix actually does this job?</p>
| mweiss | 124,095 | <p>In general, if $f(a)$ is not defined for some point $a$, but
$\lim_{x \to a}f(x)=L$
for some value $L$, then we have a <em>redefinable discontinuity</em>. Intuitively, that means that the graph has a hole in it, but the hole can be plugged by simply extending the function to a new function defined on a larger domai... |
104,549 | <p>Consider the following one-dimensional version of the game battleships. There is a battleship somewhere on $\mathbb N$, i.e., a interval $N,\ldots,N+k$. Your task is to find whether this battleship lies in the interval $1,\ldots,n$ using the minimal number of tests (on can ask if the battlship includes $i$ for any... | Axel Magnuson | 3,574 | <p>Okay, so maybe the fastest way to do this would be to keep track of position of offset, i.e. </p>
<p>$x_0 = 1, y_0 = n - 1$</p>
<p>$\begin{cases}
x_{i+1} = x_i + y_i \text{ if } x_i + y_i \leq n \newline
y_{i+1} = y_i / 2, x_{i+1} = y_{i+1} / 2 \text{ else }
\end{cases}$</p>
<p>that is to say, keep dividing t... |
104,549 | <p>Consider the following one-dimensional version of the game battleships. There is a battleship somewhere on $\mathbb N$, i.e., a interval $N,\ldots,N+k$. Your task is to find whether this battleship lies in the interval $1,\ldots,n$ using the minimal number of tests (on can ask if the battlship includes $i$ for any... | Douglas Zare | 2,954 | <p>The version of the Battleship game doesn't seem to match either the question or the motivation.</p>
<p>Suppose $n=15$ and $k=2$ so the Battleship has length $3$. There are $5$ disjoint positions $1-3, 4-6, 7-9, 10-12, 13-15$ so you have to make at least $5$ guesses to conclude the Battleship is not on the interval,... |
25,784 | <p>As many Americans know, the “traditional” high school sequence is:</p>
<p>Algebra 1</p>
<p>Geometry</p>
<p>Algebra 2</p>
<p>PreCalculus</p>
<p>Calculus</p>
<p>For those who take developmental education at the community college level, it consists of something like:</p>
<p>Developmental Algebra</p>
<p>Intermediate Alg... | meta comment guest | 20,585 | <p>The other answers well discuss how geometry is sort of "off track" for the push to calculus and the other topics (trig, algebra) are much more integral. But I don't think they make the point enough that community college students are already behind the curve in terms of time. We are talking about kids ta... |
3,189,462 | <p>Question:
<span class="math-container">$$2x^{2}y''-6xy'+6y=x^{3} $$</span></p>
<p>I have tried solving it, and obtained the roots 3 and 1. Apparently this is a case of resonance. Even so, I wasn't able to get the particular solution through variation of parameters.</p>
<p>Perhaps there are other methods to solve t... | J.G. | 56,861 | <p>Define <span class="math-container">$\phi(x):=\frac{1}{\sqrt{2\pi}}\exp-\frac{x^2}{2},\,\Phi(x):=\int_{-\infty}^x\phi(t)dt$</span> so, expanding on parsiad's answer, <span class="math-container">$$P(W_1\ge 0\,\operatorname{and}W_2\ge 0)=\int_0^\infty dw_1\phi(w_1)(1-\Phi(-w_1))=\int_0^\infty dw_1\phi(w_1)\Phi(w_1)\\... |
261,461 | <p>I would like a good <strong>hint</strong> for the following problem that takes into account the position at which I am stuck. The problem is as follows</p>
<blockquote>
<p>Let $\mathbb{Z}_n$ be the cyclic group of order $n.$ Find a simple graph $G$ such that $\mathrm{Aut}(G) = \mathbb{Z}_n.$ </p>
</blockquote>
<... | UbU | 60,108 | <p>To kill the reflection but not the rotation, you can do the following: on the edge $(i,i+1)$ in the $n$-cycle graph, insert two new vertices, say $a_i,b_i$. Create a path of length 1 emanating from $a_i$ and a path of length 2 emanating from $b_i$. The graph now has exactly the following edges for each $i$: $(i,a_... |
1,018,716 | <p>I came across a text that proves that translation operator $T_a(f):=f(x-a)$ where $a\in\mathbb{R}^n$ and $f\in L^p(\mathbb{R}^n)$ is continuous. The proof follows:
$$||f(x-a)-f(x)||_p=||f(x-a)-g(x-a)+g(x-a)-g(x)+g(x)-f(x)||_p\leq ||f(x-a)-g(x-a)||_p+||g(x-a)+g(x)||_p+||g(x)-f(x)||_p<3\varepsilon$$
Where $g$ is so... | Julián Aguirre | 4,791 | <p>First of all, in the proof you need to assume something else about the function $g$. Usually one takes $g\in C_c(\mathbb{R}^n)$, the space of continuous functions with compact support. This implies that $g$ is uniformly continuous, allowing one to prove that $\|g(x-a)-g(x)\|_p\to0$ as $a\to0$.</p>
<p>The proof show... |
2,671,173 | <p>I need to find isometry between two spaces of continuous functions <span class="math-container">$C[a,b]$</span> and <span class="math-container">$C[0,1]$</span>. That means to find function <span class="math-container">$ \phi\colon C[a,b] \longrightarrow C[0,1] $</span> which is bijection and <span class="math-conta... | José Carlos Santos | 446,262 | <p>Simply define $\phi(f)(x)=f\bigl((b-a)x+a\bigr)$. Its inverse is $\phi^{-1}(f)(x)=f\left(\frac{x-a}{b-a}\right)$. Besides\begin{align}d_\infty\bigl(\phi(f),\phi(g)\bigr)&=\sup_{x\in[0,1]}\bigl|f\bigl((b-a)x+1\bigr)-g\bigl((b-a)x+1\bigr)\bigr|\\&=\sup_{x\in[a,b]}\bigl|f(x)-g(x)\bigr|\\&=d_\infty(f,g).\end... |
3,441,200 | <p>Prove that a tangent developable has constant Gaussian curvature zero. Also compute its mean curvature.</p>
<p>I have a tangent developable as let <span class="math-container">$\gamma:(a,b)\rightarrow R^3$</span> be a regular space curve, and s>0. Then <span class="math-container">$\sigma(s,t) = \gamma(t)+s\gamma'(... | Arctic Char | 629,362 | <p>You are not given the expression for <span class="math-container">$\gamma$</span>, so everything will depends on <span class="math-container">$\gamma$</span>. </p>
<p>So you can get </p>
<p><span class="math-container">$$ \sigma_t = \gamma'(t) + s\gamma''(t), \ \ \sigma_s = \gamma'(t),$$</span></p>
<p>so </p>
<p... |
338,480 | <blockquote>
<p>Find the point where equations $x=t^2-t$ and $y= t^3 -3t-1$ cross itself.</p>
</blockquote>
<p>This's the first time I meet this kind of problem, can someone give me some idea? Thank you.</p>
| lab bhattacharjee | 33,337 | <p>As Glen has identified $t_2\ne t_1$</p>
<p>$$t_2^2-t_2=t_1^2-t_1\implies(t_2-t_1)(t_2+t_1)=t_2-t_1$$
$$\implies t_2+t_1=1\text{ as } t_2\ne t_1$$</p>
<p>$$t_2^3-3t_2-1=t_1^3-3t_1-1\implies t_2^3-t_1^3=3(t_2-t_1)$$
$$\implies t_1^2+t_1t_2+t_2^2=3\text{ as } t_2\ne t_1$$
$$\implies (t_1+t_2)^2-t_1t_2=3\implies t_1t_... |
1,188,928 | <p>I know that on a smooth projective variety any coherent sheaf has a finite locally free resolution. I read somewhere that this implies that any object in $D^b(X)$ for $X$ smooth projective is then isomorphic in $D^b(X)$ to a complex of locally free objects. It seems to me it should be a proof by induction on the len... | Brian Fitzpatrick | 56,960 | <p>Here is a rough idea of why sheaves may be identified with their projective resolutions in the derived category.</p>
<p>Recall that for an abelian category $\mathcal A$, the <strong>bounded derived category</strong> is defined as the localization of the homotopy category of bounded chain cocomplexes $K^b(\mathcal A... |
2,251,240 | <p>What is the first derivative and nth derivative of the following function $ y = \sqrt {2 +\sqrt {3 + \sqrt {x}}}$ </p>
<p>I think taking the ln for both sides will remove the first square root only?
Could anyone give me a hint ? </p>
| The Dead Legend | 433,379 | <p>First of all. Use chain rule. I'd recommend learning something new instead of relying on app.(personal opinion)</p>
<p>$$\frac{dy}{dx}=\frac{1}{2\sqrt{2+\sqrt{3+\sqrt{x}}}}.\frac{d\sqrt{3+\sqrt{x}}}{dx}$$
$$\frac{dy}{dx}=\frac{1}{2\sqrt{2+\sqrt{3+\sqrt{x}}}}.\frac{1}{2\sqrt{3+\sqrt{x}}}\frac{d\sqrt{x}}{dx}$$
$$\fra... |
3,444,500 | <p>I need to prove that <span class="math-container">$3x^6+12x^5+9x^4-24x^3+9x^2-4x+3=0$</span> does not have any real root. I tried analyzing the derivatives to see the maxima and minima but I can't compute them exactly, so I couldn't proceed further. Hints are appreciated.</p>
| Calum Gilhooley | 213,690 | <p>I wanted to see if there was a way to prove this without heavy
numerical calculation. I found such a proof, but it suggests, in
two ways, that it is better to work with the original trigonometric
form of the problem. For one thing, my proof requires a finicky
division of the range of values of <span class="math-cont... |
2,287,312 | <p>So the 12 marbles are distinct with each marble being a different color. Therefore I have to take into account, the different colors and different amount each person have. For example, person 1 can have 2 marbles, person 2 can have 7 marbles, and person 3 can have 3 marbles with each marble being a different color. ... | Caleb Stanford | 68,107 | <p>To complement the other answers for anyone who is curious, here is a <strong>proof using nonstandard analysis.</strong> (Note: $x \approx y$ means $x - y$ is infinitesimal.)</p>
<p>First, for any $\epsilon \approx 0$, note that $f(\epsilon)$ is either $0$ or $\epsilon$ -- either way $f(\epsilon) \approx 0 = f(0)$ ,... |
1,224,692 | <p>What is the most computationally efficient way to find the layer on which a ball (i) belongs when arranged in a tetrahedron or 3 dimensional triangle with a triangular base. The ball on the top layer is numbered one. The balls on the second layer are numbered 2 - 4. The fifth layer 4-10 and so on. </p>
| abel | 9,252 | <p>you will need the fact that if the columns(rows) of a square matrix are linearly dependent, if and only if the determinant is not zero. this leads to the fact that determinant is zero if and only if $Ax = 0$ has non trivial solution. it is the last fact that gets used to determine the characteristic values $\lambda$... |
1,890,496 | <p>I am so confused on how to find the domain of this function $3^\sqrt{x^2-3x}$ without graphing it. I have no idea what to do in this situation.</p>
| user7090 | 98,577 | <p>Let $f(x)= \sqrt{x^2-3x}$ and let $g(x) = 3^x$. </p>
<p>The function you describe can be written as, $(g \circ f)(x) = 3^{\sqrt{x^2-3x}}$.</p>
<p>Since $g(x)$ has domain every number on the $x$-axis, i.e $x \in \mathbb{R}$, you need only worry about those $x$ values in the domain of $f(x)$. </p>
<p>A square root ... |
87,648 | <pre><code>\[GothicCapitalR] = {{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, {{0, 1,
0}, {0, 0, 1}, {1, 0, 0}}, {{0, 0, 1}, {1, 0, 0}, {0, 1,
0}}, {{1, 0, 0}, {0, 0, 1}, {0, 1, 0}}, {{0, 0, 1}, {0, 1,
0}, {1, 0, 0}}, {{0, 1, 0}, {1, 0, 0}, {0, 0, 1}}};
i = 1;
j = 1;
det = 1;
a = Subsets[Range[6], {3}];
v = {x, y, z};
k = \... | Feyre | 7,312 | <p>Replace the Print[det] in the print with:</p>
<pre><code>Paste[det]
</code></pre>
|
87,648 | <pre><code>\[GothicCapitalR] = {{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, {{0, 1,
0}, {0, 0, 1}, {1, 0, 0}}, {{0, 0, 1}, {1, 0, 0}, {0, 1,
0}}, {{1, 0, 0}, {0, 0, 1}, {0, 1, 0}}, {{0, 0, 1}, {0, 1,
0}, {1, 0, 0}}, {{0, 1, 0}, {1, 0, 0}, {0, 0, 1}}};
i = 1;
j = 1;
det = 1;
a = Subsets[Range[6], {3}];
v = {x, y, z};
k = \... | Enrique Pérez Herrero | 27,225 | <pre><code>\[GothicCapitalR] = {{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, {{0, 1,
0}, {0, 0, 1}, {1, 0, 0}}, {{0, 0, 1}, {1, 0, 0}, {0, 1,
0}}, {{1, 0, 0}, {0, 0, 1}, {0, 1, 0}}, {{0, 0, 1}, {0, 1,
0}, {1, 0, 0}}, {{0, 1, 0}, {1, 0, 0}, {0, 0, 1}}};
i = 1;
j = 1;
det = 1;
a = Subsets[Range[6], {3}];
v = {x, y, z};
k = \... |
33,361 | <p>I have a function like this</p>
<p>$f(x,y)=c\,y\,(y-x),\ \text{for}\ 0<x<2,\;-x<y<x$</p>
<p>and I need to find the value of $c$ such that $f(x,y)$ is a PDF. </p>
<p>How can I do that? </p>
<p>I know that the condition is double integral $f(x,y)=1$, but how can I impose that condition with <em>Mathe... | Ray Koopman | 8,159 | <p>You forgot the prior condition that the density must not be negative.<br>
There are two solutions, depending on the sign of c. </p>
<p>If c is positive then <code>Reduce[y(y-x) >= 0 && 0 < x < 2 && -x < y < x, {x,y}]</code></p>
<blockquote>
<p>0 < x < 2 && -x < y ... |
4,157,288 | <p>High school student here just wanting to be better at maths and to know how to approach and solve problems and also how to think like a mathematician. Please recommend a book to me it would help a lot.</p>
| K7PEH | 175,104 | <p>When I was a senior in high-school (1964/65) I had the same question, the same desire to learn more of mathematics and how mathematicians think. My math teacher gave me a book that fulfilled my quest and I still have that book today. At the start, a lot of the book was beyond me but I would often refer to it again... |
1,264,353 | <blockquote>
<p>Evaluate the determinants given that $\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}=-6.$</p>
</blockquote>
<ol>
<li>$\begin{vmatrix} a+d & b+e & c+f \\ -d & -e & -f \\ g & h & i \end{vmatrix}$ </li>
<li>$\begin{vmatrix} a & b &am... | nbubis | 28,743 | <p>Geometrically, the fact that you can add multiples of rows to each other while keeping the determinant the same, is a reflection of the fact that the determinant can be seen as the volume of the parallelepiped with the rows or columns as it's vectors.</p>
<p>Adding a row to a row has the effect of simply skewing th... |
129,693 | <p><img src="https://i.stack.imgur.com/tFha8.png" alt="enter image description here"></p>
<p>I'm lost on what's happening here. This is regarding MinML( "an idealized programming language" ) . More pics below: Thank You Very Much</p>
<p><img src="https://i.stack.imgur.com/BYjTF.png" alt="enter image description here"... | Robert Israel | 8,508 | <p>There is no elementary antiderivative for either of those. </p>
<p>It's actually easier to deal with $e^{ix}/(1+x^2)$. As a corollary of a theorem of Liouville, if $f e^g$ has an elementary antiderivative, where $f$ and $g$ are rational functions and $g$ is not constant, then it has an antiderivative of the form $... |
423,938 | <p>I don't know if I apply for this case sin (a-b), or if it is the case of another type of resolution, someone with some idea without using derivation or L'Hôpital's rule? Thank you.</p>
<p>$$\lim_{x\to0}\frac{\sin(x^2+\frac{1}{x})-\sin\frac{1}{x}}{x}$$</p>
| Ian Mateus | 17,751 | <p>Using $\sin\left(a\right)-\sin\left(b\right)=2\sin\left(\frac{a-b}{2}\right)\cos\left(\frac{a+b}{2}\right)$ and $\cos\left(a+b\right)=\cos\left(a\right)\cos\left(b\right)-\sin\left(a\right)\sin\left(b\right)$: $$L=\lim_{x\rightarrow 0}\frac{2\sin\left(x^2/2\right)\cos\left(x^{-1}+x^2/2\right)}{x}\\=\lim_{x\rightarro... |
4,621,030 | <p>I now that this is wrong, but why?</p>
<p><span class="math-container">$$3x\log(2)+2x\log(3) = \log(6)$$</span>
<span class="math-container">$$3x\log(2)+2x\log(3) = \log(2*3)$$</span>
<span class="math-container">$$3x\log(2)+2x\log(3) = 1\log(2)+1\log(3)$$</span>
<span class="math-container">$$3xA+2xB = 1A+1B$$</spa... | Lelouch | 991,491 | <p>The general idea is to rewrite the polynomial <span class="math-container">$k^2$</span> as a sum of Hilbert's polynomial (i.e <span class="math-container">$X$</span>, or <span class="math-container">$X(X-1)$</span> , or <span class="math-container">$X(X-1)(X-2)$</span> etc ). With this idea you can compute any sum <... |
2,802,035 | <p>(a) All roots of $f(x)$ are real.<br>
(b) $f(x)$ has one real root and $2$ complex roots.<br>
(c) $f(x)$ has two roots in $(-1,1).$<br>
(d) $f(x)$ has at least one negative root.</p>
<p>I thought of solving this question using Descartes Rule. 'c' is negative and 'a' is turning out to be positive. 'b' should be nega... | fleablood | 280,126 | <p>Compare</p>
<p>$f(0) = c <0$</p>
<p>$f(1) = 1 + a + b+ c > 1 + -1 > 0$</p>
<p>$f(-1) = -1 +a - b + c > -1 + 1 > 0$.</p>
<p>So there is a root in $(-1,0)$ and in $(0, 1)$.</p>
<p>So that tells us b) is false, d) is true</p>
<p>Complex roots come in pairs and there are at most three roots so there... |
143,173 | <p>I have a small question that I think is very basic but I am unsure how to tackle since my background in computing inequalities is embarrassingly weak - </p>
<p>I would like to show that, for a real number <span class="math-container">$p \geq 1$</span> and complex numbers <span class="math-container">$\alpha, \beta$... | Davide Giraudo | 9,849 | <ul>
<li>The map $x\mapsto x^p$ for $x\geq 0$ is convex, since its second derivative is $p(p-1)x^{p-2}\geq 0$. </li>
<li>We have
$$\left|\frac{a+b}2\right|^p\leq \left(\frac{|a|+|b|}2\right)^p\leq \frac{|a|^p+|b|^p}2.$$</li>
</ul>
|
2,634,408 | <p>Here is the question:</p>
<p>Define the set WFF as follows--</p>
<p>a) Every element in set $A = \{s_1, s_2, ...\}$ is in WFF. $A$ is countably infinite.</p>
<p>b) If $a$ is in WFF, so is $(\neg a)$. If $n$ and $m$ are in WFF, so is $(n \lor m)$.</p>
<p>c) No other elements are in WFF.</p>
<p>Show that WFF is c... | William Elliot | 426,203 | <p>S = A $\cup$ {$\neg,\lor,$(,) } is a countable set of symbols.<br>
Sf, the set of finite strings of symbols from S, is countable.<br>
As WWF, the set of well formed formulas, is a subset of Sf,<br>
WWF is countable.</p>
|
4,274,290 | <p>I want to solve: <span class="math-container">$x^2\equiv 1 \pmod{20}, x^2\equiv 6 \pmod {15}, x^2\equiv 9\pmod{18}.$</span> This is a system of congruence equations, but these are not linear and moduli are not coprime. So,we cannot apply chinese remainder theorem here. However, I think I can solve for <span class="m... | Martin Argerami | 22,857 | <p>This fails even in the case <span class="math-container">$A=\mathbb C$</span> (and thus in every unital C<span class="math-container">$^*$</span>-algebra), by taking <span class="math-container">$a=-1$</span>.</p>
|
2,848,251 | <p>From: Philip Johnson-Laird <a href="https://dof.princeton.edu/about/clerk-faculty/emeritus/philip-nicholas-johnson-laird" rel="nofollow noreferrer">BA PhD Psychology (UCL)</a>, Stuart Professor of Psychology Emeritus at Princeton. (Author isn't a logician.) <a href="https://www.amazon.ca/How-We-Reason-Philip-... | Bram28 | 256,001 | <p>Here's a formal proof in Fitch:</p>
<p><a href="https://i.stack.imgur.com/PfOjZ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PfOjZ.png" alt="enter image description here"></a></p>
|
2,471,546 | <p>In order to find a shorter proof for this thread : </p>
<p><a href="https://math.stackexchange.com/questions/2470262/find-all-functions-fmfn2-1-fnfmn-fm-n/2470409?noredirect=1#comment5106478_2470409">Find all functions $f(m)[(f(n))^2-1]=f(n)[f(m+n)-f(m-n)]$</a></p>
<p><em>Rem: just to be clear, $\mathbb N^*$ stand... | marty cohen | 13,079 | <p>No.
It is fairly common.</p>
<p>An example is the
Fibonacci numbers.</p>
<p>(see <a href="https://en.wikipedia.org/wiki/Fibonacci_number#Relation_to_the_golden_ratio" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Fibonacci_number#Relation_to_the_golden_ratio</a>)</p>
<p>They satisfy</p>
<p>$F_n
=\dfrac... |
351,815 | <p>Having trouble understanding this. Is there anyway to prove it?</p>
| Jim | 56,747 | <p>An intuitive way to see this is to consider that you're trying to show
$$a^n < n!$$
for sufficiently large $n$. Take the log of both sides, you get
$$n\log(a) = \log(a^n) < \log(n!) = \sum_{i = 1}^n\log(i).$$
Now as you increase $n$ you only add $\log(a)$ to the left side, but the $\log(n + 1)$ that you add t... |
4,209,055 | <p>Let <span class="math-container">$S=\sum_{k=1}^{m}e^{2\pi ik^2/m}$</span>,if <span class="math-container">$m$</span> is odd,how to directly calculate the absolute value of <span class="math-container">$S=\sqrt{m}$</span>.Don't use Gauss sum since here it says "it's easily shown"<a href="https://i.stack.img... | Vishu | 751,311 | <p>Using the fact that <span class="math-container">$$\sum_{n=1}^L \cos(a+nd) = \csc\left(\frac d2 \right) \sin\left(\frac{dL}{2} \right) \cos\left(a+\frac d2(L+1)\right), $$</span> the inner sum can be evaluated exactly to reduce it down to <span class="math-container">$$- 2\sum_{k=1}^{m-1} \cot\left(\frac{2\pi k}{m} ... |
910,020 | <p>Show that if $G$ is a group of order $168$ that has a normal subgroup of order $4$ , then $G$ has a normal subgroup of order $28$.</p>
<p><strong>Attempt:</strong> $|G|=168=2^3.3.7$</p>
<p>Then number of sylow $7$ subgroups in $G = n_7 = 1$ or $8$.</p>
<p>Given that $H$ is a normal subgroup of order $4$ in $G$. <... | Timbuc | 118,527 | <p>Hints:</p>
<p>(a) If $\;H,K\;$ are subgroups of a group $\;G\;$ , then $\;HK:=\{hk\;:\; h\in H\,,\,k\in K\}\;$ is a subgroup of $\;G\;$ iff $\;HK=KH\;$</p>
<p>(b) If $\;H\;$ is a <strong>normal</strong> subgroup of $\;G\;$, then for <em>any</em> subgroup $\;K\;$ of $\;G\;$ we have that $\;HK=KH\;$</p>
<p>(c) In y... |
69,373 | <p>Let $N$ be the number of rolls until the same number appears $k$ consecutive times. Show the expected value $E[N]=\dfrac{6^k-1}{5}$. I've tried conditioning this on the first occurrence of the expected number, but I'm having a hard time generalizing further than 2 consecutive times. I think I need to use the conditi... | joriki | 6,622 | <p>You can prove this by induction. Let $a_k$ be the expected number of rolls until the same number appears $k$ consecutive times. Clearly $a_1=1=(6^1-1)/5$. Now assume $a_{k-1}=(6^{k-1}-1)/5$.</p>
<p>To get $k$ consecutive rolls with the same number, you first need to get $k-1$, and this is expected to take $a_{k-1}$... |
4,020,412 | <p>I was reading about ODE variable separation solving and the book says that assuming that a function can be expressed as the product of two single-variable functions loses generality, which I understand. I cannot, however, prove that it does. For example: the function <span class="math-container">$\sqrt{x+t}$</span> ... | Ian Zhang | 885,624 | <p>I think <span class="math-container">$\log$</span> and partial derivative can do it.</p>
<p><span class="math-container">$\frac{\partial}{\partial x}\log(\sqrt{x+t})=\frac{1}{2}\frac{\partial}{\partial x}\log(x+t)=\frac{1}{2(x+t)}$</span> is still related to <span class="math-container">$t$</span>.</p>
<p>If <span c... |
2,805,803 | <p>The question is as follows: </p>
<blockquote>
<p>A typical long-playing phonograph record (once known as an LP) plays for about $24$ minutes at $33 \frac{1}{3}$ revolutions per minute while a needle traces the long groove that spirals slowly in towards the center. The needle starts $5.7$ inches from the center an... | Théophile | 26,091 | <p><strong>Hint:</strong> Instead of a spiral, suppose the groove on the LP were made of $800$ concentric circles of equal width; this will make calculations easier and will provide a very accurate approximation.</p>
<p>Since the groove has a non-zero width, the circles aren't really circles; they're <a href="https://... |
44,082 | <p>I have basic questions about elliptic curves over finite fields. </p>
<ol>
<li><p>Where to find general references? Hartshorne for instance restricts to algebraically closed ground fields.</p></li>
<li><p>Over an arbitrary field $K$, is the right definition of an elliptic curve a smooth proper
curve of genus 1 with... | Leonardo | 14,809 | <p>I completely agree with the earlier answers. Just two remarks...</p>
<ul>
<li>For question 3, If $K$ is finite of cardinality $q$, then $E(K)$ is isomorphic to $\mathbf Z/n\mathbf Z\times\mathbf Z/m\mathbf Z$, where $n$ divides gcd$(q-1,m)$.</li>
<li>Concerning your last question, here is a simple example where you... |
2,531,538 | <p>I wish to show the following in equality:</p>
<p>$$\dfrac {n!}{(n-k)!} \leq n^{k}$$</p>
<p>Attempt:</p>
<p>$$\dfrac {n!}{(n-k)!} = \prod\limits_{i = 0}^{k -1} (n-i) = n\times (n-1)\times \cdots \times(n-({k-1})) $$</p>
<p>I am not sure how to make the argument that $n\times (n-1)\cdots \times (n-({k-1})) \leq n^... | Reiner Martin | 248,912 | <p>If $n-i\le n,$ then the product of $k$ of those is clearly $\le n^k.$</p>
|
981,181 | <p>I'm stuck with a logic problem like this</p>
<blockquote>
<p>I eat ice cream if I am sad.</p>
<p>I am not sad.</p>
<p>Therefore I am not eating ice cream.</p>
</blockquote>
<p>Is this conclusion logical? The first sentence can be understood both like "ice cream <span class="math-container">$\implies$</span> sad... | Ali Caglayan | 87,191 | <p>$P\implies Q$ does not necessarily mean $Q \implies P$</p>
<p>But what does this mean exactly?</p>
<p>Well think of it like this.</p>
<blockquote>
<p>All Carrots are vegetables</p>
</blockquote>
<p>but this doesn't mean that</p>
<blockquote>
<p>All vegetables are Carrots</p>
</blockquote>
|
73,219 | <p>For Riemann surfaces there are at least to possible notions of hyperbolicity. The classical one given by the Uniformization Theorem, or equivalently the type problem, which essentially says that a simply connected Riemann surfaces is conformally equivalent to one of the following:</p>
<ul>
<li>Riemann Sphere $\math... | R W | 8,588 | <p>I don't quite understand your question about surfaces as the notions of hyperbolicity you are talking about deal with two different structures: a Riemann surface is endowed with a conformal structure, whereas Gromov's hyperbolicity is a property of metric spaces (in particular, of Riemannian manifolds). </p>
<p>The... |
3,197,572 | <p>Suppose you have a 4-sided die, a 6-sided die, and a 12-sided die. You
roll the three dice and add up the numbers that show up. What is the expected value
of the sum of the rolls?</p>
<p>My attempt solution is using indicators. Here is the outline:</p>
<p>Let <span class="math-container">$I_A=\{4-sided\}$</span>, ... | Yocoxcanemitia | 480,875 | <p>Thanks to @Foobaz John, the variance is,</p>
<p><span class="math-container">$$Var(X)=Var(A)+Var(B)+Var(C)=\frac{(4-1)^2}{12}+\frac{(6-1)^2}{12}+\frac{(12-1)^2}{12}\approx12.92$$</span></p>
<p>by independence of variables.</p>
|
256,806 | <p>How can I prove or disprove that $\lim\limits_{n\to \infty} (n+1)^{1/3}−n^{1/3}=\infty$?</p>
<p>My guess is that it is false but I can't prove it.</p>
| André Nicolas | 6,312 | <p>Multiply top and (missing) bottom by $(n+1)^{2/3}+(n+1)^{1/3}n^{1/3}+n^{2/3}$.</p>
<p>We are exploiting the identity $x^3-y^3=(x-y)(x^2+xy+y^2)$. </p>
<p><strong>Another approach</strong> is to note that $(n+1)^{1/3}$ is a little less than $n^{1/3}+ \dfrac{1}{3n^{2/3}}$. To prove this, cube the last expression. ... |
256,806 | <p>How can I prove or disprove that $\lim\limits_{n\to \infty} (n+1)^{1/3}−n^{1/3}=\infty$?</p>
<p>My guess is that it is false but I can't prove it.</p>
| user1551 | 1,551 | <p>Let $f(x)=x^{1/3}$. By mean value theorem, $(n+1)^{1/3}-n^{1/3}=f'(c)=1/(3c^{2/3})\le1/(3n^{2/3})$ for some $c\in(n,\,n+1)$. Hence the difference goes to zero when $n\rightarrow\infty$.</p>
|
1,026,135 | <p>Let a, b be elements of a group G and H a normal subgroup of G. Is it true that $aH = bH$, then $a^{-1}H = b^{-1}H$? How can I prove this?</p>
| Matt Samuel | 187,867 | <p>This is true if $H$ is a normal subgroup. This is because there is a surjective homomorphism $f:G\to G/H$ with kernel $H$ such that $f(a^{-1})=f(a)^{-1}=f(b)^{-1}=f(b^{-1})$.</p>
|
2,314,744 | <p>Prove that there is no integer $n \geq 2$ for which $$\frac{3^n - 2^n}{n}$$ is an integer</p>
<p>I really don't know how to start with it except with the parity of n (n being even clearly doesn't make the fraction an integer)</p>
<p>Edit:
If I take n=pk for a prime p then all i get is to disprove the congruence 3... | Mudream | 221,395 | <p>Hint : try to pick Prime $p>3$ s.t. $n = pk$ and $\gcd(p-1, n) = 1$</p>
<p>The existence is below</p>
<blockquote class="spoiler">
<p> pick the smallest prime divisor $p$ of n, since $2, 3 \nmid 3^n - 2^n$, so $p>3$</p>
</blockquote>
<p>More steps and why pick $\gcd(p-1, n) = 1$ :</p>
<blockquote class="... |
895,325 | <p>I am dealing with some nice rings that are always isomorphic to some fairly nice quotient ring of a polynomial ring. A typical example is:</p>
<p>$$ \mathbb{C}[X,XY,XY^2] \cong \frac{\mathbb{C}[U,V,W]}{\langle V^2 - UW \rangle}. $$</p>
<p>I would like a nice way to write the Kahler differentials of such rings. Fo... | 122520253025 | 161,859 | <p>Firstly, multiplicative only means for relatively prime integers $m$ and $n$. The integers $m$ and $n$ can be expressed as a product of primes: $m = 2^{a_1} {p_2}^{a_2} {p_3}^{a_3} ... {p_n}^{a_n}$. $n = 2^{b_1}{q_2}^{b_2} {q_3}^{b_3} ... {q_n}^{b_n}$.Note that $p_i \not= q_i$ for any $i$ because m and n are relativ... |
1,598,695 | <p>Does someone know how to do the Fourier Transform of the signal </p>
<p>$$x(t) = t \cdot \frac{\sin^2(t)}{(\pi t)^2}$$</p>
<p>My first thought was:
$$x(t)= \frac{t}{\pi^2} \cdot \frac{\sin^2(t)}{t^2} = \frac{t}{\pi^2} \cdot \operatorname{sinc}^2(t)$$</p>
<p>and try it with the convolution:</p>
<p>$$X(j \omega) =... | Ben Grossmann | 81,360 | <p><strong>Hint:</strong> You split $x(t)$ in a non-helpful way. Instead, note that
$$
x(t) = \frac{1}{\pi^2} \sin t \cdot \operatorname{sinc}(t)
$$
Now, $x$ is the product of two functions whose Fourier transform you may compute.</p>
|
80,922 | <p>I am generally confused about integer valued polynomials, and how to count them. Trying to learn the subject I started by listing permutations of the rows in a lower triangular table:</p>
<p>$$\displaystyle T = \left(\begin{matrix} 1&0&0&0&0&0&0&\cdots \\ 1&1&0&0&0&0&... | 2012rcampion | 21,750 | <p>Another option for you is to use <code>ParametricNDSolve</code>, which numerically solves differential equations with one or more parameters.</p>
<pre><code>soln = ParametricNDSolveValue[
{y'[x] == 1/2 (1 - y[x]) y[x] (a - y[x] + a y[x]), y[0] == y0}, (* your equations*)
y, (* the expression you want to be return... |
3,179,874 | <p>This question came up on a recent linear algebra exam of mine, and it's been bothering me ever since. The group is defined such that every element plus the identity matrix is invertible:</p>
<p><span class="math-container">$$(G,\ast):=\{\textbf{A}\in G | \textbf{A}+\textbf{E}_{n} \text{ is invertible}\}$$</span></p... | user1551 | 1,551 | <p>You may first prove that the neutral element <span class="math-container">$E$</span> is <span class="math-container">$0$</span> by solving the equation <span class="math-container">$A\ast E=A$</span>. Then you may find <span class="math-container">$B=A^{-1}$</span> by solving <span class="math-container">$A\ast B=E(... |
1,341,231 | <p>Let $f$ be continuous on $[0,1]$, and let $\alpha>0$. Find: $\lim\limits_{x\to 0}{x^{\alpha}\int_{x}^{1}{f(t)\over t^{\alpha +1}}dt}$. I tried integration by parts, but I am not sure if $f$ is integrable and to what extent. It also gets really messy. Besides, I am not sure if I am to express the limit using $f$, ... | Tim Raczkowski | 192,581 | <p>One of the best is <em>An Introduction to the Theory of Numbers</em> by Niven, Zuckerman, and Montgomery.</p>
|
1,341,231 | <p>Let $f$ be continuous on $[0,1]$, and let $\alpha>0$. Find: $\lim\limits_{x\to 0}{x^{\alpha}\int_{x}^{1}{f(t)\over t^{\alpha +1}}dt}$. I tried integration by parts, but I am not sure if $f$ is integrable and to what extent. It also gets really messy. Besides, I am not sure if I am to express the limit using $f$, ... | Community | -1 | <p>I recommend <a href="https://global.oup.com/academic/product/number-theory-9780198846734?cc=ca&lang=en&" rel="nofollow noreferrer"><em>Number Theory: Step by Step</em> published in Dec. 2020 by Kuldeep Singh</a> for two reasons. It provides solutions <a href="https://maths-for-all.co.uk/linear-algebra/" rel=... |
970,872 | <p>There is a common brain teaser that goes like this:</p>
<p>You are given two ropes and a lighter. This is the only equipment you can use. You are told that each of the two ropes has the following property: if you light one end of the rope, it will take exactly one hour to burn all the way to the other end. But it d... | Barry Cipra | 86,747 | <p>You say "Therefore, in order to arrive at a rope that burns in time $T/2$, one would need to light each uniform segment on both ends, not simply the end of both ends of the total rope. What am I doing wrong?"</p>
<p>If I understand what you mean here, the idea is that you cut the rope into short segments, each one ... |
970,872 | <p>There is a common brain teaser that goes like this:</p>
<p>You are given two ropes and a lighter. This is the only equipment you can use. You are told that each of the two ropes has the following property: if you light one end of the rope, it will take exactly one hour to burn all the way to the other end. But it d... | Mike S | 340,069 | <p>If you light both ends of the rope, it will burn in 30 minutes. Think about it this way: if the left half of the rope takes 40 minutes to burn, then the right half should take 20 minutes, totaling 60 minutes. Now, let's light both ends. After 20 minutes, the right half of the rope is gone, but the left half has only... |
3,146,447 | <p>I have a function <span class="math-container">$y=1/(1+e^{-x})$</span>. I have been asked to use the first derivative to find any stationary points and then use the second derivative to classify them and provide points of inflection.</p>
<p>When I derive the function, I get the result <span class="math-container">$... | Michael Rybkin | 350,247 | <p>The first derivative is defined for all values of x which means you can get a y value for any x value that you plug in. Don't confuse "undefined" and "has no roots". The fact that the second derivative is never zero simply means that the function does not have any stationary points (you can't draw a horizontal tange... |
37,977 | <p>I have two lists, say <code>a</code> and <code>b</code>, both of length <code>n</code>. I'd like to compute the following:</p>
<ul>
<li>minimum of $a[i]/b[i]$ where $i=1, 2, ...n$ and $b[i]>0$</li>
</ul>
<p>I'd also like to know the index of the element where the min occurs.</p>
| bill s | 1,783 | <p>One way to enforce the positivity condition on <code>b</code> is to locate the positions of all the positive elements of <code>b</code> and use those to index into the division of <code>a</code> by <code>b</code>.</p>
<pre><code>a = RandomReal[{-10, 10}, 10];
b = RandomReal[{-10, 10}, 10];
pos = Flatten[Position[b,... |
37,977 | <p>I have two lists, say <code>a</code> and <code>b</code>, both of length <code>n</code>. I'd like to compute the following:</p>
<ul>
<li>minimum of $a[i]/b[i]$ where $i=1, 2, ...n$ and $b[i]>0$</li>
</ul>
<p>I'd also like to know the index of the element where the min occurs.</p>
| Kuba | 5,478 | <pre><code>n = 10000;
{a, b} = RandomReal[{-1, 1}, {2, n}];
</code></pre>
<p>This may be quite fast:</p>
<pre><code>r = a/b; Position[r, min = Min@Pick[r, Positive@b]]
min
</code></pre>
<blockquote>
<pre><code>{{7955}}
-850.273
</code></pre>
</blockquote>
|
37,977 | <p>I have two lists, say <code>a</code> and <code>b</code>, both of length <code>n</code>. I'd like to compute the following:</p>
<ul>
<li>minimum of $a[i]/b[i]$ where $i=1, 2, ...n$ and $b[i]>0$</li>
</ul>
<p>I'd also like to know the index of the element where the min occurs.</p>
| ubpdqn | 1,997 | <p><strong>EDIT</strong></p>
<p>After Kuba appropriate comment (and using his code for picking Min) to deal with positive denominator:</p>
<pre><code>f[a_, b_] := Quiet[{s = Position[a/b, Min@Pick[a/b, Positive@b]],
Extract[(a/b), First@s]}]
</code></pre>
<p>Note test case with two cases:</p>
<pre><code>f[{1, 2,... |
3,775,508 | <p>Given <span class="math-container">$x \in \Bbb R$</span> and</p>
<p><span class="math-container">$$P = \begin {bmatrix}1&1&1\\0&2&2\\0&0&3\end {bmatrix}, \qquad Q=\begin {bmatrix}2&x&x\\0&4&0\\x&x&6\end {bmatrix}, \qquad R=PQP^{-1}$$</span></p>
<p>show that</p>
<p><spa... | Klaus | 635,596 | <p>You used multilinearity incorrectly. It should be</p>
<p><span class="math-container">$$\left|\begin {array}&2&x&x\\0&4&0\\x&x&6\end {array}\right|=\left|\begin {array}&2&x&x\\0&4&0\\x&x&5\end {array}\right|+\left|\begin {array}&2&x&x\\0&4&0... |
3,775,508 | <p>Given <span class="math-container">$x \in \Bbb R$</span> and</p>
<p><span class="math-container">$$P = \begin {bmatrix}1&1&1\\0&2&2\\0&0&3\end {bmatrix}, \qquad Q=\begin {bmatrix}2&x&x\\0&4&0\\x&x&6\end {bmatrix}, \qquad R=PQP^{-1}$$</span></p>
<p>show that</p>
<p><spa... | Cesareo | 397,348 | <p>Hint.</p>
<p><span class="math-container">$$
\det\left|\begin {array}&2&x&x\\0&4&0\\x&x&6\end {array}\right|=\det\left|\begin {array}&0&0&4\\2&x&x\\x&6&x\end {array}\right| = \det\left|\begin {array}&0&0&4\\2&x&x\\x&5&x\end {array}\r... |
18,861 | <p>I hope I am asking my question in the right forum.</p>
<p>I am trying to introduce some mathematical problems (Better to be famous in the math community) to a group of senior high school students with a typical background in high school mathematics like (differentiation and applications - basic probability- basic pl... | Will Orrick | 1,737 | <p>Euler conjectured that there are no <a href="https://en.wikipedia.org/wiki/Mutually_orthogonal_Latin_squares" rel="nofollow noreferrer">Graeco-Latin squares</a> of size congruent to <span class="math-container">$2$</span> mod <span class="math-container">$4$</span>. He sought a proof of this for size <span class="ma... |
1,072,750 | <p>I'm looking at this solution to this problem:</p>
<p><img src="https://i.stack.imgur.com/qQTTU.png" alt="enter image description here"></p>
<p>I'm getting thrown off by the special case where $n = 2$. If $n = 2$, why must it be that $x = 1$? All that we then know is that $x^2 = 1$ or that $x = x^{-1}$. However, I ... | Aaron Maroja | 143,413 | <p>Because as $n=2$ we have that $x^2 = x$ then $x(x-1) = 0$ and $x$ is not zero. In a integral Domain what happens to $x$? </p>
|
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