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 |
|---|---|---|---|---|
3,240,240 | <blockquote>
<p>If a right circular cone has three mutually perpendicular generators
then find its semi-vertical angle.</p>
</blockquote>
<p>We see that if <span class="math-container">$ax^2+by^2+cz^2+2fyx+2gzx+2hxy=0$</span> has three mutually perpendicular generators, then <span class="math-container">$a+b+c=0$<... | Angina Seng | 436,618 | <p>If the generators are in the directions of the usual coordinate
axes, then the axis of the cone is in the direction of the vector <span class="math-container">$(1,1,1)$</span>.
So the semi-vertical angle is the angle between the vectors <span class="math-container">$(1,0,0)$</span> and <span class="math-container">$... |
265,037 | <p>Let $\mathbb{F}_p$ be a finite field, $A=\{a_1,\dots,a_k\}\subset\mathbb{F}_p^*$ a $k$-element set, for $k<p$. $\mathfrak{S}_k=$permutation gp.</p>
<blockquote>
<p><strong>Question.</strong> Is it true there is always a $\pi\in\mathfrak{S}_k$ such that the following are pair-wise distinct in $\mathbb{F}_p$?
... | Seva | 9,924 | <p>It took me some effort to find the references you were requesting in your comment, but here they are eventually: </p>
<ul>
<li><p><a href="https://mathoverflow.net/questions/164300/ordering-subsets-of-the-cyclic-group-to-give-distinct-partial-sums/203111#comment503832_203111">Ordering subsets of the cyclic group to... |
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... | QuantumSpace | 661,543 | <p>Hint: If <span class="math-container">$A = C(X)$</span> for some compact Hausdorff space, then the inequality is obvious.</p>
<p>If <span class="math-container">$A$</span> is a general unital <span class="math-container">$C^*$</span>-algebra, note that the <span class="math-container">$C^*$</span>-algebra generated ... |
647,757 | <blockquote>
<p>How many integers $n$ are there such that $\sqrt{n}+\sqrt{n+7259}$ is an integer? </p>
</blockquote>
<p>No idea on this one.</p>
| imranfat | 64,546 | <p>PERHAPS this is one approach:<br>
Let $n=t^2$ that takes care of the first square root. Then in the second square root we get $t^2+7259$. Set this square root term equal to, say $v^2$ so that we end up with $v^2-t^2=7259$ or $(v-t)(v+t)$=$7259$. Find all factors of $7259$ (which is finite) and figure out possible va... |
351,815 | <p>Having trouble understanding this. Is there anyway to prove it?</p>
| Mark McClure | 21,361 | <p>Another possibility is to use the ratio test. Then, it's easy to make the argument rigorous and to get a sense of the relative sizes of $a^n$ and $n!$. Let $x_n = a^n/n!$, then</p>
<p>$$\frac{x_{n+1}}{x_n} = \frac{a^{n+1}}{(n+1)!}\frac{n!}{a^n} =
\frac{a\,a^n}{a^n}\frac{n!}{(n+1)n!} = \frac{a}{n+1}.$$</p>
<p>Si... |
351,815 | <p>Having trouble understanding this. Is there anyway to prove it?</p>
| Mickey | 82,157 | <p>A simple visual with no fancy proof. </p>
<p>Let $n = 100$.</p>
<p>$2^n = 2\times2\times2\times2\times2\times2\times\dots \times 2$ <-- the 100th "$2$"</p>
<p>$n! = 1 \times2\times3\times4\times5\times6\times\dots\times 100$ </p>
<p>See above after the 4th multiplication $2^n$ (i.e., $2^4$) = $16$ and $4! =... |
389,269 | <p>Let <span class="math-container">$G$</span> be a nonabelian finite simple group all of whose Sylow subgroups of odd order are cyclic.</p>
<p>If we further assume that its Sylow <span class="math-container">$2$</span>-subgroup is dihedral, then due to Suzuki, we know that <span class="math-container">$G\cong \operato... | Geoff Robinson | 14,450 | <p>Such simple groups are a subset of the finite simple "thin" groups, and the latter have been classified (by Michael Aschbacher in 1976 and 1978). A <span class="math-container">$2$</span>-local subgroup of a finite group is the normalizer of some non-trivial <span class="math-container">$2$</span>-subgroup... |
2,729,116 | <p><strong>Question 1.</strong> Let $V=\mathbb{R}^3$, $T:V \rightarrow V$ be linear. Suppose that $T^3=T, T^2 \neq T, T^2 \neq Id,$ and $\dim \ker T = 2.$ Show that the matrix of $T$ with respect to some basis is</p>
<p>$$
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & -1 \\
\end{pmatr... | Disintegrating By Parts | 112,478 | <p>You are given that $T(T-I)(T+I)=0$, $T(T-I)\ne 0$ and $(T-I)(T+I)\ne 0$.</p>
<ul>
<li><p>Because $T(T-I)\ne 0$ and $(T+I)T(T-I)=0$, then $-1$ is an eigenvalue of $T$.</p></li>
<li><p>Because $(T-I)(T+I)\ne 0$ and $T(T-I)(T+I)=0$, then $0$ is an eigenvalue of $T$.</p></li>
</ul>
<p>The dimension of $\mbox{ker}(T)$ ... |
156,321 | <p>I have data that represents a cyclist’s power output in 1-second intervals, sampled while the cyclist was working to a prescribed training regime.</p>
<p>I want to find sequences within that data where the cyclist sustained a power output (plus or minus a certain percentage threshold) for fixed lengths of time. So:... | jiaoeyushushu | 35,486 | <pre><code>SeedRandom[5]; list = RandomInteger[{1, 5}, 400];
list[[##]] & /@
Select[Split[
Catenate@Position[list, n_ /; 3 < n < 7], #2 - #1 == 1 &],
Length@# == 5 &]
</code></pre>
|
507,133 | <p>I came across a problem in a book that asked us to find the first number $n$ such that $\phi(n)\geqslant 1,000$ it turns out that the answer is 1009, which is a prime number. There were several questions of this type, and our professor conjectured that it will always be the next prime. However, no one has been able ... | RghtHndSd | 86,816 | <p>Slightly too long for a comment.</p>
<p>One can create arbitrarily long series of integers which are not primes. Let $n \in \mathbb{N}$ and consider such a series $n, n+1, \dots, n+r$ where $n+r$ is the first prime above $n$. Suppose one can find a number $n+s$ with $s < r$ and $n+s = pq$ where $p$ and $q$ are p... |
482,793 | <p>As title says, how does one show that a function is continuous over some interval (let us say over some interval of real numbers?)</p>
<p>Would(Can) this involve derivative?</p>
| Mikasa | 8,581 | <p>Basically, you have to show that:</p>
<ul>
<li><p>$f(x)$ is continuous on $(a,b)$, i.e; $\forall x_0\in(a,b), \lim_{x\to x_0}f(x)=f(x_0)$</p></li>
<li><p>$\lim_{x\to a^+}f(x)=f(a),~~\lim_{x\to b^-}f(x)=f(b)$</p></li>
</ul>
<p>I assume you aregoing to show that if $f(x)$ is continuous on $[a,b]\subset\mathbb R$.</p... |
2,819,950 | <p>I am a university graduate with a B.S. in mathematics who has been developing software for the past 8 years. I recently discussed a mutual interest in topology with a friend who is just about to complete his degree. Due to time constraints both of us missed our chance to take a topology course during our undergradua... | Chris Leary | 2,933 | <p>Munkres in an excellent choice. You might also want to consider Willard's "General Topology." Two inexpensive options worth mentioning are Gemignani, "Elementary Topology" and Mendelson, "Introduction to Topology." Both are available from Dover press and are probably at a somewhat more basic level than Willard or Mu... |
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... | Alex B. | 35,416 | <p>As Xandi Tuni said, most of the answers to your questions can be found in standard references.</p>
<ol>
<li><p>Silverman, Knapp's book on elliptic curves, Milne's book, many more (just google for elliptic curves).</p></li>
<li><p>Yes</p></li>
<li><p>The number of points is bounded by the Hasse-bound. Within that bo... |
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... | Sebastian Petersen | 8,680 | <p>1) Silverman's book mentioned above is surely a very good reference. For the first steps you can also consider Silverman-Tate "Rational points on elliptic curves". As you already use Hartshorne, I assume that you are familiar with basic notations in algebraic geometry and scheme theory. So you may wish to switch to ... |
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})) $$</p>
<p>I am not sure how to make the argument that $n\times (n-1)\cdots \times (n-({k-1})) \leq n^... | Donald Splutterwit | 404,247 | <p>\begin{eqnarray*}
\underbrace{n(n-1) \cdots (n-k+1)}_{ k \text{ terms}} \underbrace{ \leq}_{ n-i \leq n} n^k.
\end{eqnarray*}</p>
|
2,246,889 | <p>$\displaystyle \lim_{\substack{x \rightarrow 4 \\ y \rightarrow \pi}} x^2 sin \frac{y}{x}$</p>
<p>I don't know in which way should I try to solve this example, can maybe the single variable limit of $sin \frac{1}{x}$ help me ?</p>
| haqnatural | 247,767 | <p>$$\lim _{ { x\rightarrow 4\\ y\rightarrow \pi } } x^{ 2 }sin\frac { y }{ x } =16\sin { \frac { \pi }{ 4 } } =16\frac { \sqrt { 2 } }{ 2 } =8\sqrt { 2 } $$</p>
|
293,257 | <blockquote>
<p>How many arrangements are there of TINKERER with two, but not three consecutive vowels?</p>
</blockquote>
<p><strong>Solution:</strong></p>
<ul>
<li>Possible number of pairs: 3 (EE, EI, IE)</li>
<li>Number of arrangements for end cases (when the pair is at end or beginning): $2\cdot5\cdot\frac{5!}{2... | André Nicolas | 6,312 | <p>Good cases analysis: your solution and answer are correct. A small modification may be a little easier. Start by pretending the letters are all different.</p>
<p>By a cases analysis like yours, there are $30$ ways to choose the <strong>slots</strong> the vowels will go into. There are then $3!$ ways to fill these s... |
293,257 | <blockquote>
<p>How many arrangements are there of TINKERER with two, but not three consecutive vowels?</p>
</blockquote>
<p><strong>Solution:</strong></p>
<ul>
<li>Possible number of pairs: 3 (EE, EI, IE)</li>
<li>Number of arrangements for end cases (when the pair is at end or beginning): $2\cdot5\cdot\frac{5!}{2... | bryan.blackbee | 45,767 | <p>I would like to propose a different approach to this question - Consider if you arrange like this $$ T \space N\space K\space R\space R$$
There are, indeed $\frac {5!}{2!}$ ways to arrange, with $2!$ to account for double-counting $R$.<br/><br/>
You mentioned that there are $6$ slots, which is correct. You can choo... |
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>, ... | Sri-Amirthan Theivendran | 302,692 | <p>The variables aren't indicator functions. Let <span class="math-container">$A\sim \text{Unif}\{1,2,3,4\}$</span>, <span class="math-container">$B\sim \text{Unif}\{1,2,3,4, 5, 6\}$</span> and <span class="math-container">$C\sim \text{Unif}\{1,2,\dotsc, 12\}$</span> be uniform random variables on their respective sets... |
2,602,438 | <p>For an embedded software implementation I would like to compute,</p>
<p>$S(b) = \sum_{i=1}^{N}log( x_i - b )$,</p>
<p>for various values of $b$. Here $x_i$ is an array of fixed numbers.</p>
<p>Is there a fast way to do this without having to recompute the sum?</p>
<p>--</p>
<p>I tried looking at the Taylor expa... | ziggurism | 16,490 | <p>If $x^2\not\geq0$ then $x$ is not a real number.</p>
|
2,190,394 | <p><a href="https://i.stack.imgur.com/X1WFP.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/X1WFP.jpg" alt="enter image description here"></a></p>
<p>The required angle can be found out by using cosine rule and then we get it as be $90^{\circ}$.
Can someone provide a bit more geometric approach? Or ... | Michael Biro | 29,356 | <p>Use the Pythagorean theorem to see that $AD = BC = 2$ and then argue that triangles $ADC$ and $ABC$ are congruent. That means that the angle $ADC$ is equal to angle $ABC$.</p>
|
2,190,394 | <p><a href="https://i.stack.imgur.com/X1WFP.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/X1WFP.jpg" alt="enter image description here"></a></p>
<p>The required angle can be found out by using cosine rule and then we get it as be $90^{\circ}$.
Can someone provide a bit more geometric approach? Or ... | quasi | 400,434 | <p>By the Pythagorean Theorem in $\Delta BDC$,
$$BC^2 = BD^2+DC^2 = 1^2 + \left(\sqrt{3}\right)^2=4$$</p>
<p>By the Pythagorean Theorem in $\Delta ABC$,
$$AC^2 = AB^2 + BC^2 = \left(\sqrt{3}\right)^2 + 4 = 7$$</p>
<p>By the Pythagorean Theorem in $\Delta ABD$, </p>
<p>$$AD^2 = AB^2 + BD^2 = \left(\sqrt{3}\right)^2 +... |
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>
| Raúl Alegre | 185,587 | <p>It is <strong>not true</strong>.</p>
<p>However...
$$aH=bH\ \Longleftrightarrow \ (b^{-1}a)H=H \ \Longleftrightarrow \ b^{-1}a\in H \ \Longleftrightarrow \ H=H(b^{-1}a) \ \Longleftrightarrow \ Ha^{-1} = Hb^{-1}$$</p>
<p>Then</p>
<blockquote>
<p>$$aH=bH\ \Longleftrightarrow \ Ha^{-1} = Hb^{-1}$$</p>
</blockquote... |
1,722,256 | <p>I'm trying to calculate the percentage difference $\Bbb{P}$ between two numbers, $x$ and $y$. I'm not given much context about how these numbers are different (for example, if one is "expected outcome" and one is "observed outcome").</p>
<p>When is each of this formulas relevant?</p>
<p>$$\Bbb{P} = \frac{|x-y|}{x}... | jameselmore | 86,570 | <p>The word 'change' implies a beginning state, and an ending state. If we have an observation $x$, that is then realized as $y$ in the future, then the percentage change of $x$ to $y$ would be $$100\%\cdot \frac{y-x}{x}$$</p>
|
1,598,695 | <p>Does someone know how to do the Fourier Transform of the signal </p>
<p>$$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) =... | Mark Viola | 218,419 | <p><strong>HINTS:</strong></p>
<p>From the <a href="https://en.wikipedia.org/wiki/Convolution_theorem" rel="nofollow">Convolution Theorem</a>, we have</p>
<p>$$\int_{-\infty}^\infty f(t)g(t)\,e^{i\omega t}\,dt=\frac{1}{2\pi}\int_{-\infty}^\infty F(\omega-\omega')G(\omega')\,d\omega'$$</p>
<p>Setting $f(t)=g(t)=\frac... |
296,259 | <p>My friends argue this $d_t( \partial_{\dot{x}} g)=1+2\dot{\dot{x}} \not = \partial_t (\partial_{\dot{x} }g)$ where $g=t\dot x + x^2 + \dot{x}^2$. Why?</p>
| Community | -1 | <p>Because they are derivatives of different functions. </p>
<p><strong>Example $g(x,t)=t x^2$</strong></p>
<blockquote>
<p>When $x$ is itself a function of $t$, we have two options: </p>
<ul>
<li>hold $x$ constant and take the partial derivative of $g(x,t)=tx^2$ with respect to $t$. This gives $\partial_t ... |
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>$$\displaystyle T = \left(\begin{matrix} 1&0&0&0&0&0&0&\cdots \\ 1&1&0&0&0&0&... | Michael E2 | 4,999 | <p>To create a function of the type you ask for, you can do the following. Note the use of <a href="http://reference.wolfram.com/language/ref/Set.html" rel="noreferrer"><code>Set</code></a> and <a href="http://reference.wolfram.com/language/ref/DSolveValue.html" rel="noreferrer"><code>DSolveValue</code></a>.</p>
<pre... |
1,396,045 | <p>$$\int{\sqrt{\frac{(1-\sin x)(2-\sin x)}{(1+\sin x)(2+\sin x)}}dx}$$</p>
<p>$$\int\sqrt{\frac{(1-\sin x)(2-\sin x)}{(1+\sin x)(2+\sin x)}}dx=\int \frac{(1-\sin x)(2-\sin x)}{\sqrt{(1-\sin x)(2-\sin x)(1+\sin x)(2+\sin x)}}dx$$</p>
<p>I am stuck. Please help me....</p>
| Empty | 174,970 | <p><strong>Hint:</strong></p>
<p><span class="math-container">$$\int\sqrt{\frac{(1-\sin x)(2-\sin x)}{(1+\sin x)(2+\sin x)}}dx$$</span></p>
<p><span class="math-container">$$=\int\frac{\cos x\sqrt{4-\sin^2 x}}{(1+\sin x)(2+\sin x)}\,dx$$</span></p>
<p>( multiplying numerator & denominator by <span class="math-conta... |
286,989 | <p>There is a very simple formulation for the character of irreducible representations of $S_n$ evaluated on an n-cycle, i.e. that it is 0 on all non-hook partitions, and $(-1)^m$ on hooks. Is there an analogous computation for irreducible characters of $B_n$, the hyperoctahedral group, evaluated on signed 2n-cycles? T... | Dima Pasechnik | 11,100 | <p>Did you try small values of $n$? A quick examination of the character table (computed using GAP, say) for $5\leq n\leq 9$ shows that in this range there always is a $2n$-class with character values $0,\pm 1$.
(For $n$ odd it is unique, so one does not even need to check that this is the class you are looking at in t... |
3,794,148 | <p><span class="math-container">$n$</span> is fixed.
<br />
<span class="math-container">$ x \in ]0,1[$</span>
<br />
Coud you help me prove, without theorem of dominated convergence :
<span class="math-container">$$ \int_{1}^{n} \frac{x^t}{t} dt \underset{x \to 1^{-}}{\rightarrow} \int_{1}^{n} \frac{1}{t} dt$$</span><... | Oliver Díaz | 121,671 | <p>It is enough to show that <span class="math-container">$x^t\xrightarrow{x\rightarrow1-}1$</span> uniformly in <span class="math-container">$t\in[1,n]$</span>.</p>
<p>One my use the mean value theorem: for all <span class="math-container">$|x-1|<\frac12$</span>,
<span class="math-container">$$|x^t-1|=|e^{t\log(x)... |
43,690 | <p>I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of question and since it is worrying me increasingly as of late I feel that I must ask it.</p>
<p>My question is: what ca... | anonymous | 10,073 | <p>I like the cannon fodder analogy -- made me laugh! I think of myself that way sometimes. Also, though, deep down I like to think I might be smarter than I realize. If I keep at it, maybe I will tap into some hidden reserves of insight. Also, remember the line from Ecclesiastes:</p>
<p>"I returned, and saw under the... |
43,690 | <p>I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of question and since it is worrying me increasingly as of late I feel that I must ask it.</p>
<p>My question is: what ca... | Gerhard Paseman | 3,402 | <p>Mathematics needs people to create, to explain, to synthesize, to apply, to teach, to learn, even to proselytize (in socially acceptable ways). If you want fame bordering on immortality, solve a very hard problem or create something that both solves and poses hard problems. If you want instead to be a great contri... |
43,690 | <p>I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of question and since it is worrying me increasingly as of late I feel that I must ask it.</p>
<p>My question is: what ca... | porton | 4,086 | <p>@muad: You say you are not strong in mathematics. I don't know whether this is really such. But let's assume that you cannot do math discoveries for the purpose of this discussion.</p>
<p>You may do something else.</p>
<p>You may consider to write software for mathematics. One such project is <a href="http://www.t... |
275,517 | <p>Some mathematical concepts are ended with "-oid", such as Matroid, greedoid, groupoid. What does that mean? Do these concepts share something in common? Thanks!</p>
| Elias Costa | 19,266 | <p>Means "bed", "incomplet","deficient" or "weak". For exemple: Grupoid is a algebric struture "deficient" that is quase a Group. </p>
|
1,961,614 | <p>I can't say I've gotten very far. You can show $3 + \sqrt{11}$ is irrational, call it $a$. Then I tried supposing it's rational, i.e.:</p>
<p>$a^{1/3}$ = $\frac{m}{n}$ for $m$ and $n$ integers.</p>
<p>You can write $m$ and $n$ in their canonical factorizations, then cube both sides of the equation...but I can't s... | KonKan | 195,021 | <p>If $(3 + \sqrt{11})^\frac{1}{3}=\frac{p}{q}$ were a rational, then $3 + \sqrt{11}=\frac{p^3}{q^3}$ would also be a rational. </p>
|
2,948,557 | <p>So I've been trying to prove that
<span class="math-container">$$\sum^{\infty}_{n=1}\frac{(-1)^{n}\sin(n)}{n}=-\frac{1}{2}$$</span></p>
<p>I've tried putting various bounds on it to see if I can "squeeze" out the result. Say something like (one of many tried examples):
<span class="math-container">$$ -\frac{1}{n}-\... | abc | 602,272 | <p>Its is also possible to calculate the series above using the residue theorem from complex analysis. It goes as follows:<span class="math-container">$$\sum_{i=-\infty}^\infty \text{Res}\left(\pi\csc(\pi z)\cdot\frac {\sin(z)}z,n\right)=-\text{Res}\left(\pi\csc(\pi z)\cdot\frac {\sin(z)}z,0\right)$$</span></p>
<p>The... |
4,567,552 | <p>Calculate the residues of <span class="math-container">$\frac{1}{z^4+1}$</span> in terms of it's poles. What I've done is reduce <span class="math-container">$(x-x^3)(x-x^5)(x-x^7)$</span> mod <span class="math-container">$(x^4+1)$</span> to get <span class="math-container">$4x^3$</span>. So we get <span class="math... | Ennar | 122,131 | <p>Let's start from definition of residue. Given Laurent series of <span class="math-container">$f(z) = \sum_n a_n(z-c)^n$</span> around <span class="math-container">$c$</span>, the residue is simply the coefficient <span class="math-container">$a_{-1}$</span>. Now, let's say that <span class="math-container">$f$</span... |
2,419,922 | <p>An involution is a permutation $P$ which is its own inverse: $P\cdot P = \text{id}$.</p>
<p>Every permutation can be written (in various ways) as a sequence of single-element swaps (transpositions). Sometimes these sequences are palindromes, in that the reversal of the sequence is the same sequence.</p>
<p>If $T$ ... | Angina Seng | 436,618 | <p>Suppose we have a product of transpositions $\tau=\pi_1\pi_2\ldots\pi_n$
with $\pi_j=\pi_{n+1-j}$ for all $j$.</p>
<p>If $n=2m$ is even then
$$\tau=\pi_1\cdots\pi_m\pi_m\cdots\pi_1$$
which cancels off from the middle to give the identity.</p>
<p>If $n=2m+1$ is odd then
$$\tau=\pi_1\cdots\pi_m\pi_{m+1}\pi_m\cdots\p... |
3,380,479 | <p>i'm doing a seminar about Galois Theory but I have a problem with the definition of purely inseparable element and the one of purely inseparable extension, I read the definitions but i would like to see an example for each one of them. If somebody can help me please. </p>
| J. W. Tanner | 615,567 | <p><span class="math-container">$2$</span> is a primitive root modulo <span class="math-container">$13;$</span> the consecutive powers of <span class="math-container">$2$</span> are <span class="math-container">$1, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7,$</span> and <span class="math-container">$1\mod 13$</span>. From her... |
541,541 | <blockquote>
<p>Assume that $f_1\colon V_1\to W_1, f_2\colon V_2\to W_2$ are $k$-linear maps between $k$-vector spaces (over the same field $k$, but the dimension may be infinity). Then the tensor product $f_1\otimes f_2\colon V_1\otimes V_2\to W_1\otimes W_2$ is defined, and it's obvious that $\ker f_1\otimes V_2+ V... | Martin Brandenburg | 1,650 | <p>Yes, that's true. Let $f_i : V_i \to W_i$ be two linear maps. Since $\mathrm{im}(f_1) \otimes \mathrm{im}(f_2)$ embeds into $W_1 \otimes W_2$, we may assume that $f_1,f_2$ are surjective. But then they are split, so that we can assume that $V_i = W_i \oplus U_i$ and that $f_i$ equals the projection $V_i \to W_i$, wi... |
904,547 | <p>I have the sum ( $M$ is any integer $> 1$ ):</p>
<p>$$
\sum_{h = 1}^{M}\left(\,\left\lfloor\, 2M + 1 \over h\,\right\rfloor
-\left\lfloor\, 2M \over h\,\right\rfloor\,\right)
$$</p>
<p>and looking for a way to simplify it. In the sense of either finding a <strong>simple closed form or a good approximation for M... | UserX | 148,432 | <p>Using the method of disks(<span class="math-container">$V=π\int_{\alpha}^{\beta} f^2(x) dx$</span>);
The function <span class="math-container">$y=\sqrt{\frac{2-x^2}{2}}$</span> has two distinct real roots <span class="math-container">$(\alpha,\beta)=(-\sqrt{2},\sqrt{2})$</span> and the solid of revolution of this ha... |
904,547 | <p>I have the sum ( $M$ is any integer $> 1$ ):</p>
<p>$$
\sum_{h = 1}^{M}\left(\,\left\lfloor\, 2M + 1 \over h\,\right\rfloor
-\left\lfloor\, 2M \over h\,\right\rfloor\,\right)
$$</p>
<p>and looking for a way to simplify it. In the sense of either finding a <strong>simple closed form or a good approximation for M... | user84413 | 84,413 | <p>I am assuming that you want to find the area of the surface generated by revolving the ellipse $\frac{x^2}{2}+y^2=1$ about the x-axis. In that case,</p>
<p>Then $y^2=1-\frac{x^2}{2}=\frac{2-x^2}{2}$, so $y=\frac{1}{\sqrt{2}}\sqrt{2-x^2}$ on the top half of the ellipse; and</p>
<p>$S=\int_{-\sqrt{2}}^{\sqrt{2}}2\p... |
2,326,551 | <p>Prove that $\varphi(n^2)=n\cdot\varphi(n)$ for $n\in \Bbb{N}$, where $\varphi$ is Euler's totient function.</p>
| Darío A. Gutiérrez | 353,218 | <p><strong>Hint</strong>: Using this identity</p>
<blockquote>
<p>$$\varphi(n^m)=n^{m-1}\varphi(n)$$</p>
</blockquote>
|
1,895,422 | <p>I would like to know if it is possible to prove the identity $\mathbb E[e^{4W_t-8t}(W_t-4t)^4]=3t^2$ without the use of a measure change and Girsanov Theorem? I only know the solution by using the exponential martingale for the first term and then simplify terms under the new measure. </p>
| carmichael561 | 314,708 | <p>Since $W_t$ is a normal random variable with mean $0$ and variance $t$, we have
$$ \mathbb{E}[e^{4W_t-8t}(W_t-4t)^4]=\frac{1}{\sqrt{2\pi t}}\int_{-\infty}^{\infty}(x-4t)^4e^{4x-8t-\frac{x^2}{2t}}\;dx$$
$$ =\frac{1}{\sqrt{2\pi t}}\int_{-\infty}^{\infty}(x-4t)^4e^{-\frac{1}{2t}(x-4t)^2}\;dx=\frac{1}{\sqrt{2\pi t}}\int... |
888,486 | <p>Q: Solve the following limits using a table of values</p>
<p>$$\lim_{x\to0}\frac {4^x-1}{8^x-1}$$</p>
<p>I tried rewriting it as</p>
<p>$$\lim_{x\to0}\frac{2^{2x}-1}{2^{3x}-1}$$</p>
<p>but I do not know where to go from this, I got this question from <a href="http://math.bard.edu/~mbelk/math141/LimitsExercises.p... | vadim123 | 73,324 | <p>You need to make a table of values, as the instructions indicate.</p>
<p>Plug in $x=0.1, x=0.01, x=0.001, x=0.0001$, calculate the fraction, and see if you can guess where the results are headed.</p>
|
4,312,353 | <p>Let <span class="math-container">$f_n$</span> and <span class="math-container">$f$</span> functions of <span class="math-container">$L_1\mathbb R^n$</span> such that
<span class="math-container">$\lim_{n \rightarrow \infty} f_n = f$</span> almost always in <span class="math-container">$\mathbb R^n$</span> and <span ... | Danny Pak-Keung Chan | 374,270 | <p>Note that
<span class="math-container">\begin{eqnarray*}
& & |f_{n}|\\
& = & |\left(f_{n}-f\right)+f|\\
& \leq & \left|f_{n}-f\right|+\left|f\right|.
\end{eqnarray*}</span>
Therefore, <span class="math-container">$|f_{n}|-|f|-|f_{n}-f|\leq0\leq2|f|$</span>. On the other hand,
<span class=... |
1,123,647 | <p>I am looking into starting an amateur project for online tutoring. I need a math/geometry program that I can use to create shapes, graph functions, and create animation. I know of many types of software (such as geogebra) but they have restrictions for commercial use. In other words, I can't use the material or feat... | Hilaire Fernandes | 327,678 | <p>You can take a look at <a href="http://drgeo.eu" rel="noreferrer"><strong>GNU Dr. Geo</strong></a> a free software of mine, there is no commercial limitation and you can even redistribute it along your work.</p>
<p>You can do classic interactive geometry with:
<a href="https://i.stack.imgur.com/nM9ZU.png" rel="nore... |
1,316,661 | <p>Let $\{ h_n :X \to Y\}_{n \in \mathbb{Z^+}} $ be a sequence of continuous functions from a topological space $X$ to another topological space $Y$, and for each $n$ let $U_n$ be an open subset of $Y$. </p>
<p>Does
$$
\bigcup_{n=1}^{\infty} {h_n}^{-1} (\overline{U_n}) \subset \overline{ \bigcup_{n=1}^{\infty} {h_n}... | ajotatxe | 132,456 | <p>Perhaps you'll find this board useful:</p>
<p>$$\begin{array}{|c|c|c|c|}
\hline\\
&y<0&y=0&y>0\\
\hline\\
0<x<1&x^y>1&x^y=1&0<x^y<1\\
\hline\\
x=1&x^y=1&x^y=1&x^y=1\\
\hline\\
x>1&0<x^y<1&x^y=1&x^y>1\\
\hline
\end{array}$$</p>
|
3,250,593 | <p>Revising the modal logic principles, I have encountered an negative introspection axiom:
<span class="math-container">$$ \vDash \neg \square \alpha \longrightarrow \square \neg \square \alpha $$</span>
with additional information, that it is a direct effect from Euclidean property of accessibility relation <span cla... | spaceisdarkgreen | 397,125 | <p>This is the same thing as <span class="math-container">$\lozenge \alpha\to \square\lozenge \alpha.$</span> If <span class="math-container">$\lozenge \alpha$</span> holds, then there is some accessible world at which <span class="math-container">$\alpha$</span> holds. By the Euclidean property, that world is accessib... |
2,326,072 | <p>We define 3 sequences $(a_n),(b_n),(c_n)$ with positive terms so that
$$ a_{n+1}\leq\frac{b_n+c_n}{3}\ ,\ b_{n+1}\leq\dfrac{a_n+c_n}{3}\ ,\ c_{n+1}\leq\dfrac{a_n+b_n}{3} $$
Check if any of $(a_n),(b_n),(c_n)$ converge, and if they do find their limit.</p>
<p>PROOF</p>
<p>My part of the proof is this: By adding the... | Vincenzo Tibullo | 6,266 | <p>To prove that $x_n$ converges to $0$ you need to take the limit
$$
0\leq x_{n+1}\leq \frac{2}{3}x_n \implies 0\leq l\leq \frac{2}{3}l \implies l=0
$$</p>
|
2,352,211 | <blockquote>
<p>Consider $n$ seats in which $k$ distinct men and $m$ distinct women
are going to be seated ($n \ge \max \{m+k,2k\}$). In how many ways is
this possible given that no two men sit next to each other?</p>
</blockquote>
<p>My approach is the following: </p>
<p>First seat the $k$ distinct men in $k!$... | N. F. Taussig | 173,070 | <p>Your solution is correct, as true blue anil's elegant alternative solution shows. </p>
<p>As a sanity check, let's verify the extreme case: $n = m + k$, with $m \geq k$</p>
<p>We can arrange $m$ women in $m!$ ways. This creates $m + 1$ gaps, $m - 1$ between successive women and two at the ends of the row. To e... |
1,012,652 | <p>I have a problem with the following question.</p>
<p>For which $n$ does the following equation have solutions in complex numbers</p>
<p>$$|z-(1+i)^n|=z $$</p>
<p>Progress so far.</p>
<ol>
<li><p>Let $z=a+bi$.</p></li>
<li><p>Since modulus represents a distance, the imaginary part of RHS has to be 0. This immedia... | Anurag A | 68,092 | <p>Since $z$ is equal to the absolute value. Therefore $z$ has to be real. Thus it has no purely imaginary solutions.</p>
|
1,012,652 | <p>I have a problem with the following question.</p>
<p>For which $n$ does the following equation have solutions in complex numbers</p>
<p>$$|z-(1+i)^n|=z $$</p>
<p>Progress so far.</p>
<ol>
<li><p>Let $z=a+bi$.</p></li>
<li><p>Since modulus represents a distance, the imaginary part of RHS has to be 0. This immedia... | bof | 111,012 | <p>Let's break the problem into two parts. (I) Find the set $S$ of all complex numbers $z_0$ such that the equation $|z-z_0|=z$ has a solution. (II) Determine the set of all integers $n$ such that $(1+i)^n\in S$.</p>
<p>(I) The equation $|z-z_0|=z$ says that $z$ is a nonnegative real number and is equidistant from $z_... |
1,253,382 | <p>I am almost sure that this would have been asked before, but how can one find
$$
\int \frac{x^2}{1+x^2} dx?
$$
If I had a $x^2 - 1$ in the denominator, then I could factor into $(x-1)(x+1)$ and use partial fractions. But here I have an irreducible $1+x^2$.</p>
<p>I am sure that this question has already been answer... | Ivo Terek | 118,056 | <p>Ok, just a hint: $$\int \frac{x^2}{x^2+1}\,{\rm d}x = \int 1 - \frac{1}{1+x^2}\,{\rm d}x,$$ and the last one is immediate.</p>
|
2,272,831 | <blockquote>
<blockquote>
<p>$$f_{x,y}(x,y) = \begin{cases}\frac{4}{3}(2-x)y, & 0\leq x\leq 1, 0\leq y\leq 1\\
0, & \text{otherwise.}\end{cases}$$
Obtain $P(X\leq 2Y)$. Sketch the region of integration.</p>
</blockquote>
</blockquote>
<p>Is this the correct way to solve it?</p>
<p>$$\int_0^1\int_0... | supinf | 168,859 | <p>In a first step we prove this for the case $n=2$ where $X\in \mathbb{R}^n$.</p>
<p>Let $A\in \mathbb{R}^{2\times 2}$ be given such that $X^TAX=0$ for all $X$.
Then chosing $X=(1,0)$ and $X=(0,1)$ yields that the diagonal of $A$ has to be zero.
So, we assume
$$
A=
\begin{bmatrix}
0 & a \\ b & 0
\end{bm... |
2,272,831 | <blockquote>
<blockquote>
<p>$$f_{x,y}(x,y) = \begin{cases}\frac{4}{3}(2-x)y, & 0\leq x\leq 1, 0\leq y\leq 1\\
0, & \text{otherwise.}\end{cases}$$
Obtain $P(X\leq 2Y)$. Sketch the region of integration.</p>
</blockquote>
</blockquote>
<p>Is this the correct way to solve it?</p>
<p>$$\int_0^1\int_0... | Widawensen | 334,463 | <p>The matrix $A$ can be presented as the sum of its symmetric and skew-symmetric part $A=A_{sym}+A_{sk}$. </p>
<p>From this and $X^TA X=0 $ we have $X^TA_{sym}X=-X^TA_{sk}x$. </p>
<p>It was proved previously (in OP) that for $A_{sk}$ for all vectors $X$ we have $x^TA_{sk}x=0$,<br>
so we should have <strong>for ... |
3,340,374 | <p>I'm writing up several proofs for myself, all of which have a particular sticking point.</p>
<p>Essentially, I want to prove that for a function <span class="math-container">$f$</span> of two real variables, we have</p>
<p><span class="math-container">$\lim_{h \to 0} \frac{f(x \;+\; h, \; y \;+\; h) \; - \; f(x, \... | Hans Lundmark | 1,242 | <p>A sufficient condition for the claim to be true is that <span class="math-container">$f'_x=\partial f/\partial x$</span> exists in a neighbourhood of <span class="math-container">$(x,y)$</span> and is continuous at <span class="math-container">$(x,y)$</span>.</p>
<p><strong>Proof.</strong> Since <span class="math-c... |
998,633 | <p>I'm having trouble to understand exactly <em>how we are using Fubini's theorem</em> in the following proof involving the distribution function, since it newer explicitly involves an integral with product measure.</p>
<p>The proof is given to show that we can calculate an integral over some measure space $X $ as an ... | Did | 6,179 | <p>You missed the hypothesis that $\phi(0)=0$ and that $f\geqslant0$ almost everywhere, then, for every nonnegative $s$, $$\phi(s)=\int_0^s\phi'(t)\mathrm dt=\int_0^\infty\phi'(t)\mathbf 1_{s\gt t}\mathrm dt.$$ Now, $f(x)\geqslant0$ for $\mu$-almost every $x$ hence $$\phi\circ f(x)=\int_0^\infty\phi'(t)\mathbf 1_{f(x)\... |
1,309,111 | <p>I am considering a collection of function of the type, $ f:[0,2\pi]\rightarrow \mathbb{R^2}$. I want to define the $L^2$ norm of the function in that space.</p>
<p>I am defining the a norm of $f(=(f_1,f_2)')$ as $\rho(f)=\int_0^{2\pi}(f^2_1(x)+f^2_2(x))^{\frac{1}{2}}dx$. </p>
<p>Could anyone please suggest me how ... | Jonas Meyer | 1,424 | <p>If $(X,\mu)$ is a measure space and $H$ is a Hilbert space, then an $L^2$ space of $H$-valued functions on $X$ is usually defined by taking $L^2(X,H)=\{f:X\to H: \int_X \|f(x)\|^2\,d\mu<\infty\}$, with norm $\|f\|=\left(\int_X \|f(x)\|^2\,d\mu\right)^{1/2}$. This norm comes from the inner product $\langle f,g\ra... |
893,259 | <p>I have tried this problem multiple times, I have the solution but not the steps. I keep getting the wrong answer. I believe it may be in the algebra after I have taken the Laplace on both sides.</p>
<p>$y''+y = \sin(t) ;\:\: y(0) = 1, \:\: y'(0) = -1 $</p>
| Pauly B | 166,413 | <p>Instead of a Laplace transform (and to be different from everyone else), why not try an annihilator? If $D$ is a differential operator we have</p>
<p>$(D^2+1)(y)=\sin(t)$</p>
<p>$(D^2+1)(D^2+1)(y)=(D^2+1)\sin(t)=0$</p>
<p>Solution of this equation is $y=C_1\sin(t)+C_2\cos(t)+C_3t\sin(t)+C_4t\cos(t)$</p>
<p>Plugg... |
594,810 | <p>I need multiple ways to solve this question. Thank you! </p>
<p><strong>There are $A$ black balls, and $B$ white balls in an urn. You select balls one by one from this urn randomly without replacement. What is the probability that the $k$-th ball you selected is black? $(1\leq k \leq A+B)$, and $A,B$ are positive i... | Henry Swanson | 55,540 | <p>What M.B is saying is that the chance that "the first person gets a black ball" is <em>exactly the same as</em> "the $k$th person gets a black ball". As surprising as this sounds, it is completely true.</p>
<p>Imagine that after you draw each marble, you put it next to the previous ones, so you have a row of marble... |
1,768,188 | <p>I used to compute complexity of an algorithm which reaches to constant value after x level because of $a^\frac{1}{x}=1$. Now I need to find $x$ to reach answer.</p>
<blockquote>
<p>To describe more : my recursive algorithm is $T(n)=2T(\sqrt{n})+log_{2}^{n}$ ; ans $T(1)=1$. I tried to solve it using binary tree wh... | Umberto P. | 67,536 | <p>Well, since $\frac 1x \log a = 0$, $a$ must be equal to $1$ and $x$ can be absolutely any real number whatsoever!</p>
|
3,243,440 | <p>Find the number k such that:</p>
<p><span class="math-container">$$det\begin{bmatrix}
3a_1 & 2a_1 + a_2 - a_3 & a_3\\\
3b_1 & 2b_1 + b_2 - b_3 & b_3\\\
3c_1 & 2c_1 + c_2 - c_3 & c_3\end{bmatrix}$$</span></p>
<p><span class="math-container">$$ = k \bullet det\begin{bmatrix}
a_1 & a_2 &... | Bernard | 202,857 | <p><strong>Hint</strong>:</p>
<p>The determinant is <code>multilinear</code>, <code>alternate</code> function w.r.t. the columns. Hence, when you add to a column a linear combination of the other column, you don't change the determinant. Whan you multiply a column by a scalar, you multiply the determinant by this scal... |
1,903,884 | <p>3 dice are rolled simultaneously. Event A: sum is multiple of 3 and event B: sum is multiple of 5. </p>
<p>Are this events independent? </p>
<p>There are 216 points in the sample space for this problem. Is there any way to do it without writing all points. </p>
<p>If they are independent or dependent..is there an... | Graham Kemp | 135,106 | <p>A sum will be a multiple of both 3 and 5 if it is a multiple of 15.</p>
<p>Let $A$ be the event of a sum being a multiple of 3. How many of the $6^3$ outcomes match this event? </p>
<p>Let $B$ be the event of a sum being a multiple of 5. How many of the $6^3$ outcomes match this event?</p>
<p>Then $... |
134,424 | <p>I would like to create a function where I can define which case I want to use to create a path.</p>
<pre><code>p1 = {40, 48}; p2 = {50, 116}; p3 = {63, 160};
listPurple = Symbol["p" <> ToString[#]] & /@ Range[3];
disksPurple = {Purple, Disk[#, 2] & /@ listPurple};
Graphics[{disksPurple}, ImageSize -&g... | kglr | 125 | <pre><code>ClearAll[f, pathF]
f[dir : v | h][p1_, p2_] := Module[{d = dir /. {v -> 2, h -> 1},
mid = {{p1[[1]], p2[[2]]}, {p2[[1]], p1[[2]]}}}, {p1, mid[[3 - d]], p2}];
pathF[p_List, dir_List] := Join@@(f[#2][## & @@ #]&@@@ Transpose[{Partition[p, 2, 1], dir}])
pts = {{40, 48}, {50, 116}, {63, 160}... |
877,850 | <p>I'm trying to calculate the expected area of a random triangle with a fixed perimeter of 1. </p>
<p>My initial plan was to create an ellipse where one point on the ellipse is moved around and the triangle that is formed with the foci as the two other vertices (which would have a fixed perimeter) would have all the ... | Tunk-Fey | 123,277 | <p>Let $y = \arctan x$, then
\begin{align}
\tan y&=x\\
\frac{1}{\tan y}&=\frac{1}{x}\\
\cot y&=\frac{1}{x}\\
\tan\left(\frac\pi2-y\right)&=\frac{1}{x}\\
\frac\pi2-y&=\arctan\left(\frac{1}{x}\right)\\
\frac\pi2-\arctan x&=\text{arccot}\ x\\
\large\color{blue}{\arctan x+\text{arccot}\ x}&\colo... |
1,879,509 | <p>Please forgive the crudeness of this diagram.</p>
<p><a href="https://i.stack.imgur.com/AoS2Q.png" rel="noreferrer"><img src="https://i.stack.imgur.com/AoS2Q.png" alt="enter image description here"></a></p>
<p>(I took an image from some psychobabble website and tried to delete the larger circle that's not relevant... | Robert Z | 299,698 | <p>If the circle has radius $r$, the area of the "black diagram" is the area of the circle minus the area of two "petals".
The area of two "petals" can be obtained as the area of a circle minus the area of the inscribed square.
Therefore the area of the black diagram is just the area of the inscribed square $4(r^2/2))... |
1,879,509 | <p>Please forgive the crudeness of this diagram.</p>
<p><a href="https://i.stack.imgur.com/AoS2Q.png" rel="noreferrer"><img src="https://i.stack.imgur.com/AoS2Q.png" alt="enter image description here"></a></p>
<p>(I took an image from some psychobabble website and tried to delete the larger circle that's not relevant... | Blue | 409 | <p>For one "petal" only, consider this:</p>
<p><a href="https://i.stack.imgur.com/nfpi7m.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nfpi7m.png" alt="enter image description here"></a></p>
<blockquote>
<p>$$\text{area of petal} = \text{area of square} = \left(\;\sqrt{2}\;r\;\right)^2 = 2 r^2... |
216,055 | <p>Does anyone have a suggestion for the best computer program to perform calculations in the 2nd Weyl algebra? </p>
| Igor Rivin | 11,142 | <p><a href="http://www.singular.uni-kl.de/" rel="nofollow">Singular</a> claims to be wise in the ways of Weyl algebra.</p>
|
14,858 | <p>Let $V$ be a complex vector space of dimension 6 and let $G\subset {\mathbb P}^{14}\simeq {\mathbb P}(\Lambda^2V)$ be the image of the Plucker embedding of the Grassmannian $Gr(2, V)$.</p>
<ol>
<li>Why the degree of $G$ is 14? or in general, how to calculate the degree of a Plucker embedding?</li>
</ol>
<p>Let ${\... | J.C. Ottem | 3,996 | <p>1) In general, the degree of the Grassmannian $G(k,n)$ in the Plucker embedding is given by
$$
(k(n-k))!\prod_{i=1}^k\frac{(i-1)!}{(n-k+i-1)!}
$$This is calculated by finding $\sigma_1^{k(n-k)}$ using Pieri's rule. See <a href="http://ocw.mit.edu/NR/rdonlyres/Mathematics/18-727Spring-2006/9EEAB391-89EE-4BBA-A61B-3B9... |
301,905 | <p>I'm having troubles to solve this integration: $\int_0^1 \frac{x^{2012}}{1+e^x}dx$</p>
<p>I've tried a lot using so many techniques without success. I found $\int_{-1}^1 \frac{x^{2012}}{1+e^x}dx=1/2013$, but I couldn't solve from 0 to 1.</p>
<p>Thanks a lot.</p>
| Imago | 174,539 | <p>Experimenting with wolfram alpha lead me to: </p>
<p>$$ \int {x^n \over 1+e^x} = (-1)^{n+1} x^{n+1} + (-1)^n * log(e^x+1)+ \sum_{k = 0}^{n-1} x^{n-i-1} (-1)^n * {n! \over (n-i)!} * Li_{i+2}(-e^x)$$</p>
<p>I got this by trying out small values for n so wolfram alpha manages to do the computations and then generali... |
691,927 | <p>Now before I begin, I know this question has been asked multiple times but all the answers but I had so many questions of my own that I figured I should make a new question as my thoughts are different than previous answers. </p>
<p>Now I will ask the question first then explain my thoughts and troubles :)</p>
<p>... | Jimmy R. | 128,037 | <p>Assume your observations are $X_1, X_2, ..., X_n$. Order them ascending so that you have your ordered sample $$X_{(1)},\ldots,X_{(n)}$$ that is $X_{(n)}=\max_{1\le n}\{X_i\}$. Now ask yourself: This $X_i$ are between $a,b$ but you do not know which numbers are $a,b$. Still $b$ is the highest limit. And you have foun... |
1,498,952 | <p>How does one compute the bias for the estimator given by the sample geometric mean for a gamma distribution with parameters ($\theta$,1)?</p>
<p>i.e:
Given $X_1,...,X_n$ are iid with distribution Gamma($\theta$,1), what is the bias of the estimator given by: $$\hat\theta = \left(\prod_{i=1}^n X_i\right)^{1/n}$$</p>... | heropup | 118,193 | <p>For a gamma random variable $X$ with shape $\alpha$ and rate $\beta$, the PDF is $$f_X (x) = \frac{\beta^\alpha x^{\alpha-1} e^{-\beta x}}{\Gamma(\alpha)}, \quad x > 0.$$ Now let $Y = g(x) = \log X$, thus $Y$ is distributed with density $$f_Y(y) = f_X(g^{-1}(y)) \left| \frac{dg^{-1}}{dy} \right| = \frac{\beta^\a... |
4,605,607 | <blockquote>
<p><strong>4.3.</strong> Identify the following rings:</p>
<ol>
<li><p><span class="math-container">$\mathbb{Z}[x] / (x^2 - 3, 2x + 4)$</span>,</p>
</li>
<li><p><span class="math-container">$\mathbb{Z}[i] / (2 + i)$</span>,</p>
</li>
<li><p><span class="math-container">$\mathbb{Z}[x] / (6, 2x - 1)$</span>,... | diracdeltafunk | 19,006 | <p>Good start! I'll first give a high-level, "intuitive" explanation of what's going on, then sketch a rigorous proof.</p>
<hr />
<p><strong>High Level Intuition</strong> <span class="math-container">$\mathbb{Z}_6[x]/(2x-1)$</span> can be thought of as <span class="math-container">$\mathbb{Z}_6[\frac{1}{2}]$<... |
3,589,926 | <p>I wanted to come up with a proof using contrapositive, so <strong>I needed the negation of the following: "one of x and y is congruent to 1 modulo 6 while the other is congruent to 5 modulo 6."</strong></p>
<p>I interpreted this statement as being the same as "one of x and y is congruent to 1 modulo 6 and the other... | heropup | 118,193 | <p>We should be clear about the difference between "calculus" and "real analysis." Based on the candidate texts, it seems you are interested in the former and not the latter. If you have familiarity with high school calculus, then either Spivak or Apostol (Volume 1) will be suitable for an undergraduate-level calculu... |
2,878,073 | <p><strong>Problem:</strong> $\{a_n\}_{n\in \mathbb N}, \quad a_{n+1}=\sqrt{2+a_n}, \quad \forall n\geq 1, \quad a_1=\sqrt{2}$</p>
<p><strong>Solution:</strong></p>
<p>We assume $a_n$ converges. Then is $\lim_{n\to\infty}a_n=\lim_{n\to\infty}a_{n+1}$</p>
<p>So we get</p>
<p>$$$a_n=\sqrt{2+a_n} \Rightarrow a_n^2-a_n... | Kavi Rama Murthy | 142,385 | <p>$a_n <2$ implies $a_{n+1} =\sqrt {2+a_n} < \sqrt {2+2}=2$. Since $a_1 <\sqrt 2$ it follows by induction that $a_n <\sqrt 2$ for all $n$. Next $a_{n+2}^{2}- a_{n+1}^{2}=(2+a_{n+1})-(2+a_n)= a_{n+1}-a_n$. This gives $(a_{n+2}-a_{n+1})(a_{n+2}+a_{n+1})=(a_{n+1}-a_n)$. Since $a_{n+2}+a_{n+1} >0$, we can ... |
3,051,281 | <p>The question is:</p>
<p>Let <span class="math-container">$n \ge 2$</span> be an integer and consider a uniformly random permutation
(<span class="math-container">$a_1$</span>, <span class="math-container">$a_2$</span>, . . . , <span class="math-container">$a_n$</span>) of the set (1, 2, . . . , n). </p>
<p>For ea... | Wuestenfux | 417,848 | <p>In view of your linear equation, let <span class="math-container">$-\pi/2<x<\pi/2$</span>. Then the linear equation gives <span class="math-container">$\tan x = -\lambda_2/\lambda_1$</span>. But this would need to hold for each such <span class="math-container">$x$</span>.</p>
|
3,051,281 | <p>The question is:</p>
<p>Let <span class="math-container">$n \ge 2$</span> be an integer and consider a uniformly random permutation
(<span class="math-container">$a_1$</span>, <span class="math-container">$a_2$</span>, . . . , <span class="math-container">$a_n$</span>) of the set (1, 2, . . . , n). </p>
<p>For ea... | Farid | 628,982 | <p>Try a proof by contradiction and follow the same approach you give here. The expression you have for <span class="math-container">$ \tan x $</span> is the contradiction. (It implies that the function is constant). </p>
|
1,164,471 | <p>I'm having trouble with the following problem:</p>
<p>A man found that $3$ out of $10$ inspected bottles were defective. What is the probability that the $2$ first defective bottles were found in the first $7$ inspected bottles? The probability that a bottle is defective is $0.1988$.</p>
<p>Let $A$ be the event th... | MooS | 211,913 | <p>The generator is $\zeta \mapsto \zeta^k$, where $k$ is a primitive root modulo $p$.</p>
|
4,560,624 | <p>Is there any elementary way to prove that <span class="math-container">$(\sin x + \cos x)(6 - \sin x)<9$</span>?</p>
<p>I've noticed that <span class="math-container">$(\sin x + \cos x)$</span> has to be positive so <span class="math-container">$x \in\left(-\dfrac{\pi}{4}, \dfrac{3\pi}{4}\right)$</span> and then ... | Glorious Nathalie | 948,761 | <p><span class="math-container">$\begin{equation} \begin{split}
(\sin x + \cos x)(6 - \sin x) & = 6 \cos x + 6 \sin x - \sin^2 x - \sin x \cos x \\ &= 6 \cos x + 6 \sin x - \dfrac{1}{2} (1 - \cos(2x) ) - \dfrac{1}{2} \sin(2x)\\
&= 6 \cos x + 6 \sin x + \dfrac{1}{2} (\cos(2x) - \sin(2x) ) - \dfrac{1}{2} \\
&... |
1,077,119 | <p>I am starting to learn about tensor products of abelian groups.</p>
<p>Why is the tensor product defined for <strong>abelian</strong> groups? In which part of the construction the commutativity of the groups is needed?</p>
| WLOG | 21,024 | <p>The tensor product in defined for $R$-modules, with $R$ unit ring, and the abelian groups are $\Bbb{Z}$-modules.</p>
|
326,501 | <p>Arrange the numbers $1,2,...,9$ in such an order that no four of them appear (adjacently or otherwise) in ascending or descending order.
Show that there is no arrangement of the numbers $1,2,...,10$ with this property.</p>
<p>Now suppose $n>1$, and find the maximum $k$ such that the numbers $1,2,...,k$ can be a... | Anubis Black | 149,753 | <p>$k = (n - 1)^2$</p>
<p>You need to arrange the $k$ numbers in $n - 1$ groups of $n - 1$ descending elements, where every following group is bigger than the previous.</p>
<p>In the case of $n = 4$ we have</p>
<p>$3\:2\:1\hspace{8 pt}6\:5\:4\hspace{8 pt}9\:8\:7$ or $3$ groups of $3$. Introducing a $10$th element wi... |
374,194 | <p>Given a metric space $(X,d)$ and a transformation $T:X\rightarrow X$, a point $x\in X$ is said to be <strong>recurrent</strong> iff it belongs to the closure of its orbit $\{T(x), T^2(x),...\}$: more precisely, there exists an increasing sequence $(n_k)$ of natural numbers with $n_k \rightarrow \infty$ such that $... | Twiceler | 54,911 | <p>Are you sure that this is true?</p>
<p>Let $X$ be the circle, and let $d(x,y)$ be the measure of the smallest angle between $x$ and $y$. Let $f$ be a rotation of the circle by $\alpha$, and take $\alpha \notin \{2 \pi n : n \in \mathbb{Z}\}$, so that the rotation is non-trivial. $f$ is continuous. But <em>none</em>... |
1,249,730 | <p>let $ x \in G$ such that $(a^{-1})*(x^2)*(a) = x^3$ for some self inverse $a.$ Prove that $x$ has order $5.$</p>
<p>I don't know how to start this proof. Seems really difficult. </p>
| Mark Bennet | 2,906 | <p>Since there is an answer up, I would do $$ax^2a=x^3$$</p>
<p>$$x^2=a^2x^2a^2=ax^3a=x^3axa$$</p>
<p>$$axa=x^{-1}$$</p>
<p>$$ax^2a=x^{-2}=x^3$$</p>
<p>Leaving you with some bits to fill in.</p>
|
391,860 | <p>If p is a prime number and x and y are integers, how do I prove "if xy and x+y are both divisible by p, then x and y is divisible by p"?</p>
<p>I started like this..</p>
<p>1) p divides xy, so p divides x or p divides y, since p is a prime number.</p>
<p>and then I'm already stuck :/</p>
<p>Help me to get to the... | Ross Millikan | 1,827 | <p>If $p$ divides $x$ but not $y$, can $p$ divide $x+y$?</p>
|
391,860 | <p>If p is a prime number and x and y are integers, how do I prove "if xy and x+y are both divisible by p, then x and y is divisible by p"?</p>
<p>I started like this..</p>
<p>1) p divides xy, so p divides x or p divides y, since p is a prime number.</p>
<p>and then I'm already stuck :/</p>
<p>Help me to get to the... | DonAntonio | 31,254 | <p>Hints:</p>
<p>$$(1)\;\;\;p\,\mid\,xy\implies\;p\,\mid\,x\;\;\vee\;\;p\,\mid\,y$$</p>
<p>Now suppose WLOG that $\,p\mid x\,$, and also we have $\,x+y=tp\,$ , so</p>
<p>$$(2)\;\;\;x=kp\implies tp=x+y=kp+y\implies\ldots\ldots$$</p>
|
2,001,441 | <p>I am having the hardest time solving the following trigonometric equation. Can anyone help please? Thank you.</p>
<p>Solve for x. [Hint: Let $\\u = \tan^{-1}(x)$ and $\\v = tan^{-1}(2x)$. Solve the equation $\\u+v = \frac{π}{4}$ by taking the tangent of each side.]</p>
<p>$\tan^{-1}(x) +\tan^{-1}(2x)= \frac{π}{... | mfl | 148,513 | <p>We have that $$\tan(a+b)=\dfrac{\tan a+\tan b}{1-\tan a\tan b}.$$ Applying this we get</p>
<p>$$\tan(\tan^{-1}x+\tan^{-1}(2x))=\dfrac{\tan (\tan^{-1}x)+\tan (\tan^{-1}(2x))}{1-\tan (\tan^{-1}x)\tan (\tan^{-1}(2x))}=\dfrac{x+2x}{1-2x^2}=\tan\frac{\pi}{4}=1.$$</p>
<p>That is, we have the equation $$3x=1-2x^2$$ or eq... |
4,418,518 | <p><span class="math-container">$X$</span> is Banach Space and <span class="math-container">$A\subseteq X$</span>.
<span class="math-container">$A$</span> is dense <span class="math-container">$G_{\delta}$</span> set. We have to show that <span class="math-container">$A-A=X$</span>.</p>
<p>Since <span class="math-conta... | Chris Sanders | 309,566 | <p>This answer speaks of subsets of a Banach space <span class="math-container">$X$</span>. The first two exercises are results that don't hold for arbitrary topological spaces.</p>
<p>An exercise to get you started:</p>
<p><strong>Suppose <span class="math-container">$U_n$</span> are all open and dense, and suppose th... |
4,299,417 | <p>Let <span class="math-container">$f: \mathbb{R} \rightarrow [0, \infty)$</span> be a function that:
<span class="math-container">$$f^2(x+y)+f^2(x-y)=2f^2(x)+2f^2(y) \forall x,y \in \mathbb{R}$$</span>
Prove that <span class="math-container">$f(x+y) \leq f(x)+f(y) \forall x,y \in \mathbb{R}$</span></p>
<p>This proble... | WhatsUp | 256,378 | <p>This is a standard result in linear algebra. Here I write a solution accessible to high school students, as the question is originally posted as a high school contest question.</p>
<p>You have already shown that <span class="math-container">$f(kz) = kf(z)$</span> for all <span class="math-container">$k \in \Bbb Z_{\... |
1,598,343 | <p>If you form a set of orthonormal polynomials on $[0,1]$, by applying the Gram-Schmidt process from monomials $\{1, x, x^2, \dots \}$ then what is required to show that this is a basis for $L^2[0,1]$?</p>
<p>$\text{Span}\{ e_n \}_{n \in \mathbb{N}} = L^2[0,1]$?</p>
<p>How can I use this to prove the above?</p>
<p>... | Justpassingby | 293,332 | <p>A Schauder basis of a Hilbert space (or more generally a Banach space) is a linearly independent set of vectors such that every vector can be uniquely written as the sum of a norm convergent series where the individual terms of the series are multiples of the basis vectors.</p>
<p>Any polynomial is a linear combina... |
1,779,843 | <p>I need help finishing this proof. I've come to a point where I don't know how to continue. I need to prove that the following inequality is true for all positive integers $n$.</p>
<p>$$\sum_{i=1}^{n-1} (-1)^{i+1}i! \leq \frac{(2n)!}{2}$$</p>
<p>It's true for $P(1)$, now I consider the inequality to be my inductive... | Vincent | 332,815 | <p>This can be solved by linear equations.</p>
<p>Note $x^{i}$ the amount of people that bought only the juice $i$ ($i=A$ for apple, $i=G$ for grape, $i=O$ for orange), $x^{i,j}=x^{j,i}$ when they bought two kinds of juices and $x^{A,O,G}$ when they bought every kind.</p>
<p>$x^O+x^{O,A}+x^{O,G}+x^{A,O,G}=400$</p>
<... |
1,779,843 | <p>I need help finishing this proof. I've come to a point where I don't know how to continue. I need to prove that the following inequality is true for all positive integers $n$.</p>
<p>$$\sum_{i=1}^{n-1} (-1)^{i+1}i! \leq \frac{(2n)!}{2}$$</p>
<p>It's true for $P(1)$, now I consider the inequality to be my inductive... | ervx | 325,617 | <p>Let $O$ be the set of people you bought orange juice.</p>
<p>Let $A$ be the set of people you bought apple juice.</p>
<p>Let $G$ be the set of people you bought grape juice.</p>
<p>Based on the information given, we know the following:</p>
<p>$|O|=400$.</p>
<p>$|A|=110$.</p>
<p>$|G|=50$. </p>
<p>$|O\cap A|=75... |
1,085,668 | <blockquote>
<p>Let $X$ and $Y$ be i.i.d. $\operatorname{Geom}(p)$, and $N = X + Y$. Find the joint PMF of $X, Y, N$.</p>
</blockquote>
<p>I have generally difficulties with such problems, as I get easily confused. Below I detailed my (most probability incorrect) approach. Besides the correct approach, I also would... | ronno | 32,766 | <p><strong>Hint:</strong>:
$$P(X=x,Y=y,N=n) = \begin{cases}P(X = x, Y = y) & \text{if }x+y=n \\ 0 & \text{otherwise}\end{cases}$$</p>
|
1,958,152 | <blockquote>
<p>I want to know why $\frac{\log4}{\log b}$ can't be simplified to $\frac4b$. </p>
</blockquote>
<p>I am a high school student. Please do not quote some theories that are too advanced for me. Thank you!</p>
| Eleven-Eleven | 61,030 | <p>You have $\log{4}$ and lets set it equal to $X$.</p>
<p>$$\log{4}=X$$ </p>
<p>and this has the equivalent exponential form of $10^X=4$.</p>
<p>Similarly, if $\log{b}=Y$, then $10^Y=b$. Thus,</p>
<p>$$\frac{4}{b}=\frac{10^X}{10^Y}\neq\frac{X}{Y}=\frac{\log{4}}{\log{b}}$$</p>
|
1,958,152 | <blockquote>
<p>I want to know why $\frac{\log4}{\log b}$ can't be simplified to $\frac4b$. </p>
</blockquote>
<p>I am a high school student. Please do not quote some theories that are too advanced for me. Thank you!</p>
| Christian Blatter | 1,303 | <p>The operation $\log\square$ is a black box that eats up the number you write into the square, processes it somehow, and outputs some other number as a result. At the moment you have no big idea what happens in the interior of that box.</p>
<p>There is no reason to assume that for arbitrary $a$ and $b$ one has
$${\l... |
2,461,773 | <p>I would like some clarification about the Cantor Set:</p>
<ul>
<li>What are the elements in the Cantor Set?</li>
<li>How do I write the Cantor Set in mathematical terms (i.e in a summation)? I have seen online a formula but I do not understand how they got it so would be grateful if you could explain why it is this... | Trevor Gunn | 437,127 | <p>Let us define the Cantor representation of a real number to be the ternary expansion of that number where we don't allow a suffix of $1000\cdots$ or $12222\cdots$. So we wouldn't write $1$ as $1.0000\cdots$, we would write it as $0.2222\cdots$. Likewise, $2/3$ is $0.2000\cdots$ not $0.1222\cdots$.</p>
<p>The Cantor... |
223,087 | <p>Given a list of numbers in decimal form, what is the most efficient way to determine if there are any consecutive 1s in the binary forms of those numbers? My solution so far:</p>
<pre><code>dim = 3;
declist = Range[0, 2^dim - 1];
consecutiveOnes[binary_] := AnyTrue[Total /@ Split[binary], # > 1 &];
consecuti... | MassDefect | 42,264 | <p>It seems like directly constructing the list might be the fastest method. Most numbers will have consecutive ones. Based on the wrap-around criteria, we already know that testing any odd number is a waste of time. Playing around with the numbers and their binary representations, it seems like there's a pattern. Any ... |
223,087 | <p>Given a list of numbers in decimal form, what is the most efficient way to determine if there are any consecutive 1s in the binary forms of those numbers? My solution so far:</p>
<pre><code>dim = 3;
declist = Range[0, 2^dim - 1];
consecutiveOnes[binary_] := AnyTrue[Total /@ Split[binary], # > 1 &];
consecuti... | yarchik | 9,469 | <p>Too late for a party. Here is a one-liner</p>
<pre><code>noZ[n_] := Map[Total[2^(Rest[FoldList[1 + #1 + #2 &, 0, #]] - 1)] &,IntegerPartitions[n]]
</code></pre>
<p>Input is the number of zeroes desired to have in the binary form. Output is the list of numbers whose binary representation has no neighboring ... |
1,032,874 | <p>Prove that $(1+2+3+\cdots + n)^2 = 1^3+2^3+3^3+\cdots + n^3$ for every $n \in \mathbb{N}$.</p>
<p>Proof. We will use mathematical induction.
If $n = 1$, then we have $(1)^2= 1^3 = 1$. We must show that $S_n$ implies $S_{n+1}$. Assume that for $n \in \mathbb{N}$, the statement $(1+2+3+\cdots + n)^2 = 1^3+2^3+3^3+\cd... | RE60K | 67,609 | <p>Consider these:</p>
<blockquote>
<ul>
<li>$$\sum_{i=1}^ni=\frac{n(n+1)}2$$</li>
<li>$$\sum_{i=1}^ni^2=\frac{n(n+1)(2n+1)}6$$</li>
<li>$$\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}4$$</li>
</ul>
</blockquote>
|
1,351,513 | <p>Express the function F in the form $f \circ g \circ h$. </p>
<p>$$F(x)=\frac {9}{( x^2 + 7)}$$</p>
<p>I'm not sure how to get $x^2+7$ in the denominator. Here is what I tried:</p>
<p>$$h(x) = (x+7)$$</p>
<p>$$g(x) = x$$</p>
<p>$$f(x) = \frac {9}{x^2}$$</p>
<p>But obviously that gives me $F(x) = \dfrac {9}{(x+... | Lubin | 17,760 | <p>Here’s what you do in general: suppose you want to compute a value of your function. What do you do, in order? Clearly you first square your value of $x$; then you add $7$; then you divide the result into $9$. Now traanslate each of these steps into functional notation, and you have it.</p>
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.