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 |
|---|---|---|---|---|
267,318 | <p>Let $G$ be a connected, reductive group over a local field $F$ of characteristic zero, and $H$ a closed subgroup of $G$ which is defined over $F$. Let $\mu_H, \mu_G$ be right Haar measures on $H(F), G(F)$ with modular functions $\delta_H, \delta_G$.</p>
<p>In papers, notes, and textbooks on automorphic forms and r... | paul garrett | 15,629 | <p>Of course, you are asking several related-but-different questions... I would grant you that it is hard to "see the general pattern", but/and by this point I think that the reason it is hard to see the general pattern is that there isn't one, except at a fairly philosophical, or else extremely utilitarian, level. Tha... |
277,060 | <p>For example I have this equation,I want to use c->a/b ,but it can not work
<span class="math-container">$$
\frac{a^2}{b^2}+\frac{b^2}{a^2}+\frac{a}{b}+e^{a/b}+\frac{b}{a}+\log \left(\frac{a}{b}\right)
$$</span></p>
<pre class="lang-mathematica prettyprint-override"><code>rule = {a/b -> c};
eq = 1/(a/b)^2 + (a/... | user64494 | 7,152 | <p>Let us write down the same rule in other formula.</p>
<pre><code>rule ={a -> c*b};eq = 1/(a/b)^2 + (a/b)^2 + 1/(a/b) + a/b + Exp[a/b] + Log[a/b];eq /. rule
</code></pre>
<blockquote>
<p><code>1/c^2 + 1/c + c + c^2 + E^c + Log[c]</code></p>
</blockquote>
<p>Similar questions were asked and answered here many,many ... |
426,129 | <p><a href="http://cds.cern.ch/record/630829/files/sis-2003-264.pdf" rel="nofollow noreferrer">Muñoz Garcia and Pérez-Marco - The product over all primes is <span class="math-container">$4\pi^2$</span></a> claims that the regularized value of product <span class="math-container">$\prod_{k=1}^\infty k$</span> is <span c... | user64494 | 35,959 | <p>The command of Mathematica 13.1</p>
<pre><code>Product[j, {j, 2, Infinity}, Regularization -> "Dirichlet"]
</code></pre>
<p>produces <span class="math-container">$\sqrt{2 \pi }$</span>. In fact, MMA finds the sum of the series of <span class="math-container">$\log k$</span>, making use of <a href="https... |
124,522 | <p><a href="https://i.stack.imgur.com/CxrDw.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/CxrDw.png" alt="enter image description here"></a></p>
<p>Greetings,</p>
<p>I have created 4 sample data and classified them to 4 classes as shown. However, when I want to test the ClassifierFunction, the ou... | mikado | 36,788 | <p>Unlike @Sascha, I assume that you are trying to do exactly what your code says: namely, classify data into four classes using one example of a length 63 feature vector for each.</p>
<p>This is challenging for some types of classifier, in particular the default (<code>Method->"LogisticRegression"</code>), which (... |
109,762 | <p>I read quite a while ago this proof of Binet's formula. (
I am not 100% sure this is the way it was presented, but it gives an idea. I'm not approving of this method or saying it is correct.)</p>
<p>Let $\hat{S}$ be an operator such that</p>
<p>$$\hat{S}a_n =a_{n+1}$$</p>
<p>Then, we can define Fibonacci's number... | Math Gems | 75,092 | <p>Hint: the key is to factor <span class="math-container">$S^2 - S - 1 = (S-\phi)(S-\phi')$</span>. Because <span class="math-container">$S-\phi,\ S-\phi'$</span> commute, both solutions <span class="math-container">$\phi^n,\ \phi'^n$</span> of <span class="math-container">$(S-\phi) a_n = 0 = (S-\phi') a_n $</span> yi... |
868,613 | <p>When I say that $p_1=2$, I mean that the first prime in the standard ring of integers $(\mathbb{Z},*,+)$ is $2$. I was wondering whether the notion of ordering the primes like this can be generalized for any ring $R$. First of all, is it even possible to consistently equip a general ring (such as a finite one) with ... | richard gayle | 164,419 | <p>You don't need to assume anything is true. You can easily show this is true. Let $x \in \emptyset$. Since there is no such $x$, all statements about $x$ are perforce true, including $x \in \{\emptyset\}$. So $\emptyset \subset \{\emptyset\}$.</p>
|
2,975,135 | <p>I want to prove that, if <span class="math-container">$m \equiv_4 n$</span> for all <span class="math-container">$m,n \in \mathbb{Z}$</span>, then <span class="math-container">$123^m \equiv_{10} 33^n$</span></p>
<p>I have no idea how to prove something like that</p>
| rtybase | 22,583 | <p>Start with <span class="math-container">$$\varphi(10)=4 \tag{1}$$</span>
where <span class="math-container">$\varphi(n)$</span> is <a href="https://en.wikipedia.org/wiki/Euler%27s_totient_function" rel="nofollow noreferrer">Euler's totient function</a>. Then, because
<span class="math-container">$$123 \equiv 33 \pm... |
3,720,635 | <p><strong>Question:</strong></p>
<blockquote>
<p>Prove that the series <span class="math-container">$$\sum_{n=1}^\infty \sin\left(\frac{x}{n^4}\right)\cos(nx)$$</span> converges absolutely, and it is continuous on <span class="math-container">$\mathbb{R}$</span>.</p>
</blockquote>
<p><strong>Attempt:</strong> I can re... | Enforce | 742,414 | <p><span class="math-container">$$\log(M_n) = (n+1)\log\left(\frac{n}{n+1}\right)$$</span></p>
<p>And also</p>
<p><span class="math-container">$$\frac{d}{dx}\left((x+1)\log\left(\frac{x}{x+1}\right)\right) = \log\left(\frac{x}{x+1}\right)+ \frac{1}{x}.$$</span></p>
<p>Since</p>
<p><span class="math-container">\begin{al... |
3,948,869 | <p>I used the definition.</p>
<p><span class="math-container">$1$</span>; <span class="math-container">$\cos(x)$</span>; <span class="math-container">$\cos^2(\frac{x}{2})$</span></p>
<p><span class="math-container">$c_1\cdot1+c_2\cdot\cos(x)+c_3\cdot\cos^2(\frac{x}{2}) = 0$</span></p>
<p>I tried converting <span class=... | azif00 | 680,927 | <p>From <span class="math-container">$\cos^2(\tfrac{x}{2}) = \tfrac12+\tfrac12\cos(x)$</span> we get <span class="math-container">$$\tfrac12 + \tfrac12\cos(x) - \cos^2(\tfrac{x}{2}) = 0$$</span> that is, <span class="math-container">$c_1\cdot1+c_2\cdot\cos(x)+c_3\cdot\cos^2(\frac{x}{2}) = 0$</span> where <span class="m... |
112,660 | <p>Let (A,m) be a local ring and M be a finitely generated A-module contained in a free module F of rank r with length(F/M) < $\infty$. Then I have the following question : Is the statement "M doesn't have a non-trivial free summand if and only if M$\subset$mF " true? I was trying around Nakayama's lemma</p>
| Chad Brewbaker | 7,359 | <p>Every finite group can be broken down into elements consisting of prime cycle permutations. Just go up the lattice. Every nontrivial subgroup has to contain at least one of these prime cycle permutations: <a href="http://oeis.org/A186202" rel="nofollow">http://oeis.org/A186202</a> </p>
|
167,934 | <p>Let $X$ be a proper algebraic variety. $X$ is said to have polynomial point count if there is a polynomial $P$ such that for all finite fields $\mathbb F_q$ with $q$ elements, $|X(_q)|=P(q)$.</p>
<p>If in addition $X$ is smooth, then by the Weil conjectures one can derive that $X$ has no odd cohomology.</p>
<p>My ... | Ben Webster | 66 | <p>You can do a bit better than a paving: it's enough to check that the cohomology is generated by algebraic cycles (or any other way of showing the Hodge filtration is pure). </p>
|
2,923,346 | <p>Let $f_1,..., f_n: \mathbb{R}\rightarrow\mathbb{R}$ be measurable functions. And $F:\mathbb{R}^n\rightarrow\mathbb{R}$ be continuous. </p>
<p>For $g:\mathbb{R}\rightarrow\mathbb{R}$, $g(x):=F(f_1(x),f_2(x),...,f_n(x))$. Is $g$ measurable?</p>
<p>I think I was be able to do that if F is measurable, but I don't know... | Michael | 155,065 | <h3>Assume:</h3>
<ul>
<li><p>Let $\mathcal{X}$ be any set. </p></li>
<li><p>Let $\Sigma$ be any sigma-algebra on $\mathcal{X}$. </p></li>
<li><p>Fix $n$ as a positive integer. </p></li>
<li><p>For each $i \in \{1, ..., n\}$ and each $c \in \mathbb{R}$, the function $f_i:\mathcal{X}\rightarrow\mathbb{R}$ has the prope... |
2,216,070 | <p>I'm having problems finding the primitive function to $\int \frac{1}{x \sqrt{8-x^2}} dx$. I've tried to use the substitution $t = x^2-8$, but then I just get stuck with $\int \frac{1}{(8-t)\sqrt{t}} dt$ instead. Using the substitution $t = \sqrt{x^2-8}$ doesn't get me much further either.</p>
<p>Any help is much ap... | Ziad Fakhoury | 295,839 | <p>Consider the substitution $x = \root \of{8}\sin u$ and $dx = \root \of{8} \cos u du$. Your integral becomes
$$\int\frac{\root \of{8} \cos u }{\root \of{8} \sin u \root \of {8(1-\sin^2 u)} } du$$
$$=\int\frac{\root \of{8} \cos u }{8 \sin u \cos u} du = \frac{\root \of 8}{8}\int \csc u du$$</p>
<p>You can take it fr... |
1,625,621 | <p>I am stuck on this limit and have no idea how to solve it and which trig identity to use. Any help would be appreciated. Thanks!</p>
<p>$\lim\limits_{x \to 0^-} \frac{\sqrt{1+2\sin^2 \frac{x}{2}-\cos^2x}}{\left\lvert x \right\rvert}$</p>
<p>Note: <strong>Without</strong> using L'Hopitals rule.</p>
| John B | 301,742 | <p>Write the first few terms of the Taylor series of what is under the square root, which gives $1+2\sin^2\frac x 2-\cos^2 x=\frac32 x^2+\cdots$. Hence the limit is $\sqrt{3/2}$.</p>
|
2,824,504 | <p>$\sum_{n=1}^{\infty} \dfrac{\arctan n + \sqrt {|x|}}{n^2}$.
Does the series converge uniformly on $R$. ?</p>
<p>Let $f_k(x) = \dfrac{\arctan k + \sqrt {|x|}}{k^2}$. Then, I have to find $M_k$ such that $|f_k(x)| \le M_k$ for all $x \in R$. Concurrently, $\sum_{k=1}^{\infty} M_k$ has to converge. I think such $M_k$... | Jean-Claude Arbaut | 43,608 | <p>Let</p>
<p>$$R_n(x)=\sum_{k=n}^{\infty}\dfrac{\arctan k+\sqrt{|x|}}{k^2}$$</p>
<p>This is the rest of the series.</p>
<p>Uniform convergence would mean that</p>
<p>$$\forall \varepsilon>0,\exists N,\forall n>N,\forall x\in\Bbb R, |R_n(x)|<\varepsilon$$</p>
<p>If you suspect that the series does not con... |
2,471,838 | <p>If I have a a vector $\vec{v}=(x,y,z)$ in cartesian coordinates and I want to rotate my coordinate system by spherical angles $\theta$ and $\phi$ how would I find the resulting vector $\vec{v'}=(x',y',z')$ in cartesian coordinates?</p>
<p><a href="https://www.uwgb.edu/dutchs/MATHALGO/sphere0.htm" rel="nofollow nore... | Tobia Marcucci | 364,521 | <p>In the link you gave it gives the generic form of a rotation matrix of angle $\theta$ around an axis $r$. There are two alternative ways to arrive to that relation: i) composing rotation matrices, ii) constructing the rotated vector geometrically.</p>
<p>i) From: Siciliano, Sciavicco, Villani, Oriolo - Robotics, Mo... |
2,141,517 | <p>I have been attempting this problem for a while, it is an assignment problem so I don't want somebody to just post the answer, I'm just looking for hints.</p>
<p>Let $\mathfrak{u}(n,\mathbb{C})$ be the Lie algebra of strictly upper triangular $n\times n$ matrices over $\mathbb{C}$.</p>
<p>I am trying to show that ... | Joshua Ruiter | 399,014 | <p>EDIT: My previous answer was wrong, so I deleted it. </p>
<p>The thing to prove is: $L^k$ has a basis given by $e_{ij}$ with $i < j - k$. </p>
<p>By definition, $L^k = [L,L^{k-1}]$, which is the span of $$\{[e_{ij},e_{ab}]:i < j, a+k< b \} $$
by using the induction hypothesis on $L^{k-1}$. After applying ... |
2,141,517 | <p>I have been attempting this problem for a while, it is an assignment problem so I don't want somebody to just post the answer, I'm just looking for hints.</p>
<p>Let $\mathfrak{u}(n,\mathbb{C})$ be the Lie algebra of strictly upper triangular $n\times n$ matrices over $\mathbb{C}$.</p>
<p>I am trying to show that ... | Dietrich Burde | 83,966 | <p>Your computations easily generalize to arbitrary $n\ge 1$. We have to compute $L^1=[L,L]$, $L^2=[L,[L,L]]$, $L^3=[L,[L,[L,L]]$ and so on. In each step the strictly upper triangular matrices of $L$ obtain a new side diagonal consisiting of zeros.
In other words, the zero diagonal is expanding to the right upper corne... |
2,767,471 | <p>You enter a metro station in a big hurry, and decide to take the first train that arrives. </p>
<p>There are two lines running through this station: one runs every five minutes (line A), the other every three (line B). To be precise, suppose the next arrival of the A train is uniformly distributed on the interval [... | Saurav | 707,178 | <p><strong>Method 1</strong>:</p>
<p>If Z = min(X,Y)</p>
<p>p(Z>z) = p(X>z and Y>z) = p(X>z). p(Y>z) Independent = 1/15. (5-z)(3-z) . (0
<p>dF= d(1-p(Z>z)) = -1/15.(2z-8)dz</p>
<p>E[z] = int [z. dF] = -1/15.(2z3/3-8z2/2) =1.2 min, for z=3</p>
<p><strong>Method 2:</strong></p>
<p>Given that both trains arrive in t... |
2,584,044 | <p>I'm learning about how use mathematical induction.
I'm tasked with proving the inequality shown in (1). It is a requirement that I use mathematical induction for the proof.</p>
<p>$(1) \quad P(n):\quad 2n+1 < 2^{n}, \quad n \ge 3$</p>
<p>I would like some feedback regarding whether my proof is valid and if my u... | Paolo Leonetti | 45,736 | <p>You know it for $n=3$. Suppose this is true for $n$, i.e., $2n+1<2^n$. Let us prove it for $n+1$, i.e., $2(n+1)+1<2^{n+1}$. You have
$$
2n+3=(2n+1)+2<2^n+2 \le 2^{n+1}.
$$
The last inequality is true because $n\ge 3$.</p>
|
2,584,044 | <p>I'm learning about how use mathematical induction.
I'm tasked with proving the inequality shown in (1). It is a requirement that I use mathematical induction for the proof.</p>
<p>$(1) \quad P(n):\quad 2n+1 < 2^{n}, \quad n \ge 3$</p>
<p>I would like some feedback regarding whether my proof is valid and if my u... | Peter Szilas | 408,605 | <p>Hypothesis: </p>
<p>1) You assume that $2n+1 < 2^n$ for an $n.$</p>
<p>Step:</p>
<p>Assuming the hypothesis : </p>
<p>Show that $2(n+1) +1 < 2^{n+1}$, I.e.</p>
<p>the formula holds for $n+1.$</p>
<p>$2n+1 + 2 =$</p>
<p>$2(n+1) +1 < 2^n +2 ;$</p>
<p>$2$ has been added to both sides of $2n+1 <2^n$ ... |
5,563 | <p>This question applies to any package, but I encountered this problem while working with graphs.
There are symbols in the <code>Combinatorica</code> package (such as <code>Graph</code>, <code>IncidenceMatrix</code>, <code>EdgeStyle</code>, and others) that have the same name as analogous symbols in <code>System</code... | Leonid Shifrin | 81 | <p>The answer of @R.M. already explains the essence of the problem. You can streamline the process of removing the <code>Combinatorica</code> from the <code>$ContextPath</code> by loading it via</p>
<pre><code>Block[{$ContextPath}, Needs["Combinatorica`"]]
</code></pre>
<p>(or use <code>Get</code> intead of <code>Ne... |
206,241 | <p>I have a strange (well to me at least) MLE problem. If we let $\{X_i\}_{i=1}^n$ be an i.i.d. sample of a random variable $X$ whose mean is $\mu$ and variance $\sigma^2$. Suppose further that $X_1\sim N(\mu,1)$. I must show that the MLE, $\max\{\bar{X}_n,1\}$for $\max\{\mu,1\}$ suffers from bias. </p>
<p>First of al... | André Nicolas | 6,312 | <p>$1.$ We can even do it without the quadratic formula. We have $a+b=4\sqrt{ab}$ and therefore<br>
$$(a+b)^2=16ab.$$
Also,
$$(a-b)^2=(a+b)^2-4ab=(a+b)^2-\frac{1}{4}(a+b)^2=\frac{3}{4}(a+b)^2.$$
Thus
$$\frac{(a+b)^2}{(a-b)^2}=\frac{4}{3},$$
and therefore
$$\frac{a+b}{a-b}=\pm\frac{2}{\sqrt{3}}.$$</p>
<p>$2.$ The roo... |
206,241 | <p>I have a strange (well to me at least) MLE problem. If we let $\{X_i\}_{i=1}^n$ be an i.i.d. sample of a random variable $X$ whose mean is $\mu$ and variance $\sigma^2$. Suppose further that $X_1\sim N(\mu,1)$. I must show that the MLE, $\max\{\bar{X}_n,1\}$for $\max\{\mu,1\}$ suffers from bias. </p>
<p>First of al... | juantheron | 14,311 | <p>if the quadratic equation:</p>
<blockquote>
<p>$(b^2+c^2)x^2-2(a+b)cx+(c^2+a^2)=0$</p>
</blockquote>
<p>Now we can write it as $\displaystyle (bx-c)^2+(cx-a)^2 = 0$</p>
<p>Means $bx-c = 0\Rightarrow bx = c$ and $cx-a = 0 \Rightarrow cx=a$</p>
<p>So $\displaystyle x = \frac{c}{b} = \frac{a}{c}\Rightarrow c^2 = ... |
39,973 | <p>Assume that we have two residually finite groups $G$ and $H$. Which properties of $G$ and $H$ could be used to show that their pro-finite (or pro-p) completions are different?</p>
<p>I asked a while ago in the group-pub mailing list whether finite presentability is such a property but Lubotzky pointed out that it i... | Ian Agol | 1,345 | <p>There's a theorem that two finitely generated residually finite groups have the same profinite completions if and only if they have the same finite quotients. A reference for the statement of this is <a href="http://www.ma.utexas.edu/users/areid/Groth_revised.pdf" rel="nofollow">Theorem 2 of this paper</a>, but they... |
425,189 | <p>I have a very brief question: if I put $M$ balls into $N$ boxes at random, what is the average number of balls in the boxes that are <strong>not</strong> empty?</p>
| Community | -1 | <p>Let $A$ be the number of non-empty boxes. Then the average number of balls in each box=$\displaystyle{\frac{M}{A}}$. </p>
<p>In random distribution, the value of $A$ may vary. </p>
<p>Probability of $A$ boxes being selected= $\displaystyle{\frac{\binom{N}{A}}{\binom{N}{1}+\binom{N}{2}+\dots \binom{N}{M}}}$</p>
<p... |
2,024,468 | <p>I need to find the points of intersection of a circle with radius $2$ and centre at $(0,0)$ and a rectangular hyperbola with equation $xy=1$. As per the topic statement is there any way to solve this without the graphical method. I have tried setting the $y$ values equal but I cant solve the resulting equation for $... | Babji | 103,742 | <p>Solve
x^2+y^2=4
xy=1</p>
<p>we get 4 solutions for x,y in 1st and 3rd quadrant and symetric around 0,0</p>
<p>(sqrt(6)+sqrt(2))/2,(sqrt(6)-sqrt(2))/2</p>
<p>(sqrt(6)-sqrt(2))/2,(sqrt(6)+sqrt(2))/2</p>
<p>-(sqrt(6)+sqrt(2))/2,-(sqrt(6)-sqrt(2))/2</p>
<p>-(sqrt(6)-sqrt(2))/2,-(sqrt(6)+sqrt(2))/2</p>
|
1,054,346 | <p>Solve for reals:</p>
<p>$a(b+c-a^3)=b(c+a-b^3)=c(a+b-c^3)=1$</p>
<p>I found cyclic relation</p>
<p>$c=(a+b)(a^2+b^2)$</p>
<p>and a solution $a=b=c=1$</p>
<p>But now I am not getting anything.</p>
| Empy2 | 81,790 | <p>A partial solution:<br>
Let $S=a+b+c$, then $a^4+a^2+1=aS$. This quartic has three solutions $a$,$b$,$c$. Suppose they are all different.<br>
The sum of the quartic's four solutions is 0, so the fourth solution is $-S$. So $S^4+S^2+1=-S^2$, and $(S^2+1)^2=0$.<br>
This can't happen in reals, so we must have at le... |
4,068,216 | <p>I know that the sequence <span class="math-container">$\sqrt[n]{n}$</span> converges to 1 and that <span class="math-container">$\text{log}(\sqrt[n]{n})$</span> thus converges to 0 as <span class="math-container">$n\to\infty$</span> since the logarithmic function is continuous. But how can I calculate the limits as ... | Limestone | 866,771 | <p><a href="https://i.stack.imgur.com/I39A1.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/I39A1.png" alt="Triangle PAT" /></a></p>
<p>Draw <span class="math-container">$PG$</span> and let its midpoint be <span class="math-container">$D$</span>. Draw <span class="math-container">$MN$</span>, <span c... |
260,512 | <p>I am stuck with the following problem:</p>
<p>Let $A$ be a $3\times 3$ matrix over real numbers satisfying $A^{-1}=I-2A.$<br>
Then find the value of det$(A).$</p>
<p>I do not know how to proceed. Can someone point me in the right direction? Thanks in advance for your time.</p>
| user1551 | 1,551 | <p>No such $A$ exists. Hence we cannot speak of its determinant.</p>
<p>Suppose $A$ is real and $A^{-1}=I-2A$. Then $A^2-\frac12A+\frac12I=0$. Hence the minimal polynomial $m_A$ of $A$ must divide $x^2-\frac12x+\frac12$, which has no real root. Therefore $m_A(x)=x^2-\frac12x+\frac12$. But the minimal polynomial and ch... |
260,512 | <p>I am stuck with the following problem:</p>
<p>Let $A$ be a $3\times 3$ matrix over real numbers satisfying $A^{-1}=I-2A.$<br>
Then find the value of det$(A).$</p>
<p>I do not know how to proceed. Can someone point me in the right direction? Thanks in advance for your time.</p>
| Patrick Da Silva | 10,704 | <p>There is a formula for the characteristic polynomial for $3 \times 3$ matrices which says that
$$
c_A(\lambda) = \det(A - \lambda I) = -\lambda^3 + c_2 \lambda^2 - c_1 \lambda + \det(A).
$$
(There is an analog for $2 \times 2$ matrices but I don't think there is a general formula for every $n$.) Since
$$
2A^2 - A ... |
3,321,367 | <p>According to Euler's Formula, <span class="math-container">$e^{ix} = \cos(x) + i\sin(x).$</span>
I'm computing the product <span class="math-container">$e^{ix} \cdot e^{iy}.$</span></p>
<p>What is the real part (that is, the term without a factor of <span class="math-container">$i$</span>)?</p>
<p>Why is it <span... | mrtaurho | 537,079 | <p>You made a slight (but crucial) mistake. The correct formula should be</p>
<blockquote>
<p><span class="math-container">$$n = \frac{2^{x+2}-(4q+3^{\color{red}b}n)}{3^{b+1}}$$</span></p>
</blockquote>
<p>And then it is indeed a trivial conclusion as</p>
<p><span class="math-container">\begin{align*}
n&=\frac... |
3,204,547 | <blockquote>
<p>If <span class="math-container">$\alpha,\beta,\gamma \in [-3,10].$</span> Then largest value of the determinant</p>
<p><span class="math-container">$$\begin{vmatrix}3\alpha^2&\beta^2+\alpha\beta+\alpha^2&\gamma^2+\alpha\gamma+\alpha^2\\\\
\alpha^2+\alpha\beta+\beta^2& 3\beta^2&\gamma^2+\... | Helmut | 540,853 | <p>The matrix is the product <span class="math-container">$AB$</span>, where
<span class="math-container">$$A=\begin{pmatrix}\alpha^2&\alpha&1\\\beta^2&\beta&1\\
\gamma^2&\gamma&1\end{pmatrix}\mbox{ and }B=\begin{pmatrix}1&1&1\\\alpha&\beta&\gamma\\
\alpha^2&\beta^2&\gamm... |
2,504,613 | <p>I have problem with solving the following equation:</p>
<blockquote>
<p>$$ty'=3y+t^5y^\frac{1}{3}$$</p>
</blockquote>
<p>I know it's easy without the $y^\frac{1}{3}$ term, but I'm confused now.</p>
<p>Any help would be appreciated.</p>
| Dr. Sonnhard Graubner | 175,066 | <p>for your Control the solution is given by $$\left( y \left( t \right) \right) ^{2/3}-2/9\,{t}^{5}-{t}^{2}{\it
\_C1}=0
$$</p>
|
1,608,734 | <p>I have absolutely no idea to inverse functions containing different functions. Apparently this is a one-to-one function with inverse $f^{-1}$ and I'm asked to calculate the inverses of the given functions in terms of $f^{-1}$</p>
| tomi | 215,986 | <p>Let $y=r^{-1}(x)$</p>
<p>Then $r(y)=x$</p>
<p>So $1-2f(3-4y)=x$</p>
<p>Rearrange to make $y$ the subject...</p>
<p>$1-x=2f(3-4y)$</p>
<p>$\frac{1-x}2=f(3-4y)$</p>
<p>$f^{-1} \left (\frac{1-x}2 \right )=3-4y$</p>
<p>$4y=3-f^{-1} \left (\frac{1-x}2 \right )$</p>
<p>$y=\frac 34- \frac 14 f^{-1} \left (\frac{1-x... |
752 | <p>This question is about how and when can I delete my own answer? I mean remove the answer, not editing it and replace the text by some remark such as "the answer was deleted".</p>
<p>Edit: <em>The "delete" link displays "<strong>vote to remove this post</strong>"</em>. I thought one would only vote to delete (remove... | Qiaochu Yuan | 232 | <p>There should be a "delete" link in the lower left between "edit" and "flag." Note that deleted answers are still visible to people with 2000+ rep (until the beta ends, then it's 10000+ rep).</p>
|
752 | <p>This question is about how and when can I delete my own answer? I mean remove the answer, not editing it and replace the text by some remark such as "the answer was deleted".</p>
<p>Edit: <em>The "delete" link displays "<strong>vote to remove this post</strong>"</em>. I thought one would only vote to delete (remove... | Jeff Atwood | 153 | <p>The behavior depends on the number of answers and votes the post has.</p>
<p>See:<br>
<a href="http://blog.stackoverflow.com/2009/01/adventures-in-delclusionism/" rel="nofollow">http://blog.stackoverflow.com/2009/01/adventures-in-delclusionism/</a></p>
|
995,551 | <p>How would one calculate the log of a number where the base isn't an integer (in particular, an irrational number)? For example:</p>
<p>$$0.5^x = 8 \textrm{ (where } x = -3\textrm{)}$$</p>
<p>$$\log_{0.5}8 = -3$$</p>
<p>How would you solve this, and how would this work for an irrational base (like $\sqrt{2}$)?</p>... | Lubin | 17,760 | <p>Yet another attempt to help. Let’s look at a somewhat different case. How do you calculate $\log_{3.8}(8)$? Again as egreg says, rewrite as $3.8^x=8$, take the (ordinary) log of both sides to get $x\log_{10}(3.8)=\log_{10}(8)$, and $x=\log_{10}(8)/\log_{10}(3.8)$</p>
<p>If you want $\log_{\sqrt2}(8)$, it’s the same... |
14,499 | <p><strong>Definition</strong>: n is said to be a highly composite number if and only if $d(n)>d(m)$ for all $m<n$, where $d(n)$ denotes number of divisors of n.</p>
<p><strong>Questions</strong>:<br>
1) Are there any theorems about highest prime that divides n, assuming that n is highly composite?. It seems tha... | Mike Spivey | 2,370 | <p>This <a href="http://www.ams.org/journals/tran/1944-056-00/S0002-9947-1944-0011087-2/S0002-9947-1944-0011087-2.pdf">paper</a> by Alaoglu and Erdős proves that if $p$ is the largest prime divisor of a highly composite number (HCN) $n$ then the exponent $k_q$ on another prime $q$ dividing
$n$ satisfies
$$\log\left(1 ... |
1,036,684 | <p>There is an inequality:</p>
<p>$$\sqrt[n]{\prod_{i = 1}^{n}{(a_i+b_i)}} \geq \sqrt[n]{\prod_{i = 1}^{n}{a_i}} + \sqrt[n]{\prod_{i = 1}^{n}{b_i}}$$</p>
<p>which I even don't know its name. I'd like to have an ask of its name and usage.</p>
| Umberto P. | 67,536 | <p>It is a special case of the <em>Brunn-Minkowski Theorem</em>. It isn't strange at all; it is a rather useful inequality in measure theory.</p>
|
2,617,286 | <p>While I was solving one problem, with natural variables $(v,x,y,z)$ and I make change to $(\varphi,\chi,\psi,\omega)$ defined as</p>
<p>$\varphi=\arctan\big(\frac{Ax-y}{z}\big) \qquad \psi=\frac{(y-Ax)^2+z^2}{2A} \qquad \chi=\frac{y}{A} \qquad \text{and} \qquad \omega=z-Av $</p>
<p>For continue solving it, I need ... | Stella Biderman | 123,230 | <p>Without a link to the specific webpage, it’s hard to know what’s going on. However, the non-attacking queens problem is usually defined as being about placing $N$ queens on an $N\times N$ board. That problem has no solution for $N=3$. Your proposal isn’t a solution to it because you only have two queens.</p>
|
279,478 | <p>I want to determinate $p$ and $q$ in RSA.</p>
<p>I know that $n = 172451$ and $\phi(n) = 171600$.</p>
<p>$$171600 = pq - (p+q) + 1 = 172451 -(p + q) + 1$$
$$p + q = 172451-171600+1 = 852$$
$$(p-q)^2 = (p+q)^2-4pq = (852)^2 - 4(172451) = 36100$$</p>
<p>Now I'm stuck at this point and don't understand how can I get... | André Nicolas | 6,312 | <p>You are almost finished. We have $(p-q)^2=36100$. Without loss of generality we may assume that $p\ge q$. So $p-q=190$ (we took the square root).</p>
<p>We now know $p+q$ and $p-q$. By adding, we find $2p$ and hence $p$.</p>
|
102,313 | <p>How do I write <em>let</em> in symbolic logic? For example, if I am in the middle of a proof and there is a variable which I can assign to an arbitrary value, what would I write? My best guess is:</p>
<p>$$ x := a $$</p>
<p>Would that be clear? Is there a better way to write "let $x$ equal $a$"?</p>
| Carl Mummert | 630 | <p>There is no symbol in first-order logic that corresponds to "let", and formal proofs in first-order logic do not have a means to assign a value to a variable. </p>
<p>The phenomenon of setting a variable equal to a value is limited to natural-language proofs, in which it hard to improve on "Let $x$ equal $a$".</p>
... |
235,871 | <p>I am writing a Fortran function that needs to receive two vectors of reals, and as an output, returns a vector. The function could be for example the sum of the two vectors:</p>
<pre><code>function sumtwovectors( n, v1, v2 )
implicit none
real(kind=8), dimension(100000) :: sumtwovectors
integer, intent(... | Denis Cousineau | 21,675 | <p>The comment from I.M. "<em>C functions do not return arrays</em>" made me try a different approach with a subroutine rather than a function.</p>
<p>Let the Fortran subroutine be</p>
<pre><code>subroutine vsum( n, v1, v2, v3 )
implicit none
integer, intent(in) :: n
real(kind=8), di... |
522,702 | <p>Let $f(x)\colon \Bbb R\to\Bbb R$ be a bounded function satisfying the following condition:</p>
<p>$$f(x+\frac{13}{42})+f(x)=f(x+\frac16)+f(x+\frac17), \forall x\in\Bbb R$$</p>
<p>Is the function $f(x)$ periodic?</p>
| Jose Garcia | 17,341 | <p>you may write the fucntion as</p>
<p>$$ f(x+13/42)+f(x)=f(x+7/42)+f(x+6/42)=g(x) $$</p>
<p>and $ 42=13.4$ $42=7.6 $ </p>
<p>if this stuff is periodic there will be a number T so $ g(x)=g(x+T) $</p>
<p>this gives $ 42T+13 $ $42T+7$ and $42T+6 $</p>
<p>now if the normal period of the function $ f(x)$ is q so $ f(... |
367,686 | <p>How many injective functions $f:[1,...,m]\to{[1,...,n]}$ has no fixed point? $(m\le n)$</p>
<p>I thought about the next thing:</p>
<p>$f(x_1)\neq x_1$, Means i can choose for $x_1$ - (n-1) options,</p>
<p>But then, for $x_2$, there are two options:</p>
<ol>
<li><p>If i choose $f(x_1)=x_2$ then for $x_2$ i still... | David Bevan | 12,778 | <p>If we represent a no-fixed-point injection $f$ by a labelled digraph with an edge from $x$ to $f(x)$ for each $x\in[m]$, then the digraph consists just of directed paths and oriented cycles (of length at least $2$). </p>
<p>Using the "symbolic method" (see <a href="http://algo.inria.fr/flajolet/Publications/book.pd... |
1,700,493 | <p>The following is an exercise from <em>Linear Analysis</em> by Bollobas.</p>
<p>Let $f:X\to X$, with $X$ a compact metric space. Suppose that for every $\epsilon>0$, there is a $\delta=\delta(\epsilon)$ such that if $d(x,f(x))<\delta$ then $f(B(x,\epsilon))\subset B(x,\epsilon)$. Let $x_0\in X$ and define $x_n... | Andrew D. Hwang | 86,418 | <p>$\newcommand{\eps}{\varepsilon}$If $x = ru$ with $r = \|x\| \geq 0$ and $u$ a point on the unit sphere, and if $\phi$ denotes the unique continuous extension to the origin, as seemed clearly intended, then
$$
\phi(x) = \left(\tan \frac{r\pi}{2\eps}\right) u.
$$
Bijectivity is immediate, as are smoothness of $\phi$ a... |
1,103,217 | <p>I know what does set of generators mean and that subset of $G$ generates some subgroup. But I have no idea how to prove the statement above in title. It's like comes from definition and there is nothing to prove.</p>
| egreg | 62,967 | <p>Consider the set $H$ of products of elements in $S$. This is nonempty and is closed under multiplication. (Note: if $S$ is empty, then $H=G=\{1\}$.)</p>
<p><strong>Theorem.</strong> <em>If a non empty subset $H$ of a finite group $G$ is closed under multiplication, then it is a subgroup.</em></p>
<p>Hint for the p... |
990,222 | <p>I want to prove that $(X \times Y) \setminus (A \times B) = (X \setminus A) \times (Y \setminus B)$ with $A \subseteq X$ and $B \subseteq Y$ is not always true. However, I have difficulties to understand how this can be shown.</p>
<p>The solution of the right side of the equation is a group of Cartesian products of... | Did | 6,179 | <p>For every $i$, let $F_i=\sigma(B_s,s\leqslant t_i)$ then each $e_i$ is $F_i$-measurable and each $\Delta B_i$ is centered, independent of $F_i$, and $F_{i+1}$-measurable. </p>
<p>Now, fix $i\lt j$. Then $e_i$, $e_j$ and $\Delta B_i$ are all $F_j$-measurable and $\Delta B_j$ is independent of $F_j$ hence $e_ie_j\Del... |
990,222 | <p>I want to prove that $(X \times Y) \setminus (A \times B) = (X \setminus A) \times (Y \setminus B)$ with $A \subseteq X$ and $B \subseteq Y$ is not always true. However, I have difficulties to understand how this can be shown.</p>
<p>The solution of the right side of the equation is a group of Cartesian products of... | Martingalo | 127,445 | <p>You can see it using the "tower property" for instance,</p>
<p>Assume $i<j$, then</p>
<p>$$E[e_i e_j \Delta B_i \Delta B_j] = E\left[ E[e_i e_j \Delta B_i \Delta B_j |\mathcal{F}_{t_j}]\right].$$</p>
<p>Now observe that since $i<j$ then $\mathcal{F}_{t_i}\subset \mathcal{F}_{t_j}$ and by hypothesis $e_i$ i... |
3,348,532 | <p>I've some confusions about this textbook question: </p>
<p><a href="https://i.stack.imgur.com/C2v0x.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/C2v0x.png" alt="enter image description here"></a></p>
<ol>
<li><p>Firstly, isn't this question describing a unit sphere (and not a unit circle)? Be... | Paul Sinclair | 258,282 | <ol>
<li>As Joe has already mentioned in the comments, this is about the unit circle in <span class="math-container">$\Bbb R^2$</span>. The Right half is where <span class="math-container">$x \ge 0$</span>, the Bottom half is where <span class="math-container">$y \le 0$</span>.</li>
<li>Actually, the first one is integ... |
3,348,532 | <p>I've some confusions about this textbook question: </p>
<p><a href="https://i.stack.imgur.com/C2v0x.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/C2v0x.png" alt="enter image description here"></a></p>
<ol>
<li><p>Firstly, isn't this question describing a unit sphere (and not a unit circle)? Be... | Mohammad Riazi-Kermani | 514,496 | <p>For each of the given integrls the region is part of a unit disk and the integrand on that region is either positive, negative or mixed. </p>
<p>For integrals of problems <span class="math-container">$8$</span> and <span class="math-container">$9$</span> the integrand is positive so the integral will be positive.<... |
22,638 | <p>Exporting Image files (intermittently) doesn't work:</p>
<p><img src="https://i.stack.imgur.com/C18mr.png" alt="enter image description here"></p>
<p>I get messages back indicating the error is to do with escaping characters:</p>
<pre><code>Syntax::stresc: Unknown string escape \\U.
</code></pre>
| Alejandro Lujambio | 57,565 | <p>Put double slashes instead of single slashes:</p>
<p>Not:</p>
<blockquote>
<p>C:\Users\Blabla</p>
</blockquote>
<p>Rather:</p>
<blockquote>
<p>C:\\Users\\Blabla</p>
</blockquote>
|
1,184,795 | <p>I looked at this as saying that $P(Y=k) = P(X < 10) + P(X=10) + 1 - P(X \le 10)$.</p>
<p>Then for each pmf of X I just put in the summation of each of those according to the geometric distribution. Is that the correct path? </p>
<p>Or Should it be that $P(\min(X, 10) = k) = P(10 \ge X)$?
The ten is throwing me ... | Prahlad Vaidyanathan | 89,789 | <p>An easy way to deal with minima is
$$
P(Y \geq y) = P(X \geq y, 10 \geq y) = \begin{cases}
P(X\geq y) &: y \leq 10 \\
0 & : y>10
\end{cases}
$$
Now write $P(Y=y) = P(Y\geq y) - P(Y\geq y+1)$, so
$$
P(Y=y) = \begin{cases}
P(X=y) &: y \leq 9 \\
P(X\geq 10) &: y=10 \\
0 &: y > 10
\end{cases}
$... |
625,162 | <p>If you know a coupon collector problem, you will know what I am talking about. But if you are not familiar with I will try to explain what is the coupon collector problem.
I have $n$ bins. I throw balls consecutively into these bins. Each bin is choosen independently and with the same probability.
Let's suppose that... | Lubin | 17,760 | <p>Divide the whole mess by $b$, to get a monic real polynomial with constant term $1$. So the two roots are (a) reciprocals of each other; (b) conjugates of each other. But if $1/z=\overline z$, then $|z|=1$.</p>
|
1,619,103 | <p>I'm trying to find the infinite sum that is defined by:</p>
<p>$$
3 \cdot \frac{9}{11} + 4 \cdot \left(\frac{9}{11}\right)^2 + 5 \cdot \left(\frac{9}{11}\right)^3 + \cdots
$$</p>
<p>However, I do not know of any known formula to do this. Am I missing something really simple? Thanks!</p>
| Alan Wang | 165,867 | <p>You can write the series as
$$\sum_{n=3}^{\infty}n\left(\frac{9}{11}\right)^{n-2}$$
By ratio test you would find that this series converge and you can find the sum by using geometric series.</p>
|
1,252,955 | <p>I'm trying to find the sum of the following series:</p>
<p>$$\sum^\infty_{n=1}\frac {(x-3)^{2n}}{2n}$$</p>
<p>I tried to "convert" it to a simple geometrical series, but with no luck. Has someone any idea?</p>
<hr>
<p>Thanks for inspiration! My solution:
$$\sum^\infty_{n=1}\frac {(x-3)^{2n}}{2n} = \sum^\infty_{n... | Jack D'Aurizio | 44,121 | <p>Since for any $|z|<1$ we have:
$$ -\log(1-z)=\sum_{n\geq 1}\frac{z^n}{n}, \tag{1} $$
$$ -\log(1+z)=\sum_{n\geq 1}\frac{(-1)^n\, z^n}{n}, \tag{2} $$
(you can check them both by integrating termwise the usual geometric series for $\frac{1}{1-z}$ or $\frac{1}{1+z}$) by just summing $(1)$ and $(2)$ one gets:
$$ -\fra... |
1,535,057 | <p>I am confused with the inductive step of this very basic induction example in the book <a href="http://rads.stackoverflow.com/amzn/click/0072899050" rel="nofollow">Discrete Mathematics and Its Applications</a>:</p>
<p>$$1 + 2+· · ·+k = k(k + 1) / 2$$</p>
<p>When we apply $k+1$, the equation becomes:</p>
<p>$$1 + ... | Cameron Buie | 28,900 | <p>The kicker, here, is to <strong>assume</strong> $$1+\cdots+k=\frac{k(k+1)}{2}\tag{$\star$}$$ for some $k,$ then use it to <strong>prove</strong> that $$1+\cdots+k+(k+1)=\frac{(k+1)(k+2)}{2}.\tag{$\heartsuit$}$$ To do so, we use $(\star)$ to substitute $\frac{k(k+1)}2$ for $1+\cdots+k,$ which turns $1+\cdots+k+(k+1)$... |
1,535,057 | <p>I am confused with the inductive step of this very basic induction example in the book <a href="http://rads.stackoverflow.com/amzn/click/0072899050" rel="nofollow">Discrete Mathematics and Its Applications</a>:</p>
<p>$$1 + 2+· · ·+k = k(k + 1) / 2$$</p>
<p>When we apply $k+1$, the equation becomes:</p>
<p>$$1 + ... | skyking | 265,767 | <p>I start with the answer in plain text as the OP seem to use screen reader that can't read TeX, the TeX formatted answer follows.</p>
<p>As you seem to have understood you want to prove (in the induction step that)</p>
<p>1+2+...+k+(k+1) = (k+1)(k+2)/2</p>
<p>but already in the assumption in the induction step you... |
1,354,491 | <p>If I wanted to have a die that rolled, for example:</p>
<pre><code>| Roll | Prob (in %) |
|------|-------------|
| 1 | 60 |
| 2 | 25 |
| 3 | 12 |
| 4 | 4 |
| 5 | 1 |
| 6 | 0.2 |
| 7 | 0.04 |
| ... | ... |
</code></pre>
<p>(... | Dleep | 240,562 | <p>If you have an $f: \mathbb{N} \rightarrow [0,1]$ function such that $n$ is a side and $f(n)$ the probability of getting that side, and if we can produce a function $g: \mathbb{N} \rightarrow [0,1]$ such that $g$ is invertible and:</p>
<p>$$ g(n) = \sum_{i=1}^n f(n) $$</p>
<p>Then you could just produce a random fl... |
19,876 | <p>Very important in integrating things like $\int \cos^{2}(\theta) d\theta$ but it is hard for me to remember them. So how do you deduce this type of formulae? If I can remember right, there was some $e^{\theta i}=\cos(\theta)+i \sin(\theta)$ trick where you took $e^{2 i \theta}$ and $e^{-2 i \theta}$. While I am draf... | Arturo Magidin | 742 | <p>In my experience, almost all trigonometric identities can be obtained by knowing a few values of $\sin x$ and $\cos x$, that $\sin x$ is odd and $\cos x$ is even, and the addition formulas:
\begin{align*}
\sin(\alpha+\beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta,\\
\cos(\alpha+\beta) = \cos\alpha\cos\beta - \si... |
714,679 | <p>Could anyone point me a program so i can calculate the roots of</p>
<p>$$ K_{ia}(2 \pi)=0 $$</p>
<p>here $ K_{ia}(x) $ is the modified Bessel function of second kind with (pure complex)index 'k' :D</p>
<p>My conjecture of exponential potential means that the solutions are $ s=2a $ with </p>
<p>$$ \zeta (1/2+is)=... | Antonio Vargas | 5,531 | <p>I can quantify somewhat Raymond's suggestion that the zeros of $K_{ia}(2\pi)$ are much more regular than the zeros of $\zeta(1/2+i2a)$. The calculations below are rough and I didn't verify the details, so this is perhaps more of a comment than an answer.</p>
<p>The Bessel function in question has the integral repr... |
62,471 | <p>Given a minimal parabloic subgroup we know that conjugation by the longest element in the weyl group takes it to the opposite parabolic. </p>
<p>Can we do the same thing if we choose a standard parabolic subgroup? Can we always find an element in the weyl group such that conjugation by this element takes it to the ... | Dahni | 17,550 | <p>For real semisimple Lie groups with finite center the answer is YES: any two parabolic subgroups are conjugate by an element of the Weyl group.</p>
|
6,695 | <p>The standard approach for showing <span class="math-container">$\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$</span> is to multiply by <span class="math-container">$\dfrac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$</span> and then do a substitution with <span class="math-cont... | Aryabhata | 1,102 | <p>A useful technique is to use the half angle formulas in terms of $\tan (\theta/2)$ in order to convert trigonometric (rational) functions into rational functions.</p>
<p>For example if $t = \tan(\theta/2)$ we have that $\sec \theta = \frac{1+t^2}{1-t^2}$</p>
<p>We have $2\,\mathrm dt = (1 + \tan^2(\theta/2))\,\mat... |
6,695 | <p>The standard approach for showing <span class="math-container">$\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$</span> is to multiply by <span class="math-container">$\dfrac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$</span> and then do a substitution with <span class="math-cont... | Américo Tavares | 752 | <p>Instead of presenting another way of evaluating this integral, I justify a more general case in an approach which uses partial fractions and trigonometric identities, at the level of a Calculus class, I think:</p>
<p>$$\int \dfrac{1}{a+b\cos x}dx=\dfrac{1}{\sqrt{b^{2}-a^{2}}}\ln \left\vert
\dfrac{\sqrt{a+b}+\sqrt{... |
6,695 | <p>The standard approach for showing <span class="math-container">$\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$</span> is to multiply by <span class="math-container">$\dfrac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$</span> and then do a substitution with <span class="math-cont... | Martin Gales | 2,323 | <p>Here is a way an electrician solves the problem. Since $\cos(x)=\sin(\frac{\pi}{2} + x)$ it is easier consider the integral $$ I=\int \csc x \, dx = \int \dfrac1{\sin x} \, \mathrm dx$$ </p>
<p>Now: $$ \frac1{\sin x} \, \mathrm dx= \frac1{2\sin \frac{x}{2}\cos\frac{x}{2}} \, \mathrm dx=\frac1{2\tan\frac{x}{2}\co... |
6,695 | <p>The standard approach for showing <span class="math-container">$\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$</span> is to multiply by <span class="math-container">$\dfrac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$</span> and then do a substitution with <span class="math-cont... | Michael Hardy | 11,667 | <p>These articles exist:</p>
<p><a href="http://en.wikipedia.org/wiki/Integral_of_the_secant_function">http://en.wikipedia.org/wiki/Integral_of_the_secant_function</a></p>
<p><a href="http://en.wikipedia.org/wiki/Weierstrass_substitution">http://en.wikipedia.org/wiki/Weierstrass_substitution</a></p>
<p>V. Frederick ... |
6,695 | <p>The standard approach for showing <span class="math-container">$\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$</span> is to multiply by <span class="math-container">$\dfrac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$</span> and then do a substitution with <span class="math-cont... | Idris Addou | 192,045 | <p>Here is another way to compute $$\int \sec x\,dx $$</p>
<p>First, we need a trig identity
\begin{eqnarray*}
\cos^2 x &=&(1-\sin x)(1+\sin x) \\
\frac{1-\sin x}{\cos x} &=&\frac{\cos x}{1+\sin x} \\
\sec x &=&\tan x+\frac{\cos x}{1+\sin x}
\end{eqnarray*}
Next, it suffices to integrate each s... |
6,695 | <p>The standard approach for showing <span class="math-container">$\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$</span> is to multiply by <span class="math-container">$\dfrac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$</span> and then do a substitution with <span class="math-cont... | Michael Hardy | 11,667 | <p>Here's the argument in my less-than-one-page paper in the <em>Monthly</em> in June 2013:$^\dagger$
\begin{align}
x & = \tan \left( \frac \pi 4 + \frac \theta 2 \right) \\[10pt]
\frac{x^2-1}{x^2+1} & = \sin\theta \quad (\text{But we won't use this line, so move on to the next.}) \\[10pt]
\frac{2x}{x^2+1} &... |
2,269,144 | <p>Background: this is Arfken et al mathematical methods 12.5.4 and the answer is 1.</p>
<p>Using the infinite sin product we need the alternating terms in red to cancel when $\pi$ is plugged into z but I don't know how to do that:</p>
<p>$$\frac{\sin(z)}{z(1-z^2/\pi^2)}=\prod_{\color{red}{n=2}}^{\infty}(1-\frac{z^2}... | MathAdam | 266,049 | <p>Given $$a=b$$ you want to know if $$a^n = b^n$$ for some negative value of $n.$ <p> Since $a=b$, let's replace them <em>both</em> with $c$ in the first equation: $$c=c$$ Now let's put both sides of the equation to the power of $n$: $$c^n=c^n$$ Finally, let's replace the first $c$ with $a$ (because $a=c$) and the sec... |
356,749 | <p>I have seen many references in the (geometric representation theory, symplectic geometry, etc) literature to "infinity local systems". </p>
<p>From what I've been told, given a good cover <span class="math-container">$\{U_i\}$</span> of <span class="math-container">$X$</span>, an infinity local system on a connecte... | Jonny Evans | 10,839 | <p>Since you tagged this with "symplectic geometry", I'm going to give an answer from a symplectic geometry perspective, which may not be what you're looking for, but (as a symplectic person) I find it is a helpful point of view. This will use the language of <span class="math-container">$A_\infty$</span>-cat... |
1,575,253 | <p>I need to prove that if $A$ is an $n \times n$ matrix, then $\lambda $ is an eigenvalue of $A$ if and only if $\lambda^k $ is an eigenvalue of $A^k$ for any positive integer $k \geq 1$. I am assuming $\lambda \in \mathbb R$. Otherwise I think the propositions is false.</p>
<p>The first part ($\implies$) is very eas... | Rajat | 177,357 | <p>First, I think this result is true for $\lambda \in \mathbb{C}$ also.This state is also true for any $p(A)$.</p>
<p>So, here I am showing how to prove the $(\Leftarrow)$ part.</p>
<p>What I have is- $p(\lambda_j)$ is a eigenvalue of $p(A)$ and assume that $p$ is a $k$ degree polynomial.</p>
<p>$(p(A)-p(\lambda_j... |
3,166,359 | <p><a href="https://i.stack.imgur.com/KytwH.png" rel="noreferrer"><img src="https://i.stack.imgur.com/KytwH.png" alt="enter image description here"></a></p>
<p>There is black semicircle, which radius is <span class="math-container">$ R $</span>. The red circle is tangentially inward to the semicircle and to the diamet... | orangeskid | 168,051 | <p>Consider a coordinate system with origin <span class="math-container">$O$</span> at the center of the semicircle. Let <span class="math-container">$(x,y)$</span> be the coordinate of the center <span class="math-container">$O'$</span> of a circle of radius <span class="math-container">$r$</span> that is tangent to t... |
301,889 | <p>For example, in a homework assignment I need to prove that a metric space $M$ is isometric to itself. If I say that $M$ maps to $M$ is that the same as saying that theres a function that takes elements from $M$ and I get a result in $M$? Or what is the "literal" meaning behind saying "map/maps/mapping".</p>
| Asaf Karagila | 622 | <p>As sets these are different sets. One set is a set of functions, and the other is a set of ordered pairs of functions. </p>
<p>To see that the cardinality is equal, note that:</p>
<ol>
<li><p>If $I$ is finite then this is really just by definition of cardinal multiplication.</p></li>
<li><p>If $I$ is infinite (and... |
4,171,804 | <p>I proved that it's convergent.But I have a doubt in proving its limit.
I thought like this.</p>
<p>Let <span class="math-container">$x_n$</span> converges to <span class="math-container">$0$</span>.Then for every <span class="math-container">$\varepsilon>0, x_n$</span> should satisfy the condition <span class="ma... | NirF | 782,124 | <p>Since <span class="math-container">$\lim_{n \rightarrow \infty} \space x_n = \lim_{n \rightarrow \infty} \space x_{n+1}$</span> and since the sequence converges, we can set <span class="math-container">$\lim_{n \rightarrow \infty} \space x_n = L$</span>.<br></p>
<p><span class="math-container">$$x_{n+1}=\sqrt{3x_n} ... |
1,221,487 | <p>Problem: Let V be the subspace of all 2x2 matrices over R, and W the subspace spanned by:</p>
<p>\begin{bmatrix}
1 & -5 \\
-4 & 2 \\
\end{bmatrix}
\begin{bmatrix}
1 & 1 \\
-1 & 5 \\
\end{bmatrix}
\begin{bmatrix}
2 & -4 \\
-5 & 7 \\
\end{bmatrix}
\begin{bmatrix}
1 & -7 \\
-5 & 1 \\
\e... | celtschk | 34,930 | <p>Well, to start with, the claim is not <em>completely</em> true: If the number contains of only the digits 0 and 9 (such as 9099090), then clearly ignoring all of them gives you a wrong result (namely 0 instead of 9). So the actual rule is: You can ignore a 9 if there's at least one non-zero non-ignored digit remaini... |
2,433,705 | <p>Does $X\otimes X$ equal $(X\otimes I)(I\otimes X)$? (By the parentheses I mean to signify normal matrix multiplication.)</p>
<p>$X$ is any unitary matrix of the same dimensions as $I$.</p>
| RideTheWavelet | 394,393 | <p>Using block matrix notation, $X\otimes X=[x_{i,j}X]_{i,j}$. $X\otimes I=[x_{i,j}I]_{i,j}$ and $I\otimes X=[\delta_{i,j}X]_{i,j},$ where $\delta_{i,j}$ is the Kronecker delta. Then the product of the latter two matrices may be computed using block matrix multiplication, giving: $$((X\otimes I)(I\otimes X))_{i,j}=\sum... |
325,018 | <p>If $n\in \Bbb N $ such that $\gcd(n,6)=1$ and $a_1,\ldots,a_{\phi( n)}$ are relatively prime with $n$ and smaller than $n$, how to prove : $$n\mid {a_1}^2+\cdots +{a}^2_{\phi (n)}$$</p>
| coffeemath | 30,316 | <p>The result will follow if we take a particular scaling in which the length $XY$ is 2. We approach it using complex numbers for the points, and the fact that four noncollinear complex numbers $a,b,c,d$ are on a circle if and only if the cross-ratio
$$R(a,b,c,d)=\frac{(a-c)(b-d)}{(a-d)(b-c)}$$ is a real number (imagin... |
309,851 | <p>I was considering the integral $\int_{-1}^1 \frac{1}{x^3}\, \mathrm{d} x$. At first, I suspected that it diverged due to the singularity present at $x = 0$, and WolframAlpha verified my hypothesis. However, I attempted to prove this more rigorously, but was unable. This was my reasoning:</p>
<p>$$\int_{-1}^1 \frac{... | André Nicolas | 6,312 | <p>The integral diverges. What you have discovered is the <a href="http://en.wikipedia.org/wiki/Cauchy_principal_value" rel="nofollow">Cauchy Principal Value,</a> which is occasionally useful. It is defined as you did, by finding
$$\lim_{b\to 0^+}\left(\int_{-1}^{-b}\frac{dx}{x^3}+\int_b^1 \frac{dx}{x^3}\right).$$
Do ... |
2,477,817 | <p>The first step in calculating the variance of a Binomial Random Variable is calculating the second moment. </p>
<p><strong><a href="https://i.stack.imgur.com/KsqSQ.png" rel="noreferrer"><img src="https://i.stack.imgur.com/KsqSQ.png" alt="enter image description here"></a></strong></p>
<p>I have no idea as to how t... | Siong Thye Goh | 306,553 | <p>First term is obtained as follows:</p>
<p>\begin{align}
\sum_{j=0}^{n} \frac{n!(j^2-j)}{(n-j)!j!}p^j(1-p)^{n-j} &= \sum_{j=2}^{n} \frac{n!(j^2-j)}{(n-j)!j!}p^j(1-p)^{n-j}\tag{1}\\&=\sum_{j=2}^{n} \frac{n!(j(j-1))}{(n-j)!j!}p^j(1-p)^{n-j} \tag{2}\\
&= \sum_{j=2}^{n} \frac{n!}{(n-j)!(j-2)!}p^j(1-p)^{n-j}... |
1,459,830 | <p>$$r>1$$</p>
<p>The following is applying gauss' law by explicit integration.</p>
<p>$$\int_0^{\pi } \frac{\sin (\theta )}{\sqrt{r^2-2 r \cos (\theta )+1}} \, d\theta=\frac{2}{r}$$</p>
<p>The following is finding the potential of a charged spherical shell by explicit integration. Mathematica takes forever to so... | Harish Chandra Rajpoot | 210,295 | <p><strong>Substitution Method</strong></p>
<p>Let $r^2-2r\cos \theta+1=u^2\implies 2r\sin\theta d\theta=2udu$ or $\sin\theta d\theta=\frac{udu}{r}$
$$\int_{0}^{\pi}\frac{\sin\theta}{\sqrt{r^2-2r\cos \theta+1}}d\theta=\int_{r-1}^{r+1}\frac{1}{u}\frac{udu}{r}$$
$$=\frac{1}{r}\int_{r-1}^{r+1}du=\frac{2}{r}$$</p>
|
347,175 | <p>We all know that a mgf of a random variable <span class="math-container">$m_X(t)$</span> is positive and <span class="math-container">$m(0)=1$</span>. My question is: if a positive real function <span class="math-container">$f(t)$</span> satisfies <span class="math-container">$f(0)=1$</span> and the function is smoo... | SashaP | 39,304 | <p>For any such lift <span class="math-container">$\widetilde{FG}:\mathrm{pTop}\to \mathrm{Gp}$</span> the induced map <span class="math-container">$pTop(X,Y)\to Gp(\widetilde{FG}(X),\widetilde{FG}(Y))$</span> must factor through the set of homotopy classes of the maps between <span class="math-container">$X$</span> an... |
110,377 | <p>How can I extract a single dimension from an <code>InterpolatingFunction</code>? As an example:</p>
<pre><code>ClearAll[x];
s = NDSolve[
Evaluate[Derivative[1][x][t] == -x[t]] && x[0] == {10, -10, 4},
x, {t, 0, 5}]
x = x /. First@s
Plot[x[t], {t, 0, 5}]
</code></pre>
<p><a href="https://i.stack.imgur.... | Dr. belisarius | 193 | <pre><code>x = NDSolveValue[x'[t] == -x[t] && x[0] == {10, -10, 4}, x, {t, 0, 5}]
y[t_?NumericQ] := x[t][[1]] + x[t][[3]]
sum = FunctionInterpolation[y[t], {t, 0, 5}];
Plot[{x[t], sum@t}, {t, 0, 5}]
</code></pre>
<p><img src="https://i.stack.imgur.com/TidM7.png" alt="Mathematica graphics"></p>
<p>But I believ... |
56,942 | <p>If $ x $ and $ y $ have $ n $ significant places, how many significant places do $ x + y $, $ x - y $, $ x \times y $, $ x / y $, $ \sqrt{x} $ have?</p>
<p>I want to evaluate expressions like $ \frac{ \sqrt{ \left( a - b \right) + c } - \sqrt{ c } }{ a - b } $ to $ n $ significant places, where $ a $, $ b $, $ c $ ... | Fox | 14,514 | <p>$ \left[ a , b \right] + \left[ c , d \right] = \left[ a + c , b + d \right] $</p>
<p>$ \left[ a , b \right] - \left[ c , d \right] = \left[ a - d , b - c \right] $</p>
<p>$ \left[ a , b \right] \times \left[ c , d \right] = \left[ \min \left( a \times c , a \times d , b \times c , b \times d \right) , \max \left... |
2,752,646 | <p>$\lim_{n \rightarrow \infty}$ $ n^{b}/a^{n} $ </p>
<p>I have tried to approach it using L Hopital but it is not working. Maybe using sandwich could work but i cant think of the function to enclose it </p>
| user284331 | 284,331 | <p>Assume that $b>0$ and $c>1$, then $c^{n}=(1+p)^{n}\geq\dfrac{n(n-1)}{2}p^{2}$, where $p=1-c>0$, and we have if $a>1$ and $c=a^{1/b}$ that $n^{b}/a^{n}=(n/(a^{1/b})^{n})^{b}=(n/c^{n})^{b}\leq(2/(n-1)p^{2})^{b}\rightarrow 0$ as $n\rightarrow\infty$.</p>
|
4,438,577 | <p>This question was on a math competition.</p>
<blockquote>
<p>Is there a triangle, which is not equilateral, whose sides form a geometric sequence and whose angles form an arithmetic sequence?</p>
<p>If such a triangle exists, find its sides and angles.</p>
</blockquote>
<p>My attempt:</p>
<p>Assume a triangle with s... | Gareth Ma | 948,125 | <p>Following your work, we have</p>
<p><span class="math-container">$$r = \frac{\sin\frac{\pi}{3}}{\sin\left(\frac{\pi}{3} - \phi\right)} = \frac{\sin\left(\frac{\pi}{3} + \phi\right)}{\sin\frac{\pi}{3}}$$</span></p>
<p>The last two terms give <span class="math-container">$\sin\left(\frac{\pi}{3} - \phi\right)\sin\left... |
2,222,616 | <p>Let $V = P_{2}$ be the vector space of polynomials of degree less than or equal to $2$ with real coefficients, and let $W$ be the subset of polynomials $p(x)$ in $P_{2}$ such that:</p>
<p>$$\int_{-2}^{0}p(x)\,dx = 4\int_{0}^{2}p(x)\,dx.$$</p>
<p>$b)$ Find a basis for $W$, and compute $\dim(W)$.</p>
<p>For $b)$, I... | lab bhattacharjee | 33,337 | <p>As $-a\le x\le b\iff-2a\le2x\le2b\iff-(a+b)\le2x+a-b\le(a+b)$</p>
<p>Assuming $a+b>0,$ WLOG $2x+a-b=(a+b)\cos2y$ where $0\le2y\le\pi$</p>
<p>$\sqrt2f(x)=\sqrt{2x+2a}+\sqrt{2b-2x}$</p>
<p>$=\sqrt{(a+b)\cos2y+b-a+2a}+\sqrt{2b-(a+b)\cos2y+a-b}$</p>
<p>$=\sqrt{2(a+b)}(\cos y+\sin y)=2\sqrt{a+b}\sin\left(\dfrac\... |
1,017,026 | <blockquote>
<p><strong>Cauchy-Schwarz Inequality:</strong></p>
<p>If <span class="math-container">$\textbf{u}$</span> and <span class="math-container">$\textbf{v}$</span> are vectors in a real inner product space <span class="math-container">$V$</span>, then <span class="math-container">$$|\left\langle\textbf{u},\text... | dustin | 78,317 | <p>In programming, atan2 is used to determine the correct angle and quadrant. With $\arctan(x)$, the domain is $x\in(-\pi/2, \pi/2)$. This causes problems if $x = \frac{-a}{-b}$. With this case, you get an angle that is off by $\pi$. atan2 will come up with angle $- \pi$.</p>
<hr>
<p><strong>Here is the definition fr... |
935,000 | <p>Let's start considering a simple fractions like $\dfrac {1}{2}$ and $\dfrac {1}{3}$.</p>
<p>If I choose to represent those fraction using decimal representation, I get, respectively, $0.5$ and $0.3333\overline{3}$ (a repeating decimal).</p>
<p>That is where my question begins.</p>
<p>If I multiply either $\dfrac ... | LearningMath | 114,222 | <p>$0.9999$ repeating is equal to $1$. There is some crucial thing you should understand: Every finite decimal of the form $0.999$ is $not$ equal to $0.999$ repeating. The latter represents the number one and no other number. Watch this video and the others related to the construction of reals by the professor Francis ... |
527,568 | <p>I need to integrate, $\int\limits_{|z| = R} \frac{|dz|}{|z-a|^2}$ where $a$ is a complex number such that $|a|\ne R$. </p>
<p>So first I tried polar coordinates, which gives something I cannot continue.</p>
<p>Then I tried to write $|dz| = rd\theta = dz/ie^{i\theta}$ and I have
$$\int\limits_{|z| = R} \frac{dz}{i... | Ron Gordon | 53,268 | <p>First sub $z=R e^{i \phi}$, $dz = i R e^{i \phi} d\phi$, $|dz|=R d\phi$. Then realize that</p>
<p>$$|z-a|^2 = R^2 + |a|^2 - 2 R |a| \cos{\phi}$$</p>
<p>(I set an arbitrary phase to zero - it won't matter for the integration.)</p>
<p>The integral then becomes</p>
<p>$$R \int_0^{2 \pi} \frac{d\phi}{R^2 + |a|^2 - ... |
107,355 | <p>I am looking for a solver that allows me to solve an optimization problem of the form</p>
<p><span class="math-container">$$\begin{array}{ll} \text{minimize} & x_1 x_2 \cdots x_n\\ \text{subject to} & \color{gray}{\text{(some linear constraints)}}\end{array}$$</span></p>
<p>I've used Gurobi before. However, ... | Sándor Kovács | 10,076 | <p>The canonical divisor of the blow up $\pi: X\to \mathbb P^2$ at $k$ ordinary points is
$$
K_X = -3\pi^*L +\sum_{i=1}^k E_i,
$$
where $L\subset \mathbb P^2$ is a hyperplane and $E_i$ is an exceptional curve of the first kind. Choosing the representatives right and an easy computation shows that
$$
K_X^2 = 9 - k,
$$... |
729,352 | <p>I am trying to prove that $a_1$, $a_2$, $a_3$ are linearly independent.</p>
<p>I am asked to use vector product and prove that if $c_{1}a_{1} + c_{2}a_{2} + c_{3}a_{3} = 0$ then $c_1 = c_2 = c_3 = 0$</p>
<p>I am completely stuck on where to go with this problem. I would think that linearly independent then the nul... | 0xC00005 | 917,438 | <p>For any orthonormal vector, we know the following 2 properties:</p>
<ol>
<li><span class="math-container">$a_i^Ta_j=0$</span> when <span class="math-container">$i\neq j$</span>, because they are perpendicular to each other.</li>
<li><span class="math-container">$a_i^Ta_j=1$</span> when <span class="math-container">$... |
2,452,143 | <p>I need to simplify this expression further:</p>
<p>$$ \sum_{m=1}^N (-1)^{m-1} m \binom{N}{m} $$</p>
| Sri-Amirthan Theivendran | 302,692 | <p>Use the identity
$$
\binom{N}{m}\binom{m}{1} =\binom{N}{1}\binom{N-1}{m-1}
=N\binom{N-1}{m-1}
$$
to get that
$$
\sum_{m=1}^N (-1)^{m-1} m \binom{N}{M}
=\sum_{m=1}^N (-1)^{m-1}N\binom{N-1}{m-1}
=N\sum_{u=0}^{N-1} (-1)^{u}\binom{N-1}{u}=0
$$
by considering the binomial expansion of $(1-1)^{N-1}$.</p>
|
3,378,104 | <blockquote>
<p>Let <span class="math-container">$\mathcal{F} \subseteq \mathcal{G}$</span>.</p>
<p>Show that <span class="math-container">$$E((E(X\mid \mathcal{G})-E(X\mid \mathcal{F}))^2=E(E(X\mid \mathcal{G}))^{2}-E(E(X\mid \mathcal{F}))^{2}$$</span></p>
</blockquote>
<p>My idea:</p>
<p><span class="math-container">... | Mark Fischler | 150,362 | <p>Let's transform <span class="math-container">$f(x,y) = xy\log(x^2+y^2)$</span> into a more useful form: Let
<span class="math-container">$$
x = r \cos \theta\\ y = r\sin \theta \\ f(x,y) = r^2\cos\theta\sin\theta \log(r^2) = r^2\cos (2\theta)\log r
$$</span></p>
<p><span class="math-container">$\cos(2\theta)$</span... |
4,066,653 | <p>We are given two non-intersecting circles centered at <span class="math-container">$O_{1}$</span> and <span class="math-container">$O_{2}$</span>. For simplicity sake, let <span class="math-container">$O_{1} = (0,0)$</span> and <span class="math-container">$r_{1} = r_{2} = 10$</span>. Let <span class="math-containe... | William M. | 464,801 | <p>So I came back to this a few days ago and have an answer for my own question.<br />
As a reminder, <strong>a few assumptions</strong>:</p>
<ul>
<li><span class="math-container">$r$</span> is the same for both circles. I do not try to generalize at all</li>
<li>without loss of generality, both circles are translated ... |
727,664 | <p>I need to evaluate $I = \int^\pi_{-\pi} \cos^3(x) \cos(ax)~dx$ where $a$ is some integer. </p>
<p>I get:
$\dfrac{2a(a^2-7)\sin(\pi a)}{a^4 - 10a^2 + 9}$. However $\sin(\pi a)$ is $0$ for all $a$ so $I=0$. But as noted by an answer, there are answers for $a = 1,3,-1,-3$. </p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>We can use $$\cos3x=4\cos^3x-3\cos x\implies \cos^3x=\frac{\cos3x+3\cos x}4$$</p>
<p>Then use <a href="http://mathworld.wolfram.com/WernerFormulas.html" rel="nofollow">Werner</a> Formulas</p>
<p>Also, observe that $ax+x, ax-x$ have same parity like $3x+ax,3x-ax$</p>
|
1,580,634 | <blockquote>
<p>Given i.i.d random variable with mean $\mu$ and variance 1, $\bar{X}_n =
\frac{1}{n}(X_1+\cdots+X_n)$,
use the CLT to approximate the following probability:</p>
</blockquote>
<p>$$P(|\bar{X}_n - \mu| \ge \frac{2}{\sqrt {n}})$$</p>
<p><strong>My attempt:</strong></p>
<p>$$P(|\bar{X}_n - \mu| \ge ... | Em. | 290,196 | <p><strong>EDIT:</strong>
Since last I read this, there has been a change in the value of $\mu$.
So, I will let $\bar X = \bar X_n$. I think you can determine that $\mu = E[\bar X] $, and $\sigma^2 = \text{Var}[\bar X] = \frac{1}{n}$. Then
\begin{align*}
P\left[|\bar X - \mu| > \frac{2}{\sqrt n}\right]
&= P\left... |
774,485 | <p>So I'm trying to solve this problem but not sure how. Consider the vectors u=(1,2,3) and v=(2,3,1) in r3. Find k so that w=(1,k,4) is a linear combination of u and v. </p>
<p>I'm not sure what to do. Any help would be greatly appreciated. Sorry if format isn't correct, asking this from my tablet.</p>
| Community | -1 | <p>You look for $\alpha$ and $\beta$ such that
$$w=\alpha u+\beta v$$
so we find for the components:
$$1=\alpha+2\beta\quad;\quad k=2\alpha+3\beta\quad;\quad 4=3\alpha+\beta$$
now solve the first and the last equations for $\alpha$ and $\beta$ and then you find $k$ in the second equality.</p>
|
766,359 | <p>$12$ women and $10$ men are on the faculty. How many ways are there to pick a
committee of $7$ if </p>
<p>(a) Claire and Bob will not serve together, </p>
<p>(b) at least one woman must be chosen</p>
<p>I'm not sure how to start a. Essentially I need to remove a woman and remove a man?</p>
<p>For b, $\binom{22}{... | user145252 | 145,252 | <p>For part a, I would use a sum:
(without Bob and Claire) + (with only Bob) + (with only Claire) = all possibilities where both Bob and Claire are not together. For the groups formed with only one of them, say Bob, there's only 6 selections left to make up the committee of 7 so r = 6. Additionally, once we've decid... |
423,334 | <h3>Set up</h3>
<p>Suppose <span class="math-container">$\gamma$</span> a simple closed curve, oriented in a counterclockwise direction. <span class="math-container">$f(z)$</span> is a complex polynomial
<span class="math-container">$$
f(z)=a_nz^{n}+a_{n-1}z^{n-1}+\cdots+a_0.
$$</span>
We already know that the integral... | Kevin Casto | 5,279 | <p>This is just an extended comment to answer your concerns. Maybe it's helpful to treat <span class="math-container">$f$</span> as a black-box function, where all we know is that it's meromorphic on <span class="math-container">$\mathbb CP^1$</span> (of course, this means it's actually rational, but let's suppress tha... |
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