qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,531,493 | <blockquote>
<p>How many different integers can be expressed as the sum of three distinct numbers from the set{$13$,$10$,$23$,$28$,$33$,$36$,$43$,$48$}?</p>
</blockquote>
<p><strong>MyApproach</strong></p>
<p>Out of $8$ numbers, Select $3$ distinct numbers.</p>
<p>So Ans would be $8$C$3$=$56$-2=$54$.</p>
<p>Becau... | Brian Risk | 181,857 | <p>These kinds of questions seem good to start with a programming approach. Maybe that's gauche, but call me gauche! This Java snippet loops through all 3-tuple and uses the uniqueness constraint of HashMaps to find the unique answers:</p>
<pre><code>public static void main(String [] args) {
int [] array = {13,... |
2,391,931 | <p>The rate of data transfer, $r$, over a particular network is directly proportional to the bandwidth, $b$, and inversely proportional to the square of the number of networked computers, $n$.</p>
<p><strong>Quantity A</strong> = The resulting rate of data transfer if the bandwidth is quadrupled and the number of net... | symplectomorphic | 23,611 | <p>You cannot say $r=\alpha b$ if you know $r$ depends on another variable as well (in this case, $n$). For example, consider that the volume of a cylinder is directly proportional to both the height and the square of the radius. Then it does not follow that $V=\alpha h$ for some constant $\alpha$. Rather, $\alpha$ mus... |
2,130,076 | <p>The problem is
$$\lim_{x\to 0} \frac{(x \sqrt{1 + \sin x} - \ln{\sqrt{(1 + x^2)}-x)}}{\tan^3{x}} $$</p>
<p>I know that
$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!}... $$
got this
$$\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15}...$$ </p>
<p>and suppose to use taylor expansion for $\ln {(1 + x)}$</p>
<p>Trie... | Ennar | 122,131 | <p>Note that $$x\sqrt{1+\sin x} = x\sqrt{1+x+O(x^3)} = x(1+\frac x2-\frac{x^2}8+O(x^3)) = x+\frac{x^2}{2}-\frac{x^3}8+O(x^4)$$</p>
<p>and</p>
<p>$$\ln\sqrt{1+x^2} = \ln(1+\frac{x^2}2+O(x^4)) = \frac{x^2}{2}+O(x^4)$$</p>
<p>so we have</p>
<p>$$\lim_{x\to 0}\frac{x\sqrt{1+\sin x}-\ln\sqrt{1+x^2}-x}{\tan^3x} = \lim_{x... |
3,860,982 | <p>How do I prove that if:
<span class="math-container">$$\cos^3(x) + \sin^3(x) = 1$$</span>
then:
<span class="math-container">$$\cos(x) = 0 ; \sin(x)=1 \text{ or } \cos(x)=1 ; \sin(x)=0?$$</span></p>
<p>Starting from the first expression, I couldn't figure out how to reach the conclusion. I replaced 1 by <span class=... | CyclotomicField | 464,974 | <p>Rather than carry the notation we write <span class="math-container">$s=\sin(x)$</span> and <span class="math-container">$c=\cos(x)$</span>. Subtracting the equation from <span class="math-container">$s^2+c^2=1$</span> yields <span class="math-container">$s^3-s^2 + c^3-c^2=0$</span> or <span class="math-container">$... |
3,860,982 | <p>How do I prove that if:
<span class="math-container">$$\cos^3(x) + \sin^3(x) = 1$$</span>
then:
<span class="math-container">$$\cos(x) = 0 ; \sin(x)=1 \text{ or } \cos(x)=1 ; \sin(x)=0?$$</span></p>
<p>Starting from the first expression, I couldn't figure out how to reach the conclusion. I replaced 1 by <span class=... | orangeskid | 168,051 | <p>Write <span class="math-container">$u = \cos x$</span>, <span class="math-container">$v=\sin x$</span>. From <span class="math-container">$u^2 + v^2=1$</span> we get <span class="math-container">$u$</span>, <span class="math-container">$v$</span> <span class="math-container">$\le 1$</span> so <span class="math-con... |
181,532 | <blockquote>
<p>Find an ideal $I$ of $\mathbb{Z}[i]$ such that $\mathbb{Z}[i]/{I}$ is a field.</p>
</blockquote>
<p>How can one justify the answer in the shortest number of lines?</p>
| William | 13,579 | <p>If $R$ is a ring and $I$ is a maximal ideal of $R$, then $R / I$ is a field. </p>
<p>$\mathbb{Z}[i]$ is a Euclidean domain, so prime ideals are maximal. Since $\mathbb{Z}[i]$ is a Euclidean domain, it is a PID. Thus it suffices to find the prime elements of $\mathbb{Z}[i]$. </p>
<p>A well-known result is that: </p... |
181,532 | <blockquote>
<p>Find an ideal $I$ of $\mathbb{Z}[i]$ such that $\mathbb{Z}[i]/{I}$ is a field.</p>
</blockquote>
<p>How can one justify the answer in the shortest number of lines?</p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> $\ \Bbb Z[i]/(i\!-\!1)\cong \Bbb Z/2\:$ is the unique extension of <em>parity</em> from $\,\Bbb Z,\:$ given by defining $\,i\,$ to be odd.</p>
<p><strong>Remark</strong> $\ $ Any ring having $\,\Bbb Z/2\,$ as an image has parity structure induced by such. Generally it not be unique, e.g. see <... |
222,759 | <p>Here is the given series 3/(9n+1), decide whether it converges or diverges.
I used the ratio test only to end up with the ratio=1.
I know this is harmonic series but it is smaller than 1/n, therefore i cannot conclude it diverges.
Please help!!</p>
| M. Strochyk | 40,362 | <p><em>Hint</em></p>
<p>$$\dfrac{3}{9 n+1} \geqslant \dfrac{3}{12n}=\dfrac{1}{4}\cdot \dfrac{1}{n}$$</p>
|
2,565,880 | <p>Number of ways in which 5 boys and 4 girls can be seated around a circular table such that no two girls sit together and two particular boys are always together ?</p>
<p>The answer to this question is $3!2!4!$ . It is done by considering $2$ boys as one unit and the the number of units (of boys) is $4$ so they can ... | Asinomás | 33,907 | <p>You want to minimize $f(x,y)$ (maybe the minimum doesn't exist.</p>
<p>Notice that we can work only inside $B$ where $B$ is a closed ball of big radius, because $f$ is large outside $B$.</p>
<p>We conclude that the minimum does exist because $B$ is compact.</p>
<p>We know that the minimum occurs in the interior o... |
17,335 | <p>Starting with a representation <span class="math-container">$\rho:G \to \mathrm{GL}(V)$</span>. Then we can build the tensor product of <span class="math-container">$V$</span> with itself by defining <span class="math-container">$g(v_1 \otimes v_2) = g(v_1) \otimes g(v_2)$</span>. Then by saying <span class="math-... | Douglas Zare | 2,954 | <p>$V \wedge V$ is bad notation when $V$ is a representation, just as $V~ Sym ~V$ would be. $\wedge^2 V$ is less misleading. </p>
<p>You could try to define $V \wedge W = \wedge^2 (V \oplus W)$. For $v\in V, w\in W$, we can naturally identify $v\wedge w$ and $w \wedge v$ with elements of $\wedge^2 (V \oplus W)$, and $... |
628,409 | <p>$$f(x)=\max\{x,0\}$$</p>
<p>I want to check whether this function is continuous in its domain $\mathbb{R}$ or not, but unfortunately I have no idea how to start.</p>
| copper.hat | 27,978 | <p>Slightly more generally, let $\phi:\mathbb{R}^n \to \mathbb{R}$ be given by $\phi(x) = \max_k x_k$.
Let $I(x) = \{ k | x_k = \phi(x) \}$.</p>
<p>If $k \in I(x), k' \in I(y)$, we have $\phi(x)-\phi(y) = x_k-y_{k'} \le x_k -y_k \le \|x-y\|_\infty$. Reversing the roles of $x,y$ gives
$| \phi(x)-\phi(y) | \le \|x-y\|_\... |
307,144 | <p>Are there irrational numbers for which we know that computing its nth digit would take (at least) linear/polynomial/exponential/superexponential time (wrt to length of n and with "big enough" n)?</p>
| Michael Joyce | 17,673 | <p>Every solution to $Ax = b$ is a solution to $A^T A x = A^T b$, but not the other way around. A solution to $A^T A x = A^T b$ need not be a solution to $Ax = b$, unless $A^T$ is invertible (which is often not the case in applications).</p>
|
307,144 | <p>Are there irrational numbers for which we know that computing its nth digit would take (at least) linear/polynomial/exponential/superexponential time (wrt to length of n and with "big enough" n)?</p>
| abel | 9,252 | <p>here is an example of $A^TAx = b$ has a solution but $Ax = b$ has none. take $$A= \pmatrix{1\\0}, b = \pmatrix{1\\1}.$$ then $$A^\top A=1, A^\top b=1, A^\top Ax = b\to x = 1 $$ but $$ Ax = b \to \pmatrix{x\\0} = \pmatrix{1\\1}$$ has no solution.</p>
|
3,341,059 | <p>In design controller for a first order system such as:
<span class="math-container">$$\dot{x}=-ax+bu$$</span>
they assume that value of <span class="math-container">$a>0$</span>. I need to know is it possibile to neglect this assume an design the controller even if <span class="math-container">$a=0$</span> as sa... | Lee Mosher | 26,501 | <p>You don't have to worry about <span class="math-container">$x < 3$</span> because it is not in the domain. Irrational powers are not defined when the base is negative. </p>
<p>You might remember that for an irrational number <span class="math-container">$a$</span>, by definition the exponentiation function with ... |
3,341,059 | <p>In design controller for a first order system such as:
<span class="math-container">$$\dot{x}=-ax+bu$$</span>
they assume that value of <span class="math-container">$a>0$</span>. I need to know is it possibile to neglect this assume an design the controller even if <span class="math-container">$a=0$</span> as sa... | Alexdanut | 629,594 | <p>Hint: For <span class="math-container">$x<3$</span>, <span class="math-container">$(x-3)^{\sqrt 2}$</span> is not defined, so you must impose the condition <span class="math-container">$x\ge 3$</span> from the start. Aftwerwards you may proceed as you did. </p>
|
43,688 | <p>The nuclear norm of a matrix is defined as the sum of its singular values, as given by the singular value decomposition (SVD) of the matrix itself. It is of central importance in Signal Processing and Statistics, where it is used for matrix completion and dimensionality reduction. </p>
<p>A question I have is wheth... | dmuir | 4,830 | <p>This is just a guess, but since the sum of the singular values of A is the maximimum of tr(A*U) where U is orthogonal, might one be able to estimate this by computing the maximum sweeping over the <a href="http://en.wikipedia.org/wiki/Givens_rotation" rel="nofollow">givens rotations</a> ? </p>
<p>The maximum, over ... |
1,466,662 | <p>Hi i have the answer but don't understand so please explain your answer... it should be $-yx^{(y-1)}$</p>
| Konstantinos Michailidis | 50,350 | <p>It should be $\frac{d}{{dx}}\left( {{x^y}} \right) = y{x^{y - 1}}$</p>
|
34,657 | <p>In section III.1 of P.M. Cohn's <a href="http://books.google.co.uk/books?id=vZsHZ1YP4KkC&lpg=PA108&ots=GIztdoRc2E&dq=universal%20functor&pg=PA108#v=onepage&q=universal%20functor&f=false" rel="nofollow">Universal Algebra</a> a notion of <em>universal functor</em> ${\cal L} \rightarrow {\cal K}... | Andres Koropecki | 19,393 | <p>I'm late by a year, but just in case, the result from this one-page paper seems to answer your question (but it has nothing to do with the Wada property):</p>
<p>M. Brown and J. M. Kister, Invariance of complementary domains of a fixed point set, Proc. Amer. Math. Soc. 91 (1984), no. 3, 503–504. MR 744656</p>
<p>T... |
3,393,655 | <p>I want to show that:
<span class="math-container">$$
\lim_{n \rightarrow \infty} \int_0^{2\pi} \sin(x)^n \, dx=0
$$</span></p>
<p>and my idea was to use DCT (dominated convergenece theorem).</p>
<p>However, my textbook has the requirement that <span class="math-container">$u(x)=\lim_{n \rightarrow \infty} u_n(x)$<... | Marios Gretsas | 359,315 | <p><span class="math-container">$(\sin{x})^n \to 0$</span> a.e on <span class="math-container">$[0,2\pi]$</span></p>
<p>And <span class="math-container">$|\sin{x}|^n \leq 1 \in L^1([0,2\pi])$</span></p>
<p>So from Dominated Convergence theorem you have the conclusion.</p>
|
1,810,348 | <p>v= R^4 is an <strong>Inner product space</strong> and u=span{(1,0,-1,0)} subspace.
how can I find a base for the vectors which orthogonal to U(the complement of U)?</p>
<p>Thanks!</p>
| mvw | 86,776 | <p>$n = e_1 - e_3$ is the base vector for $U$.</p>
<p>If we had three other vectors, such that all four are linear independent, we could use <a href="https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process" rel="nofollow">Gram-Schmidt</a> starting with $n$ to come up with a orthonormal basis for $V$. The other thre... |
4,155,766 | <p>The area of a triangle is <span class="math-container">$14\sqrt{3}$</span> <span class="math-container">$cm^2$</span>. The lengths of two sides of the triangle are <span class="math-container">$7$</span> <span class="math-container">$cm$</span> and <span class="math-container">$8$</span> <span class="math-container"... | Math Lover | 801,574 | <p>As you rightly evaluated, <span class="math-container">$\angle C = 120^0$</span> or <span class="math-container">$60^0$</span>.</p>
<p>And yes, WLOG <span class="math-container">$ \ a = 7, b = 8$</span>.</p>
<p><a href="https://i.stack.imgur.com/yIcSi.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.co... |
5,238 | <p>For my homework, I have been asked to rationalise and simplify this surd;</p>
<p><span class="math-container">$$\frac{11}{3\sqrt{3}+7}$$</span></p>
<p>Each time I do this I get the wrong answer. The method I am using is;</p>
<p><span class="math-container">$$ \frac{11}{3\sqrt3+7} \times \frac{3\sqrt3-7}{3\sqrt3-7... | Rasmus | 367 | <p><strong>HINT:</strong> The general trick is
$$
\frac{1}{\sqrt{a}+b}=\frac{\sqrt{a}-b}{(\sqrt{a}+b)(\sqrt{a}-b)}=\frac{\sqrt{a}-b}{a-b^2}.
$$</p>
|
5,238 | <p>For my homework, I have been asked to rationalise and simplify this surd;</p>
<p><span class="math-container">$$\frac{11}{3\sqrt{3}+7}$$</span></p>
<p>Each time I do this I get the wrong answer. The method I am using is;</p>
<p><span class="math-container">$$ \frac{11}{3\sqrt3+7} \times \frac{3\sqrt3-7}{3\sqrt3-7... | Bill Dubuque | 242 | <p>You're mistakenly multiplying $\rm\; a * b\sqrt{3} \ =\ ab + a\sqrt{3}\:\,\;$ but $\rm\; ab\:\sqrt{3}\;$ is correct. </p>
<p>In other words $\rm\; b\:\sqrt{3}\;$ means $\rm b * \sqrt{3}\:,\;$ not $\rm\; b + \sqrt{3}\:.$</p>
<p>Also, to rationalize the denominator use $\rm\; (a+b\sqrt 3)\:(a-b\sqrt 3)\ =\ a^2 - 3 ... |
1,655,992 | <p>Give an example of a equicontinuous sequence of functions ($f_n$) over a non-compact set $S\subset\Bbb R^n$ converging pointwise to a function $f$ at each $x\in S$, but $f_n$ does not converge uniformly to $f$ over $S$.</p>
<p>I'm really stuck on this problem, and I thought about the cases of $f_n(x) = x^n$ with th... | Martín-Blas Pérez Pinilla | 98,199 | <p>$$f_1(x) = 0$$
except in a "hump" between 0 and 1.
$$f_n(x) = f_1(x-n).$$</p>
|
221,279 | <blockquote>
<p>How does one show that $\chi_{[1, \infty)}1/x$ is not (Lebesgue) integrable?</p>
</blockquote>
<hr>
<p>What I could think of is as follows:</p>
<p>Letting $f(x)=1/x$ (defined for $x\geq 1$), define
$$
f_n(x)=f\chi_{[1, n)}(x).
$$ </p>
<p>Each $f_n$ is, therefore, Riemann integrable on $[1, n)$ wi... | Julien | 38,053 | <p>By definition, $\int_{[1,+\infty)}|f(x)|dx=\int_{[1,+\infty)}1/|x|dx$ is the sup, over all simple functions $0\leq s \leq |f|$, of the $\int_{[1,+\infty)}s(x)dx$. Now for each $N\geq 1$. Observe that
$$
0\leq s_N(x):=\sum_{n=1}^N \frac{1}{n+1} 1_{(n,n+1)}(x)\leq\frac{1}{|x|}\qquad\forall x\geq 1.
$$
Hence
$$
\int_{[... |
3,603,312 | <p>there are two urns with White balls and Black balls. first urn has 21 whites and 5 blacks, second one has 8 whites and 9 blacks. we take 7 balls from first urn and put them into the second one. afterwards, out of the second urn we take one ball. what is the probability that it's white?</p>
<p>I've been struggling o... | drhab | 75,923 | <p><strong>Method</strong>: Calculate for every white ball the probability to become the chosen ball and add these probabilities (of mutual exclusive events).</p>
<hr>
<p>A white ball that was at the start of this in the first urn has probability
<span class="math-container">$\frac{7}{26}\frac{1}{24}$</span> to be ch... |
2,706,358 | <p>Is the function $f(x)=9-x^2$ continuous? </p>
<p>1.Lets say for $x=1$ </p>
<p>$f(1)= 9-1=8$</p>
<p>2.$\lim\limits_{x \to 1} (9-x^2)= 9-1 = 8$</p>
<p>3.So $f(a) = \lim\limits_{x \to a} f(x) $</p>
<p>Does this mean that the function is continuous ?</p>
| JetJet13 | 364,556 | <p>I don't want to explicitly walk you through the answer because this looks like a homework question and it's important that the solution you come up with is your own so I'm only going to provide you with some direction.</p>
<p>In regards to the steps you walked through, you proved that a limit exists at $x=1$, which... |
424,978 | <blockquote>
<p><img src="https://i.stack.imgur.com/VMq02.png" alt="enter image description here"></p>
<p><img src="https://i.stack.imgur.com/eEZlE.png" alt="enter image description here"></p>
</blockquote>
<p>How does he know that $f^{-1}$ is one-one? Doesn't he have to prove that? Or is he applying his first ... | Will Orrick | 3,736 | <p>To answer the additional question in your edit: if $f$ is decreasing then $f(b)<f(a)$ whenever $b>a.$ Therefore $-f(b)>-f(a)$ whenever $b>a.$ Hence $-f$ is increasing. By the previous result, the inverse of $-f$ is therefore increasing. This implies that $f^{-1}$ is decreasing.</p>
<p>To understand ... |
4,612,286 | <p>This is a bit of a soft question, but I am interested in a list of classes of structures (in the sense of model theory) which are "surprisingly" first-order axiomatizable classes. Meaning, the class of structures is defined in such a way that it is not at all obvious that it is in fact first-order axiomati... | Noah Schweber | 28,111 | <p>Here's a class of examples which connects to a currently-0pen large-cardinals-flavored question:</p>
<p>Given a first-order theory <span class="math-container">$T$</span>, let <span class="math-container">$NE(T)$</span> be the class of structures <strong>not</strong> embeddable in a model of <span class="math-contai... |
2,373,109 | <blockquote>
<p>Let $\{x,y,z\}\subset[0,+\infty)$,and $x+y+z=6$. Show that:
$$xyz(x-y)(x-z)(y-z)\le 27$$</p>
</blockquote>
<p>I tried AM -GM but without success.
$$xyz\le\left(\dfrac{x+y+z}{3}\right)^3=8$$
maybe $$(x-y)(x-z)(y-z)\le \dfrac{27}{8}$$ it doesn't always true。</p>
| River Li | 584,414 | <p><strong>Update</strong> 2019/10/05 </p>
<p>(add the proof of <span class="math-container">$a = b = c = 1$</span> and <span class="math-container">$ABC(A-B)(A-C)(B-C) = 27$</span>)</p>
<p><span class="math-container">$\phantom{2}$</span></p>
<p>Note that <span class="math-container">$xyz(x-y)(x-z)(y-z)$</span> is... |
2,016,588 | <p>I'd really appreciate a push in the right direction for solving this. I just can't get it. Thanks</p>
<p>Prove $ ({x+1})^{1/3} < 1 + {\frac13}x $ for x > 0 </p>
| Lai | 732,917 | <p>We are going to evaluate the integral by auxiliary angle. <span class="math-container">$$
\begin{aligned}
\int \frac{2 d x}{(\cos x-\sin x)^{2}} &=\int \frac{2 d x}{\left[\sqrt{2} \cos \left(x+\frac{\pi}{4}\right)\right]^{2}} \\
&=\int \sec ^{2}\left(x+\frac{\pi}{4}\right) d x \\
&=\tan \left(x+\frac{\pi... |
2,117,784 | <p>A <strong>multilinear form</strong> is a mapping</p>
<p>\begin{align}
\Delta: V^n \rightarrow K
\end{align}</p>
<p>where $V$ is a finite-dimensional vector space over field $K$.</p>
<p>It must meet the following requirements:</p>
<ul>
<li>First:</li>
</ul>
<p>\begin{align}
&\Delta\left(a_1, \dots, a_{i-1}, ... | egreg | 62,967 | <p>If you have a bilinear form $\varphi\colon U\times V\to K$, you can define a map $\varphi_v\colon U\to V^*$, for each $u\in U$, by
$$
\varphi_u(v)=\varphi(u,v)
$$
and this is a linear map.</p>
<p>Conversely, if you have a linear map $f\colon U\to V^*$, you can define a bilinear map $\hat{f}\colon U\times V\to K$ by... |
1,025,548 | <p>Can one help finding this limit</p>
<p>$$\lim_{x \to 0}\frac{x^{3}-\sin^{2}x\tan x}{\tan(\sin x) - \sin (\tan x)}$$</p>
<p>L'Hospital's rule is permited.</p>
<p>(<a href="https://math.stackexchange.com/questions/516483/find-lim-lim-x-to0-frac-tan-tan-x-sin-sin-x-tan-x-sin-x">Find lim:$\lim_{x\to0} \frac{\tan(\tan... | Idris Addou | 192,045 | <p>Seven months ago, I have posted this question of computing the limit
\begin{equation*}
\lim_{x\rightarrow 0}\frac{x^{3}-\sin ^{2}\tan x}{\tan (\sin x)-\sin (\tan x)
},
\end{equation*}
without making use of Taylor series. After that, I have wrote the fraction
as a product
\begin{equation*}
\frac{x^{3}-\sin ^{2}\tan x... |
1,625,306 | <p>I have to evaluate an integral $I(a) = \sin(ax)\cos(x)$ from $0$ to $\pi/2$.The variable of $a$ is not is greater than $1$:</p>
<p>$$\int_0^{\pi/2} \sin(ax)\cos(x)\,dx$$
I attempted to change the function to $[\sin(ax+x)+\sin(ax-x)]/2$ and then integrate, but I am left with (-)cosines with a zero in the denominato... | LeviathanTheEsper | 184,454 | <p>When you change the function and integrate you get</p>
<p>$$\frac{1}{2}\int\sin(ax+x)+\sin(ax-x)dx$$
Then the integral of each term is, for example:</p>
<p>$$\int\sin(ax+x)dx=-\frac{\cos(ax+x)}{a+1}.$$
Unless $a+1=0$ or $a-1=0$ (For the other integral) you shouldn't have any problem integrating it with respect to ... |
1,625,306 | <p>I have to evaluate an integral $I(a) = \sin(ax)\cos(x)$ from $0$ to $\pi/2$.The variable of $a$ is not is greater than $1$:</p>
<p>$$\int_0^{\pi/2} \sin(ax)\cos(x)\,dx$$
I attempted to change the function to $[\sin(ax+x)+\sin(ax-x)]/2$ and then integrate, but I am left with (-)cosines with a zero in the denominato... | Sasha | 11,069 | <p>Your approach is good. Integrate the following identity term-wise:
$$
\sin(a x) \cos(x) = \frac{1}{2} \sin\left(\left(1+a\right)x\right) - \frac{1}{2} \sin\left(\left(1-a\right) x\right)
$$
getting
$$
\int_0^{\pi/2} \sin(a x) \cos(x) \mathrm{d}x = \frac{1}{2} \int_0^{\pi/2} \sin\left(\left(1+a\right)x\right)\m... |
2,701,182 | <p>I must once again resort to the advice of this great community.</p>
<p>As I was reading about the pigeonhole principle something about its proof struck me as odd. Allow me to explain:</p>
<p>After reading the "The Foundations: Logic and Proofs" chapter in Rosen's "Discrete mathematics and its applications" book I ... | David | 119,775 | <p>Here is a way to put more symbols into the proof.</p>
<blockquote>
<p>Let $t$ be the total number of objects, and for $j=1,2,\ldots,k$ let $n_j$ be the number of objects in box $j$. Suppose $\forall j\ n_j\le1$. Then
$$t=\sum_{j=1}^k n_j\le\sum_{j=1}^k1=k\ .$$
This is a contradiction since $t=k+1$. So we h... |
2,701,182 | <p>I must once again resort to the advice of this great community.</p>
<p>As I was reading about the pigeonhole principle something about its proof struck me as odd. Allow me to explain:</p>
<p>After reading the "The Foundations: Logic and Proofs" chapter in Rosen's "Discrete mathematics and its applications" book I ... | Graham Kemp | 135,106 | <p>Don't worry, be happy. It is okay. There are symbols that can be used, but using more of them doesn't really make the logic better. $$\begin{array}{|l}\lVert C\rVert=k\\\lVert\,\bigcup_{B\in C} B\,\rVert=k+1\\\hline~\begin{array}{|l}\forall B\in C~(\lVert B\rVert\leq 1)\\\hline\lVert\,\bigcup_{B\in C}B\,\rVert \leq ... |
2,701,182 | <p>I must once again resort to the advice of this great community.</p>
<p>As I was reading about the pigeonhole principle something about its proof struck me as odd. Allow me to explain:</p>
<p>After reading the "The Foundations: Logic and Proofs" chapter in Rosen's "Discrete mathematics and its applications" book I ... | Sarah | 544,203 | <p>I have a PhD in Mathematics, when I started school I always thought I wanted to use more symbols than words. However, often you cannot do this. A proof is not solving a problem, it is much more! It should certainly contain many words. When you are learning to write proofs, you should actually use more words to make ... |
2,701,182 | <p>I must once again resort to the advice of this great community.</p>
<p>As I was reading about the pigeonhole principle something about its proof struck me as odd. Allow me to explain:</p>
<p>After reading the "The Foundations: Logic and Proofs" chapter in Rosen's "Discrete mathematics and its applications" book I ... | Matthew Leingang | 2,785 | <p>I'm going to agree with Kevin that it depends on what you're trying to do.</p>
<ul>
<li><p>If you're trying to work out the proof of a statement, symbols and shorthand is very handy to express your thoughts succinctly on paper, or on a board. Then you can (hopefully) assemble them into a proof. Symbols are also g... |
2,744,299 | <p>I am working on the following questions:</p>
<blockquote>
<p>Let <span class="math-container">$X$</span> be a compact Riemann surface of genus <span class="math-container">$3$</span> with two points <span class="math-container">$p\neq q$</span>.</p>
<ul>
<li><p>Find a non-constant meromorphic function on <span class... | Aolong Li | 304,775 | <p>Take the Riemann surface $X$ as the Klein's surface, i.e. the Riemann surface associated to the algebraic function
$$w^7=z(z-1)^2$$
Applying Riemann-Hurwitz formula, we have the genus of $X$ is $3$.</p>
<p>To construct the holomorphic $1$-form, you can see the section 7.2 of </p>
<p><a href="http://www.indiana.edu... |
2,744,299 | <p>I am working on the following questions:</p>
<blockquote>
<p>Let <span class="math-container">$X$</span> be a compact Riemann surface of genus <span class="math-container">$3$</span> with two points <span class="math-container">$p\neq q$</span>.</p>
<ul>
<li><p>Find a non-constant meromorphic function on <span class... | Georges Elencwajg | 3,217 | <p>a) Consider the divisor <span class="math-container">$D=-2\cdot p+N\cdot q$</span> on <span class="math-container">$X$</span>. We have <span class="math-container">$L(D)=\Gamma (X,\mathcal O(D))\neq 0$</span> as soon as <span class="math-container">$N\geq 5$</span> (by Riemann-Roch).
Choose such an <span class="math... |
63,974 | <p>Let $E$ be a spectrum. For any CW complex $X$, define $h_*=\pi_i(E\wedge X)$. Then we know that $h_*$ form a homology theory. In other words, there functors satisfy the homotopy invariance, maps a cofiber sequence of spaces to a long exact sequence of abelian groups, also satisfy the wedge axiom in the definition of... | Tilman | 4,183 | <p>For homology theories on CW-complexes or homology theories that map weak equivalences to isomorphisms, that's Brown's representability theorem, which you can find in any textbook on stable homotopy theory. You forgot the important axiom of excision, by the way. The short answer is yes.</p>
|
6,541 | <p>Here is <code>e^{-\lvert\frac{x-\mu}{\sigma}\rvert}</code>: <span class="math-container">$e^{-\lvert\frac{x-\mu}{\sigma}\rvert}$</span></p>
<p>The bars on the absolute value are too small, so I decided to make them bigger. Using <code>\left</code> and <code>\right</code> made them look pretty good:</p>
<p><code>e^{-... | Davide Cervone | 7,798 | <p>There are actually two factors involved here. Mythealias has already discussed the TeX language issue of the way that <code>\left...\right</code> work in comparison to <code>\bigl</code> and <code>\bigr</code>, which explains the difference in height. MJD also complains about the thickness of the characters, which... |
1,865,735 | <p>I'm not sure about the use of the Theorem.
I have:</p>
<p>$$f(x)=\int_0^{x^2}(t-1)g(t)dt$$
I need the derivative of $f$. I know i have to apply the chain rules, but i'm not sure about the results.
My result is:
$$f'(x)=(4x*g(x^2))+((x^2-1)*g'(x^2))$$
Is the correct way?</p>
| Community | -1 | <p>The general form of the type of integral you have in your question is this:
$$ f(x) = \int_a^{b(x)} h(t)\, dt $$</p>
<p>The derivative of $f$ can be obtained by using the FTC and the chain rule:
$$ f'(x) = h(b(x)) \cdot b'(x)$$</p>
<p>In your specific case, you have:
$$ f(x) = \int_0^{\overbrace{x^2}^{\color{red}{... |
9,758 | <p>What is an intuitive explanation of a positive-semidefinite matrix? Or a simple example which gives more intuition for it rather than the bare definition. Say $x$ is some vector in space and $M$ is some operation on vectors.</p>
<p>The definition is:</p>
<p>A $n$ × $n$ Hermitian matrix M is called <em>positive-sem... | Community | -1 | <p>One intuitive definition is as follows. Multiply any vector with a positive semi-definite matrix. The angle between the original vector and the resultant vector will always be less than or equal $\frac{\pi}{2}$. The positive definite matrix tries to keep the vector within a certain half space containing the vector. ... |
9,758 | <p>What is an intuitive explanation of a positive-semidefinite matrix? Or a simple example which gives more intuition for it rather than the bare definition. Say $x$ is some vector in space and $M$ is some operation on vectors.</p>
<p>The definition is:</p>
<p>A $n$ × $n$ Hermitian matrix M is called <em>positive-sem... | Abhimanyu Pallavi Sudhir | 78,451 | <p>Positive-definite matrices are matrices that are <strong>congruent to the identity matrix</strong>, i.e. that can be written as <span class="math-container">$P^HP$</span> for invertible <span class="math-container">$P$</span> (for some reason, a lot of authors define congruence as <span class="math-container">$N=P^T... |
379,554 | <p>How can you fit a equilateral triangle on three arbitrary parallel lines with an edge and compass?</p>
<p><img src="https://i.stack.imgur.com/x8s9a.png" alt="enter image description here"></p>
| ccorn | 75,794 | <p>For brevity, I will not go into details about how to construct perpendiculars and 30-degree angles using straightedge and compass. Same for doubling lengths. Translating those operations to compass-and-straightedge primitives is left to the reader. Given the three parallel lines and a vertex C arbitrarily fixed on o... |
379,554 | <p>How can you fit a equilateral triangle on three arbitrary parallel lines with an edge and compass?</p>
<p><img src="https://i.stack.imgur.com/x8s9a.png" alt="enter image description here"></p>
| David | 297,532 | <p>I'm not going to go into the details of a ruler-and-compass construction, but I'll indicate the main idea, which is very simple.</p>
<p>Referring to the figure in the question, $CDE$ being equilateral is equivalent to saying that $D$ is obtained by rotating $C$ by an angle of $60^{\circ}$ about $E$. </p>
<p>Theref... |
1,449,306 | <p>I found an unclear part in derivation of PCA in the <a href="https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/resources/mit18_s096f15_ses2_4/" rel="nofollow noreferrer">lecture notes</a> of A. Bandeira for <a href="https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-s... | Adrian Cousot | 273,219 | <p>I think I found a way to demonstrate it. Restating the minimisation of error like a least square regression, adding the translation part into matrix V, a column of 1s, and normalised, calling the new matrix A. The last column of $AA^{T}$ has to be perpendicular to affine subspace, and that is enough to demonstrate t... |
304,422 | <p>Let $E$ be an elliptic curve over $\mathbb Q$. Let's look at the group of points of this elliptic curve over $\mathbb Q(1^{1/\infty})$ which we get after adding all roots of unity to $\mathbb Q$. It is easy to prove that it is not finitely generated and the theorem of K.Ribet asserts that its torsion is finite. What... | Will Sawin | 18,060 | <p>$E (\mathbb Q^{ab})/\operatorname{tors}$ is a sum of countably many copies of $\mathbb Z$. </p>
<p>To prove this, take a countable basis $x_1,x_2,\dots$ of $E (\mathbb Q^{ab})/\operatorname{tors}\otimes \mathbb Q$. Suppose we show that $\oplus_{i=1}^n x_i \mathbb Q \cap E (\mathbb Q^{ab})/\operatorname{tors}$ is ... |
304,422 | <p>Let $E$ be an elliptic curve over $\mathbb Q$. Let's look at the group of points of this elliptic curve over $\mathbb Q(1^{1/\infty})$ which we get after adding all roots of unity to $\mathbb Q$. It is easy to prove that it is not finitely generated and the theorem of K.Ribet asserts that its torsion is finite. What... | Vesselin Dimitrov | 26,522 | <p>Since you ask more generally for results on $E(\mathbb{Q}^{\mathrm{ab}})$, let me expand my comment into a short answer. </p>
<p>Amoroso and Dvornicich discovered (<em>A lower bound on the height in abelian extensions</em>, JNT 2000) that the absolute logarithmic height on $\mathbb{G}_m(\bar{\mathbb{Q}})$ is bounde... |
304,422 | <p>Let $E$ be an elliptic curve over $\mathbb Q$. Let's look at the group of points of this elliptic curve over $\mathbb Q(1^{1/\infty})$ which we get after adding all roots of unity to $\mathbb Q$. It is easy to prove that it is not finitely generated and the theorem of K.Ribet asserts that its torsion is finite. What... | Álvaro Lozano-Robledo | 4,180 | <p>As for the torsion subgroup, Michael Chou has classified the possible subgroups that may occur as $E(\mathbb{Q}^{\text{ab}})_{\text{tors}}$ for an elliptic curve $E/\mathbb{Q}$. You can find <a href="https://arxiv.org/abs/1711.00412" rel="nofollow noreferrer">a preprint here</a>.</p>
|
2,981,554 | <p>This might be silly, but I am not sure:</p>
<p>Does there exist a Lebesgue measurable subset <span class="math-container">$E \subseteq (0,1)$</span> such that</p>
<ol>
<li><p><span class="math-container">$E$</span> and <span class="math-container">$(0,1) \setminus E$</span> both have positive Lebesgue measure.</p>... | José Carlos Santos | 446,262 | <p>Let <span class="math-container">$E$</span> be a fat Cantor set <span class="math-container">$C$</span> together with the <span class="math-container">$C^\complement\cap\mathbb Q$</span>.</p>
|
2,391,676 | <p>I have a difficulty when reading "Lectures in Abstract Algebra, II. Linear Algebra" by Nathan Jacobson (GTM #31). The section is excerpted at the end of the question. The sentence I have difficulty is underlined with red. My problem is: the definition of a bilinear form should be given based on a left and a right ve... | Will Nelson | 62,773 | <p>This text is a slightly difficult to follow, but I think it's quite unambiguous. The first paragraph is basically saying "You already know what a bilinear form is. In this chapter, we'll define a similar idea, a product group". The second paragraph actually defines what a product group is. That's the heart. The thir... |
3,123,120 | <p>Prove that given sequence <span class="math-container">$$\langle f_n\rangle =1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+.....+\frac{(-1)^{n-1}}{n}$$</span> </p>
<p>is a Cauchy sequence </p>
<p>My attempt :
<span class="math-container">$|f_{n}-f_{m}|=\Biggl|\dfrac{(-1)^{m}}{m+1}+\dfrac{(-1)^{m+1}}{m+2}\cdots\dots+\dfra... | MachineLearner | 647,466 | <p><span class="math-container">$$f(\boldsymbol{w}) = \dfrac{1}{1+\exp\left[-\boldsymbol{w}^T\boldsymbol{x}\right]}$$</span></p>
<p><span class="math-container">$$\implies \dfrac{\partial f}{\partial w_i} = \dfrac{0\cdot(1+\exp\left[-\boldsymbol{w}^T\boldsymbol{x}\right])-1\cdot\dfrac{\partial}{\partial w_i}(1+\exp\le... |
691,497 | <p>Suppose that $M$ and $F$ are real matrices. Let $A$ be the block-matrix
$$
A=
\begin{pmatrix}
M & F \\
F & M
\end{pmatrix}
$$
If $\det(M)=0$ is $\det(A)\leq0$? If not, what conditions need to be satisfied?</p>
<p>Also, </p>
<p>Does A have a non-positive eigenvalue?</p>
| user1551 | 1,551 | <p>$A$ can possess a full set of positive eigenvalues (and hence a positive determinant). For instance, <a href="https://www.wolframalpha.com/input/?i=eigenvalues%20%7B%7B10,%200,-1,-9%7D,%7B%200,%200,%201,%200%7D,%7B-1,-9,10,%200%7D,%7B%201,%200,%200,%200%7D%7D" rel="nofollow">according to Wolfram Alpha</a>, all four ... |
317,904 | <p>Given, in 3D space: a point $P$ and a direction $v$, a point $Q$ and a direction $w$. So, two lines, $L_1 = P + tv$, $L_2 = Q + tw$.</p>
<p>I am looking for two points, one on each line, say P' and Q'. My requirement is that the distance from $P$ to $P'$ plus $Q$ to $Q'$ equals the distance from $P'$ to $Q'$:</p>
... | Community | -1 | <p>Here, I complete @milo's try: (we use $\ln(1+x)\sim_0x)
$$\lim_{n\to \infty} \left(1+\frac{1}{n!}\right)^{2n} = \lim_{n\to \infty} e^{\ln{\left(1+\frac{1}{n!}\right)^{2n}}} = \lim_{n\to \infty} e^{2n\ \ln{\left(1+\frac{1}{n!}\right)}}=\lim_{n\to \infty} e^{\frac{2n}{n!}}=e^0=1.$$</p>
|
2,553,073 | <p>For the question:</p>
<blockquote>
<p>For what values of $c$ does $8x+5y = c$ have exactly one strictly positive solution? </p>
</blockquote>
<p><a href="https://i.stack.imgur.com/LmheC.png" rel="nofollow noreferrer">The solution is this</a></p>
<p>So I have 3 questions.</p>
<p>I understand everything up unti... | Peter Smith | 35,151 | <p>What happens if you simply let $A$ and $B$ be two distinct atomic (<em>non-logical</em>) formulae of the relevant language -- 'non-logical' to rule out the likes of $\top$ and $\bot$?</p>
|
1,698,257 | <p>How to find a function whose curl is $(7e^y,8x^7 e^{x^8},0)$?</p>
<p>I've tried several integration but can't find a trivial form. </p>
| Arthur | 15,500 | <p>$8x^7e^{x^8}$ looks a whole lot like $\frac{\partial}{\partial x}e^{x^8}$, so let's say that this contribution comes from the $z$-component of the original field.</p>
<p>Also, $7e^y$ is $\frac{\partial}{\partial y}7e^y$, so this could also come from the $z$-component of the original field. This takes care of every ... |
1,698,257 | <p>How to find a function whose curl is $(7e^y,8x^7 e^{x^8},0)$?</p>
<p>I've tried several integration but can't find a trivial form. </p>
| Christopher Carl Heckman | 261,187 | <p>Here's a way to do it without guessing, a la: <a href="https://www.physicsforums.com/threads/operation-that-will-undo-a-curl-operation.14580/" rel="nofollow">https://www.physicsforums.com/threads/operation-that-will-undo-a-curl-operation.14580/</a></p>
<p>Let $F = (f,g,h)$. It turns out that one function is arbitra... |
4,129,851 | <p>This was a problem in my textbook.</p>
<p>Suppose we had a bag with <span class="math-container">$2$</span> balls, an orange and a blue ball. If we pick a blue ball, we simply put it back. If we select an orange ball, we put it back but add <strong>another</strong> orange ball. Suppose we do this <span class="math-c... | Oliver Díaz | 121,671 | <p>If you are familiar with the basic of Fourier series, one can approach this though either the Riemann-Lebesgue lemma, to by Féjer's formula.</p>
<p>For example, since <span class="math-container">$\mathbb{1}_{[0,1]}(x)e^{x^2}$</span> and <span class="math-container">$x\mapsto\sin x$</span> is <span class="math-con... |
2,309,864 | <p>Prove that $f$ is a homeomorphism iff $f[\overline A] = \overline {f[A]}$, I know how to prove that $f$ is continuous iff $f[\overline A] \subset \overline {f[A]}$, but how can I complete?</p>
| Henno Brandsma | 4,280 | <p>$f$ is continuous iff $\forall A \subseteq X: f[\overline{A}] \subseteq \overline{f[A]}$</p>
<p>$f$ is closed iff $\forall A \subseteq X: \overline{f[A]} \subseteq f[\overline {A}]$ (if $f$ is closed, for every $A\subseteq X$: $f[\overline{A}]$ is closed and contains $f[A]$ hence also $\overline{f[A]}$, and if the ... |
184,210 | <p>Consider the set $S=\{1,2,\ldots, n\}$, and let $a<b<n$. What is the minimum number $f(a,b)$ such that there exist $f(a,b)$ subsets of $S$ of size $a$ for which any subset of $S$ of size $b$ contains at least one of the chosen subsets?</p>
<p>It is not hard to obtain a bound $f(a,b)\leq \dbinom{n}{a}-\dbinom{... | Włodzimierz Holsztyński | 8,385 | <p>For starters, let's have the upper bound for $\ f(2\ 3);\ $ then it will be easy to generalize this approach. Partition $\ S\ $ into two sets $\ S=X\cup Y\ $ of cardinalities $\ \lfloor\frac n2\rfloor\ $ and $\ \lceil\frac n2\rceil.\ $ Let $\ A\ $ be the family of all $2$-subsets $\ \alpha\subseteq S\ $ such that $\... |
541,761 | <p>Can we show that the ring of Gaussian integers $$\mathbb{Z}[\sqrt{17}]:=\{a+b\sqrt{17}:a,b\in\mathbb{Z}\}$$
$$\mathbb{Z}[\sqrt{11}]:=\{a+b\sqrt{11}:a,b\in\mathbb{Z}\}$$</p>
<p>equipped with standard addition and multiplication are not isomorphic?</p>
| Kyle | 97,353 | <p>You made a small mistake on the last step: $-4 x^2 + (x^2 + 12) \not= -4 x^2$.</p>
|
541,761 | <p>Can we show that the ring of Gaussian integers $$\mathbb{Z}[\sqrt{17}]:=\{a+b\sqrt{17}:a,b\in\mathbb{Z}\}$$
$$\mathbb{Z}[\sqrt{11}]:=\{a+b\sqrt{11}:a,b\in\mathbb{Z}\}$$</p>
<p>equipped with standard addition and multiplication are not isomorphic?</p>
| Mikasa | 8,581 | <p>You've got your error as indicated in the comments, but I suggest you to consider the fraction as below:</p>
<p>$$y'=(-48x)(x^{12}+2)^{\color{red}{-2}}$$ So you have touse just this rule that $(uv)'=u'v+uv'$ and $(u^n)'=nu^{n-1}\times u'$.</p>
|
20,807 | <p>我想要讓我的抽象代數班級的學生能夠透過網路向我詢問問題，
我可以建置一個私人的社群，
並且讓他們可以在這裡用中文問問題，
並且讓我用中文回答他們嗎？</p>
<hr>
<p>Google translate produces:</p>
<blockquote>
<p>I want to let my abstract algebra class of students through the Internet to be able to ask me questions, I can build a private community, and so that they can ask questions here ... | user21820 | 21,820 | <p>I've noticed that some users post their questions in both their native Chinese language as well as attempt an English translation. I think that is the best option here, since not only can English readers try to read the question, the Chinese readers here can easily edit to improve the translation. Similarly for answ... |
2,835,802 | <p>I'm struggling to come up with the method to find the number of ways to take $k$ objects from $n$ groups which at least one object from each group is taken and order matters.</p>
<p>More specifically, I'm trying to order $8$ digits from the digit pool of $5-10$ ($6$ digits) and each digit must appear at least once ... | awkward | 76,172 | <p>Since you are interested in generating functions, here is an exponential generating function approach. Let $a_r$ be the number of strings of length $r$ with characters taken from a six-character alphabet, with each character used at least once. We define the exponential generating function of $a_r$ by
$$f(x) = \su... |
1,860,782 | <p>I have the following recurrence relation that I'm trying to solve:</p>
<p>$$f(n)=2f(n-1)-f(n-2)-2$$</p>
<p>The homogeneous part is easy:</p>
<p>The characteristic polynomial $r^2-2r+r=0$ has root $r=1$ with multiplicity 2, so the general solution is:</p>
<p>$$f(n)=An+B$$</p>
<p>for some initial conditions.</p>
... | Claude Leibovici | 82,404 | <p>Starting with $$f(n)=2f(n-1)-f(n-2)-2$$ define $f(n)=g(n)+n-n^2$. This gives $$g(n)+n-n^2=2g(n-1)+2(n-1)-2(n-1)^2-g(n-2)-(n-2)+(n-2)^2-2$$ Develop and simplify to get $$g(n)=2g(n-1)-g(n-2)$$ which looks more pleasant.</p>
|
3,826,237 | <p>The question asks to prove directly from the definition of a Cauchy sequence that <span class="math-container">$b_k$</span> is Cauchy, but I am hopelessly confused, these are evidently series approaching infinity</p>
| Oliver Díaz | 121,671 | <p>Hint:</p>
<p>Since <span class="math-container">$a_n\rightarrow\frac12$</span>, there is <span class="math-container">$N$</span> such that <span class="math-container">$|a_n-1|>\frac{1}{3}$</span> for all $n\geq N. (details left to you)</p>
<p><span class="math-container">$\Big|\frac{1}{1-a_n}-\frac{1}{1-a_m}\Bi... |
1,808,222 | <p>While solving PhD entrance exams I have faced the following problem:</p>
<blockquote>
<p>Minimize the function $f(x)=- \sum_{i=1}^n \ln(\alpha_i +x_i)$ for fixed $\alpha_i >0$ under the conditions: $\sum_{i=1}^n x_i =1$ and $x_i \ge0$.</p>
</blockquote>
<p>I was trying to use <a href="https://en.wikipedia.org... | Mark | 76,963 | <p>Let's do it in a few steps.</p>
<p>Firstly, there is a solution: you minimize a continuous function over a nonempty compact domain, so a solution exists by the Extreme Value Theorem.</p>
<p>Secondly, it is unique by looking at the convexity properties of the goal function and domain.</p>
<p>Thirdly, it must satis... |
975,076 | <p>I have three transforms: $C$, $T$, and $P$. Each of these transforms consists of 3D rotations and translations. I know $T$ and $P$, and I would like to solve for $C$. They are related by $T = C^{-1} P C$. Is there a good way to solve this? Is this even guaranteed to have a solution?</p>
<p>The only way I can think ... | Ben Grossmann | 81,360 | <p>Assuming all three are rigid transformations:</p>
<p>If such a $C$ exists, then $T$ and $P$ are said to be <a href="https://en.wikipedia.org/wiki/Matrix_similarity" rel="nofollow">orthogonally similar</a>. In fact, given that $T$ and $P$ are rigid transformations, we know that the following are equivalent:</p>
<u... |
931,851 | <p>In trying to prove that every tree, <em>T</em>, has at most one perfect matching, I came across this idea:</p>
<blockquote>
<p>Since the matchings are perfect, each vertex has degree $0$ or $2$ in the symmetric difference, so every component is an isolated vertex or a cycle.</p>
</blockquote>
<p>Why is this true... | user521739 | 521,739 | <p>Let <span class="math-container">$M$</span> and <span class="math-container">$M_0$</span> be perfect matchings in a tree.
Form the symmetric difference of the edge sets, <span class="math-container">$M \bigtriangleup M_0$</span>. Since the matchings
are perfect, each vertex has degree <span class="math-container">$0... |
2,528,227 | <p>Prove Number in decimal representation $N=abc,def,ghi,\cdots ,xyz$ is divisible by $7$.
Iff $abc-def+ghi-\cdots+xyz$, alternating sum of numbers formed by dividing the string $N$ into $3$ digit pairs of consecutive digits. Is divisible by $7$.</p>
| Donald Splutterwit | 404,247 | <p>Note that $7$ divides $10^{3(2n+1)}+1$ and $10^{3(2n)}-1$, this is easy to see by Fermat
\begin{eqnarray*}
10^{6n} \equiv 1 \pmod{7} \\
10^{6n+3} \equiv -1 \pmod{7}.
\end{eqnarray*}
So $7 \mid 10^{3n} a_n b_n c_n + 10^{3(n-1)} a_{n-1} b_{n-1} c_{n-1} +\cdots +10^{3} a_1 b_1 c_1 + a_0 b_0 c_0 $ if and only if $7 \mid... |
215,752 | <p>Let $A$ (or $X$) be </p>
<p>$\log A \sim N(\mu,\sigma^2)$, (lognormal distribution) </p>
<p>I have to show</p>
<p>$$E[A] = \exp[\mu + (\sigma^2/2)]\mbox{ and }E[A^2] = \exp[2\mu + 2\sigma^2].$$ </p>
<p>Do I have to use mgf of the normal dist. ?</p>
<p>It is easy to show E[$A^2$] since it is the second order d... | Did | 6,179 | <p>One may start with $\log A=\mu+\sigma Z$ and $Z$ standard gaussian. Thus $A^x=\mathrm e^{x\mu}\cdot\mathrm e^{x\sigma Z}$ for every real number $x$, hence $\mathbb E(A^x)=\mathrm e^{x\mu}\cdot\psi(x\sigma)$ where $\psi(t)=\mathbb E(\mathrm e^{t Z})$. Furthermore,
$$
\psi(t)=\int_{-\infty}^{+\infty}\mathrm e^{tz}\cdo... |
3,516,494 | <p><span class="math-container">$$2x^2 + 3x + 1$$</span></p>
<p>applying quadratic formula:</p>
<p><span class="math-container">$$x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$</span></p>
<p><span class="math-container">$$a=2, b=3, c=1$$</span></p>
<p><span class="math-container">$$x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \c... | user729424 | 729,424 | <p>Algebra students learn that the quadratic formula says</p>
<p><span class="math-container">$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$$</span></p>
<p>When is this true? It's helpful to remember that the quadratic formula comes from the following statement, which is true for any complex numbers <span class="math-container... |
2,485,482 | <p>$4*3^x - 9*2^x = 5* 3^\frac x2 * 2^\frac x2$ </p>
<p>I did not understand this equality how to solve it for $x$?</p>
| Siong Thye Goh | 306,553 | <p>Hint: </p>
<p>Solve
$$4m^2-5mn-9n^2=0$$</p>
<p>by factorization. </p>
<p>Notice that $9-4=5$.</p>
|
2,714,418 | <p>Let $(f,g):\mathbb R\to \mathbb R$, $f(x) = x^2 - \frac{\cos x}{2}$ and $g(x)= \frac{x\sin x}{2}$</p>
<p>These are the given options :-</p>
<p>(A) $f(x) = g(x)$ for more than two values of $x$</p>
<p>(B) $f(x) \neq g(x)$, for all $x \in \mathbb R$</p>
<p>(C) $f(x) = g(x)$ for exactly one value of $x$</p>
<p>(D)... | TheSimpliFire | 471,884 | <p>Let $$h(x)=f(x)-g(x)=x^2-\frac12x\sin x-\frac12\cos x$$ We know that $h$ is continuous over $\mathbb{R}$ so we find its derivative $$h'(x)=2x-\frac12\sin x-\frac12x\cos x+\frac12\sin x=x\left(2-\frac12\cos x\right)=0$$ for stationary points so either $x=0$ or $\cos x=4$ which is impossible. </p>
<p>Now $$h''(x)=2-\... |
2,080,644 | <p>I know that a function is odd when
$$f(-x) = -f(x)$$
Therefore I can say that if for a function $$-f(x) + f(x) = f(-x) + f(x) = 0$$</p>
<p>Then the function is odd!</p>
<p>I tried to use this <em>trick</em> to prove that $f(x) = \ln\left(x+\sqrt{x^2 + 4}\right) - \ln2$ is odd.</p>
<p>However, I would want to prov... | jugglingmike | 341,620 | <p>Hint: Rationalise the denominator of the fraction inside the ln. </p>
<p>I.e. multiply top and bottom by $x+\sqrt{x^2+4}$.</p>
|
2,080,644 | <p>I know that a function is odd when
$$f(-x) = -f(x)$$
Therefore I can say that if for a function $$-f(x) + f(x) = f(-x) + f(x) = 0$$</p>
<p>Then the function is odd!</p>
<p>I tried to use this <em>trick</em> to prove that $f(x) = \ln\left(x+\sqrt{x^2 + 4}\right) - \ln2$ is odd.</p>
<p>However, I would want to prov... | Jack D'Aurizio | 44,121 | <p>A faster approach. If we set $f(x)=\log\left(\frac{x+\sqrt{x^2+4}}{2}\right)$ and $x=2\sinh\theta$ we have
$$f(2\sinh\theta) = \log\left(\frac{2\sinh\theta+2\cosh\theta}{2}\right) = \log(e^\theta) = \theta $$
and since $\theta\mapsto 2\sinh\theta$ is a bijective odd function from $\mathbb{R}$ to $\mathbb{R}$,
$$ f(... |
2,216,418 | <p>Liz and Sara start new jobs on the same day. Liz works three days in a row followed by $1$ rest day. Sara works $7$ days in a row followed by $3$ rest days. How many days between Day $1$ and Day $1000$ will they both have a rest day ? I know the answer is $100$, but how does this come about ?</p>
| David K | 139,123 | <p>We could answer this with number theory, but this particular
problem it is easy enough to solve without invoking a lot of equations.</p>
<p>First, write down the first $20$ days of the combined schedule showing which days are rest days for Liz, which are rest days for Sara, and which are rest days for both.</p>
<p... |
2,583,015 | <p>So, from my understanding there are two versions of this theorem:</p>
<p>Version one states that, if $\displaystyle F(x)= \int_a^xf(t)~dt$, then $$\frac{dF}{dx}=\frac{d}{dx}\left[\int_a^xf(t)~dt\right]=f(x)$$whereas the second version states that $$\int_a^bf(x)~dx=F(b)-F(a)$$what I'm hoping to establish is this: I ... | Christian Blatter | 1,303 | <p>Define integrals $\int_{[a,b]} f(t)\>dt$ as limits of Riemann sums in your favorite way. The two versions of the FTC are relating such integrals to the notion of derivative, which comes as a miracle. The two formulas are:
$${d\over dx}\int_{[a,x]}f(t)\>dt=f(x)\ ,\tag{1}$$
$$\int_{[a,b]} F'(t)\>dt=F(b)-F(a)\... |
2,946,986 | <p>Suppose <span class="math-container">$(P, \leq)$</span> is a partially ordered set. </p>
<p>For <span class="math-container">$x \in P$</span>, define <span class="math-container">$U_x := \{ y \in P \ | \ y \geq x\}$</span>.</p>
<p>Is it true that for any <span class="math-container">$x,y \in P$</span>, either <spa... | Ethan Bolker | 72,858 | <p>You can get an approximate answer by figuring out how large the side of a little cube should be in order to stack <span class="math-container">$10^{17}$</span> of them in your cubic meter. That number will be (about) the distance between your atoms.</p>
|
1,681,134 | <p>This is a problem from Harvard Stat 110 Probability Homework set 2, and Blitzstein's <em>Introduction to Probability</em> (2019 2 ed) Ch 1, Exercise 54, p 51.</p>
<blockquote>
<p>Alice attends a small college in which each class meets only once a week. She is deciding between 30 non-overlapping classes. There are 6 ... | Brian M. Scott | 12,042 | <p>You’re not alone: it’s one of the most common overcounting errors. To see exactly what the problem is, let’s say that she ends up with classes <span class="math-container">$M_1,M_2,M_3,T_1,W_1,\Theta_1$</span>, and <span class="math-container">$F_1$</span>, where the first three meet on Monday, and the remaining fou... |
1,877,632 | <p>My integral calculus is rusty.
How do I calculate the interior area (blue region) of four bounding circles?<br><br>
<a href="https://i.stack.imgur.com/VtQIy.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VtQIy.png" alt="enter image description here"></a></p>
| haqnatural | 247,767 | <p>Let radius of circle is $a$,then "area is=area of square -area of a cirle"
$$S={ 4a }^{ 2 }-\pi { a }^{ 2 }={ a }^{ 2 }\left( 4-\pi \right) $$
<a href="https://i.stack.imgur.com/O2pxG.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/O2pxG.png" alt="enter image description here"></a></p>
|
2,217,890 | <p>The singular integrands I will ask about in this post about appear a lot in numerical integration so I would really like to understand theoretically what makes some of these integrands 'more integrable' (whatever that means) than others.</p>
<p><strong>Question 1:</strong></p>
<p>I experimented in Wolfram Alpha by... | Pawel | 355,836 | <p>This is rather a matter of preference. I know mathematicians that remember a lot of "tricks," but I also know mathematicians that do not care about memorizing "tricks" so much. Thus, answer to your question will largely depend on personal preference.</p>
<p>In my opinion, remembering basic and useful tricks comes i... |
2,217,890 | <p>The singular integrands I will ask about in this post about appear a lot in numerical integration so I would really like to understand theoretically what makes some of these integrands 'more integrable' (whatever that means) than others.</p>
<p><strong>Question 1:</strong></p>
<p>I experimented in Wolfram Alpha by... | Stella Biderman | 123,230 | <p>Talking about computational tricks more generally than just integrals, it wildly depends on the field of mathematics. Some have a lot of things that turn out to be fundamental that, to an outsider, seem like computational tricks more than deep theorems, or even just very useful tools. Most fields will have <strong>s... |
195,898 | <p>I can't understand what is wrong with this paradox. How we should strictly mathematically explain it? </p>
<p>Mathematical induction:</p>
<p><strong>1. The basis:</strong> </p>
<p>$n=1,n=2$. Through any two (one) points on a plane we can draw a straight line.</p>
<p><strong>2. The inductive step:</strong></p>
... | Brian M. Scott | 12,042 | <p>The argument in the induction step doesn’t work in going from $n=2$ to $n=3$: the two lines aren’t forced to be the same line. Try drawing a picture of such an example.</p>
|
195,898 | <p>I can't understand what is wrong with this paradox. How we should strictly mathematically explain it? </p>
<p>Mathematical induction:</p>
<p><strong>1. The basis:</strong> </p>
<p>$n=1,n=2$. Through any two (one) points on a plane we can draw a straight line.</p>
<p><strong>2. The inductive step:</strong></p>
... | Clive Newstead | 19,542 | <p>Since you've tagged this as homework I won't just give you the answer, but think about this: your stage <strong>3.</strong> relies on $P_2 \ne P_k$. When might this not happen?</p>
<p><strong>Edit</strong> (in response to comment): Let $\phi$ be a statement about natural numbers; we say $\phi(n)$ if $\phi$ holds fo... |
1,333,605 | <p>Suppose we have a vector space $V = (K, +, \cdot)$. Let $B$ be a basis for $V$. Now we take an arbitrary square matrix $S \neq 0$.</p>
<p>$BS$ is just a linear combination of $B$. Thus $BS$ should be a new basis. Am I right?</p>
| Martin Sleziak | 8,297 | <p>Your question is open to several interpretations. (At least the <a href="https://math.stackexchange.com/revisions/1333605/3">current revision</a>.) And we could write quite a lot about each of these interpretations.</p>
<p>You are using notation $BS$, which is (as far as I can say), most frequently use to denote th... |
3,919,925 | <p>If we randomly pick a real number from the number line, the probability of picking a number (say x) is 0. This is true for all real numbers x and it makes sense to me why this must be true. But seemingly there is a paradox lurking here. Suppose we pick the number y. This number also had probability 0 but was still c... | Michael Stachowsky | 337,044 | <p><strong>Infinity doesn't work like you think it does</strong>. There are several ways to interpret probability. One of those ways is the "frequentist" interpretation - you divide the number of outcomes you're looking for by the number of outcomes possible, and that's the probability. The issue is that, ... |
2,394,613 | <p>Minimizing the following function</p>
<p>$f(x_1,x_2,\cdots,x_n)=\prod\limits_i^n x_i^{x_i}$ </p>
<p>such that </p>
<p>$x_1+x_2+\cdots+x_n=P, 2\le x_i$ and $x_i$ are integers.</p>
<p>My attempt: In my opinion we obtain the result when all $x_i's$ are almost equal i.e. $|x_i-x_j|\le 0$ for all $i$ and $j$. I am tr... | MAN-MADE | 422,850 | <p>Using weighted AM-GM inequality, $$\frac{P}{\dfrac{x_1}{x_1}+\dfrac{x_2}{x_2}+\dots+\dfrac{x_n}{x_n}}\leq\Big(\prod_{i=1}^{n}x^{x_i}_i\Big)^\dfrac{1}{P}\leq \frac{x_1\cdot x_1+x_2\cdot x_2+\dots+x_n\cdot x_n}{P}\\\Rightarrow\frac{P}{n}\leq \Big(\prod_{i=1}^{n}x^{x_i}_i\Big)^\dfrac{1}{P}\leq\frac{x_1^2+x^2_2+\dots+x_... |
1,769,494 | <p>Say I have a function </p>
<p>$$ f(x) = \dfrac 1x$$</p>
<p>and I'm looking at its $n^{th}$ derivative and trying to come up with a formula.</p>
<p>I can easily get it because if forms a very consistent pattern and it somewhat reminds me of harmonic series.</p>
<p>The formula is </p>
<p>$$ f'^n (x) = (-1)^n {n!... | Community | -1 | <p>Besides the excellent approach by @Element118, you can simply split in four cases, noting that $f''''(x)=-4f(x)$,</p>
<p>$$\begin{align}f^{(4n)}(x)&=(-4)^ne^x\cos(x),\\
f^{(4n+1)}(x)&=(-4)^ne^x(\cos(x)-\sin(x)),\\
f^{(4n+2)}(x)&=(-4)^ne^x(-2\sin(x)),\\
f^{(4n+3)}(x)&=(-4)^ne^x(-2\sin(x)-2\cos(x)).
\... |
2,411,890 | <blockquote>
<p>If $x$ is a real number, then $|x+1| \leq 3$ implies that $-4 \leq x \leq 2$.</p>
</blockquote>
<p>I've tried to prove this by exhaustion, is that the right way to prove it? </p>
| Michael Rozenberg | 190,319 | <p>You can use the definition of $|.|$.</p>
<p>The absolute value of real number $a$ (we'll write it $|a|$) it's the distance between the point, </p>
<p>which corresponds to $a$ in the $x$-axis and the origin.</p>
<p>In our case the distance between $x+1$ and $0$ less or equal to $3$.</p>
<p>Thus, we have $$-3\leq ... |
2,411,890 | <blockquote>
<p>If $x$ is a real number, then $|x+1| \leq 3$ implies that $-4 \leq x \leq 2$.</p>
</blockquote>
<p>I've tried to prove this by exhaustion, is that the right way to prove it? </p>
| Graham Kemp | 135,106 | <p>For real valued $x$ we have two cases.</p>
<ul>
<li><p>Either $x+1\geq 0$, in which case $\lvert x+1\rvert =(x+1)$ so... $0\leq x+1 \leq 3$</p></li>
<li><p>Or $x+1<0$, in which case $\lvert x+1\rvert=-(x+1)$, so... $0 < -(x+1)\leq 3$.</p></li>
</ul>
<p>Thus $(\lvert x+1\rvert \leq 3 )\iff ( -3\leq x+1 \leq ... |
3,395,177 | <p>I'm interested in knowing whether or not <span class="math-container">$\mathbb{C}[x,y]/\langle x^2+y^2\rangle$</span> is a field, where <span class="math-container">$\langle x^2+y^2\rangle$</span> denotes the ideal generated by the polynomial <span class="math-container">$x^2+y^2\in\mathbb{C}[x,y]$</span> and <span ... | lhf | 589 | <p><em>Hint:</em> <span class="math-container">$x^2 + y^2 = (x-iy)(x+iy)$</span> implies that there are zero divisors in the quotient ring.</p>
|
3,561,805 | <p>Problem: Let <span class="math-container">$W$</span> equal the weight of laundry soap in a 1-kilogram box that is distributed in Southeast Asia. Suppose <span class="math-container">$P(W<1)=0.02$</span> and <span class="math-container">$P(W>1.072)=0.08$</span>. Call a box of soap light, good, or heavy dependin... | J. W. Tanner | 615,567 | <p>Let <span class="math-container">$n=2k+1$</span>. Then <span class="math-container">$n^2-1=(n+1)(n-1)=(2k+2)(2k)=4(k+1)k.$</span></p>
|
2,445,809 | <p>I want to calculate the derivative of an integral of a two-variable function, so</p>
<p>$\frac{d}{dy}\int_{0}^1f(x,y)\,dx$.</p>
<p>I am sorry if this is a basic question but a google search yields unusable results. I am 90% sure that the derivative can simply go under the integral, but I would like to be sure. Tha... | Community | -1 | <p>Define <span class="math-container">$$F(x,y)= \int^1_0 f(x,y)dx$$</span> and use the chain rule on <span class="math-container">$F(x,y)$</span>.</p>
<p>This is a special case of the <a href="http://en.wikipedia.org/wiki/Leibniz_integral_rule#General_form_with_variable_limits" rel="nofollow noreferrer">Leibniz integr... |
2,964,439 | <p>How do I prove the following by induction?</p>
<blockquote>
<p>Given a finite collection of numbers, the sums/products obtained by adding/multiplying them in any order are all equal.</p>
</blockquote>
| RhythmInk | 242,446 | <p>Okay. So, we proceed by way of induction. You need to show the base case is true which in this case is <span class="math-container">$n=3$</span>. So, suppose we have <span class="math-container">$3$</span> arbitrary numbers <span class="math-container">$m_1,m_2,m_3$</span>. We know that for any two numbers <span cla... |
2,763,466 | <p>I know there is no non-vanishing vector field on $S^2$, so I cannot comb the hair on a ball.
(I am treating $S^2$ as a manifold without the ambient space $\mathbb R^3$, which amounts to demanding that the vector field is tangential to $S^2$ at every point if you prefer the Euclidean point of view.)
Is this still imp... | Aloizio Macedo | 59,234 | <p>An alternative way to see why this isn't true is to fix a Riemannian metric. If such a subbundle existed, then restricting to those vectors of norm $1$ would give a two-sheeted covering of $S^2$. If it were disconnected, there would be a nowhere zero vector field on $S^2$, a contradiction. If it were connected, $S^2... |
316,672 | <p>I do not see how this is even valid. Could someone point this out to me:</p>
<p>Assume that $x_n$ is a cauchy sequence of rational numbers satisfying $|x_n| \geq r$ for all $n\in\mathbb{N}$. Show that there is $N\in\mathbb{N}$ s.t. either $x_n > r$ for all $n \geq N$ or $x_n < -r$ for all $n\geq N$.</p>
<p>H... | Jim | 56,747 | <p>First you need to assume that $r$ is positive and the condition needs to be $|x_n| > r$, not $\geq$. Then, as you always have $|x_n| > r$ each term $x_n$ satisfies either $x_n > r$ or $x_n < -r$. What you need to show is that after a point it stops jumping back and forth and settles on always being one... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.