qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,029,708 | <p>Suppose there is a vector <span class="math-container">$U \in \mathbb{R}^n$</span>. How would you find the derivative of:</p>
<p><span class="math-container">$$
F(U)=trace\left(diag(U) A\ diag(U) \right)
$$</span>
where <span class="math-container">$A \in \mathbb{R}^{n \times n} \succ 0 $</span> and where <span cl... | greg | 357,854 | <p>Here are two useful identities involving the Hadamard <span class="math-container">$(\odot)$</span> product of a matrix and two vectors.<br>
<span class="math-container">$$\eqalign{
A\odot xy^T &= {\rm Diag}(x)\cdot A\cdot{\rm Diag}(y) \\
{\rm Tr}\Big(A\odot xy^T\Big) &= x^T\left(I\odot A\right)y \\
}$$</spa... |
3,215,381 | <p>When we write <span class="math-container">$\mathbb{Z}_3$</span>, does it mean <span class="math-container">$\mathbb{Z}/3\mathbb{Z}$</span>? Also, does <span class="math-container">$3\mathbb{Z}_3$</span> mean <span class="math-container">$0 \pmod 3$</span>?</p>
| PrincessEev | 597,568 | <p><span class="math-container">$\Bbb Z_n$</span> is a bit of an ambiguous notation without further context. I've seen it used both as a means of denoting the integers mod <span class="math-container">$n$</span>, and as a means of denoting the <span class="math-container">$n$</span>-adic integers - two very different t... |
1,464,747 | <p>I am trying to solve this question:</p>
<blockquote>
<p>How many ways are there to pack eight identical DVDs into five indistinguishable boxes so that each box contains at least one DVD?</p>
</blockquote>
<p>I am very lost at trying to solve this one. My attempt to start this problem involved drawing 5 boxes, an... | André Nicolas | 6,312 | <p>You are nearly finished. We have three DVD to dump into boxes. Maybe we put all $3$ in the same box. Maybe we use a $2$-$1$ split. Or maybe we use a $1$-$1$-$1$ split. Since the boxes are indistinguishable, we have a total of three possibilities.</p>
|
58,060 | <p>I have been looking at Church's Thesis, which asserts that all intuitively computable functions are recursive. The definition of recursion does not allow for randomness, and some people have suggested exceptions to Church's Thesis based on generating random strings. For example, using randomness one can generate str... | Joseph O'Rourke | 6,094 | <p>I asked <a href="https://cstheory.stackexchange.com/questions/1263/truly-random-number-generator-turing-computable">a related question at CS Theory</a>, which ended with this question:</p>
<blockquote>
<p>Is it the case that a TM [Turing Machine] with access to a pure source of randomness (an oracle?), can compute a... |
2,764,221 | <p>Let $A$ be a symmetric invertible $n \times n$ matrix, and $B$ an antisymmetric $n \times n$ matrix. Under what conditions is $A+B$ an invertible matrix? In particular, if $A$ is positive definite, is $A+B$ invertible? </p>
<p>This isn't homework, I am just curious. Assume all matrices have entries in $\mathbb{R}$... | N. S. | 9,176 | <p>Pick $B$ any anti-symmetric matrix which is not nilpotent, and $\lambda \neq 0$ an eigenvalue of $B$.</p>
<p>Set $$A=-\lambda I$$</p>
|
3,399,195 | <p>So I've seen various questions with the limit 'equal' to <span class="math-container">$\infty$</span> or that the limit doesn't exist in a case where the function tends to <span class="math-container">$\infty$</span>.</p>
<p>For example, the limit of <span class="math-container">$\sqrt{x}$</span> as <span class="mat... | Clark Driscoll | 1,064,598 | <p>If you could show some examples of the problems you talking about that would help greatly.</p>
<p>For cases where <span class="math-container">$x \to \infty$</span>, if the limit is increases with out bounds then you would say the limit equals infinity.</p>
<p><span class="math-container">$\lim_{x\to\infty}\sqrt{x}=... |
3,009,362 | <p>I need to find
<span class="math-container">$$\lim_{x\rightarrow -5} \frac{2x^2-50}{2x^2+3x-35}$$</span></p>
<p>Looking at the graph, I know the answer should be <span class="math-container">$\frac{20}{17}$</span>, but when I tried solving it, I reached <span class="math-container">$0$</span>.</p>
<p>Here are the... | Peter Szilas | 408,605 | <p>Numerator</p>
<p><span class="math-container">$2x^2-50=2(x-5)(x+5)$</span>.</p>
<p>Denominator</p>
<p><span class="math-container">$2x^2+3x -35 =(2x-7)(x+ 5)$</span></p>
<p><span class="math-container">$\dfrac{2(x-5)(x+5)}{(2x-7)(x+5)}=$</span></p>
<p><span class="math-container">$\dfrac{2(x-5)}{2x+7}.$</span><... |
3,169,142 | <p>The question goes like this:</p>
<p>If <span class="math-container">$f(x)$</span> is a non-constant, continuous function defined on a closed interval <span class="math-container">$[a,b]$</span> Then by the Extreme Value Theorem, there exist an absolute minimum <span class="math-container">$m$</span> and an absolute... | Simon Goodwin | 364,836 | <p>Consider the Intermediate value theorem, let <span class="math-container">$x\in [m,M]$</span> and then since there are <span class="math-container">$x_1,x_2$</span> such that <span class="math-container">$f(x_1)=m, f(x_2)=M$</span> we get the desired result.</p>
|
1,843,662 | <p>Let C be that part of the circle $z=e^{i\theta}$, where $0\le\theta\le\frac\pi2$. Evaluate $\int_{c}\frac{z}{i}dz$.</p>
<p>This is my first time posting my question here. I'm really poor in writing English. for that reason please understand my bad explanation. proceed to the main issue I have no idea on solving thi... | J.-E. Pin | 89,374 | <p>According to Eilenberg [1, Chap. IV, Prop. 1.1], the following result holds:</p>
<blockquote>
<p><strong>Proposition</strong>. For any nonempty subset $L$ of $A^*$, the following conditions are equivalent:</p>
<ol>
<li>for all $u, v \in L$, $u^{-1}L = v^{-1}L$,</li>
<li>the minimal automaton of $L$ has a... |
912,217 | <p>Let $X$ be a R.V whose pdf is given by
$$f(x)=p\frac{1}{\sqrt{2\pi\sigma_1^2}}\exp\left(-\frac{(x-\mu_1)^2}{2\sigma_1^2}\right)+
(1-p)\frac{1}{\sqrt{2\pi\sigma_2^2}}\exp\left(-\frac{(x-\mu_2)^2}{2\sigma_2^2}\right)$$</p>
<p>clearly $X\sim pN(\mu_1,\sigma_1^2)+(1-p)N(\mu_2,\sigma_2^2)=N(p\mu_1+(1-p)\mu_2,p^2\sigma_... | Snufsan | 122,989 | <p>Note that if $f$ is differentiable in $x$ then $f'(x)$ exsists, i.e. the limit exist, which exactly means that the side-limits are equal: $f'(x)=f'_{-}(x)=f'_{+}(x)$</p>
|
912,217 | <p>Let $X$ be a R.V whose pdf is given by
$$f(x)=p\frac{1}{\sqrt{2\pi\sigma_1^2}}\exp\left(-\frac{(x-\mu_1)^2}{2\sigma_1^2}\right)+
(1-p)\frac{1}{\sqrt{2\pi\sigma_2^2}}\exp\left(-\frac{(x-\mu_2)^2}{2\sigma_2^2}\right)$$</p>
<p>clearly $X\sim pN(\mu_1,\sigma_1^2)+(1-p)N(\mu_2,\sigma_2^2)=N(p\mu_1+(1-p)\mu_2,p^2\sigma_... | Paramanand Singh | 72,031 | <p>An alternative way of looking at this problem (different from your class notes):</p>
<p>What happens when $f'(x) < 0$ even at a single point $c$? Clearly this would mean that the there will be an interval $I$ containing $c$ such that $f(x) > f(c) $ for all $x \in I, x < c$ and $f(x) < f(c)$ for all $x \... |
2,832,374 | <p>I'm given the following definition asked to prove the following theorem:</p>
<p>Definition: Let $X$ be a set and suppose $C$ is a collection of subsets of $X$. Then $\cup \mathbf{C}=\{x \in X : \exists C\in \mathbf{C}(x\in C)\}$</p>
<p>Theorem: Let $\mathbf{C,D}$ be collections of subsets of a set $X$. Prove that ... | Community | -1 | <p>Suppose $f_n \to f$ with $\Vert f_n \Vert \leq 1$, then also $\Vert f_n \Vert \to \Vert f \Vert$ by the reverse triangle inequality, and limits preserve inequalities.</p>
|
3,156,570 | <p>I need to evaluate the following limit:
<span class="math-container">$$
\lim_{x\downarrow 0} \dfrac{(1 - e^x)^{-1}}{x^c}
$$</span>
for different values of the constant <span class="math-container">$c$</span>.</p>
<p><em>What I've tried thus far:</em></p>
<p>We have that
<span class="math-container">$$
\lim_{x\down... | farruhota | 425,072 | <p>Write it as:
<span class="math-container">$$\lim_{x\to 0}\dfrac{1}{x^c(1 - e^x)}=\lim_{x\to 0}\dfrac{x^{-c}}{1 - e^x}\stackrel{L'H}=\lim_{x\to 0}\dfrac{-cx^{-c-1}}{-e^x}=\\
\lim_{x\to 0}\dfrac{c}{x^{c+1}}=\begin{cases}-1, c=-1\\ \ \ \ 0, c<-1\\
-\infty, -1<c<0\\
+\infty, c>0\\
\end{cases}$$</span>
If <sp... |
3,314,561 | <p>Consider the triangle <span class="math-container">$PAT$</span>, with angle <span class="math-container">$P = 36$</span> degres, angle <span class="math-container">$A = 56$</span> degrees and <span class="math-container">$PA=10$</span>. The points <span class="math-container">$U$</span> and <span class="math-contain... | Blazej | 155,834 | <p>Reason 1. Even if you actually care only about smooth solutions, it some cases it is much easier to first establish that a weak solution exists and separately show that the structure of the PDE actually enforces it to be smooth. Existence and regularity are handled separately and using different tools.</p>
<p>Reaso... |
122,468 | <p>I know how to find the number of solutions to the equation:</p>
<p><span class="math-container">$$a_1 + a_2 + \dots + a_k = n$$</span></p>
<p>where <span class="math-container">$n$</span> is a given positive integer and <span class="math-container">$a_1$</span>, <span class="math-container">$a_2$</span>, <span class... | Marc van Leeuwen | 18,880 | <p>While there is no closed form for the number of partitions of $n$ into $k$ parts, these numbers are not hard to compute. First let's get rid of the awkward initial strict inequality: by setting $a'_i=a_i-1$ we get $0\leq a'_1\leq\cdots\leq a'_k$ and $a'_1+\cdots+a'_k=n-k$. This means we want to count weak partitions... |
21,491 | <p>The question is prompted by change of basis problems -- the book keeps multiplying the bases by matrix $S$ from the left in order to keep subscripts nice and obviously matching, but in examples bases are multiplied by $S$ (the change of basis matrix) from whatever side. So is matrix multiplication commutative if at ... | Ofir | 2,125 | <p>In general, two matrices (invertible or not) do not commute. For example
$$\left(\begin{array}{cc}
1 & 1\\
0 & 1\end{array}\right)\left(\begin{array}{cc}
1 & 0\\
1 & 1\end{array}\right) = \left(\begin{array}{cc}
2 & 1\\
1 & 1\end{array}\right)
$$
$$
\left(\begin{array}{cc}
1 & ... |
3,394,050 | <p>I'm having trouble with this problem.</p>
<blockquote>
<p>Using logical equivalencies prove that <span class="math-container">$(p \land q)\implies (p \lor q)$</span> is a tautology.</p>
</blockquote>
| RyRy the Fly Guy | 412,727 | <p><span class="math-container">$T$</span></p>
<p><span class="math-container">$\Leftrightarrow T \vee T$</span></p>
<p><span class="math-container">$\Leftrightarrow (p \vee \neg p) \vee (q \vee \neg q) \Leftrightarrow T$</span></p>
<p>We drop the parentheses because the statement is well defined...</p>
<p><span cl... |
1,026,506 | <p>If $I_n=\int _0^{\pi }\:sin^{2n}\theta \:d\theta $, show that
$I_n=\frac{\left(2n-1\right)}{2n}I_{n-1}$, and hence $I_n=\frac{\left(2n\right)!}{\left(2^nn!\right)2}\pi $</p>
<p>Hence calculate $\int _0^{\pi }\:\:sin^4tcos^6t\:dt$</p>
<p>I knew how to prove that $I_n=\frac{\left(2n-1\right)}{2n}I_{n-1}$ ,, but I am... | Community | -1 | <blockquote>
<p>what does it mean Hence $I_n = \frac{(2n)!\pi}{(2^n n!)2}$ do we need to prove this part as well or is it just a hint to use? </p>
</blockquote>
<p>"Hence" means "from here"; it is an archaic word. It survives in mathematical English, where it means "because of what was just said", or "using what wa... |
3,007,443 | <p>I've heard the words "internal" and "external" generalization of concepts in category theory.</p>
<p>Specifically, i heard the idea that the concept of 'power set' has an internal and an external generalization in category theory.</p>
<p>What is the difference between these two?</p>
| Musa Al-hassy | 80,406 | <p>Categorial internalisation is about taking a statement
that involves “points”, which is the usual Set theoretic rendition,
and turning it “point free” so that it is soley rendered in the
language of category theory.</p>
<p>For example, an adjoint between preorders is a pair <span class="math-container">$f, g$</span... |
2,524,890 | <p>I know that if matrix $a$ is similar to matrix $b$ then $\operatorname{trace} a=\operatorname{trace} b$.</p>
<p>Does it go to the other side?</p>
<p>Thanks.</p>
| John Doe | 399,334 | <p>The equation is actually $$(b-4)^2=16(1-a)$$</p>
<p>Your LHS is a square number, your RHS is $16 x$ where $x$ is an integer. When can your RHS be made into a square number? </p>
<p>It already has $2^4=(2^2)^2$ as a factor, so all you need is for $x$ to be a square number.</p>
<p>So $x=1-a=n^2$ for $n\in \Bbb N$. ... |
2,524,890 | <p>I know that if matrix $a$ is similar to matrix $b$ then $\operatorname{trace} a=\operatorname{trace} b$.</p>
<p>Does it go to the other side?</p>
<p>Thanks.</p>
| nonuser | 463,553 | <p>First we see that $8\mid b^2$ so $b=4c$ for some integer $c$. Thus we get $$c^2=2c-a\;\;\;\;\Longrightarrow \;\;\;\;c\mid a$$
So $a=cd$ for some integer $d$ and we have now $$c=2-d$$
So all solution are $(a,b)= (2d-d^2, 8-4d)$, where $d\in \mathbb{Z}$.</p>
|
2,600,776 | <blockquote>
<p>A continuos random variable $X$ has the density
$$
f(x) = 2\phi(x)\Phi(x), ~x\in\mathbb{R}
$$
then</p>
<p>(<em>A</em>) $E(X) > 0$</p>
<p>(<em>B</em>) $E(X) < 0$</p>
<p>(<em>C</em>) $P(X\leq 0) > 0.5$</p>
<p>(<em>D</em>) $P(X\ge0) < 0.25$</p>
<p>\begin{eqnarray}... | Gono | 384,471 | <p>It holds $$\left(\Phi^2(x)\right)' = 2\Phi(x)\varphi(x)$$</p>
<p>So we have $$P(X \le x) = \Phi^2(x)$$</p>
<p>And we can conclude $$P(X \le 0) = 0.25$$
$$P(X \ge 0) = 1 - P(X < 0) = 0.75$$</p>
<p>For the expectation we get:</p>
<p>$$\begin{align*} E[X] &= \int_0^\infty (1-\Phi^2(x)) dx - \int_{-\infty}^0 ... |
2,600,776 | <blockquote>
<p>A continuos random variable $X$ has the density
$$
f(x) = 2\phi(x)\Phi(x), ~x\in\mathbb{R}
$$
then</p>
<p>(<em>A</em>) $E(X) > 0$</p>
<p>(<em>B</em>) $E(X) < 0$</p>
<p>(<em>C</em>) $P(X\leq 0) > 0.5$</p>
<p>(<em>D</em>) $P(X\ge0) < 0.25$</p>
<p>\begin{eqnarray}... | Ant | 66,711 | <p>An intuitive way to see that A) holds is the following:</p>
<p>Forget about the $2$ in front, it won't change the sign of the integral. Now, if there was only $\phi(x)$, you would get zero, because we know what </p>
<p>$$\int _\mathbb R x\phi(x) dx = 0$$</p>
<p>Now you multiply the integrand by $\Phi(x)$. Being a... |
3,535,088 | <p>A function <span class="math-container">$\phi:X\rightarrow Y$</span> between two topological space <span class="math-container">$(X,\tau)$</span> and <span class="math-container">$(Y,\sigma)$</span> is continuous in <span class="math-container">$x\in X$</span> if and only if for any open set <span class="math-contai... | daw | 136,544 | <p>Take <span class="math-container">$w\in L^2(\Omega)$</span>. Then
<span class="math-container">$$
\int_\Omega(u_nv_n - uv)w =
\int_\Omega u(v_n - v)w +
\int_\Omega v_n(u_n - u)w .
$$</span>
The first integral vanishes because <span class="math-container">$uw\in L^1(\Omega)$</span> and <span class="math-container">... |
1,921,101 | <p>$∃x.P(x) \Rightarrow ∀x.P(x) $</p>
<p>How can I read this in simple English? I translated it as: There exists an element x for which P(x) implies that for all elements x, P(x) is true - but I feel like this doesn't make much sense. What am I doing wrong here?</p>
| James | 246,902 | <p>Your translation is right: suppose that there is an element $x$ that makes $P(x)$ true. Then all elements make $P(x)$ true.</p>
<p>This does sound a little weird, but just because it is weird, that doesn't make the translation incorrect. (Logic is pretty weird.) There are predicates $P(x)$ that make $\exists x. P(x... |
2,657,301 | <p>On the set of natural numbers$\mathbb { N} $, define the operations $a \oplus b := \max(a,b)$
and $a\otimes b := a+b$ Is $(\mathbb {N},\oplus,\otimes)$ is ring? commutative ring with unity? Field?</p>
<p>My solution :</p>
<p>1- $(\mathbb {N},\oplus) $ is abelian groub because : </p>
<p>a. It is comutative
$a \... | Adam | 67,429 | <p>The identity for $(\mathbb{N}, \oplus)$ should be $e=0$.</p>
|
303,933 | <p>How can I prove that
$$ \lim_{n\to\infty} \frac{(\ln(n))^a}{n^b} = 0 \;\forall a,b > 0 $$
? Intuitively it is clear to me because of the behavior of the functions. Thanks for all.</p>
<p><strong>Edit</strong> I'm not able to use L'Hopital rule. Sorry.</p>
| Julien | 38,053 | <p>Afer the appropriate change of variable ($x=\ln n$ and then $y=bx$), this boils down to
$$
\lim_{y\rightarrow +\infty}\frac{y^a}{e^y}=0.
$$</p>
<p>Let $n$ be an integer such that $n>a$.
Then, for $y\geq 0$,
$$
e^y=\sum_{k=0}^{+\infty}\frac{y^k}{k!}\geq \frac{y^n}{n!}.
$$
So
$$
0\leq \frac{y^a}{e^y}\leq \frac{n!y... |
478,517 | <blockquote>
<p>Construct a topological mapping of the open disk $|z|<1$ onto the whole plane.</p>
</blockquote>
<p>I represent $z=re^{i\theta}$. I thought about the bijection from $(0,1)$ to $(0,\infty)$, which is given by $x\rightarrow \dfrac1x-1$. Applying this to the norm, we will get the mapping $re^{i\theta... | Brian M. Scott | 12,042 | <p>Don’t flip it: just expand it. To do this, you must multiply $z$ by a factor that increases without bound as $|z|\to 1$; one natural choice is $(1-|z|)^{-1}$.</p>
|
3,786,654 | <blockquote>
<p>Let <span class="math-container">$x_1, x_2, x_3 \in \Bbb R$</span>, satisfy <span class="math-container">$0 \leq x_1 \leq x_2 \leq x_3 \leq 4$</span>. If their squares form an arithmetic progression with common difference <span class="math-container">$2$</span>, determine the minimum possible value of <... | boojum | 882,145 | <p><a href="https://i.stack.imgur.com/xG0Bj.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xG0Bj.png" alt="enter image description here" /></a></p>
<p>( vertical scale compressed )</p>
<p>A way to see this intuitively is to consider the parabolic curve <span class="math-container">$ \ y \ = \ x^2 \ ... |
4,068,314 | <p>I do know that double negation and LEM are equivalent, but can we prove
<span class="math-container">$$\vdash \neg \neg (p \vee \neg p)$$</span>
without using either of them, in a Fitch-style proof?</p>
| Mauro ALLEGRANZA | 108,274 | <p>Yes, we can.</p>
<p><em>Hint</em>: assume <span class="math-container">$\lnot (p \lor \lnot p)$</span> and derive a contradiction.</p>
<p>The conclusion follows by <span class="math-container">$(\lnot \text I)$</span>, which is intuitionistically valid.</p>
|
3,762,174 | <p>I have some confusion in integration . My confusion marked in red and green circle as given below<a href="https://i.stack.imgur.com/4fc1I.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/4fc1I.png" alt="enter image description here" /></a></p>
<p>Im not getting why <span class="math-container">$$\... | zkutch | 775,801 | <p>Hint: Try to plot these areas and look firstly from <span class="math-container">$Ox$</span>, then from <span class="math-container">$Oy$</span> axes.</p>
<p>More formally: we should prove</p>
<p><span class="math-container">$$\left\{ \begin{array}{}
0 \leqslant x \leqslant 1 \\
0 \leqslant y \leqslant x
\end{arra... |
454,040 | <p>I need to know whether There exists any continuous onto map from $(0,1)\to (0,1]$</p>
<p>could any one give me any hint?</p>
| rurouniwallace | 35,878 | <p>Find a polynomial that:</p>
<ol>
<li>Crosses the x-axis at $x=0$ and $x=1$.</li>
<li>Has an absolute maximum of $f(x)=1$.</li>
</ol>
<p>$$f(x)=-4(x^2-x),x\in(0,1)$$</p>
|
1,654,545 | <p>My teacher explained this problem to us - "There are $3$ mailboxes. $3$ people put letters in at random. There is no preference for any of the $3$ mailboxes. Compute the probability that each mailbox contains $1$ letter."</p>
<p>I tried this problem on my own and got the wrong answer. I understand the teacher's so... | Jan van der Vegt | 298,403 | <p>The problem with your approach is that not every one of those outcomes have the same probability. In your sample space the probability of $300$ is not $\frac{1}{10}$, it is $\frac{1}{27}$. This is because there is no preference, you can multiply the probabilities of each event individually. The first letter has $\fr... |
484,117 | <p>What is the simplest $\Bbb{R}\to\Bbb{R}$ function with two peaks and a valley?</p>
<p>I have a set of points in $\Bbb{R^2}$ and I would like to fit a curve to the points, the points approximately lie on a curve like the one depicted in the following figure:</p>
<p><img src="https://i.stack.imgur.com/8tVLu.png" alt... | Bazin | 93,122 | <p>You will not find a close solution, except in some very particular cases such as $f$ is a constant. The uniqueness is not true in general for such a problem since $\sqrt{f(t)-y}$ is only Hölder continuous with exponent $1/2$. However, uniqueness holds say near $t=0, y=0$ provided $f(0)>0$, say with $f$ continuous... |
4,644,904 | <p>Find the greatest and the least values of the function <span class="math-container">$f(x)=\sin x\sin2x$</span> on the interval <span class="math-container">$(-\infty,\infty)$</span>.</p>
<p>The solution presented is as follows:</p>
<blockquote>
<p>Represent the function <span class="math-container">$y=f(x)=\sin x\si... | Enkidu | 455,216 | <p>It is also periodic on <span class="math-container">$[-\pi,\pi]$</span>, and so by eveness the solutions are symmetric along <span class="math-container">$0$</span> and so we can focus on <span class="math-container">$[0,\pi]$</span>.</p>
|
259,308 | <p>The output of <code>ListPointPlot3D</code> is shown below:
<a href="https://i.stack.imgur.com/ypt73.png" rel="noreferrer"><img src="https://i.stack.imgur.com/ypt73.png" alt="enter image description here" /></a>
I only want to connect the dots in such a way that it forms a ring-like mesh. However, when I use <code>Li... | N.J.Evans | 11,777 | <p>You can partition the data into units of three or four and create polygons. There's probably a better way to do it, but here's one option:</p>
<pre><code>dat = {{15.,-4.33154,0.015},{15.4114,-3.89839,0.0154114},{15.7792,-3.46523,0.0157792},{16.1037,-3.03208,0.0161037},{16.3847,-2.59893,0.0163847},{16.6224,-2.16577,0... |
2,448,696 | <p>Show that $\frac{1}{n}<\ln n$, for all $n>1$ where n is a positive integer</p>
<p>I've tried using induction by multiplying both sides by $\ln k+1$ and $\frac{1}{k+1}$ but but all it does is makes it more complicated, I've tried using the fact that $k>1$ and $k+1>2$ during the inductive $k+1$ step, but ... | Jihoon Kang | 452,346 | <p>Suppose $\frac1{k}<\ln(k)$. Then: $$\frac1{k+1}<\frac1{k}<\ln(k)<\ln(k+1)$$</p>
<p>Now you just have to show $\frac12<\ln(2)$.</p>
|
2,448,696 | <p>Show that $\frac{1}{n}<\ln n$, for all $n>1$ where n is a positive integer</p>
<p>I've tried using induction by multiplying both sides by $\ln k+1$ and $\frac{1}{k+1}$ but but all it does is makes it more complicated, I've tried using the fact that $k>1$ and $k+1>2$ during the inductive $k+1$ step, but ... | Fred | 380,717 | <p>For $n \ge 2$ we have:</p>
<p>$1/n \le 1/2 < \ln 2 \le \ln n$.</p>
|
4,089,114 | <p>I'm a newbie for mathematics and now I'm learning PDE and stuck on that. Could anyone help me out to understand this elimination from PDE. The equation is similar to solve <span class="math-container">$$(D^2 -6DD'+9D'^2)u = y\cos x$$</span></p>
| Summand | 754,593 | <p>From Frobenius inequality we have <span class="math-container">$$\operatorname{rank}(ABC)+\operatorname{rank}(B) \geq \operatorname{rank}(AB)+\operatorname{rank}(BC)$$</span></p>
<p>Set <span class="math-container">$A=A, B=A^k, C=A$</span> so that <span class="math-container">$ABC=A^{k+2},AB=BC=A^{k+1}$</span>. But ... |
1,163,033 | <p>I want to calculate $ 8^{-1} \bmod 77 $ </p>
<p>I can deduce $ 8^{-1} \bmod 77$ to $ 8^{59} \bmod 77 $ using Euler's Theorem.</p>
<p>But how to move further now. Should i calculate $ 8^{59} $ and then divide it by $ 77 $ or is there any other theorem i can use ? </p>
| Fermat | 83,272 | <p>$$8\equiv 1 \bmod 7\implies 8^{-1}\equiv 8 \bmod 7$$
also
$$7\times 8\equiv 1 \bmod 11\implies 8^{-1}\equiv 7 \bmod 11$$
Now apply the chinese remainder theorem. Or without this theorem you can write
$$11\times 8^{-1}\equiv 88 \bmod 77$$
and
$$7\times8^{-1}\equiv 49 \bmod 77$$
multiplying the latter by 2 and subtrac... |
1,561,370 | <p>Is there any graphical interface in <a href="http://gap-system.org" rel="noreferrer">GAP</a>? Something like <a href="https://www.rstudio.com/" rel="noreferrer">RStudio</a> for <a href="https://www.r-project.org/" rel="noreferrer">R</a> or <a href="http://andrejv.github.io/wxmaxima/" rel="noreferrer">WxMaxima</a> fo... | Olexandr Konovalov | 70,316 | <p>There were several attempts, one started as Max Neunhöffer's GAP package <a href="http://www.gap-system.org/Packages/xgap.html" rel="nofollow noreferrer">XGAP</a> which works only on Linux, others as external software (see at the bottom of <a href="http://www.gap-system.org/Packages/undep.html" rel="nofollow norefer... |
2,526,716 | <p>Define, by structural induction, a function $f : A^* \to A^*$ that removes all occurrences of the letter $a$. For instance, we should have
$f(abcbab) = bcbb$ and $f(bc) = bc$.</p>
<p>I came up with this:</p>
<p>$f(\lambda) = \lambda$ (empty word)</p>
<p>$f(xw) = xf(w)$ if $x$ is not $a$</p>
<p>$f(w)$, otherwise.... | zoli | 203,663 | <p>The minimum distance is at the $x$ at which the tangent to the parabola is $4$. Where the tangent is parallel to the straight line:</p>
<p><a href="https://i.stack.imgur.com/kGTYq.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/kGTYq.png" alt="enter image description here"></a></p>
<p>So, one ha... |
316,866 | <p>Suppose $(a_n)$ is a real sequence and $A:=\{a_n \mid n\in \Bbb N \}$ has an infinite linearly independent subset (with respect to field $\Bbb Q$). Is $A$ dense in $\Bbb R?$</p>
| user642796 | 8,348 | <p>Not necessarily. We can use recursion to construct a linearly independent set $A = \{ a_n : n \in \mathbb{N} \}$ so that $a_n \in ( n , n+1 )$ for all $n$. Such a set is clearly not dense, but even more it is a closed discrete subset of $\mathbb{R}$.</p>
<p>Even a Hamel basis $B$ for $\mathbb{R}$ may not be dense... |
1,622,779 | <p>I have been asked to calculate $\frac{1+i \tan \alpha}{1-i \tan \alpha}$, where $\alpha \in \mathbb{R}$.</p>
<p>So, I multiplied top and bottom by the complex conjugate of the denominator:</p>
<p>$\frac{(1+i \tan \alpha)(1+ i \tan \alpha)}{(1-i \tan \alpha)(1+ i \tan \alpha) } = \frac{1 + 2 i \tan \alpha + i^{2} \... | Shaswata | 68,110 | <p>Well what you really should've tried is multiplying the numerator and denominator by $\cos \alpha$ which would give you,
$$\frac{\cos\alpha +i \sin \alpha}{\cos\alpha -i \sin \alpha}=\frac{e^{i\alpha}}{e^{-i \alpha}}=e^{i 2 \alpha} = \cos 2\alpha + i \sin 2\alpha$$</p>
|
732,996 | <p><img src="https://i.stack.imgur.com/kXJEt.png" alt="enter image description here"></p>
<p>Hi! I am working on some ratio and root test online homework problems for my calc2 class and I am not sure how to completely solve this problem. I guessed on the second part that it converges, but Im not sure how to solve of t... | Ted Shifrin | 71,348 | <p>You aren't expected to figure out what the value of the series is (although in time you might figure out it has something to do with $e+e^{-1}$). Did you actually compute $\rho$? What is $$\lim_{n\to\infty} \frac{(2n)!}{(2n+2)!}?$$</p>
|
605,772 | <p>Solving $x^2-a=0$ with Newton's method, you can derive the sequence $x_{n+1}=(x_n + a/x_n)/2$ by taking the first order approximation of the polynomial equation, and then use that as the update. I can successfully prove that the error of this method converges quadratically. However, I can't seem to prove this for th... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\ds}[1]{\displayst... |
135,426 | <p>$$\frac{d}{dq}\int_{s_{1}-z-q}^{z-s_{1}} \varphi(w) \, dw$$</p>
<p>(if it helps, in my setting $\varphi$ is the CDF of some arbitrary uniform distribution). So I want to get a nice expression for this integral and it seems to suggest FTC, but I tried a change of variable and ended up with a $q$ inside the integrand... | Jyrki Lahtonen | 11,619 | <p>You may be familiar with Leibniz' result on an alternating monotonically decreasing (in absolute value) series: the partial sum is above (resp. below) the sum, if the last included term was positive (resp. negative). That gives us a clue. Compute
$$
\frac12-\frac13+\frac14-\frac15+\frac16-\frac17+\frac18=\frac{307}{... |
3,498,199 | <p>Suppose if a matrix is given as</p>
<p><span class="math-container">$$ \begin{bmatrix}
4 & 6\\
2 & 9
\end{bmatrix}$$</span></p>
<p>We have to find its eigenvalues and eigenvectors.</p>
<p>Can we first apply elementary row operation . Then find eigenvalues.</p>
<p>Is their any relation on the matrix if ... | CroCo | 103,120 | <p>First compute the eigenvalues as follows:
<span class="math-container">$$
\begin{align}
\det(\lambda I-A)&=\det\left(
\begin{bmatrix}
\lambda & 0 \\
0 & \lambda
\end{bmatrix}
-
\begin{bmatrix}
4 & 6 \\
2 & 9
\end{bmatrix}
\right)
= \det \left(
\begin{bmatrix}
\lambda-4 & -6 \\
-2 & \lamb... |
2,697,069 | <p>Two series of functions are given in which I cannot figure out how to find $M_n$ of the second problem. $$1.\space \sum_{n=1}^{\infty} \frac{1}{1+x^n}, x\in[k,\infty)\\ 2. \space \sum_{n=1}^{\infty} (\cos x)^n, x\in(0,\pi)$$.. </p>
<p>I have determined the $M_n$ for problem no. $1.$ [$\space|\sum_{n=1}^{\infty} \f... | Rebellos | 335,894 | <p>It is : $y=\frac{2}{5}x$ so simply by substituting :</p>
<p>$$\frac{\frac{2}{5}x}{3} = \frac{z}{4} \Leftrightarrow \frac{x}{z} = \frac{15}{8}$$</p>
|
2,416,424 | <p>It is known that the collection of finite mixtures of Gaussian Distributions over $\mathbb{R}$ is dense in $\mathcal{P}(\mathbb{R})$ (the space of probability distributions) under convergence in distribution metric.</p>
<p>I'm interested to know the following:</p>
<p>Let $P_X$ be a random variable with finite $p$ ... | Theo Bendit | 248,286 | <p>Finding the roots is a good idea when your inequality is in the form $f(x) > 0$, where $f$ is a <strong>continuous</strong> function (the definition of "continuous" is a little technical, but it means, in essence, that the graph can be drawn without lifting pencil off paper).</p>
<p>There is a theorem about cont... |
3,547,529 | <p>I did the following: I set <span class="math-container">$3^m+3^n+1=x^2$</span> where <span class="math-container">$x\in\Bbb{N}$</span> and assumed it was true for positive integer exponents and for all whole numbers x so that I can later on prove it's invalidity with contradiction. Since <span class="math-container"... | Yiorgos S. Smyrlis | 57,021 | <p>If <span class="math-container">$A(y)>0$</span>, such that
<span class="math-container">$$
a\big(A(y)\big)^{2n}+b\big(A(y)\big)^{2n-1}+\cdots+q=y, \tag{1}
$$</span>
then <span class="math-container">$A(y)\to\infty$</span>, as <span class="math-container">$yas\to\infty$</span>, and hence <span class="math-contain... |
3,547,529 | <p>I did the following: I set <span class="math-container">$3^m+3^n+1=x^2$</span> where <span class="math-container">$x\in\Bbb{N}$</span> and assumed it was true for positive integer exponents and for all whole numbers x so that I can later on prove it's invalidity with contradiction. Since <span class="math-container"... | Paramanand Singh | 72,031 | <p>Let's replace <span class="math-container">$A(y), B(y) $</span> by <span class="math-container">$A, B$</span> to simplify typing and let <span class="math-container">$A>0>B$</span> and we write <span class="math-container">$C=-B$</span> so that <span class="math-container">$C>0$</span>. Then we have <span c... |
2,571,909 | <p>$$\left|\frac{-10}{x-3}\right|>\:5$$</p>
<ul>
<li>Find the values that $x$ can take. </li>
</ul>
<p>I know that</p>
<p>$$\left|\frac{-10}{x-3}\right|>\:5$$
and
$$\left|\frac{-10}{x-3}\right|<\:-5$$</p>
| haqnatural | 247,767 | <p>$$\left| \frac { -10 }{ x-3 } \right| >\: 5\\ \frac { 10 }{ \left| x-3 \right| } >5\\ \left| x-3 \right| <2\\ -2<x-3<2\\ 1<x<5\\ \left( 1;5 \right) -\left\{ 3 \right\} \\ \\ $$</p>
|
1,789,373 | <p>I'm trying to figure out why the following is true:</p>
<p>Let $ \kappa $ be an uncountable, regular cardinal. Suppose we turn it into a group (i.e. there are operations $ (\cdot, ^{-1}, e) $ with which $ \kappa $ is a group. My aim is to prove that the set</p>
<p>$$ \{ \alpha \in \kappa : \alpha \text{ is a subgr... | zyx | 14,120 | <p>The geometry is the same as in 3 dimensions, when taking the plane section $x+y+z = c$ of a sphere $x^2+y^2+z^2=C$. Extrema of $z$ are when $x=y$. The only remaining thing to understand is why the answer for the parameters given in the problem does not involve square roots.</p>
<p>Let $n$, which equals $4$ in the... |
4,506,026 | <p>Consider the set of equations:
<span class="math-container">$$
\begin{cases}
x^2 &= -4y-10\\y^2 &= 6z-6\\z^2 &= 2x+2\\
\end{cases}$$</span></p>
<p>With <span class="math-container">$x,y,z$</span> being real numbers.</p>
<p>By adding the three equations, after simple manipulations, we easily obtain
<span ... | Sambo | 454,855 | <p>This is how I think about it. When you use your initial set of three equations, which I'll call (A), to obtain your final equation, which I'll call (B), what you're saying is the following: "If <span class="math-container">$x,y,z$</span> satisfy (A), then they satisfy (B)".</p>
<p>Importantly, this reasoni... |
3,182,802 | <p>Show that if <span class="math-container">$ \sigma $</span> is a solution to the equation <span class="math-container">$ x^2 + x + 1 = 0 $</span> then the following equality occurs:</p>
<p><span class="math-container">$$ (a +b\sigma + c\sigma^2)(a + b\sigma^2 + c\sigma) \geq 0 $$</span></p>
<p>I looked at the solu... | Donald Splutterwit | 404,247 | <p>First note that <span class="math-container">$ \sigma ^3=1$</span>. So the expression can be rewritten as
<span class="math-container">\begin{eqnarray*}
a^2+b^2+c^2+(ab+bc+ca)(\sigma+\sigma^2)=a^2+b^2+c^2-(ab+bc+ca) \\ =\frac{1}{2}((a-b)^2+(b-c)^2+(c-a)^2)
\end{eqnarray*}</span>
which is clearly non-negative.</p>
|
35,688 | <p>I'm looking for a fun (not too many tedious calculations) calculus one problem that uses the concept that, after subsitution, you have two integrals of diffent functions with different limits, but equal area. For example:</p>
<p><a href="http://www.wolframalpha.com/input/?i=int%20%28sin%28%28pi%5E2%29/x%29%29/%28x%... | Shai Covo | 2,810 | <p>How about calculating the area, $S$, of one quarter of the unit circle? </p>
<p>First note that $S = \int_0^1 {\sqrt {1 - x^2 } \,{\rm d}x} $. The substitution $x=\sin(u)$ (hence, ${\rm d}x = \cos(u) \,{\rm d}u$) gives
$$
\int_0^1 {\sqrt {1 - x^2 } \,{\rm d}x} = \int_0^{\pi /2} {\sqrt {1 - \sin ^2 (u)} \cos (u)\,{... |
3,613,235 | <p>I know such integral: <span class="math-container">$\int_0^{\infty}\frac{\ln x}{e^x}\,dx=-\gamma$</span>. What about the integral <span class="math-container">$\int_0^{\infty}\frac{\ln x}{e^x+1}\,dx$</span>? </p>
<p>The answer seems very nice: <span class="math-container">$-\frac{1}{2}{\ln}^22$</span> but how it co... | Mark Viola | 218,419 | <p><strong>HINT:</strong></p>
<p>Note that we have</p>
<p><span class="math-container">$$\frac1{\sqrt n+\sqrt{n+2}}=\frac{\sqrt{n+2}-\sqrt n}{2}$$</span></p>
<p>Now telescope.</p>
|
3,029,446 | <p>If </p>
<ul>
<li><p><span class="math-container">$A$</span> is <span class="math-container">$m \times n$</span> (<span class="math-container">$m<n$</span>), and its rows are independent</p></li>
<li><p><span class="math-container">$B$</span> is <span class="math-container">$n \times p$</span> (<span class="math-... | AnyAD | 107,693 | <p>For this we need <span class="math-container">$AB $</span> to have rank <span class="math-container">$p $</span> (since <span class="math-container">$p$</span> columns). All we are garanteed however is that the rank of <span class="math-container">$AB$</span> is less orequal to the minimum of <span class="math-conta... |
4,468,112 | <p>Let <span class="math-container">$a,b\in\mathbb R$</span> with <span class="math-container">$a<b$</span>, <span class="math-container">$$\mathcal D_{[a,\:b]}:=\{(t_0,\ldots,t_k):k\in\mathbb N\text{ and }a=t_0<\cdots<t_k\}$$</span> and <span class="math-container">$$\mathcal T_\varsigma:=\{(\tau_1,\ldots,\ta... | Sangchul Lee | 9,340 | <p>The following claims hold:</p>
<blockquote>
<p><strong>Theorem 1.</strong> Let <span class="math-container">$g$</span> be absolutely continuous on <span class="math-container">$[a, b]$</span>, and let <span class="math-container">$f$</span> be continuous on <span class="math-container">$[a, b]$</span>. Then <span cl... |
962,691 | <p>I'm trying to integrate $ \int_0^1\frac {u^2 + 1}{u - 2}du$</p>
<p>I've calculated that this equates to $ [\frac{u^2}{2}+2u +5ln(u-2)]_0^1 $</p>
<p>But then I have to evaluate $ln(-1)$ and $ln(-2)$ which are obviously not defined in the real plane. I have drawn the graph and I know for certain that this integral e... | BaronVT | 39,526 | <p>Recall that the antiderivative of $\frac{1}{u}$ is $\ln|u|$, not $\ln u$</p>
|
323,665 | <p>Given the base case <span class="math-container">$a_0 = 1$</span>, does <span class="math-container">$a_n = a_{n-1} + \frac{1}{\left\lfloor{a_{n-1}}\right \rfloor}$</span> have a closed form solution? The sequence itself is divergent and simply goes {<span class="math-container">$1, 2, 2+\frac{1}{2}, 3, 3+\frac{1}{3... | Ale De Luca | 7,320 | <p>A similar, possibly simpler closed form is the following: set <span class="math-container">$$b_n=\left\lfloor\frac{1+\sqrt{8n-7}}{2}\right\rfloor,$$</span>
then <span class="math-container">$$a_n=\frac{b_n+1}{2}+\frac{n-1}{b_n}.$$</span>
It is not hard to derive this from the observation that whenever <span class="m... |
1,649,907 | <p>Please kindly forgive me if my question is too naive, i'm just a <em>prospective</em> undergraduate who is simply and deeply fascinated by the world of numbers.</p>
<p>My question is: Suppose we want to prove that $f(x) > \frac{1}{a}$, and we <em>know</em> that $g(x) > a$, where $f,g$ and $a$ are all positive... | fleablood | 280,126 | <p>The whole thing boils done to:</p>
<p>if $a > b$ (a and b positive) then $1/a < 1/b$. The reciprical "<em>flips</em>" the inequality.</p>
<p>$f(x)g(x) < 1$ (or greater than) implies $f(x) < 1/g(x)$ (or greater than). To get a result $g(x) ?? a$ must be the <em>OPPOSITE</em> inequality because $1/g(x)... |
362,881 | <p>I am going to try to explain this as easily as possible. I am working on a computer program that takes input from a joystick and controls a servo direction and speed. I have the direction working just fine now I am working on speed. To control the speed of rotation on the servo I need to send it so many pulses per s... | Shuhao Cao | 7,200 | <p>Depends on how you define the Dirac delta, if it is defined as a linear functional acting on an $f\in C^{\infty}_0(\mathbb{R})$:
$$
\delta[f](x) = \int^{\infty}_{-\infty} \delta(x)f(x) \,dx = f(0)
$$
Then we can see for any test function $\phi$:
$$
(e^{x}\delta(\cdot) - \delta(\cdot))[\phi](x) = \int^{\infty}_{-\inf... |
2,487,234 | <p>I want to prove that the following only have one solution, for $\zeta\in[0,1]$, at $\zeta =1$.</p>
<p>$$f(\zeta)=\frac{1}{1+\zeta}$$</p>
<p>$$g(\zeta) =
\frac{
(1-\zeta)(2-\zeta)\zeta
-
(2-\zeta)^2\log\left(2 - \zeta \right)
}
{\zeta(\zeta-4)(\zeta-1)^2}$$</p>
<p>These are plotted below. Note that $f(0)=1$, $\lim... | Lerigorilla | 450,010 | <p>we want to solve g(x)=f(x) for x in [0,1]
by doing the calculations we end up in this equation
$$ \ log(2-x)= \frac{4x^3-10x^2+6x}{(x+1)(2-x)^2}$$
$$h(x)=log(2-x)$$ and
$$ k(x)=\frac{4x^3-10x^2+6x}{(x+1)(2-x)^2}$$
h(x) is convex and k(x) is concave in [0,1] (1)
$$ k'(x)=\frac{-2(x^3+8x^2-17x+6)}{(x-2)^3(x+1)^2} $$
... |
2,399,710 | <p>I am asking for what, if any, the preferred study skills of approaching classes are like in higher math education are like <strong>in general,</strong> and I am not asking necessarily for personal anecdotes for this question (though they are welcome if it's all you have to share).</p>
<p>My question is what are the ... | Merkh | 141,708 | <p>You question is waay too many questions in one for this website. Just FYI. Anyway...</p>
<p>"... real mathematics is usually done where you understand 2 or 3 pages a day of a text on your first reading of it. Is this true for graduate-level work for the average student??"</p>
<ul>
<li>This completely depends on w... |
2,399,710 | <p>I am asking for what, if any, the preferred study skills of approaching classes are like in higher math education are like <strong>in general,</strong> and I am not asking necessarily for personal anecdotes for this question (though they are welcome if it's all you have to share).</p>
<p>My question is what are the ... | Joe | 61,666 | <p>I found benefit in reading two books by Lara Alcock:</p>
<p>How to Study as a Mathematics Major</p>
<p>and</p>
<p>How to Think About Analysis</p>
|
1,662,876 | <p>Now we have some examples of what I mean $$\int_0^{2\pi} \sin x~dx=0$$
$$\int_0^{8\pi} \cos 4x~dx=0$$</p>
<p>$$\int_{\pi}^{2\pi} \sin^3 10x~dx=0$$</p>
<p>Looking at the graph of $f(x)=\sin (x)$ for example it makes some sense to me that $$\int_0^{2\pi} \sin x~dx=0$$ because the region below the $x$ axis will "canc... | John Hughes | 114,036 | <p>Roughly speaking, a trig integral (sine or cosine) that includes an integer number of cycles will be zero. The same goes for $\sin nx$ or $\cos nx$. Your "area above the line is the same as area below" is a perfectly good explanation. </p>
<p>As for powers of sines, any odd power of sine can be simplified by the ru... |
1,662,876 | <p>Now we have some examples of what I mean $$\int_0^{2\pi} \sin x~dx=0$$
$$\int_0^{8\pi} \cos 4x~dx=0$$</p>
<p>$$\int_{\pi}^{2\pi} \sin^3 10x~dx=0$$</p>
<p>Looking at the graph of $f(x)=\sin (x)$ for example it makes some sense to me that $$\int_0^{2\pi} \sin x~dx=0$$ because the region below the $x$ axis will "canc... | Surb | 154,545 | <p><strong>Hint</strong></p>
<p>If $f$ is periodic and odd, you will have $$\int_a^{a+T}f(x)dx=0$$
for all $a\in\mathbb R$.</p>
<p>For $$\int_\pi^{2\pi}\sin^3(10 x)dx$$</p>
<p>you have that $x\longmapsto \sin^3(10x)$ is odd and $\frac{\pi}{5}-$periodic. Moreover, $$\int_\pi^{2\pi}\sin^3(10x)dx=\sum_{k=0}^4\underbrac... |
1,662,876 | <p>Now we have some examples of what I mean $$\int_0^{2\pi} \sin x~dx=0$$
$$\int_0^{8\pi} \cos 4x~dx=0$$</p>
<p>$$\int_{\pi}^{2\pi} \sin^3 10x~dx=0$$</p>
<p>Looking at the graph of $f(x)=\sin (x)$ for example it makes some sense to me that $$\int_0^{2\pi} \sin x~dx=0$$ because the region below the $x$ axis will "canc... | MPW | 113,214 | <p>If $p$ is a period for $f$, then $p$ is also a period for $f^n$.</p>
<p>If the integral is over an interval whose length is a whole number of periods for the integrand, and if the integral over a single period is zero, then that integral is also zero.</p>
<p>A period for $\sin 10x$ is $p=2\pi/10 = \pi/5$. So $\pi/... |
4,206,205 | <p>I came across a theorem in algebraic number theory:</p>
<blockquote>
<p><strong>Theorem</strong> Let <span class="math-container">$A$</span> be a Dedekind ring and <span class="math-container">$M, N$</span> two modules over <span class="math-container">$A$</span>. If <span class="math-container">$M_\mathfrak{p} \sub... | Stefan4024 | 67,746 | <p>In fact, the claim is true for any unital ring <span class="math-container">$R$</span>. Indeed, suppose that <span class="math-container">$M_\mathfrak p \subseteq N_\mathfrak p$</span> for all prime ideals <span class="math-container">$\mathfrak p$</span> of <span class="math-container">$R$</span>.</p>
<p>Let <span ... |
3,129,372 | <p>I want to compute if exists <span class="math-container">$\lim_{r\rightarrow +\infty} \int_{B(0,r)} \frac{y}{1+\sqrt{(x^2+y^2)^5}}$</span> dx dy</p>
<p>I use polar coordinates and I found <span class="math-container">$4\lim_{r\rightarrow +\infty} \int_{0}^{r} \frac{\rho^2}{1+\rho^5} d\rho$</span>.</p>
<p>This inte... | TheSilverDoe | 594,484 | <p>How many possibilities do you have for the basket <span class="math-container">$A$</span> ? You have to choose <span class="math-container">$3$</span> fruits among a total of <span class="math-container">$5$</span> fruits. By definition, the number of possibilities is <span class="math-container">${5 \choose 3}$</sp... |
3,129,372 | <p>I want to compute if exists <span class="math-container">$\lim_{r\rightarrow +\infty} \int_{B(0,r)} \frac{y}{1+\sqrt{(x^2+y^2)^5}}$</span> dx dy</p>
<p>I use polar coordinates and I found <span class="math-container">$4\lim_{r\rightarrow +\infty} \int_{0}^{r} \frac{\rho^2}{1+\rho^5} d\rho$</span>.</p>
<p>This inte... | user298173 | 504,775 | <p>Consider filling Basket A first. For this, you have five fruit and must choose three of them. There are 5 choices for the first fruit, and once this is chosen there are 4 for the next fruit and finally 3 choices for the last fruit and so <span class="math-container">$5\cdot 4\cdot 3$</span> choices. Since the order ... |
1,022,523 | <blockquote>
<p>Make a series expansion of $f(z)=\dfrac{1}{z^2+z-6}$ valid in the region $2<|z|<3$.</p>
</blockquote>
<p>By partial fractions,</p>
<p>$$f(z) = \frac{1}{(z-2)(z+3)} = \frac{1}{5(z-2)}-\frac{1}{5(z+3)}.$$</p>
<p>From here, how are these fractions expanded into a geometric series?</p>
| Mister Benjamin Dover | 196,215 | <p>A very informative, brief paper is P. Deligne, <em>Quelques idées maîtresses de l'œuvre de A. Grothendieck</em> (Matériaux pour l'histoire des mathématiques au XXe siècle. Actes du colloque à la mémoire de Jean Dieudonné, Séminaires et Congrès 3), which focuses on Grothendieck's contribution to the theory of schemes... |
3,811,498 | <p>Solve, by bringing the equation to Bernoulli form:</p>
<p><span class="math-container">$$
y’ = \frac{2-xy^3}{3x^2y^2}
$$</span></p>
<hr />
<p>Therefore we want to bring it to a form like:</p>
<p><span class="math-container">$$
y’ + p(x)y = q(x)y^n
$$</span></p>
<p>So working with the equation i get:</p>
<p><span cla... | Physor | 772,645 | <p>you have <span class="math-container">$n = -2$</span></p>
<p><span class="math-container">$$
y^\prime +\frac{1}{3x}y = \frac{2}{3x^2}y^{-2}
$$</span></p>
|
1,033,383 | <p>$ABCD$ is a rectangle and the lines ending at $E$, $F$ and $G$ are all parallel to $AB$ as shown. </p>
<p>If $AD = 12$, then calculate the length of $AG$.<img src="https://i.stack.imgur.com/OUQZ8.png" alt="enter image description here"></p>
<p>Ok, I started by setting up a system of axes where $A$ is the origin an... | Sam Dittmer | 194,283 | <p>A solution without coordinates:</p>
<p>Let $E',F'$, and $G'$ be the intersection points on $AC$ of the lines parallel to $AB$ through $E,F$, and $G$ respectively. By symmetry, $AE=ED=6$. And $\triangle ABD \sim \triangle EE'D$, so $EE' = \tfrac 12 AB$.</p>
<p>Then $\triangle ABF' \sim \triangle E'EF'$, and $EE' = ... |
876,310 | <p>So I <em>think</em> I understand what differentials are, but let me know if I'm wrong.</p>
<p>So let's take $y=f(x)$ such that $f: [a,b] \subset \Bbb R \to \Bbb R$. Instead of defining the derivative of $f$ in terms of the differentials $\text{dy}$ and $\text{dx}$, we take the derivative $f'(x)$ as our "primitive"... | Hola_Mundo | 166,011 | <p>OP here. I think I've figured this out:</p>
<p>This definition does seem to hold for differentiation and integration.</p>
<p><strong>Differentiation</strong><br>
My worry here was that because $\Delta x = dx$ and $\Delta y$ is a function of $\Delta x = dx$, that $\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$ w... |
1,794,072 | <p>My attempt :</p>
<p>If $n$ is odd, then the square must be 2 (mod 3), which is not possible.</p>
<p>Hence $n =2m$</p>
<p>$2^{2m}+3^{2m}=(2^m+a)^2$</p>
<p>$a^2+2^{m+1}a=3^{2m}$</p>
<p>$a (a+2^{m+1})=3^{2m} $</p>
<p>By fundamental theorem of arithmetic, </p>
<p>$a=3^x $</p>
<p>$3^x +2^{m+1}=3^y $</p>
<p>$2^{... | lulu | 252,071 | <p>Work $\pmod 3$. We see that $$2^n+3^n\equiv 2^n\pmod 3$$ from which we quickly deduce that your expression is only a square $\pmod 3$ if $n$ is even.</p>
<p>Now work $\pmod 5$. We see that $$2^n+3^n\equiv 2^n+(-2)^n\equiv 2^n(1+(-1)^n)\pmod 5$$ from which we deduce that your expression is only a square $\pmod 5$ ... |
105,750 | <p>Given a <code>ContourPlot</code> with a set of contours, say, this:</p>
<p><a href="https://i.stack.imgur.com/cKoyo.jpg"><img src="https://i.stack.imgur.com/cKoyo.jpg" alt="enter image description here"></a></p>
<p>is it possible to get the contours separating domains with the different colors in the form of lists... | Sumit | 8,070 | <p>A simple way could be using the <code>PlotRange</code>.</p>
<pre><code>Table[
ContourPlot[x*Exp[-x^2 - y^2], {x, 0, 2}, {y, -1, 1},
PlotRange -> {i, j}, ColorFunction -> "Rainbow",
PlotLegends -> True, PlotLabel -> {i, j}]
, {i, 0.1, 0.3, 0.1}, {j, i + 0.1, 0.4, 0.1}] // TableForm
</code></... |
881,282 | <p>Same as above, how to simplify it. I am to calculate its $n$th derivative w.r.t x where t is const, but I can't simplify it. Any help would be appreciated. Thank you.</p>
| Deathkamp Drone | 56,720 | <p>It seems like you're right, Kasper. A trigonometric substituition would solve the equation completely. I'll write the solution I arrived at:</p>
<p>Finding inequalities is helpful to motivate the right subtituition. We know that both $\frac{1}{x^2}$ and $\frac{1}{(4-\sqrt{3}x)^2}$ are positive, so we can say that:
... |
2,783,423 | <p>If I have a line formed by points A and B, how can I find the distance of another point from that line. Also, whether that line is clockwise or CCW from point A.
<a href="https://i.stack.imgur.com/dpazD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/dpazD.png" alt="enter image description here"><... | Community | -1 | <p>$$\frac{AB}{\|AB\|}\times AZ$$ gives you the distance with a positive or negative sign depending on which side $Z$ lies.</p>
|
3,294,082 | <p>The exercise is to prove that the minimum value between <span class="math-container">$a^{1/b}$</span> and <span class="math-container">$b^{1/a}$</span> is no greater than <span class="math-container">$3^{1/3}$</span>, where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are posit... | Jihoon Kang | 452,346 | <p>Without loss of generality, let <span class="math-container">$a \leq b$</span> and keep <span class="math-container">$b$</span> be a random fixed value, then since <span class="math-container">$a, b \geq 1$</span> and <span class="math-container">$1/b \leq 1/a$</span>,</p>
<p><span class="math-container">$$\min{(a^... |
2,339,101 | <blockquote>
<p>There are six socks in a drawer. The socks are of two colors: black and white. If you draw two socks randomly, the probability that you get white socks is $\frac{2}{3}$. What is the probability of getting black socks, when two socks are drawn at a time?</p>
</blockquote>
<p>There is no detail about t... | Leo163 | 185,102 | <p>The answer is $0$: out of $15$ possible unordered couples of socks, you know that $10$ are made by white socks. It means that only $5$ possible pairs of socks contain a black sock. It can only be true if there is one black sock and $5$ white socks.</p>
|
619,477 | <blockquote>
<p>Alice opened her grade report and exclaimed, "I can't believe Professor Jones flunked me in Probability." "You were in that course?" said Bob.
"That's funny, i was in it too, and i don't remember ever seeing you there."
"Well," admitted Alice sheepishly, "I guess i did skip class a lot." "Yeah, ... | Hagen von Eitzen | 39,174 | <p>Let $n$ be the total number of lectures, $n_{Ab}$ the number of lectures Alice attended and Bob didn't, $n_{aB}$ the number of lectures Bob attended and Alice didn't, $n_{AB}$ the number they both attended and $n_{ab}$ neither of thenm attended.
Then $n=n_{ab}+n_{aB}+n_{Ab}+n_{AB}$ and Alice missed $n_{ab}+n_{aB}$, ... |
3,578,357 | <p>The problem is like this : How do you solve <span class="math-container">$$ \lim _{x\to 0}\:\:\frac{x^m-sin^n(x)}{x^{n+2}} $$</span> for different values of <span class="math-container">$ n \in \Bbb N $</span>
Now, what i've started doing is to add <span class="math-container">$$ \lim _{x\to 0}\:\:\frac{x^m-x^n+x^n... | marty cohen | 13,079 | <p>If <span class="math-container">$f \ne 0$</span> then,
since <span class="math-container">$(1/f)' = -f'/f^2$</span>,
<span class="math-container">$(1/f)' = -1$</span> so
<span class="math-container">$1/f = -x+c$</span> so
<span class="math-container">$f(x) = 1/(-x+c)$</span>.</p>
<p>Then
<span class="math-containe... |
3,578,357 | <p>The problem is like this : How do you solve <span class="math-container">$$ \lim _{x\to 0}\:\:\frac{x^m-sin^n(x)}{x^{n+2}} $$</span> for different values of <span class="math-container">$ n \in \Bbb N $</span>
Now, what i've started doing is to add <span class="math-container">$$ \lim _{x\to 0}\:\:\frac{x^m-x^n+x^n... | Behnam Esmayli | 283,487 | <p>Note that your <span class="math-container">$f$</span> is a solution to the ODE
<span class="math-container">$$
dy/dx = y^2 \ ,
$$</span>
which can be solved using separation of variable. ALL solutions to this ODE other than the obvious solution <span class="math-container">$y = 0 $</span> are of the form
<span clas... |
3,753,060 | <blockquote>
<p>If <span class="math-container">$\int f(x)dx =g(x)$</span> then <span class="math-container">$\int f^{-1}(x)dx $</span> is equal to</p>
<p>(1) <span class="math-container">$g^{-1}(x)$</span></p>
<p>(2) <span class="math-container">$xf^{-1}(x)-g(f^{-1}(x))$</span></p>
<p>(3) <span class="math-container">... | Z Ahmed | 671,540 | <p>Given <span class="math-container">$y=f(x) \implies x=f^{-1}(y)$</span>
Then <span class="math-container">$$\int f^{-1}(x) dx= \int f^{-1}(y) dy= \int x dy= \int x \frac{dy}{dx} dx=\int xf'(x) dx $$</span> <span class="math-container">$$=xf(x)-\int f(x)dx=xf(x)-g(x)+C.$$</span> Lastly, we have done integration by p... |
1,615,883 | <p>A cubic polynomial with real coefficients, $a x^3 + b x^2 + c x + d$, has either three real roots, or one real root and a pair of complex conjugate ones. If the latter happens, what is the explicit formula for this real solution, and what conditions can be placed on $a,b,c$ and $d$ to guarantee that the real root is... | Mike Earnest | 177,399 | <p>Your answer is correct, assuming that each of the books are placed independently, and each shelf is equally likely.</p>
<p>However, if the phrase "randomly tidied up" instead meant "choose a random arrangement of books", then you get a different answer.</p>
<ul>
<li>There are $5$ ways where they are all on the sam... |
2,555,399 | <p>The question is to find out the coefficient of $x^3$ in the expansion of $(1-2x+3x^2-4x^3)^{1/2}$</p>
<p>I tried using multinomial theorem but here the exponent is a fraction and I couldn't get how to proceed.Any ideas?</p>
| Jack D'Aurizio | 44,121 | <p>It is quite practical to exploit the identity
$$ (1+x)^2 (1-2x+3x^2-4x^3) = 1-5x^4-4x^5 \tag{A}$$
from which
$$\begin{eqnarray*} \sqrt{1-2x+3x^2-4x^3} &=& \frac{\sqrt{1-5x^4-4x^3}}{1+x} \\&=&\left(1+O(x^4)\right)(1-x+x^2-x^3+O(x^4))\tag{B}\end{eqnarray*}$$
and the coefficient of $x^3$ in the RHS of $... |
45,911 | <p>I've been wondering for some time now about the difference between a point and a vector. In high school, it was very important to distinguish them from each other, and we used the notation $(x,y,z)$ for points and $[x,y,z]$ for vectors. We always had to translate the point $P=(a,b,c)$ to the vector $\overrightarrow{... | Qiaochu Yuan | 232 | <p>A point in Euclidean space is properly regarded as an element of an <a href="http://en.wikipedia.org/wiki/Affine_space">affine space</a> rather than a vector space. That's because vector spaces have a distinguished origin, and "space" in the general sense doesn't: you can move the origin anywhere you want. Affine sp... |
45,911 | <p>I've been wondering for some time now about the difference between a point and a vector. In high school, it was very important to distinguish them from each other, and we used the notation $(x,y,z)$ for points and $[x,y,z]$ for vectors. We always had to translate the point $P=(a,b,c)$ to the vector $\overrightarrow{... | Robert Israel | 8,508 | <p>In general, mathematicians would distinguish between points and vectors in a context where that distinction is important, and might not bother to distinguish between them in a context where it isn't important. </p>
|
588,802 | <p>The problem is: $\sum_{n=1}^{\infty} \frac{1}{n(n+3)}$</p>
<p>The first thing I did was use the divergence test which didn't help since the result of the limit was 0.</p>
<p>If I multiply it through, the result is $\sum_{n=1}^{\infty} \frac{1}{n^2+3n}$</p>
<p>I'm wondering if I can consider this as a p-series and... | ILoveMath | 42,344 | <p>First, use estimations</p>
<p>$$ n^2 + 3n \geq n^2 \implies \frac{1}{n^2 + 3n } \leq \frac{1}{n^2} $$</p>
<p>Secondly, show that $\sum \frac{1}{n^2}$ converges. In fact, it does. More generally,</p>
<p>$$ \sum \frac{1}{n^p} \; \; \text{converges when} \; \; p > 1 $$</p>
<p>Third, use the comparison theorem:... |
588,802 | <p>The problem is: $\sum_{n=1}^{\infty} \frac{1}{n(n+3)}$</p>
<p>The first thing I did was use the divergence test which didn't help since the result of the limit was 0.</p>
<p>If I multiply it through, the result is $\sum_{n=1}^{\infty} \frac{1}{n^2+3n}$</p>
<p>I'm wondering if I can consider this as a p-series and... | Zhoe | 99,231 | <p>$$n^2 + 3n > n^2 \implies \frac{1}{n^2 +3n} < \frac{1}{n^2}$$</p>
<p>Use the Comparison Test which states that if $\sum a_n$ and $\sum b_n$ are such that $0 \le a_n \le b_n$, if $\sum b_n$ converges, then $\sum a_n$ converges. </p>
<p>Since $0 < \sum \frac{1}{n^2 +3n} < \sum \frac{1}{n^2}$ and $\sum \f... |
2,369,431 | <p>I took a course some years ago and in it was a treatment of how to associate a field to an abstract geometry. </p>
<p>I would very much appreciate some reading on this, as I have been unsuccessful on where to find any resources on such ideas, and I have since lost my notes!</p>
<blockquote>
<p>Is there a book wh... | Somos | 438,089 | <p>You may want to look at articles on Veblen-Wedderburn systems, Hall planes and Moufang planes to see if that helps your research. The book by Dembowski on <em>Finite Geometries</em> may interest you.</p>
|
1,379,283 | <p>In my notes it shows how to calculate by using the unit circle. But I do not know why the value of sin is the y coordinate and the value of cos is the x coordinate.</p>
| AlexR | 86,940 | <p>I'm not sure what your difficulty is, so I'll use the geometrical definition to get the requested values. This definition means that when going a distance of $\theta$ counter-clockwise along the unit circle starting at $(1,0)$, the coordinates of the point on the circle are $(\cos \theta, \sin \theta)$.</p>
<p>Note... |
3,358,592 | <p>In the following quote, what does the notation <span class="math-container">$\{a_n\}$</span> mean?</p>
<blockquote>
<p>Дана последовательность Фибоначчи <span class="math-container">$\{a_n\}$</span>.</p>
</blockquote>
<p><strong>Translation:</strong> "You are given the Fibonacci sequence <span class="math-contai... | G Cab | 317,234 | <p>Your function can be approximated through two shifted ramp functions, as
<span class="math-container">$$
\eqalign{
& f(x) = x\,H(x) - \left( {x - 1} \right)\,H(x - 1) \approx \cr
& \approx {x \over 2}\left( {1 + {x \over {\sqrt {x^{\,2} + \varepsilon ^{\,2} } }}} \right) - {{\left( {x - 1} \right)}... |
285,114 | <blockquote>
<p>Find the solution of the differential equation
$$\frac{dy}{dx}=-\frac{x(x^2+y^2-10)}{y(x^2+y^2+5)}, y(0)=1$$</p>
</blockquote>
<p>Trial: $$\begin{align} \frac{dy}{dx}=-\frac{x(x^2+y^2-10)}{y(x^2+y^2+5)} \\ \implies \frac{dy}{dx}=-\frac{1+(y/x)^2-10/x^2}{(y/x)(1+(y/x)^2+5/x^2)} \\ \implies v+x\... | Mikasa | 8,581 | <p>Hint: It is an exact equation. Assume there is an differentiable function $f(x,y)$ such that $$f_x=x^3+xy^2-10x,~~~f_y=x^2y+y^3+5y$$ and then find the function. The solution is as $$f(x,y)=C$$</p>
|
3,897,361 | <p>Find the GS of the following system of DE's where the independent variable is <span class="math-container">$t$</span> and <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are the dependent variables</p>
<p><span class="math-container">\begin{cases}
x' = x-3y\\
y'=3x+7y
\end{cases}<... | Z Ahmed | 671,540 | <p>Your method is correct.</p>
<p>Another way is to write the system as <span class="math-container">$X'=MY$</span></p>
<p>Where <span class="math-container">$M$</span> is the matrix <span class="math-container">$$\begin{pmatrix} 1 & -3 \\ 3 & 7 \end{pmatrix}$$</span>
its eigen values are <span class="math-cont... |
28,811 | <p>There are lots of statements that have been conditionally proved on the assumption that the Riemann Hypothesis is true.</p>
<p>What other conjectures have a large number of proven consequences?</p>
| Dan Ramras | 4,042 | <p>Resolution of singularities for algebraic varieties in positive characteristic is another example. Many statements in algebraic K-theory have been proven to follow from this conjecture. </p>
|
118,406 | <p>I have a single flat directory with over a million files. I just wanted to take a sample of the first few files but <code>FileNames</code> doesn't include a "only the first n" option, and so it took over a minute:</p>
<p><a href="https://i.stack.imgur.com/s5cBS.png" rel="nofollow noreferrer"><img src="https://i.sta... | george2079 | 2,079 | <p>a linux version using <code>find</code> to list files and <code>head</code> to take the first <code>n</code>.</p>
<pre><code>dir = "/path"
StringSplit[
RunProcess[
{"/bin/sh",
"-c",
"find "<>dir<>" -maxdepth 1 -type f | head -10"
},"StandardOutput"]]
</code></p... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.