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 |
|---|---|---|---|---|
2,646,890 | <blockquote>
<p>If <span class="math-container">$p(x)$</span> is a polynomial of degree <span class="math-container">$n$</span> such that <span class="math-container">$$p(-2)=-15,\ p(-1)=1,\ p(0)=7,\ p(1)=9,\ p(2)=13,\ p(3)=25.$$</span>
Then smalest possible value of <span class="math-container">$n$</span> is</p>
<p>Op... | trancelocation | 467,003 | <p>You do not need to calculate any interpolation polynomial. Here, it is about the minimum degree. So the given points should have a property that makes it possible to find the degree quickly.</p>
<p>When you look at the points you see that the given $x$-values are consecutive integers.</p>
<p>So, you can make a qui... |
95,126 | <p>Consider the finite sum</p>
<pre><code>rs[x_, n_] := x/n Sum[n^2/(i + (n - i) x)^2, {i, 1, n}]
</code></pre>
<p>Is there a way to bring <em>Mathematica</em> to calculate the limit for <code>n -> ∞</code>?</p>
<p>I have tried <code>Limit[]</code> as well as <code>NLimit[]</code> without success.</p>
| Anton Antonov | 34,008 | <p>For a fixed numerical value of x the sum limit can be found using <a href="https://en.wikipedia.org/wiki/Shanks_transformation">Shanks transformation</a> or <a href="https://en.wikipedia.org/wiki/Richardson_extrapolation">Richardson extrapolation</a>. The Richardson extrapolation transformation of the sequence gives... |
95,126 | <p>Consider the finite sum</p>
<pre><code>rs[x_, n_] := x/n Sum[n^2/(i + (n - i) x)^2, {i, 1, n}]
</code></pre>
<p>Is there a way to bring <em>Mathematica</em> to calculate the limit for <code>n -> ∞</code>?</p>
<p>I have tried <code>Limit[]</code> as well as <code>NLimit[]</code> without success.</p>
| Dr. Wolfgang Hintze | 16,361 | <p>See EDIT #3 for a valid answer.</p>
<p><strong>First answer</strong></p>
<p>This is not an answer to my question but just a mathematical derivation of the limit the existence of which was even doubted in some comments and answers.
I hope this does not spoil the creativity. The task for Mathematica is still open.</... |
321,916 | <p>In order to define Lebesgue integral, we have to develop some measure theory. This takes some effort in the classroom, after which we need additional effort of defining Lebesgue integral (which also adds a layer of complexity). Why do we do it this way? </p>
<p>The first question is to what extent are the notions d... | Dave L Renfro | 15,780 | <p>The "area under a graph" approach is used in Wheeden/Zygmund's 1977 text <a href="https://rads.stackoverflow.com/amzn/click/com/0824764994" rel="noreferrer" rel="nofollow noreferrer"><strong>Measure and Integral. An Introduction to Real Analysis</strong></a>, a book that was used (among other possible places) in the... |
321,916 | <p>In order to define Lebesgue integral, we have to develop some measure theory. This takes some effort in the classroom, after which we need additional effort of defining Lebesgue integral (which also adds a layer of complexity). Why do we do it this way? </p>
<p>The first question is to what extent are the notions d... | Nate Eldredge | 4,832 | <p>Suppose your main interest is in constructing the Lebesgue integral over a general abstract measure space <span class="math-container">$(X,\mu)$</span>. From the usual definitions via simple functions, this is fairly straightforward, and one can prove the standard theorems (dominated convergence, etc) without too m... |
321,916 | <p>In order to define Lebesgue integral, we have to develop some measure theory. This takes some effort in the classroom, after which we need additional effort of defining Lebesgue integral (which also adds a layer of complexity). Why do we do it this way? </p>
<p>The first question is to what extent are the notions d... | Max M | 375 | <p>There are several references to various books defining the Lebesgue integral this way in the answers, but the first person to define it this way is ... Lebesgue. In his thesis "Intégrale, Longueur, Aire" (1902) he first discusses the Darboux treatment of Riemann's integral, and then defines his own integrals geometr... |
230,416 | <p>I'm looking for numerical graph invariants that are bounded by a constant either for a graph $G$ or its complement $\bar{G}$. (The complement graph $\bar{G}$ has the same set of vertices as $G$ but the edges are complemented.) More specifically I’m looking for what numerical “Invariant $X$” is out there for which ... | David Eppstein | 440 | <p>Another one: girth. By Ramsey's theorem, for every graph $G$ on six or more vertices, either it or its complement has girth at most three.</p>
|
1,728,595 | <p>Here is the claim I'm trying to understand: Given that $N$ is an integer-valued random variable, why is it true that</p>
<p>$$Var(N) = \sum_{i=1}^\infty Var(1_{N\ge i})$$</p>
<p>For context, this is a step in the answer to exercise 4.5.10 in Rosenthal, <em>A First Look at Rigorous Probability Theory</em>, 2nd ed.,... | MohsenSoltanifar | 330,952 | <p>@ MarshalFarrier, @Jimmy, @ Joriki:</p>
<p>The corrected solution (including correlation) and an additional solution has been posted here:</p>
<p><a href="http://probability.ca/jeff/ftpdir/Ex4.5.10sol.pdf" rel="nofollow">http://probability.ca/jeff/ftpdir/Ex4.5.10sol.pdf</a></p>
|
1,251,537 | <p>$f:[a,b] \to R$ is continuous and $\int_a^b{f(x)g(x)dx}=0$ for every continuous function $g:[a,b]\to R$ with $g(a)=g(b)=0$. Must $f$ vanish identically?</p>
<hr>
<p>Using integration by parts I got the form:
$\int_a^bg(x)f(x)-g'(x)F(x)=0$. Where $F'(x)=f(x)$.</p>
| agha | 118,032 | <p>Suppose that there exists $x_0$ that $f(x_0)=\varepsilon \neq 0$, we can assume that $\varepsilon>0$ without loosing of generality. $f$ is continuous, so there exists $\delta$ such that for $x_1 \in (x_0-\delta,x_0+\delta) \subset (a,b)$ we have $f(x_1) > \frac{\varepsilon}{2}$. Now you can find such a functio... |
4,316,876 | <p>I want to prove that given <span class="math-container">$a,b,c\in\mathbb{R}$</span> we have <span class="math-container">$|a+b|\leq|a|+|b|$</span> using an absurd and reaching a contradiction.</p>
<p>So, I state, by absurd, that <span class="math-container">$|a+b|>|a|+|b|$</span>, but I can't reach the contradict... | soupless | 888,233 | <p>Our claim is that <span class="math-container">$|a| + |b| \geq |a + b|$</span> for real <span class="math-container">$a$</span> and <span class="math-container">$b$</span> and we want to prove it by contradiction. Hence, we assume that <span class="math-container">$|a| + |b| < |a + b|$</span>.</p>
<hr />
<p>Here ... |
3,696,371 | <p>I was rolling stats for a set of characters (main + backup) with my DM, and he told me I could choose between 3 sets of two rolls. One rolled by him, one rolled by another player, and one rolled by me. Him and the other player use physical dice, rolling three dice, then rerolling the lowest value twice. Myself, I ro... | heropup | 118,193 | <p>There is a bit of ambiguity in Scenario 2. If we interpret the procedure literally, it is possible to roll the maximum outcome of <span class="math-container">$18$</span> before the two re-rolls occur. If the re-rolls are required no matter the earlier result, then we have what I call Scenario <span class="math-co... |
2,669,524 | <p>I am reading <strong>Algebraic Geometry</strong>, Vol 1, <em>Kenji Ueno</em>. My problem is that
$$k\left[ x,y,t\right]/\left(xy-t\right)\otimes_{k\left[t\right]}k\left[t\right]/\left(t-a\right) \simeq k\left[x,y\right]/\left(xy-a\right) $$
where $k$ is a field and $a$ is an element in $k$. I don't understand how i... | stochastic | 491,395 | <p>An integral is defined by the limit of Riemann sum as <span class="math-container">$$\int_{V_1}^{V_2}f(V) dV = \lim_{N\to\infty}\sum_{\substack{V=V_1\\\text{with increments }\frac{\Delta V}{N}}}^{V_2} f(V)\; \frac{\Delta V}{N},$$</span>
where <span class="math-container">$\Delta V=V_2-V_1$</span>. The founders of ca... |
4,004,978 | <blockquote>
<p>For all <span class="math-container">$a, b, c, d > 0$</span>, prove that
<span class="math-container">$$2\sqrt{a+b+c+d} ≥ \sqrt{a} + \sqrt{b} + \sqrt{c} + \sqrt{d}$$</span></p>
</blockquote>
<p>The idea would be to use AM-GM, but <span class="math-container">$\sqrt{a} + \sqrt{b} + \sqrt{c} + \sqrt{d}... | radekzak | 876,772 | <p>You can use QM-AM for <span class="math-container">$\sqrt{a},\ \sqrt{b},\ \sqrt{c},\ \sqrt{d}$</span>; I'm leaving the details for you.</p>
|
2,429,164 | <p>I'd like to prove
$$\lim_{n\rightarrow+\infty} \int_0^\pi n\sqrt{n^2+x^2-2nx\cos\theta}-n^2 \mathrm{d}\theta = \frac{\pi}{4}x^2$$
for $x\in[0,\infty)$. I checked it numerically and derived it with Matlab Symbolic Toolbox, but cannot prove it by calculus.</p>
<p>I cannot use the Dominated Convergence Theorem since I... | Claude Leibovici | 82,404 | <p>It seems that you are entering the world of elliptic integrals.</p>
<p>$$I=\int\sqrt{n^2+x^2-2nx\cos\theta} \,d\theta = \sqrt{n^2+x^2} \int \sqrt{1- \frac {2nx}{n^2+x^2}\cos\theta}\,d\theta $$ and $$\int \sqrt{1-k\cos\theta}\,d\theta=-2 \sqrt{k-1} E\left(\frac{\theta}{2}|\frac{2 k}{k-1}\right)$$ making the definite... |
621,409 | <p>I need some help with the following question:</p>
<p>We have $H$ acting by automorphisms on $N$, and let $\rho:H\to Aut(N)$ the associated representation by automorphisms.</p>
<p>Suppose that $G=H[N]_{\rho}$ is a semidirect product, and $K=\ker(\rho)$.</p>
<p>Prove that $K\unlhd G$ and that $G/K$ is also a semid... | amWhy | 9,003 | <p>Using polar coordinates, and substituting $r^2 = x^2 + y^2$, we have $$\lim_{(x, y) \to (0, 0)}\frac{1-\cos(x^2+y^2)}{(x^2+y^2)^2}=\lim_{r \to 0} \dfrac {1 - \cos(r^2)}{(r^2)^2} = \lim_{r \to 0} \frac{1 - \cos(r^2)}{r^4}$$</p>
<p>You can use l'hopital, now, repeatedly (three times)..</p>
|
748,325 | <p>In order to prove non-uniqueness of singular vectors when a repeated singular value is present, the book (Trefethen), argues as follows: Let $\sigma$ be the first singular value of A, and $v_{1}$ the corresponding singular vector. Let $w$ be another linearly independent vector such that $||Aw||=\sigma$, and construc... | Ewan Delanoy | 15,381 | <p>This is actually very simple. The main point (seemingly missed by the OP) is that $Av_1$ and $Av_2$ must be orthogonal (this is something obvious to people familiar with proofs of SVD, because the induction decomposes into orthogonal spaces).</p>
<p>For every $z\in{\mathbb C}$, you have </p>
<p>$$
||A(v_2+zv_1)||^... |
3,197,262 | <p>When I was solving <span class="math-container">$ \operatorname{Cov}(X,E(X\mid Y)) = \operatorname{var}(E(X\mid Y))$</span>, I notice that <span class="math-container">$E(X\mid Y)$</span> was treated as a function of <span class="math-container">$Y$</span>.
My thinking is <span class="math-container">$E(X\mid Y)$</... | dnqxt | 651,088 | <p>The answer is already given. Here is an example.</p>
<p>Let a discrete r.v. <span class="math-container">$X$</span> have values <span class="math-container">$x\in \{ -6,-3,7,14\}$</span>, and <span class="math-container">$P(X=x)=(0.1,0.2,0.3,0.4)$</span> respectively. Define a r.v. <span class="math-container">$Y$<... |
1,043,094 | <p>I have to find the limit of following</p>
<p><span class="math-container">$$\lim_{x \to 0}\left(\frac{1}{x} - \frac{1}{x^2}\right)$$</span></p>
<p>I have no idea how to start this one off.
How would I do it?</p>
<p>Do I just substitute the <span class="math-container">$0$</span>? It doesn't look that easy and sim... | Timbuc | 118,527 | <p>An idea. We take $\;x\;$ very close to zero, say $\;|x|<10^{-4}\;$ :</p>
<p>$$x>0:\;\;\frac1x-\frac1{x^2}=\frac{x-1}{x^2}<\frac{-\frac12}{x^2}=-\frac1{2x^2}$$</p>
<p>and now you only have to show the rightmost expresion is unbounded below, which I think is pretty easy.</p>
|
1,043,094 | <p>I have to find the limit of following</p>
<p><span class="math-container">$$\lim_{x \to 0}\left(\frac{1}{x} - \frac{1}{x^2}\right)$$</span></p>
<p>I have no idea how to start this one off.
How would I do it?</p>
<p>Do I just substitute the <span class="math-container">$0$</span>? It doesn't look that easy and sim... | k170 | 161,538 | <p>Here's another way to prove the following statement
<span class="math-container">$$\lim\limits_{x \to 0}\left(\frac{1}{x} - \frac{1}{x^2}\right)=-\infty$$</span>
Which is equivalent to the following
<span class="math-container">$$\forall N\gt0,\exists\delta\gt 0:0\lt\left|x\right|\lt\delta\Rightarrow\frac{1}{x}-\fra... |
864,237 | <p>Let's take a short exact sequence of groups
$$1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$$
I understand what it says: the image of each homomorphism is the kernel of the next one, so the one between $A$ and $B$ is injective and the one between $B$ and $C$ is surjective. I get it. But other than being a so... | Community | -1 | <p>"The kernel of $B \to C$" is often not a satisfactory description of a group. Having a group $A$ that it is isomorphic to (along with a given isomorphism) is very useful.</p>
<p>"The quotient $B/A$" is often not a satisfactory description of a gruop. Having a group $C$ that it is isomorphic to (along with a given i... |
949,512 | <p>How do mathematicians define inner product on a vector space. </p>
<p>For example: $a = (x_1,x_2)$ & $ b =(y_1,y_2) $ in $ \mathbb{R}^2.$ </p>
<p>Define $\langle a,b\rangle= x_1y_1-x_2y_1-x_1y_2+4x_2y_2$. It's an inner product.</p>
<p>But how does one motivate this inner product? I think there is some sort of... | Benjamin | 118,815 | <p>The abstract definition of an inner product of real-valued vectors is a function $\langle \, , \rangle: \mathbb R^n \times \mathbb R^n \to \mathbb R$ satisfying the following axioms, where $\alpha$ and $\beta$ are scalars and the $x$'s and $y$'s are vectors.</p>
<ol>
<li>$\langle \alpha x_1 + \beta x_2, y \rangle =... |
949,512 | <p>How do mathematicians define inner product on a vector space. </p>
<p>For example: $a = (x_1,x_2)$ & $ b =(y_1,y_2) $ in $ \mathbb{R}^2.$ </p>
<p>Define $\langle a,b\rangle= x_1y_1-x_2y_1-x_1y_2+4x_2y_2$. It's an inner product.</p>
<p>But how does one motivate this inner product? I think there is some sort of... | Dan Rust | 29,059 | <p>Every positive-definite matrix $M$ is associated to an inner product when viewed as a quadratic form. In this case the matrix is $M=\begin{pmatrix}1&-1\\-1&4\end{pmatrix}$ and the inner product is given by $\langle x,y\rangle=x^TMy$.</p>
|
633,223 | <p><img src="https://i.stack.imgur.com/xVS2C.png" alt="enter image description here"></p>
<p>This one has a great degree of self-evidence. Paradoxically, I find it difficult to deduce it from primitive propositions. The book only hinted ❋4.21 and ❋4.22.</p>
| Mauro ALLEGRANZA | 108,274 | <p>I think that we may simplify a little bit Albert's proof.</p>
<p>From :</p>
<p>$*4.22. \vdash (p \equiv q) \land (q \equiv r) \supset (p \equiv r)$</p>
<p>applying :</p>
<p>$*3.3.\vdash ((p \land q) \supset r) \supset (p \supset (q \supset r))$</p>
<p>we get directly :</p>
<blockquote>
<p>$\vdash (p \equiv q... |
1,370,576 | <p>I am working on a trigonometry question at the moment and am very stuck. I have looked through all the tips to solving it and I cant seem to come up with the right answer. The problem is </p>
<blockquote>
<p>What is exact value of<br>
$$\cot \left(\frac{7\pi}{6}\right)? $$</p>
</blockquote>
| Narasimham | 95,860 | <p>$$\cot \left(\frac{7\pi}{6}\right) = \cot \left(\frac{\pi}{6}\right) = \sqrt 3 $$</p>
<p>because $$ \tan ( \theta + \pi) = \tan ( \theta ) $$</p>
|
274,961 | <p>I want to calculate the determinant along the last slice of a 3-dimensional array. So for I do this by slow the <code>Table</code> command. I know that for time reasons I should use <code>Map</code> or <code>Apply</code>, however couldn't successful solve the problem.</p>
<pre><code>m = 2; n = 3; o = 10;
SeedRandom[... | Daniel Huber | 46,318 | <p>You must rearrange the matrix so that the blocks you want the Det from, appear at level 1:</p>
<p>Consider the first block or matrix:</p>
<pre><code>x[[All, All, 1]] // MatrixForm
</code></pre>
<p><a href="https://i.stack.imgur.com/F1AXt.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/F1AXt.png" a... |
650,450 | <p>Suppose that $a_n$ and $b_n$ are Cauchy sequences, and that $a_n < b_n$ for all n. Prove that $\lim_{x \to \infty}a_n \le \lim_{x \to \infty}b_n$ for all n.</p>
<p>Is it sufficient to say that we know both Cauchy sequences must converge to the limit, and since $a_n$ is always less than $b_n$, the limits will fol... | Community | -1 | <p>Let $a_n$ and $b_n$ be your two sequences such that $a_n<b_n$ for all $n$. Consider the
sequence $\{b_n-a_n\}$ which is greater than $0$ for all $n$. This sequence can also be seen to be Cauchy. Of course, all Cauchy sequences in $\mathbb{R}$ converge. Call its limit $x$. Suppose its limit was strictly
less than ... |
452,803 | <p>Test the convergence of improper integrals :</p>
<p>$$\int_1^2{\sqrt x\over \log x}dx$$</p>
<p>I basically have no idea how to approach a problem in which log appears. Need some hint on solving this type of problems.</p>
| Mhenni Benghorbal | 35,472 | <p>First make the change of variables $\ln(x)=u$</p>
<p>$$\int_1^2{\sqrt x\over \log x}dx =\int _{0}^{\ln \left( 2 \right) }\!{\frac {{{\rm e}^{3/2\,u}}}{u}}{du}.$$</p>
<p>Now, you can see that the integrand behaves as</p>
<p>$$ {\frac {{{\rm e}^{3/2\,u}}}{u}}\sim _{u\to 0} \frac{1}{u} $$</p>
<p>which is not integ... |
378,953 | <p><strong>Problem:</strong> Give an example of a permutation of the first $n$ natural numbers from which it is impossible to get to the standard permutation $1,2,\ldots,n$ after less than $n-1$ transposition operations (i.e switching the place of two elements).</p>
<p><strong>My attempt</strong></p>
<p>Suppose we ha... | zyx | 14,120 | <p>There is a quantifiable sense in which permutations with more motion require more transpositions to achieve.</p>
<p>For a sequence of transpositions of pairs in a set $S$, build an undirected graph, with vertices $S$ and an edge between every pair that is exchanged at some time in the sequence. </p>
<p>The vertex ... |
4,330,991 | <p>I do understand that if:</p>
<p><span class="math-container">$a=b \Rightarrow a^2 = b^2 $</span></p>
<p>But clearly, the graph representing these two equations won't be the same. So, (correct me if I'm wrong) this would suggest that if you square both sides of the equation, you essentially get a different set of ans... | José Carlos Santos | 446,262 | <p>In general, <span class="math-container">$a=b$</span> is not equivalent to <span class="math-container">$a^2=b^2$</span>; for instance <span class="math-container">$1^2=(-1)^2$</span>, but <span class="math-container">$1\ne-1$</span>.</p>
<p>However, if <span class="math-container">$a,b\geqslant0$</span>, then, yes ... |
4,140,956 | <p>I'm trying to determine the order of the pole in the complex expression</p>
<p><span class="math-container">$$f(z)=\frac{1}{(1-\cos(z))^2}$$</span></p>
<p>I have determined the pole to be <span class="math-container">$z=2\pi n, n\in \mathbb{Z}$</span>.</p>
<p>However, when I use the equation <span class="math-conta... | TheSimpliFire | 471,884 | <p>If a function <span class="math-container">$g$</span> has a pole of order <span class="math-container">$k$</span> then <span class="math-container">$g^2$</span> will have a pole of order <span class="math-container">$2k$</span>. Using your method, we have by L'Hopital, <span class="math-container">$$\lim_{z\to2\pi n... |
2,250,469 | <p>Let n $\geq$ 4 be an integer. Determine the number of permutations of
$\{1, 2, . . . , n\}$, in which $1$ and $2$ are next to each other, with $1$ to the left of $2$.<br>
I can't make sense of this problem statement. The way I see it, if $n$ is an integer, then the pair $1,2$ could be formed by any pair with the for... | user247327 | 247,327 | <p>Think of "12" as a single object, say "a". Then the problem becomes to determine the number of permutations of the n- 1 objects, "a, 3, 4, ..., n". There are (n-1)! such permutations.</p>
|
335,651 | <p>I'm having trouble proving $$\left(\frac{\sin(\frac{n\theta}{2})}{\sin(\frac{\theta}{2})}\right)^2=\left|\sum_{k=1}^{|n|}e^{ik\theta}\right|^2$$ where $n\in\mathbb{Z}$ and $\theta\in\mathbb{R}$. Can anyone suggest a hint?</p>
| prob_noob | 67,596 | <p>Using the sum of sines and cosines with arguments in arithmetic progression as given above: if $\theta\ne0$ and let $\varphi =0$, then we have,
\begin{align} &S =\sin{(\theta)} + \sin{(2\theta)} + \cdots + \sin{(n\theta)} = \frac{\sin{\left(\frac{(n+1) \theta}{2}\right)} \cdot \sin{(\frac{n \theta}{2})}}{\sin{\f... |
335,651 | <p>I'm having trouble proving $$\left(\frac{\sin(\frac{n\theta}{2})}{\sin(\frac{\theta}{2})}\right)^2=\left|\sum_{k=1}^{|n|}e^{ik\theta}\right|^2$$ where $n\in\mathbb{Z}$ and $\theta\in\mathbb{R}$. Can anyone suggest a hint?</p>
| lab bhattacharjee | 33,337 | <p>Using <a href="http://mathworld.wolfram.com/EulerFormula.html" rel="nofollow">Euler's Formula</a> , $e^{ix}= \cos x+i\sin x$</p>
<p>So, $e^{-ix}= \cos(-x)+i\sin (-x)=\cos x-i\sin x\implies 2i\sin x=e^{ix}-e^{-ix}$ </p>
<p>$$\text{ If }n>0,\sum_{k=1}^{|n|}e^{ik\theta}= \sum_{k=1}^n e^{ik\theta}=e^{i\theta}\left(... |
2,262,371 | <p>If $a,b,c$ are positive real numbers, prove that
$$\frac{2}{a+b}+\frac{2}{b+c}+ \frac{2}{c+a}≥ \frac{9}{a+b+c}$$</p>
| Darth Geek | 163,930 | <p>Let $a+b+c = s$. Then we have to prove</p>
<p>$$\dfrac{1}{s-a} + \dfrac{1}{s-b} + \dfrac{1}{s-c} \geq \dfrac{9}{2s},$$</p>
<p>or, equivalently, </p>
<p>$$\dfrac{3}{\dfrac{1}{s-a} + \dfrac{1}{s-b} + \dfrac{1}{s-c}} \leq \dfrac{2s}{3}.$$</p>
<p>Note that the LHS is the harmonic mean of $s-a,s-b,s-c$ and the RHS ... |
4,004,827 | <p>I need to calculate:
<span class="math-container">$$\displaystyle \lim_{x \to 0^+} \frac{3x + \sqrt{x}}{\sqrt{1- e^{-2x}}}$$</span></p>
<p>I looks like I need to use common limit:
<span class="math-container">$$\displaystyle \lim_{x \to 0} \frac{e^x-1}{x} = 1$$</span></p>
<p>So I take following steps:</p>
<p><span c... | José Carlos Santos | 446,262 | <p>Note that<span class="math-container">\begin{align}\lim_{x\to0^+}\frac{\left(3x+\sqrt x\right)^2}{1-e^{-2x}}&=\lim_{x\to0^+}\frac{9x^2+6x\sqrt x+x}{1-e^{-2x}}\\&=-\lim_{x\to0^+}\frac x{e^{-2x}-1}\left(9x+6\sqrt x+1\right)\\&=-\frac1{\lim_{x\to0^+}\frac{e^{-2x}-1}x}\times\lim_{x\to0^+}\left(9x+6\sqrt x+1\... |
200,093 | <p>I have a BLDC electric motor, I'm currently trying to control via a <code>PIDTune</code>. This is mostly an attempt to reduce (remove) a small run away drift that ends up showing up in the motor signal <code>u[t]</code>.</p>
<p>I've modelled this via:</p>
<pre><code>ssm = StateSpaceModel[\[ScriptCapitalJ] \[Phi]''... | JJBK | 59,726 | <p>Here's an example of a basic PD controller based on frequency domain tuning. The technique is described in any basic control book.
Basic tuning rule-of-thumbs that have been applied (no further advanced tuning)</p>
<p>Let's write <code>ssm</code> as a transfer function:</p>
<pre><code>tf = TransferFunctionModel[ss... |
128,666 | <p>If we start with a number like 1234 and produce the following sum 1234 + 123 + 12 + 1 = 1370.</p>
<p>If we are given the 1370 can I retrieve the 1234? A similar question was migrated over to the Math.SE because the OP did not in any way relate it to MMa. The math given over there is in no way too tough for anyone o... | J. M.'s persistent exhaustion | 50 | <p>A ploddingly procedural implementation:</p>
<pre><code>n = 308460277;
FromDigits[Reap[Do[{q, n} = QuotientRemainder[n, (10^k - 1)/9]; Sow[q],
{k, IntegerLength[n], 1, -1}]][[-1, 1]]]
277614253
</code></pre>
|
2,120,539 | <p>Find the points of local maximum and minimun of the function:
$$f(x)=\sin^{-1}(2x\sqrt{1-x^2})~~~~;~~x\in (-1,1)$$
I know
$$f'(x)=-\frac{2}{\sqrt{1-x^2}}$$</p>
<p>How to find the local maximum and minimum? I have drawn the fig and seen the points of local maximum and minimum. But how to find then analytically?
<a h... | Community | -1 | <p><strong>Hint</strong>: We know that $$\arcsin x + \arcsin y = \arcsin (x\sqrt{1-y^2} + y\sqrt {1-x^2}) $$ If $x=y $, then $$2\arcsin x = \arcsin (2x\sqrt {1-x^2}) $$</p>
<p>Hope you can take it from here. </p>
|
2,942,204 | <p>I was reviewing <span class="math-container">$\mathbb{R}-$</span>analisys with a friend and I'm thinking about one of the questions...</p>
<blockquote>
<p>Prove by the <span class="math-container">$\epsilon-\delta$</span> limit definition that <span class="math-container">$\lim_{x\rightarrow 2}\frac{x^2-1}{x-3}=-... | Hector Blandin | 170,571 | <p>Hint: Recall that <span class="math-container">$0<a<b$</span> implies that <span class="math-container">$\frac{1}{a}>\frac{1}{b}>0$</span>. So,</p>
<p><span class="math-container">$$0<\vert x-2 \vert<\delta$$</span>
<span class="math-container">$$\Longrightarrow \ 1<\vert x-2 \vert+1<\delta... |
2,942,204 | <p>I was reviewing <span class="math-container">$\mathbb{R}-$</span>analisys with a friend and I'm thinking about one of the questions...</p>
<blockquote>
<p>Prove by the <span class="math-container">$\epsilon-\delta$</span> limit definition that <span class="math-container">$\lim_{x\rightarrow 2}\frac{x^2-1}{x-3}=-... | Nosrati | 108,128 | <p>I think (i) part is correct, and assume we choose <span class="math-container">$x$</span> from <span class="math-container">$|x-2|<\dfrac12$</span> then
<span class="math-container">$$\dfrac32<x<\dfrac52$$</span>
<span class="math-container">$$-\dfrac32<x-3<-\dfrac12$$</span>
<span class="math-contain... |
634,132 | <p>Let $G$ be a cyclic group with $N$ elements. Then it follows that</p>
<p>$$N=\sum_{d|N} \sum_{g\in G,\text{ord}(g)=d} 1.$$</p>
<p>I simply can not understand this equality. I know that for every divisor $d|N$ there is a unique subgroup in $G$ of order $d$ with $\phi(d)$ elements. But how come that when you add all... | Doug Spoonwood | 11,300 | <p>If we assume that $\forall$x(x=x), then yes.</p>
<p>Suppose that b=c.</p>
<p>f(a,b)=f(a,b) since we can derive it from substitution and that $\forall$x(x=x).</p>
<p>Now since b=c we can replace just the second "b" by "c" and obtain</p>
<p>f(a,b)=f(a,c).</p>
|
806,476 | <p>In Milnor's book <em>Topology from the Differentiable Viewpoint</em> there's the following problem:</p>
<p><strong>Problem $6$</strong> (Brouwer). Show that any map $S^n\to S^n$ with degree different from $(-1)^{n+1}$ must have a fixed point.</p>
<p><strong>My solution:</strong> Assume that the map $f:S^n\to S^n$ ... | Behnam Esmayli | 283,487 | <p>Best solution: Notice that $\|f(x)-(a \circ f) (x)\| < 2$ for every $x.$ But then a problem above this one in the book, asks to prove that such maps must be homotopic (smoothly so if maps themselves are smooth.) Then your computation goes thru with nothing on conscience!</p>
|
3,173,242 | <p><strong>Context:</strong></p>
<p>In the context of circuit theory and graph theory, suppose we have a graph <span class="math-container">$G,$</span> then <a href="https://en.wikipedia.org/wiki/Laplacian_matrix" rel="nofollow noreferrer">the Laplacian (Kirchhoff) matrix</a> <span class="math-container">$L$</span> is... | Romi | 606,615 | <p>For your first question, I think you can easily prove that <span class="math-container">$L_{BB}$</span> and <span class="math-container">$L_{CC}$</span> are
<a href="https://en.wikipedia.org/wiki/Weakly_chained_diagonally_dominant_matrix" rel="nofollow noreferrer">weakly chained diagonally dominant matrices</a> from... |
3,911,548 | <p><strong>If a,b,c,d are real numbers and <span class="math-container">$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=17$</span> and <span class="math-container">$\frac{a}{c}+\frac{c}{a}+\frac{b}{d}+\frac{d}{b}=20$</span>, then find the sum of all possible valuse of <span class="math-container">$\frac{a}{b}$</span>+... | Ng Chung Tak | 299,599 | <p><span class="math-container">\begin{align}
17 &= \frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a} \\
&= \frac{a}{b}+\frac{c}{d}+\frac{b}{c}+\frac{d}{a} \\
&= \color{red}{\frac{ad+bc}{bd}}+\color{blue}{\frac{ab+cd}{ac}} \\
&= \color{red}{x}+\color{blue}{y} \\
20 &= \frac{a}{c}+\frac{c}{a}... |
83,246 | <p>Let H be a separable and infinite-dimensional Hilbert space and let B be the closed ball
of H having unit radius, whose center is at the origin h of H. Suppose one would like to
know how much of B can be "filled up" by any of its compact subsets-since B itself
(although closed and bounded) is not compact. Let E be t... | Fabian Lenhardt | 18,256 | <p>Assume $C \subset H$ is compact such that each point of the unit disk has distance at most $1-\epsilon$ from $C$ for some $\epsilon > 0$. There are finitely many points $p_1,...p_n$ such that the balls with radius $\frac{\epsilon}{4}$ around the $p_i$ cover all of $C$. Now pick an orthonormal basis $e_i$ of your... |
192,095 | <p>Suppose I have a convex program which has only two variables, the objective function is strictly convex, and the constraints are linear functions. </p>
<p>I think removing all non-tight constraints doesn't change the optimal solution.</p>
<p>However, when there are more than 2 tight constraints, I am not sure if r... | Dirk | 9,652 | <p>No, you can't. Even in linear optimization with only two variables you may have several linear constraints that form a corner of your feasible domain. Removing some constraints may make the corner less sharp and lead to an unbounded objective. Imagine that removing some constraint adds new feasible directions at the... |
2,713,937 | <p>I need this lemma for another proof I'm doing, but I can't crack it. I want something of the structure:</p>
<p>$$\frac{pq}{(p-1)(q-1)} < \dots = \frac{pq}{\frac{1}{2}pq} = 2,$$
but I can't figure out what to do with the denominator. </p>
| Przemysław Scherwentke | 72,361 | <p>As $p\geq3$, $q\geq5$, we have $p/(p-1)\leq3/2$, $q/(q-1)\leq5/4$.</p>
|
2,713,937 | <p>I need this lemma for another proof I'm doing, but I can't crack it. I want something of the structure:</p>
<p>$$\frac{pq}{(p-1)(q-1)} < \dots = \frac{pq}{\frac{1}{2}pq} = 2,$$
but I can't figure out what to do with the denominator. </p>
| Barry Cipra | 86,747 | <p>If $p\le q$ are odd primes, then $p=3+a$ and $q=5+b$ for some $a,b\ge0$. We have</p>
<p>$$\begin{align}
{pq\over(p-1)(q-1)}={(3+a)(5+b)\over(2+a)(4+b)}\lt2
&\iff15+5a+3b+ab\lt2(8+4a+2b+ab)\\
&\iff0\lt1+3a+b+ab
\end{align}$$</p>
<p>The inequality has very little to do with $p$ and $q$ being either distinct... |
1,261,977 | <p>I tried doing this the same way you would find the Fourier transform for $1/(1+x^2)$ but I guess I'm having some trouble dealing with the 2x on top and I could really use some help here.</p>
| Tryss | 216,059 | <p>Another way to solve this. As you know, $$-ix \mathcal{F}[g](x) = \mathcal{F}[g'](x)$$ and $$\mathcal{F}[\mathcal{F}[g]] = cg$$</p>
<p>You have (with some coefficients)</p>
<p>$$ \mathcal{F}[ x \mathcal{F}[g](x) ] (\xi)= \mathcal{F}[i\mathcal{F}[g'] ] (\xi) = i g'(\xi)$$</p>
<p>Now call $f = \mathcal{F}[g]$, and ... |
2,861,867 | <p>Let $A$ be any commutative ring (with $1$) and $x,y \in A$ such that $x+y = 1$. Then it follows that for any $k,l$, there exist $a,b \in A$ such that $ax^k+by^l = 1$. </p>
<p>(Proof: Suppose otherwise. Then, $(x^,k,y^l) \subset \mathfrak p$ for some prime ideal $\mathfrak p$. But then this implies that $x,y \in \ma... | Mohan | 245,104 | <p>You have $(x+y)^{k+l}=1$, so using binomial theorem you get, letting $m=k+l$ (actually $k+l-1$ would do), $\sum \binom{m}{r} x^ry^{m-r}=1$. Thus, you have $\sum_{r\leq k}\binom{m}{r} x^ry^{m-r}+\sum_{r>k} \binom{m}{r} x^ry^{m-r}=1$. Notice that the first sum, every term is a multiple of $y^l$ and the second has a... |
3,393,193 | <p>I am asked to answer the following:</p>
<p>Let <span class="math-container">$f:Z->Z$</span> be defined by <span class="math-container">$f(x) = 2x$</span>.</p>
<ul>
<li>Write down infinitely many functions <span class="math-container">$g:Z->Z$</span> such that <span class="math-container">$g(f(x)) = Id_z$</sp... | dan_fulea | 550,003 | <p><strong>Discussion, notations and a first solution:</strong></p>
<p>We will use the following (general) stopping strategy, depending on the sets <span class="math-container">$A_1,A_2,A_3\subset\{1,2,3,4,5,6\}=:\Omega_0$</span>. The whole modeling space is <span class="math-container">$\Omega=\Omega_0^{\times 4}$</s... |
5,586 | <p>I'm in my last year of highschool. And I'm aiming for a perfect grade in maths. The problem is that this year is the hardest year of maths I have ever faced in my entire life. Especially derivation and limits as its the first time I am studying it. Here are the lessons that are required to study for the first semest... | JPBurke | 759 | <p>Welcome to the site!</p>
<p>It's great that you're motivated and want to get a perfect grade in your studies! There is no actual formula for getting a perfect grade. One goal among all the goals of your math educators is to deepen your understanding of the mathematics you are studying in class and in whatever books... |
5,586 | <p>I'm in my last year of highschool. And I'm aiming for a perfect grade in maths. The problem is that this year is the hardest year of maths I have ever faced in my entire life. Especially derivation and limits as its the first time I am studying it. Here are the lessons that are required to study for the first semest... | markt | 3,245 | <p>In college I had a friend who was also a Math major, and had been in the Marine Corp. His advice was a little coarse and echoed his military background: "Work problems until you puke, then wipe off the puke and work some more". I followed this advice and was an A student in college. I found that in Calculus in par... |
649,239 | <p>By <a href="http://en.wikipedia.org/wiki/Post%27s_theorem" rel="nofollow">Post's Theorem</a> we know that a set $A\subseteq\mathbf{N}$ is recursively enumerable iff it is definable by a $\Sigma_1$-formula, i.e. there exists a $\Sigma_1$-formula $\varphi(x)$ with $x$ free such that for every number $n$:
\[
n\in A\lon... | Wouter Stekelenburg | 27,375 | <p>By the axiom of choice there is a function that maps each r.e. set to one of its definitions. We can 'find' the definition in that sense. This function cannot be too nice, however, because that would decide the extensional equivalence of two definitions.</p>
<p>Let $f$ be a function that maps each set to one of its... |
4,498,801 | <p>I am trying to deeply understand the similarities between these two theorems; the first being a generalization of the second.</p>
<blockquote>
<p><strong>Theorem 16.13.</strong> If <span class="math-container">$f$</span> is nonnegative, then
<span class="math-container">$$
\int_{\Omega} f(T \omega) \mu(d \omega)=\in... | Ruy | 728,080 | <p>I have a lot of sympathy for this question since it bothered me for a long time as well. After meditating about it for
years I eventually settled for the interpretation that these two results cannot be stated in a unified way because they
have a fundamental difference!</p>
<p>We will see that, contrary to what was... |
3,059,695 | <p>Let <span class="math-container">$A$</span> be a subset of a compact topological space such that every point of <span class="math-container">$A$</span> is an isolated point of <span class="math-container">$A$</span>. Is <span class="math-container">$A$</span> necessarily finite?</p>
| TonyK | 1,508 | <p>No. Take for instance the compact space <span class="math-container">$[0,1]$</span>, and let <span class="math-container">$A=\{1/n:n\in\Bbb N_{>0}\}$</span>.</p>
|
530,484 | <p>Let $f:\mathbb{R}\rightarrow\mathbb{R}^2$ be a $C^1$ function. Prove that the image of $f$ contains no open set of $\mathbb{R}^2$.</p>
<p>So say $f(x)=(g(x),h(x))$. Since $f$ is $C^1$, we have that $g'(x),h'(x)$ both exist and are continuous functions in $x$. To show that $f$ contains no open set of $\mathbb{R}^2$,... | copper.hat | 27,978 | <p>Here is a more prosaic approach that relies on $f$ being Lipschitz on compact intervals.</p>
<p>Choose an interval $[t_0,t_1]$. Since $f$ is $C^1$, it is uniformly lipschitz on this interval with some rank $L$.
Let $D = f([t_0,t_1])$.
We can use $L$ to find an upper bound on the measure of $D$.</p>
<p>Let $R(\tau_... |
2,569,096 | <p>The problem goes as follows:
$$
P=\left(
\begin{matrix}
a & 0.6\\
1-a & 0.4\\
\end{matrix}
\right)
$$</p>
<blockquote>
<p>Determine the value of the parameter $a \in [0,1]$ for which $P$ does <strong>not</strong> have an inverse.</p>
</blockquote>
<p>So then I know the value of $a$ lies ... | Community | -1 | <p><strong>Hint:</strong></p>
<p>A matrix does not have an inverse when its <strong>determinant $=0$</strong>. The value of $a$ that does this is: $$a\times 0.4 - (1-a)\times 0.6 =0\implies a=\, ?$$</p>
|
1,701,260 | <p>My textbook states the following:<br>
i) If $ f : [a,b] \rightarrow \mathbb{R} $ is bounded and is continuous at all but finitely many points of $[a,b]$, then it is integrable on $[a,b]$.<br>
ii) Any increasing or decreasing function on $[a,b]$ is integrable on $[a,b]$.</p>
<p>The proof for (i) is clear to me. I fo... | NicNic8 | 24,205 | <p>I suspect that by increasing and decreasing they mean that it's increasing / decreasing on the entire domain, which is <span class="math-container">$\mathbb{R}$</span>. So it's defined everywhere on <span class="math-container">$\mathbb{R}$</span>. Since it's defined everywhere, it's bounded on any compact subset ... |
372,064 | <p>can someone explain me why</p>
<p>$\dot{a}\ddot{a}=\frac{1}{2}\frac{d}{dt}\left(\dot{a}^{2}\right)$</p>
<p>Many thanks</p>
| André Nicolas | 6,312 | <p>Use the Product Rule. The derivative of $\dot{a}\dot{a}$ with respect to $t$ is $\dot{a}\ddot{a}+\dot{a}\ddot{a}$, which is<br>
$2\dot{a}\ddot{a}$. </p>
|
54,311 | <p>I found <a href="http://www.rle.mit.edu/dspg/documents/HilbertComplete.pdf" rel="nofollow">this paper</a> on Hilbert Transform, which is a very nice read. I've studied signal processing, but from a more practical than mathematical perspective. Can someone explain to me how we arrive at equation (2) in this paper?</p... | anon | 11,763 | <p>By symmetry if the first equality of (2) holds then the second equality also holds. Expand the integrand using the series expansions:</p>
<p>$$X(v)H(z/v)v^{-1} = \left(\sum_{n=-\infty}^\infty x(n)v^{-n}\right)\left(\sum_{m=-\infty}^\infty y(m)z^{-m}v^m \right)v^{-1} $$</p>
<p>Note that $\oint_\gamma v^{-k}dv=2\pi ... |
3,829,431 | <blockquote>
<p>If the area of equilateral triangle is <span class="math-container">$3\sqrt3$</span> cm<span class="math-container">$^2$</span> , then what is the height of the equilateral triangle?</p>
</blockquote>
<p>I am stuck with this question
<br>I solved it like this:
<br>Area of equilateral triangle is <span c... | Georges Elencwajg | 3,217 | <p>According to Zariski's Nullstellensatz (Qing Liu's Corollary 1.12, page 30) the extension <span class="math-container">$\mathbb R\subset A/\mathfrak m$</span> is finite, hence of degree <span class="math-container">$2$</span> (degree <span class="math-container">$1$</span> is excluded because <span class="math-conta... |
2,507,247 | <p>Given a bimatrix game of
<span class="math-container">$$\left(\begin{matrix}(0,-1) & (0,0)\\(-90,-6)&(10, -10)\end{matrix}\right)$$</span>
<a href="https://i.stack.imgur.com/uY44c.jpg" rel="nofollow noreferrer">Source</a></p>
<p>How to find the nash equilibrium strategy for both players?</p>
| quasi | 400,434 | <p>I'll follow the method described in the text
<p>
$\qquad$Thomas$\,-\,$Games, Theory and Applications (1984)
<p>
on pages 59-61.
<p>
Suppose players $1$ and $2$ use mixed strategies $(x,1-x)$ and $(y,1-y)$, respectively, where </p>
<ul>
<li>The probability that player $1$ chooses row $1$ is $x$.$\\[2pt]$
<li>The pro... |
4,610,394 | <p>Clearly, none of the roots are in <span class="math-container">$\mathbb{Q}$</span> so <span class="math-container">$f(x) = x^4 + 1$</span> does not have any linear factors. Thus, the only thing left to check is to show that <span class="math-container">$f(x)$</span> cannot reduce to two quadratic factors.</p>
<p>My ... | orangeskid | 168,051 | <p>The only non-trivial decomposition over <span class="math-container">$\mathbb{R}$</span> of <span class="math-container">$x^4+1$</span> is <span class="math-container">$x^4 + 1 = (x^2 + \sqrt{2} x + 1)(x^2 - \sqrt{2} x + 1)$</span>. Since <span class="math-container">$\mathbb{Q} \subset \mathbb{R}$</span>, if there ... |
4,610,394 | <p>Clearly, none of the roots are in <span class="math-container">$\mathbb{Q}$</span> so <span class="math-container">$f(x) = x^4 + 1$</span> does not have any linear factors. Thus, the only thing left to check is to show that <span class="math-container">$f(x)$</span> cannot reduce to two quadratic factors.</p>
<p>My ... | Bob Dobbs | 221,315 | <p>Clearly, <span class="math-container">$x+a$</span>, <span class="math-container">$a\in\Bbb{Z}$</span> can not be a factor since then <span class="math-container">$a^4=-1$</span>. On the other hand, <span class="math-container">$x^2+ax+b$</span>, <span class="math-container">$a,b\in\Bbb{Z}$</span> can not be a factor... |
2,803,398 | <p>We know that in category of $\mathbb{Set}$ the inverse limit is the direct product. But I am looking for specific category in which inverse limit does not exist. Any comments would be highly appreciated.</p>
| David C. Ullrich | 248,223 | <p>Not quite. The correspondence between infinite decimals and elements of $(0,1)$ is not itself a bijection because some numbers have more than one decimal expansion.</p>
<p>You solve that by restricting to the expansion with infinitely many non-zero digits, fine. But now the mapping from $(0,1)^2$ to $(0,1)$ is not ... |
278,368 | <p><strong>Problem:</strong></p>
<p>Assume the number of cars passing a road crossing during an hour satisfies a Poisson distribution with parameter $\mu$, and that the number of passengers in each car satisfies a binomial distribution with parameters $n \in \mathbb{N}$ and $p \in (0,1)$. Let $Y$ denote the total numb... | leonbloy | 312 | <p>One way of doing the first is using conditioned probability. $N$ and $X_i$ are both random (independent) variables. Now, what you computed first is actually the conditional expectation:</p>
<p>$$E[Y|N] = \sum_{i=1}^N E(X_i) = N n p$$</p>
<p>but $E[Y] = E[ E[Y | N]]$ (<a href="http://en.wikipedia.org/wiki/Law_of_t... |
3,975,895 | <p>Let <span class="math-container">$a,b,c\in\mathbb{Z}$</span>, <span class="math-container">$1<a<10$</span>, <span class="math-container">$c$</span> is a prime number and <span class="math-container">$f(x)=ax^2+bx+c$</span>. If <span class="math-container">$f(f(1))=f(f(2))=f(f(3))$</span>, find <span class="mat... | cosmo5 | 818,799 | <p>(Edited to add details)</p>
<p>Due to the symmetry of vertical parabola, for distinct <span class="math-container">$x_1, x_2,$</span> <span class="math-container">$f(x_1)=f(x_2) \Rightarrow x_1+x_2=-b/a$</span> and <span class="math-container">$f'(x_1)+f'(x_2)=0$</span></p>
<p>For a quadratic, w cannot have <span cl... |
787,894 | <p>Find the values of $x,y$ for which $x^2 + y^2$ takes the minimum value where $(x+5)^2 +(y-12)^2 =14$.</p>
<p>Tried Cauchy-Schwarz and AM - GM , unable to do.</p>
| Community | -1 | <p>HINT: For this case where the curve is a circle, the value you seek is square of the distance of the centre of the circle from origin minus the radius. (Draw a diagram to see why?)
$$x^2+y^2 = (13-\sqrt{14})^2.$$</p>
<p>You can also find the maximum of $x^2+y^2$ using this trick.</p>
|
595,552 | <p>Let $R$ be a ring. Prove that each element of $R$ is either a unit or a nilpotent element iff the ring $R$ has a unique prime ideal.</p>
<p>Help me some hints.</p>
| BIS HD | 73,067 | <p>Hint for $\Leftarrow$:</p>
<p>Every ring with identiy has a prime ideal. Let $P$ be a prime ideal of $R$. Then it contains all the nilpotent elements (why?). It does not contain a unit (why?), so it is in fact the set of nilpotent elements of $R$ and hence because of the condition the unique prime ideal. </p>
|
3,086,218 | <p>The second order differential equation is given by -</p>
<p><span class="math-container">$ \frac{d^{2}y}{dx^{2}} + \sin (x+y) = \sin x$</span> </p>
<p>Is this a homogeneous differential equation <span class="math-container">$?$</span></p>
<p>Well, I guess this is not a homogeneous differential equation since the ... | doraemonpaul | 30,938 | <p>Let <span class="math-container">$u=x+y$</span> ,</p>
<p>Then <span class="math-container">$\dfrac{du}{dx}=1+\dfrac{dy}{dx}$</span></p>
<p><span class="math-container">$\dfrac{d^2u}{dx^2}=\dfrac{d^2y}{dx^2}$</span></p>
<p><span class="math-container">$\therefore\dfrac{d^2u}{dx^2}+\sin u=\sin x$</span></p>
<p>Thi... |
3,224,455 | <p>I derived the volume of a cone using two approaches and compared the results.</p>
<p>First I integrated a circle of radius <span class="math-container">$r$</span> over the height <span class="math-container">$h$</span> to get the expression: <span class="math-container">$$V_1=\frac{1}{3}\pi r^2 h$$</span></p>
<p>T... | st.math | 645,735 | <p>You cannot simply replace expressions that tend to infinity by that same symbol; then you would lose information on how fast something tends to infinity, for example.</p>
<p>Take <span class="math-container">$a_n=n$</span>, <span class="math-container">$b_n=2n$</span>. Then clearly, both tend to infinity. But <span... |
3,224,455 | <p>I derived the volume of a cone using two approaches and compared the results.</p>
<p>First I integrated a circle of radius <span class="math-container">$r$</span> over the height <span class="math-container">$h$</span> to get the expression: <span class="math-container">$$V_1=\frac{1}{3}\pi r^2 h$$</span></p>
<p>T... | nmasanta | 623,924 | <p><span class="math-container">$\lim_{n \to \infty} (n\tan{\frac{180°}{n}}) = \pi$</span></p>
<p>Take <span class="math-container">$\frac{1}{n}= x$</span>, then <span class="math-container">${n \to \infty} \implies {x \to 0}$</span></p>
<p>Also we have <span class="math-container">$$\lim_{n \to \infty} {\frac{\tan x... |
1,614,989 | <p>A portion of a $30$m long tree is broken by
tornado and the top struck up the ground
making an angle $30^{\circ}$ with ground
level. The height of the point where the tree
is broken is equal to:</p>
<p>$a.)\ \dfrac{30}{\sqrt{3}}m$ $~~~~~~~~~~$ $\color{green}{b.)\ 10m} \\$
$~~~~~~~~~~$ $c.)\ 30\sqrt{3}m$ ... | angryavian | 43,949 | <p>The height of the tree is $30$ meters. You have defined $x$ to be the length of the part of the tree that is still standing, so $30-x$ is the length of the part of the tree that has fallen.</p>
<hr>
<p>You want to solve for $x$. Using the definition of sine, we have $\sin 30^\circ = \frac{x}{30-x}$. You probably l... |
3,252,765 | <p>We are trying to codify in terms of modern algorithm the works of the ancient Indian mathematician <em>Udayadivakara</em> (CE 1073). In his work <em>Sundari</em>, he quotes one <em>Acarya Jayadeva</em> who has given methods to solve Pell's equations. In these methods, one can find the the cyclic <em>Chakravala</em> ... | Joebloggs | 206,669 | <p>Here is a program in pari-gp which solves all solvable instances of the Pell Equation in reasonably quick time (it's competitive with the best, but a lot simpler), as follows:</p>
<pre><code>Pell_1(N,f)=
</code></pre>
<p>{</p>
<pre><code>/*** Set square root Precision ***/
/*********************************/
/... |
2,438,111 | <p>I hope my title somehow encapsulates my problem.</p>
<p>Let's say we have a 1-D Grid with the values 2,1,5,8,1,1. Imagine those values are of some physical quantity $\alpha$.
The mean of this would be $(2+1+5+8+1+1)/6 = 3$</p>
<p>Now let's say we have some function $f(x) = x^2$, which computes another quantity $\b... | Bernard | 202,857 | <p><strong>Hint:</strong></p>
<p>$$\sqrt{3}\cos (x)+\sin(x)=2\biggl(\frac{\sqrt 3}2\cos x+\frac12\sin x\biggr)=\dotsm$$</p>
|
2,346,804 | <p>Please help me finish this problem.</p>
<p>$xy''+(3x-1)y'-(4x+9)y=0$ where $y(0)=0$</p>
<p>$L[xy'']+L[(3x-1)y']-L[(4x+9)y]=L[0]$</p>
<p>$L[xy'']=\frac{d}{dp}(p^2Y)$</p>
<p>$L[(3x-1)y']=-3\frac{d}{dp}(pY)$</p>
<p>$L[(4x+9)y]=-4\frac{dY}{dp}$</p>
<p>$-\frac{d}{dp}(p^2Y)-3\frac{d}{dp}(pY)+4\frac{dY}{dp}=0$</p>
<... | Community | -1 | <p>You have a population of $10$ where $3$ are true and $7$ are false,</p>
<p>A random draw from this population therefore has $3/10$ chance of being true. </p>
|
2,879,035 | <p>$f(x) = \int_{1}^{\infty} \frac{2}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2} dx$</p>
<p>find $P(X > 1)$</p>
<p>This is $X$ ~ $Norm(0, 1)$.</p>
<p>$P(X > 1) = 1 - P(X \leq 1) = 1 - 2 \phi(1) = 1-2(1-\phi(-1)) = 1 - 2(1-0.1587) = -0.6826$. </p>
<p>Yikes. Negative number. What am I doing wrong? </p>
| Deepesh Meena | 470,829 | <p><a href="https://i.stack.imgur.com/Gd2kt.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Gd2kt.png" alt="enter image description here"></a>
thus $$P(X\le1)=\phi(1)$$</p>
<p>$$your answer =1-\phi(1)$$</p>
|
3,244,193 | <p>Here's what I did: <span class="math-container">$$\lim_{n\rightarrow +\infty}(2+3^n)^{\frac{1}{2n}}=\lim_{n\rightarrow +\infty}e^{\frac{1}{2n}\ln(2+3^n)}$$</span>
What should I do next in order to solve it?</p>
| Yiorgos S. Smyrlis | 57,021 | <p>Note that
<span class="math-container">$$
3^n \le 2+3^n\le 2\cdot3^n
$$</span>
Hence
<span class="math-container">$$
3^{1/2} \le (2+3^n)^{1/2n}\le 2^{1/2n}\cdot3^{1/2}\to 3^{1/2}.
$$</span></p>
|
990,930 | <p>Graphing this function is difficult as many overlaps exist and finding a viewing window is hard.</p>
<p>What's a good algebraic method to solve this problem? </p>
| CLAUDE | 118,773 | <p>$2\sin^2(x)-3=3\cos(x)\rightarrow 2(1-\cos^2(x))-3=3\cos(x)\rightarrow 2\cos^2(x)+3\cos(x)+1=0\rightarrow (\cos(x)+1)(2\cos(x)+1)=0\rightarrow$</p>
<p>So we can say $\cos(x)=-1$ or $\cos(x)=\frac{-1}{2}$</p>
<p>$\cos(x)=-1\rightarrow x=2k\pi+\pi\rightarrow x=\pi$</p>
<p>$\cos(x)=\frac{-1}{2}\rightarrow x=\frac{2\... |
1,149,561 | <p>I've tried using mods but nothing is working on this one: solve in positive integers $x,y$ the diophantine equation $7^x=3^y-2$.</p>
| Qiaochu Yuan | 232 | <p>The Euler characteristic is an extremely weak invariant: since it's the alternating sum of the ranks of the homology groups, there's a lot of cancellation possible in that sum, and so there's a lot of ways that two spaces can have the same Euler characteristic. For example, $S^1$ and $S^1 \times S^1$ have the same E... |
3,568,050 | <blockquote>
<p>Let <span class="math-container">$R$</span> be an equivalence relation in the set <span class="math-container">$A$</span> and <span class="math-container">$a,b \in A$</span>. Show that <span class="math-container">$R(a)=R(b)$</span> <strong>iff</strong> <span class="math-container">$aRb$</span>.</p>
<... | Derek Luna | 567,882 | <p>If <span class="math-container">$aRb$</span>, then <span class="math-container">$x \in R(a)$</span> implies that <span class="math-container">$xRa$</span> since <span class="math-container">$R$</span> is symmetric so by transivity <span class="math-container">$xRb$</span> and <span class="math-container">$x \in R(b)... |
3,568,050 | <blockquote>
<p>Let <span class="math-container">$R$</span> be an equivalence relation in the set <span class="math-container">$A$</span> and <span class="math-container">$a,b \in A$</span>. Show that <span class="math-container">$R(a)=R(b)$</span> <strong>iff</strong> <span class="math-container">$aRb$</span>.</p>
<... | fleablood | 280,126 | <blockquote>
<p>If b is an element of both of these equivalency classes, then certainly b=a</p>
</blockquote>
<p>I can not see how you think that. That makes no sense. Unless this class has only one element, then (because they are equal) <em>all</em> the elements are in both and all the elements can't all be equal... |
3,575,804 | <p>The question goes as follows. My attempts are below it. </p>
<p>A motel has ten rooms, all located on the same side of a single corridor and numbered 1 to 10 in numerical order. The motel always randomly allocates rooms to its guests. There are no other guests besides those mentioned.</p>
<p>a) Friends Molly and P... | 0Ketrav0 | 766,652 | <p>Now, 10 rooms can be allocated to 3, in 10P3 number of ways, ie. 10*9*8 number of ways; as there are 10 room options for the first person, 9 for the second person and 8 for the third person.</p>
<p>So, all possible cases</p>
<p>=10*9*8</p>
<p>=720.</p>
<p>Now, in 10 rooms, there are 6 combinations of 5 consecuti... |
1,715,265 | <p>I've tried a method similar to showing that $\mathbb{Q}(\sqrt2, \sqrt3)$ is a primitive field extension, but the cube root of 2 just makes it a nightmare.</p>
<p>Thanks in advance </p>
| Community | -1 | <p>$\mathbb{Q}(\alpha,\beta)$ will equal $\mathbb{Q}(\alpha + s\beta)$ if you choose any rational number $s$ that is not $-\frac{\alpha_i - \alpha}{\beta_j - \beta}$ for any of the conjugates $\alpha_i$ of $\alpha$ and $\beta_j$ of $\beta.$</p>
<p>In your case you can take $s = 1.$</p>
|
1,758,159 | <p>A is symmetric(skew-symmetric) matrix and B is nonsingular matrix .
What can i say about this $$BAB^T$$
???</p>
| Robert Lewis | 67,071 | <p>Note that</p>
<p>$(BAB^T)^T = B^{TT}A^TB^T = BA^TB^T, \tag 1$</p>
<p>since</p>
<p>$B^{TT} = B. \tag 2$</p>
<p>If $A$ is symmetric, $A = A^T$, so</p>
<p>$(BAB^T)^T = BAB^T, \tag 3$</p>
<p>that is, $BAB^T$ is symmetric. Likewise if $A$ is skew-symmetric, $A^T =-A$, so</p>
<p>$(BAB^T)^T = B(-A)B^T= -BAB^T; \tag... |
387,505 | <p>Let <span class="math-container">$f$</span> be a non-invertible bounded outer function on the unit disk. Does <span class="math-container">$f$</span> has radial limit <span class="math-container">$0$</span> somewhere? Note that such a property holds for singular inner functions.</p>
| Mateusz Kwaśnicki | 108,637 | <p>The answer is negative: <span class="math-container">$f$</span> may have a non-zero radial limit everywhere.</p>
<hr />
<p><em>Part 1.</em> Let us start with definitions and notation. A holomorphic function <span class="math-container">$f$</span> in the unit disk <span class="math-container">$D$</span> is an outer f... |
739,960 | <ol>
<li><p>$ \log_a{b} \times \log_b{a} = $ ?</p></li>
<li><p>$ \log_a{b} + \log_b{a} = \sqrt{29} $</p></li>
</ol>
<p>What is $ \log_a{b} - \log_b{a} = $ ?</p>
<p>3.</p>
<p>What is b in the following:</p>
<p>$$ \log_b{3} + \log_b{11} + \log_b{61} = 1 $$</p>
<p>and</p>
<p>4.</p>
<p>$$ \frac{1}{log_2{x}} + \frac... | nadia-liza | 113,971 | <p>2 . let $ x=\log_a{b} =\frac{1}{ \log_b{a}} $
$$\log_a{b}+\frac{1}{\log_a{b}}=\sqrt{29}$$
Then $ \log_a{b}$ and $ \log_a{b}$ are solutions of the following equation</p>
<p>$$x+\frac{1}{x}=\sqrt{29}$$</p>
<p>$$x_1=\frac{\sqrt{29}-5}{2}=\log_b{a}$$
$$x_2=\frac{\sqrt{29}+5}{2}=\log_a{b}$$ (We assume that $a<b$ )... |
1,821,582 | <blockquote>
<p>Find all solutions of $$\{x^3\}+[x^4]=1$$
where $[x]=\lfloor x\rfloor$</p>
</blockquote>
<p>$$$$</p>
<p>I know that $0\le\{x^3\}<1\Rightarrow 0<[x^4]\le 1$. Thus $[x^4]=1$. I couldn't get any further though since I'm having trouble with $x^4$ in the term $[x^4]$. $$$$As an example, in anoth... | triple_sec | 87,778 | <p>Suppose that $\{x^3\}+\lfloor x^4\rfloor=1$. Clearly, $\lfloor x^4\rfloor$ is an integer, so that $$\{x^3\}=1-\lfloor x^4\rfloor$$ must be an integer, too. But the fractional part of any number is always contained in $[0,1)$. This implies that $\{x^3\}=0$, so that $x^3$ is an integer.</p>
<p>Going back to the origi... |
213,405 | <p>So here's the question:</p>
<blockquote>
<p>Given a collection of points $(x_1,y_1), (x_2,y_2),\ldots,(x_n,y_n)$, let
$x=(x_1,x_2,\ldots,x_n)^T$, $y=(y_1,y_2,\ldots,y_n)^T$,
$\bar{x}=\frac{1}{n} \sum\limits_{i=1}^n x_i$, $\bar{y}=\frac{1}{n} \sum\limits_{i=1}^n y_i$.<br>
Let $y=c_0+c_1x$ be the linear funct... | murage kibicho | 873,735 | <p>I saw this in the book, Applied Algebra- Codes, Ciphers and Discrete Logarithms by Darel W.Hardy. It's on page 138 of the second edition, in the section about the Chinese Remainder Theorem and Extended Precision Arithmetic. The accepted answer gives a wonderful summary of the book's content.</p>
<p>In the general ca... |
2,191,360 | <blockquote>
<p>Show that
$$ f(x,y)= \begin{cases} \dfrac{xy^2}{x^2+y^4} & (x,y) ≠ (0,0) \\ 0 & (x,y) = (0,0) \end{cases}$$
is bounded.</p>
</blockquote>
<p>I thought about splitting it up into different cases like $x<y$ but it turned out to be too many and I could not cover all of them.
As a hint I... | Matthew Leingang | 2,785 | <p>The insight here is that the fraction is of the form $\frac{ab}{a^2+b^2}$, with $a=x$ and $b=y^2$.<br>
\begin{align*}
(x-y^2)^2 &\geq 0 \\
\implies x^2 - 2xy^2 + y^4 &\geq 0 \\
\implies x^2 + y^4 &\geq 2xy^2 \\
\implies\frac{xy^2}{x^2 + y^4} &\leq \frac{1}{2}
\end{align*}</p>
<p>You can also app... |
3,913,244 | <p>My attempt :</p>
<p><span class="math-container">$A=2^3×5^2×7^3$</span></p>
<p>Let's determine number of numbers primes with A, and
Smaller than A</p>
<p><span class="math-container">$\rho (A) = 2^2 ×4×5×6×7^2 =23520$</span></p>
<p>23520 is a number of numbers primes with A and smaller than A</p>
<p><span class="mat... | Bob Dobbs | 221,315 | <p>Let <span class="math-container">$F(x)=\sum_{k=0}^{m}a_kx^k$</span> where <span class="math-container">$m\leq p-2.$</span>
<span class="math-container">$$\sum_{i=0}^{p-1}F(i)=\sum_{i=0}^{p-1}\sum_{k=0}^{m}a_ki^k=\sum_{k=0}^{m}a_k\sum_{i=0}^{p-1}i^k=\sum_{k=0}^{m}a_k\frac{B_{k+1}(p)-B_{k+1}(0)}{k+1}=\sum_{k=0}^{m}a_k... |
572,541 | <blockquote>
<p>Let $L$ be the set of all lines in the plane. Prove that $L$ is uncountable, but only countably many of the lines in $L$ contain more than one rational point.</p>
</blockquote>
<p><strong>Attempt</strong>: Well, I was trying to define $L$ using linear combinations of points since a line is a linear c... | Kile Kasmir Asmussen | 72,934 | <p>All lines in the plane are a superset of all lines perpendicular to a given line.</p>
<p>How is it with perpendicular lines, points on their parent line and the number of points on a line?</p>
|
572,541 | <blockquote>
<p>Let $L$ be the set of all lines in the plane. Prove that $L$ is uncountable, but only countably many of the lines in $L$ contain more than one rational point.</p>
</blockquote>
<p><strong>Attempt</strong>: Well, I was trying to define $L$ using linear combinations of points since a line is a linear c... | GeorgeMoreno | 109,822 | <p>How about this? Let L be a line in the plane and v some vector that does not generates that line. You can prove that for every pair of real numbers (a,b), L+bv and L+av have empty intersection. For the second part, try this idea:
Every line with a irrational slope contains only one rational point (prove it by contra... |
457,977 | <p>I am trying to use residues to compute $$\int_0^\infty\frac{\log x}{(1+x)^3}\,\operatorname d\!x.$$My first attempt involved trying to take a circular contour with the branch cut being the positive real axis, but this ended up cancelling off the term I wanted. I wasn't sure if there was another contour I should use.... | Tunk-Fey | 123,277 | <h2>This approach is not using residues method but I'd like to post the general solution.</h2>
<hr>
<p>Let
$$\mathcal{I}=\int_0^\infty\frac{x^{m-1}}{(1+x)^{m+n}}\ln x\ dx\tag1$$
Consider <a href="http://en.wikipedia.org/wiki/Beta_function" rel="nofollow">beta function</a>
$$
\text{B}(m,n)=\int_0^\infty\frac{x^{m-1}}{... |
1,719,568 | <p>Can we say that $k$ grows faster than $\sqrt{k}$ when term is large? But what is the formal way write it ?</p>
| Gabe Cunningham | 759 | <p>You can factor a $\sqrt{k}$ out of the bottom to get:
$\lim_{k \to \infty} \frac{1}{\sqrt{k}(\sqrt{k}-2)}$.
Now it should be clear that the bottom goes to infinity.</p>
|
1,447,547 | <p>$$x^3>x$$</p>
<p>Steps I took:</p>
<p>$$x^{ 3 }-x>0$$</p>
<p>$$x(x^{ 2 }-1)>0$$</p>
<p>$$x(x-1)(x+1)>0$$</p>
<p>Now I see that all three linear factors must equal a positive value when multiplied. </p>
<p>I took each linear factor and set each to either greater than or less than $0$ since the solut... | Mark Viola | 218,419 | <p>Either all $3$ factors, $x+1$, $x$, and $x-1$ are positive or one and only one is positive. </p>
<p>That observation facilitates analysis.</p>
<p>Either $x>1$, in which case all $3$ factors are positive or $-1<x<0$, in which case only one, $x+1$, is positive and the other two are negative. In both cases... |
3,063,651 | <p>i am currently looking out for some possible topics i could study for my research project in high school. Algebra, trigonometry, Pythagoras’ theorem, geometry, circles and their properties, etc. and perhaps combined with a little knowledge from Physics i.e. Kinematics, Gravity, etc. could interest me. </p>
<p>Anoth... | Wuestenfux | 417,848 | <p>Welcome to MSE! The rubic's cube is an interesting object from the group theory point of view. Below you will find the enumeration of the faces of the cube. The central faces remain stationary. There six basic rotations of 90 degrees as shown: Left, Right, Up, Down, Front, Back.</p>
<p>These rotations can be describ... |
718,166 | <p><strong>Question:</strong></p>
<blockquote>
<p>let $a\in(0,1)$, and such $f(x)\geq0$, $x\in R$ is continuous on $R$,</p>
<p>if
$$f(x)-a\int_x^{x+1}f(t)dt,\forall x\in R $$ is constant,</p>
<p>show that</p>
<p>$f(x)$ is constant;</p>
<p>or $$f(x)=Ae^{bx}+B$$ where $A\ge 0,|B|\le A$ and $A,B$... | Karl Marx | 144,187 | <p>I need help proving the following result:</p>
<p>If $f(x)-a\int_x^{x+1}f(t)dt$ is constant, then $f(x)$ is constant or $f(x)=Ae^{bx}+B$.</p>
<ol>
<li>Firstly, we transform this ODE to the form:
$$\frac{dy}{dx}+P(x)y=Q(x).$$</li>
</ol>
<p>Such equation has the solution:
$$y=e^{-\int{Pdx}}$$[\int{Qe^{\int{Pdx}}+C}]... |
2,904,912 | <p>$$24a(n)=26a(n-1)-9a(n-2)+a(n-3)$$
$$a(0)=46, a(1)=8, a(2)=1$$
$$\sum\limits_{k=3}^{\infty}a(k)=2^{-55}$$
How can I prove it?</p>
| mengdie1982 | 560,634 | <p>Consider the <strong>generating function</strong> $$f(x)=a_0x^0+a_1x^1+a_2x^2+\cdots+a_nx^n+\cdots.\tag1$$</p>
<p>We have $$f(x)\cdot x^3=a_0x^3+a_1x^4+a_2x^5+\cdots+a_nx^{n+3}+\cdots;\tag2$$
$$f(x)\cdot x^2=a_0x^2+a_1x^3+a_2x^4+\cdots+a_nx^{n+2}+\cdots;\tag3$$
$$f(x)\cdot x=a_0x^1+a_1x^2+a_2x^3+\cdots+a_nx^{n+1}+\... |
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