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 |
|---|---|---|---|---|
3,289,401 | <p>As an example in MATLAB</p>
<pre><code>[U,S,V]=svd(randn(3,2)+1j*randn(3,2))
assert(isreal(V(1,:)))
</code></pre>
<p>Why is the first row of V purely real?</p>
| Community | -1 | <p>In Cartesian coordinates it's <span class="math-container">$(x_1,\dots,x_{n+1})\to (\frac{Rx_1}{R-x_{n+1}},\dots,\frac {Rx_n}{R-x_{n+1}},0)$</span>.</p>
<p>See <a href="https://en.m.wikipedia.org/wiki/Stereographic_projection" rel="nofollow noreferrer">here</a>.</p>
|
2,968,235 | <p><span class="math-container">$\log_3 4$</span> and <span class="math-container">$\log_7 10$</span>: which of these two logarithms is greater?</p>
<p>I figured out that both are between <span class="math-container">$1$</span> and <span class="math-container">$2$</span>, then between <span class="math-container">$1$<... | Michael Rozenberg | 190,319 | <p>We'll show that <span class="math-container">$$\log_34>\log_710$$</span> or
<span class="math-container">$$4>3^{\log_710},$$</span> which is true because <span class="math-container">$$\log_710<1.2$$</span> and <span class="math-container">$$4>3^{1.2}.$$</span></p>
|
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... | cgodfrey | 113,296 | <p>As others have pointed out, the Lebesgue integral is still computing the area under the graph. I'd just like to point out how it computes that area in a different way than the Riemann integral. For the sake of example let <span class="math-container">$f: I \to \mathbb{R}_{\geq 0}$</span> be a non-negative function o... |
1,296,230 | <p>This is from Lang's <em>Algebra</em> (page 251)</p>
<blockquote>
<p><strong>Proposition 6.11</strong> <em>Let <span class="math-container">$E/F$</span> be a normal field extension. Let <span class="math-container">$E^G$</span> be the fixed field of <span class="math-container">$\operatorname{Aut}(E/F)$</span>. Th... | reuns | 276,986 | <p>Let <span class="math-container">$a \in \overline{F}$</span>, <span class="math-container">$F(a) \cong F[x]/(h(x))$</span>.</p>
<p>If <span class="math-container">$F(a)/F$</span> is not separable then <span class="math-container">$\gcd(h ,h') \ne 1$</span>, thus <span class="math-container">$h' = 0$</span> (since o... |
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>
| Crostul | 160,300 | <p>The answer is yes. To prove it, define for $n$ big enough
$$g_n:\left[ a+\frac{1}{n} , b-\frac{1}{n}\right] \longrightarrow \Bbb{R} \quad \quad g_n(x)=f(x)$$
and then define $f_n:[a,b]\longrightarrow \Bbb{R}$ extending $g_n$ in a suitable way that $f_n(a)=0=f_n(b)$ and $f_n$ are uniformly bounded.</p>
<p>Then $$0=\... |
1,251,537 | <p>$f:[a,b] \to R$ is continuous and $\int_a^b{f(x)g(x)dx}=0$ for every continuous function $g:[a,b]\to R$ with $g(a)=g(b)=0$. Must $f$ vanish identically?</p>
<hr>
<p>Using integration by parts I got the form:
$\int_a^bg(x)f(x)-g'(x)F(x)=0$. Where $F'(x)=f(x)$.</p>
| celtschk | 34,930 | <p>Assume that for some $x_0\in(a,b)$ we have $f(x_0)>0$. Since $f$ is assumed to be continuous, this means that there is an $\epsilon>0$ such that $f(x)>0$ in $(x_0-\epsilon,x_0+\epsilon)$. Now let
$$g(x)=\begin{cases}
0 & x\notin(x_0-\epsilon,x_0+\epsilon)\\
x-x_0+\epsilon & x_0-\epsilon \le x \le x_... |
162,630 | <p>Let $\mathbb{G}$ be a reductive group defined over a number field $K$, let $Z$ be its center, and let $\mathbb{A}:=\mathbb{A}_K$ be the ring of adeles of $K$. Reasonably, we care about the $\mathbb{G}(\mathbb{A})$-representation: $L^2(\mathbb{G}(K)\backslash \mathbb{G}(\mathbb{A}))$. It naturally contains the sub-re... | paul garrett | 15,629 | <p>First, one should be a little careful about saying that $L^2(G_k\backslash G_\mathbb A)$ has $L^2(Z_\mathbb A G_k\backslash G_\mathbb A,\omega)$ inside it... since appearing as direct integral "integrands" is not a very strong commitment. If $G$ has non-compact center, $L^2(G_k\backslash G_\mathbb A)$ will have no d... |
162,630 | <p>Let $\mathbb{G}$ be a reductive group defined over a number field $K$, let $Z$ be its center, and let $\mathbb{A}:=\mathbb{A}_K$ be the ring of adeles of $K$. Reasonably, we care about the $\mathbb{G}(\mathbb{A})$-representation: $L^2(\mathbb{G}(K)\backslash \mathbb{G}(\mathbb{A}))$. It naturally contains the sub-re... | Marc Palm | 10,400 | <p>In addition to Paul Garrett's answer, I address your last paragraph in a special example:</p>
<p>Strong approximation gives a homeomorphism
$SL_2(Z) \backslash H \cong Z(A) GL_2(Q) \backslash GL_2(A) / \prod_p GL_2(Z_p) \times O(2)$.</p>
<p>Lets $f$ corresponds to $\tilde{f}$. This translates</p>
<p>$$ \int_{0}^1... |
746,180 | <p>I'm working through Stephen Abbott's wonderful <em>Understanding Analysis</em> in preparation for entering a math undergrad degree this fall. A personal note about me: Friends and family tell me I tend to be periphrastic; if there's a long-winded, inelegant way of explaining myself, I'll find it. As I work through A... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Using <a href="http://en.wikipedia.org/wiki/Trigonometric_substitution" rel="nofollow">Trigonometric Substitution</a> $$x=2\tan\theta$$</p>
|
746,180 | <p>I'm working through Stephen Abbott's wonderful <em>Understanding Analysis</em> in preparation for entering a math undergrad degree this fall. A personal note about me: Friends and family tell me I tend to be periphrastic; if there's a long-winded, inelegant way of explaining myself, I'll find it. As I work through A... | Community | -1 | <p>Put $x = 2 \tan t $, then $dx = 2 \sec^2 t dt $. and $\sqrt{x^2 +4} = \sqrt{ 4 \tan^2 t + 4 } = 2 \sec t$ hence,</p>
<p>$$ \int \frac{dx}{x^2 \sqrt{x^2+4}} = \int \frac{2 \sec^2 t dt}{4 \tan^2 t 2 \sec t} = \frac{1}{4} \int \frac{ \sec t dt }{\tan^2 t} = \frac{1}{4} \int \frac{\frac{1}{\cos t}}{\frac{\sin^2t}{\cos^... |
746,180 | <p>I'm working through Stephen Abbott's wonderful <em>Understanding Analysis</em> in preparation for entering a math undergrad degree this fall. A personal note about me: Friends and family tell me I tend to be periphrastic; if there's a long-winded, inelegant way of explaining myself, I'll find it. As I work through A... | Artem | 29,547 | <p>Hint:
$$
x^2\sqrt{x^2+4}=x^3\sqrt{1+\frac{4}{x^2}}\,,\quad d\left(\frac{1}{x^2}\right)=-\frac{2}{x^3}d x
$$</p>
|
506,152 | <p>Is $$\frac{a+b}{c+d}<\frac{a}{c}+\frac{b}{d}$$
for $a,b,c,d>0$</p>
<p>If it is true, then can we generalize?</p>
<p>EDIT:typing mistake corrected.</p>
<p>EDIT, WILL JAGY. Apparently the <strong>real question</strong> is
Is $$\color{magenta}{\frac{a+b}{c+d}<\frac{a}{c}+\frac{b}{d}}$$
for $a,b,c,d>0,$ ... | Bob Anderson | 97,156 | <p>A slightly different approach:</p>
<p>Multiply both sides by (c+d), which we can do without altering the inequality because c and d are positive:</p>
<p>$$ a+b < \frac{a(c+d)}{c} +\frac{b(c+d)}{d}$$
$$ a+b < \frac{ac}{c} +\frac{ad}{c} +\frac{bc}{d} +\frac{bd}{d}$$
$$ a+b < a + \frac{ad}{c} +\frac{bc}{d} +... |
506,152 | <p>Is $$\frac{a+b}{c+d}<\frac{a}{c}+\frac{b}{d}$$
for $a,b,c,d>0$</p>
<p>If it is true, then can we generalize?</p>
<p>EDIT:typing mistake corrected.</p>
<p>EDIT, WILL JAGY. Apparently the <strong>real question</strong> is
Is $$\color{magenta}{\frac{a+b}{c+d}<\frac{a}{c}+\frac{b}{d}}$$
for $a,b,c,d>0,$ ... | Hypergeometricx | 168,053 | <p>Let <span class="math-container">$a=\lambda c$</span> and <span class="math-container">$b=\mu d$</span>, where <span class="math-container">$\lambda, \mu>0$</span>.
<span class="math-container">$$\frac {a+b}{c+d}=\frac {\lambda c+\mu d}{c+d}=\frac {\lambda (c+d)+\mu (c+d)-(\lambda d+\mu c)}{c+d}=\lambda + \mu -\f... |
3,162,464 | <p>I need help making an OGF for <span class="math-container">$1 + x^i + x^{2i}+...+x^{ki}$</span>. I already know how to verify that <span class="math-container">$1 +x +x^2+...+x^k$</span> can be written by <span class="math-container">$({1-x^{k+1}})/({1-x})$</span>. I'm wondering if there is any correlation between t... | Peter Foreman | 631,494 | <p>Hint: make the substitution <span class="math-container">$u=x^i$</span>.</p>
|
858,952 | <p>related to <a href="https://math.stackexchange.com/questions/830599/one-sided-limit-lim-x-rightarrow-0-fx-where-wolfram-alpha-does-not-hel">this question</a>:</p>
<p>Is there an easy closed-form term for</p>
<p>$$\sum_{j=k}^{\infty} \frac{x^j}{j!}e^{-x},$$</p>
<p>thus when the sum starts at a constant $k$ instead... | mookid | 131,738 | <p>Yes:</p>
<p>$$e^{-x}\sum_{j=k}^\infty \frac 1{j!} x^j =
e^{-x}\left[\exp x - \sum_{j=0}^{k-1} \frac 1{j!} x^j \right]
=e^{-x}\int_0^x \frac{(x-t)^{k-1}}{(k-1)!} e^{t} dt
$$</p>
|
858,952 | <p>related to <a href="https://math.stackexchange.com/questions/830599/one-sided-limit-lim-x-rightarrow-0-fx-where-wolfram-alpha-does-not-hel">this question</a>:</p>
<p>Is there an easy closed-form term for</p>
<p>$$\sum_{j=k}^{\infty} \frac{x^j}{j!}e^{-x},$$</p>
<p>thus when the sum starts at a constant $k$ instead... | Did | 6,179 | <p>You just made a mistake when differentiating $$f(x)=\sum_{j=k}^{\infty} \frac{(xp)^j}{j!} e^{-xp}.$$
The actual derivative is
$$f'(x)=\sum_{j=k}^{\infty} \left( \color{red}{j}\,\frac{p\,(xp)^{j-1}}{j!} e^{-xp} -\frac{p\,(xp)^j}{j!} e^{-xp} \right),$$
that is, provided $k\geqslant1$, using the change of indices $\ell... |
1,075,215 | <p>Question: An actuary is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of women. For each of the three factors, the probability is 0.1 that a woman in the population only has this risk factor (and no others). For any two of three factors, the probability is 0.12 tha... | turkeyhundt | 115,823 | <p>You can divide the sample into 8 pools.
$$
\begin{array}{|c|c|} \hline
\text{Factor}& \text{Probability} \\ \hline
\text{A} & .1 \\ \hline
\text{B} & .1 \\ \hline
\text{C} & .1 \\ \hline
\text{AB} & .12 \\ \hline
\text{AC} & .12 \\ \hline
\text{BC} & .12 \\ \hline
\text{ABC} & .06 \... |
3,197,262 | <p>When I was solving <span class="math-container">$ \operatorname{Cov}(X,E(X\mid Y)) = \operatorname{var}(E(X\mid Y))$</span>, I notice that <span class="math-container">$E(X\mid Y)$</span> was treated as a function of <span class="math-container">$Y$</span>.
My thinking is <span class="math-container">$E(X\mid Y)$</... | Surb | 154,545 | <p>An easy way to clarify : if <span class="math-container">$f(y)=\mathbb E[X\mid Y=y]$</span>, then <span class="math-container">$\mathbb E[X\mid Y]=f(Y)$</span>.</p>
|
1,780,797 | <p>According to <a href="https://en.wikipedia.org/wiki/Null_set#Lebesgue_measure" rel="nofollow noreferrer">Wikipedia</a>, the straight line <span class="math-container">$\mathbb{R}$</span> is a null set in <span class="math-container">$\mathbb{R}^2$</span>.</p>
<p>That means, the line <span class="math-container">$\ma... | Benjamin Lindqvist | 96,816 | <p>Here's an attempt.</p>
<p>$$D(P_X||P_{X+Y}) = \mathbb{E}[\log \frac{P_X(X)}{P_{X+Y}(X+Y)}] = \mathbb{E}[\log \frac{P_X(X)P_Y(Y)}{P_{X+Y}(X+Y)P_Y(Y)}] \\
= \mathbb{E}[\log \frac{P_X(X)}{P_Y(Y)}] + \mathbb{E}[\log \frac{P_Y(Y)}{P_{X+Y}(X+Y)}] = \infty$$</p>
<p>because $g$ is uniform and has $g(x)=0$ for some $x$ suc... |
2,590,165 | <p>How to show $f(x)$=$\frac{1}{1+x^2}$ is uniform continuous on $\Bbb R$. </p>
<p>Although, of course for any interval $[a,b]$, this function is continuous and bounded, therefore also uniformly continuous. Following <strong>Continuous Extension Theorem</strong> it is uniformly continuous on any $(a,b)$. Therefore pr... | Andreas | 317,854 | <p>So you need (the values in parantheses are now positive)</p>
<p>$$
{e^{xB}}(1-xB) \le {e^{xA}}(1-xA)
$$</p>
<p>Since $B>A$, you want that $
f(q) = {e^{xq}}(1-xq)$ is decreasing with $q$. The derivative is </p>
<p>$
f'(q) = -x^2q \; {e^{xq}}$</p>
<p>and this is clearly negative for $x,q \in (0,1)$. That's it.... |
3,819,658 | <p>Calculate, <span class="math-container">$$\lim\limits_{(x,y)\to (0,0)} \dfrac{x^4}{(x^2+y^4)\sqrt{x^2+y^2}},$$</span> if there exist.</p>
<p>My attempt:</p>
<p>I have tried several paths, for instance: <span class="math-container">$x=0$</span>, <span class="math-container">$y=0$</span>, <span class="math-container">... | user | 505,767 | <p>Assuming <span class="math-container">$y<1$</span> and using polar coordinates we have</p>
<p><span class="math-container">$$\dfrac{x^4}{(x^2+y^4)\sqrt{x^2+y^2}} \le \dfrac{x^4}{(x^2+y^2)\sqrt{x^2+y^2}} =r \sin^4\theta \to 0$$</span></p>
|
1,052,180 | <p>I need to find connected graph $G = (V, E), |V| \geq 3$ such that every power of his adjacency matrix contains zeroes.</p>
<p>I know that that graph will be path and adjacency matrix for even and odd powers would look like this (lets say for $|V| = 3$):</p>
<p>$M=
\left[ {\begin{array}{ccccc}
0 & 1 & ... | Robert Israel | 8,508 | <p>Hint: if the graph is bipartite (with parts $A$ and $B$), any walk starting in part $A$ will be in part $B$ after an odd number of steps and in part $A$ after an even number of steps.</p>
|
3,454,682 | <p>I know that <span class="math-container">${\langle x, y \rangle}$</span> means the inner product but I've stumbled upon the notation <span class="math-container">${\langle x, y \rangle}_a$</span> with <span class="math-container">$a \in \mathbb{R}$</span> and I can't figure out what it means. Usually what's in the s... | Masacroso | 173,262 | <p>In your context the notation <span class="math-container">$\langle f,g \rangle_1$</span> or <span class="math-container">$\langle f,g \rangle_2$</span> is just a way to name two distinct inner products, this is all. The numbers doesn't have a "mathematical" meaning, it just a name, a tag.</p>
<p>We could say also t... |
2,250,469 | <p>Let n $\geq$ 4 be an integer. Determine the number of permutations of
$\{1, 2, . . . , n\}$, in which $1$ and $2$ are next to each other, with $1$ to the left of $2$.<br>
I can't make sense of this problem statement. The way I see it, if $n$ is an integer, then the pair $1,2$ could be formed by any pair with the for... | JMoravitz | 179,297 | <p>The problem statement requires that we count the number of arrangements such that</p>
<ul>
<li><p>The arrangement is a permutation of $\{1,2,\dots,n\}$ (<em>i.e. it uses each and every number from $\{1,2,\dots,n\}$ exactly once</em>)</p></li>
<li><p>$1$ and $2$ are adjacent, i.e. they appear right next to one anoth... |
2,262,371 | <p>If $a,b,c$ are positive real numbers, prove that
$$\frac{2}{a+b}+\frac{2}{b+c}+ \frac{2}{c+a}≥ \frac{9}{a+b+c}$$</p>
| Lazy Lee | 430,040 | <p>By Cauchy-Schwartz. $$\sum_{cyc}\frac{1}{a+b}\sum_{cyc}(a+b)\geq (1+1+1)^2=9\implies 2\cdot\sum_{cyc}\frac{1}{a+b}\geq2\cdot \frac{9}{\sum_{cyc}(a+b)}=\frac{9}{a+b+c}$$</p>
|
1,073,628 | <p>I am trying to find generating functions which will give me a power logarithm. </p>
<p>I am trying to find generating sums in the form</p>
<p>$$\sum_{n=1}^{\infty} a_n\,x^n = -\frac{\log^2(1-x)}{1-x}$$</p>
<p>or </p>
<p>$$\sum_{n=1}^{\infty} a_n\,x^n = \frac{\log^2(x)}{x}.$$</p>
<p>Something, which will return ... | Dr. Wolfgang Hintze | 198,592 | <p><strong>Extension to general powers $(-\log (1-x))^n$</strong></p>
<p>Exploiting the powerful idea of paramteric differentiation exposed here by Felix Marin I was able to find expressions for arbitrary powers $n$ and provide examples for $n = 1..8$. </p>
<p>The results for $n\ge 4$ seem to be new (please correct m... |
2,300,382 | <p>I cannot think of a non-constructible algebraic number of degree $4$ over $\Bbb Q$ so far. I wish if I can find such an example. Could some one tell me some such numbers with justification? Also I would like to know the track of working out such an example. Any help or reference would be appreciate. Thanks in advanc... | José Carlos Santos | 446,262 | <p>One problem which leads to such a number is <a href="https://en.wikipedia.org/wiki/Alhazen%27s_problem" rel="nofollow noreferrer">Alhazen's billiard problem</a>.</p>
|
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]''... | Suba Thomas | 5,998 | <p>The first thing is that I find it odd that your model has current as the input and speed as the output? Typically, it's <a href="https://reference.wolfram.com/language/MicrocontrollerKit/workflow/MotorSpeedControl.html" rel="nofollow noreferrer">voltage to speed</a>, and also <a href="https://reference.wolfram.com/l... |
789,407 | <p>If the roots of the equation $$ax^2-bx+c=0$$ lie in the interval $(0,1)$, find the minimum possible value of $abc$. </p>
<p><strong>Edit:</strong> I forgot to mention in the question that $a$, $b$, and $c$ are natural numbers. Sorry for the inconvenience.<br>
<strong>Edit 2:</strong> As Hagen von Eitzen said abo... | apt1002 | 106,285 | <p>The answer is $a=4$, $b=4$, $c=1$, giving $x = \frac12$ (twice), and a product $abc=16$. Exhaustive search through all $1 \leq a,b,c \leq 16$ gave no better answer.</p>
|
2,877,085 | <p>I think that there could be used Abel and Dirichlet method, but I have no idea how</p>
<p>$$ \sum_{n=1}^{\infty} (-1)^n\frac{3n-2}{n+1}\frac{1}{n^{1/2}} .$$</p>
| marty cohen | 13,079 | <p>Write the fraction as
$3-5/(n+1)$.</p>
<p>The sum of the first converges by Leibnitz
and the second converges absolutely.</p>
|
1,392,205 | <p>The equation of line $A$ is $3x + 6y - 1 = 0$. Give the equation of a line that passes through the point $(5,1)$ that is</p>
<ol>
<li><p>Perpendicular to line $A$.</p></li>
<li><p>Parallel to line $A$.</p></li>
</ol>
<p>Attempting to find the parallel,</p>
<p>I tried $$y = -\frac{1}{2}x + \frac{1}{6}$$</p>
<p>$$... | heropup | 118,193 | <p>You have a problem with your geometric PMF: the sum of from $x = 0$ to $\infty$ is not equal to $1$. As such, you must write either</p>
<p>$$\Pr[X = x] = (1/4)^x (3/4), \quad x = 0, 1, 2, \ldots,$$ or $$\Pr[X = x] = (1/4)^{x-1} (3/4), \quad x = 1, 2, 3, \ldots.$$ Which one you mean, I cannot tell, and because th... |
587,077 | <p>Given any prime $p$. Prove that $(p-1)! \equiv -1 \pmod p$.</p>
<p>How to prove this?</p>
| Akshaj Kadaveru | 100,840 | <p>Consider the set of residues $\{1, 2, 3, \cdots , p-1\}$, and consider an arbitrary element $\alpha$. </p>
<p><strong>Lemma 1: There exist $\gamma$ such that $\alpha \gamma \equiv 1 \pmod p$</strong></p>
<p>Proof: Consider $$1, p+1, 2p+1, 3p+1, \cdots , (\alpha-1)p + 1$$ If two are equivalent modulo $\alpha$, we w... |
257,978 | <p>Is there any non-monoid ring which has no maximal ideal?</p>
<p>We know that every commutative ring has at least one maximal ideals -from Advanced Algebra 1 when we are study Modules that makes it as a very easy Theorem there.</p>
<p>We say a ring $R$ is monoid if it has an multiplicative identity element, that if... | Julian Kuelshammer | 15,416 | <p>Not every non-unital ring (or rng) has a maximal ideal. For example take $(\mathbb{Q},+)$ with trivial multiplication, i.e. $xy=0$ for all $x,y\in \mathbb{Q}$, then a maximal ideal is nothing more than a maximal subgroup. See <a href="https://math.stackexchange.com/questions/234995/mathbbq-has-no-maximal-subgroup">t... |
3,173,242 | <p><strong>Context:</strong></p>
<p>In the context of circuit theory and graph theory, suppose we have a graph <span class="math-container">$G,$</span> then <a href="https://en.wikipedia.org/wiki/Laplacian_matrix" rel="nofollow noreferrer">the Laplacian (Kirchhoff) matrix</a> <span class="math-container">$L$</span> is... | Misha Lavrov | 383,078 | <p>For your second question, if we think of <span class="math-container">$M^{\mathsf T}$</span> as a linear transformation, it takes an element of <span class="math-container">$\mathbb R^V$</span> (an assignment of a real scalar to every vertex) to an element of <span class="math-container">$\mathbb R^E$</span> (an ass... |
4,549,070 | <p>How can I prove this without using Stirling's formula?</p>
<p><span class="math-container">$${n\choose an} \le 2^{nH(a)}$$</span>
<span class="math-container">$$H(a) := -a\log_2a -(1-a)\log_2(1-a)$$</span></p>
| Qiaochu Yuan | 232 | <p>Let <span class="math-container">$X = \text{Bin}(n, \frac{1}{2})$</span> be a binomial random variable with <span class="math-container">$p = \frac{1}{2}$</span>. The <a href="https://math.stackexchange.com/a/4546041/232">Chernoff bound</a> applied to <span class="math-container">$X$</span> gives that for <span clas... |
4,364,686 | <p>I have a point in a 3D coordinate system 1 (CS1). There can be two situations: the point is constant or the point is moving along a straight line from one known position to another at constant speed.</p>
<p>The CS1 is rotating in another (static) 3D coordinate system (CS2). The rotations of CS1 are known, i.e. the s... | bubba | 31,744 | <p>I’d recommend that you just calculate a large number of points and then interpolate them with a polyline. If you need more accuracy, use more points. The nice thing about polylines is that their arclength are very easy to calculate.</p>
<p>Your idea of using a (cubic) spline won’t help very much, because computing t... |
3,661,474 | <p><span class="math-container">$ h:R^{N+1} \to [0 , \infty)$</span> , <span class="math-container">$ h $</span> is measurable</p>
<p><span class="math-container">$ g:R^{N+1} \to [0 , \infty)$</span> , <span class="math-container">$ g $</span> is measurable</p>
<p><span class="math-container">$x,y \in R^N$</span></p... | ibnAbu | 334,224 | <p>Excluding the trivial case for <span class="math-container">$h=0$</span> ae, then it must be that <span class="math-container">$h$</span> is positive on some set of positive
measure</p>
<p>The functional equation can be coverted to</p>
<p><span class="math-container">$$f (x, x^2) f (y, y^2)= G (x+y, x^2+y^2) \text{w... |
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... | Pietro Majer | 6,101 | <p>What you are describing is exactly the ball measure of non-compactness of the closed bounded set $B$:
$$\alpha(B):=\inf \{r>0 \, : \exists F\subset B, \mathrm {\, F\, finite\, ,s.t.} B\subset \cup_{x\in F} B(x,r) \}\, .$$
It can be viewed as the point-set distance $\inf_{F\in \mathcal{F}}d(B,F)$ between $B$ (as... |
3,528,237 | <p>I am just being introduced to quantifiers in logic and my lecturer was going through the following two statements. The question is to determine which, if any, is/are true.</p>
<ol>
<li><span class="math-container">$(\forall x \in \mathbb{R})(\exists y \in \mathbb{R})[x + y = 0]$</span></li>
<li><span class="math-co... | Community | -1 | <p>If you translate the sentences into English, you can see the differences between them. </p>
<p>The first sentence says "Every real number has an additive inverse." That is for every real number <span class="math-container">$x$</span>, there is real number <span class="math-container">$y$</span> such that <span c... |
4,386,952 | <p>Informally, mathematicians treat Integers like a subset of rational numbers.</p>
<p>But according to the standard, formal construction of <span class="math-container">$\mathbb{Q}$</span>, <span class="math-container">$\mathbb{Q}$</span> is an equivalence class over <span class="math-container">$\mathbb{Z} \times \ma... | Lazy | 958,820 | <p>One approach to this would be: If we knew that <span class="math-container">$c$</span> is differentiable then we knew from the product rule that <span class="math-container">$(fg)'(a) = f'(a)g(a) + f(a)g'(a) = f'(a)g(a)$</span> by <span class="math-container">$f(a)=0$</span>. Since this does not depend on <span clas... |
2,514,236 | <p>For example, the matrix could have finitely many rows and columns, but each row/column has uncountably many elements and you can do the standard matrix multiplication by taking care to match up the entries with corresponding pairs of real number indices. </p>
<p>Do such objects exist and has there been any work on ... | rschwieb | 29,335 | <p>Infinite matrix rings are pretty interesting. I know several interesting examples.</p>
<p>The simplest is the ring of column-finite matrices over a field $F$, which is isomorphic to the ring of linear transformations of an infinite dimensional $F$ vector space. <a href="http://ringtheory.herokuapp.com/rings/ring/15... |
39,466 | <p>I could not solve this problem:</p>
<blockquote>
<p>Prove that for a non-Archimedian field $K$ with completion $L$, $$\left\{|x|\in\mathbb R \mid x\in K\right\} =\left\{|x|\in\mathbb R \mid x\in L\right\}$$</p>
</blockquote>
<p>I considered a Cauchy sequence in $K$ with norms having limit $l$, but I could not co... | Eric Naslund | 6,075 | <p><strong>Hint:</strong> If the sequence $a_k$ is bounded, then there exists $M$ such that $|a_k|\leq M$ for all $k$.</p>
<p>Then, what can you say about $|\sum_{k=1}^\infty a_k x^k|$ and $\sum_{k=1}^\infty Mx^k$? What is the radius of convergence of the second one?</p>
|
102,427 | <p>I just coded a simple simulation module that looks at the evolution of a continuous trait in a haploid asexually reproducing population under density dependent competition in discrete time (i.e. non-overlapping generations, using recurrence equations). What I am interested in is finding out whether evolution would a... | Simon Woods | 862 | <p>Here is a modified version of your code. On my PC it completes your example run in about 5 seconds.</p>
<p>I won't try to describe every change but will point out the major features. Some of the changes are stylistic rather than performance-based. This is not a criticism of your style but a reflection of the way I ... |
2,795,777 | <p>I encountered this problem in one of my linear algebra homeworks (Linear Algebra with Applications 5th Ed 1.3.44):</p>
<p>Consider a $n \times m$ matrix $A$, such that $n > m$. Show there is a vector $b$ in $\mathbb{R}^{n}$ such that the system $Ax=b$ is inconsistent.</p>
<p>I have a strong intuition as to why ... | Hagen von Eitzen | 39,174 | <p>Note that $A^{10}=10^{10}I$, which rules out most answers.</p>
|
2,795,777 | <p>I encountered this problem in one of my linear algebra homeworks (Linear Algebra with Applications 5th Ed 1.3.44):</p>
<p>Consider a $n \times m$ matrix $A$, such that $n > m$. Show there is a vector $b$ in $\mathbb{R}^{n}$ such that the system $Ax=b$ is inconsistent.</p>
<p>I have a strong intuition as to why ... | mechanodroid | 144,766 | <p>Your matrix is the transpose of the <a href="https://en.wikipedia.org/wiki/Companion_matrix" rel="nofollow noreferrer">companion matrix</a> so its characteristic and minimal polynomial are equal to $x^{10} - 10^{10}$.</p>
<p>Therefore, the spectrum of $A$ is precisely the set of real roots of $x^{10} - 10^{10}$, wh... |
529,886 | <p>In the context of learning about comparison theorem, using integrals to determine convergence and learning about exponential series (That's what $n^p$ is called right?).</p>
| newzad | 76,526 | <ol>
<li>Say $m(EAB)=\alpha$ then $m(DAF)=30-\alpha$, $m(DFA)=60+\alpha$ and $m(FEC)=30+\alpha$.</li>
<li>Join $M$ and $E$.</li>
<li>$m(MEA)=m(MEF)=30$, therefore $m(MEC)=60+\alpha$</li>
<li>We see that $m(MEC)=m(MFD)=60+\alpha$ and $m(EMA)=m(ECF)=90$.</li>
<li>As a result we can say that a circle pass through the poin... |
890,313 | <p>Say the probability of an event occurring is 1/1000, and there are 1000 trials.</p>
<p>What's the expected number of events that occur? </p>
<p>I got to an answer in a quick script by doing the above 100,000 times and averaging the results. I got 0.99895, which seems like it makes sense. How would I use math to ge... | drhab | 75,923 | <p>Give the trials the numbers $1,2,\dots,1000$.</p>
<p>Let $X_i$ take value $1$ if the event is occurring at the $i$-th trial and value $0$ otherwise. </p>
<p>$$X:=X_1+\cdots+X_{1000}$$ is the number of events that occur. This with:</p>
<p>$$\mathbb E(X_i)=1\times P(X_i=1)+0\times P(X_i=0)=1\times\frac{1}{1000}+0\t... |
1,348,099 | <p>We know that cross product gives a vector that is orthogonal to other two vectors. Let this vector denoted by $$|\vec{v} \times \vec{u}| = \vec{n}$$
Then $$\vec{n}\cdot \vec{u} = 0 $$ Everything okay up to here. Then how we choose a vector from two possible orthogonal vectors, $$\vec{n}$$ or $$\vec{-n}$$ Why followi... | Vincent | 147,033 | <p>Let $u:=mg+kx \implies x= \frac{u-mg}{k}$ <em>and</em> $du=k\,dx$ then $$\int\frac{x\cdot dx}{mg+kx}=\int\frac{\frac{u-mg}{k}\cdot \frac{du}{k}}{u}=\frac{1}{k^2}\int\frac{u-mg}{u}\,du=\frac{1}{k^2}\int 1-\frac{mg}{u}\,du=\frac{1}{k^2}[u-mg\ln(u)]+C,$$ where $C$ is an arbitrary constant. Substituting $u=mg+kx$ back i... |
3,393,193 | <p>I am asked to answer the following:</p>
<p>Let <span class="math-container">$f:Z->Z$</span> be defined by <span class="math-container">$f(x) = 2x$</span>.</p>
<ul>
<li>Write down infinitely many functions <span class="math-container">$g:Z->Z$</span> such that <span class="math-container">$g(f(x)) = Id_z$</sp... | Floris Claassens | 638,208 | <p>You continue if the expected value of continuing is higher than your current role. </p>
<p>On the third roll, the expected value of continuing equals
<span class="math-container">$$C_{3}=\frac{1}{6}(1+2+3+4+5+6)=\frac{7}{2}=4-\frac{1}{2},$$</span>
so you continue if you roll lower than <span class="math-container"... |
1,887,856 | <p>When is a group homomorphism</p>
<p>$$\varphi:\Bbb Z/2\Bbb Z\to \Bbb Z/2\Bbb Z\oplus \Bbb Z/2\Bbb Z$$</p>
<p>an "isomorphism onto the first summand" ?</p>
<p>Is the map $\varphi:1\mapsto (1,1)$ an isomorphism onto the first summand?</p>
| Mr. Chip | 52,718 | <p>Such a $\varphi$ is an isomorphism in case it is bijective onto its image and has its image contained in the first summand, which is the subset $\mathbb{Z}/2\mathbb{Z} \oplus 0 = \{ (0,0), (1,0) \}$. So your map is not such an isomorphism.</p>
|
1,887,856 | <p>When is a group homomorphism</p>
<p>$$\varphi:\Bbb Z/2\Bbb Z\to \Bbb Z/2\Bbb Z\oplus \Bbb Z/2\Bbb Z$$</p>
<p>an "isomorphism onto the first summand" ?</p>
<p>Is the map $\varphi:1\mapsto (1,1)$ an isomorphism onto the first summand?</p>
| Zev Chonoles | 264 | <p>To say that a map
$$\varphi:A\to B\oplus C$$
is an isomorphism onto the first summand means that there is an isomorphism $\psi:A\to B$ for which $\varphi(a)=(\psi(a),0)$. </p>
<p>In other words, $\varphi$ is an isomorphism onto the first summand when both of the following are true:</p>
<ol>
<li>the image of $\varp... |
2,569,096 | <p>The problem goes as follows:
$$
P=\left(
\begin{matrix}
a & 0.6\\
1-a & 0.4\\
\end{matrix}
\right)
$$</p>
<blockquote>
<p>Determine the value of the parameter $a \in [0,1]$ for which $P$ does <strong>not</strong> have an inverse.</p>
</blockquote>
<p>So then I know the value of $a$ lies ... | kam | 514,050 | <p>There is no inverse to a matrix with determinant 0. so when Det(<em>P</em>)=0, $P^{-1}$ doesn't exist.</p>
<pre><code>i.e 0.4a-0.6(1-a)=0,
so 0.4a+0.6a-0.6=0,
so a=0.6
</code></pre>
|
213,665 | <p><strong>I've tried 3 methods but all failed to do that.</strong></p>
<p>1st Method</p>
<pre><code>Apply[Flatten, {1, {2, {3, 4}, 5}, 6}, {2}]
</code></pre>
<p>2nd Method</p>
<pre><code>Map[Flatten, {1, {2, {3, 4}, 5}, 6}, {2}]
</code></pre>
<p>3rd Method</p>
<pre><code>Flatten[{1, {2, {3, 4}, 5}, 6}, {2}]
</co... | kglr | 125 | <pre><code>lst = {1, {2, {3, 4}, 5}, 6};
FlattenAt[lst, {2, 2}]
</code></pre>
<blockquote>
<p>{1, {2, 3, 4, 5}, 6}</p>
</blockquote>
<p>Also</p>
<pre><code>Map[## & @@ # &, lst, {2}]
</code></pre>
<blockquote>
<p>{1, {2, 3, 4, 5}, 6}</p>
</blockquote>
<pre><code>Replace[lst, List -> Sequence, {3}, H... |
2,966,392 | <p>Suppose <span class="math-container">$\lim_{n\rightarrow\infty }z_n=z$</span>.<br>
Prove <span class="math-container">$\lim_{n\rightarrow\infty}\operatorname{Re}(z_n)=\operatorname{Re}(z)$</span></p>
<p>Where, <span class="math-container">$z\in\mathbb{C}$</span> and <span class="math-container">$z_n$</span> is a c... | José Carlos Santos | 446,262 | <p>What you should do is<span class="math-container">\begin{align}\bigl\lvert\operatorname{Re}(z_n)-\operatorname{Re}(z)\bigr\rvert&=\bigl\lvert\operatorname{Re}(z_n-z)\bigr\rvert\\&\leqslant\lvert z_n-z\rvert\\&<\varepsilon.\end{align}</span></p>
|
1,177,583 | <p>I have a question similar to <a href="https://math.stackexchange.com/questions/757525/picture-behind-so3-so2-simeq-s2/" title="one">this one</a>, but that question is not answered. The question is to show that $SO(3)/SO(2)$ is isomorphic to the 2-sphere:
$$
SO(3)/SO(2)\cong S^2
$$
How does one establish the isomorph... | Daniel Valenzuela | 156,302 | <p>Consider $SO(R^3)$ which acts on $R^3$ by rotations, but restricts to an action of $S^2$. For every point $x\in S^2$ we have a unique orthogonal plane $V$, hence $SO(V)\subset SO(R^3)$ will fix $x$. It is easy to see that in fact $Stab(x)=SO(V) \cong SO(2)$. Hence we have a fiber bundle
$$
SO(3) \to S^2
$$</p>
<p>... |
1,177,583 | <p>I have a question similar to <a href="https://math.stackexchange.com/questions/757525/picture-behind-so3-so2-simeq-s2/" title="one">this one</a>, but that question is not answered. The question is to show that $SO(3)/SO(2)$ is isomorphic to the 2-sphere:
$$
SO(3)/SO(2)\cong S^2
$$
How does one establish the isomorph... | Analogue Multiplexer | 241,272 | <p>$\bullet \space \mathbf{SO(3) / SO(2) \simeq S^2}:$</p>
<p>Consider a fundamental representation of the Lie group $G := SO(3)$. Any element $M$ of $G$ can be written as a linear map $M : \mathbb{R}^3 \rightarrow \mathbb{R}^3$ such that $M^{-1} = M^T$ and $\det(M) = 1$. We can easily restrict to $M : S^2 \rightarrow... |
1,260,260 | <blockquote>
<p>Find, with proof, the smallest value of $N$ such that $$x^N \ge \ln x$$ for all $0 < x < \infty$. </p>
</blockquote>
<p>I thought of adding the natural logarithm to both sides and taking derivative. This gave me $N \ge \frac 1{\ln x}$. However, is there a better way to this?</p>
<p>Please note... | shalop | 224,467 | <p>Hint for one possible approach:</p>
<p>Consider the function $f:(0,\infty) \to \mathbb{R}$ defined by $f(x) = \frac{\log x}{x^N}$. Take the derivative, and you find that $f$ attains a global maximum value at $x=e^{\frac{1}{N}}$.</p>
<p>Now solve the following equation for $N$: $$f(e^{\frac{1}{N}})=1$$ Solving this... |
163,672 | <p>Is there a characterization of boolean functions $f:\{-1,1\}^n \longrightarrow \{-1,1\}$,
so that $\mathbf{Inf_i}[f]=\frac{1} {2}$, for all $1\leq i\leq n$? Is it known how many such functions there are? </p>
| karpasi | 35,660 | <p>Apparently such functions have been studied before, in cryptography. This condition is called the Strict Avalanche Criterion, and a guy called Daniel K. Biss proved a lower bound of $2^{2^{n-o(1)}}$.</p>
<p><a href="http://ac.els-cdn.com/S0012365X97001805/1-s2.0-S0012365X97001805-main.pdf?_tid=5d5aa18e-fd17-11e3-91... |
3,975,895 | <p>Let <span class="math-container">$a,b,c\in\mathbb{Z}$</span>, <span class="math-container">$1<a<10$</span>, <span class="math-container">$c$</span> is a prime number and <span class="math-container">$f(x)=ax^2+bx+c$</span>. If <span class="math-container">$f(f(1))=f(f(2))=f(f(3))$</span>, find <span class="mat... | Thehx | 469,873 | <p>Okay here's the outline of how you solve this.</p>
<p>first, you write down the system of equations
<span class="math-container">$$
0=f(f(2))-f(f(1))=(3a+b)*(b+5a^2+3ab+2ac) \\
0=f(f(3))-f(f(2))=(5a+b)*(b+13a^2+5ab+2ac)
$$</span>
and this leaves you with four possibilities:</p>
<p>(a) <span class="math-container">$... |
787,894 | <p>Find the values of $x,y$ for which $x^2 + y^2$ takes the minimum value where $(x+5)^2 +(y-12)^2 =14$.</p>
<p>Tried Cauchy-Schwarz and AM - GM , unable to do.</p>
| Macavity | 58,320 | <p>Another way is triangle inequality (essentially think of the triangle between the origin, the centre of the circle and any point on the circle):</p>
<p>$$\sqrt{x^2+y^2} +\sqrt{(x+5)^2+(y-12)^2} \ge \sqrt{(-5)^2+(12)^2} \implies x^2+y^2 \ge \left(13 - \sqrt{14}\right)^2$$</p>
|
3,844,235 | <p>Suppose a matrix <span class="math-container">$A \in \text{Mat}_{2\times 2}(\mathbb{F}_5)$</span> has characteristic polynomial <span class="math-container">$x^2 - x +1$</span>. Is <span class="math-container">$A$</span> diagonalizable over <span class="math-container">$\mathbb{F}_5$</span>?</p>
<p>Normally, I would... | Misha Lavrov | 383,078 | <p>If your matrix were diagonalizable, then there would be a diagonal matrix <span class="math-container">$$\Lambda = \begin{bmatrix}\lambda_1 & 0 \\ 0 & \lambda_2\end{bmatrix} \in \text{Mat}_{2 \times 2}(\mathbb F_5)$$</span> with the same characteristic polynomial. But then, <span class="math-container">$\Lam... |
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>
| rschwieb | 29,335 | <p>$F[x]/(x^3)$ consists of units and nilpotent elements, but has four ideals, so this suggests you meant something more like <em>unique prime ideal</em>.</p>
<p>This is indeed true for commutative rings. The hypothesis that nonunits are nilpotent means that the nilradical is a maximal ideal. But considering that all ... |
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>
| Truong | 100,751 | <p><a href="http://am-solutions.wikispaces.com/Solutions+to+Chapter+1" rel="noreferrer">http://am-solutions.wikispaces.com/Solutions+to+Chapter+1</a></p>
<p>"Let $A$ be a ring, $R$ its nilradical. Show that the following are equivalent:</p>
<p>1) $A$ has exactly one prime ideal;</p>
<p>2) every element of $A$ is eit... |
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 ... | Lutz Lehmann | 115,115 | <p>You are correct, as it is not a linear ODE, it is neither homogeneous nor inhomogeneous.</p>
<p>The cited characterization is most likely based on the fact that <span class="math-container">$y=0$</span> is a solution, but that is only a necessary condition for linearity, not a sufficient one.</p>
|
947,191 | <p>Show that $\sum _{n=1 } ^{\infty } (n \pi + \pi/2)^{-1 } $ diverges.</p>
<p>Both the root test and the ratio test is inconclusive. Can you suggest a series for the series comparison test?</p>
<p>Thanks in advance!</p>
| Community | -1 | <p>$$\frac{1}{n\pi +\frac{\pi}{2}} \geq \frac{1}{n\pi +n\pi}\geq \frac{1}{8n} =\frac{1}{8}\cdot\frac{1}{n}$$</p>
|
653,319 | <p>I understand that $\lim_{\theta\to0}(\sin(θ)/θ) = 1$ but what is $x$ when,
$\lim_{\theta\to0}(\tan(θ)/θ) = x$ where $x$ is a real constant value. </p>
<p>Please help me, I will be eternally great full :D</p>
| amWhy | 9,003 | <p>We have that $\dfrac{\tan\theta}{\theta}=\dfrac{1}{\cos\theta}\cdot \dfrac {\sin\theta}{\theta}.$ </p>
<p>Then recall that the limit of a product is equal to the product of the limits (when those limits do in fact exist.) </p>
<p>$$\lim_{\theta\to 0} \dfrac{\tan\theta}{\theta}= \lim_{\theta\to 0} \dfrac{\sin\theta... |
653,319 | <p>I understand that $\lim_{\theta\to0}(\sin(θ)/θ) = 1$ but what is $x$ when,
$\lim_{\theta\to0}(\tan(θ)/θ) = x$ where $x$ is a real constant value. </p>
<p>Please help me, I will be eternally great full :D</p>
| Billie | 48,863 | <p>I'll use $x$ instead of $\theta$.</p>
<p>Use the identity:</p>
<p>$$\tan(x) = \frac{\sin x}{\cos x}$$</p>
<p>By limit rules,</p>
<p>$$\lim_{x \ \to 0} \frac{f(x)}{g(x)} = \frac{\lim_{x \ \to 0} f(x)}{\lim_{x \ \to 0} g(x)}$$</p>
<p>Thus:</p>
<p>$$\lim_{x \to 0} \sin x = 0.$$
$$\lim_{x \to 0} \cos x = 1.$$</p>
... |
2,791,863 | <p>We need to calculating the limit
$$
\lim _{n\rightarrow \infty}((4^n+3)^{1/n}-(3^n+4)^{1/n})^{n3^n}
$$</p>
<p>I have tried taking the logarithm, but the limit doesnt seem to arrive at any familiar form.</p>
| user | 505,767 | <p>Following the hint by <a href="https://math.stackexchange.com/a/2791877/505767">Alex</a> with some more detail, we have that</p>
<ul>
<li>$(4^n+3)^{1/n}=4\left(1+\frac3{4^n}\right)^\frac1n=4e^{\frac1n\log \left(1+\frac3{4^n}\right)}=4\left(1+\frac3{n4^n}+o\left(\frac1{n4^n}\right)\right)=\\=4+\frac{12}{n4^n}+o\left... |
239,202 | <p>Let $\Gamma$ be a $C^2$ compact submanifold of $\mathbb{R}^n$. Consider the distance function $\delta$ from $\Gamma$. It is well known that, for sufficiently small $\varepsilon>0$, $\delta$ is $C^2$ on $\{ 0<\delta < \varepsilon\}$, and that it satisfies the eikonal equation</p>
<p>$$ \| \nabla \delta \| =... | Anton Petrunin | 1,441 | <p>If $u$ is smooth, the left-hand-side in $(\star\star)$ could be rewritten as $(\nabla u)(\Delta u)$. Assuming that $u$ is $C^2$ this expression $(\nabla u)(\Delta u)$ is always defined, while your original expression $\nabla u \cdot \nabla \Delta u$ might be undefined.</p>
<p>Indeed assume $\gamma$ is a unit-speed ... |
760,032 | <p>Consider the integral
\begin{equation}
I(x)= \frac{1}{\pi} \int^{\pi}_{0} \sin(x\sin t) \,dt
\end{equation}
show that
\begin{equation}
I(x)= \frac{2x}{\pi} +O(x^{3})
\end{equation}
as $x\rightarrow0$.</p>
<p>=> I Have used the expansion of McLaurin series of $I(x)$ but did not work.
please help me.</p>
| DeepSea | 101,504 | <p>$sin(x\cdot sint) = x\cdot sint - \dfrac{(x\cdot sint)^3}{3!} + ...$, and integrate term by term should give the answer.</p>
|
1,532,275 | <p>The kernel of a monoid homomorphism $f : M \to M'$ is the submonoid $\{m \in M : f(m)=1\}$. (This should not be confused with the kernel pair, which is often also named the kernel.)</p>
<p><em>Question.</em> Which submonoids $N$ of a given monoid $M$ arise as the kernel of a monoid homomorphism? (If necessary, let ... | Thomas Andrews | 7,933 | <p>Hint: $$\frac{m}{k\cdots (k+m)} = \frac {1}{k\cdots (k+m-1)} - \frac{1}{(k+1)\cdots (k+m)}$$</p>
|
2,879,035 | <p>$f(x) = \int_{1}^{\infty} \frac{2}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2} dx$</p>
<p>find $P(X > 1)$</p>
<p>This is $X$ ~ $Norm(0, 1)$.</p>
<p>$P(X > 1) = 1 - P(X \leq 1) = 1 - 2 \phi(1) = 1-2(1-\phi(-1)) = 1 - 2(1-0.1587) = -0.6826$. </p>
<p>Yikes. Negative number. What am I doing wrong? </p>
| callculus42 | 144,421 | <p>Let´s say the pdf is $$f(x)=\frac{2}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}$$</p>
<p>(without the integral)</p>
<p>Now you want to calculate $P(X> 1)$ which is </p>
<p>$\int_{1}^{\infty} \frac{2}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2} dx$</p>
<p>First we can factor out $2$. </p>
<p>$2\cdot \int_{1}^{\infty} \frac{1}{\sqr... |
4,498,263 | <p>I know that a group action is transitive when there is one orbit. Say that <span class="math-container">$G$</span> is a group acting on the set <span class="math-container">$A$</span>. The identity element of <span class="math-container">$G$</span> will clearly create <span class="math-container">$|A|$</span>-many o... | JBL | 1,080,305 | <p>The right answer to the question is "Have you tried looking at any concrete examples?" I would start with some obvious ones like the symmetric group <span class="math-container">$S_3$</span> acting on <span class="math-container">$\{1, 2, 3\}$</span>, the alternating group <span class="math-container">$A_... |
2,007,224 | <p>Analysis problem:</p>
<p><strong>Let $f$ and $g$ be differentiable on $ \mathbb R$. Suppose that $f(0)=g(0)$ and that $f' (x)$ is less or equal than $g' (x)$ for all $x$ greater or equal than $0$ Show that $f(x)$ is less or equal than$g(x)$ for all $x$ greater or equal than $0$.</strong></p>
<p>Is my proof correct... | Learnmore | 294,365 | <p>Let $h(x)=f(x)-g(x);x\in [0,\infty)$</p>
<p>$h^{'}(x)=f^{'}(x)-g^{'}(x)\le 0\implies h$ is decreasing on $[0,\infty)\implies h(x)\le h(0)\forall x\in [0,\infty)\implies f(x)\le g(x)$</p>
|
29,766 | <p>I'm looking for a news site for Mathematics which particularly covers recently solved mathematical problems together with the unsolved ones. Is there a good site MO users can suggest me or is my only bet just to google for them?</p>
| Willie Wong | 3,948 | <p>arXiv.org</p>
<p>Any paper worth reading <em>should</em> include some background material and a description of general progress in its introduction section. This is especially true of papers that actually <em>solve</em> a problem, rather than chipping away at some small technicality. </p>
|
821,845 | <p>As the title says, why are those two equivalent? I can find a simple derivation (using natural deduction) of $\bot$ from $\neg\neg\bot$, but i fail at proving the other implication.</p>
| dtldarek | 26,306 | <p>The $\bot \to \neg\neg\bot$ implication is a special case of $\forall \alpha.\ \forall \beta.\ \alpha \to (\beta \to \alpha)$.<br>
Below there are both derivations using <a href="http://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence" rel="nofollow">Curry-Howard isomorphism</a>. </p>
<p>Formula
$$((\bot ... |
925,140 | <p>$$f(x)=\frac { x }{ x+4 } $$</p>
<p>I am not sure how to go about solving this but here is what I have done so far:</p>
<p>$$y=\frac { x }{ x+4 } $$</p>
<p>$$(x+4)y=\frac { x }{ x+4 } (x+4)$$</p>
<p>$$yx+4y=x$$</p>
<p>I feel stuck now. Where do I go from here?</p>
| Adi Dani | 12,848 | <p>$$y=\frac { x }{ x+4 },x\neq-4 $$
$$y(x+4)=x$$
$$xy+4y=x$$
$$xy-x=-4y$$
$$x(y-1)=-4y,y\neq1$$
$$x=\frac{-4y}{y-1}$$
$$x=\frac{4y}{1-y}$$</p>
|
47,890 | <p>hey so i'm programming something that finds an angle of a line between 0 and 180 degrees based on two points.... </p>
<p>the equation to find the answer is <code>Angle = sin-1((1/Hypotenuse)*B)</code>where B is the vertical side of the triangle formed and the hypotenuse is the distance between point 1 and 2. </p>
... | Arturo Magidin | 742 | <p>If $B$ is the length of the opposite side, and $H$ is the length of the hypothenuse, then $B/H$ is the sine of the angle. This is <em>not</em> measured in either degrees or radians; it's the value of the sine.</p>
<p>If you take $\arcsin(B/H)$, this will be given in radians. To convert to degrees, you multiply be ... |
3,578,191 | <p>Without tables or a calculator, find the value of <span class="math-container">$\displaystyle\frac{(\sqrt5 +2)^6 - (\sqrt5 - 2)^6}{8\sqrt5}$</span>.</p>
<p>I do not understand how the positive/negative signs are obtained as shown in the book; is there a formula for expanding these kind of things (what kind of expre... | J. W. Tanner | 615,567 | <p>The book solution used the formulas for the sum and difference of two cubes, </p>
<p><span class="math-container">$x^3+y^3=(x+y)(x^2-xy+y^2)$</span> and <span class="math-container">$x^3-y^3=(x-y)(x^2+xy+y^2),$</span></p>
<p>with <span class="math-container">$x=\sqrt5+2$</span> and <span class="math-container">$y=... |
3,691,692 | <p>Find all real values of a such that <span class="math-container">$x^2+(a+i)x-5i=0$</span> has at least one real solution. </p>
<p><span class="math-container">$$x^2+(a+i)x-5i=0$$</span></p>
<p>I have tried two ways of solving this and cannot seem to find a real solution.</p>
<p>First if I just solve for <span cla... | Barry Cipra | 86,747 | <p>Taking a different approach entirely, note that</p>
<p><span class="math-container">$$\left({b\over a}+{d\over c}\right)\left({a\over b}+{c\over d}\right)=1+{ad\over bc}+{bc\over ad}+1$$</span></p>
<p>Thus, letting <span class="math-container">$ad/bc=x$</span> and noting that <span class="math-container">$x\gt0$</... |
4,506,151 | <p>Determine all functions <span class="math-container">$f:\mathbb{R} \to \mathbb{R}$</span> such that <span class="math-container">$$f(x f(x+y))+f(f(y) f(x+y))=(x+y)^{2}, \forall x,y \in \mathbb{R} \tag1)$$</span></p>
<p>My approach:
Let <span class="math-container">$x=0$</span>, we get
<span class="math-container">$$... | RDK | 979,028 | <blockquote>
<p><span class="math-container">$f: \Bbb{R \to R}, f(xf(x+y))+f(f(y)f(x+y))=(x+y)^2.$</span></p>
</blockquote>
<p>My attempt was to show that <span class="math-container">$f(0)=0$</span>.. And I did it.</p>
<p><span class="math-container">\begin{align}
P(0, y): \; & f(0)+f(f(y)^2)=y^2. \\
P(0, 0): \; &... |
2,256,534 | <p>As I just started learning the different rules of differentiation, I have some burning question marks in my head as such in the picture . I'm required to differentiate the following with respect to $x$.</p>
<blockquote>
<p><strong>1)</strong>
$$\frac{2x^2+4x}{x}$$
<strong>2)</strong>
$$\frac{(1-x)(x-2)}{x}$... | projectilemotion | 323,432 | <p>I think this is what you mean by "I was told not to change the question".</p>
<p>This is to answer question <strong>2)</strong>. I assumed that you know how to do <strong>1)</strong>. Please correct me if I am wrong.</p>
<hr>
<p>Since you cannot expand the $(1-x)(x-2)$, start by applying the <a href="https://en.w... |
3,743,743 | <p>I have this condition:</p>
<p><strong>(A is true OR B is true OR C is true) OR (A is false AND B is false AND C is false)</strong></p>
<p><em>(edit: It's been pointed out that this formula is wrong for what I want)</em></p>
<p>So as the title says, I want the condition to be true if only 1 of A, B or C is true, or i... | mathreadler | 213,607 | <p>To invert any matrix <span class="math-container">$\bf X$</span> you can reformulate it into</p>
<p>"Find the matrix which if multiplied by <span class="math-container">$\bf X$</span> gets closest to <span class="math-container">$\bf I$</span>".</p>
<p>To solve that problem, we can set up the following equ... |
1,652,758 | <p>the question (not homework) I am trying to answer is, in part:</p>
<blockquote>
<p><em>Let $f$ be an analytic function that maps the open unit disk $D$ into
itself and vanishes at the origin. Prove that the inequality $$|f(z)| + |f(−z)| ≤ 2 |z^2| $$ is strict, except at the origin, unless f has the
form $f(z... | Memo Flota | 928,734 | <p>You alredy have that <span class="math-container">$f(z) +f(-z) = 2\lambda z^2$</span>. Let <span class="math-container">$h:D \longrightarrow \mathbb{C}$</span> be a function defined by <span class="math-container">$h(z) = \lambda z^2 -f(-z)$</span>. Hence, <span class="math-container">$h$</span> is analytic and <spa... |
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>
| Mathmo123 | 154,802 | <p>Let $\alpha=\sqrt2+\sqrt[3]2$. It's clear that $\mathbb Q(\alpha)$ is a subextension of $\mathbb Q(\sqrt2,\sqrt[3]2)$. All that remains is to show that $\mathbb Q(\alpha)$ has degree $6$ over $\mathbb Q$.</p>
<p>You could do this by explicitly calculating the minimal polynomial of $\alpha$ over $\mathbb Q$, or by o... |
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 Israel | 8,508 | <p>Hint: the transpose of a product is the product of transposes in reverse order.</p>
|
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>$$ \frac{1}{log_2{x}} + \frac... | Asimov | 137,446 | <p>The way to start all of these and turn them into simple algebra is that $\log_ab=\frac{\log_x b}{\log_x a}$ Using that formula, all of these become basic algebra. Give it a try and comment what you get.</p>
|
1,329,078 | <p>I am having problems in classifying the differential equation $y''=y(x^2)$ in categories like homogeneous, exact, bernoulli, separable and non-exact so I could have the general solution. </p>
<p>Or would someone help me find the solution </p>
| user2661923 | 464,411 | <p><strong>Tools</strong><br></p>
<ul>
<li><p><a href="https://en.wikipedia.org/wiki/Binomial_theorem" rel="nofollow noreferrer">Binomial Theorem</a> <br>
For <span class="math-container">$\displaystyle k \in \Bbb{Z_{\geq 0}} ~: ~2^k = (1 + 1)^k = \sum_{r=0}^k \binom{k}{r}$</span>.</p>
</li>
<li><p><a href="https://en... |
2,079,822 | <p>I am asked to find the maximum velocity of a mass. </p>
<p>I know that the equation for maximum acceleration is </p>
<p>$$a = w^2A$$</p>
<p>However I do not know how to find the maximum velocity. Is velocity just the same as acceleration? </p>
| Noah Schweber | 28,111 | <p>Velocity is not the same as acceleration. Acceleration is a measure of how your velocity changes over time: speed up, and acceleration is positive, etc. This is similar to how velocity measures how your <em>position</em> changes over time. Indeed, velocity is the derivative of position, and acceleration is the deriv... |
2,079,822 | <p>I am asked to find the maximum velocity of a mass. </p>
<p>I know that the equation for maximum acceleration is </p>
<p>$$a = w^2A$$</p>
<p>However I do not know how to find the maximum velocity. Is velocity just the same as acceleration? </p>
| mvw | 86,776 | <p>If the position is given by $x = x(t)$ then the velocity is $v = \dot{x}$ and the acceleration $a = \dot{v} = \ddot{x}$. The dot is the Newton-style notation for the derivative regarding time. </p>
|
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... | Community | -1 | <p>Your textbook <em>surely</em> states what they mean by the phrase.</p>
<p>But to take a guess, they're probably referring to a "residue number system", where you represent not-too-large integers as a sequence of residue classes modulo a set of moduli (usually primes).</p>
<p>(to do the reverse conversion, from the... |
3,648,485 | <p>I want to determine all the points where <span class="math-container">$g(x) = |\sin(2x)|$</span> is differentiable. </p>
<p>A function is differentiable at a point if the left and right limits exist and are equal.</p>
<p>So it follows that <span class="math-container">$g(x)$</span> is differentiable for all <span ... | Robert Israel | 8,508 | <p>It will depend on the starting point.<br>
If <span class="math-container">$x_1 > x_0$</span>, then you can show by induction that <span class="math-container">$x_{n+1} > x_n$</span>. But in that case it won't converge.</p>
|
3,648,485 | <p>I want to determine all the points where <span class="math-container">$g(x) = |\sin(2x)|$</span> is differentiable. </p>
<p>A function is differentiable at a point if the left and right limits exist and are equal.</p>
<p>So it follows that <span class="math-container">$g(x)$</span> is differentiable for all <span ... | Aman Pandey | 469,000 | <p>First send complete question.
What is the initial value <span class="math-container">$x_0$</span>?
Put <span class="math-container">$x_{n+1}=x_n=A$</span>.Then solve the equation. Which point is convergent point depend on the nature of the sequence. This is the method of this sort of problems.</p>
|
2,435 | <p>I'm not sure we already have something similar, but I'm working on more code inspections for the IntelliJ plugin and it's always a good idea to ask the community. Since it doesn't really fit on main, I'm posting it here on Meta.</p>
<p>Linting is an excellent way to point the developer to probable errors that he mi... | b3m2a1 | 38,205 | <blockquote>
<h1>Status Completed</h1>
</blockquote>
<p>I often accidentally include a <code>;</code> at the end of a line in the variable declaration for <code>Module/Block/DynamicModule/etc</code> or screw them up somehow. It'd be nice if IntelliJ could catch fundamental, silly errors like that.</p>
<p>E.g. I do ... |
3,977,687 | <p>A coin of radius 1 cm is tossed onto a plane surface that has been tessellated by right triangles whose sides are 8 cm, 15 cm, and 17 cm long. Find the probability that the coin lands within a triangle.</p>
<p>I know that this has to do with similarity because the inner triangle that is formed by the area where the ... | cr001 | 254,175 | <p><a href="https://i.stack.imgur.com/zJiPK.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/zJiPK.png" alt="enter image description here" /></a></p>
<p>First since <span class="math-container">$I$</span> is the incenter of <span class="math-container">$\triangle ABC$</span>, you can find <span class=... |
2,300,613 | <p>I tried to calculate few derivatives, but I cant get $f^{(n)}(z)$ from them. Any other way? </p>
<p>$$f(z)=\frac{e^z}{1-z}\text{ at }z_0=0$$</p>
| sharding4 | 254,075 | <p>Standard fact the coefficient for the series $\frac{f(z)}{1-z}$ is $\sum_{n=0}^{\infty}(\sum_{k=0}^{n}a_k)x^n$ where $f(z)$ has the expansion $\sum_{n=0}^{\infty}a_nx^n$</p>
|
598,962 | <p>I have to determine the following:</p>
<p>$$\lim_{x \rightarrow 0}\frac{9}{x}\left(\frac{3}{(x+3)^3}-\frac{1}{9}\right)$$</p>
<p>I've got so far:</p>
<p>$$\lim_{x \rightarrow 0}\frac{9}{x}\left(\frac{3}{(x+3)^3}-\frac{1}{9}\right)= \lim_{x \rightarrow 0}\left(\frac{27}{x(x+3)^3}-\frac{1}{x}\right)=\lim_{x \righta... | Shahar | 114,474 | <p>When you plug in $0$ to $x$, you see that the answer is $0/0$. You have to use L'Hospital's Rule, which says</p>
<p>$$\lim_{x\to a} \frac{f(x)}{g(x)} = \lim_{x\to a} \frac{f'(x)}{g'(x)}.$$</p>
<p>This applies only to $0/0$ or $\infty/\infty$.</p>
<p>Hence, you just need to take the derivative of the top and botto... |
4,385,908 | <p>For an ideal <span class="math-container">$I$</span> in <span class="math-container">$A = \mathbb{C}[x, y, z]$</span> set <span class="math-container">$$Z_{xy}(I) = \{(a, b) \in \mathbb{C}^2: f(a, b, z) = 0\text{ for all }f \in I\}.$$</span></p>
<p>Let
<span class="math-container">$$J = \{f(x, y): f(a, b) = 0\text{ ... | clay | 157,069 | <p>Conditional expectation is defined with the property such that for any <span class="math-container">$A \in \mathcal{F}_n$</span>, <span class="math-container">$\int_A X_n \, dP = \int_A Y \, dP$</span>.</p>
<p>Consider sets of the form <span class="math-container">$A = \{k \} \in \mathcal{F}_n$</span> which exist fo... |
91,739 | <p>I have 2 groups: </p>
<ul>
<li>general linear $ k \times k $ with $\cdot$</li>
<li>top-triangle matrix $ n \times n $ with 1 on main diagonal. Operation is $\cdot$ too</li>
</ul>
<p>Is there isomorphism for any any non-trivial $n,k$ i.e $n \neq 2 \ or \ k \neq 1$ over $\mathbb{R}$ or $\mathbb{Q}$?</p>
<p>If no... | tungprime | 20,973 | <p>Another reason is that if the field is of characteristics $0$ then all elements (except the identity matrix) in the set upper triangle matrix with 1 on the main diagonal do not have finite order. However, there are lots of matrix in $ GL_k(F)$ has finite order. For instance, those have $-1$ or $1$ on the main diagon... |
434,290 | <p>According to the <a href="http://arxiv.org/abs/0910.5922" rel="nofollow">equation 4</a>,
$$\phi(0,t)= \frac{A_0}{(1+\frac{2t^2}{R^4})^{3/4}}\cos \left(\sqrt2 t+ \frac{3}{2}\tan^{-1}\left[\frac{\sqrt2 t}{R^2}\right]\right)\tag{1}$$
what conditions makes, $$\cos \left(\sqrt2 t+ \frac{3}{2}\tan^{-1}\left[\frac{\sqrt2 ... | Start wearing purple | 73,025 | <p>One possible way is to introduce
$$ I(s)=\frac{1}{16}\int_0^{\infty}\frac{y^{s-\frac34}dy}{1+y}.\tag{1}$$
The integral you are looking for is obtained as $I'(0)$ after the change of variables $y=x^4$.</p>
<p>Let us make in (1) another change of variables: $\displaystyle t=\frac{y}{1+y}\Longleftrightarrow y=\frac{t}... |
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