qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,077,248 | <p>Prove:
For every positive integer $n, n^2 + 4n + 3$ is not a prime.</p>
<p>I tried to disprove the statement, which I could not using several number examples with constructive proof.</p>
<p>However I am not sure how to correctly step by step prove it.</p>
<p>Thank you in advance!</p>
| Edward Jiang | 179,276 | <p>Hint:
$$n^2+ 4n+3=(n+1)(n+3)$$</p>
|
2,540,954 | <p>Let $T$ be a (countable) partition of $X$ and let $\sigma(T)$ be the generated $\sigma$-algebra of our interest.</p>
<p>I'm trying to figure out whether $\sigma(T)$ is a complete lattice?</p>
<p><strong>Def.</strong> a <em>complete lattice</em> is a partially ordered set in which all subsets have both a supremum (... | drhab | 75,923 | <p>Since $T$ is a countable partition $\sigma(T)$ will have as elements exactly the sets that can be written as a union of elements of $T$. </p>
<p>You can prescribe $F:\wp(T)\to\sigma(T)$ by $S\mapsto\cup S$.</p>
<p>(Here $\cup S:=\bigcup_{R\in S}R$)</p>
<p>It is not really difficult to verify that $F$ is bijective... |
90,864 | <p>I'm being asked to integrate $f(z) = \frac{e^z}{z^2 + 2z + 1}$ around a $5 \times 5$ square, centered at $i$, in the counter clockwise direction. It seems to me that applying Cauchy's integral formula for the first derivative directly yields an answer, but I am concerned the approach is incorrect due to a double pol... | Sasha | 11,069 | <p><a href="http://en.wikipedia.org/wiki/Cauchy%27s_integral_formula" rel="nofollow">Cauchy's differentiation formula</a> states:
$$
f^{(n)}(a) = \frac{n!}{2 \pi i} \oint_\gamma \frac{f(z)}{(z-a)^{n+1}} \mathrm{d} z
$$
In the case at hand, $a=-1$, $n=1$, $f(z) = \mathrm{e}^z$. Hence
$$
\oint_\gamma \frac{\mat... |
90,864 | <p>I'm being asked to integrate $f(z) = \frac{e^z}{z^2 + 2z + 1}$ around a $5 \times 5$ square, centered at $i$, in the counter clockwise direction. It seems to me that applying Cauchy's integral formula for the first derivative directly yields an answer, but I am concerned the approach is incorrect due to a double pol... | yoyo | 6,925 | <p>$$\oint f(z)dz=e^{-1}\oint \sum_{n=0}^{\infty}\frac{(z+1)^{n-2}}{n!}=2\pi ie^{-1}$$</p>
|
103,925 | <p>One can form a polygon of $4 n$ sides by intersecting $n$ congruent squares (treated as closed sets, i.e., filled squares):
<br />
<img src="https://i.stack.imgur.com/TQAVo.jpg" alt="Three Squares"><br /></p>
<blockquote>
<p><b>Q1</b>. For which of the $k=3,4,\ldots... | Christian Rau | 10,932 | <p>The reason your formula for $\mathbf{b}$ (shouldn't it be $\mathbf{n}$, anyway?) doesn't work is, that your curve is not parameterized by arc-length. The formula</p>
<p>$\mathbf{n}(s)=\frac{\mathbf{x}\verb|"|(s)}{\|\mathbf{x}\verb|"|(s)\|}$</p>
<p>only holds for an arc-length parameterization (notice the... |
847,266 | <p>Assume a pdf $f(x)$ is continuous along $-\infty$ to $+\infty$.
Does this assumption guarantee that $f(+\infty)=f(-\infty)=0$? How to prove?
Thanks in advance.</p>
| Did | 6,179 | <p>Here is a setting worth keeping in mind when looking for examples. Consider some sequences $(\mu_n)$, $(\sigma^2_n)$ and $(p_n)$ with $\sigma^2_n\gt0$, $p_n\gt0$, and $\sum\limits_np_n=1$, and the function
$$
f=\sum_np_n\,g_{\mu_n,\sigma^2_n},
$$
where, for every $(\mu,\sigma^2)$, $g_{\mu,\sigma^2}$ is the gaussian ... |
1,900,333 | <p>If $\frac ab$ rounded to the nearest trillionth is $0.008012018027$, where $a$ and $b$ are positive integers, what is the smallest possible value of $a+b$?</p>
<p>I don't see any strategies here for solving this problem, any help? Thanks in advance!</p>
| Doug M | 317,162 | <p>$\frac 1{0.008012018027} = 124.8125000006\\
0.8125 = \frac {13}{16}\\
124.8125 = \frac {1997}{16}\\
a+b = 1997+16=2013$</p>
|
2,536,791 | <p>I am taking a basic complex analysis course and I'm trying to understand the differences between different forms of convergence.</p>
<p>Specifically, I am trying to distinguish normal convergence from pointwise convergence. I searched around for a similar question, but I was only able to find a comparison between n... | Robert Lewis | 67,071 | <p>Take another look at</p>
<p>$\displaystyle\lim_{p\to \infty} \int_0^pdx\;\beta e^{-\beta x}; \tag 1$</p>
<p>Note that</p>
<p>$\displaystyle \lim_{p\to \infty} \int_0^pdx\;\beta e^{-\beta x} = \lim_{p\to \infty} \beta \int_0^pdx\; e^{-\beta x} = \beta \lim_{p\to \infty} \int_0^pdx\; e^{-\beta x}, \tag 2$</p>
<p>w... |
217,863 | <p>Find the congruence of $4^{578} \pmod 7$.</p>
<p>Can anyone calculate the congruence without using computer?</p>
<p>Thank you!</p>
| Jill Clover | 43,252 | <p>Using <strong>Fermat's Little theorem</strong>: </p>
<p><em>If p is a prime and a is an integer, then $a^{p-1}\equiv1$ (mod p),
if p does not divide a.</em></p>
<p>$4^{6}\equiv1 (mod 7)$ </p>
<p>Since $4^{578}=(4^{6})^{96}\cdot4^{2}$, we can conclude that$4^{578}\equiv1^{96}\cdot4^{2}(mod 7)$.</p>
<p>Hence$4^{5... |
2,411,812 | <blockquote>
<p>When <span class="math-container">$c$</span> is real and in the interval <span class="math-container">$[-1,1]$</span>, the roots <span class="math-container">$z$</span> of <span class="math-container">$z^2-2cz+1=0$</span> have <span class="math-container">$|z|=1$</span>; when <span class="math-container... | dxiv | 291,201 | <p>Hint: $z=0$ is not a root, so dividing by $z \ne 0$ the equation becomes $z+1/z = 2c\,$. A root $z_1$ has modulus $1$ iff $\bar z_1 = 1 / z_1\,$, which then gives $z_1 + \bar z_1 = 2c\,$. Draw your conclusions on $c$ from here.</p>
|
2,102,326 | <blockquote>
<p>Let $a,n$ be positive integers. Find all $a$ such that for some $n$ the largest power of $2$ dividing $(n+1)(n+2) \cdots (an)$ is greater than $2^{(a-1)n}$.</p>
</blockquote>
<p>Since I thought there were no such $a$, I thought about proving this by contradiction. That is, assume that $2^{(a-1)n+1}$ ... | Doug M | 317,162 | <p>There are $(a-1)n$ factors in the original expression.</p>
<p>$\lfloor \frac {(a-1)n + 1}{2}\rfloor$ are divisible by $2.$
$\lfloor\frac {(a-1)n + 1}{4}\rfloor$ are divisible by $4$.
$\lfloor\frac {(a-1)n + 1}{2^i}\rfloor$ are divisible by $2^i$</p>
<p>The largest power of $2$ that divides $(n+1)\cdots(an) = 2^{\l... |
2,154,513 | <p>Suppose that $U$ is an open set containing $0.$ $f,g:U\to\mathbb{R}$ are continuos functions such that</p>
<ul>
<li>$g\in C^1(U)$</li>
<li>$g(0)=0$</li>
<li>$f|_{U\setminus 0}\in C^1(U\setminus 0)$ ($f$ is of $C^1$ class away of $0$)</li>
</ul>
<blockquote>
<p><strong>Question</strong>. Does product $fg$ is of $... | Fred | 380,717 | <p>No. Let $g(x)=x$ and $f(x)=x \sin(1/x)$ for $x \ne 0$ and $f(0)=0$</p>
<p>Then $(fg)(x)=x^2 \sin(1/x)$ for $x \ne 0$ and $(fg)(0)=0$.</p>
<p>Show: the derivative of $fg$ is not continuous at $x=0$</p>
|
3,255,737 | <blockquote>
<p>If <span class="math-container">$\phi$</span> is the solution of the integral equation <span class="math-container">$$\phi(x)=1-2x-4x^2+\int_0^x[3+6(x-t)-4(x-t)^2]\phi(t)dt$$</span></p>
<p>Then the value of <span class="math-container">$\phi(\log 2)$</span> is</p>
<p>(a). 2</p>
<p>(b). ... | user97357329 | 630,243 | <p><strong>A solution by Cornel Ioan Valean</strong></p>
<p>If we set <span class="math-container">$a_n=H_n/n$</span> and <span class="math-container">$b_n=\zeta(2)-H_{2n}^{(2)}$</span>, and then apply Abel's summation, we arrive at
<span class="math-container">$$\sum_{n=1}^{\infty}\frac{H_n}{n}\left(\zeta(2)-H_{2n}^{(... |
1,396,324 | <p>Im trying to calculate the eigenvalues and eigenvectors of the following matrix:</p>
<p>$\begin{bmatrix}1 & 1 & 0\\1& 1 & 1\\0 &1 &1\end{bmatrix}$</p>
<p>so far I worked out:</p>
<p>$A-λI=\begin{bmatrix}1-λ & 1 & 0\\1& 1-λ & 1\\0 &1 &1-λ\end{bmatrix}$</p>
<p>I know... | Emilio Novati | 187,568 | <p>Hint:</p>
<p>You can calculate the determinant developing by the first row:
$$
\det
\begin {bmatrix}
1-\lambda&1&0\\
1&1-\lambda&1\\
0&1&1-\lambda
\end{bmatrix}=
(1-\lambda)\left(\det\begin {bmatrix}
1-\lambda&1\\
1&1-\lambda
\end{bmatrix}\right)-1\left( \det
\begin {bmatrix}
1&1... |
1,396,324 | <p>Im trying to calculate the eigenvalues and eigenvectors of the following matrix:</p>
<p>$\begin{bmatrix}1 & 1 & 0\\1& 1 & 1\\0 &1 &1\end{bmatrix}$</p>
<p>so far I worked out:</p>
<p>$A-λI=\begin{bmatrix}1-λ & 1 & 0\\1& 1-λ & 1\\0 &1 &1-λ\end{bmatrix}$</p>
<p>I know... | Flo Ryan | 239,723 | <p>You need to solve:</p>
<p>$\det\left(A-λI\right)=\det \left(\begin{bmatrix}1-λ & 1 & 0\\1& 1-λ & 1\\0 &1 &1-λ\end{bmatrix}\right) \overset{!}{=} 0$</p>
<p>In order to calculate the determinant, I suggest you use Sarrus's rule <a href="https://en.wikipedia.org/wiki/Rule_of_Sarrus" rel="nofol... |
30,728 | <p>Is this graph in the list among the so-called "standard" structures used in <code>GraphData</code>? However, I have not found yet anything like "Carpet" or "Sponge" in the list of the objects that can be built. Maybe, this graph has a different name? </p>
<p>For me, using <code>GraphData</code> helps to save time f... | Szabolcs | 12 | <p>As of version 11, this is built in:</p>
<pre><code>GraphData["SierpinskiCarpet"]
(* {{"Cycle", 8}, {"SierpinskiCarpet", 2}, {"SierpinskiCarpet", 3}, {"SierpinskiCarpet", 4}} *)
GraphData /@ %
</code></pre>
<p><img src="https://i.stack.imgur.com/ouqgo.png" alt="Mathematica graphics"></p>
<hr>
<p>The latest versi... |
114,447 | <p>Is there a relation between the idempotent elements of a ring $R$ and those of $M_{n}(R)$ - the ring of $n \times n$ matrices over $R$?</p>
| Ray Hoobler | 20,674 | <p>Yes. You can see the relationship easily in the following way. Suppose the ring $R=R_1\times R_2$ so that there are two primitive idempotents $e_1$ and $e_2$ with $1=e_1 +e_2$. Next note that $M_n(R)\approxeq Hom_R(R^n,R^n)$. Then $M_n(R)\approxeq Hom_{R_1\times R_2}(\(R_1\times R_2\)^n,\(R_1\times R_2\)^n)\approxeq... |
114,447 | <p>Is there a relation between the idempotent elements of a ring $R$ and those of $M_{n}(R)$ - the ring of $n \times n$ matrices over $R$?</p>
| Chris Heunen | 10,368 | <p>Any idempotent $e$ of $R$ induces an idempotent $E=\mathop{diag}(e,\ldots,e)$ of $M_n(R)$. In fact, if $e_i$ are idempotents in $R$, then $E=\mathop{diag}(e_1,\ldots,e_n)$ is an idempotent of $M_n(R)$.</p>
<p>Conversely, if $R$ is nice enough, an idempotent $E$ in $M_n(R)$ can be diagonalized to $E=U^{-1} \cdot \ma... |
2,337,117 | <p>I'm trying to find the mean out of a cumulative density function (cdf). I found <a href="https://math.stackexchange.com/questions/2154001/finding-the-mean-of-a-cdf">this question</a> but it was no use because I didn't cover the explanation I was expecting.
Here's my function:
$$
F(x) =
\begin{cases}
0, & \text{... | Doug M | 317,162 | <p>$F(x) = \int_{0}^{x} f(x) dx\\
\mu = \int_{0}^{6} xf(x) dx$</p>
<p>Integration by parts.</p>
<p>$\mu = xF(x)|_0^6 - \int_{0}^{6} F(x) dx\\
\mu = 6 - \int_{-\infty}^{\infty} F(x) dx\\
\mu = 6 - \int_{0}^{\frac 12} x^2 dx - \int_{\frac 12}^{3} \frac 14 dx- \int_{3}^{6} \frac {x-2}{4} dx$</p>
<p>However you could al... |
4,373,464 | <p>Let a and b be a positive integers. Proof that if number <span class="math-container">$ 100ab -1 $</span> divides number <span class="math-container">$ (100a^2 -1)^2 $</span> then also divides number <span class="math-container">$ (100b^2 -1)^2 $</span>.</p>
<p>My attempt:</p>
<p>Let's notice that <span class="math-... | Community | -1 | <p>For a modular arithmetic argument:</p>
<p>Let <span class="math-container">$N=100ab-1$</span> and work modulo <span class="math-container">$N$</span>.</p>
<p>Starting with <span class="math-container">$(100a^2 -1)^2\equiv 0$</span>, multiply by <span class="math-container">$10^4b^4$</span>:
<span class="math-contai... |
216,815 | <p>I have the following string dataset:</p>
<pre><code>{{22/03 updating, 55.218 (+1.640), 44.321 (+1.640), 4.825 (+0), 6.072 (+0), details},
{21/03, 53.578 (+6.557), 42.681 (+4.821), 4.825 (+793), 6.072 (+943), details},
{20/03, 47.021 (+5.986), 37.860 (+4.670), 4.032 (+627), 5.129 (+689), details},
{19/03, 41.035 (... | Akku14 | 34,287 | <pre><code>Cases[sol, {_?PrimeQ, _?PrimeQ}]
(* {{2063, 853}, {2069, 857}} *)
</code></pre>
|
344,479 | <p>Suppose that $a = 2^kb,$ where $b$ is odd. If $\phi(x) = a,$ prove that $x$ has at most $k$ odd prime divisors.</p>
| Inceptio | 63,477 | <p><strong>Hint:</strong></p>
<p>Let $x= p_1^{a_1} \times p_2^{a_2} \times \dots p_m^{a_m} $</p>
<p>$\phi (x)=\phi( p_1^{a_1}) \times \phi( p_2^{a_2}) \times \dots \phi( p_m^{a_m})$</p>
<p>$ \phi(x)= \prod _{i=1}^m p_i^{a_i}(1- \dfrac{1}{a_i})$</p>
<p>$\prod _{i=1}^m p_i^{a_i}(1- \dfrac{1}{a_i}) =n \prod _{i=1}^m (... |
4,438,491 | <p><span class="math-container">$$\sum _{k=2}^{\infty }\:\frac{1}{\sqrt{k}\left(\ln k\right)^{\ln k}}$$</span></p>
<p>I've tried limit comparison with <span class="math-container">$\frac{1}{\sqrt{n}}$</span> and <span class="math-container">$\frac{1}{n^2}$</span> and also Cauchy condensation test but nothing seems to w... | José Carlos Santos | 446,262 | <p>For each <span class="math-container">$k\in\Bbb N$</span>,<span class="math-container">$$\frac{2^k}{k\sqrt{2^k}c^k}=\frac1k\left(\frac{\sqrt2}c\right)^k\to\infty,$$</span>since <span class="math-container">$\frac{\sqrt2}c>1$</span>. Therefore, your series diverges.</p>
|
1,578,717 | <p>What is the inverse Laplace transform of $$\frac{e^{\frac{-2}{s}}}{s}$$ I have seen an answer using Maclaurin series expansion of this function. This function is not analytic at $0$, so, is such expansion legal? </p>
| doraemonpaul | 30,938 | <p>$\mathcal{L}^{-1}\left\{\dfrac{e^{\frac{-2}{s}}}{s}\right\}$</p>
<p>$=\mathcal{L}^{-1}\left\{\sum\limits_{n=0}^\infty\dfrac{(-1)^n2^n}{n!s^{n+1}}\right\}$</p>
<p>$=\sum\limits_{n=0}^\infty\dfrac{(-1)^n2^nt^n}{(n!)^2}$</p>
<p>$=J_0(2\sqrt{2t})$</p>
|
3,631,212 | <p>I am trying to show that the functors <span class="math-container">$h^n(X)=\text{Hom}(H_n(X),\Bbb Z)$</span> do not define a cohomology theory on CW complexes. If a contravariant functor <span class="math-container">$h^n(X)$</span> is a cohomology theory, by definition it must satisfy the followings:</p>
<p>(1) If ... | Maxime Ramzi | 408,637 | <p>You have to find an example where <span class="math-container">$H^n(X)$</span> is in fact different from <span class="math-container">$\hom(H_n(X),\mathbb Z)$</span>. The easiest example is, I think, <span class="math-container">$X = \mathbb RP^2$</span>.</p>
<p>You can then take something like <span class="math-co... |
194,813 | <p>Please help me solving $\displaystyle\lim_{x\to a}\frac{a^{a^{x}}-{a^{x^{a}}}}{a^x-x^a}$</p>
| Ross Millikan | 1,827 | <p>For the denominator, you can write $x^a-a^x=x^a-a^a+a^a-a^x$ and use the derivative of each.</p>
|
194,813 | <p>Please help me solving $\displaystyle\lim_{x\to a}\frac{a^{a^{x}}-{a^{x^{a}}}}{a^x-x^a}$</p>
| Did | 6,179 | <p>The ratio is $R(x)=\dfrac{u(t)-u(s)}{t-s}$ with $u(z)=a^z$, $t=a^x$ and $s=x^a$. When $x\to a$, $t\to a^a$ and $s\to a^a$ hence $R(x)\to u'(a^a)$. Since $u(z)=\exp(z\log a)$, $u'(z)=u(z)\log a$. In particular, $u'(a^a)=u(a^a)\log a$. Since $u(a^a)=a^{a^a}$, $\lim\limits_{x\to a}R(x)=a^{a^a}\log a$.</p>
|
252,957 | <p>I used to think naively the construction is straightforward, which is, if we add one layer innermost each time, then we could have one that corresponds to $\omega$ in Neumann's representation, which is. of course, constructible in <strong>ZF</strong> excluding axiom of foundation .</p>
<p>But I was told I can't def... | Community | -1 | <p>The correct construction is to apply <a href="http://en.wikipedia.org/wiki/Aczel%27s_anti-foundation_axiom" rel="nofollow">AF</a> to the pointed
<br>graph with exactly one vertex and exactly one self-loop.</p>
|
1,657,931 | <p>What is the smallest value for the expression, $a^2+b^2+ab-a-2b$? </p>
<p>Please explain this.</p>
| Henry | 6,460 | <p>You might try expanding $$\tfrac14(a+2b-2)^2+\tfrac34 a^2 - 1$$
and then consider why $a=0, b=1$ gives a minimum</p>
|
1,518,804 | <p>In real analysis we showed that if $\displaystyle \lim_{x\to x_0}|f(x)|=|L|$, then not necessarily $\displaystyle \lim_{x\to x_0}f(x)=L$ (the converse is true). </p>
<p>I want to find a counter example in complex analysis, i.e, if $\displaystyle \lim_{z\to z_0}|f(z)|=|L|$, then not necessarily $\displaystyle \lim_{... | Hagen von Eitzen | 39,174 | <p>For simplicity, let $f$ be <em>any</em> continuous (at $z_0$) function with $f(z_0)\ne 0$ and let $L=-f(z_0)$.</p>
|
3,441,225 | <p>Let <span class="math-container">$S=1-1/3+1/5-1/7+\cdots$</span>. As each term in the series is decreasing and tends to <span class="math-container">$0$</span>, it is known that their sum exists and is finite by alternating series test. And by considering <span class="math-container">$\int_0^11/(1+x^2)dx$</span>, it... | Robert Israel | 8,508 | <p>The sum is always <span class="math-container">$\ge$</span> each even-numbered partial sum and <span class="math-container">$\le$</span> each odd-numbered one. So <span class="math-container">$S \ge 1 - 1/3 = 2/3 > 0$</span>. </p>
|
3,056,075 | <p>As far as I understand, according to linear algebra, linear functions, both single and multivariable, can be represented in vector form.</p>
<p>For instance, </p>
<p><span class="math-container">$$z = aw + bx + cy + d$$</span></p>
<p>can be rewritten as</p>
<p><span class="math-container">$$z = \begin{bmatrix}a... | amd | 265,466 | <p>In the same vein as your example, one can write <span class="math-container">$$w^2+x^2+y^2+8=\begin{bmatrix}w&x&y&1\end{bmatrix}\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&8\end{bmatrix}\begin{bmatrix}w\\x\\y\\1\end{bmatrix}.$$</span></p>
|
562,166 | <p>I am trying to prove that Axiom of choice implies well-ordering principle.</p>
<p>Proof: If $A = \emptyset,$ then, take $\alpha = 0$ and' $\alpha \cong $ </p>
<p>So given a nonempty set $A$, $\exists a_0 \in A$. Define by recursion, the following function $F:ON \rightarrow V$ </p>
<p>$$F(\alpha)=\begin{cases}
\al... | Macavity | 58,320 | <p><strong>Hints:</strong></p>
<p>1) If $p(x)$ is an odd polynomial, it has a real root (say $c$) and hence $0 = |p(c )| \le |f(x)|$.</p>
<p>2) If $p(x)$ has even degree, WLOG let the leading coefficient be positive. Then we see that $p(x)$ approaches $\infty$ on both ends of the axis. This implies that if we pick ... |
900,326 | <p>Let $G=(V,E)$ be a connected graph which is not a tree. Prove that if for every cycle $C$ of the graph G and for any $v \in V(G)- V(C)$ there are more than $\frac{|C|}{2}$ edges from $v$ to $V(C)$ then G is Hamiltonian.</p>
<p>My proof:</p>
<p>I will show that every vertex $w \in V$ is part of some cycle.
Proof b... | N. S. | 9,176 | <p>Since the graph is not a tree, it has at least a cycle.</p>
<p>Pick a cycle $C$ with the most number of vertices. I claim, it is a Hamiltonian cycle.</p>
<p>Indeed, assume by contradiction is not Hamiltonian. Let $v$ be a vertex outside the cycle. Use the pigeonhole principle, and the given condition to show that ... |
35,014 | <p>I want to display a rational number in <em>Mathematica</em> in periodic style.
<code>PeriodicForm</code> isn't working anymore. It worked in <em>Mathematica</em> 5 and now I'm using <em>Mathematica</em> 9.</p>
<p>I want to display the number $3.13678989898989898989\ldots$, where the repeating $89$ part should be di... | DavidC | 173 | <p><a href="https://stackoverflow.com/questions/5200617/how-can-i-make-a-working-repeating-decimal-representation-of-a-rational-number">Here</a> you will find a discussion of issues related to the production of repeated decimals.</p>
<p>The code is reproduced below for convenience.</p>
<pre><code>repeatingDecimal[n_I... |
2,416,597 | <blockquote>
<p>Which of the following is the largest?</p>
<p>A. <span class="math-container">$1^{200}$</span></p>
<p>B. <span class="math-container">$2^{400}$</span></p>
<p>C.<span class="math-container">$4^{80}$</span></p>
<p>D. <span class="math-container">$6^{300}$</span></p>
<p>E. <span class="math-container">$10^... | Kenny Lau | 328,173 | <p>$$2^{400} = 1024^{40} < 10000^{40} = 10^{160} < 10^{250}$$</p>
<p>See the other answers for $6^{300}$ and $10^{250}$.</p>
|
4,101,139 | <blockquote>
<p>Calculate the volume bounded by the surface <span class="math-container">$x^n + y^n + z^n = a^n$</span> <span class="math-container">$(x>0,y>0,z>0)$</span>.</p>
</blockquote>
<p><span class="math-container">$$\iiint\limits_{x^n+y^n+z^n \le a^n \\ \ \ \ \ \ \ x,y,z > 0}\mathrm dx~ \mathrm dy ... | TomKern | 908,546 | <p><span class="math-container">$x^n+y^n+z^n$</span> is not the same as <span class="math-container">$r^n$</span>, and I suspect that transforming to spherical or cylindrical coordinates will just make this problem more gross.</p>
<p>These shapes are a specific type of superellipsoid (<a href="https://en.wikipedia.org/... |
3,824,959 | <p>In reading a recent paper, I came across the inequality: <span class="math-container">$e^x - 1 \le e x$</span> for <span class="math-container">$x \in [0, 1]$</span>.</p>
<p>I tried to prove this using (the reverse) Bernoulli's inequality i.e. <span class="math-container">$(1 + y)^r \le 1 + ry$</span>, for <span cla... | Michael Rozenberg | 190,319 | <p>For any <span class="math-container">$x\in[0,1]$</span> we have: <span class="math-container">$$(1+ex-e^x)'=e-e^x\geq0$$</span>
Thus, <span class="math-container">$$1+ex-e^x\geq1+e\cdot0-e^0=0.$$</span></p>
<p>You can use Bernoulli for the following.
<span class="math-container">$$(1+ex)^{\frac{1}{x}}\geq e$$</span>... |
317,531 | <p>I wonder why the polynomial $x^p-x$ has $p$ distinct zeros in $\mathbb Z_p$ for any prime $p$, i.e. $x^p-x=x(x-1)\cdots(x-p+1)$.
Do I need to expand the polynomial in order to get the conclusion?</p>
| Math Gems | 75,092 | <p>Use <a href="http://en.wikipedia.org/wiki/Fermat%27s_little_theorem" rel="nofollow">little Fermat;</a> or <a href="http://en.wikipedia.org/wiki/Freshman%27s_dream" rel="nofollow">Freshman's Dream</a> (a.k.a. <a href="http://en.wikipedia.org/wiki/Frobenius_endomorphism" rel="nofollow">Frobenius map</a>) $\rm\color{#C... |
317,531 | <p>I wonder why the polynomial $x^p-x$ has $p$ distinct zeros in $\mathbb Z_p$ for any prime $p$, i.e. $x^p-x=x(x-1)\cdots(x-p+1)$.
Do I need to expand the polynomial in order to get the conclusion?</p>
| awllower | 6,792 | <p>Since it is tagged <strong>ring theory</strong>, I assume that OP already knows some group theory.<br>
As $p$ is a prime, every number in {$1,\cdot\cdot\cdot, p-1$} is prime to $p$, hence, <a href="http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity" rel="nofollow">by Bézout identity</a>, the set $\mathbb {Z/pZ}=$... |
1,057,295 | <p>Sorry if this has already been asked before. Is there any formula for such questions?</p>
<p><img src="https://i.stack.imgur.com/rhWTj.png" alt="How many triangles in the picture?"></p>
<p>EDIT:</p>
<p>I have numbered the smallest triangles in the picture and marked the pentagon as x.
Then I listed all the possib... | Ayan | 191,307 | <p>No formula. I simply counted them and got 22.</p>
<p>There are 9 individual parts, out of which 8 are triangles themselves.</p>
<ul>
<li>Triangles made by 1 part - 8</li>
<li>Made by 2 parts - 5</li>
<li>Made by 3 parts - 6</li>
<li>Made by 4 parts - 0</li>
<li>Made by 5 parts - 3</li>
</ul>
<p>Total 22.</p>
|
2,223,031 | <p>Can someone explain to me how trig functions work in the complex plane? I'm trying to show that $f(z) = \cos(1-\frac{1}{z})$ has an essential singularity at $z=0$, and part of doing that requires I find a sequence $z_n$ so that when $z = z_n$, $\lim\limits_{z \to 0} f(z)=\infty$. Originally I assumed $\cos(z)$ would... | The_Sympathizer | 11,172 | <p>To explain the deeper answer to the "why" of why it will happen is first off to point out that one <em>cannot</em> and <em>should not</em> expect that the behavior of at least analytic complex functions should mirror that on the real number line.</p>
<p>In particular, one very important theorem of complex analysis ... |
599,394 | <p>A pack contains $n$ card numbered from $1$ to $n$. Two consecutive numbered cards are removed from</p>
<p>the pack and sum of the numbers on the remaining cards is $1224$. If the smaller of the numbers on</p>
<p>the removing cards is $k$, Then $k$ is.</p>
<p>$\bf{My\; Try}::$ Let two consecutive cards be $k$ and ... | John | 7,163 | <p>A couple of observations that may give some hints to you:</p>
<ol>
<li>You can get in the right ballpark by looking for values of $n$ that give you something a bit above $1224$ for your formula for $n$.</li>
<li>Since the sum after the cards are removed is even, the sum before the cards were removed is odd. This m... |
599,394 | <p>A pack contains $n$ card numbered from $1$ to $n$. Two consecutive numbered cards are removed from</p>
<p>the pack and sum of the numbers on the remaining cards is $1224$. If the smaller of the numbers on</p>
<p>the removing cards is $k$, Then $k$ is.</p>
<p>$\bf{My\; Try}::$ Let two consecutive cards be $k$ and ... | Brian M. Scott | 12,042 | <p>You have $n^2+n-4k-2450=0$. Treat this as a quadratic in $n$ with constant term $-4k-2450$. Clearly we need a positive value, so</p>
<p>$$n=\frac{-1+\sqrt{1+16k+9800}}2=\frac{-1+\sqrt{16k+9801}}2\;,$$</p>
<p>$9801=99^2$; $100^2-99^2=199$, which is not a multiple of $16$, but $$101^2-99^2=2\cdot200=400=16\cdot25\;.... |
2,584,968 | <p>I know that if X is locally compact and Hausdorff, then any non-empty open set $S$ contains a non-empty closed set. I know this to be the case because a locally compact space is a regular space, in which the claim holds.</p>
<p>But why does any open $S$ contain a <em>non-empty open set whose closure is compact and ... | Ivo Terek | 118,056 | <p>It is more basic: because every compact subset of an Hausdorff space is closed. Follow your nose: if $K$ is said compact and $X$ is the space, take $p \in X \setminus K$. By Hausdorffness, for each $x \in K$ there are open sets $U_x$ and $V_x$ such that $x \in U_x$, $p \in V_x$ and $U_x \cap V_x = \varnothing$. Then... |
27,794 | <p>Hi.</p>
<p>I want to know if for $f:X\to S$ a proper flat holomorphic map with n-dimensionnal fibers over reduced complex space S, the relative canonical sheaf $w_{X/S}:=H^{-n}(f^{!}O_{S})$ is a dualizing sheaf which imply that the two functor, on Coh(S), $G\to H^{-n}(f^{!}G)$ and $G \to f^{*}G\otimes w_{X/S}$ agre... | kaddar | 6,742 | <p>If the relative canonical sheaf $w_{X/S}:=H^{-n}(f^{!}O_{S})$ is a dualizing sheaf then necessarily the two functors $G-->H^{-n}(f^{!}G)$ and $G-->f^{*}G\otimes w_{X/S}$ agree as well.</p>
<p>Remark that the first functor is left exact and the second is right exact which imply immediately that $w_{X/S}$ is f... |
1,800,821 | <p>Consider an $n \times n$ matrix of the form
$$
A = \begin{bmatrix}
a_1 & a_2 & \ldots & a_{n-1} & a_n \\
1 \\
& 1 \\
& & \ddots \\
& & & 1
\end{bmatrix}
$$
for certain $a_1, \ldots, a_n \in \mathbb{R}$ (and zeroes in all blank spaces). Can this matrix ever (for some $n \in \ma... | Chan Hunt | 218,186 | <p>You can use the following property of a matrix:</p>
<p>Given some n by n matrix, A, the trace of the matrix is equivalent to the sum of its eigenvalues.</p>
<p>The sum of the eigenvalues of the matrix is:</p>
<p>trace (A) = $a_{1}$ + 0(n-1)</p>
<p>= $a_{1}$ = $\lambda$</p>
<p>Using the above theorem, we can th... |
730,130 | <p>I have data which are visualized in this chart:
<img src="https://i.stack.imgur.com/6ekLT.png" alt="enter image description here"></p>
<p>I need to compute slope of increasing / decreasing parts of the curve. I can't use any 2 points because of noise in data. Maybe numerical derivative can help but I don't know how... | Shuzheng | 58,349 | <p>We have </p>
<p>$$ | |\frac {g(x)-g(x_0)} {x-x_0}| - |g^{'}(x_0)||\le | \frac {g(x)-g(x_0)} {x-x_0} - g^{'}(x_0)|\le \frac {|g^{'}(x_0)|} 2$$</p>
<p>by using the triangle inequlity for complex numbers $\mathbb C$.</p>
<p>Thus $$ -\frac {|g(x_0)|} 2 \le|\frac {g(x)-g(x_0)} {x-x_0}| - |g^{'}(x_0)|\le \frac {|g^{'}(... |
658,815 | <p>I'm looking for a good way to remember/understand part of the <a href="https://en.wikipedia.org/wiki/Convergence_of_measures" rel="nofollow noreferrer">Portmanteau theorem</a>. Specifically, let <span class="math-container">$X$</span> be a metric space. The part of the Portmanteau theorem I'm asking about says that ... | grand_chat | 215,011 | <p>Of the two possibilities for <span class="math-container">$\le$</span>:
<span class="math-container">$$
\limsup_n \mu_n(A)\le \mu(A)\qquad\text{vs.}\qquad\liminf_n\mu_n(A)\le\mu(A)
$$</span>
remember that Portmanteau has the <strong>stronger</strong> condition, since <span class="math-container">$\liminf\le\limsup$<... |
1,369,669 | <p>We are given the following:</p>
<p>$$\int \sin(xy)dy$$</p>
<p>We start by assigning anything algebraic into our first variable, $u$. Recalling LIATE (Logarithmic, Inverse-Trig., Algebra, Trig., Exponential) we start with algebra.</p>
<p>If I assign $$u=xy$$ then</p>
<p>$$\int \sin(u)\frac{du}{x}dy$$</p>
<p>Henc... | Mark | 169,199 | <p>You are integrating with respect to $y$, so treat $x$ as a constant. </p>
<p>Your $u$-sub is done incorrectly: if $u=xy$, then applying $d/dy$ to both sides yeilds $(du)/(dy)=x$ or $du/x=dy$, which would yeild</p>
<p>$$\int sin(u)du/x=(\int sin(u)du)/x$$</p>
<p>since $x$ is a constant.</p>
<p>(also, putting doll... |
4,283,075 | <p>I have a vector <span class="math-container">$[d] = [d_1, d_2, ..., d_i]$</span>. All elements of <span class="math-container">$d$</span> are always <span class="math-container">$0$</span> except for one of them which can be either <span class="math-container">$+1$</span> or <span class="math-container">$-1$</span>.... | Sayan Dutta | 943,723 | <p>Define <span class="math-container">$\mathbb N_i=\{1,2,3,\dots ,i\}$</span> and say
<span class="math-container">$$(\forall j\in\Bbb N_i\setminus\{k\}.d_j=0)\land d_k=\pm1$$</span></p>
<p>However, writing what you wrote (in words) is also absolutely fine I guess.</p>
|
4,283,075 | <p>I have a vector <span class="math-container">$[d] = [d_1, d_2, ..., d_i]$</span>. All elements of <span class="math-container">$d$</span> are always <span class="math-container">$0$</span> except for one of them which can be either <span class="math-container">$+1$</span> or <span class="math-container">$-1$</span>.... | JMoravitz | 179,297 | <p>Commonly, we let <span class="math-container">$e_1 = [1,0,0,0,\dots,0], e_2 = [0,1,0,0,\dots,0],\dots, e_k = [0,0,\dots,0,\underbrace{1}_{k\text{'th position}},0,\dots,0]$</span> etc... be the common notation for <a href="https://en.wikipedia.org/wiki/Standard_basis#:%7E:text=In%20mathematics%2C%20the%20standard%20b... |
437,645 | <p>I am trying to gain a better understanding of Schur-Weyl duality specifically applied to symmetric functions. My motivating example is trying to understand the Frobenius character of the multilinear component of the free Lie algebra (Theorem 8.1 in Reutenauer's book on the subject), but my general confusion is more ... | Vladimir Dotsenko | 1,306 | <p>In your context, you want to think of the Schur-Weyl duality as a way to construct representations of <span class="math-container">$GL(V)$</span> out of representations of symmetric groups.</p>
<p>To give a precise answer along these lines that works well for your original motivation: let us denote by <span class="m... |
3,640,171 | <p><strong>If <span class="math-container">$A_{1}$</span>,...,<span class="math-container">$A_{m}$</span> are independent events and P(<span class="math-container">$A_{i}$</span>)=p, where (P=probability measure) for i=1,2,...,m find the probability that an even number of <span class="math-container">$A_{i}$</span> occ... | Sri-Amirthan Theivendran | 302,692 | <p>Let <span class="math-container">$N$</span> be the number of events that occur i.e. <span class="math-container">$N=\sum_{k=1}^m I(A_k)$</span> where <span class="math-container">$I$</span> is the indicator function. Since the <span class="math-container">$A_k$</span> are independent it follows that <span class="mat... |
1,302,738 | <p>I'm trying to evaluate this sum </p>
<p>$$\sum_{n=1}^{\infty}\sum_{k=n}^{\infty}\frac{1}{(n^2+n-1)(k^2+k-1)}$$</p>
<p>I have no idea how to deal with it. </p>
<p>With one sum I can, with partial-fraction decomposition, express it as a function of Digamma function and I stuck there.</p>
| Thomas Andrews | 7,933 | <p>Suggesting too long for a comment:</p>
<p>When $f(n)$ is a positive function with $\sum_1^\infty f(n)$ converging: $$\left(\sum_{n=1}^\infty f(n)\right)^2=\sum_{n=1}^{\infty}\sum_{k=1}^\infty f(n)f(k)=2\sum_{n=1}^\infty \sum_{k=n}^{\infty} f(n)f(k) - \sum_{n=1}^{\infty}f(n)^2$$</p>
<p>with $f(n) = \frac{1}{n^2+n-1... |
517,904 | <blockquote>
<p>Let $A,B$ be non-empty sets and $f:A\to B$ a function. Proof that $f$ is injective, iff $f\circ g=f\circ h$ implies that $g=h$ for all functions $g,h:Y\to A$, for every set $Y$?</p>
</blockquote>
<p>I can see why this is. But how do I prove this? I get confused by the if and only if part.</p>
| njguliyev | 90,209 | <p>Hint: Consider $\lfloor\dfrac{y}{x}\rfloor$. Also note that this is <em>not</em> the answer.</p>
|
113,370 | <p>Let $p<q$ be positive integers (with the allowance that $q$ may be $\infty$). How can we show that the sum of $L^p$ and $L^q$ is a Banach space under the norm $\|f\|=\inf\{\|g\|_p+\|h\|_q: g+h=f\}$?</p>
<p>Let $\{f_n\}$ be a sequence in $L^p+L^q$, such that $$\sum_{n=1}^\infty \|f_n\|<\infty.$$</p>
<p>We wou... | Nate Eldredge | 822 | <p>Hint: for each $n$, choose $g_n \in L^p$, $h_n \in L^q$ such that $f_n = g_n + h_n$ and $||g_n||_p + ||h_n||_q \le ||f_n|| + 2^{-n}$.</p>
|
3,828,586 | <p>Four cards are face down on a table. You are told that two are red and two are black, and you need to guess which two are red and which two are black. You do this by pointing to the two cards you’re guessing are red (and then implicitly you’re guessing that the other two are black). Assume that all configurations ar... | Victor Hugo | 322,450 | <p>The function <span class="math-container">$f(x)=1-x^2$</span> is not bounded at infinity, that is, for all <span class="math-container">$A>0$</span> and <span class="math-container">$k>0$</span> there exists <span class="math-container">$x_0 \in \mathbb{R}$</span> such that <span class="math-container">$x_0&g... |
3,828,586 | <p>Four cards are face down on a table. You are told that two are red and two are black, and you need to guess which two are red and which two are black. You do this by pointing to the two cards you’re guessing are red (and then implicitly you’re guessing that the other two are black). Assume that all configurations ar... | TheSilverDoe | 594,484 | <p><span class="math-container">$$\lim_{|x| \rightarrow +\infty} f(x) = -\infty$$</span></p>
<p>so <span class="math-container">$$\liminf_{|x| \rightarrow +\infty} f(x) = \limsup_{|x| \rightarrow +\infty} f(x) =-\infty$$</span></p>
|
1,878,519 | <blockquote>
<p>$$\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\frac{\sec^2x}{\sqrt[3]{\tan\ x}}dx$$</p>
</blockquote>
<p>$$f(x) = (\tan \ x)^{\frac{2}{3}}, \ f'(x) = \frac{2}{3} \cdot (\tan \ x)^{-\frac{1}{3}} \cdot \sec^2x$$</p>
<p>$$\therefore \int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\frac{\sec^2x}{\sqrt[3]{\tan\ x}}dx = \fra... | DeepSea | 101,504 | <p>Put $u = \tan x\implies I = \displaystyle \int_{1}^{\sqrt{3}} u^{-\frac{1}{3}}du= \left[\frac{3}{2}u^{\frac{2}{3}}\right]|_{u=1}^{u = \sqrt{3}}= \dfrac{3}{2}\left(\sqrt[3]{3}-1\right)$</p>
|
2,653,503 | <p>How to find a <strong>bijective</strong> function $f: 3\mathbb{N}+1\to 4\mathbb{N}+1$ such that $$f(xy)=f(x)f(y),\forall x,y\in 3N+1$$</p>
<p>If i let $x,y\in 3\mathbb{N}+1$ then there exists $n,m\in \mathbb{N}$ such that $x=3n+1,y=3m+1$ </p>
<p>but I have no idea how I can find a such $f$, Is there a method pleas... | Tiago Emilio Siller | 526,875 | <p>$a= \overbrace{11\ldots11}^{2018} = \dfrac{10^{2018}-1}{9}$</p>
<p>$b= \overbrace{22\ldots22}^{1009} = 2\cdot\dfrac{10^{1009}-1}{9}$</p>
<p>$\Rightarrow a-b = \dfrac{10^{2018}-1}{9}-2\cdot\dfrac{10^{1009}-1}{9} = \dfrac{10^{2018}-2\cdot10^{1009}+1}{9} = \left(\dfrac{10^{1009}-1}{3} \right)^2$</p>
<p>$\Rightarrow... |
1,950,701 | <p>I was asked to prove or disprove the above question and "If there are functions g: A⟹B and f: B⟹C, if f◦g and f are one-to-one then g is one-to-one"</p>
<p>I think I have found a way to disprove it using the counterexample:
if f(x) = √x
and g(x) = x^2
then (f ◦ g)(x) = x, f and f◦g are injective, but g is not.</p>
... | Evan Aad | 37,058 | <p>The statement in the title is false. Take, for instance the functions $f:\{1,2\}\rightarrow\mathbb{R}$, $g:\{1\}\rightarrow\{1, 2\}$ defined as follows.
$$
\begin{align}
f(1) &:= f(2) := 1 \\
g(1) &:= 1.
\end{align}
$$</p>
|
1,950,701 | <p>I was asked to prove or disprove the above question and "If there are functions g: A⟹B and f: B⟹C, if f◦g and f are one-to-one then g is one-to-one"</p>
<p>I think I have found a way to disprove it using the counterexample:
if f(x) = √x
and g(x) = x^2
then (f ◦ g)(x) = x, f and f◦g are injective, but g is not.</p>
... | Bernard | 202,857 | <p>Actually $f\circ g$ one-to-one alone ensures $g$ is one-to-one.</p>
<p>Proof by contrapositive: if $g$ is not one-to-one, $f\circ g$ can't be one-to-one.</p>
<p>For the question in the title, $f\circ g$ and $g$ one-to-one don't ensure $f$ is.</p>
<p>As a counter-example, let$f(x)=x^2$, which is not one-to-one (i... |
9,840 | <p>The formula for finding the roots of a polynomial is as follows</p>
<p>$$x = \frac {-b \pm \sqrt{ b^2 - 4ac }}{2a} $$
what happens if you want to find the roots of a polynomial like this simplified one
$$ 3x^2 + x + 24 = 0 $$
then the square root value becomes
$$ \sqrt{ 1^2 - 4\cdot3\cdot24 } $$
$$... | Juan S | 2,219 | <p>100% correct, and good observation.</p>
<p>To solve this, we define $\sqrt{-1}=i$ where $i$ is the <a href="http://en.wikipedia.org/wiki/Imaginary_unit" rel="nofollow">imaginary unit</a></p>
<p>Then $\sqrt{-287}=\sqrt{287}i$, and we can solve as per the general quadratic formula. Numbers of the form $a+bi$ are kno... |
40,181 | <p>Currently, my math training includes Calc 1-3, linear algebra, and some introduction to set theory/discrete math.
What would you recommend that I study over summer in preparation for the Putnam? Real analysis, topology, abstract algebra (all of the above)? What would be the most pertinent? Thanks!</p>
| GeoffDS | 8,671 | <p>There are lots of good books.</p>
<ul>
<li>William Lowell Putnam Mathematical Competition: Problems & Solutions: 1938-1964 </li>
<li>William Lowell Putnam Mathematical Competition: Problems & Solutions: 1965-1984 </li>
<li>William Lowell Putnam Mathematical Competition: Problems & Solutions: 1985-200... |
1,883,047 | <p>I have this logical statement</p>
<p>$$\neg x\lor (x \wedge y)$$</p>
<p>However I do not know what is considered a valid transformation. Normally if there is an $\wedge$ in the middle I treat it like multiplication and pull out some "shared" piece but here I don't know how to use distributive properties. </p>
| Graham Kemp | 135,106 | <blockquote>
<p>I am familiar with those laws but not how they would be applied here. NOT always messes me up when it comes to distributive manipulations. </p>
</blockquote>
<p>Practice it slowly, one step at a time, until perfect. When distributing, carefully keep the "nots" stuck to their variables.</p>
<p... |
199,916 | <p>Let $\mathbb F_q$ be the finite field with $q$ elements. Suppose $V$ is a linear space of dimension $n$ over $\mathbb F_q$, and $r<n$. What is the maximal $k$ such that for arbitrary $k$ subspaces $W_1,W_2,\dots,W_k$ of $V$ of dimension $r$, there always exists a subspace $U$ of $V$ of dimension $n-r$ which satis... | Andreas Blass | 6,794 | <p>EDIT: This answers an earlier version of the question:</p>
<p>Your question seems to be equivalent to fixing $V$ and an $(n-r)$-dimensional subspace $U$ and then asking how many $r$-dimensional subspaces $W$ have $\{0\}$ intersection with that $U$. The number is (unless I've made a silly mistake)
$$
\frac
{(q^n-q^... |
3,956,963 | <p>What is the difference between countable infinity and uncountable infinity? Are there any examples? How can I imagine it?
Can you offer some assistance? please.</p>
| Wuestenfux | 417,848 | <p>A denumerable set is in one-to-one correspondence with the set of natural numbers. Such sets may also be called countably infinite, see <a href="https://mathworld.wolfram.com/DenumerableSet.html" rel="nofollow noreferrer">https://mathworld.wolfram.com/DenumerableSet.html</a></p>
<p>Such a set has the form <span clas... |
3,462,835 | <p>I am supposed to evaluate the following limit:</p>
<p><span class="math-container">$$\lim_{\left ( x,y \right )\rightarrow \left ( 0,0 \right )}\frac{x^{2}+y^{2}}{x+y}$$</span> </p>
<p>My solution:</p>
<p><span class="math-container">$$\lim_{\left ( x,y \right )\rightarrow \left ( 0,0 \right )}\frac{x^{2}+y^{2}}{... | user | 505,767 | <p>We have that</p>
<ul>
<li>for <span class="math-container">$x=y=t \to 0$</span></li>
</ul>
<p><span class="math-container">$$\frac{x^{2}+y^{2}}{x+y}=t \to 0$$</span></p>
<ul>
<li>for <span class="math-container">$x=t \to 0$</span> and <span class="math-container">$y=t^2-t$</span></li>
</ul>
<p><span class="math-... |
132,936 | <p>In propositional logic, a <strong>theory</strong> <span class="math-container">$T$</span> consists of a set of logical symbols and statements which we call axioms. A logical statement <strong><span class="math-container">$A$</span> can be proven by <span class="math-container">$T$</span></strong> if there is a proof... | Rudy the Reindeer | 5,798 | <p>Here's an example: take your language to be the group language $L_G = \{ e, \cdot \}$ and your theory to be the three group axioms:
$$ (i) \exists e \in G: \forall g \in G: eg = ge = g$$</p>
<p>$$ (ii) \forall g \in G \exists g^{-1} \in G: gg^{-1} = g^{-1}g = e$$</p>
<p>$$ (iii) \forall a,b,c \in G: a(bc) = (ab)c$... |
1,331,010 | <p>I am trying to solve $\dfrac{d\mathbf{x}}{dt} = \left[\begin{array}{cc}
-4 &1\cr
-6 &1
\end{array}\right] \mathbf{x}$ and I need to find the general solution of the system in the form $x=c_1x_1+c_2x_2$.</p>
<p>Finding the eigenvalues, I have $\det(P - \lambda I) = 0 \implies \lambda = \dfrac{-5 \pm \sqrt{21... | Jared | 138,018 | <p>OK, I'll give the "dumb" way to do this--the way I used to do it in my Diff. EQ. class because I didn't understand eigenvalues or linear algebra at the time. Assume you have the following:</p>
<p>$$
x = Ae^{rt} \\
y = Be^{rt}
$$</p>
<p>Plug in:</p>
<p>$$
\dot{x} = rAe^{rt} = -4Ae^{rt} + Be^{rt} \\
\dot{y} = rBe^... |
3,933,241 | <p>I am studying maths as a hobby. I have come across this problem:</p>
<p>Find a general solution for the equation cos 3x = sin 5x</p>
<p>I have said, <span class="math-container">$\sin 5x = \cos(\frac{\pi}{2} - 5x)$</span></p>
<p>so</p>
<p><span class="math-container">$\cos 3x = \sin 5x \implies 3x = 2n\pi\pm(\frac{\... | user3482749 | 226,174 | <p>No, the two are equivalent. In particular, if <span class="math-container">$m$</span> = <span class="math-container">$-n$</span>, then <span class="math-container">$$\dfrac{\pi}{2}(1 - 4m) = \dfrac{\pi}{2}(4n + 1),$$</span>
so all that's really happened is tha tyou've listed the solutions in a different order.</p>
|
821,654 | <p>I have a Taylor series problem, well more precisely a Maclaurin series.</p>
<p>I am trying to find convergence of: $f(x) = e^{x^3} + e^{{2x}^3}$</p>
<p>Okay here goes:</p>
<p>$$f'(x) = 3xe^{x^3} + 6x e^{{2x}^3}$$
$$f''(x) = 9x^2e^{x^3} + 3e^{x^3} + 36x^2e^{{2x}^3} + 6e^{{2x}^3}=e^{x^3}(9x^2+3) + e^{{2x}^3}(36x^2+... | Hakim | 85,969 | <blockquote>
<p><a href="http://en.wikipedia.org/wiki/Quadratic_formula" rel="nofollow"><strong>Quadratic Formula:</strong></a> We consider the equation $ax^2+bx+c=0$ where $a,b,c\in\mathbb R$ and $a\neq0$, then its solutions are given by the formula: $$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}.$$
where we have 3 cases:</... |
3,770,506 | <p><span class="math-container">$\newcommand{\rank}{\operatorname{rank}}$</span>Suppose <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are <span class="math-container">$n\times n$</span> matrices. Let <span class="math-container">$\rank(A)=s$</span> and <span class="math-container">... | Please correct GrammarMistakes | 740,776 | <p>I find the conclusion of <span class="math-container">$R(A+B)\leq R(A)+R(B)$</span> on page 69 of <a href="http://book.ucdrs.superlib.net/views/specific/2929/bookDetail.jsp?dxNumber=000004937296&d=7EC730F56E3D971C8F377B4263F41927&fenlei=18060101" rel="nofollow noreferrer">this book</a>. As for the lower boun... |
185,295 | <p>I would like to solve the following equation <span class="math-container">$y^2=x^2+ax^2y^2+by^2x^3+cy^3x^2$</span> where <span class="math-container">$a,b,c$</span> are small, so <span class="math-container">$y\approx x+O(x^3)$</span>. I would like to have a series approximation of the solution rather than an exact ... | Carl Woll | 45,431 | <p>The new in M12 function <a href="http://reference.wolfram.com/language/ref/AsymptoticSolve" rel="nofollow noreferrer"><code>AsymptoticSolve</code></a> can be used to find the series:</p>
<pre><code>AsymptoticSolve[y^2 == x^2 + a x^2 y^2 + b y^2 x^3 + c y^3 x^2, {y, 0}, {x, 0, 4}]
</code></pre>
<blockquote>
<p>{{... |
2,470,463 | <p>If $x+iy=\sqrt\frac{1+i}{1-i}$, where $x$ and $y$ are real, prove that $x^2+y^2=1$</p>
<p>I tried multiplying $\sqrt{(\frac{1+i}{1-i})(\frac{1+i}{1+i})}=\sqrt{i}$ but I'm not sure what to do after</p>
<p>thanks in advance :)))</p>
| Fred | 380,717 | <p>Let $z=x+iy$. Then $z^2=\frac{1+i}{1-i}$, hence $1=|z|^2$ and so $x^2+y^2=|z|^2=1$.</p>
|
4,141,512 | <p>I have got the answer, seeking intuition from
<a href="https://www.doubtnut.com/question-answer/find-the-equations-of-the-projection-of-the-line-x-1-2y-1-3z-2-4-on-the-plane-2x-y-4z1-1116852" rel="nofollow noreferrer">https://www.doubtnut.com/question-answer/find-the-equations-of-the-projection-of-the-line-x-1-2y-1-... | Math Lover | 801,574 | <p>Finding direction vector of the line using the method you started with is possible. But we need additional steps to find a point on the line. But to answer your question, here is how we can find direction vector using the method you started with.</p>
<p>Say unit direction vector of the line is <span class="math-cont... |
162,349 | <p>Is there any conventional notation for variables that can only take the value 0 or 1? (I'm looking for something of the nature of an overbar, a caret, etc.)</p>
| Asaf Karagila | 622 | <p>I don't know of such notation. You can always define that $\dot x$ means that $x$ is a Boolean variable with values in $\{0,1\}$. </p>
<p>Of course the dot can be replaced by other symbol. Be forewarned, though, that there are many many different contexts in which these symbols already have meaning. If you specify ... |
1,651,227 | <p>I was searching for pythagorean triples where $b=a+1$, and I found using a python program I made the first 10 integer solutions:</p>
<ol>
<li>$0^2+1^2=1^2$</li>
<li>$3^2+4^2=5^2$</li>
<li>$20^2+21^2=29^2$</li>
<li>$119^2+120^2=169^2$</li>
<li>$696^2+697^2=985^2$</li>
<li>$4059^2+4060^2=5741^2$</li>
<li>$23660^2+236... | Tito Piezas III | 4,781 | <blockquote>
<p><strong>I. Silver ratio</strong></p>
</blockquote>
<p>What you have discovered is the square of the <em><a href="https://en.wikipedia.org/wiki/Silver_ratio" rel="noreferrer">silver ratio</a></em>, </p>
<p>$$S=1+\sqrt{2} = 2.414213\dots$$ </p>
<p>It is a cousin of the <em>golden ratio</em>, $\phi=\... |
3,487,989 | <p>Slope of line <span class="math-container">$PQ$</span> is
<span class="math-container">$$m=\frac{1}{1-k}$$</span>
The slope perpendicular to it will be
<span class="math-container">$k-1$</span></p>
<p>Since the line is a bisector of PQ it will pass through
<span class="math-container">$(\frac{1+k}{2},\frac 72)$<... | Peter Szilas | 408,605 | <p>Option:</p>
<p>The perpendicalur bisector intersects the <span class="math-container">$y$</span>-axis in <span class="math-container">$(0,4)$</span>.</p>
<p>A circle with center <span class="math-container">$(0,4)$</span> passes through <span class="math-container">$P(1,4)$</span> and <span class="math-container">... |
1,118,269 | <p>I'm reading: <em>Mathematical thought from ancient to modern times by Kline</em>. My question is about this pasasge:</p>
<blockquote>
<p>Beyond its achievements in subject matter, the nineteenth century
reintroduced rigorous proof. No matter what individual mathematicians may
have thought about the soundness ... | franz lemmermeyer | 23,365 | <p>In mathematics, we prove results by deducing them from accepted truths that we call axioms. This is what Euclid taught us, and this is how mathematics is done until today. In my opinion, this wasn't substantially different between Euclid and Weierstrass, it's just that the mathematicians in the times of Fermat, Eule... |
263,349 | <p>If secant and the tangent of a circle intersect at a point outside the circle then <strong>prove that the area of the rectangle formed by the two line segments corresponding to the secant is equal to the area of the square formed by the line segment corresponding to the tangent</strong><br>
I find this question hig... | Clayton | 43,239 | <p>Let $LC(X)$ denote the set of points where $X$ is locally connected. If $LC(X)=\varnothing$, it clearly is a $G_\delta$-set as $\varnothing=\bigcap\varnothing$ and $\varnothing$ is open. </p>
<p>Suppose $LC(X)$ is nonempty. Then there is a point $x\in LC(X)$ such that for every open set containing $U$ such that $x\... |
2,340,204 | <p>I searched extensively for an answer, but couldn't find one that specifically explained what I was looking for. In working through a problem in my textbook, part of it involves simplifying an expression using power reduction. This is the step:</p>
<p>$$
\cos^{2}(2\theta) = \frac{1+\cos(2(2\theta))}{2}
$$</p>
<p>I ... | user1551 | 1,551 | <p>You are essentially asking whether the conditions that $\sum_jE_jAE_j^\dagger$ is real for every real symmetric $A$ and that $\sum_jE_jE_j^\dagger=I$ would imply that $E_jAE_j^\dagger$ is also real for every $j$ and for every real symmetric $A$. The answer is negative. Here is a counterexample. Let
$$
E_1=\frac1{\sq... |
8,236 | <p>What are the axioms of four dimensional Origami.</p>
<p>If standard Origami is considered three dimensional, it has points, lines, surfaces and folds to create a three dimensional form from the folded surface. This standard origami has seven axioms which have been proved complete.</p>
<p>The question I have is whe... | Greg Kuperberg | 1,450 | <p>In the $n$-dimensional origami question, you start with a generic set of hyperplanes and their intersections, which can then be some collection of $k$-dimensional planes. An "axiom" is a set of incidence constraints that determines a unique reflection hyperplane, or conceivably a reflection hyperplane that is an is... |
1,821,318 | <p>Generally a function is shown continuous by directly taking left hand or right hand limit.But sometimes the same can be shown continuous by letting h tend to zero.What is the difference,would somebody please explain?</p>
| John | 7,163 | <p>Just to start from the beginning, the AM-GM inequality is this:</p>
<p>$$\frac{m+n}{2} \geq \sqrt{mn}.$$</p>
<p>We can replace $(m,n)$ with $(a^4, b^4)$:</p>
<p>$$\frac{a^4+b^4}{2} \geq \sqrt{a^4 b^4},$$</p>
<p>or</p>
<p>$$a^4 + b^4 \geq 2a^2b^2.$$</p>
<p>Then adding $8$ to both sides doesn't change anything:<... |
3,770,004 | <p>Not a duplicate of</p>
<p><a href="https://math.stackexchange.com/questions/2401434/suppose-a-b-and-c-are-sets-prove-that-c-%e2%8a%86-a-b-iff-c-%e2%8a%86-a-%e2%88%aa-b-and">Suppose $A$, $B$, and $C$ are sets. Prove that $C ⊆ A △ B$ iff $C ⊆ A ∪ B$ and $A ∩ B ∩ C = ∅$.</a></p>
<p><a href="https://math.stackexchange.c... | Poorwelsh | 808,225 | <p>You can shorten you proof by writing <span class="math-container">$A \cup B=(A \bigtriangleup B) \cup (A \cap B)$</span>.
First assume that <span class="math-container">$C \subseteq A \bigtriangleup B$</span>.
Since, <span class="math-container">$A \bigtriangleup B \subseteq A\cup B$</span>, then <span class="math-c... |
3,770,004 | <p>Not a duplicate of</p>
<p><a href="https://math.stackexchange.com/questions/2401434/suppose-a-b-and-c-are-sets-prove-that-c-%e2%8a%86-a-b-iff-c-%e2%8a%86-a-%e2%88%aa-b-and">Suppose $A$, $B$, and $C$ are sets. Prove that $C ⊆ A △ B$ iff $C ⊆ A ∪ B$ and $A ∩ B ∩ C = ∅$.</a></p>
<p><a href="https://math.stackexchange.c... | Dan Velleman | 414,884 | <p>I think the "(<span class="math-container">$\to$</span>)" direction of your proof is fine. The "(<span class="math-container">$\leftarrow$</span>)" direction is correct but could be shortened. There was no need to break case 1 into cases 1.1, 1.2, 1.2.1, and 1.2.2. You could have completed cas... |
4,416,238 | <p>From Serge Lang's Introduction to Linear Algebra, page 152:</p>
<blockquote>
<p>Let <span class="math-container">$L:V\rightarrow V$</span> be a linear map [<span class="math-container">$V$</span> is a vector space]. Suppose that there exists a
basis <span class="math-container">$\{v_1,...,v_n\}$</span> and numbers <... | 5xum | 112,884 | <p><span class="math-container">$v_i$</span> is a vector, because it is an element of <span class="math-container">$V$</span>.</p>
<hr />
<p>You know that <span class="math-container">$\{v_1,\dots, v_n\}$</span> is a basis for <span class="math-container">$V$</span>. Well, a <em>basis</em> for a vector space is always ... |
1,543,722 | <p>We are learning about inequalities. I originally assumed it would be the same as equations, except with a different sign. And so far, it has been - except for this.</p>
<p>Take the simple inequality:
$-5m>25$ To solve it, we divide by $-5$ on both sides, as expected.
$m>-5$.</p>
<p>But, I have been told that... | user293034 | 293,034 | <p>It's just a matter of equation reforming.</p>
<p>If you have $x < y$ this is equivalent to $f(x) < f(y)$ if and only if $f$ is a strictly monotonically increasing function. If $f$ is a strictly monotonically decreasing function (like multiplying with a negative number is), it flips the inequality. If the fu... |
33,710 | <p>Let $K$ be a non-Archimedean local field, i.e., complete with respect to a non-trivial, non-archimedean discrete absolute value, with finite residue field $k$ of characteristic $p\neq 0$. Also let $K_s$ be a fixed separable closure of $K$, and $K_{un}$ (resp. $K_t$) the maximal unramified (resp. tamely ramified) ext... | Keenan Kidwell | 4,351 | <p>As all the responses indicate, the answer to my question is "yes." The most direct route seems to be the one suggested by KConrad. Explicitly, if $F/K_{un}$ is Galois of degree $e$ (inside $K_s$), then the ring of integers of $F$ is a DVR, and if $\Pi$ is a uniformizer for $O_F$, then because $F/K_{un}$ is totally r... |
2,681,107 | <p>Given that $f : \mathbb{R} \rightarrow \mathbb{R}$ satisfies </p>
<p>$2f^3(x)-3=2x-3f(x)$ , $x\in \mathbb{R}$, show that $f$ is continuous on $\mathbb{R}$.</p>
<p>How can we handle this problem?</p>
| Fimpellizzeri | 173,410 | <p><strong>Hint:</strong> Try to show that $f(x) \geq x + f(0)$ and then apply the intermediate value theorem.</p>
|
3,777,856 | <p>I have the following equation that I need to solve:</p>
<p><span class="math-container">$$1000.00116=\frac{1000}{\left(1+x\right)^{16}}+\frac{1-\left(1+x\right)^{-16}}{x}$$</span></p>
<p>However, software I use is refusing to do it. Which software/web is capable of solving it? Or could you please show me the answer?... | Cesareo | 397,348 | <p>Hint.</p>
<p><span class="math-container">$$
1000.00116-\frac{1-(x+1)^{16}}{x}-\frac{1000}{(x+1)^{16}}\approx 16.0012+16120 x-135440 x^2+O\left(x^3\right)
$$</span></p>
<p>solving for <span class="math-container">$x$</span> we have</p>
<p><span class="math-container">$$
x = \cases{
-0.000984485\\
0.120004
}
$$</span... |
1,089,193 | <p>The theorem is that every reflection $R_{S}$ in an affine subspace $S$ of $\mathbb{E}^{n}$ is an isometry:</p>
<p>$R_S:\ \mathbb{E}^{n} \rightarrow \mathbb{E}^{n}:\ x \mapsto R_{S}(x) = x + 2 \overrightarrow{x\pi_{S}(x)}$</p>
<p>I'm horrendously stuck with the proof.
I get that I'm trying to prove that $R_{S}$ pre... | Ted Shifrin | 71,348 | <p>Pick an origin for $\Bbb E^n$, and call the resulting vector space $\Bbb R^n$. To save notation, let's put the origin in $S$, so that $S$ now becomes a (vector) subspace. Write $\Bbb R^n=S\oplus S^\perp$. Now, identifying points $x\in\Bbb E^n$ with the corresponding vectors in $\Bbb R^n$, write $x=x_1+x_2$, where $x... |
22,078 | <p>It is a standard and important fact that any smooth affine group scheme $G$ over a field $k$ is a closed $k$-subgroup of ${\rm{GL}}_n$ for some $n > 0$. (Smoothness can be relaxed to finite type, but assume smoothness for what follows.) The proof makes essential use of $k$ being a field, insofar as it uses freen... | George McNinch | 4,653 | <p>This doesn't answer the question posed, but maybe speaks to the one you didn't ask...
In Bruhat-Tits [Groupes Reductifs sur un corps local II] 1.4.5 shows for Dedekind A
that an affine A-group scheme which is flat and of finite type has a faithful linear representation.</p>
|
22,078 | <p>It is a standard and important fact that any smooth affine group scheme $G$ over a field $k$ is a closed $k$-subgroup of ${\rm{GL}}_n$ for some $n > 0$. (Smoothness can be relaxed to finite type, but assume smoothness for what follows.) The proof makes essential use of $k$ being a field, insofar as it uses freen... | A Stasinski | 2,381 | <p>The result in SGA 3, VIB Remarque 11.11.1, implies that an affine flat group scheme of finite type over a <s>local Artinian principal ideal ring</s> field $A$ (or over a Dedekind ring) has a closed embedding into $\mathrm{GL}(V)$, for some module $V$ locally free and of finite type over $A$. Over a Dedekind ring, V ... |
22,078 | <p>It is a standard and important fact that any smooth affine group scheme $G$ over a field $k$ is a closed $k$-subgroup of ${\rm{GL}}_n$ for some $n > 0$. (Smoothness can be relaxed to finite type, but assume smoothness for what follows.) The proof makes essential use of $k$ being a field, insofar as it uses freen... | naf | 519 | <p>(This is more of a comment to the question than an answer, but there are already many comments...)</p>
<p>For the ring of dual numbers A over k is it clear that there exist smooth affine group schemes over A which are not trivial i.e. do not come from k by base change? By smoothness we know that as schemes they are... |
369,722 | <p>Let <span class="math-container">$\mathcal{H}$</span> denote a Hilbert space and <span class="math-container">$B(\mathcal{H})$</span> denote the algebra of all bounded operators on <span class="math-container">$\mathcal{H}$</span>. By recognizing the (Banach) dual of <span class="math-container">$B(\mathcal{H})$</sp... | Stefaan Vaes | 159,170 | <p>Also the answer to the second question is <strong>yes</strong>, and the approximation may be chosen to converge in the point-ultrastrong<span class="math-container">$^*$</span> topology.</p>
<p>First, by choosing a net of finite rank orthogonal projections <span class="math-container">$p_i \in B(\mathcal{H})$</span>... |
2,620,571 | <p>Let $\Omega = \{(x,y)\in (0,\infty)^2 | 4 < x^2+4y^2<16\}$, so the area between two ellipses in the first quadrant. I need to calculate the following integral: $$\int_{\Omega}\frac{xy}{x^2+4y^2}d(x,y)$$ I tried using normal polar coordinates, however the integral gets really messy after the transformation. Doe... | Community | -1 | <p>Draw the line $CD$ through O perpendicular to the parallel lines: $C$ on the line going through $A$ and $D$ on the line going through $B$. Those parallel lines touch the circle at $C$ and $D$. Also, let $E$ be the point where line $AB$ touches the circle.</p>
<p>We have $\triangle OAC\cong\triangle OAE$ (because $O... |
2,465,064 | <p><a href="https://i.stack.imgur.com/7lBit.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7lBit.png" alt="enter image description here"></a>
Can anyone help me with this. I honestly have no clue where to begin.</p>
| Sri-Amirthan Theivendran | 302,692 | <p><strong>Hint</strong></p>
<p>Suppose we have to choose a committee of $r$ people from a pool of $m$ men and $n$ women. The LHS counts the number of ways this can be done. Now classify the committees based on the number of men, say $k$, in the committee to interpret the RHS.</p>
|
237,741 | <p>Assume you toss a fair coin 25 times with the outcome of each toss being
independent of the outcomes of any other toss.</p>
<p>How many completed runs do you expect to observe?</p>
<p>By definition, completed runs are a run that have been terminated by the occurrence of another symbol, and a run is defined as a sequ... | André Nicolas | 6,312 | <p><strong>Hint:</strong> For $i=1$ to $25$, let $X_i=1$ if the $i$-th toss <strong>completes</strong> a run, and let $X_i=0$ otherwise. Then $Y=X_1+X_2+\cdots+X_{25}$ is the number of runs. We want $E(Y)$. </p>
<p>By the linearity of expectation, we have
$$E(Y)=E(X_1)+E(X_2)+\cdots +E(X_{25}).\tag{$1$}$$</p>
<p>Cal... |
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