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 |
|---|---|---|---|---|
4,096,885 | <p>For example if I have two sets <em>A</em> and <em>B</em>, where I take the Cartesian Product of both, does it matter if I perform the operation in this order <em>A</em>x<em>B</em> or whether I perform the operation in this order or the order <em>B</em>x<em>A</em>, or does the order not even matter? I am enquiring be... | Floridus Floridi | 841,808 | <p>The cartesian product is a set, and in a set as such there is no order ( due to the extensionality principle).</p>
<p>So , <em><strong>in the cartesian product</strong></em> the " order" of the couples ( i.e. the elements of the cartesian product) does not matter.</p>
<p>Let <span class="math-container">... |
74,086 | <p>Given a projective surface $S$, and a smooth projective curve $C\subset S$ over $\mathbb{C}$. Furthermore we have a locally free sheaf $E$ of rank $r$ on $S$.</p>
<p>Then for any $l\geq 1$, the projective scheme $\operatorname{Quot}(E,l)$ classifies quotients $E\to T$, such that $\operatorname{dim}_{\mathbb{C}}(H^0... | Brendan McKay | 9,025 | <p>Two bounds I found in my old files:</p>
<p>Grimmett, Disc Math 16 (1976) 323-324. Two bounds:
$$ \frac{1}{n}\left( \frac{2m}{n-1} \right)^{n-1}.$$
$$ \left( \frac{2m-d}{n-1} \right)^{n-1},$$
where $d$ is the maximum degree. This second one is exact for $K_{1,n-1}$.</p>
<p>Grone and Merris, Disc Math 69 (1988) 97... |
158,687 | <p>I have 3 matrices, each of size $101 \times 101$.</p>
<pre><code>List6 = Table[{x1[[i, j]]}, {i, 101}, {j, 101}]
List7 = Table[{y1[[i, j]]}, {i, 101}, {j, 101}]
List8 = Table[{xDisp1[[i, j]]}, {i, 101}, {j, 101}]
</code></pre>
<p>and I want to use <code>ListContourPlot</code></p>
<pre><code>ListContourPlot[{{x1[[... | kglr | 125 | <p>Use your own matrices <code>x1</code>, <code>y1</code> and <code>xDisp1</code> in place of the random ones below:</p>
<pre><code>m = 30;
x1 = RandomReal[1, {m, m}];
y1 = RandomReal[1, {m, m}];
xDisp1 = RandomReal[10, {m, m}];
data = Join @@
Table[{x1[[i, j]], y1[[i, j]], xDisp1[[i, j]]}, {i, 1, m}, {j, 1, m}];... |
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... | J. M. ain't a mathematician | 498 | <p>Alright, I'll work on two of the vectors; I'll leave the third one to you.</p>
<p>You have the vector-valued function</p>
<p>$$\mathbf c(t)=\begin{pmatrix}2t^3 + t^2 - 3t + 6\\t^3 - 2t^2 + 4t\\-3t^3 + 4t^2 + 2t - 9\end{pmatrix}$$</p>
<p>The first two derivatives are easily done:</p>
<p>$$\mathbf c^\prime(t)=\beg... |
454,325 | <p>Which is the technically correct definition?</p>
<p>I) An interior point of a set $B$ is a point that is the centre of some $\epsilon$-ball in $B$.</p>
<p>II) An interior point of a set $B$ is a point that is in a set $A\subset B$ in which every point is the centre of some $\epsilon$-ball in $A$.</p>
<p>The two d... | tomasz | 30,222 | <p>Clearly, $x^6-1$ splits over ${\bf Q}(\alpha)$, as does $x^2-x+1$. Furthermore, $\alpha$ is a root of $x^6-1$, so it must belong to its splitting field.</p>
<p>What are roots of $x^2-x+1$? Can you produce a sixth root of unity from one of them?</p>
|
452,011 | <p>Consider the set $A = \{\theta(f) \mid f : \mathbb{R} \rightarrow \mathbb{R},\;f\,\text{non-decreasing}\}$ where $\theta(f)$ denotes the set of functions which are asymptotically within a constant factor of $f$. Define an order on $A$ by setting $\theta(f) \le \theta(g)$ if $f \in O(g)$.</p>
<p>What is the order st... | Narasimham | 95,860 | <p>( Explanation of yesterday's post implication reg constant of integration)</p>
<p>These are Limit Cycles in Control theory. There are three limiting (asymptotic) loci as sketched by Amzoti. However, caution is needed in choosing boundary conditions if the integrate is desired to be in a neat, closed or elegant form... |
2,671,992 | <p>I have had trouble proving that the interior of the following set is empty. I have tried to do it by definition, but haven't managed to figure out the proof.</p>
<p>$$
C := \{ (x,y) \in \mathbb{R^2} \mid x \in (-1,1) \text{ and } y=x^3 \}
$$</p>
| Community | -1 | <p>Because differentiation is not well defined on a closed interval. In fact we need to know what happen on the right and on the left of all point of the interval.</p>
<p>Hence in order to prove that a continuous function admits maximum or minimum we do not need the differentiability at the extreme points of the inter... |
1,099,885 | <p>How can you calculate</p>
<p>$$\lim_{x\rightarrow \infty}\left(1+\sin\frac{1}{x}\right)^x?$$</p>
<p>In general, what would be the strategy to solving a limit problem with a power?</p>
| idm | 167,226 | <p>$$\left(1+\sin\frac{1}{x}\right)^x=e^{x\ln\left(1+\sin \frac{1}{x}\right)}.$$</p>
<p>You have that for $x\to \infty $, $\sin \frac{1}{x}\sim \frac{1}{x}$, therefore</p>
<p>$$\lim_{x\to \infty }x\ln\left(1+\sin \frac{1}{x}\right)=\lim_{x\to \infty }\frac{\ln\left(1+ \frac{1}{x}\right)}{\frac{1}{x}}=\lim_{u\to 0}\fr... |
2,372,925 | <p>A book has 500 pages on which typographical errors could occur. Suppose that there are exactly 10 such errors randomly located on those pages. </p>
<p>$a)$ Find the probability that a random selection of 50 pages will contain no errors. </p>
<p>$b)$ Find the probability that 50 randomly selected pages will contain... | hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>$$u=\frac {x+y}{2} $$
$$v=\frac {x-y}{2} $$</p>
<p>$$0\le x<+\infty \land 0\le y<+\infty$$
$$\implies 0\le u<+\infty \land -\infty <v <+\infty$$</p>
|
1,124,078 | <p>Is $\ln|x+2|=\ln|2x+4|$? Is this right? I saw something earlier saying this was correct; my first instinct was no.</p>
| ChocolateBar | 161,284 | <p>Its wrong. Take $x=-1$. You get $$\ln(1) = 0 \neq \ln(2).$$</p>
|
2,215,409 | <p>$$T(0)=1$$
$$(N\gt0)\;T(N)\;=\;\sum_{k=0}^{N-1}(k+1)T(k)$$</p>
<p>How can I find the closed representation of this function?
I don't need to know the answer, I just need someone to point me in the right direction.</p>
| marty cohen | 13,079 | <p>Since,
for $n \ge 1$,
$t(n)
=\sum_{k=0}^{n-1}(k+1) t(k)
$,
$t(n+1)-t(n)
=(n+1)t(n)
$
so
$t(n+1)
=(n+2)t(n)
$
for $n \ge 1$.</p>
<p>Note that
$t(1) = 1$.</p>
<p>Therefore,
for
$n \ge 1$,
$\dfrac{t(n+1)}{t(n)}
=n+2
$
so that</p>
<p>$\begin{array}\\
\dfrac{t(n+1)}{t(1)}
&=\prod_{k=1}^{n}\dfrac{t(k+1)}{t(k)}\\
&a... |
2,411,812 | <blockquote>
<p>When <span class="math-container">$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... | zipirovich | 127,842 | <p>By Vieta's formulas, $z_1z_2=1$ $\implies$ $|z_1||z_2|=1$ $\implies$ either both $|z_1|=|z_2|=1$, or one of them is $>1$ and the other $<1$.</p>
<p>In the case $|z_1|=|z_2|=1$, since $z_1\bar{z}_1=z_1z_2=1$, we can easily see that $z_2=\bar{z}_1$, and so by Vieta's formulas again $2c=z_1+z_2=z_1+\bar{z}_1$ is... |
188,880 | <p>Here solutions is array .
I need to know why this code is written.Please help me.</p>
<pre><code>solutionsMod = Mod[solutions, n];
For[j = 1, j <= Length@solutions, j++,
For[i = 1, j <= Length@solution[[1]], i++,
If[solutionsMod[[j, i]] == 0, solutionsMod[[i, i]] =n;
];];];
Export[Tostr... | Somos | 61,616 | <p>The answer by Henrik Schumacher is brief and to the point. However, the user may actually want to use a modified version instead.</p>
<pre><code>solutions = Mod[solutions, n] + 1;
</code></pre>
<p>The reason why is that in order to map <code>0..n-1</code> to <code>1..n</code> in a contiguous way you need <code>i -... |
425,713 | <blockquote>
<p>We say that a metric space <span class="math-container">$M$</span> is <em>totally bounded</em> if for every <span class="math-container">$\epsilon>0$</span>, there exist <span class="math-container">$x_1,\ldots,x_n\in M$</span> such that <span class="math-container">$M=B_\epsilon(x_1)\cup\ldots\cup B... | PJ Miller | 79,379 | <p>The backward direction is proved in the comment. Here's the forward direction:</p>
<p>Suppose $M$ is conditionally compact. Fix $\epsilon$. Every sequence has a Cauchy subsequence. Choose $x_1\in M$. If $B_\epsilon(x_1)$ doesn't cover $M$, choose $x_2$ in $M-B_\epsilon(x_1)$. Note that $d(x_2,x_1)\ge\epsilon$. If $... |
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... | J. M.'s persistent exhaustion | 50 | <p>Version 12.1 now has the function <a href="https://reference.wolfram.com/language/ref/MeshConnectivityGraph.html" rel="nofollow noreferrer"><code>MeshConnectivityGraph[]</code></a> which can be used on <code>MengerMesh[]</code>:</p>
<pre><code>MeshConnectivityGraph[MengerMesh[4, 2], PlotTheme -> "LargeNetworkDef... |
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:
$$
F(x) =
\begin{cases}
0, & \text{... | Sean Roberson | 171,839 | <p>It is said in an answer that the user made an error in the arithmetic.</p>
<p>For your case, I assume the sample space is $[0, 6].$ Here's what we do:</p>
<p>Integrate the CDF as follows:</p>
<p>$$ \int_0 ^6 1 - F(x) \ dx = 6 - \left( \int_0^\frac{1}{2} x^2 \ dx + \int_\frac{1}{2} ^3 \frac{1}{4} \ dx + \int_3 ^6 ... |
2,337,856 | <p>Is there an easy way to show, that the only solution of this system of non-linear equations has only the solution a=0, b=0, c=0</p>
<p>$a+b+c=0$</p>
<p>$ax+by+cz=0$</p>
<p>$ax^2+by^2+cz^2=0$</p>
<p>For $x,y,z\neq 0$ and different.</p>
<p>Solving this gets really ugly.
Is there an elegant way?
It is obvious that... | Glorfindel | 228,959 | <p>The derivative of $x^{k-1} e^{-x^2}$ is $((k-1)x^{k-2}-2x^k)e^{-x^2}$. That means that, given that we have primitives for $e^{-x^2}$ (the error function) and $xe^{-x^2}$, we can express the integral for a given $k$ by descending:</p>
<p>$$\int x^k e^{-x^2}dx = -\frac12x^{k-1} e^{-x^2} + C + \int\frac{k-1}{2}x^{k-2}... |
1,804,146 | <p>My question is: Give me a field <span class="math-container">$K$</span>. Can we always find two <span class="math-container">$K$</span>-vector space <span class="math-container">$V_{1}$</span>, <span class="math-container">$V_{2}$</span> and a map <span class="math-container">$f:V_{1}\rightarrow V_{2}$</span> such t... | Community | -1 | <p>Inspired by Mariano Suárez-Alvarez♦'s:</p>
<p>Just like Mariano Suárez-Alvarez♦ comments: Given me a non-prime field $K$. $K$ has some proper subfield $L$ and $K$ can be viewed as an $L$-vector space.</p>
<p>Every vector space has a basis. Say $\{a_{i}:i\in I\}$ is a basis of $K$ over $L$. Consider a map $f:\{a_{1... |
1,305,236 | <p>A (probably simple) question I encountered but I don't know how to approach:</p>
<blockquote>
<p>Let $K$ be a field of prime characteristic $p>0$.
Show every $f(x) \in K[x]$ can be represented as $g(x^{p^e})$ for some $e \ge 0$ and $g \in K[x]$ with $g'(x) \neq 0$.</p>
</blockquote>
<p>I saw <a href="https:... | quid | 85,306 | <p>Note that you need $f$ non-constant, otherwise this is just not true.</p>
<hr>
<p>Now, first note, if $f'(x)\neq 0$ then set $e=0$ and $f=g$ and you are done. </p>
<p>Second, if $f'(x)= 0$, then writing $f(x)= \sum_{i=0}^n a_i x^i $, you have $f'(x)=\sum_{i=1}^ ia_i x^{i-1}$. So that $ia_i = 0$ for all $i=0, \dot... |
344,479 | <p>Suppose that $a = 2^kb,$ where $b$ is odd. If $\phi(x) = a,$ prove that $x$ has at most $k$ odd prime divisors.</p>
| Zev Chonoles | 264 | <p><strong>Hint:</strong> If the prime factorization of $x$ is $p_1^{a_1}\cdots p_r^{a_r}$, then
$$\phi(x)=\phi(p_1^{a_1})\cdots\phi(p_r^{a_r})=p_1^{a_1-1}(p_1-1)p_2^{a_2-1}(p_2-1)\cdots p_r^{a_r}(p_r-1).$$</p>
|
4,050,571 | <p>Suppose I have a video that plays for 60 minutes at normal "1x" speed. I know that if I set the video to play at "2x" speed, then it should play for 30 minutes.</p>
<p>Now, what if the video is set to play at "1.5x" speed? Intuition leads me to two answers:</p>
<ol>
<li><p>Since 1.5x is... | J.G. | 56,861 | <p>The second. Your first idea obtains the completion time from the playback rate given by a <a href="https://en.wikipedia.org/wiki/Harmonic_mean" rel="nofollow noreferrer">harmonic mean</a> of <span class="math-container">$1\times,\,2\times$</span>, viz. <span class="math-container">$\frac{2}{1+\tfrac12}=\tfrac43$</sp... |
194,813 | <p>Please help me solving $\displaystyle\lim_{x\to a}\frac{a^{a^{x}}-{a^{x^{a}}}}{a^x-x^a}$</p>
| DonAntonio | 31,254 | <p>Since this is calculus why not try with L'Hospital?</p>
<p>$$\lim_{x\to a}\frac{a^{a^x}-a^{x^a}}{a^x-x^a}=\lim_{x\to a}\frac{a^xa^{a^x}\log^2a-ax^{a-1}\log a\cdot a^{x^a}}{a^x\log a-ax^{a-1}}=$$</p>
<p>$$=\frac{a^aa^{a^a}\log^2a-a^a\log a\cdot a^{a^a}}{a^a\log a-a^a}=\frac{a^{a^a}\log^2a-\log a\cdot a^{a^a}}{\log ... |
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>
| Claude Leibovici | 82,404 | <p>Consider that you have the function $$f=a^2+b^2+ab-a-2b$$ Compute its derivatives $$f'_a=2 a+b-1$$ $$f'_b=2 b+a-2$$ You search for an extremum; if it exists, at a point $f'_a=f'_b=0$. So two equations for two unknowns.</p>
<p>I am sure that you can take it from here.</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... | Stinking Bishop | 700,480 | <p>If a sequence converges to a limit, then every subsequence converges, to the same limit. In the case of an infinite convergent sum, this implies that the partial sums with an <em>even</em> number of terms will converge to the same value. However, those partial sums are of positive numbers such as <span class="math-c... |
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">$$z = \begin{bmatrix}a... | Mohammad Riazi-Kermani | 514,496 | <p>If you let <span class="math-container">$$ u=<x,y,w>$$</span>
Then <span class="math-container">$$z=u.u+8=||u||^2 +8$$</span></p>
|
267,442 | <p>Suppose $a,b\in\Bbb N$ are odd coprime with $a,b>1$ then is it true that if all four of $$x_1a+x_2b,\mbox{ }x_2a-x_1b,\mbox{ }x_1\frac{(a+b)}2+x_2\frac{(a-b)}2,\mbox{ }x_2\frac{(a+b)}2-x_1\frac{(a-b)}2$$ are in $\Bbb Z$ for some $x_1,x_2\in\Bbb R$ then $x_1,x_2\in\Bbb Z$ should hold?</p>
| js21 | 21,724 | <p>Yes. If we set $\alpha = (a+b)/2$ and $\beta=(a-b)/2$, then the lattice generated by the four vectors</p>
<p>$$
\binom{\alpha}{\beta},\binom{-\beta}{\alpha},\binom{\alpha+\beta}{\alpha -\beta},\binom{\beta -\alpha}{\alpha +\beta}
$$
is contained in the set of $(y_1,y_2) \in \mathbb{Z}^2$ such that $x_1 y_1 + x_2 y_... |
2,969,710 | <blockquote>
<p>Use mathematical induction to prove that
<span class="math-container">$$
\frac12 + \frac16 + \ldots + \frac{1}{n(n+1)} = 1 - \frac{1}{n+1}
$$</span></p>
</blockquote>
<p>I am unsure about the prove n+1 step!
I let
<span class="math-container">$$
\frac12 + \frac16 + \ldots + \frac{1}{n(n+1)} = 1 -... | gt6989b | 16,192 | <p><strong>HINT</strong></p>
<p>General induction proofs look something like the following.</p>
<p><strong>Base Case.</strong> Let <span class="math-container">$n=1$</span>, can you validate it?</p>
<p><strong>Inductive Step.</strong> Assume the statement holds for some fixed <span class="math-container">$n$</span>,... |
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^... | Crostul | 160,300 | <p>$$6^{300} = (6^6)^{50} \ \ ; \ \ 10^{250} = (10^5)^{50}$$
so it is enough to check what is larger between $6^6$ and $10^5$. Now,
$$6^6 = (6^3)^2 = 216^2 < 300^2 = 90000 < 100000 = 10^5$$</p>
|
2,011,261 | <p>A pizza restaurant has 3 crust options, 2 cheese options and 10 choices of toppings. On Saturday nights, the restaurant offers a special deal on 2-toppings pizzas including pizzas with double portions of one toppings. How many distinct special deal pizzas are possible.</p>
<p>My approach:
I assumed (not too sure if... | Bernard | 202,857 | <p>It is enough to prove $\displaystyle\Bigl(1+\frac1n\Bigr)^n\Bigl(1-\frac1n\Bigr)^n$ has a limit.
$$\Bigl(1+\frac1n\Bigr)^n\Bigl(1-\frac1n\Bigr)^n=\Bigl(1-\frac1{n^2}\Bigr)^n<1$$
and by <em>Bernoulli's inequality</em>, we have
$$\Bigl(1-\frac1{n^2}\Bigr)^n\ge 1-n\cdot\frac1{n^2}=1-\frac1n.$$
The <em>squeeze theore... |
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... | QuantumSpace | 661,543 | <p>Here is a proof using mean value theorem. The inequality is trivial for <span class="math-container">$x=0$</span>. For <span class="math-container">$0 < x \leq 1, $</span> define
<span class="math-container">$$f: [0,x] \to \Bbb{R}: y \mapsto ey - e^y +1$$</span>
Then <span class="math-container">$f'(y) = e-e^y \... |
3,824,959 | <p>In reading a recent paper, I came across the inequality: <span class="math-container">$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... | Hagen von Eitzen | 39,174 | <p>The perhaps most versatile inequality about the exponential is
<span class="math-container">$$\tag1 e^t\ge 1+t$$</span>
for all <span class="math-container">$t\in \Bbb R$</span>.
For <span class="math-container">$x\ge0$</span> we can apply <span class="math-container">$\int_0^x\mathrm dx$</span> to both sides and ob... |
284,996 | <p>For every $f\in C[0,1]$ there is a sequence of even polynomials which converges uniformly on $[0,1]$ to f ? </p>
<p>What I have tried:</p>
<p>f is continuous on $D:=[0,1]$, let $(x_k)_{k\in \mathbb{N}} \in D$ converge to $y \in D$, then it must hold that (sequence definition of continuity): $$\lim _{k \rightarr... | cats | 58,458 | <p>Kahen basically has it, but here it is explicitly. There's a sequence of polynomials $\{p_n(x)\}$ converging to the even function $g(x),$ which is simply the natural extension of $f$ to $[-1, 1].$ But $\{p_n(-x)\}$ converges to $g(-x) = g(x).$ Add up $p_n(-x)$ and $p_n(x)$ and divide by $2.$</p>
|
1,140,527 | <p>How many ways to arrange the books? </p>
<p>I tried to compute by using combination with repetitions that is the 3C22 but I am not so sure.</p>
| Satish Ramanathan | 99,745 | <p>Out of the 8R's and 81's, only two best hands are possible for any two players. The probability of that would be</p>
<p>$$\frac{ {8\choose3}{8\choose1}{5\choose3}{7\choose1}{4\choose2}{32\choose4}{28\choose4}}{{40\choose4}{36\choose4}{32\choose4}{28\choose4}}$$</p>
<p>Reasoning, any one player gets the best hand i... |
1,040,442 | <p>How do I find one value of $x$ in these equations?
$$
\begin{cases}
x \equiv 3 \pmod{5}\\
x \equiv 4 \pmod{7}
\end{cases}
$$</p>
| Barney Chambers | 169,234 | <p>I reckon I've worked it out, so I'll just answer my own question.</p>
<p>Firstly, make sure the GCD of your mods are equal to one. ie in this example:</p>
<pre><code> gcd(5,7) = 1
</code></pre>
<p>Then solve the question in 2 parts since there are 2 simultaneous equations.</p>
<pre><code>x = _______ + ________
<... |
3,319,010 | <p>I would like to learn more about combinatorics of finite sets (including theorems such as Sperner, Erdos-Ko-Rado theorems, LYM inequality). Is there any good book or article for this topic (if possible with problems and exercises)?</p>
| Zbigniew | 11,995 | <p><span class="math-container">$x$</span> and <span class="math-container">$x^2$</span> are considered to be elements of the vectorial space <span class="math-container">$\mathbb{R}[X]$</span> or <span class="math-container">$\mathbb{R}[X]_2 $</span> endowed with the basis <span class="math-container">$1,x,x^2$</span... |
69,900 | <p>Hello!</p>
<p>Given $n$ I would like to find a lower bound (or a tight asymptotics) for the number $s(n)$ of solutions to $$ p_1 + \ldots + p_k \leq n \quad (1) $$ where $k$ is arbitrary and $p_1 \leq \ldots \leq p_k$ are odd prime numbers. I have edited the answer and gave three attempts I tried to use in order t... | David Hansen | 1,464 | <p>If $T(n)$ denotes the number of partitions of $n$ into sums of primes, then </p>
<p>$T(n) = \exp{((\frac{2\pi}{\sqrt{3}}+o(1))\sqrt{\frac{n}{\log{n}}})}$. </p>
<p>See e.g. page 260 of the AMS-Chelsea edition of Ramanujan's collected papers.</p>
|
599,394 | <p>A pack contains $n$ card numbered from $1$ to $n$. Two consecutive numbered cards are removed from</p>
<p>the pack and sum of the numbers on the remaining cards is $1224$. If the smaller of the numbers on</p>
<p>the removing cards is $k$, Then $k$ is.</p>
<p>$\bf{My\; Try}::$ Let two consecutive cards be $k$ and ... | Michael Hoppe | 93,935 | <p>Your last equation becomes $(n+1/2)^2=(16k+9801)/4$. The numerator must be a perfect square.</p>
|
599,394 | <p>A pack contains $n$ card numbered from $1$ to $n$. Two consecutive numbered cards are removed from</p>
<p>the pack and sum of the numbers on the remaining cards is $1224$. If the smaller of the numbers on</p>
<p>the removing cards is $k$, Then $k$ is.</p>
<p>$\bf{My\; Try}::$ Let two consecutive cards be $k$ and ... | Steven Stadnicki | 785 | <p>The key additional piece of information is that you know that $k$ is between $1$ and $n-1$. Since $n(n+1)=2448+(4k+2)$, this means that $n(n+1) \geq 2448+(4\cdot1+2) = 2454$ and that $n(n+1) \leq 2448+4(n-1)+2$, or (rearranging the terms) that $n(n-3) \leq 2446$. You can complete the squares to solve these quadrat... |
3,226,067 | <p>I have a triangle ABC and I know that <span class="math-container">$\tan\left(\frac{A}{2}\right)=\frac{a}{b+c}$</span>, where <span class="math-container">$a,b,c$</span> are the sides opposite of the angles <span class="math-container">$A,B,C$</span>. Then this triangle is:</p>
<p>a. Equilateral</p>
<p>b. Right tr... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: Use that <span class="math-container">$$\tan(\frac{\alpha}{2})=\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}$$</span> where <span class="math-container">$$s=\frac{a+b+c}{2}$$</span>
Using this we get <span class="math-container">$$-{\frac { \left( {a}^{2}+{b}^{2}-{c}^{2} \right) \left( {a}^{2}-{b}^{
2}+{c}^{2} \right) }{ ... |
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... | Leo Alonso | 6,348 | <p>If $f\colon X \to S$ a proper flat of schemes map with $n$-dimensionnal fibers over a noetherian scheme $S$, the relative canonical sheaf $\omega_{X/S}:=H^n(f^!{\mathcal{O}_S})$ is a dualizing sheaf. I guess that this should imply what you want by a GAGA-type argument.</p>
<p>Indeed, being $f$ flat we have that $f... |
3,951,593 | <p>Suppose we have 2 normal distributions <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> with mean <span class="math-container">$u_1$</span> and <span class="math-container">$u_2$</span> and variance <span class="math-container">$\sigma_1^2$</span> and <span class="math-container">$... | angryavian | 43,949 | <p>Let us take for granted that
<span class="math-container">$$\frac{X-\mu_1}{\sigma_1} = \rho \frac{Y-\mu_2}{\sigma_2} + \sqrt{1-\rho^2} Z \tag{$*$}$$</span>
where <span class="math-container">$Z \sim N(0,1)$</span> is independent of <span class="math-container">$Y$</span>. See the end of my answer for an explanation.... |
3,020,024 | <blockquote>
<p>Let <span class="math-container">$f: \Bbb R \to \Bbb R$</span> be a function such that <span class="math-container">$f'(x)$</span> exists and is continuous over <span class="math-container">$\Bbb R$</span>. Moreover, let there be a <span class="math-container">$T > 0$</span> such that <span class="... | Robert Fan | 70,942 | <p>Since $f$ is periodic, it has maximum and minimum. Choose the period $[a,b]$, such that $f(a)=f(b)=M$, where $M$ is the maximum.</p>
<p>It can be seen that, there is a point $f(c)=m$, where $m$ is the minimum. c is golbal minimum, and thus local minimum, so:</p>
<p>$$f'(c)=0$$</p>
<p>from your condition.</p>
<p>... |
3,020,024 | <blockquote>
<p>Let <span class="math-container">$f: \Bbb R \to \Bbb R$</span> be a function such that <span class="math-container">$f'(x)$</span> exists and is continuous over <span class="math-container">$\Bbb R$</span>. Moreover, let there be a <span class="math-container">$T > 0$</span> such that <span class="... | p4sch | 530,357 | <p>We don't even need that <span class="math-container">$f'(x)$</span> is continuous! It is enough to show that <span class="math-container">$f(x) \ge 0$</span> on <span class="math-container">$[0,T]$</span>, because of periodicity. Since <span class="math-container">$[0,T]$</span> is compact, <span class="math-contain... |
1,610,798 | <p>If two numbers are less than a given number, how can we algebraically show that their difference is also less than the given number .
Both numbers are greater than zero and in $\mathbb{Q}$.</p>
| Kevin Quirin | 267,868 | <p>Let $x,y\in\mathbb Q^+$, such that $x < z$, $y< z$. WLOG, suppose $x \leq y$.</p>
<p>As $x$ is positive, $y-x \leq y$ (because $y - (y-x) = x \geq 0$), and $y< z$. By transitivity, $y-x < z$.</p>
|
222,320 | <p>Let $f(x)$ be a continuous real function s.t $f(x_0) > 0$</p>
<p><strong>Prove</strong>: There is some interval of the form $(x_0 -\delta, x_0 + \delta)$ where $f$ is positive.</p>
<p><strong>Proof</strong>:</p>
<p>Since $f$ is continuous: $\forall \,{\epsilon > 0}\,\, \exists \,{\delta>0}$ s.t. $|x- x_0... | Community | -1 | <p>Your proof by contradiction is incorrect. Specifically, the following statements are incorrect.</p>
<blockquote>
<p>This means that $f(x_0) - \epsilon < f(x) < f(x_0) + \epsilon < 0$. Hence we have a contradiction since $\epsilon$ and $f(x_0)$ are both greater than zero.</p>
</blockquote>
<p>You can arg... |
1,995,165 | <p>The Bauer–Fike theorem states that if $A = VDV^{-1}$ for $D = \operatorname{diag}(\lambda_{1},\ldots,\lambda_{n})$ is a diagonalizable matrix and $\tilde{A}=A+E$ is a perturbed matrix than the eigenvalues of $\tilde{A}$ lie in the union of the discs $D_{i}$ where every $D_{i}$ is centered at $\lambda_{i}$ and has ra... | Frits Veerman | 273,748 | <p>Empty, no; vanishing radius, yes. By definition, $\kappa$ cannot vanish (see <a href="https://en.wikipedia.org/wiki/Bauer%E2%80%93Fike_theorem" rel="nofollow noreferrer">here</a>), but a trivial perturbation $E=0$ yields $||E||=0$, and therefore the disks have zero radius. Note that the Bauer-Fike theorem allows the... |
4,401,567 | <p>The question is that find all numbers <span class="math-container">$x$</span> such that
<span class="math-container">$x+3^x <4$</span>.</p>
<p>This is a question from problem <span class="math-container">$4$</span> from Spivak Calculus book,</p>
<p>My attempt:</p>
<p>It easy to see that <span class="math-containe... | egreg | 62,967 | <p>Once you show that the function is strictly increasing, you're almost done. You just need to find <span class="math-container">$x_0$</span> such that <span class="math-container">$f(x_0)=4$</span> and you'll have that</p>
<ul>
<li>for <span class="math-container">$x<x_0$</span>, <span class="math-container">$f(x)... |
2,093,152 | <p>$A$ is a $2\times3$ matrix, $B$ is a $3\times2$ matrix, $\text{rank}(A)=\text{rank}(B)=2$</p>
<p>Does always $\text{rank}(AB)$ equal to $2$?</p>
| Sachchidanand Prasad | 249,258 | <p>See this is Legender's Equation, and the Legender's polynomial has some special properties. See <a href="http://mathworld.wolfram.com/LegendreDifferentialEquation.html" rel="nofollow noreferrer">Legendre's Differential Equation.</a></p>
<p><a href="https://i.stack.imgur.com/ps6NP.jpg" rel="nofollow noreferrer"><img... |
573,619 | <blockquote>
<p>I want to show that the $x$-axis is closed. </p>
</blockquote>
<p>Below is my attempt - I would appreciate any tips on to improve my proof or corrections:</p>
<p>Let $(X,d)$ be a metric space with the usual metric.<br>
Want to Show: $\{(x,y) | x ∈ \Bbb R, y = 0\}$ is closed</p>
<p>Claim: $\{(x,y) |... | Tom | 103,715 | <p>Another fun way you might approach this problem is to let $f : \mathbb{R}^2 \to \mathbb{R}$ be defined by $f(x,y) = y^2$. If you know/can show that $f$ is continuous, then it will imply that $f^{-1}(\{0\})$ is closed.</p>
|
573,619 | <blockquote>
<p>I want to show that the $x$-axis is closed. </p>
</blockquote>
<p>Below is my attempt - I would appreciate any tips on to improve my proof or corrections:</p>
<p>Let $(X,d)$ be a metric space with the usual metric.<br>
Want to Show: $\{(x,y) | x ∈ \Bbb R, y = 0\}$ is closed</p>
<p>Claim: $\{(x,y) |... | Darry | 411,975 | <p>You can as well show that the interior of $R$ is empty in the standard topology. Since any open ball around every point of $R$ contains points in $R$ and outside $R$ then the boundary of $R$ is $R$. By definition, closure of is the union of $R$ and its boundary. Therefore Closure of $R$ is just $R$.</p>
<p>Please c... |
4,198,478 | <p>Here is the question
<a href="https://i.stack.imgur.com/OcAuN.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/OcAuN.png" alt="enter image description here" /></a>
I am on my second week of learning inferential statistics at a high pace, so I apologize if this is a trivial question.</p>
<p>How exac... | Bernard Pan | 800,888 | <p>Your idea is correct. The most important thing here is to figure out the bounds for integration.</p>
<p>To calculate <span class="math-container">$f_X(x)$</span> at a fixed point <span class="math-container">$x_0$</span>, we need to integrate <span class="math-container">$f_{X,Y}(x,y)$</span> over <span class="math-... |
318,754 | <p>This is more of a philosophical or historical question, and I can be totally wrong in what I am about to write next.</p>
<p>It looks to me, that complex-analytic geometry has lost its relative positions since 50's, especially if we compare it to scheme theory. <em>Are there internal mathematical reasons for why tha... | Vincenzo Zaccaro | 80,084 | <p>For what regards Intersection Theory in Analytic Geometry, in my (nonexpert!) view it is important to take a look at the work of D. Barlet on algebraic cycles on analytic spaces. In his paper <em>“Espace analytique r´eduit des cycles analytiques complexes
compacts d’un espace analytique complexe de dimension finie”,... |
1,237,425 | <p>I have to prove the following:</p>
<p>If $a, b \in \mathbb{C}$ and are both algebraic over $\mathbb{Z}$, then:</p>
<ol>
<li><p>$a + b$ is algebraic over $\mathbb{Z}$ </p></li>
<li><p>$a - b$ is algebraic over $\mathbb{Z}$</p></li>
<li><p>$ab$ is algebraic over $\mathbb{Z}$</p></li>
</ol>
<p>I tried this for the f... | lhf | 589 | <p>The easiest way is by using these facts, which are easily proved:</p>
<ul>
<li><p><span class="math-container">$a\in\mathbb C$</span> is algebraic iff <span class="math-container">$\mathbb Q[a]$</span> is finite-dimensional over <span class="math-container">$\mathbb Q$</span>.</p>
</li>
<li><p>If <span class="math-c... |
2,220,872 | <p>Let $R \subseteq A \times A$ be a irreflexlive relation($\forall a \in A: (a,a) \notin R)$. I want to proof that if $R$ is dense, then $A$ cannot be finite. A relation $R$ is dense if $\forall (a,b) \in R$: there is $(a,c) \in R$ and $(c,b) \in R$.<br>
I am not exactly sure where to start on this proof. I tried to ... | user3865391 | 499,516 | <p>No proof exists since your claim is false. Given a finite set $A$ with $|A|\geq 3$, the relation $R$ given by:</p>
<p>$$R=\{(x,y)\in A\times A:x\neq y\}$$</p>
<p>Serves as a contradiction. As it is finite, dense, and irreflexive.
For more details, see this <a href="https://math.stackexchange.com/questions/2921328... |
3,163,123 | <p>I am trying to prove that the determinant of a magic square, where all rows, columns and diagonal add to the same amount, is divisible by 3. </p>
<p>I proved it for magic squares which have entries <span class="math-container">$1,\ldots, 9$</span>, but it turns out I need to show it for magic squares which can have... | FredH | 82,711 | <p>Let the three rows of the magic square be <span class="math-container">$r_1$</span>, <span class="math-container">$r_2$</span>, and <span class="math-container">$r_3$</span>. Since the determinant is unchanged by row operations that add a multiple of one row to another, the matrix with rows <span class="math-contai... |
2,825,704 | <p>Dear optimization experts,</p>
<p>My apologies for asking probably the well-known relation between the Huber-loss based optimization and $\ell_1$ based optimization. However, I am stuck with a 'first-principles' based proof (without using Moreau-envelope, e.g., <a href="https://math.stackexchange.com/questions/1030... | Alex Shtof | 5,073 | <p>The idea is much simpler. Use the fact that
$$\min_{\mathbf{x}, \mathbf{z}} f(\mathbf{x}, \mathbf{z}) = \min_{\mathbf{x}} \left\{ \min_{\mathbf{z}} f(\mathbf{x}, \mathbf{z}) \right\}.$$
In your case, the solution of the inner minimization problem is exactly the Huber function.</p>
|
2,825,704 | <p>Dear optimization experts,</p>
<p>My apologies for asking probably the well-known relation between the Huber-loss based optimization and $\ell_1$ based optimization. However, I am stuck with a 'first-principles' based proof (without using Moreau-envelope, e.g., <a href="https://math.stackexchange.com/questions/1030... | Steph | 993,428 | <p>Consider the proximal operator of the <span class="math-container">$\ell_1$</span> norm
<span class="math-container">$$
z^*(\mathbf{u})
=
\mathrm{argmin}_\mathbf{z}
\
\left[
\frac{1}{2}
\| \mathbf{u}-\mathbf{z} \|^2_2
+
\lambda \| \mathbf{z} \|_1
\right]
=
\mathrm{soft}(\mathbf{u};\lambda)
$$</span></p>
<p>In your... |
897,324 | <p>How many boundary values are needed to uniquely determine an automorphism of unit disk?</p>
<p>More precisely, find the $n \in \mathbb{N} $ such that for any two sets $ \{ x_1, x_2, ..., x_n \} $ and $ \{ y_1, y_2, ..., y_n \} $ of $n$ distinct points on $ \mathbb{S}^1 $, there exists a unique automorphism $ f: \ma... | Emrys-Merlin | 153,873 | <p>I realize I am perhaps a little late to answer this question, but I was wondering about the same thing and it took me quite some time to figure it out. So I hope other people might find this solution in the future.</p>
<p>I find the question easier to answer in the upper half plane model $\mathbb{H}$. Using for exa... |
3,336,886 | <p>How do I factorize this thing?<br>
<span class="math-container">$x^4+x^2+1$</span><br>
I tried to solve the integral <span class="math-container">$\int{\frac{1}{x^4+x^2+1}}$</span> and after trying some substitutions that did not work, I plugged the integral into an integral calculator and it turns out that <span cl... | BallBoy | 512,865 | <p>It's a (somewhat hidden) difference of squares:</p>
<p><span class="math-container">$$\begin{align}
x^4+x^2+1&=(x^4+2x^2+1)-x^2\\&=(x^2+1)^2-x^2\\&=((x^2+1)+x)((x^2+1)-x)\end{align}$$</span></p>
|
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{\... | Lion Heart | 809,481 | <p><span class="math-container">$$\sin 5x = \cos (\frac{\pi}{2}-5x)= \cos 3x $$</span></p>
<p><span class="math-container">$$3x=\frac{\pi}{2}-5x+2k\pi$$</span></p>
<p><span class="math-container">$$x=\frac{\pi}{16}+\frac{k\pi}{4}=\frac{\pi}{16}(1+4k)$$</span></p>
<p>or</p>
<p><span class="math-container">$$3x=-(\frac{\... |
821,654 | <p>I have a Taylor series problem, well more precisely a Maclaurin series.</p>
<p>I am trying to find convergence of: $f(x) = e^{x^3} + e^{{2x}^3}$</p>
<p>Okay here goes:</p>
<p>$$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+... | JPi | 120,310 | <p>A somewhat different way of putting this is that if you know that $x=20$ is a solution to $x^2-x=380$ then that implies that you can write</p>
<p>$$x^2-x-380 = 0 \Leftrightarrow (x-20) f(x) =0,$$</p>
<p>for some function $f$. In this case $f(x)=x+19$ so the other value of $x$ that makes $x^2-x-380$ equal to zero ... |
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-... | thecuriousguy | 928,686 | <p>A straightforward method is:</p>
<ol>
<li>Take the normal of the plane : <span class="math-container">$\vec{i}$</span> + <span class="math-container">$\vec{j}$</span> + 2.<span class="math-container">$\vec{k}$</span> ( A line with direction cosines <1, 1, 2>)</li>
<li>Take two points on the given line; for exa... |
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... | individ | 128,505 | <p>Actually, you should write a general formula, and these issues are constantly emerging.
Solutions of the equation: <span class="math-container">$$x^2+(x\pm{a})^2=z^2$$</span>
Can be written using the solutions of Pell's equation: <span class="math-container">$p^2-2s^2=\pm{a}$</span></p>
<p>And have the form:</p>
... |
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">$(\frac{1+k}{2},\frac 72)$<... | Alexey Burdin | 233,398 | <p><span class="math-container">$$(\frac{k+1}{2},\frac{7}{2})+t(1,k-1)=(0,4)$$</span>
<a href="https://www.wolframalpha.com/input/?i=%7B%28k%2B1%29%2F2%2C7%2F2%7D%2Bt*%7B1%2Ck-1%7D%3D%7B0%2C4%7D" rel="nofollow noreferrer"><span class="math-container">$$\begin{cases}
k+1+2t=0,\\
7-t(k-1)=4
\end{cases}$$</span></a>
Since... |
21,318 | <p>Let $\mathcal V$ be a monoidal category and let $\mathcal C$ be a $\mathcal V$-category. Let's denote the $\mathcal V$-valued hom-functor $[-,-]$. Now for every object $X\in\mathcal C$ we have it's endomorphism object $\mathcal End(X):=[X,X]$ - it is actually a monoid in $\mathcal V$. Can the assignment
$$X\mapsto \... | Alexander Kurz | 68,799 | <p>One way to get a handle on the fact that <span class="math-container">$X\mapsto [X,X]$</span> is of 'mixed variance' was invented by Dana Scott around 1970. He showed how to solve the 'domain equation' <span class="math-container">$X\cong [X,X]$</span> to obtain a model of untyped lambda calculus.</p>
<p>The basic i... |
1,673,399 | <p>Let the ring of $p$-adic integers be the projective limit
$$
\mathbb{Z}_p = \varprojlim_{n\in\mathbb{Z}_{\geq 1}}(\mathbb{Z}/p^n\mathbb{Z}),
$$
and denote an element $x\in\mathbb{Z}_p$ as a sequence $x=(x_n)_{n\in\mathbb{Z}_{\geq 1}}$ with $x_n \in \mathbb{Z}/p^n\mathbb{Z}$.</p>
<p>I want to prove the following:</... | Community | -1 | <p>The mean and variance of the distribution are</p>
<p>$$m=e^{\mu+\sigma^2/2},v=(e^{\sigma^2}-1)e^{2\mu+\sigma^2}.$$</p>
<p>Hence,</p>
<p>$$\sigma^2=\ln\left(\frac v{m^2}+1\right),\mu=\ln(m)-\frac{\sigma^2}2.$$</p>
<p>These are not perfect estimators, but a good start.</p>
|
1,951,733 | <blockquote>
<p>Can any polynomial $P\in \mathbb C[X]$ be written as $P=Q+R$ where $Q,R\in \mathbb C[X]$ have all their roots on the unit circle (that is to say with magnitude exactly $1$) ? </p>
</blockquote>
<p>I don't think it's even trivial with degree-1 polynomials... In this supposedly simple case, with $P(X)=... | Aneesh Saripalli | 374,577 | <p>Numbers are conceptually independent of their digits (as shown by the difference in bases but congruence if applied to counting sticks). The purpose of divisibility tricks is meant only to apply to the numbers of base 10, effectively ignoring the true value of the number and focusing plainly on the external digits i... |
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... | Anton Petrunin | 1,441 | <p>For usual flat origami, essentially you have one rule (you should not use word axiom here, although everyone does). That is 2 points go to 2 lines --- the rest can be obtained applying this rule few times. </p>
<p>My guess is in higher dimension, everything can be made out of 3 points to 3 planes (but there might b... |
412,407 | <p>Need help showing that if $f$ is analytic and not identically zero on $A$ then if $f(z_0)=0$, there is an integer $k$ such that $f(z_0) = 0 = \dots = f^{(k-1)}(z_0)$ and $f^{(k)}(z_0) \neq 0$. Any hints on how to get it started? Thanks. </p>
| Hagen von Eitzen | 39,174 | <p>Otherwise, $f^{(k)}(z_0)=0$ for all $k$, hence the Taylor series around $z_0$ is $0$.</p>
|
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>
| Jack's wasted life | 117,135 | <p>You need two applications of AM-GM
$$
\color{blue}{a^4+b^4}+8\ge\color{green}{2a^2b^2}+8\ge2[\color{blue}{(ab)^2+2^2}]\ge2\times\color{green}{2ab\times2}=8ab
$$</p>
|
3,581,475 | <p>We know that for any prime P, the radical R(P)=P. However is the converse of this Statement true. That is, if we know that radical of an ideal I is itself, i.e. R(I)=I, is I prime? I presume it is not but couldn't come with a counterexample. </p>
| orangeskid | 168,051 | <p>For <span class="math-container">$I$</span>, <span class="math-container">$J$</span> ideals we have </p>
<p><span class="math-container">$$R(I\cap J)\subset R(I)\cap R(J) \subset R(I J)$$</span>
so
<span class="math-container">$$R(I\cap J)=R(I)\cap R(J)$$</span></p>
<p>Therefore, if <span class="math-container">$R... |
553,297 | <p>Please help me to evaluate this integral:
$$\large\int_0^{\pi/2}\frac{x}{\sin x}\log^2\left(\frac{1+\cos x-\sin x}{1+\cos x+\sin x}\right)dx$$</p>
| Cody | 13,295 | <p>I have been dabbling with this one off and on and thought I would post some results.</p>
<p>L.T., may I ask where you found this monster?. </p>
<p>Using Integrals idea of using $\displaystyle \frac{1+\cos(x)-\sin(x)}{1+\cos(x)+\sin(x)}=\tan(\frac{\pi}{4}-\frac{x}{2})$, one can make the sub $t=\tan(\frac{\pi}{4}-\f... |
2,451,327 | <p>I have the following propositions and statement</p>
<p>\begin{align}
1. & &q \to \neg p\\
2. & &p\vee s\\
3. && \neg q \to \neg r\\
4. && r\\
\therefore &&s
\end{align}</p>
<p>I need to demonstrate that is true.</p>
| Community | -1 | <p>Choose every possible $m$ of the form $\sum {2^r}$, where $r$´s are less than $n$, and $n$ is travelling through natural numbers. With this you covered an infinite number of numbers in $[0,1]$ and around every one there is an infinite number of other. Now shift $\sum {2^r}$ by $1$ to arrive at $\sum {2^r}+1$. Shift... |
4,541,065 | <p>I need help with calculating the sum of this arithmetic series:<br>
<span class="math-container">$9-6+4- \frac 83 + ... + \frac{256}{729}-\frac{512}{2187}$</span> <br><br>
I watched this math video to try to solve it: <a href="https://youtu.be/BA0uxIaMtMs" rel="nofollow noreferrer">https://youtu.be/BA0uxIaMtMs</a> <... | Esgeriath | 1,021,258 | <p>Arithmetic progression is sequence <span class="math-container">$a_n$</span> such that <strong>difference of consecutive terms</strong> is constant. That is for any <span class="math-container">$n, m$</span>,
<span class="math-container">$ a_{n + 1} - a_n = a_{m + 1} - a_m$</span>. Formulas you gave are true for any... |
168,619 | <blockquote>
<p>Can $a^2+b^2+2ac$ be a perfect square if $c\neq \pm b$? </p>
</blockquote>
<p>$a,b,c \in \mathbb{Z}$.<br>
I have tried some manipulations but still came up with nothing. Please help. </p>
<p>Actual context of the question is:<br>
Let say I have an quadratic equation $x^2+2xf(y)+25$ that I have to m... | N. S. | 9,176 | <p>Just a short observation: we want</p>
<p>$$d^2=a^2+b^2+2ac=(a+b)^2-2ab+2ac $$</p>
<p>Write $d=a+b+e$ then we want</p>
<p>$$(a+b)^2+2ae+2be+e^2=(a+b)^2-2ab+2ac$$</p>
<p>or</p>
<p>$$c= \frac{2ae+2be+e^2+2ab}{2a}=e+b+\frac{2be+e^2}{2a} $$</p>
<p>This tells us that whenever $2a|2be+e^2$ we have a solution, in part... |
533,315 | <p>What is the dimension of vector space of $R$ over $Z_p$ ? I think it is $p$.
Fruitful suggestion on how to look at it would be great. </p>
| egreg | 62,967 | <p>A vector space $V$ over $F=\mathbb{Z}/p\mathbb{Z}$ is, first of all an abelian group with respect to addition. But not any abelian group can be made into a vector space over $F$, for a very simple reason: for each vector $v$ you can write $v=1v$ (where $1\in F$) and do
$$
\underbrace{v+v+\dots+v}_{\text{$p$ summands... |
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... | markvs | 454,915 | <p>The first inclusion follows from the fact that the symmetric difference is inside the union. The second condition foolows from the fact that symmetric difference is disjoint from the intersection.</p>
<p>As for your proof, it is correct but too long.</p>
|
2,385,599 | <p>ABC is a triangle. D is the center of BC . AC is perpendicular to AD. prove that $$\cos(A)\cdot \cos(C)=\frac{2(c^2-a^2)}{3ac}$$
problem and my attempts are shown in images. I cannot find the exact way to the answer.</p>
<p><a href="https://i.stack.imgur.com/PUcka.jpg" rel="nofollow noreferrer"><img src="https://i... | asheshkebab | 327,561 | <p>Simply note that $(1+\frac{1}{n})^n)$ tends to $e$ as $n$ goes to infinity, hence the expression is approximate to $e^{n+1}$ as $n$ becomes sufficiently large. The rest of the proof can be done through an epsilon-delta argument, but it is sufficient to note that this exponential clearly diverges as $n$ goes to infin... |
2,202,473 | <p>Can you please help me for this problem?</p>
<p>Show that the map $x:M\rightarrow \mathbb{R}$, $\mathbb{R}$ has group structures under addition, is defined by $x\left( \left[ \begin{matrix} a& b\\ o& c\end{matrix} \right] \right)$ =$ \log \left( \dfrac {a} {c}\right) $ is a group homomorphism. </p>
<p>Also... | Yahya Fidouh | 207,061 | <p>I'll assume that $M \subset \mathrm{GL}_2(\mathbb{R})$ , where all of the elements in $M$ are upper triangular matrices.</p>
<p>To show that $x$ is a homomorphism, we just need to show that it preserves the identity and the operations, i.e. $x\left( I \right)= 0$ and for any $a,b \in M$, $x(ab)=x(a)+x(b)$.</p>
<p>... |
2,250,932 | <p>I learned in an algebra class that given any category $\mathcal{C}$ of algebras of signature $(\Omega,E)$ where $\Omega$ is the set of function symbols and $E$ is the collection of identities, colimits exists in $\mathcal{C}$. Specifically, the colimit of some diagram $\mathcal{I}$ in $\mathcal{C}$ will be the quoti... | Tony Blair's Witch Project | 445,451 | <p>Direct limits have lots of very nice properties, but their presentations by generators and relations are not typically very transparent. We can take as a generating system the disjoint union of a set of generators of the algebra structures whose direct limit we are looking at; and as a set of relations the union of ... |
2,250,932 | <p>I learned in an algebra class that given any category $\mathcal{C}$ of algebras of signature $(\Omega,E)$ where $\Omega$ is the set of function symbols and $E$ is the collection of identities, colimits exists in $\mathcal{C}$. Specifically, the colimit of some diagram $\mathcal{I}$ in $\mathcal{C}$ will be the quoti... | Community | -1 | <p>You've sort of given a description of generators of the colimit already: you get a generating set by taking (the images of) the elements of objects of $\mathcal{I}$.</p>
<p>In fact, the colimit is precisely the algebra presented by:</p>
<ul>
<li>Generators are the elements of objects of $\mathcal{I}$ (with the obj... |
1,397,991 | <p>Consider the real interval $[0,1)$, this is partially ordered set (totally ordered actually). This set has a upper bound like $1$, and according to Zorn's Lemma each partially ordered set with a upper bound should have at least one maximal element. However, in this set there is no maximal element, i.e., element that... | Community | -1 | <p>You are actually not using Zorn's Lemma which states that if each chain in $[0,1)$ has an upper bound, then $[0,1)$ has a maximal element. However, the chain $\{1- \frac 1 n \}_{n \in \mathbb N}$ has no upper bound in $[0,1)$.</p>
|
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... | whacka | 169,605 | <p>The act of multiplying by a positive scalar is to stretch the number line outward from the origin (or shrink inward if the scaling factor is less than one). If one point on the number line is to the left of another, that fact remains true after stretching. Multiplying by a negative not only stretches/shrinks it but ... |
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... | CiaPan | 152,299 | <p>Imagine two points on a number axis, say $1$ and $3$. Certainly $1$ is to the left of $3$, which we write $$1<3$$ Now, let's multiply both sides by $2$. That means we <em>scale</em> the situation to a twice bigger: $1$ becomes $2$ and $3$ lands at $6$. Of course what was on the left in the pair, is still on the l... |
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... | Tony Scholl | 5,480 | <p>Are you asking whether the structure of the tame inertia group is the same in the henselian case as in the complete case? The answer is "yes", and in fact similar results hold for local henselian rings of higher dimension (with the appropriate definition of tame ramification). See for example SGA1, Expose XIII, Appe... |
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>
| Martín-Blas Pérez Pinilla | 98,199 | <p>By continuity, the image of $f$ will be an interval. Suppose that this interval is $\ne\Bbb R$. Suppose wlog that the interval is bounded above and let be $M = \sup\{f(x) : x\in\Bbb R\} < +\infty$. Take $x_0\in\Bbb R$ s.t. $f(x_0) > M - 1$. As $f$ is increasing.
$$M - 1 < f(x_0) < f(x_0 + 1)\le M.$$
Now,... |
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... | Syd Kerckhove | 108,291 | <p>Let $\{u_1,\dotsc,u_k\}$ be an orthonormal basis for $T_x$.
Now pick a point $p$ on $S$.
$\pi_S(x)$ can then be written as follows:
$$ \pi_S(x)$ = x + \sum_{i=1}^k ((p-x) \cdot u_i)u_i$$</p>
<p>For all points $x$ and $y$ of $\mathbb{E}^n$, watch what happens to the distance:</p>
<p>First (for simplicity later), I ... |
3,290,047 | <p>I understand the solution of <span class="math-container">$m^{2}+1=0$</span> is <span class="math-container">$\iota$</span>. However for sure this solution (<span class="math-container">$(m^{2}+1)^2=0$</span>) should contain four roots. The answer reads <span class="math-container">$\pm \iota$</span> and <span class... | TheSilverDoe | 594,484 | <p>Why should it have four roots ? A <span class="math-container">$4-$</span>degree polynomial has <strong>at most</strong> four distinct roots... </p>
<p>Here <span class="math-container">$(m^2+1)^2=0$</span> implies that <span class="math-container">$m^2+1=0$</span>, then you have <span class="math-container">$m=i$<... |
87,463 | <p>Hi all,</p>
<p>I am trying to slove the recursion equation: $x_{n+1}x_{n−1}=x_n^2(1−4x_n)$ in the form of $x_n=x_n(x_1,x_2)$ or $x_n=x_n(c_1,c_2)$, and finally get the limit of the ratio: $\dfrac{x_n}{x_{n+1}}$.</p>
<p>I tried the way of setting: $x_n=f(n)$, and use the 1st order taylor expansion of $f(n+1)$ and $... | Cristina Serpa | 20,989 | <p>If your goal is get the limit of the ratio there's no need to explicitly solve the recursion equation.
When $n$ is enough big we have $\frac{x_{n+1}}{x_{n+2}}=\frac{x_{n}}{x_{n+1}}=\frac{x_{n-1}}{x_{n}}$.
Then $\frac{x_{n+1}}{x_{n+2}}=\frac{\frac{x_{n}^{2}}{x_{n-1}}\left(1-4x_{n}\right)}{\frac{x_{n+1}^{2}}{x_{n}}\l... |
1,290,444 | <p>It seems pretty clear to me that both of these are at least uncountable (which I think I could prove with some work). It also seems that you should be able to make some diagonal argument about the two, but I'm not really sure how to make that. I've been trying to think of functions between groups and rings and ways ... | Noah Schweber | 28,111 | <p>There are proper-class-many groups and rings - in fact, there are proper class many $X$s, for any reasonable kind of mathematical structure $X$. One way to prove this is the following: given a cardinal $\kappa$, the direct product of $\kappa$-many rings is still a ring, and ditto for groups.</p>
<p>So is there a se... |
1,290,444 | <p>It seems pretty clear to me that both of these are at least uncountable (which I think I could prove with some work). It also seems that you should be able to make some diagonal argument about the two, but I'm not really sure how to make that. I've been trying to think of functions between groups and rings and ways ... | Censi LI | 223,481 | <p>Hint: In the setting of Grothendieck Universe, we can talk about the size the of the set of <strong>small</strong> groups (denoted by <strong>Grp</strong>) and that of <strong>small</strong> rings (denoted by <strong>Rng</strong>), though not very meaningful I think. Then the mapping which sends each small group to ... |
585,924 | <p>Given a function $$F(x) = \int_0^x \frac{t + 8}{t^3 - 9}dt,$$ is $F'(x)$ different when $x<0$, when $x=0$ and when $x>0$? </p>
<p>When $x<0$, is $$F'(x) = - \frac{x + 8}{x^3 - 9}$$ ... since you can't evaluate an integral going from a smaller number to a bigger number? That's what I initially thought, but ... | Shubham | 103,522 | <p>Why do you presume we can't evaluate an integral going from a smaller number to a bigger number? The restriction on being positive is for areas, not integrals. The upper and lower limits can be any numbers, within the specified set of numbers(real or complex).</p>
<p>And no, $F'(x)$ has the same expression for all ... |
3,117,111 | <p>For a given <span class="math-container">$n \times n$</span>-matrix <span class="math-container">$A$</span>, the characteristic polynomial of <span class="math-container">$A$</span> is <span class="math-container">$\lambda^n+a_{n-1}\lambda^{n-1}+\cdots+a_1\lambda+a_0$</span>. I am curious to know if we can upper bou... | Ryan Cory-Wright | 603,945 | <p>Generally speaking, you should be able to obtain bounds via the Gershgorin circle theorem (see <a href="https://en.wikipedia.org/wiki/Gershgorin_circle_theorem" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Gershgorin_circle_theorem</a>, or Horn and Johnson's book for a more detailed analysis).</p>
<p>For... |
1,095,334 | <p>My question is about a cubic graph $G$ that is the edge-disjoint union of subgraphs isomorphic to the graph $H$ that is as below:</p>
<p><img src="https://i.stack.imgur.com/K0fEM.png" alt="enter image description here"></p>
<p>I want to prove that $0$ is an eigenvalue of the adjacency matrix of $G$.</p>
<p>I thin... | kpax | 97,229 | <p>for the adjacency matrix of $G$ we can build an eigenvector $X$ where $AX=0$,for any vertex of degree 3,I mean similar to 1 and 2 put 2 and for others put -1,this vector will be the eigenvector of $\lambda=0$.</p>
|
600,063 | <p>If the function $f(x)$ is such that
$$f^2(x)=x+f(x+1),$$
find a closed-form expression for $f$.</p>
<p>I found
$$f(x)=\sqrt{x+\sqrt{x+1+\sqrt{x+2+\sqrt{x+3+\cdots}}}}$$
is such an $f$. Does anyone have other solutions? Thank you.</p>
| GEdgar | 442 | <p>You can always take either square-root, so
$$
f(x)=\pm\sqrt{x\pm\sqrt{x+1\pm\sqrt{x+2\pm\sqrt{x+3\pm\cdots}}}}
$$
Gives you uncountably many solutions...</p>
|
3,065,659 | <p>I have been looking online and on lecure notes and I have observed that there are 2 definitions for the completeness axiom and I cannot relate them together.</p>
<p>These are:</p>
<p>1) Every non-empty set of real numbers that is bounded above has
a supremum. Every non-empty set of real numbers that is bounded
bel... | Mohammad Riazi-Kermani | 514,496 | <p>If an increasing sequence has a supremum, then it is bounded above and by (2) it does converge to the supremum. </p>
<p>Similarly if a decreasing sequence has an infimum, then it is bounded below and it converges to its infimum. </p>
<p>Thus your statement is valid for monotonic sequences that the existence of sup... |
4,224,471 | <p>These two statements are equivalent.</p>
<p>a) x(a + b) = 8 + 5a</p>
<p>b) xa + xb = 8 + 5a</p>
<p>So why is that if we solve X for both of them... They both have different answers. For example a) will equal to x = 8 + 5a / a + b</p>
<p>But b) will equal to x + x = 8 + 5a / a + b, because essentially in b), we will ... | John Dawkins | 189,130 | <p>(I find it more convenient to work with <span class="math-container">$\alpha :=1/\lambda$</span>.)</p>
<p>Heuristically, the transition operators of the Feller process are related to <span class="math-container">$\mathcal L$</span> by
<span class="math-container">$$
P_t =e^{t\mathcal L}\qquad\qquad (1)
$$</span>
The... |
656,791 | <p>For all $n\geq 1$, prove with mathematical induction </p>
<p>$\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$</p>
<p>So far.. I have substituted 1 and saw that the statement is true and I have plugged in n+1 to show that the proof is true for all integers but I don't know how to g... | Daniel W. Farlow | 191,378 | <p>This exercise shows that the sum of the reciprocals of the squares converges to something at most $2$; in fact, the series converges to $\frac{\pi^2}{6}$. </p>
<p>For $n\geq 1$, denote the statement in the exercise by
$$
S(n) : 1 + \frac{1}{4} + \frac{1}{9} + \cdots + \frac{1}{n^2} \leq 2 - \frac{1}{n}.
$$</p>
<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... | Henry | 6,460 | <p>A completed run finishes when a different face turns up next. </p>
<p>There are $24$ times a different face can turn up next and the probability for each is $\frac{1}{2}$ so the expected number of completed runs is $24 \times \frac12$. </p>
|
2,426,659 | <p>I have a very basic question on wether or not we use the axiom of choice when we prove the very simple fact that the union of open sets of $\mathbb{R}$ (defined as unions of open intervals) is an open set of $\mathbb{R}$. </p>
<p>Say that $(U_i)_{i\in I}$ is a family of open sets of $\mathbb{R}$. So each $U_i$ is o... | Asaf Karagila | 622 | <p>Hagen wrote correctly, that we don't need to choose, and we can take all of the intervals.</p>
<p>But actually, more is true. Every open set of real numbers has a unique decomposition into pairwise disjoint intervals. Simply look at the connected components of the open set.</p>
<p>So in this case there is absolute... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.