qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
2,419,057 | <p>Trying to assimilate the meaning of the differential I have looked for different examples of functions which:</p>
<ol>
<li>Admits all directional derivatives but are not continuous $(f: \mathbb{R}^2 \rightarrow \mathbb{R} \quad ,\quad (x,y) \mapsto
\begin{cases}
0 & \text{for } (x,y)=(0,0) \\
\frac... | Sangchul Lee | 9,340 | <p>Define the function $f:\mathbb{R}^2 \to \mathbb{R}$ such that $f(0) = 0$ and</p>
<p>$$ f(r\cos\theta, r\sin\theta) = r \left| \cos\theta \right|^{1/r}.$$</p>
<p>Then $f$ is continuous everywhere and satisfies</p>
<p>$$ D_v f(0) = \begin{cases}
|v|, & \text{if } v = (v_1, 0) \\
0, & \text{otherwise}
\end{c... |
1,717,149 | <p>Is it true or false that if $V$ is a vector space and $T:V \to W$ is a linear transformation such that $T^2 = 0$, then $Im(T) \subseteq Ker(T)$ ?<br>
I don't understand it that much. It doesn't seem related... I can have a vector $v$ from $V$ that its power by 2 equals zero but $T(v) \neq 0_{v}$ </p>
| Community | -1 | <p>I recommend the Marcus du Sautoy's book: <a href="http://rads.stackoverflow.com/amzn/click/0007214626" rel="nofollow noreferrer"><strong>Finding Moonshine</strong> : Mathematician's Journey Through Symmetry</a><br>
<a href="https://i.stack.imgur.com/wTUyG.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgu... |
1,717,149 | <p>Is it true or false that if $V$ is a vector space and $T:V \to W$ is a linear transformation such that $T^2 = 0$, then $Im(T) \subseteq Ker(T)$ ?<br>
I don't understand it that much. It doesn't seem related... I can have a vector $v$ from $V$ that its power by 2 equals zero but $T(v) \neq 0_{v}$ </p>
| Guru | 743 | <p>I would suggest to read, Simon Singh's Fermat's Last Theorem. <a href="http://simonsingh.net/books/fermats-last-theorem/" rel="nofollow">http://simonsingh.net/books/fermats-last-theorem/</a></p>
<p>It starts with the introduction to simple problem, solutions. And entire history as it evolves to solve the problem su... |
468 | <p>Textbook writers are blessed with only solving problems with neat answers. Numerical coefficients are small integers, many terms cancel, polynomials split into simple factors, angles have trigonometric functions with known values. Pure bliss.</p>
<p>The "real life" is different (as any of us knows).</p>
<p>Giving ... | quid | 143 | <p>I agree that such exercises have a built in risk of giving the students the wrong idea what is the generic situation and what is quite a special case. </p>
<p>However, "the risk that the student knows she is wrong when some crooked formula/value shows up" you mention is more a feature in my opinion. It is not a bad... |
201,807 | <p>I heard this problem, so I might be missing pieces. Imagine there are two cities separated by a very long road. The road has only one lane, so cars cannot overtake each other. $N$ cars are released from one of the cities, the cars travel at constant speeds $V$ chosen at random and independently from a probability di... | joriki | 6,622 | <p>There are already two answers that show that under a certain interpretation of the question the answer is the $N$-th harmonic number. This can be seen more directly by noting that the $k$-th car is the "leader" of a group iff it is the slowest of the first $k$ cars, which occurs with probability $1/k$. Thus the expe... |
1,377,412 | <p>I am brand new to ODE's, and have been having difficulties with this practice problem. Find a 1-parameter solution to the homogenous ODE:$$2xy \, dx+(x^2+y^2) \, dy = 0$$assuming the coefficient of $dy \ne 0$
The textbook would like me to use the subsitution $x = yu$ and $dx=y \, du + u \, dy,\ y \ne 0$
Rewriting t... | Michael Hardy | 11,667 | <p>The variables $y$ and $u$ can be separated from each other:
$$
2u(y \, du + u \, dy) + (u^2+1 ) \, dy=0
$$
$$
2u\left( du + u \, \frac{dy} y \right) + (u^2+1 ) \, \frac{dy} y = 0
$$
$$
2u\left( \frac{du} u + \frac{dy} y \right) + \left( u + \frac 1 u \right) \, \frac{dy} y = 0
$$
$$
2 \left( \frac{du} u + \frac{dy} ... |
2,432,213 | <p>I am having a really hard time understanding this problem. I know that for uniqueness we need that the derivative is continuous and that the partial derivative is continuous. I also know that the lipschitz condition gives continuity. I can't figure out what to do with this problem though. </p>
<p><a href="https://i... | Hans H | 481,395 | <p>Here is a hint. Consider the operator $T$ defined below that maps continuous functions into continuous functions. For any $h\in C(-\infty,r]$, let</p>
<p>$$
(Th)(t) := \phi(0) + \int_0^t f(\tau, h(\tau), h(g(\tau)))\ d\tau
$$
when $t>0$ and let $(Th)(t)= \phi(t)$ when $t\leq0$. Notice that $T: C(-\infty,r] ... |
4,617,031 | <p>How would I order <span class="math-container">$x = \sqrt{3}-1, y = \sqrt{5}-\sqrt{2}, z = 1+\sqrt{2} \ $</span> without approximating the irrational numbers? In fact, I would be interested in knowing a general way to solve such questions if there is one.</p>
<p>What I tried to so far, because they are all positive ... | GohP.iHan | 151,481 | <p>Suppose <span class="math-container">$ x \geqslant y $</span>, then</p>
<p><span class="math-container">$$ \begin{array} { r c l }
\sqrt3 + \sqrt2 &\geqslant& \sqrt5 + 1 \\
(\sqrt3 + \sqrt2)^2 &\geqslant& (\sqrt5 + 1)^2 \\
5 + 2 \sqrt6 &\geqslant& 6 + 2 \sqrt5 \\
2( \sqrt6 - \sqrt5) &\geq... |
1,614,274 | <p><br>
I need to find the splitting field of $\; x^2+1 \in \mathbb Z_7 [x] \;$ over $\mathbb Z_7 $.<br><br> The roots of the polynomial are $-i \;$ and $i$. Therefore I would conclude that the splitting field is $\mathbb Z_7(i)$ but that would be more like a guess.<br><br> I don´t really understand splitting field ove... | Hagen von Eitzen | 39,174 | <p>The smallest field extension of $\Bbb F_7$ is $\Bbb F_{49}$. As $\Bbb F_{49}^\times $ is a cyclic group of order $48$, it has an element of order $4$, let's call that $i$. Then $i^4= 1$ but $i^2\ne 1$, meaning that $i^2=-1$, as desired. </p>
<p>Note however that even though I introduced the name $i$ for an element ... |
1,614,274 | <p><br>
I need to find the splitting field of $\; x^2+1 \in \mathbb Z_7 [x] \;$ over $\mathbb Z_7 $.<br><br> The roots of the polynomial are $-i \;$ and $i$. Therefore I would conclude that the splitting field is $\mathbb Z_7(i)$ but that would be more like a guess.<br><br> I don´t really understand splitting field ove... | AdLibitum | 210,743 | <p>Since $x^2+1$ has no roots in $\Bbb Z_7$ and the <em>polynomial is quadratic</em> the splitting field of $X^2+1$ is the quadratic extension
$$
\Bbb Z_7[X]/(X^2+1)
$$
which you can concretely model as the degree $\leq1$ polynomials $aX+b$ with coefficients in $\Bbb Z_7$ with the multiplication rule given (unsurprisin... |
19,373 | <p>I posted this question earlier today on the Mathematics site (<a href="https://math.stackexchange.com/q/3988907/96384">https://math.stackexchange.com/q/3988907/96384</a>), but was advised it would be better here.</p>
<p>I had a heated argument with someone online who claimed to be a school mathematics teacher of man... | Jan | 15,298 | <p>When I was in school, I once got an answer marked as error for having too many digits. IIRC it was in trigonometry and I had just written down as many digits as the calculator displayed. (I was able to discuss it away, but was told to avoid unreasonable amounts of digits in the future)</p>
<p>That was in the 1990s i... |
747,519 | <p><img src="https://i.stack.imgur.com/jYzfz.png" alt="enter image description here"></p>
<p>I tried this problem on my own, but got 1 out of 5. Now we are supposed to find someone to help us. Here is what I did:</p>
<p>Let $f:[a,b] \rightarrow \mathbb{R}$ be continuous on a closed interval $I$ with $a,b \in I$, $a \... | Faqir Chand | 812,679 | <p>Closed intervals are compact in <span class="math-container">$ \mathbb{R}$</span>. <span class="math-container">$f$</span> continuous implies image of <span class="math-container">$f$</span> is compact. By Heine-Borel, we know that compact sets are closed and bounded in <span class="math-container">$\mathbb{R}$</spa... |
102,721 | <p>This is probably a very simple question, but I couldn't find a duplicate.</p>
<p>As everybody knows, <code>{x, y} + v</code> gives <code>{x + v, y + v}</code>. But if I intend <code>v</code> to represent a vector, for example if I am going to substitute <code>v -> {vx, vy}</code> in the future, then the result <... | Edmund | 19,542 | <p>You may use <code>Indexed</code> to represent <code>v</code> as its components in the calculation.</p>
<pre><code>{x, y} + (Indexed[v, #] & /@ Range[2])
(* {x + Indexed[v, {1}], y + Indexed[v, {2}]} *)
</code></pre>
<p>Later when <code>v</code> is assigned the <code>Indexed</code> components will become <code>... |
102,721 | <p>This is probably a very simple question, but I couldn't find a duplicate.</p>
<p>As everybody knows, <code>{x, y} + v</code> gives <code>{x + v, y + v}</code>. But if I intend <code>v</code> to represent a vector, for example if I am going to substitute <code>v -> {vx, vy}</code> in the future, then the result <... | John Doty | 27,989 | <p>I like to wrap lists in some suitable function to block this evaluation. <code>MatrixForm</code> is good because it looks pretty, but an undefined function would also work.</p>
<pre><code>a = MatrixForm[{x, y}] + MatrixForm[v]
</code></pre>
<p>Then, once you have concrete definitions for your arrays, take away the... |
167,575 | <p>I have 6 sets of 4D points. Here is an example of one set :</p>
<pre><code>{{30., 5., 111.925, 113.569}, {30., 7.5, 114.7, 158.286}, {30., 10., 115.625, 206.023},
{30., 12.5, 115.625, 257.528}, {30., 15., 117.475, 294.663}, {30., 17.5, 119.325, 328.03},
{30., 20., 121.175, 357.982}, {30., 22.5, 122.1, 393.646}, {... | J. M.'s persistent exhaustion | 50 | <p>Using the given data:</p>
<pre><code>data = {{30., 5., 111.925, 113.569}, {30., 7.5, 114.7, 158.286},
{30., 10., 115.625, 206.023}, {30., 12.5, 115.625, 257.528},
{30., 15., 117.475, 294.663}, {30., 17.5, 119.325, 328.03},
{30., 20., 121.175, 357.982}, {30., 22.5, 122.1, 393.646},
{3... |
3,167,571 | <p>Let consider a square <span class="math-container">$10\times 10$</span> and write in the every unit square the numbers from <span class="math-container">$1$</span> to <span class="math-container">$100$</span> such that every two consecutive numbers are in squares which has a... | Cesareo | 397,348 | <p>With the help of the slack variables <span class="math-container">$\epsilon_i$</span> and calling</p>
<p><span class="math-container">$$
L(p,\mu,\epsilon,\lambda)= p_1 p_2 p_3 p_4 p_5+\mu _5 \left(p_1-p_2-\epsilon _5^2\right)+\mu _4 \left(p_2-p_3-\epsilon _4^2\right)+\mu _3 \left(p_3-p_4-\epsilon _3^2\right)+\mu _2... |
209,869 | <p>I am very interested in the maximum number of triangles could a connected graph with $n$ vertices and $m$ edges have. For example, if $m\leq n−1$, this number is $0$, if $m=n$, this number is $1$, if $m=n+1$, this number is $2$, and if $m=n+2$, this number is $4$. </p>
| Shahrooz | 19,885 | <p>It is a bound and since it is very long, I wrote it an answer, may be it can be helpful.</p>
<p>Let $G$ be a connected graph with $n$ vertices and $m$ edges. Suppose the eigenvalues of this graph are $\lambda_1\geq \lambda_2\geq\ldots\geq\lambda_n$. We know that $\sum{\lambda_i^3}=6\Delta_G$, where $\Delta_G$ count... |
65,059 | <p>I have two points ($P_1$ & $P_2$) with their coordinates given in two different frames of reference ($A$ & $B$). Given these, what I'd like to do is derive the transformation to be able to transform any point $P$ ssfrom one to the other.</p>
<p>There is no third point, but there <em>is</em> an extra constra... | davidlowryduda | 9,754 | <p>So given two points, an initial frame of reference, and the fact that the 'new y axis', which I will designate $y'$, is in the $xy$ plane, we do not have enough information to determine a unique new frame of reference around the new point. Even if $y'$ were parallel to $y$, we still don't quite have enough i... |
1,319,767 | <p>If we know that $\frac{2^n}{n!}>0$ for every $n\in \mathbb{N}$ and $$\frac{2^n}{n!}=\frac{2}{1}\frac{2}{2}...\frac{2}{n}$$ how to bound this sequence above?</p>
| Timbuc | 118,527 | <p>$$a_n:=\frac{2^n}{n!}\implies\frac{a_{n+1}}{a_n}=\frac{2^{n+1}}{(n+1)!}\frac{n!}{2^n}=\frac2{n+1}\xrightarrow[n\to\infty]{}0$$</p>
<p>so by the ratio test (d'Alembert's), we get that the series</p>
<p>$$\sum_{n=1}^\infty\frac{2^n}{n!}\;\;\;\text{converges}\;\;\;\implies\;\;\;\lim_{n\to 0}\frac{2^n}{n!}=0$$</p>
|
2,115,484 | <p>I am not too sure how to prove that a hyperplane in $\mathbb{R}^{n}$ is convex? So far I know the definition of what convex is, but how do we add that hyperplane in $\mathbb{R}^{n}$ is convex?</p>
<p>Thanks in advance!</p>
| Kuifje | 273,220 | <p>Roughly speaking, you need to show that any two points of the hyperplane of $\mathbb{R}^n$ can be joined by a line segment.</p>
<p>First, lets define what this hyperplane $H$ is:
$$
H=\left\{ \pmatrix{x_1\\ \vdots\\ x_n} \in \mathbb{R}^n \;|\;
a_1x_1+\cdots+a_n x_n = c \right\}
$$
where $a_1,\cdots,a_n \neq 0$ and ... |
1,802,020 | <p>Let $S$ be a set such that if $A,B\in S$ then $A\cap B,A\triangle B\in S,$ where $\triangle$ denotes the symmetric difference operator. I would like to show that if $S$ contains $A$ and $B$, then it also contains $A\cup B, A\setminus B$.</p>
<p>The difference was easy to find, but I am not succeeding with the union... | Bérénice | 317,086 | <p>$$A\cup B=((A \triangle B) \cap A)\triangle B$$</p>
|
2,124,068 | <p>I came across the following problem in a book I was reading on continuous probability distributions:-</p>
<p>$Q.$ Let $Y$ be uniformly distributed on $(0,1)$. Find a function $\phi$ such that $\phi(Y )$ has the gamma density $\Gamma(\frac12,\frac12)$.</p>
<p>I know that the probability density represented by $\Gam... | David G. Stork | 210,401 | <pre><code>Manipulate[
Column[
{TextGrid[{{"x", "y", "z"},
v = {Cos[tt], Sin[tt], tt/10}},
Frame -> All],
Show[
ParametricPlot3D[{Cos[t], Sin[t], t/10}, {t, 0, 30}],
Graphics3D[{Red, PointSize[0.05], Point[v]}]]},
Alignment->Center],
{tt, 0, 30}]
</code></pre>
|
729,054 | <p>Let $f$ be continuous and $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$.</p>
<p>Suppose $|f(x)-x| \leq 2$ holds for all $x$. Is $f$ surjective?</p>
| Thomas Andrews | 7,933 | <p>A homotopy proof (outline.) Not sure how to do it with only analysis.</p>
<p>Assume $x_0\in\mathbb R^2$ is not in the range of $f$. consider $S=\{x\in\mathbb R^2: \left|x-x_0\right|=3\}$ be the circle of radius $3$ around $x_0$. </p>
<p>Then $f:S\to\mathbb R^2\setminus\{x_0\}$ is homotopic with the inclusion map $... |
422,948 | <p>How could I/is it possible to take a fourier transform of text? i.e. What domain would/does text exist in? Any help would be great.</p>
<p>NOTE: I do not mean text as an image. I understand it's value, but I'm wondering if it is possible to map text to some domain and transform text on the basis of letters. This is... | Seth | 229,250 | <p>Not quite in the frequency domain, but there is a way to look for periodic structures in text -- the Index of Coincidence. For normal text the IoC will be pretty much flat. But for text encrypted with, say, an 8-letter key and the Vigenère cipher, the IoC will show a pattern of 7 low values and a spike every 8th.</p... |
4,106,273 | <p>In how many ways can a committee of four be formed from 10 men (including Richard) <br>
and 12 women (including Isabel and Kathleen) if it is to have two men and two women <br></p>
<p>a) Isabel refuses to serve with Richard,</p>
<p>b) Isabel will serve only if Kathleen does, too</p>
<p>My Thoughts : <br>
a) Total nu... | IanJ | 854,548 | <p>You could divide the problem into all the committees without Isabel and all those with her. <span class="math-container">${10 \choose 2} {11 \choose 2}$</span> for those without her. And <span class="math-container">${9 \choose 2} * 1$</span> for those with her.</p>
|
1,255,376 | <blockquote>
<p>For all $q \in \Bbb Q$ there exists $n \in \Bbb Z$ so that $q + n = 271$.</p>
</blockquote>
<p>This is true? Because both $q$ and $n$ are rational numbers and $271$ is an integer thus it's a rational number? </p>
<p>Also,</p>
<blockquote>
<p>For all $n \in \Bbb Z$ there exists $q \in \Bbb Q$ so t... | Newb | 98,587 | <blockquote>
<p>For all $q \in \Bbb Q$ there exists $n \in \Bbb Z$ such that $q + n = 271$.</p>
</blockquote>
<p>False. </p>
<p>$\Bbb Q$ is the set of rationals, i.e. fractions: $\Bbb Q = \{\ldots, -\frac 1 3,- \frac 2 2,-\frac 1 2, 0, \frac 1 2, \frac 2 2, \frac 1 3, \ldots\}$.</p>
<p>$\Bbb Z$ is the set of integ... |
4,253,598 | <p>My textbook states that if <span class="math-container">$f(x) \to 0$</span> as <span class="math-container">$x \to 0$</span> <span class="math-container">$$\lim_{x \to 0} (1+f(x))^\frac{1}{g(x)} = e^l$$</span> where <span class="math-container">$$l=\lim_{x \to 0} \frac{f(x)}{g(x)}$$</span><br />
I try to do this as... | Alessio K | 702,692 | <p>You are taking the limit with respect to <span class="math-container">$x$</span>, so the RHS should not depend on <span class="math-container">$x$</span>. Then you arrive at <span class="math-container">$\lim_{x \to 0} ((1+f(x))^\frac{1}{f(x)})^\frac{f(x)}{g(x)}$</span> and took the limit inside the bracket, which i... |
1,382,479 | <p>I would like to know why $a^p \equiv a \pmod p$ is the same as $a^{p-1} \equiv 1 \pmod p$, and also how Fermat's little theorem can be used to derive Euler's theorem, or vice versa. </p>
<p>Please keep in mind that I have little background in math, and I am trying to understand these theorems to understand the math... | almahed | 258,478 | <p>They are <strong>not</strong> the same, if $\gcd(a,p)>1$, that is, if $p\mid a$, then $a^p\equiv a\pmod p$ is true while $a^{p-1}\equiv 1\pmod p$ is false. It follows that the first is always true for any $a\in\mathbb{Z}$, but the second only in the case where $\gcd(a,p)=1$.</p>
<p>As for the second question, as... |
24,550 | <p>Let <span class="math-container">$H$</span> and <span class="math-container">$K$</span> be affine group schemes over a field <span class="math-container">$k$</span> of characteristic zero. Let <span class="math-container">$\varphi:H\to Aut(K)$</span> be a group action. Then we can form the semi-direct product <span ... | David Ben-Zvi | 582 | <p>The algebraic stack BG is the quotient of BK by the action of H, induced by its action on K. So we can describe coherent sheaves on it (aka reps of G) via descent from BK - i.e. reps of G are H-equivariant sheaves on BK, or H-equivariant objects in Rep K. Now this is not yet the answer you want, since it involves Re... |
416,253 | <p>I am playing around a bit with <span class="math-container">$W^*$</span>-algebras, and I'm trying to come up with a definition for the <span class="math-container">$W^*$</span>-algebraic tensor product. Here is my first attempt:</p>
<p><a href="https://i.stack.imgur.com/WE797.png" rel="noreferrer"><img src="https://... | Matthew Daws | 406 | <p>Something similar was studied by Wiersma in <a href="https://arxiv.org/abs/1506.01671" rel="noreferrer">arXiv:1506.01671 [math.OA]</a>. However, that paper takes a different definition of the universal property: it wants the individual maps <span class="math-container">$\sigma$</span> and <span class="math-containe... |
3,644,870 | <p><strong>Give an example or argue that it is impossible.</strong></p>
<p>I argue that it is impossible because, if <span class="math-container">$(x_n)_n$</span> is a sequence which converges to 0, then <span class="math-container">$(x_n)_n$</span> must be bounded above or below by 0. As <span class="math-container">... | Raoul | 761,097 | <p>Compactness is not needed only the fact that <span class="math-container">$K$</span> is closed: if <span class="math-container">$(x_n)$</span> is a sequence in a closed set that converges to <span class="math-container">$\ell$</span>, then <span class="math-container">$\ell$</span> belongs to said closed set.</p>
|
1,477,871 | <p>If the space $X$ is banach , then I want to show that any linear map $T:X \to X$ is continuous iff the null space is closed. I could show that if $T$ is continuous then the null space is closed. But I am unable to prove the converse. Any hints are appreciated. Thanks</p>
| Arpit Kansal | 175,006 | <p>This is not the solution of your problem but its a interesting result in case of $f$ is linear functional.</p>
<p>Here is a sketch of more general result.Try to fill the gaps in the proof.</p>
<p><strong>Lemma:</strong> Let $f$ be a linear functional on $X$ .A hyperplane $H= \{ x\in X : f(x) = \alpha\}$ is closed ... |
25,778 | <p>I am going to teach a Calculus 1 course next semester, and I have 15 weeks for the course material. The class meets MWF for 50 minutes each. I have taught this class before using the same syllabus, but my colleague shared concerns that my pacing is too fast:</p>
<p>Week 1: Review of Functions</p>
<p>Week 2: Limits a... | David E Speyer | 51 | <p>As fedja says, if your students are doing as well as you say, and you believe the students are similar this year, there isn't a reason to change.</p>
<p>It seems hard to believe, though. I've taught calculus at U Michigan and UC Berkeley, which are generally considered to be good schools, and in 14 weeks with 4 hour... |
1,186,516 | <p>Please the highlighted part in the image below. I don't understand why w(c2) must be larger than s(c1, c2) considering s(c1, c2) is counting the position where c1 + c2 = 0, c1 != 0 and c2 != 0 while w(c2) is only counting position where w(c2) != 0.</p>
<p>Should s(c1, c2) be larger than w(c2)?</p>
<p>Thanks for he... | Surb | 154,545 | <p>Since $|h|\leq H$ we have $\frac{|h|}{H}\leq 1$ and thus $$\frac{|h|^k}{H^k}\leq \frac{|h|^2}{H^2}$$ for every $k\geq 2$. Moreover adding two nonnegative terms (since $0\leq H$) in the sum will also make it larger so that</p>
<p>$$\sum_{k=2}^n\binom{n}{k}|x|^{n-k}\frac{|h|^k}{H^k}H^k \leq \sum_{k=2}^n\binom{n}{k}|x... |
4,380,748 | <p>I suppose by contradiction that <span class="math-container">$x+a$</span> is a factor of <span class="math-container">$x^n-a^n$</span> for all odd <span class="math-container">$n$</span>. In particular for <span class="math-container">$n=1$</span>, we have that <span class="math-container">$x+a$</span> is a factor o... | Brendan Connery | 1,082,137 | <p>Humans always tell the truth
Werewolves always lie.</p>
<p>Each option is a statement identifying how many of them are liars in the group. meaning, that each statement is contradictory to one another, so only one of them can be true. There can only be 1 2 or 3 werewolves, but a werewolf cannot tell the truth. This i... |
2,859,463 | <blockquote>
<p>Prove or disprove. All four vertices of every regular tetrahedron in $ \mathbb{R}^3$ have at least two irrational coordinates.</p>
</blockquote>
<p><a href="https://i.stack.imgur.com/hYrWv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/hYrWv.png" alt="enter image description here"... | Christian Blatter | 1,303 | <p>Choose four suitable vertices of the unit cube $[0,1]^3$, and you have a regular tetrahedron.</p>
|
3,757,222 | <p>Let <span class="math-container">$n_{1}, n_{2}, ... n_{k} $</span> be a sequence of k consecutive odd integers. If <span class="math-container">$n_{1} + n_{2} + n_{3} = p^3$</span> and <span class="math-container">$n_{k} + n_{k-1} + n_{k-2} + n_{k-3} + n_{k-4} = q^4$</span> where both p and q are prime, what is k?<... | quasi | 400,434 | <p>Hints:</p>
<p>The sum of <span class="math-container">$3$</span> consecutive odd integers is always a multiple of <span class="math-container">$3$</span>.</p>
<p>What does that imply about <span class="math-container">$p$</span>?</p>
<p>Can you then find <span class="math-container">$n_1,n_2,n_3$</span>?</p>
<p>The ... |
3,757,222 | <p>Let <span class="math-container">$n_{1}, n_{2}, ... n_{k} $</span> be a sequence of k consecutive odd integers. If <span class="math-container">$n_{1} + n_{2} + n_{3} = p^3$</span> and <span class="math-container">$n_{k} + n_{k-1} + n_{k-2} + n_{k-3} + n_{k-4} = q^4$</span> where both p and q are prime, what is k?<... | salfaris | 453,441 | <p>The idea is to figure out <span class="math-container">$p$</span>, then <span class="math-container">$n_1$</span>, then <span class="math-container">$q$</span> and finally <span class="math-container">$k$</span>.</p>
<p><span class="math-container">$3n_1 + 6 = 3(n_1 + 2) = p^3$</span> implies that <span class="math-... |
584,171 | <blockquote>
<p>Show that every graph can be embedded in $\mathbb{R}^3$ with all
edges straight. </p>
</blockquote>
<p>(Hint: Embed the vertices inductively, where should you
not put the new vertex?)</p>
<p>I've also received a tip about using the curve ${(t, t^2 , t^3 : t \in \mathbb{R} )}$ but I'm not sure ho... | alejopelaez | 1,318 | <p><strong>Hint:</strong> To expand on the hint you were given. If you are able to show that you can put the vertices in such a way that no four of them are coplanar then you are done (Can you see why?).</p>
<p>Now, place the first three vertices, in light of my previous comment, where should you not place the fourth?... |
120,992 | <p>An algorithm book <a href="http://rads.stackoverflow.com/amzn/click/1849967202" rel="nofollow">Algorithm Design Manual</a> has given an description:</p>
<blockquote>
<p>Consider a graph that represents the street map of Manhattan in New York City. Every junction of two streets will be a vertex of the graph. Neigh... | El'endia Starman | 10,537 | <p>Every junction between an avenue and a street is a vertex. As there are $15$ avenues and (about) $200$ streets, there are (about) $15*200=3000$ vertices. Furthermore, every vertex has an edge along an avenue and an edge along a street that connect it to two other vertices. Hence, there are (about) $2*3000 = 6000$ ed... |
120,992 | <p>An algorithm book <a href="http://rads.stackoverflow.com/amzn/click/1849967202" rel="nofollow">Algorithm Design Manual</a> has given an description:</p>
<blockquote>
<p>Consider a graph that represents the street map of Manhattan in New York City. Every junction of two streets will be a vertex of the graph. Neigh... | FangyuanJ | 27,056 | <p>This is so simple, I mean 15 avenues crossing 200 streets, which means there are 15 * 200 = 3000 crossings, i.e. 3000 nodes. </p>
<p>each nodes have upper, lower, left and right neighbors, so for each node, there are 4 edges connecting to the neighbors. However, each edge has been counted twice since node 1 has an ... |
546,123 | <p>Let <span class="math-container">$X$</span> be any uncountable set with the cofinite topology. Is this space first countable?</p>
<p>I don't think so because it seems that there must be an uncountable number of neighborhoods for each <span class="math-container">$ x \in X$</span>. But I am not sure if this is true.... | Brian M. Scott | 12,042 | <p>You are correct: it is not first countable. However, this is not because each point of $X$ has uncountably many nbhds: each point of $\Bbb R$ also has uncountably many nbhds, but $\Bbb R$, being a metric space, is certainly first countable.</p>
<p>To prove that $X$ is not first countable, you must show that some po... |
1,282,545 | <p>If $G$ is a infinite group, then $G$ must have an element of infinite order.</p>
<p>Is this true?</p>
<p>I know that if $G$ is infinite cyclic, then it's isomorphic to $\mathbb Z$. </p>
<p>(I guess fact is irrelevant now)</p>
| Hagen von Eitzen | 39,174 | <p>Consider $(\mathbb Z/2\mathbb Z)^{\mathbb N}$ or $\mathbb Q/\mathbb Z$.</p>
|
1,282,545 | <p>If $G$ is a infinite group, then $G$ must have an element of infinite order.</p>
<p>Is this true?</p>
<p>I know that if $G$ is infinite cyclic, then it's isomorphic to $\mathbb Z$. </p>
<p>(I guess fact is irrelevant now)</p>
| David Holden | 79,543 | <p>in an abelian group the elements of finite order form its <i>torsion</i> subgroup. a torsion group is infinite iff it is not finitely generated. $\Bbb Q / \Bbb Z$, cited by Hagen in his answer is a particularly clear example. </p>
|
1,369,076 | <p>Are there any good "analysis through problems" type books? I've tried reading analysis books but I literally get bored to death, and, until I manage to concoct a way of transforming a normal textbook into a problem book (maybe by trying to prove all the theorems myself, but that probably requires more math maturity ... | Danny | 76,017 | <p>One book that you can get for free online is <em>Introductory Single Variable Real Analysis: A Learning Approach Through Problem Solving</em> by Marcel Finan.</p>
<p>One book that I'd particularly recommend if you're looking for really unique and interesting analysis problems is <em>Real Mathematical Analysis</em> ... |
1,724,419 | <p>I can create a large collection of normalized real valued $n$-dimensional vectors from some random process which I hypothesis should be equidistributed on the unit sphere. I would like to test this hypothesis.</p>
<ul>
<li>What is a good way numerically to test if vectors are equidistributed on the unit sphere? I ... | Raaja_is_at_topanswers.xyz | 286,483 | <p>In the whole vector space defined by your normalised vectors in $\mathbb{R}^n$, you can try to find the inner product of the vectors (in $\mathcal{L}_2$ space) and with the output, you can decide whether it is equidistributed on the unit sphere (n-dimension).</p>
<p>This is one of the numerically reliable method.</... |
3,203,678 | <blockquote>
<p>Find all prime numbers <span class="math-container">$p$</span>, for which there are positive integers <span class="math-container">$m$</span> and <span class="math-container">$n$</span> such that <span class="math-container">$p=m^2+n^2$</span> and <span class="math-container">$p \mid m^3+n^3-4$</span>... | TBTD | 175,165 | <p>This problem is from 2004 Silkroad Mathematical Competition. By the way, if you are into this problems, I think you should really visit AoPS forums, audience here at stack exchange are not very experienced with olympiadic problems.</p>
<p>Here is a short reasoning (similar to above). Check that, <span class="math-c... |
3,772,534 | <p>Tangents to a circumference of center O, drawn by an outer point C, touch the circle at points A and B. Let S be any point on the circle. The lines SA, SB and SC cut the diameter perpendicular to OS at points A ', B' and C ', respectively. Prove that C 'is the midpoint of A'B'.</p>
<p><a href="https://i.stack.imgur.... | Mikael Helin | 418,258 | <p>Use <span class="math-container">$(-1)^{\frac{1}{4}}$</span> to be able to use the series, so you get
<span class="math-container">$$\frac{\cos (\frac{(1+i)x}{\sqrt 2})+\cosh (\frac{(1+i)x}{\sqrt 2})}{2}=\sum_{n=0}^\infty\frac{(-1)^nx^{4n}}{(4n)!}
$$</span>
Then use
<span class="math-container">$$
\cos(a+ib)=\cos a\... |
1,972,927 | <p>What's the function determined by the series $1+\sin(x)\cos(x) + \sin(x)^2 \cos(x)^2 + \cdot \cdot \cdot$?</p>
<hr>
<p>Note, the series converges uniformly.</p>
| lab bhattacharjee | 33,337 | <p>As $\sin x\cos x=\dfrac{\sin2x}2$ and $-1\le\sin2x\le1$ for real $x$</p>
<p>$$\sum_{r=0}^\infty ar^n=\dfrac a{1-r}$$ for $|r|<1$</p>
|
1,690,346 | <p>How can we solve this integral?
$\int_0^1\:\frac{\ln(x)\:\Big[1+x^{-\frac{1}{3}}\Big]}{(1-x)\sqrt[3]{x}}\:dx$</p>
| Ron Gordon | 53,268 | <p>Subbing $x=u^3$ is very natural here and leads to a simple sum...</p>
<p>$$I = \int_0^1 dx \frac{\left (1+x^{-1/3} \right ) \log{x}}{x^{1/3} (1-x)} = 9 \int_0^1 du \, \frac{(1+u) \log{u}}{1-u^3}$$</p>
<p>Expand the denominator...</p>
<p>$$\begin{align}I &= 9 \sum_{k=0}^{\infty} \int_0^1 du \, (1+u) u^{3 k} \l... |
2,522,342 | <p>So far I have only got 9 from just guess and check. I am thinking of using Vieta's Formula, but I am struggling over the algebra. Can someone give me the first few steps?</p>
| Ovi | 64,460 | <p>If $x$ is odd, there we can apply the factoring formula of $a^x + b^x$ to show that $2^x+1$ is divisible by $3$. If $x$ is even, you can factor $2^x-1$ and see that $x$ must be $2$.</p>
|
171,364 | <p>So I'm looking for a function that takes in the degree of the polynomial and the range of coefficients from -c to c, and outputs a list of all the monic polynomials of that degree and with coefficients in that range.</p>
<p>I already have code to numerically compute the roots and plot in the complex plane, I just n... | ulvi | 1,714 | <p><code>toMonicpol[lis_] :=
x^(Length[lis]) + Dot[lis, Table[x^r, {r, 0, Length[lis] - 1}]]</code></p>
<p><code>pols[deg_, c_] := Map[toMonicpol[#] &, Tuples[Range[-c, c], deg]]</code></p>
<p><code>pols[2, 3]</code></p>
<p><code>{-3 - 3 x + x^2, -3 - 2 x + x^2, -3 - x + x^2, -3 + x^2, -3 +
x + x^2, -3 + 2 ... |
1,111,041 | <p>Given: $y=\log(1+x)$</p>
<p>Show that $y≈x$ if $x$ gets small (less than 1).</p>
<p>I don't think we're supposed to use Taylor series (because they were never formally introduced in class), but I do think we have to differentiate and show that the derivative of $\log(1+x)$ is approximately equal to $\log(1+x)$ on ... | Barry Cipra | 86,747 | <p>By definition of the (natural) logarithm,</p>
<p>$$\log(1+x)=\int_1^{1+x}{du\over u}$$</p>
<p>If $x\approx0$, then ${1\over u}\approx1$ for $1\le u\le 1+x$, in which case</p>
<p>$$\log(1+x)\approx\int_1^{1+x}du=u\Big|_1^{1+x}=(1+x)-1=x$$</p>
<p>(Remark: I wrote $1\le u\le 1+x$ with $x\gt0$ in mind. A more prec... |
2,935,693 | <p>I am trying to prove that </p>
<p><span class="math-container">$(a\to(b\to c))\to((a\to b)\to(a\to c))$</span></p>
<p>holds in natural deduction, in particular when I work backwards from a Fitch style proof I can only get so far:</p>
<p><a href="https://i.stack.imgur.com/w2BYf.png" rel="nofollow noreferrer"><img ... | Mauro ALLEGRANZA | 108,274 | <p>You have only to remove the unnecessary LEM application : lines 3,9,10,11, and you will get the correct derivation :</p>
<p>1) <span class="math-container">$a \to (b \to c)$</span> --- assumed [a]</p>
<p>2) <span class="math-container">$a \to b$</span> --- assumed [b]</p>
<p>3) <span class="math-container">$a$</s... |
1,025,321 | <p>$\ln(1+xy) = xy$</p>
<p>When I try to implicitly differentiate this I get</p>
<p>$\frac{1}{1+xy}(y + xy')$ = (y + xy')</p>
<p>At which point the $(y + xy')$ terms cancel out, leaving no $y'$ to solve for.</p>
<p>However, the answer to this is $-\frac{y}{x}$... How do you get this?</p>
| orangeskid | 168,051 | <p>Let $\phi \colon [0,1) \to [0, \infty)$ a diffeomorphism with inverse $\psi$. Some possible choices: $t \mapsto \frac{t}{1-t}$, $t \mapsto \tan (\frac{\pi}{2}\cdot t)$.</p>
<p>The map
$$x \mapsto \phi(||x||) \cdot \frac{x}{||x||}$$</p>
<p>is a diffeomorphism from $B^n$ to $ \mathbb{R}^n$ with inverse
$$y \mapsto... |
1,025,321 | <p>$\ln(1+xy) = xy$</p>
<p>When I try to implicitly differentiate this I get</p>
<p>$\frac{1}{1+xy}(y + xy')$ = (y + xy')</p>
<p>At which point the $(y + xy')$ terms cancel out, leaving no $y'$ to solve for.</p>
<p>However, the answer to this is $-\frac{y}{x}$... How do you get this?</p>
| Jesus RS | 203,197 | <p>Or you can try $f(x)=x/\sqrt{1-|x|^2}$ for $x\in B^n$.</p>
|
2,010,434 | <p>Say we are given n piles of stones.
Sizes are $s_{1}, s_{2}, .. , s_{n}$, they can be any positive integer numbers.
The game is played by two players, they alternate their moves.
The allowed moves are:
1. Take exactly 1 stone from 1 pile.
2. Take all stones from 1 pile.</p>
<p>Wins the player who mades the last mo... | Reese Johnston | 351,805 | <p>Assuming we only care about positive integers for the time being, notice that if $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ is an integer then it must be at least $1$. If three numbers add up to at least $1$, then at least one of them must be at least $\frac{1}{3}$ - so at least one of $a$, $b$, and $c$ must be no mo... |
355,489 | <p>What are suggestions for reducing the transmission rate of the current epidemics?</p>
<p>In summary, my best one so far is (once we are down to the stay home rule) to discretize time, i.e., to introduce the following rule for the general populace not directly involved in necessary services:</p>
<p><em>If members o... | Steven Landsburg | 10,503 | <p>This is just a slight expansion of my comment.</p>
<p>When the environment changes, behavioral parameters (that is, parameters like <span class="math-container">$M$</span> that describe people's behavior --- in this case the behavior of public servants deciding how many clients to serve each day) are going to chang... |
355,489 | <p>What are suggestions for reducing the transmission rate of the current epidemics?</p>
<p>In summary, my best one so far is (once we are down to the stay home rule) to discretize time, i.e., to introduce the following rule for the general populace not directly involved in necessary services:</p>
<p><em>If members o... | Gerhard Paseman | 3,402 | <p>Here is a suggestion that may get downvoted. I make it not because it is a good idea, but because some modification may lead to a good idea.</p>
<p>Get everyone sick to get better.</p>
<p>If you hunt down my WordPress blog (grpaseman) you will see a mild expansion of this idea. The crux is to interrupt the repli... |
808,144 | <p>Here is a fun looking one some may enjoy. </p>
<p>Show that:</p>
<p>$$\int_{0}^{1}\log\left(\frac{x^{2}+2x\cos(a)+1}{x^{2}-2x\cos(a)+1}\right)\cdot \frac{1}{x}dx=\frac{\pi^{2}}{2}-\pi a$$</p>
| Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\... |
2,467,228 | <p>I attempted to start with the $L_p$ norm and raise it to the power of $p$ but got stuck because I realized that I have no idea how to eliminate the integrand.</p>
<hr>
<p><strong>$L_p$ norm:</strong>
$||f||_p = ||f||_{L_p[a,b]} = (\int_{a}^{b}~|f(x)|^p~~dx)^{\frac{1}{p}}$</p>
<p>$\\$</p>
<hr>
<p><strong>C-norm:... | Fred | 380,717 | <p>We have $|f_n(t)-f(t)| \le ||f_n-f||$ for all $f,f_n \in C$ , all $n \in \mathbb N$ and all $t \in [a,b]$. Hence</p>
<p>$|f_n(t)-f(t)|^p \le ||f_n-f||^p$ for all $f,f_n \in C$ , all $n \in \mathbb N$ and all $t \in [a,b]$.</p>
<p>This gives</p>
<p>$ \int_a^b|f_n(t)-f(t)|^p dt \le \int_a^b ||f_n-f||^p dt =(b-a)||f... |
119,552 | <p>I'm trying to create a function that does the cross product between two lists of the same dimensions, element by element. Each entry of the list is a 3D vector. Something like this:</p>
<pre><code>a = {{a1, a2, a3}, {b1, b2, b3}};
b = {{c1, c2, c3}, {d1, d2, d3}};
listCross[a, b] = {Cross[{a1, a2, a3}, {c1, c2, c3... | Simon Woods | 862 | <p>How about this</p>
<pre><code>furious[a_, b_] := Module[{a1, a2, a3, b1, b2, b3, c},
{a1, a2, a3} = Transpose[a, {2, 3, 4, 1}];
{b1, b2, b3} = Transpose[b, {2, 3, 4, 1}];
c = {-a3 b2 + a2 b3, a3 b1 - a1 b3, -a2 b1 + a1 b2};
Transpose[c, {4, 1, 2, 3}]]
</code></pre>
<p>Timing results (from march's answer) f... |
119,552 | <p>I'm trying to create a function that does the cross product between two lists of the same dimensions, element by element. Each entry of the list is a 3D vector. Something like this:</p>
<pre><code>a = {{a1, a2, a3}, {b1, b2, b3}};
b = {{c1, c2, c3}, {d1, d2, d3}};
listCross[a, b] = {Cross[{a1, a2, a3}, {c1, c2, c3... | march | 29,734 | <p>Interestingly enough, <code>MapThread</code>ing <code>Cross</code> works but is much slower:</p>
<p>Using sample lists:</p>
<pre><code>list1 = Array[c, {20, 20, 20, 3}];
list2 = Array[d, {20, 20, 20, 3}];
</code></pre>
<p>We can perform this operation in the following two ways, using <code>MapThread</code>:</p>
... |
28,104 | <p>It occured to me that the Sieve of Eratosthenes eventually generates the same prime numbers, independently of the ones chosen at the beginning. (We generally start with 2, but we could chose any finite set of integers >= 2, and it would still end up generating the same primes, after a "recalibrating" phase).</p>
<p... | Noldorin | 602 | <p>By missing out only the first prime (2), your statement is indeed correct, but it is a special case. This is because all even numbers > 4 are guaranteed to have a divisor > 2 (i.e. in the set of tested numbers). Likewise, all odd numbers that are not primes will have a divisor within your set, since you start with t... |
117,432 | <p>All varieties are over $\mathbb{C}$. Notions related to weights etc. refer to mixed Hodge structures (say rational, but I would be grateful if the experts would point out any differences in the real setting).</p>
<p>I am trying to get some intuition for the geometric meaning of/when to expect the weight filtration ... | Dan Petersen | 1,310 | <p>I don't have a general answer, but let me add some more examples.</p>
<ol>
<li><p>For your second question, examples of smooth varieties with $H^i$ pure of the 'wrong' weight, a good example is the complement of an affine arrangement of hyperplanes in $\mathbf C^n$. In this case, the Hodge structure on $H^i$ is pur... |
71,822 | <p><em>I have moved this question here from MSE, because I did not receive any answers as of yet over there.</em></p>
<p>I know that there are statements that are neither provable nor disprovable within some set of axioms, and I also know that such statements are called undecidable. Please allow me to call these state... | Emil Jeřábek | 12,705 | <p>Let $T$ be a fixed theory (recursively axiomatized, extending $I\Delta_0+\mathrm{EXP}$, sound). I read your first question as:</p>
<blockquote>
<p>Q1: Is there a sentence $A$ such that $T$ proves that “$T$ does not prove ‘$T$ proves $A$ or $T$ proves $\neg A$’ and $T$ does not prove ‘$T$ does not prove $A$ and $T... |
2,138,241 | <p>I tried to prove $$\lim_{x\to \infty}\frac 1x = 0$$
I started as thus
$$\lim_{x\to \infty}\frac 1x=\lim_{x\to \infty}\frac x{x^2}$$
Applying <a href="https://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule" rel="nofollow noreferrer">L'Hospital's Rule</a>
$$\lim_{x\to \infty}\frac 1x=\lim_{x\to \infty}\frac x{x^2... | Theorem | 346,898 | <p>This is incorrect, as you can only use L'Hospital's Rule when you know the limit of the derivative ratio exists.</p>
|
2,138,241 | <p>I tried to prove $$\lim_{x\to \infty}\frac 1x = 0$$
I started as thus
$$\lim_{x\to \infty}\frac 1x=\lim_{x\to \infty}\frac x{x^2}$$
Applying <a href="https://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule" rel="nofollow noreferrer">L'Hospital's Rule</a>
$$\lim_{x\to \infty}\frac 1x=\lim_{x\to \infty}\frac x{x^2... | Ovi | 64,460 | <p><a href="https://i.stack.imgur.com/sKQRt.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sKQRt.png" alt=""></a></p>
<p>The horizontal lines in the picture are $y = \pm \dfrac 12$. As you can see, after $P = 3$ on the $x$ axis, the values of $f(x)$ are contained on the interval $\left(-\dfrac 12, ... |
2,138,241 | <p>I tried to prove $$\lim_{x\to \infty}\frac 1x = 0$$
I started as thus
$$\lim_{x\to \infty}\frac 1x=\lim_{x\to \infty}\frac x{x^2}$$
Applying <a href="https://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule" rel="nofollow noreferrer">L'Hospital's Rule</a>
$$\lim_{x\to \infty}\frac 1x=\lim_{x\to \infty}\frac x{x^2... | Jones | 414,995 | <p>Your expression In other words:</p>
<p>As x approaches infinity, then 1/x approaches 0 so its answer is 0
Try to think in that way...</p>
<p>Your method is wrong as you can only use L'Hospital's Rule when you know the limit of the derivative ratio exists.</p>
|
3,890,064 | <p>When Max is 8 m from a lamp post which is 6 m high his shadow is 2 m long. When Max is 3 m from the lamp post, what is the length of his shadow?</p>
| Toby Mak | 285,313 | <p>Let the top of the lamp post be at <span class="math-container">$A = (0,6)$</span>, and Max be at <span class="math-container">$B = (8,0)$</span>, with his head being at height <span class="math-container">$h$</span> or <span class="math-container">$H = (8, h)$</span>.</p>
<p>The gradient of line <span class="math-c... |
1,388,565 | <p>Given for example $\omega_1$ coin tosses (i.e. a mapping from the elements of $\omega_1$ to $\{H,T\}$ with independent probabilities half), what is the probability that there is an infinite <del>subsequence</del> subinterval [<em>corrected following comments</em>] consisting only of heads?</p>
<p>Is this question e... | Eric Wofsey | 86,856 | <p>Taking "subsequence" to mean "subinterval", the set you are describing is not measurable, and in fact has outer measure $1$ and inner measure $0$. Indeed, let $S\subset\{0,1\}^{\omega_1}$ be the set of sequences which are constant with value $1$ on some infinite interval in $\omega_1$. Suppose $B\subset S$ is a Bo... |
2,784,784 | <p>Let $f(x)=$$x-1 |x \in \mathbb{Q} \brace 5-x| x \in \mathbb{Q}^c$</p>
<p>Show that $\lim_{x \to a}f(x)$ does not exists for any $a \not= 3$</p>
<p>I first showed that $lim_{x \to 3}f(x)=2$. </p>
<p>I don't know how to approach this part. Can anyone please guide? I was thinking of using density theorem at first t... | Community | -1 | <p>As $(x-1)-(5-x)=2x-6$, you will always find values in the neighborhood of $x$ that have their images $2x-6$ apart so that not all $\epsilon$ can be met. Unless $x=3$.</p>
|
2,495,555 | <p>I want to use Konig's theorem to show that the pictured graph $G$ has no perfect matching. By this theorem it suffices to find a vertex cover of size $|G|/2-1= 20$, but so far I have only been able to find vertex covers of size 21. I'm just doing this by inspection as opposed to using any algorithms, and it's not im... | Donald Splutterwit | 404,247 | <p>Must write at least $30$ charcters.</p>
<p><a href="https://i.stack.imgur.com/kXcag.jpg" rel="noreferrer"><img src="https://i.stack.imgur.com/kXcag.jpg" alt="enter image description here"></a></p>
|
2,908,993 | <p>There are $6967$ notes labelled $1,2,3,4,...,6967$. If you choose $K$ notes at random, what is the smallest number $K$ that would guarantee that you pick <strong>two notes labelled by consecutive numbers</strong>? Use the pigeonhole principle to explain</p>
<p>I'm not quite sure in this question what the pigeons an... | Rushabh Mehta | 537,349 | <p>The <a href="https://en.wikipedia.org/wiki/Pigeonhole_principle" rel="nofollow noreferrer">pigeonhole principle</a> is summarized as the following</p>
<blockquote>
<p>If $n$ items are put into $m$ containers, with $n>m$, then at least one container must contain more than one item.</p>
</blockquote>
<p>It's qu... |
2,908,993 | <p>There are $6967$ notes labelled $1,2,3,4,...,6967$. If you choose $K$ notes at random, what is the smallest number $K$ that would guarantee that you pick <strong>two notes labelled by consecutive numbers</strong>? Use the pigeonhole principle to explain</p>
<p>I'm not quite sure in this question what the pigeons an... | hartkp | 23,399 | <p>Picking all the odd numbers gives a maximal set without neighbours and $3484$ elements.
Divide the set in $\{1,2\}$, $\{3,4\}$, ..., $\{6965,6966\}$, and $\{6967\}$; that's $3484$ pigeon holes, so if you pick $3485$ numbers at least one pigeon hole gets picked from twice.</p>
|
4,116,252 | <p>I'm trying to prove (or disprove) the following:</p>
<p><span class="math-container">$$ \sum_{i=1}^{N} \sum_{j=1}^{N} c_i c_j K_{ij} \geq 0$$</span>
where <span class="math-container">$c \in \mathbb{R}^N$</span>, and <span class="math-container">$K_{ij}$</span> is referring to a <a href="https://en.wikipedia.org/w... | Gunnar Þór Magnússon | 3,225 | <p>I have some comments in a different direction from g g. Maybe they'll be useful to someone.</p>
<p>First, like g g noted, <span class="math-container">$K \geq 0$</span> if <span class="math-container">$n = 1$</span>, so the kernel matrix is semipositive there. For <span class="math-container">$n = 2$</span>, the ker... |
4,150,776 | <p>Let <span class="math-container">$(a_n)_{n=1}^\infty$</span> Let be a positive, increasing, and unbounded sequence. Prove that the series:</p>
<p><span class="math-container">$$\sum_{n=1}^\infty\left(\frac{1}{a_{2n-1}}-\frac{1}{a_{2n}}\right)$$</span></p>
<p>convergent.</p>
<hr />
<p>We know that since <span class=... | Martin R | 42,969 | <p>The fact that <span class="math-container">$(a_n)$</span> is unbounded is actually not needed, only that the sequence is positive and increasing.</p>
<p>First note that all terms <span class="math-container">$b_n = \frac{1}{a_{2n-1}}-\frac{1}{a_{2n}}$</span> are non-negative. Then verify that
<span class="math-conta... |
172,124 | <p>$$\int \frac{1}{x^{10} + x}dx$$</p>
<p>My solution :</p>
<p>$$\begin{align*}
\int\frac{1}{x^{10}+x}\,dx&=\int\left(\frac{x^9+1}{x^{10}+x}-\frac{x^9}{x^{10}+x}\right)\,dx\\
&=\int\left(\frac{1}{x}-\frac{x^8}{x^9+1}\right)\,dx\\
&=\ln|x|-\frac{1}{9}\ln|x^9+1|+C
\end{align*}$$</p>
<p>Is there completely ... | Generic Human | 26,855 | <p>Not really different, but even simpler:
$$\begin{align}
\int\frac{1}{x^{10}+x} dx=&\int\frac{x^{-10}}{1+x^{-9}} dx
=-\frac 1 9 \log |1+x^{-9}| + C
\end{align}$$</p>
|
2,730,407 | <p>How many different numbers must be selected from the first 25 positive integers to be certain that at least one of them will be twice the other ?</p>
| saulspatz | 235,128 | <p>Here's how I did it. Group the integers we get by doubling</p>
<p>A:$1,2,4,8,16$ </p>
<p>B:$3,6,12,24$</p>
<p>C:$5,10,20$</p>
<p>D:$7,14$</p>
<p>E:$9,18$</p>
<p>F:$11,22$</p>
<p>We have $7$ odd numbers from $13$ to $25$ not included and we can take them all. We can take $1$ each from groups $F$, $E$, and $D... |
4,576,868 | <p>I was provided the following generating function, and was unsure how to use it. I have never seen an example where the function “involved” itself.
The generating function is
<span class="math-container">$F(z)^8$</span>
Where
<span class="math-container">$$F(z)=z+z^6 F(z)^5+z^{11} F(z)^{10}+z^{16} F(z)^{15}+z^{21} F(... | Alexander Burstein | 499,816 | <p>Use Lagrange inversion. From the functional equation that <span class="math-container">$F(z)$</span> satisfies, we get, by multiplying through by <span class="math-container">$z$</span>,
<span class="math-container">$$
zF(z)=z^2\left(1+(zF(z))^5+(zF(z))^{10}+(zF(z))^{15}+(zF(z))^{20}\right).
$$</span>
Thus,
<span cl... |
3,003,672 | <p>Say I have an infinte 2D grid (ex. a procedurally generated world) and I want to get a unique number for each integer coordinate pair. How would I accomplish this?</p>
<p>My idea is to use a square spiral, but I cant find a way to make a formula for the unique number other than an algorythm that just goes in a squa... | Michael Stachowsky | 337,044 | <p>This is an interesting question. I will provide a method to simplify your algorithm, but not necessarily a formula just yet (I'm sure that what I'm about to show you will lead to a formula...probably).</p>
<p>Let's begin with a point in the center. That is, we are not starting at the top corner of a semi-infinite... |
2,482,341 | <p>I have tried to solve $\frac{\mathrm{d}}{\mathrm{dx}}\int_{0}^{x^2}e^{x+t}\mathrm{dt}$ by two different ways and I'm getting two answers. Please let me know the mistake: </p>
<p><strong>Method One</strong><br>
Let $F(t)$ be the antiderivative of $e^{x+t}$.<br>
Thus $F^{'}(t)=e^{x+t}$ </p>
<p>So </p>
<p>\begi... | Koshinder | 493,809 | <p>you can also use Heron's formula.
Heron's formula states that the area of a triangle whose sides have lengths a, b, and c is :-
<a href="https://i.stack.imgur.com/1dN0i.png" rel="nofollow noreferrer">Put the upper limits of a=2,b=3,c=4</a></p>
|
2,273,506 | <p>I was able to simplify a boolean expression $$(\neg a*\neg b*c)+(a*\neg b*\neg c)+(a*b*\neg c)+(a*b*c)$$into the form $$\neg b*(a\oplus c)+a*b$$ where $*$ is the logical and, $+$ is the logical or, and $\oplus$ is the logical XOR.</p>
<p>Apparently, from Wolfram Alpha, this expression can be simplified to $$\left(a... | G Tony Jacobs | 92,129 | <p>That expression you got from WA isn't the same as your original. It includes, for example, $(\neg a * b * \neg c)$, which is not included in your original expression. I figured that out by looking at a Venn diagram.</p>
<hr>
<p>Edit: With that correction, WA's solution actually matches up. As for how they obtained... |
2,984,918 | <p>How can I prove this? </p>
<blockquote>
<p>Prove that for any two positive integers <span class="math-container">$a,b$</span> there are two positive integers <span class="math-container">$x,y$</span> satisfying the following equation:
<span class="math-container">$$\binom{x+y}{2}=ax+by$$</span></p>
</blockquote... | nonuser | 463,553 | <p>Because of simmetry we can assume WLOG that <span class="math-container">$a\geq b$</span></p>
<ul>
<li>If <span class="math-container">$a>b$</span>:</li>
</ul>
<p>Let <span class="math-container">$m=2a+1$</span> and <span class="math-container">$n=2b+1$</span> and <span class="math-container">$k= m-n$</span>. S... |
2,984,918 | <p>How can I prove this? </p>
<blockquote>
<p>Prove that for any two positive integers <span class="math-container">$a,b$</span> there are two positive integers <span class="math-container">$x,y$</span> satisfying the following equation:
<span class="math-container">$$\binom{x+y}{2}=ax+by$$</span></p>
</blockquote... | Community | -1 | <p>If <span class="math-container">$a=b$</span> then let <span class="math-container">$(x,y)=(a,a+1)$</span>. </p>
<p>Otherwise, w.l.g. suppose <span class="math-container">$a>b$</span> and let <span class="math-container">$x+y=2t(a-b)$</span> for some positive integer <span class="math-container">$t$</span>. Then ... |
90,812 | <p>How do I use the Edmonds-Karp algorithm to calculate the maximum flow? I don't understand this algorithm $100\%$. What I need to know is about flow with minus arrow. Here is my graph: </p>
<p><img src="https://i.stack.imgur.com/nep5F.jpg" alt="the graph">. </p>
<p>Our $1-6-11-12$, the flow is $4$. On the next iter... | Jesko Hüttenhain | 11,653 | <p>You augment</p>
<ol>
<li>$1-6-11-12$ by $4$</li>
<li>$1-2-4-11-6-7-9-12$ by $3$</li>
<li>$1-3-5-11-6-8-10-12$ by $ 1$</li>
</ol>
<p>and you are done: $12$ is no longer reachable from $1$ in the residual graph. You have already <em>found</em> the maximal flow.</p>
|
3,310,193 | <p>I want to show:
<span class="math-container">$$\mu(E)=0 \rightarrow \int_E f d\mu=0$$</span> </p>
<p><span class="math-container">$f:X \rightarrow [0, \infty] $</span> is measurable and <span class="math-container">$E \in \mathcal{A} $</span></p>
<p>Consider a step function <span class="math-container">$s=\sum_i ... | Tsemo Aristide | 280,301 | <p>The map <span class="math-container">$H_t(x)=tF(x), t\in [0,1]$</span> restricted to the unit ball defines a deformation retract of the unit ball to a point.</p>
|
4,498,296 | <p>Is there any subtle way to compute the following integral?</p>
<p><span class="math-container">$$\int \frac{\sqrt{u^2+1}}{u^2-1}~ \mathrm{d}u$$</span></p>
<p>The solution i had in mind was substituting <span class="math-container">$u=\tan (\theta)$</span>,then after a few calculations the integral became <span class... | Quanto | 686,284 | <p>Note that <span class="math-container">$\int \frac1{\sqrt{u^2+1}}du=\sinh^{-1}u$</span>. Then <span class="math-container">\begin{align}
&\int \frac{\sqrt{u^2+1}}{u^2-1}du
- \int \frac1{\sqrt{u^2+1}}du\\
=&\int \frac2{(u^2-1)\sqrt{u^2+1}} \ du
=2\int \frac{d(\frac u{\sqrt{1+u^2}})}{\frac{2u^2}{u^2+1}-1}
= -... |
3,861,319 | <p>I have 3 features <code>age</code>, <code>income</code> and <code>rating</code>.</p>
<p>In case of age I have 3 buckets.</p>
<p>for income I have 4 buckets.</p>
<p>and for rating I have 2 buckets.</p>
<p>If one could filter data where a person could select 1 or more than 1 bucket from each feature what would be the ... | Prasiortle | 554,727 | <p>Write the expression as <span class="math-container">$4(x+y)+x$</span>, so we can see that it is even precisely when <span class="math-container">$x$</span> is even. Thus the statement becomes "there exists <span class="math-container">$x$</span> such that for all <span class="math-container">$y$</span>, <span ... |
1,548,159 | <p>This is a question asked in India's CAT exam: <a href="http://iimcat.blogspot.in/2013/08/number-theory-questions-and-solutions.html" rel="nofollow noreferrer">http://iimcat.blogspot.in/2013/08/number-theory-questions-and-solutions.html</a> </p>
<blockquote>
<p>How many numbers with distinct digits are possible pr... | Bharathwaj | 561,197 | <p>Two digit numbers; The two digits can be 4 and 7: Two possibilities 47 and 74.</p>
<p>Three-digit numbers: The three digits can be 1, 4 and 7: 3! Or 6 possibilities.</p>
<p>We cannot have three digits as (2, 2, 7) as the digits have to be distinct.</p>
<p>We cannot have numbers with 4 digits or more without repea... |
654,198 | <p>$6x^3 -11x^2 + 6x + 5 \equiv (Ax-1)(Bx - 1)(x - 1) + c$</p>
<p>Find the value of A, B and C.</p>
<p>I started it like this: </p>
<p>$6x^3 -11x^2 + 6x + 5 \equiv (Ax-1)(Bx - 1)(x - 1) + c$</p>
<p>Solving the right hand side:</p>
<p>$ (ABx^2 - Ax - Bx + 1)(x - 1) + C$</p>
<p>$ ABx^3 - ABx^2 - Ax^2 + Ax - Bx^2 + ... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\ x=1\,\Rightarrow\,c = 6.\,$ Cancelling $\,x-1\,$ yields $6x^2-5x+1 = (Ax-1)(Bx-1)\ $ so $\,\ldots$</p>
|
2,065,639 | <p>$\displaystyle \int_a^b (x-a)(x-b)\,dx=-\frac{1}{6}(b-a)^3$</p>
<p>$\displaystyle \int_a^{(a+b)/2} (x-a)(x-\frac{a+b}{2})(x-b)\, dx=\frac{1}{64}(b-a)^4$ </p>
<p>Instead of expanding the integrand, or doing integration by part, is there any faster way to compute this kind of integral?</p>
| Community | -1 | <p>You can first get rid of the integration bounds by the linear transform $a+(b-a)t$:</p>
<p>$$\int_a^b (x-a)(x-b)\,dx=(b-a)^3\int_0^1t(t-1)\,dt.$$</p>
<p>Mentally expanding the polynomial, the integral is $\frac13-\frac12=-\frac16$.</p>
<p>For the other case, $a+(b-a)t/2$:
$$ \int_a^{(a+b)/2} (x-a)(x-\frac{a+b}{2}... |
947,626 | <p>What are the conditions under which the center of a group will have a cyclic subgroup? (with proof, of course)</p>
| orangeskid | 168,051 | <p>Every nontrivial subgroup has a nontrivial cyclic subgroup. The condition is : the center is nontrivial. </p>
|
478,713 | <p>I have this logic statement:</p>
<pre><code> (A and x) or (B and y) or (not (A and B) and z)
</code></pre>
<p>The problem is that accessing A and B are rather expensive. Therefore I'd like to access them only once each. I can do this with an if-then-else construct:</p>
<pre><code>if A then
if x then
tru... | Argon | 27,624 | <p>$x$-interscepts are when the function intersects the $x$ axis, i.e. when $f(x)=0$. Thus</p>
<p>$$0=(x-3)^2 \implies 0=x-3\implies {x=3}$$</p>
|
1,476,982 | <p>I'm trying to understand why the volume of a parallelepiped whos sides are $s,u,w$ is $ V = s \cdot(u \times w)$.</p>
<p>Even the units of measurement don't add up. The length of the vectors $s,u,w$ is measured in centimeters, the volume is measured in cubic cm.</p>
<p>$u\times w$ is a vector. It is a vector that ... | Community | -1 | <p>The norm of the vector $u\times v$ is <a href="https://math.stackexchange.com/questions/1395970/what-is-the-logic-rationale-behind-the-vector-cross-product/1471129#1471129"><em>defined</em> as the area of the parallelogram</a> (scroll down to <em>Geometric Definition</em> under <em>The Cross Product</em> if you clic... |
4,173 | <p>I asked this question on mathoverflow, but it was deemed too simple, so I'm posting here instead -- </p>
<p>Is there a nice way to characterize an orthonormal basis of eigenvectors of the following $d\times d$ matrix?</p>
<p>$$\mathbf{I}-\frac{1}{d} \mathbf{v}\mathbf{v}'$$</p>
<p>Where $\mathbf{v}$ is a $d\times ... | Agustí Roig | 664 | <p>If you write $P= I - \frac{1}{d}vv^t$ (I assume your $v'$ means the transpose of $v$), then you certainly have a symmetric matrix</p>
<p>$$
P^t = \left( I- \frac{1}{d}vv^t\right)^t = I -\frac{1}{d}v^{tt}v^t = I - \frac{1}{d}vv^t = P \ .
$$</p>
<p>So it diagonalizes and has an orthonormal basis of eigenvectors. Tha... |
4,173 | <p>I asked this question on mathoverflow, but it was deemed too simple, so I'm posting here instead -- </p>
<p>Is there a nice way to characterize an orthonormal basis of eigenvectors of the following $d\times d$ matrix?</p>
<p>$$\mathbf{I}-\frac{1}{d} \mathbf{v}\mathbf{v}'$$</p>
<p>Where $\mathbf{v}$ is a $d\times ... | Unkz | 326 | <p>While I haven't taken the time to prove it, if you look at the numbers I think you'll see a very nice pattern if you multiply out all your irrational denominators.</p>
<p>(-1,0,0,0,1) norm=$\sqrt{1 \cdot 2}$</p>
<p>(-1,0,0,2,-1) norm=$\sqrt{2 \cdot 3}$</p>
<p>(-1,0,3,-1,-1) norm=$\sqrt{3 \cdot 4}$</p>
<p>(-1,4,-... |
2,000,013 | <p>On an NFA, how can the empty set ∅ and {ϵ} be considered regular languages? Does it make sense that a machine that accepts no symbols or a machine that takes the empty symbol exist? I could think of a machine (laptop) that is in off mode, where no entries (symbols) are accepted, but, is that of interest? or could it... | MJD | 25,554 | <p>We will play a game. There is a machine with buttons and a green lamp. Your job is to press the right sequence of buttons that make the green lamp light up.</p>
<p>For example, perhaps the machine has only one button, labeled <code>a</code>. When you press it, the lamp lights up! Then you press the button again an... |
2,109,197 | <p><strong>Update:</strong><br>
(because of the length of the question, I put an update at the top)<br>
I appreciate recommendations regarding the alternative proofs. However, the main emphasis of my question is about the correctness of the reasoning in the 8th case of the provided proof (with a diagram).</p>
<p><stro... | Nosrati | 108,128 | <p>The basic concept for this answer is the area of a triangle. The area of a triangle which is made on vectors $z$ and $w$ is $A=\dfrac12{\bf Im}(\bar{z}w)$. for three points in complex plane, we make a triangle on them with sides $z_1-z_2$, $z_2-z_3$ and $z_3-z_1$, we have
\begin{eqnarray}
\dfrac12{\bf Im}\overline{(... |
1,247,185 | <p>I already know, and so ask NOT about, the proof of: <a href="https://math.stackexchange.com/a/463407/53259">$A$ only if $B$ = $A \Longrightarrow B$</a>.<br>
Because I ask only for intuition, please do NOT prove this or use truth tables. </p>
<p><strong>My problem:</strong> I try to avoid memorisation. So wh... | Matt Samuel | 187,867 | <p>There have been some comments about this requiring memorization or it being different from the way the word is used in normal conversation, but this simply isn't true. It's English.</p>
<p>You can walk the dog only if you have a leash. Therefore, if you can walk the dog, it follows logically that you must have a le... |
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