qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,014,253 | <p>I am looking for examples of non-abelian groups of arbitrarily large size with the following properties </p>
<ol>
<li>Have order <span class="math-container">$p^a$</span>, where <span class="math-container">$a$</span> is a positive integer and <span class="math-container">$p$</span> is prime.</li>
<li>Contain an ab... | user3482749 | 226,174 | <p>Dihedral groups of order <span class="math-container">$2^a$</span> have both properties (the subgroup being the cyclic one generated by the square of a highest-order element), but they also have a larger abelian subgroup of order <span class="math-container">$2^{a-1}$</span>, so might not be what you're after. </p>
|
3,014,253 | <p>I am looking for examples of non-abelian groups of arbitrarily large size with the following properties </p>
<ol>
<li>Have order <span class="math-container">$p^a$</span>, where <span class="math-container">$a$</span> is a positive integer and <span class="math-container">$p$</span> is prime.</li>
<li>Contain an ab... | user10354138 | 592,552 | <p>Take, for example, the direct product of a nonabelian group of order <span class="math-container">$p^3$</span> with an abelian group of order <span class="math-container">$p^{a-3}$</span>.</p>
|
4,269,434 | <p>I want to compute the gradient of the vector function <span class="math-container">$f(\vec{x}) = \|\vec{x} - \vec{a}\|$</span>, I have a try, but the result is kind of strange to me.</p>
<p>so here is my steps
<span class="math-container">\begin{align*}
\nabla\|\vec{x} - \vec{a}\| & = \nabla\sqrt{\sum_{i = 1}^... | Ted Shifrin | 71,348 | <p>This is perfectly correct. If you remember the conceptual interpretation of the gradient, it makes good sense: The norm increases most rapidly, and with rate 1, if you move from <span class="math-container">$x$</span> radially outward from <span class="math-container">$a$</span>.</p>
|
1,794,855 | <p>I need to prove that for every three integers $(a,b,c)$, the $\gcd(a-b,b-c) = \gcd(a-b,a-c)$. Assuming that a $a \ne b$.</p>
<p>Having:</p>
<p>$d_1 = \gcd(a-b,b-c)$</p>
<p>$d_2 = \gcd(a-b,a-c)$</p>
<p>How do i prove $d_1 = d_2$?</p>
| Maria | 308,011 | <p>The trick is the using the identity $k { n \choose k} = n {n-1 \choose k-1}$.
$$\begin{align*}
&\sum_{k=1}^n k { n \choose k } p^k (1-p)^{n-k}\\
&= \sum_{k=1}^n n { n-1 \choose k-1} p^k (1-p)^{n-k}\\
&=np \sum_{k=1}^n {n-1 \choose k-1} p^{k-1} (1-p)^{(n-1)-(k-1)}\\
&=np (p+(1-p))^n\\
&=np
\end{a... |
4,238,241 | <blockquote>
<p>Let tangents <span class="math-container">$PA$</span> and <span class="math-container">$PB$</span> on hyperbola from any point <span class="math-container">$P$</span> on the Director Circle of hyperbola such that <span class="math-container">$d(P,AB).d(C,AB)=4d(S_1,PA).d(S_2,PA)$</span> and <span class=... | user10354138 | 592,552 | <p><strong>Claim</strong>: Assuming a hyperbola has a (real) director circle and satisfies <span class="math-container">$d(P,AB)\cdot d(C,AB)=K\cdot d(S_1,PA)\cdot d(S_2,PA)$</span> for all <span class="math-container">$P$</span> on the director circle (<span class="math-container">$A,B,C,S_1,S_2$</span> same definitio... |
3,065,572 | <p>Function <span class="math-container">$f: (0, \infty) \to \mathbb{R}$</span> is continuous. For every positive <span class="math-container">$x$</span> we have <span class="math-container">$\lim\limits_{n\to\infty}f\left(\frac{x}{n}\right)=0$</span>. Prove that <span class="math-container">$\lim\limits_{x \to 0}f(x)=... | Yanko | 426,577 | <p>The only way to answer this question is to ask yourself at every given point "what does it mean?"</p>
<p>We're given that <span class="math-container">$A\subseteq B$</span>. <strong>What does it mean?</strong></p>
<p>It means that every element in <span class="math-container">$A$</span> belongs to <span class="mat... |
2,091,589 | <p>The remainder when a polynomial $f(x)$ is divided by $(x-2)(x+3)$ is $ax+b$. When $f(x)$ is divided by $(x-2)$, then remainder is $5$. $(x+3)$ is a factor of $f(x)$. Find the values of $a$ and $b$. I am thinking of using the remainder and factor theorem to solve this however their quotients are different. Can anyone... | Andreas Caranti | 58,401 | <p>Your assumptions are that there are polynomials $q,s,t$ such that
$$
\begin{cases}
f(x) = (x-2)(x+3) q(x) &+ a x + b\\
f(x) = (x -2) s(x) &+ 5\\
f(x) = (x + 3) t(x)
\end{cases}
$$</p>
<p>Now calculate $f(2)$ and $f(-3)$.</p>
|
2,091,589 | <p>The remainder when a polynomial $f(x)$ is divided by $(x-2)(x+3)$ is $ax+b$. When $f(x)$ is divided by $(x-2)$, then remainder is $5$. $(x+3)$ is a factor of $f(x)$. Find the values of $a$ and $b$. I am thinking of using the remainder and factor theorem to solve this however their quotients are different. Can anyone... | Bernard | 202,857 | <p>Simple: first write $\;f(x)=(x+3)g(x)$, and perform the <em>Euclidean division</em> of $g(x)$ by $x-2$:
$$g(x)=(x-2)q(x)+g(2).$$
Now $f(2)=5g(2)$, so that
$$f(x)=(x-2)(x+3)q(x)+ g(2)(x+3)=(x-2)(x+3)q(x)+ \underbrace{\frac{f(2)}5(x+3)}_{\text{remainder}}. $$</p>
|
1,330,858 | <p>Suppose I'm at $(x=0,y=0)$ and I want to get to $(x=1,y=1)$. The shortest path is the diagonal and it has length $\sqrt{2}$. But what if I'm only allowed to make moves in coordinate directions---e.g., $1/2$ along $x$, $1/2$ along $y$, another $1/2$ along $x$, and a final $1/2$ along $y$. Then the length of my path i... | Zach Stone | 38,565 | <p>It comes down to what kind of approximation you're working with. The sum of sides of a right triangle do NOT approximate it's hypotenuse. Even it the triangle is really small, the sides are still a bad approximation. To more specific, in an isocolese right triangle, with sides lengths $x$ and hypotenuse, $y$, we hav... |
2,063,380 | <p>I am absolutely stuck, reading Bott and Tu isomorphism of de Rham cohomology. Please help. On page 92,</p>
<p><a href="http://www.maths.ed.ac.uk/~aar/papers/botttu.pdf" rel="nofollow noreferrer">http://www.maths.ed.ac.uk/~aar/papers/botttu.pdf</a></p>
<p>Step 2. $r^{*}$ is injective. </p>
<p>$$ r(\omega)=D\phi^{... | Pedro | 23,350 | <p>The crucial step is that one can always replace a cochain in $K^{**}$ by one with one with last component zero. </p>
<p>To see that $r*$ is onto, take a cocycle in $K^{**}$. By the first remark, this is represented by a pair of forms $(\eta_1,\eta_2)$, and the cocycle condition means in the vertical direction, that... |
2,063,380 | <p>I am absolutely stuck, reading Bott and Tu isomorphism of de Rham cohomology. Please help. On page 92,</p>
<p><a href="http://www.maths.ed.ac.uk/~aar/papers/botttu.pdf" rel="nofollow noreferrer">http://www.maths.ed.ac.uk/~aar/papers/botttu.pdf</a></p>
<p>Step 2. $r^{*}$ is injective. </p>
<p>$$ r(\omega)=D\phi^{... | Pedro | 23,350 | <p><em>To address the edit</em>: you're misunderstanding what they are doing. A generic element in the total complex <span class="math-container">$C^*(\mathfrak U)$</span> of <span class="math-container">$K^{**}$</span> is of the form <span class="math-container">$(a,b)$</span> with <span class="math-container">$a\in \... |
4,416,001 | <p>Given <span class="math-container">$f(xy) = f(x+y)$</span> and <span class="math-container">$f(11) = 11$</span>, what is <span class="math-container">$f(49)$</span>?</p>
| Adrian Anelta Kacaribu | 1,042,585 | <p>f(11) = 11</p>
<p>f(49)=f(49.1)=f(50)
f(50)=f(25.2)=f(27)
f(27)=f(27.1)=f(28)
f(28)=f(4.7)=f(11)</p>
<p>f(49)=f(11)</p>
<p>So, f(49)= 11</p>
|
2,709,273 | <blockquote>
<p>Determine the kernel of the following group homomorphism:
$$
\phi\colon\mathbb Z/270\mathbb Z\to\mathbb Z/270\mathbb Z\colon\overline x\mapsto\overline{6x}.
$$
Then find the solutions of the following system of equations in $\mathbb Z/270\mathbb Z$:
\begin{align}
6x=3\mod 27\\
6x=2\mod 10
\end{a... | Dr. Sonnhard Graubner | 175,066 | <p>we can write $$6x=3+27m$$ and $$6x=2+10n$$ where $m,n$ are integers, from here we get
$$1=10n-27m$$ solving this Diophantine equation we get $$m=7+10k,n=19+27k$$ and from here you will get $x$</p>
|
251,559 | <p>First of all, this is a very silly question, but I finished high school a long time ago and I really don't remember much about some basic stuff.</p>
<p>I have:</p>
<p>$$T(x) = -\frac1{10}x^2 + \frac{24}{10}x - \frac{44}{10}$$</p>
<p>$T$ = temperature</p>
<p>$x$ = hour</p>
<p>And they ask me to find the hour in ... | Cameron Buie | 28,900 | <p>Let's multiply by $-10$ to clear the fractions and get $$0=x^2-24x+44.$$ It follows, then, that $$\begin{align}100 &= x^2-24x+144\\ &=x^2+2(-12)x+(-12)^2\\ &= (x-12)^2.\end{align}$$ The two numbers whose square is $100$ are $\pm 10$, so we have $x-12=\pm 10$, from which we have $x=2$ or $x=22$.</p>
|
234,477 | <p>In non-standard analysis, assuming the continuum hypothesis, the field of hyperreals $\mathbb{R}^*$ is a field extension of $\mathbb{R}$. What can you say about this field extension?</p>
<p>Is it algebraic? Probably not, right? Transcendental? Normal? Finitely generated? Separable?</p>
<p>For instance, I was th... | ShyPerson | 1,884 | <p>Briefly, one way to approach this question would be to try to construct alternative models of the hyperreals via the Compactness Theorem that either satisfy or fail to satisfy the properties you want. The Enderton text, <em>A Mathematical Introduction to Logic</em>, uses this kind of construction and has a very rigo... |
234,477 | <p>In non-standard analysis, assuming the continuum hypothesis, the field of hyperreals $\mathbb{R}^*$ is a field extension of $\mathbb{R}$. What can you say about this field extension?</p>
<p>Is it algebraic? Probably not, right? Transcendental? Normal? Finitely generated? Separable?</p>
<p>For instance, I was th... | Community | -1 | <p>There isn't any trouble constructing $\mathbb{R}(\epsilon)$ as a <em>formally real field</em> that is isomorphic (as a field) to $\mathbb{R}(x)$, and with $\epsilon$ a positive infinitesimal. (it may be easier to see the ordering by writing a rational function as a formal Laurent series in $\epsilon$ about 0)</p>
<... |
2,861,293 | <p>I found this statement with the proof:</p>
<p><a href="https://i.stack.imgur.com/bGRiZ.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/bGRiZ.jpg" alt="enter image description here" /></a></p>
<p>But I don't understand the proof. Where is the contradiction? We have a nonempty interval <span class="... | Paul Frost | 349,785 | <p>There have been two answers, but the comments have shown that the arguments <em>at first glance</em> appeared a little obscure or cumbersome. I believe that everything is clarified now, either by enhancing the answers or by additional comments. But let us give another proof which is hopefully more transparent from t... |
4,198,496 | <p>If the value of <span class="math-container">$\int_{1}^2{e^{x^{2}}}dx$</span> is <span class="math-container">$\alpha$</span>, then what is <span class="math-container">$$\int_{e}^{e^{4}}{\sqrt{\log x}}dx$$</span></p>
<p>It just seems that substitution of any sort does not help. Is there some other way in which thes... | David | 950,170 | <p><span class="math-container">$$y=e^{x^2}\iff log y =x^2\iff \sqrt{\log y}=x$$</span>
The rectangle bounded by <span class="math-container">$(1,0), (2,0),(1,e^4), (2,e^4)$</span> has area <span class="math-container">$e^4.$</span>
The regions represented by the two integrals fill this rectangle with
an excess of a <s... |
1,225,359 | <p>Q: The sum of all the coefficients of the terms in the expansion of $(x+y+z+w)^{6}$ which contain $x$ but not $y$ is:</p>
<p>What I tried to do was make pairs of two terms and the expand it as a binomial expression and then again expand the binomial in the resulting series which gave me an expression with lot of un... | Jack D'Aurizio | 44,121 | <p><strong>Hint:</strong> What happens if you evaluate your expression in $x=1,y=0,z=1,w=1$?
And in $x=0,y=0,z=1,w=1$?</p>
|
2,410,648 | <p>$$\left(\begin{array}{ccc|c}
-1 & 2 & 1 & 3\\
3 & \alpha & -2 & \beta\\
-1 & 5 & 2 & 9
\end{array}\right)$$</p>
<p>I am struggling to solve this system $Ax=b$. I understand the basics of Gauss elimination but am not sure how to handle it with the alpha and beta. It needs to be so... | Martin Sleziak | 8,297 | <p>Let me start with some steps. Perhaps you'll be able to finish.</p>
<p>$$\left(\begin{array}{ccc|c}
-1 & 2 & 1 & 3\\
3 & \alpha & -2 & \beta\\
-1 & 5 & 2 & 9
\end{array}\right)\sim
\left(\begin{array}{ccc|c}
1 &-2 &-1 &-3\\
-1 & 5 & 2 & 9\\
3 & \alp... |
2,410,648 | <p>$$\left(\begin{array}{ccc|c}
-1 & 2 & 1 & 3\\
3 & \alpha & -2 & \beta\\
-1 & 5 & 2 & 9
\end{array}\right)$$</p>
<p>I am struggling to solve this system $Ax=b$. I understand the basics of Gauss elimination but am not sure how to handle it with the alpha and beta. It needs to be so... | Raffaele | 83,382 | <p>$$\det\left(
\begin{array}{rrr}
-1 & 2 & 1 \\
3 & a & -2 \\
-1 & 5 & 2 \\
\end{array}
\right)=-3-\alpha$$
if $-3-\alpha\ne 0$ that is $\alpha\ne -3$ there exists <strong>one and only one</strong> solution</p>
<p>$$\left\{\frac{3 a-b+6}{a+3},\frac{b+3}{a+3},\frac{3 (2 a-b+3)}{a+3}\right\}... |
162,520 | <p>This is a cross-post from <a href="https://math.stackexchange.com/questions/738094/good-book-on-analytic-continuation">MSE</a>.</p>
<p>For my Bachelor's thesis, I am investigating divergent series summation methods. One of those is analytic continuation. There are quite a few books on complex analysis that include ... | Alexandre Eremenko | 25,510 | <p>English books are Hardy, Divergent series,
and P. Dienes, Taylor series:
an introduction to the theory of functions of a complex variable. Dover Publications, Inc., New York, 1957. (The title is somewhat misleading. This is a large book that indeed
contains an "introduction to complex variables" but it is also the m... |
1,116,435 | <p>How do I get the value of </p>
<p>$$\lim_{x \rightarrow \frac{1}{4} \pi } \frac{\tan x-\cot x}{x-\frac{1}{4} \pi }?$$ </p>
<p>I need the steps without using L'hospital.</p>
| Dustan Levenstein | 18,966 | <p>idm's answer is good, but L´Hôpital's rule is often more directly an instance of a limit actually being the limit of a difference quotient defining a derivative, or, in this case, a pair of difference quotients defining a pair of derivatives:</p>
<p>$$\frac{\tan x-\cot x}{x-\frac{1}{4} \pi } = \frac{\tan x - 1}{x-\... |
3,489,642 | <blockquote>
<p>Let <span class="math-container">$D_{2\cdot 8}$</span> be given by the group presentation <span class="math-container">$\langle x,y\mid xy = yx^{-1} , y^2 = e, x^8 = e\rangle$</span>. Let <span class="math-container">$G = F_{\{x,y\}}$</span> be the free group on two generators and <span class="math-cont... | Community | -1 | <p>If I remember correctly, <span class="math-container">$N$</span> should be <em>defined</em> as the smallest normal subgroup containing <span class="math-container">$\langle x^8,y^2, xyx^{-1}y\rangle $</span>. That is <span class="math-container">$N$</span> is its normalizer; hence normal.</p>
<p>The homomorphism ... |
463,619 | <p>let us consider following problem:</p>
<p>Roger sold a watch at a profit of $10$%. If he had bought it at $10\%$ less and sold it for $13$ dollar less,then he would have made a profit of $15$%. What is the cost price of the watch?</p>
<p>suppose that price of watch is $x$ dollar, $profit=sell -cost $</p>
<p>so ... | TZakrevskiy | 77,314 | <p>We have costs $c$ and sell price $p$. We know that he had $10\%$ profit, so $$p-c=0.1c$$
On the other hand, $$(p-13) - 0.9c = 0.15\times0.9c$$
Can you take it from here?</p>
<p><strong>Edit</strong></p>
<p>On the calulation of profit:</p>
<p>Profit is the difference between the selling price of the good and costs... |
4,341,297 | <p>Evaluate the given expression <span class="math-container">$$\sqrt[n]{\dfrac{20}{2^{2n+4}+2^{2n+2}}}$$</span> The given answer is <span class="math-container">$\dfrac{1}{4}$</span>. My attempt:
<span class="math-container">$$\sqrt[n]{\dfrac{20}{2^{2n+4}+2^{2n+2}}}=\sqrt[n]{\dfrac{20}{2^{2n}\cdot2^4+2^{2n}\cdot2}}\\=... | CHAMSI | 758,100 | <p>You made a small mistake :<span class="math-container">\begin{aligned}\sqrt[n]{\dfrac{20}{2^{2n+4}+2^{2n+2}}}&=\sqrt[n]{\dfrac{20}{2^{2n}\cdot2^4+2^{2n}\cdot2^{\color{red}{2}}}}\\&=\sqrt[n]{\dfrac{20}{2^{2n}\cdot\color{red}{20}}}=\sqrt[n]{\dfrac{1}{2^{2n}}}=\frac{1}{4}\end{aligned}</span></p>
|
700,673 | <p>I saw this notation many times, but I don't understand why the $y$ variable is missing in the first term of the first equation below.</p>
<p>$$
\frac{\mathrm{d}y(x)}{\mathrm{d}x} = f(x,y)
$$</p>
<p>It just mean:</p>
<p>$$
\frac{\mathrm{d}y(x,y)}{\mathrm{d}x} = f(x,y)
$$</p>
<p>?</p>
<hr>
<h2>Tell me if I'm wro... | Christian Blatter | 1,303 | <p>Your difficulty stems from the use of the letter $y$ for two different purposes: (a) as <em>coordinate variable</em> in the $(x,y)$-plane, and (b) as <em>variable for (unknown) functions</em> $x\mapsto y(x)$ whose graphs are lying in the $(x,y)$-plane.</p>
<p>When dealing with ODEs for the first time we are given a... |
3,933,011 | <p>I know how to solve for the basic about this. But in this problem</p>
<p><span class="math-container">$\displaystyle{\frac{1}{2\pi i} \oint_{|z|=1} \frac{\cos \left(e^{-z}\right)}{z^2} dz}$</span>,</p>
<p>I don't know how to start. Can somebody help me or guide me about this? Or just give me a hint. Thanks. I wo... | Raghav | 164,237 | <p>First, recall the following form of Cauchy's integral formula. Let <span class="math-container">$f$</span> be a function holomorphic on some neighborhood of unit disk, then
<span class="math-container">$$f^{(n)}(w)=\frac{n!}{2i\pi}\oint_{|z|=1}\frac{f(\zeta)}{(\zeta-w)^{n+1}}d\zeta.$$</span></p>
<p>The above theorem... |
454,426 | <blockquote>
<p>In set theory and combinatorics, the cardinal number $n^m$ is the size of the set of functions from a set of size m into a set of size $n$.</p>
</blockquote>
<p>I read this from this <a href="http://en.wikipedia.org/wiki/Empty_product#0_raised_to_the_0th_power" rel="nofollow noreferrer">Wikipedia pag... | user84413 | 84,413 | <p>You can think of constructing your function by choosing the images of each element in the domain, one at a time. For the first element of the set, you have n choices (any element in the target space); for the second element of the set, you again have n choices, and so on. Therefore the number of ways to construct ... |
63,348 | <p>This question arises from a discussion with my friends on a commonly encountered IQ test questions: "What's the next number in this series 2,6,12,20,...". Here a "number" usually means an integer. I was wondering whether there is a systematical way to solve such problems.Let us call a point on a plane integer point ... | Lubin | 11,417 | <p>It's a topic I liked to cover when I was still teaching junior-level Algebra, even if it didn't fit in well with the other topics. You start with a function defined on the set of integers from $0$ to $n$ inclusive, and end with a polynomial of degree $\le n$ that agrees at those $n+1$ points, and if the values you s... |
2,094,243 | <p>Given the norm $||(x,y)|| = 2|x| +\frac{1}{3}|y|$.
Sketch the open ball at the on the origin $(0,0)$, and radius $1$.</p>
<p>I understand that the sketch of an open ball withina set looks like the image attached, <a href="https://i.stack.imgur.com/j9HN4.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur... | pancini | 252,495 | <p>The example you drew is not a general case at all. That is what an open ball looks like in $\Bbb R^2$ under the euclidian metric.</p>
<p>We want to cover any point in the plane $(x,y)$ such that $2|x|+\frac{1}{3}|y|<1$.</p>
<p>In the first quadrant, $x$ and $y$ are positive, so we have $2x+y/3<1$, or the are... |
1,577,174 | <p>If $a, b \in \mathbb{C}$, then we have the standard triangle inequality for the difference:</p>
<p>$$||a| - |b|| \le |a - b|.$$</p>
<p>I am wondering if this inequality generalizes to exponents greater that one.</p>
<blockquote>
<p>My question is, for $1 < p < \infty$, does there exists a constant $C_p$ s... | PhoemueX | 151,552 | <p>This is false for <span class="math-container">$p>1$</span>. In fact, by the mean value theorem, there is for <span class="math-container">$n\in \Bbb {N} $</span> some <span class="math-container">$\xi_n \in (n,n+1) $</span> with
<span class="math-container">$$
(n+1)^p - n^p = p \cdot \xi_n^{p-1}\to \infty
$$</s... |
677,859 | <p>$f(x)= f(x+1)+3$ and $f(2)= 5$, determine the value of $f(8)$.</p>
<p>I don't understand how $f(x)$ can equal $f(x+1)+3$</p>
| MPW | 113,214 | <p>This just means that if the point $(x,y)$ is on the graph of $f$, then so is the point $(x+1, y-3)$. So if you have a starting point, you can move to some specific other points on the graph by stepping $1$ unit to the right and $3$ units down. Then, since you are at a new point now, you can repeat this procedure: ri... |
3,489,280 | <p>If <span class="math-container">$f_n → f$</span> and <span class="math-container">$g_n → g$</span>, does <span class="math-container">$f_n g_n → fg$</span> in the space <span class="math-container">$C[0, 1]$</span> for the norms <span class="math-container">$||.||_1$</span> and <span class="math-container">$||.||_∞$... | Wlod AA | 490,755 | <blockquote>
<p>In my earlier answer, I've addressed only the uniform norm; as, @zhw. noted, I sloppily missed the <span class="math-container">$\|.\|_1$</span>-norm (shame on me) which zhw. did solve. And still, to make up for my misdeed, let me present another example which shows that <span class="math-container">$\|... |
1,914,752 | <p>dividing by a whole number i can describe by simply saying split this "cookie" into two pieces, then you now have half a cookie. </p>
<p>does anyone have an easy way to describe dividing by a fraction? 1/2 divided by 1/2 is 1</p>
| ZirconCode | 278,375 | <p>Dividing by a fraction is the same as multiplying by the reciprocal of the fraction. I think this is the most intuitive approach when trying to teach this concept.</p>
<p>For multiplication we first split the cookie up into denominator-number of pieces, and then take numerator-number of pieces of those pieces.</p>
|
162,611 | <p>I am working with the square-roots of square symmetric matrices. The answers are to be binary symmetric matrices.</p>
<p>If we take the matrix $$M = \begin{pmatrix}1&1&1&0&0&0&0\\1&0&0&1&1&0&0\\1&0&0&0&0&1&1\\0&1&0&0&1&0&... | Will Jagy | 10,400 | <p>Had to screw around with the field of four elements. Not that bad. The pattern is different, there are the expected five null vectors, but the vector $(1,1,1)$ is orthogonal to all five, so I put that one first. The result is a symmetric matrix $P,$ such that $P^2$ is a 21 by 21 matrix with 5's on the main diagonal ... |
28,456 | <p>I built a <code>Graph</code> based on the permutations of city's connections from :</p>
<pre><code>largUSCities =
Select[CityData[{All, "USA"}], CityData[#, "Population"] > 600000 &];
uScityCoords = CityData[#, "Coordinates"] & /@ largUSCities;
Graph[#[[1]] -> #[[2]] & /@ Permutations[largUSCitie... | bobthechemist | 7,167 | <p>Revised:</p>
<p>First, let's get some map data. This is an 8MB file and you might want to save it locally.</p>
<pre><code>map = First@
Import["http://www2.census.gov/geo/tiger/TIGER2009/tl_2009_us_\
state.zip", "Data"];
</code></pre>
<p>Then, look at the answers that are better than yours (nod nod <a href="ht... |
2,300,855 | <p>Do there exist two non-equivalent knots which are indistinguishable by <a href="https://en.wikipedia.org/wiki/Fox_n-coloring" rel="nofollow noreferrer">Fox $n$-colouring</a> for every positive integer $n$?</p>
<p>That is, do there exist non-oriented knots $K_0$ and $K_1$ which are different (here I would like to ex... | Zuriel | 23,173 | <p>I just realised that the knot <a href="http://www.indiana.edu/~knotinfo/diagram_display/diagram_display_10_124.html" rel="nofollow noreferrer">$10_{124}$</a> and <a href="http://www.indiana.edu/~knotinfo/diagram_display/diagram_display_10_153.html" rel="nofollow noreferrer">$10_{153}$</a> both have <a href="http://w... |
2,300,855 | <p>Do there exist two non-equivalent knots which are indistinguishable by <a href="https://en.wikipedia.org/wiki/Fox_n-coloring" rel="nofollow noreferrer">Fox $n$-colouring</a> for every positive integer $n$?</p>
<p>That is, do there exist non-oriented knots $K_0$ and $K_1$ which are different (here I would like to ex... | N. Owad | 85,898 | <p>These are not prime knots, but the square knot and the granny knot I believe are indistinguishable by coloring. That is, $3_1\#\bar{3}_1$ and $3_1\#3_1$. You need peripheral subgroups (or at least that was the way it was done first). And they can be colored, if that helps your work.</p>
|
253,746 | <p>I am currently aware of the following two versions of the global Cauchy Theorem. Which one is stronger?</p>
<p>1.)If the region $U$ is simply connected, then for every closed curve contained therein, the integral of the holomorphic function $f$ defined on $U$ over the curve is zero.</p>
<p>2.) If the domain $D$ is... | Applied mathematician | 42,887 | <p>The way I learned it is:</p>
<blockquote>
<p><span class="math-container">$$\int_{\Gamma}f(z)dz=0 $$</span></p>
<p>Iff</p>
<p>1.) <span class="math-container">$f$</span> is analytic in a simply connected domain D</p>
<p>2.) <span class="math-container">$\Gamma$</span> is any closed contour in D,</p>
</blockquote>
<p... |
984,852 | <p>Let $p$ be a univariate polynomial over a field $F$, and let $K$ be an extension of $F$. </p>
<p>If $p(x) = 0$ for all $x \in F$, does this imply that $p(x) = 0$ for all $x \in K$? How about if $p$ is multivariate?</p>
<p>For context, I'm trying to understand if doing Schwartz-Zippel-style arithmetic circuit ident... | orangeskid | 168,051 | <p>All you know is this: a nonzero polynomial of degree $d$ cannot have more than $d$ roots in some (extension) field. </p>
|
497,546 | <p>Let $A$ be an infinite set.</p>
<p>Then, we can construct an injective function $f:\omega \rightarrow A$. </p>
<p>But how do i construct this via orginal statement of $AC_\omega$? (i.e. $\forall countable X, [\emptyset \notin X \Rightarrow \exists f:X\rightarrow \bigcup X \forall A\in X, f(A)\in A$)</p>
<p>So my ... | user88595 | 88,595 | <p>I do not know any quick proofs from scratch to show this formula but a general formula to find generating functions is the following: $$G(x,y) = \frac{\omega(z)}{\omega(x)}\frac1{1-y\cdot p'(z)}$$
Where $z = x + y\cdot p(z)$ and $p(z)$ is from the Legendre ODE $\frac{d}{dx}[p(x)y_n'(x)] + q(x)y_n'(x) + \lambda_ny_n(... |
497,546 | <p>Let $A$ be an infinite set.</p>
<p>Then, we can construct an injective function $f:\omega \rightarrow A$. </p>
<p>But how do i construct this via orginal statement of $AC_\omega$? (i.e. $\forall countable X, [\emptyset \notin X \Rightarrow \exists f:X\rightarrow \bigcup X \forall A\in X, f(A)\in A$)</p>
<p>So my ... | 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{\ds}[1]{\displayst... |
3,540,045 | <p>The definite integral, <span class="math-container">$$\int_0^{\pi}3\sin^2t\cos^4t\:dt$$</span></p>
<p><strong>My question</strong>: for the trigonometric integral above the answer is <span class="math-container">$\frac{3\pi}{16}$</span>. What I want to know is how can I compute these integrals easily. Is there more... | user5713492 | 316,404 | <p>I see <span class="math-container">$2$</span> quick methods. Firstly, one could use <span class="math-container">$\sin t\cos t=\frac12\sin2t$</span> and <span class="math-container">$\cos^2t=\frac12(1+\cos2t)$</span> to get
<span class="math-container">$$\int_0^{\pi}3\sin^2t\cos^4t\,dt=\frac38\int_0^{\pi}\left(\sin^... |
3,540,045 | <p>The definite integral, <span class="math-container">$$\int_0^{\pi}3\sin^2t\cos^4t\:dt$$</span></p>
<p><strong>My question</strong>: for the trigonometric integral above the answer is <span class="math-container">$\frac{3\pi}{16}$</span>. What I want to know is how can I compute these integrals easily. Is there more... | Deepak | 151,732 | <p>Here's a method that can tackle more general problems of this nature.</p>
<p>First, let's consider the integral <span class="math-container">$I = \int \sin^2 x \cos^4 x dx$</span> (I just omitted the constant multiplier).</p>
<p>We can write <span class="math-container">$I = \int \sin x \cdot \sin x \cos^4 x dx$</... |
893,822 | <p>If $p(x)$ has integer coefficients and $p(100)$ equals $100$ what is the maximum number of integer solutions $k$ to the equation $p(k)=k^3$.</p>
<p>I have tried hard to solve this problem but I could not figure it out. I tried some particular cases but got nowhere, could someone please show me how to get the answer... | Jack D'Aurizio | 44,121 | <p>As pointed out by r9m in the comments, by setting $x=\cosh t$ the inequality is equivalent to:
$$ \cosh(n t)\leq \left(\cosh t+(n-1)(\cosh t-1)\right)^n \tag{1}$$
or to:
$$ \cosh(n t)\leq \left(1+2n\sinh^2\frac{t}{2}\right)^n \tag{2}$$
or to:
$$ 1+2\sinh^2(nz)\leq \left(1+2n\sinh^2 z\right)^n \tag{3}$$
or to:
$$ \fr... |
4,363,327 | <p>Consider the one-layer neural network <span class="math-container">$y=\mathbf{w}^T\mathbf{x} +b$</span> and the optimization objective <span class="math-container">$J(\mathbf{w}) = \mathbb{E}\left[ \frac12 (1-y\cdot t) \right]$</span> where <span class="math-container">$t\in\{-1,1\}$</span> is the label of our data ... | Steph | 993,428 | <p>Forgetting the expectation operator (for a while),
the differential of the cost function writes
<span class="math-container">$$
d\phi
= t(t\cdot y-1)dy
= (y-t) \mathbf{x}^Td\mathbf{w}
$$</span>
since <span class="math-container">$dy=\mathbf{x}^Td\mathbf{w}$</span>.</p>
<p>The gradient is <span class="math-container"... |
467 | <p>Moderators have started incorporating the old faq material in the new <a href="https://mathoverflow.net/help">help system</a>. It wasn't a perfect fit, a lot of stuff is no longer relevant, redundant, missing or broken. You can help by going through the <a href="https://mathoverflow.net/help">help center</a> and pos... | Community | -1 | <p>Some quick observations on in <a href="https://mathoverflow.net/help/on-topic">https://mathoverflow.net/help/on-topic</a> </p>
<ol>
<li><p>The part Where's the rule that says I have to wear pants? seems not good there anymore it should be merged into <a href="https://mathoverflow.net/help/behavior">https://mathove... |
467 | <p>Moderators have started incorporating the old faq material in the new <a href="https://mathoverflow.net/help">help system</a>. It wasn't a perfect fit, a lot of stuff is no longer relevant, redundant, missing or broken. You can help by going through the <a href="https://mathoverflow.net/help">help center</a> and pos... | TRiG | 6,395 | <p>Under <a href="https://mathoverflow.net/help/on-topic">On Topic</a>:</p>
<blockquote>
<p>If you're just really interested in how the underlying Stack Exchange software works, consider visiting meta.stackoverflow or meta.stackexchange.</p>
</blockquote>
<p>This may change in <em>ahem</em> six to eight weeks, but for ... |
467 | <p>Moderators have started incorporating the old faq material in the new <a href="https://mathoverflow.net/help">help system</a>. It wasn't a perfect fit, a lot of stuff is no longer relevant, redundant, missing or broken. You can help by going through the <a href="https://mathoverflow.net/help">help center</a> and pos... | Kaveh | 7,507 | <p>List of moderators on help/on-topic page needs to be updated.
It might be better to remove the list from there and
replace it with a link to
<a href="https://mathoverflow.net/users?tab=moderators">https://mathoverflow.net/users?tab=moderators</a></p>
|
2,506,182 | <p>The question speaks for itself, since the question comes from a contest environment, one where the use of calculators is obviously not allowed, can anyone perhaps supply me with an easy way to calculate the first of last digit of such situations?</p>
<p>My intuition said that I can look at the cases of $3$ with an ... | Guy Fsone | 385,707 | <p>the last digit correspond to
$$3^{2011}\equiv x\mod 10$$
We have,
$$3^4\equiv 1\mod 10$$</p>
<p>But, $$2011 = 4 * 502+3$$</p>
<p>Thus,
$$3^{2011}=3^{4*502+3} \equiv 3^3\mod 10~~~\text{ that is }~~
3^{2011} \equiv 7\mod 10 $$</p>
<p>Hence $x=7 $ is the last digit</p>
|
104,592 | <p>Hello everyone,</p>
<p>Suppose that I am defining a function which embeds a surface (manifold) in $\mathbb{R}^3$.</p>
<p>Is there a standard symbol or letter that is used for this function?</p>
<p>Additionally, is there any other classical or standard notation (such as the hooked arrow for inclusion maps) of whic... | Wolfgang Loehr | 15,327 | <p>I usually use $\iota$ (\iota) for all kinds of embeddings. </p>
|
9,010 | <p>I hear people use these words relatively interchangeably. I'd believe that any differentiable manifold can also be made into a variety (which data, if I understand correctly, implicitly includes an ambient space?), but it's unclear to me whether the only non-varietable manifolds should be those that don't admit smo... | Community | -1 | <p>If you are French, a variety <em>is</em> a manifold.</p>
<p>If not, the connection is a bit more subtle. The correct analogue of a manifold is a scheme of finite type over a field. That is, it has a compatible covering by ring spectra (which is similar to the requirement that a manifold have a covering by compati... |
3,784,484 | <p>Is there a bounded linear operator <span class="math-container">$T \in l^2(\mathbb{N})$</span> have for the essential spectrum the unit disk of <span class="math-container">$\mathbb{C}$</span>; i.e, such that <span class="math-container">$\sigma_{e}(T)=\textbf{D}(0, 1)$</span>; where <span class="math-container">$\... | Stephen Montgomery-Smith | 22,016 | <p>Consider <span class="math-container">$T$</span> on <span class="math-container">$\ell^2(\mathbb N \times \mathbb N)$</span>, with <span class="math-container">$T(a)_{m,n} = a_{m,n+1}$</span>. Then for <span class="math-container">$|\lambda| < 1$</span>, elements of the form <span class="math-container">$(b_m \l... |
1,725,945 | <p>I'm reading the proof of "Fundamental Theorem of Finite Abelian Groups" in Herstein Abstract Algebra, and I've found this statement in the proof that I don't see very clear.</p>
<p>Let $A$ be a normal subgroup of $G$. And suppose $b\in G$ and the order of $b$ is prime $p$, and $b$ is not in $A$. Then $A \cap (b)=(e... | Martín Vacas Vignolo | 297,060 | <p>Suppose that exists $0<k<p$ such that $b^k\in A\to b^{km}\in A \forall m\in\mathbb{Z} *$.</p>
<p>Now, $p$ is prime, $(k,p)=1\to ak+cp=1$ for any $a,c$. </p>
<p>If we put $m=a$ in $*$, obtain that $b^{ka}=b^{1-bp}=b.(b^p)^{-b}=b\in A$, an absurd.</p>
|
50,547 | <p>let $X$ be a $n$-manifold. let $A=\{(x,y,z) \, |\,x=y\}$. I want to see if $A$ is a submanifold of $X^3$.</p>
<p>Consider the map $\Delta\times 1:X\times X\rightarrow X \times X\times X;\, (x,y)\mapsto (x,x,y)$. </p>
<p>If $U_x$ and $U_y$ are neighborhoods of $x$ and $y$ in $X$ then $\Delta\times 1 (U_x\times U_y)... | Listing | 3,123 | <p>Those integrals were discussed by us in detail already.</p>
<p>To be more explicit you are asking for the special case of <a href="https://math.stackexchange.com/questions/44928/interesting-integral-formula">Interesting integral formula</a> for $m=2$, $n=4$ and $a=1$. Just directly plugging in those values in the p... |
2,494,232 | <p>I'm trying to find the out if $\sum_{n=1}^\infty {{1\over \sqrt{n}}-{1\over{\sqrt{n+1}}}}$ is divergent or convergent.</p>
<p>Here are some rules my book gives that I will try to follow:</p>
<p><a href="https://i.stack.imgur.com/TNV6x.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/TNV6x.png" al... | operatorerror | 210,391 | <p>Telescoping series, while not mentioned in your list, are important and I hope were taught to you.</p>
<p>You can write the $N$th partial sum as
$$
\bigg(\frac{1}{1}-\frac{1}{\sqrt{2}}\bigg)+\bigg(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}\bigg)+\cdots+\bigg(\frac{1}{\sqrt{N}}-\frac{1}{\sqrt{N+1}}\bigg)
$$
where it isn... |
2,494,232 | <p>I'm trying to find the out if $\sum_{n=1}^\infty {{1\over \sqrt{n}}-{1\over{\sqrt{n+1}}}}$ is divergent or convergent.</p>
<p>Here are some rules my book gives that I will try to follow:</p>
<p><a href="https://i.stack.imgur.com/TNV6x.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/TNV6x.png" al... | Guy Fsone | 385,707 | <p>The series obviously converges to $1$ by telescoping the partial sum: </p>
<p>$$\sum_{n=1}^k \frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}
=1-\frac{1}{\sqrt{k+1}}$$</p>
|
2,991,719 | <p>How to prove/disprove</p>
<blockquote>
<p>If <span class="math-container">$f : (0, \infty) \to \mathbb{R}^n$</span> is continuous on <span class="math-container">$[a, \infty) , \forall a>0$</span> then <span class="math-container">$f$</span> is continuous on <span class="math-container">$(0, \infty)$</span></p... | NoChance | 15,180 | <p>Sometimes you can't tell if matrix <span class="math-container">$B$</span> can be generated from matrix <span class="math-container">$B$</span> or not. In such a case, one may expand each determinant and compare the result of each. However, this approach requires careful attention to signs!</p>
<p>Let <span class=... |
738,122 | <p>I have been looking at Kolmogorov's book "Introductory Real Analysis" and have become stuck at the problem 4a) on page 301.</p>
<p>In this problem we are given $f$ a nonnegative and integrable function on $A$, a set of finite measure. We are asked to show that</p>
<p>$\int_A f(x) d \mu \ge 0$</p>
<p>However, how... | Ellya | 135,305 | <p>Simply state that you are restricting to a sequence simple functions that are non negative.</p>
<p>Or use the fact that $f\ge g\Rightarrow \int f\ge \int g $</p>
<p>Then $\int f = \sup\{\int s|s, simple\}=\sup\{\int s|s, simple,positive\}$</p>
|
3,647,772 | <p>In positional representations, there are always some rational numbers which have multiple representations. For example, in base 10, 1 can be written as 1 or as <span class="math-container">$0.\overline{9}$</span>. Do there exist any numerical representations in which all rationals have exactly one representation?</p... | Andrew Chin | 693,161 | <p>The fundamental theorem of arithmetic states that every natural number greater than <span class="math-container">$1$</span> can be written as a unique product of powers of prime numbers. Extending this to the rationals simply requires us to allow the negative powers of prime factors. For example, <span class="math... |
3,647,772 | <p>In positional representations, there are always some rational numbers which have multiple representations. For example, in base 10, 1 can be written as 1 or as <span class="math-container">$0.\overline{9}$</span>. Do there exist any numerical representations in which all rationals have exactly one representation?</p... | user780256 | 780,256 | <p>You can express rational numbers uniquely by terminating continued fractions. Every rational number can be expressed as a finite sequence of integers <span class="math-container">$[a_0, a_1, \ldots, a_n]$</span> containing at least one term, where <span class="math-container">$a_i \in \Bbb{Z}$</span> for <span class... |
744,787 | <p>I need the equation of the line passing through a given point $A(2,3,1)$ perpendicular to the given line
$$
\frac{x+1}{2}= \frac{y}{-1} = \frac{z-2}{3}.
$$</p>
<p>I think there must bee some kind of rule to do this, but I can't find it anyway.</p>
| Alan | 92,834 | <p><img src="https://i.stack.imgur.com/h1zw8.jpg" alt="enter image description here"></p>
<p>The system looks over determined by the line's equation. However, one chooses the line between (2,3,1) and (-1,0,2). (I minimized the distance and got a result in agreement.) I thought it a bit of a trick question. </p>
|
285,719 | <p>Why is $(-3)^4 =81$ and $-3^4 =-81 $?This might be the most stupidest question that you might have encountered,but unfortunately i'am unable to understand this.</p>
| A.D | 37,459 | <p>Note that $(-3)^4=((-1)\cdot3)^4=(-1)^4\cdot3^4=81$</p>
<p>Again note that $-3^4=(-1)\cdot3^4=-81$</p>
|
285,719 | <p>Why is $(-3)^4 =81$ and $-3^4 =-81 $?This might be the most stupidest question that you might have encountered,but unfortunately i'am unable to understand this.</p>
| guest196883 | 43,798 | <p>Because multiplication doesn't distribute into exponentiation. That is, $a(b^c) = (ab)^c$ doesn't always hold. And there's no reason why it should, unless $c = 1$. So $(-3)^4 = (-1 \cdot3)^4 = (-1)^4 \cdot (3)^4 = 81$, and not $-81$, as the multiple of $-1$ doesn't factor out. </p>
|
3,456,351 | <pre><code>Apples: 1
Apple Value: 2500
Pears: lowest = 1, highest = 10
</code></pre>
<p>If I have 1 apple and my apple is worth <strong>2500</strong> if I have 1 pear, how can I calculate the value of my apple if I have X pears, at a maximum of 10 pears and a minimum of 1 pear, where 1 pear represents 100% value and 1... | Joseph Desaulniers | 1,111,320 | <p>Here is an (maybe) easier approach.</p>
<p>To prove <span class="math-container">$\lfloor\sqrt n+\sqrt {n+1}\rfloor=\lfloor\sqrt{4n+2}\rfloor$</span>, and it's easy to notice that <span class="math-container">$\lfloor\sqrt{4n+2}\rfloor>\lfloor\sqrt n+\sqrt {n+1}\rfloor$</span> and that <span class="math-container... |
2,696,579 | <p>So I understand that in order to have a particular solution you have to have a non homogenous second order differential equation. However I have a slightly difficult time comprehending how to pick the particular solution given $g(t)$. </p>
<p>The reason I ask is our teacher skimmed over it and hardly covered it in ... | José Carlos Santos | 446,262 | <p>It is easy to prove that, for all $x\in\mathbb Q$, $f(x)=ax$, with $a=f(1)$. It follows from this and from the fact that $f$ is monotonic that, for each $y\in\mathbb R$,$$\lim_{x\to y}f(x)=f(y).$$Putting all this together, one gets that $(\forall x\in\mathbb{R}):f(x)=ax$.</p>
|
165,328 | <p>What is the difference between $\cap$ and $\setminus$ symbols for operations on sets?</p>
| JMP | 210,189 | <p>We have:</p>
<p>$$A\cap B = \{x|x\in A \land x\in B\}$$
$$A\setminus B = \{x|x\in A \land x\not\in B\}$$</p>
<p>from here we can say:</p>
<p>$$A\cap B=A\setminus B^c=B\setminus A^c$$
$$A\cap B^c=A\setminus B=B^c\setminus A^c$$</p>
<p>where:</p>
<p>$$B^c=\{x|x\not\in B\}$$</p>
|
361,740 | <p>Spivak's <em>Calculus on Manifolds</em> asks the reader to prove this (problem 1-8, pp.4-5):</p>
<blockquote>
<p>If there is a basis $x_1, x_2, ..., x_n$ of $\mathbb{R}^n$ and numbers $\lambda_1, \lambda_2, ..., \lambda_n$ such that $T(x_i) = \lambda_i x_i$, $1 \leq i \leq n$, prove that $T$ is angle-preserving i... | Alexander Jones | 86,853 | <p>I noticed this too. If you ignore the absolute value on lambda, it's a true statement which can be easily proven.</p>
|
2,071,332 | <p>I am working on implementation of a machine learning method that in part of the algorithm I need to calculate the value of $\Gamma (\alpha) / \Gamma (\beta) $. $\alpha$ and $\beta$ are quite large numbers (i.e. bigger than 200) and it causes the python $gamma$ function to overflow. However, as the difference of $\al... | Claude Leibovici | 82,404 | <p>I think that a good solution would be Stirling approximation that is to say $$\log(\Gamma(x))=x (\log (x)-1)+\frac{1}{2} \left(-\log \left({x}\right)+\log (2 \pi
)\right)+\frac{1}{12 x}+O\left(\frac{1}{x^3}\right)$$ Now, consider $$y=\frac{\Gamma(\alpha)}{\Gamma(\beta)}\implies \log(y)=\log(\Gamma(\alpha))-\log(\... |
290,527 | <p>What would be a good metric on $C^k(0,1)$, space of $k$ times continuously differentiable real valued functions on $(0,1)$ and $C^\infty(0,1)$, space of infinitely differentiable real valued functions on $(0,1)$? </p>
<p>It is of course open to interpretation what good would mean, I want it to bring a good notion o... | Davide Giraudo | 9,849 | <p>We can define the metrics
$$d(f,g):=\sum_{n=1}^{+\infty}2^{-n}\min\left\{1,\max_{0\leqslant k\leqslant n}\max_{n^{-1}\leqslant x\leqslant 1-n^{-1}}|f^{(k)}(x)-g^{(k)}(x)|\right\}\quad\mbox{on } C^\infty(0,1)$$
$$d_N(f,g):=\sum_{n=1}^{+\infty}2^{-n}\min\left\{1,\max_{0\leqslant k\leqslant N}\max_{n^{-1}\leqslant x\l... |
290,527 | <p>What would be a good metric on $C^k(0,1)$, space of $k$ times continuously differentiable real valued functions on $(0,1)$ and $C^\infty(0,1)$, space of infinitely differentiable real valued functions on $(0,1)$? </p>
<p>It is of course open to interpretation what good would mean, I want it to bring a good notion o... | mdg | 64,184 | <p>Ask yourself if the spaces $C^k(0,1)$ are of interest. </p>
<p>EDIT: I also suggest thinking more about and therefore formalising the notion of a 'good' metric on a space. You might find that a good metric is one that induces a topology on the space that has desirable properties for doing analysis, like local con... |
2,725,839 | <p>The question is below.<a href="https://i.stack.imgur.com/k3UMf.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/k3UMf.png" alt="enter image description here"></a></p>
<p>I was able to solve part (a) because the $x$-coordinate would just be the circumference of the circle, which is $2\pi$. Therefor... | Marin | 549,731 | <p>The centre point travels 2π<em>r</em> for every complete turn along <em>x</em>. Similarly it travels (1/4)2π<em>r</em> for a quarter turn.</p>
<p>Then turn the figure vertically with the <em>x</em> axis pointing down. Imagine the angle of reference as the second hand of a clock turning clockwise, starting at 9 o’cl... |
3,090,879 | <p>Is there prime number of the form <span class="math-container">$1+11+111+1111+11111+...$</span>. I've checked it up to first 2000 repunits, but i found none. If <span class="math-container">$R_1=1$</span>, <span class="math-container">$R_2=1+11$</span>, <span class="math-container">$R_3=1+11+111$</span>, <span cla... | David G. Stork | 210,401 | <p>Here's the smallest such prime, found using this code:</p>
<pre><code>Select[Accumulate[Table[
Sum[10^i, {i, 0, n}],
{n, 0, 10000}]], PrimeQ]
</code></pre>
<p><span class="math-container">$$1234567901234567901234567901234567901234567901234567901234567901234567
901234567901234567901234567901234567901234567901... |
3,148,094 | <p>In class, my professor computed the density of a gamma random variable by taking the derivative of its CDF, but he skipped many steps. I am trying to go through the derivation carefully but cannot reproduce his final result.</p>
<p>Let <span class="math-container">$k$</span> be the shape and <span class="math-conta... | MarianD | 393,259 | <p><a href="https://i.stack.imgur.com/lIbHV.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/lIbHV.jpg" alt="enter image description here"></a>
<a href="https://i.stack.imgur.com/d6JoF.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/d6JoF.jpg" alt="enter image description here"></a>... |
1,557,039 | <h2>Background</h2>
<p>I am a software engineer and I have been picking up combinatorics as I go along. I am going through a combinatorics book for self study and this chapter is absolutely destroying me. Sadly, I confess it makes little sense to me. I don't care if I look stupid, I want to understand how to solve the... | Peter | 82,961 | <p>Base case : $9\equiv 1\ (\ mod\ 4\ )$ is true.</p>
<p>Suppose $9^n\equiv 1\ (\ mod\ 4\ )$</p>
<p>Then $9^{n+1}=9\times 9^n\equiv 9\equiv 1\ (\ mod\ 4\ )$</p>
|
1,278,442 | <p>I was wondering if the singular homology functor preserve injectivity and surjectivity? I've been trying to figure out a proof or counterexample for ages now and I just can't.</p>
<p>This came up when I was looking at the reduced homology $H_p(S^{n},S^{n-1})$. To calculate it, I have looked at the canonical injecti... | Qiaochu Yuan | 232 | <p>I don't understand what you mean by relative homology preserving injectivity and surjectivity, so consider instead the following general point. </p>
<p>No interesting homotopy-invariant functor on spaces can preserve injections or surjections. The reason is that every map is homotopy equivalent to an injection and ... |
1,569,476 | <p>We throw fair dice until $6$ will appear. Let $X$ denote total number of throws and $Y$ - number of $5$ we received.</p>
<ol>
<li>Find distribution $(X,Y)$</li>
<li>Are variables $X$ and $Y$ independent?</li>
</ol>
<p>I have to say that I have utterly no idea how to proceed with this question, detailed explanation... | BGM | 297,308 | <p>For the first part you need the following:
$$X \sim \text{Geometric}\left(\frac {1} {6}\right) $$ and
$$Y|X = x \sim \text{Binomial}\left(x, \frac {1} {6}\right)$$ </p>
<p>and the joint pmf is just the product of these pmf.</p>
|
3,797,724 | <blockquote>
<p>Find all real continuous functions that verifies :
<span class="math-container">$$f(x+1)=f(x)+f\left(\frac{1}{x}\right) \ \ \ \ \ \ (x\neq 0) $$</span></p>
</blockquote>
<p>I found this result <span class="math-container">$\forall x\neq 1 \ \ f(x)=f\left(\frac{x}{x-1} \right)$</span> and I tried ... | nonuser | 463,553 | <p>Some partial results:</p>
<ul>
<li>Letting <span class="math-container">$x+1= {1\over x} $</span> we get <span class="math-container">$$x_{1,2}= {-1\pm \sqrt{5}\over 2}$$</span> are zeroes for <span class="math-container">$f$</span>.</li>
<li>If <span class="math-container">$x\to {1\over x}$</span> we get <span clas... |
3,237,242 | <p>I have the following problem:</p>
<p>I need to prove that given the following integral</p>
<p><span class="math-container">$\int_{0}^{1}{c(k,l)x^k(1-x)^l}dx = 1$</span>,</p>
<p>we the constant <span class="math-container">$c(k,l) = (k+l+1) {{k+l}\choose{k}} = \frac{(k+l+1)!}{k!l!}$</span>,</p>
<p>with the use of... | G Cab | 317,234 | <p>Let me give you an intuitive hint on how to deal with 2D induction</p>
<p><a href="https://i.stack.imgur.com/l3CTh.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/l3CTh.png" alt="2D_Induction_1"></a> </p>
<p>You have demonstrated that the hypothesis is true on the axis <span class="math-conta... |
4,000,089 | <p>Let <span class="math-container">$x$</span> be an element of a Banach Algebra. Let <span class="math-container">$\lambda \in \rho(x)$</span>, where <span class="math-container">$\rho(x)$</span> is the resolvent of <span class="math-container">$x$</span>.</p>
<p>Let <span class="math-container">${d(\lambda, \sigma(x)... | Ng Chung Tak | 299,599 | <p><strong>Equation of tangent</strong></p>
<p><span class="math-container">$$0=T(x,y)\equiv \frac{x\cos t}{a}+\frac{y\sin t}{b}-1$$</span></p>
<p><strong>Equation of normal</strong></p>
<p><span class="math-container">$$0=N(x,y) \equiv \frac{ax}{\cos t}-\frac{by}{\sin t}-(a^2-b^2)$$</span></p>
<p>Note that <span clas... |
2,074,276 | <p>How would I go about computing $\displaystyle\int_{10}^{16}\sin(\cosh^{-1}(x)+7)\mathrm dx$?</p>
<p>I haven't attempted anything yet, because I don't even know how to integrate the inverse hyperbolic cosine.</p>
| Fawad | 369,983 | <p><a href="https://i.stack.imgur.com/A052G.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/A052G.png" alt=""></a></p>
<p>Let's assume you have a number,for example 5. When you are asked 5 is in which circle you will say it is in B. Then by this statement one can say your number is also in A.</p>
<... |
2,074,276 | <p>How would I go about computing $\displaystyle\int_{10}^{16}\sin(\cosh^{-1}(x)+7)\mathrm dx$?</p>
<p>I haven't attempted anything yet, because I don't even know how to integrate the inverse hyperbolic cosine.</p>
| Chas Brown | 167,790 | <p>Let's agree that 'if you are in Paris, then you are in France' ($A \implies B$). (We could get picky, and say maybe you're in Paris, Texas; but let's not!).</p>
<p>But then the <strong>converse</strong> ($B \implies A$) is not automatically true: for example, we can't then deduce from 'if you are in Paris, then you... |
4,568,221 | <p><span class="math-container">$$\iint\limits_{D} \left(x^2+y^2\right)\mathrm{d}x \mathrm{d}y$$</span>
where <span class="math-container">$D$</span> is given each time by
<span class="math-container">$D=x^2-y^2=1,\hspace{0.5cm} x^2-y^2=9,\hspace{0.5cm} xy=2,\hspace{0.5cm} xy=4$</span></p>
<p>I try to use Polar coordin... | user170231 | 170,231 | <p><span class="math-container">$D$</span> is made of two disconnected but symmetric regions. <span class="math-container">$xy$</span> is positive, so either both <span class="math-container">$x,y$</span> are positive or they are both negative. Let <span class="math-container">$D_+$</span> be the part of <span class="m... |
2,289,777 | <p>I have a function $f(x)=(8x^2+7)^3(x^3-7)^4$</p>
<p>I have differentiated it using the chain rule and arrived at:</p>
<p>$3(8x^2+7)^2 \cdot 16x \cdot 4(x^3-7)^3 \cdot 3x^2$ And apparently this is wrong?</p>
<p>What am I missing here? </p>
| Claude Leibovici | 82,404 | <p>When you face problem like this one which only contains products, ratios and powers, your best friend is logarithmic differentiation $$f=(8x^2+7)^3(x^3-7)^4\implies \log(f)=3\log(8x^2+7)+4\log(x^3-7)$$ Differentiate both sides $$\frac{f'}f=3 \frac{16x}{8x^2+7}+4\frac{3x^2}{x^3-7}=\frac{12 x \left(12 x^3+7 x-28\right... |
449,296 | <p>I am trying to deduce how mathematicians decide on what axioms to use and why not other axioms, I mean surely there is an infinite amount of available axioms. What I am trying to get at is surely if mathematicians choose the axioms then we are inventing our own maths, surely that is what it is but as it has been pra... | Greebo | 64,519 | <p>There's one important thing that any choice of axioms should at least do, which is to be consistent, i.e. you cannot prove both a theorem and it's inverse, because if you could, you could prove anything at all, which would not give you any interesting theory.</p>
<p>That said, thanks to Gödel, we know that ZFC cann... |
449,296 | <p>I am trying to deduce how mathematicians decide on what axioms to use and why not other axioms, I mean surely there is an infinite amount of available axioms. What I am trying to get at is surely if mathematicians choose the axioms then we are inventing our own maths, surely that is what it is but as it has been pra... | Doug Spoonwood | 11,300 | <p>I will speak boldly and assert that the very purpose of an axiomatic system lies in developing a theory deductively. This implies that the axiomatic system will basically help you or your machine generate theorems which require little to no insight on your part (though of course, any insight you have, if correct, w... |
4,122,419 | <blockquote>
<p>From the triangle <span class="math-container">$\triangle ABC$</span> we have <span class="math-container">$AB=3$</span>, <span class="math-container">$BC=5$</span>, <span class="math-container">$AC=7$</span>. If
the point <span class="math-container">$O$</span> placed inside the triangle <span class="m... | JMP | 210,189 | <p>The starting position for <span class="math-container">$B$</span> for a draw with <span class="math-container">$A$</span> can be found by comparing the ratios of the race lengths for <span class="math-container">$A$</span> and <span class="math-container">$B$</span>:</p>
<p><span class="math-container">$$\frac{\text... |
908,083 | <p>I'd like to know what methods can I apply to simplify the fraction $\frac{4x + 2}{12 x ^2}$ </p>
<p>Is it valid to divide above and below by 2? (I didn't know it but Geogebra's Simplify aparantly does this)</p>
<p>Thanks in advance</p>
| Cookie | 111,793 | <p>Factor out the $2$ from both the numerator and denominator. That is,
$$\require{cancel}\frac{4x+2}{12x^2}=\frac{\cancel{2}(2x+1)}{\cancel{2}\cdot6x^2}=\frac{2x+1}{6x^2}.$$</p>
|
4,013,796 | <p>If we make a regular polygon with n vertices (n edges) and triangulate on the inside with n-3 edges, then triangulate on the outside with (n-3) edges (or draw dotted lines inside again), a Maximal Planar Graph is formed. Edges shouldn't be repeated and there's no loops or directions.</p>
<p>How many distinct graphs... | John Hunter | 721,154 | <p>Here is a partial answer.</p>
<p>In what follows the number of distinct Hamiltonian Maximal Planar Graphs with n vertices is <span class="math-container">$X(n)$</span>. It should follow A000109 <a href="https://oeis.org/A000109/list" rel="nofollow noreferrer">https://oeis.org/A000109/list</a> up to <span class="math... |
34,775 | <p><strong>What is the goal of MSE? Is it to get a repository of interesting questions and well-written answers. Or are we instead an online math tutoring site where we help anyone as long as they seem to be trying. These two goals are often in contradiction with each other!</strong></p>
<p>I am afraid that we are head... | Tryst with Freedom | 688,539 | <p>There are three main categories of questions which I feel are worth talking about:</p>
<ol>
<li><p>Conceptually deep questions which can't be answered from standard books</p>
</li>
<li><p>Conceptually deep questions which can be answered from standard books</p>
</li>
<li><p>Standard problems which come as exercises<... |
4,394,676 | <p>Solve</p>
<p><span class="math-container">$$\frac{dy}{dx}=\cos(x-y)$$</span></p>
<p>So I know I need to make the substitution <span class="math-container">$u=x-y$</span> but then what's <span class="math-container">$du$</span>, is it <span class="math-container">$du=dx-dy$</span>?</p>
<p>Or do I rewrite <span class=... | PierreCarre | 639,238 | <p>You just need to note that when <span class="math-container">$u = x - y$</span> (<span class="math-container">$x$</span> being the independent variable) you have that <span class="math-container">$u' = 1-y'$</span>, i.e. <span class="math-container">$y' = 1-u'$</span>. The equation then becomes <span class="math-con... |
2,832,614 | <blockquote>
<p>Prove by induction that
$$\lim_{x \to a} \frac{x^n-a^n}{x-a}=na^{n-1}.$$</p>
</blockquote>
<p>I did a strange proof using two initial results: We know that result is true for $n=1$ and $n=2$. Assuming the result is true for $n=k-1$ and $n=k$, I can prove the result for $n=k+1$. For this I used my a... | Bumblebee | 156,886 | <p>$$x^{n+1}-a^{n+1}=(x-a)(x^n+a^n)+ax(x^{n-1}-a^{n-1})$$</p>
|
77,136 | <p>I have a multidimensional-variable list, suppose for example {{x1,y1},{x2,y2}..}. I have duplicate values for the 'x' coordinates and I need to find for the duplicate 'x' elements the corresponding minimum 'y'. A sample of this list is the following:</p>
<pre><code>l1={{1, 1.43E-46}, {21, 2.79E-48}, {41, 3.22E-45},... | Jens | 245 | <p>This works:</p>
<pre><code>l1 = {{1, 1.43 10^-46}, {21, 2.79 10^-48}, {41, 3.22 10^-45}, {41,
1.74 10^-46}, {81, 2.77 10^-46}, {121, 9.97 10^-48}, {161,
1.24 10^-45}, {181, 1.19 10^-45}};
Map[First[SortBy[#, Last]] &, SplitBy[l1, First]]
(*
==> {{1, 1.43*10^-46}, {21, 2.79*10^-48}, {41, 1.74*10^-... |
563,927 | <p>Show that $\mathbb{R}$
is not a simple extension of $\mathbb{Q}$
as follow:</p>
<p>a. $\mathbb{Q}$
is countable.</p>
<p>b. Any simple extension of a countable field is countable.</p>
<p>c. $\mathbb{R}$
is not countable.</p>
<p>I 've done a. and c. Can anyone help me a hint to prove b.?</p>
| Johannes Kloos | 26,325 | <p>Here is a <strong>hint</strong>: Let $F$ be a countable field, $F(a)$ a simple extension. Then every element in $F(a)$ can be written as a quotient $\frac{p(a)}{q(a)}$, where $p$ and $q$ are polynomials. Now, combine known results on countability.</p>
|
2,329,751 | <p>I have this definition: $f:R^n → R^m$ is differentiable at $a∈R^n$, if there exists a linear transformation $μ:R^n→R^m$ such that</p>
<p>$\lim_{h \to 0} \frac{|f(a+h)-f(a)-\mu(h)|}{|h|} = 0$.</p>
<p>My questions are what's the linear transformation $μ(h)$ for? What does it mean and where does it come from? Why is ... | Peter | 409,941 | <p>I'll try to give you an equivalent definition, which I think is a bit clearer, I will leave the equivalence to you in first instance. I might add more explanation later.</p>
<p>First let's consider the case that $n=2$ and $m=1$, (actually what I will say works equally well for arbitrary $n$, but $n=2$ is the simple... |
1,748,914 | <blockquote>
<p>Describe the structure of the Galois group of the splitting field $L$ of the polynomial $X^4-7$ over $\mathbb{Q}$</p>
</blockquote>
<p>I believe that $L=\mathbb{Q}(\sqrt[4]{7}, i)$ is the splitting field of the polynomial</p>
<p>Now I need to describe the structure of $Gal(L, \mathbb{Q})$</p>
<hr>
... | Noble Mushtak | 307,483 | <p>We know that the minimal polynomial of $\sqrt[4] 7$ has degree $4$ and the minimal polynomial of $i$ has degree $2$, so $[\Bbb{Q}(\sqrt[4] 7, i) : \Bbb{Q}]=[\Bbb{Q}(\sqrt[4] 7, i) : \Bbb{Q}(i)][\Bbb{Q}(i) : \Bbb{Q}]=4*2=8$. Thus, the Galois group has order $8$ because of the Fundamental Theorem of Galois Theory.</p>... |
1,748,914 | <blockquote>
<p>Describe the structure of the Galois group of the splitting field $L$ of the polynomial $X^4-7$ over $\mathbb{Q}$</p>
</blockquote>
<p>I believe that $L=\mathbb{Q}(\sqrt[4]{7}, i)$ is the splitting field of the polynomial</p>
<p>Now I need to describe the structure of $Gal(L, \mathbb{Q})$</p>
<hr>
... | Pedro | 23,350 | <p>There's a general tool to deal with this type of extensions. Suppose you have a diamond of extensions $F \subseteq L,L' \subseteq E$ (make the drawing) where $L\cap L'=F,LL'=E$. </p>
<p>Suppose moreover that $L/F$ is normal, and $E/F$ is Galois. Let $G={\rm Gal}(E/F)$, $H={\rm Gal}(E/L)$ and $K={\rm Gal}(E/L')$. Th... |
109,922 | <p>Let $B_n$ denote the group of signed permutations on $n$ letters. Is there a good explanation or understandable way to see why
$$
\sum_{w\in B_n}q^{\text{inv}(w)}=(2n)_q(2n-2)_q\cdots(2)_q?
$$</p>
<p>I've been thinking about it on and off while reading through Taylor's <em>Geometry of the Classical Groups</em>, b... | hoyland | 14,722 | <p>I can only offer a rough idea (and hope that I have the same definition of inv as you do). The proof in type A is on page 36 of the PDF version of EC 1 available on Stanley's website (<a href="http://www-math.mit.edu/~rstan/ec/ec1/" rel="nofollow">here</a>). Basically, any permutation can be encoded via its inversio... |
4,057,255 | <p>Suppose we have 10 items that we will randomly place into 6 bins, each with equal probability. I want you to determine the probability that we will do this in such a way that no bin is empty. For the analytical solution, you might find it easiest to think of the problem in terms of six events Ai, i = 1, . . . , 6 wh... | user2661923 | 464,411 | <p>Addendum added to examine the danger of overcounting from a different perspective.</p>
<hr />
<p>This is a reaction to the answers of Math Lover and true blue anil.</p>
<p>I definitely regard Inclusion-Exclusion as the preferred approach. Math Lover's answer completely covers that.</p>
<p>If you wish to consider th... |
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