qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,063,638 | <p>A student is standing in each cell of an $n\times n$ grid, looking at one of the four directions: up, down, left, right. It turns out that no student is at the border and looking out of the grid, and no student directly sees the face of another student (every student sees either the ear or the back of another studen... | aes | 194,204 | <p>The minimum is $n+2$, with the construction in chubakueno's comment.</p>
<p>Proof that this is minimal:</p>
<p>Split the students into lines (maximal subsets of students in a straight line all looking the same direction).</p>
<p>The last student in each line must be looking at an ear (in fact, the number of lines... |
896,079 | <blockquote>
<p>for real numbers like <span class="math-container">$a,b,c,x,y,z$</span> that <span class="math-container">$ax-2by+cz=0$</span> and <span class="math-container">$ac-b^2>0$</span> Prove:<span class="math-container">$$zx-y^2\leq0$$</span></p>
<p>Additional info: The Proof should be by contradiction.we c... | Noam D. Elkies | 93,983 | <p>It seems <strong>user2838619</strong> and <strong>Ewan Delanoy</strong>
already gave correct solutions with elementary inequalities.
Still the following viewpoint may be of interest, connecting the result with the
<a href="http://en.wikipedia.org/wiki/Schur_product_theorem" rel="nofollow">Schur
product theorem</a>.... |
896,079 | <blockquote>
<p>for real numbers like <span class="math-container">$a,b,c,x,y,z$</span> that <span class="math-container">$ax-2by+cz=0$</span> and <span class="math-container">$ac-b^2>0$</span> Prove:<span class="math-container">$$zx-y^2\leq0$$</span></p>
<p>Additional info: The Proof should be by contradiction.we c... | Michael Rozenberg | 190,319 | <p>Consider the following equation.
$$axt^2-2byt+cz=0.$$
Since $ax-2by+cz=0$, we see that this equation has root $1$, which gives
$$b^2y^2-acxz\geq0.$$
Since $ac>b^2$, we obtain $ac>0$ and since
$$acy^2\geq b^2y^2\geq aczx,$$ we obtain
$$acy^2\geq aczx,$$
which gives $y^2\geq zx$.</p>
<p>Done!</p>
|
1,104,450 | <p>The truth table for a logical disjunction shows that there is only one situation where the result can be false, being when both statements are false. As long as one statement is true, the result is also true. </p>
<p>This makes perfect sense to me, but I am wondering why this is the case. Could it not have been equ... | zoli | 203,663 | <p>True OR False is True in every language. Simply because every statements are TRUE OR False; at least in the two valued logic.</p>
|
2,554,060 | <p>The book said: "if $f$ is a bounded linear functional then ker $f$ is a closed subspace of the Hilbert space $H$."</p>
<p>But I do not know why, could anyone clarify this for me please? </p>
| Dr. Sonnhard Graubner | 175,066 | <p>prove that $$x_1=1$$ or $$x_2=\frac{c(a-b)}{a(b-c)}$$ are Solutions of your equation
an $AP$ is not possible, if we assume that $$x_1=1,x_2=qx_1=q$$ then we get
$$q={\frac{c(a-b)}{a(b-c)}}$$</p>
|
1,769,297 | <p>I have been reading <em>Representations and Characters of Groups</em> by Gordon James and Martin Liebeck. I encountered the following construction of an $\mathbb{R}G$-module from a $\mathbb{C}G$-module.</p>
<p><a href="https://i.stack.imgur.com/gXuGt.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.co... | Marc van Leeuwen | 18,880 | <p>Indeed the argument used is not quite clear as it stands, because context is missing. Apparently something is already known about uniqueness of decomposition of characters into irreducibles. What is known here is that we have a $\Bbb R G$-module that is reducible, and whose character is $\chi+\overline\chi$; its cha... |
468,437 | <p>If $A\times B$ is closed in $X\times Y$, then are $A$ and $B$ closed in $X$ and $Y$ respectively?</p>
| A Googler | 89,679 | <p>Actually you need to calculate $$(1/2 * 10^{-6} )^{455}$$
Which is by no means easy :<br>
$$1.07486017721073420028655449423203634073126755189808458... × 10^{-2867}$$
(Using <a href="http://www.wolframalpha.com/input/?i=%281%2F2+%2a+10%5E-6%29%5E455+" rel="nofollow">Wolfram alpha</a>)</p>
|
2,500,723 | <blockquote>
<p>A price $p$ (in dollars) and demand $x$ for a product are related by $$\left(2x^2\right)-2xp+50p^2 = 20600$$ If the price is increasing at a rate of $2$ dollars per month when the price is $20$ dollars, find the rate of change of the demand.</p>
</blockquote>
<p>I was a little confused on how to proc... | gen-ℤ ready to perish | 347,062 | <p>The question is asking you to compute $\left.\dfrac{dx}{dt}\right|_{p=20}$ given that $\left.\dfrac{dp}{dt}\right|_{p=20}=2$ (where $t$ is time in months). The giveaway of that was the keyword “rate” and the unit “dollars per month.”</p>
<p>The first thing we need to do is implicitly differentiate so that our equat... |
278,511 | <p>What is the best unconditional upper bound for $p_{n^2}-p_{(n-1)^2}$ such that $p_n$ is the $n$-th prime number?</p>
<p>Asymptotics suggest it's somewhere near $4 n \ln n$, but how to prove this?</p>
<p>Edit: it's known that there is a prime between $[x,x+x^{13/23}]$, so one can bound $p_{n^2}-p_{(n-1)^2} \leq 2 n... | kodlu | 17,773 | <p>Using unconditional results cited by Dusart <a href="https://arxiv.org/pdf/1002.0442.pdf" rel="nofollow noreferrer">in this paper</a> in their simple versions obtained by Robin and Salvy which are of the form
$$
p_k \leq k\left( \log k+\log\log k-1+\frac{\log\log k-2}{\log k}\right),
$$
and
$$
p_k \geq k\left( \log ... |
3,988,390 | <p>I am trying to solve the problem: <span class="math-container">$x^{2}+xy+y^{3}=0$</span> using implicit differentiation.</p>
<p>My workings:</p>
<p><span class="math-container">$(1)\,\,\,{d\over dx}{[x^2]}\,+{d\over dx}{[xy]}\,+\,{d\over dx}{[y^3]}={d\over dx}{[0]}$</span></p>
<p><span class="math-container">$(2)\,\... | greg | 357,854 | <p>Differentials will make the process much more intuitive.
<span class="math-container">$$\eqalign{
&0 = d\Big(x^2 + xy + y^3\Big) \\
&0 = 2x\,dx + \big(x\,dy+y\,dx\big) + 3y^2dy \\
&\big(x+3y^2\big)\,dy = -\big(2x+y\big)\,dx \\
&\frac{dy}{dx} = -\left(\frac{2x+y}{x+3y^2}\right) \\
}$$</span></p>
|
561,210 | <p>I have a simple part of a question to solve. The problem is my answer is different to the solution in my textbook.</p>
<p>The equation is: $$\frac{5v}{6} = \frac{(\frac{1}{2}a+b+\frac{1}{2} c)v}{a+b+c}$$</p>
<p>I am supposed to get $$\frac{2}{3}(a+b+c) = b$$</p>
<p>But I simply get: $$b=2a +2c$$</p>
<p>I get my... | beardhatcode | 52,532 | <p>$b=2a+2c$</p>
<p>Adding $2b$ on both sides gives</p>
<p>$b + 2b =2a+2b+2c$</p>
<p>Or better</p>
<p>$3b =2a+2b+2c$</p>
<p>And if you throw the $3$ to the other side you get:
$b =\frac{2}{3}(a+b+c)$</p>
|
60,486 | <p>I have noted several instances <a href="https://mathematica.stackexchange.com/a/51377/7167">here</a> and <a href="https://mathematica.stackexchange.com/a/55412/7167">here</a> where <code>ConvexHullMesh</code>, introduced in v10 has been used to greatly simplify some geometry problems. Determining the volume of a re... | RunnyKine | 5,709 | <p>Here is another way to compute the surface area of a 3D convex hull:</p>
<pre><code>SeedRandom[0]
pts = RandomReal[5, {50, 3}];
chull = ConvexHullMesh @ pts;
</code></pre>
<p>Then:</p>
<pre><code>Tr @ PropertyValue[{chull, 2}, MeshCellMeasure]
</code></pre>
<blockquote>
<p>86.8845076</p>
</blockquote>
<p><str... |
127,645 | <p>In Lay's Linear Algebra and Its Applications textbook, he defines the matrix
$$A=\begin{bmatrix}
4 & 11 & 14\\8 & 7 & -2
\end{bmatrix}$$
and claims that the transformation $T(x)=A x$ maps the sphere to an ellipse.</p>
<p><a href="https://i.stack.imgur.com/grGkv.png" rel="nofollow noreferrer"><img sr... | chuy | 237 | <p>Another visualization you might find helpful:</p>
<pre><code>sp = MeshCoordinates[
DiscretizeGraphics[Sphere[], MaxCellMeasure -> 0.02]];
el = Map[(A.#)~Join~{-10} &, sp];
</code></pre>
<p>I used <a href="http://reference.wolfram.com/language/ref/DiscretizeGraphics.html" rel="noreferrer"><code>Discretize... |
2,055,695 | <p>Use mathematical induction to prove each of the following statements.
let <span class="math-container">$$g(n) = 1^3 + 2^3 + 3^3 + ... + n^3$$</span></p>
<p>Show that the function <span class="math-container">$$g(n)= \frac{n^2(n+1)^2}{4}$$</span> for all n in <strong>N</strong></p>
<p>so the base case is just g(1) ... | barak manos | 131,263 | <p><strong>First, show that this is true for $n=1$:</strong></p>
<p>$\sum\limits_{k=1}^{1}k^3=\frac{1^2(1+1)^2}{4}$</p>
<p><strong>Second, assume that this is true for $n$:</strong></p>
<p>$\sum\limits_{k=1}^{n}k^3=\frac{n^2(n+1)^2}{4}$</p>
<p><strong>Third, prove that this is true for $n+1$:</strong></p>
<p>$\sum... |
2,055,695 | <p>Use mathematical induction to prove each of the following statements.
let <span class="math-container">$$g(n) = 1^3 + 2^3 + 3^3 + ... + n^3$$</span></p>
<p>Show that the function <span class="math-container">$$g(n)= \frac{n^2(n+1)^2}{4}$$</span> for all n in <strong>N</strong></p>
<p>so the base case is just g(1) ... | John | 7,163 | <p>You want to show that $$\sum_{j=1}^n j^3 = \frac{n^2(n+1)^2}{4}.$$</p>
<p>You've shown that it holds for the base case, which is good.</p>
<p>Now, you assume that it holds for $n=k$:</p>
<p>$$\sum_{j=1}^k j^3 = \frac{k^2(k+1)^2}{4}.$$</p>
<p>Then, a strategy is to write out one part or the other for the $n=k+1$ ... |
2,055,695 | <p>Use mathematical induction to prove each of the following statements.
let <span class="math-container">$$g(n) = 1^3 + 2^3 + 3^3 + ... + n^3$$</span></p>
<p>Show that the function <span class="math-container">$$g(n)= \frac{n^2(n+1)^2}{4}$$</span> for all n in <strong>N</strong></p>
<p>so the base case is just g(1) ... | Vepir | 318,073 | <p>Show that: $$1^3 + 2^3 + 3^3 + ... + n^3= \frac{n^2(n+1)^2}{4}$$</p>
<hr>
<p>$(1)$ First step is to evaluate the expression for $n=1$, $$1^3=\frac{1\times(1+1)^2}{4}$$
Which is $1=1$, which means we can proceed.</p>
<hr>
<p>$(2)$ Now assume it is true for $n$, $$1^3 + 2^3 + 3^3 + ... + n^3= \frac{n^2(n+1)^2}{4}$... |
1,317,206 | <p>Let $f$ be a function defined on $\mathbb{R}$ which is differentiable at every point, and such that $f (x + 1) = f (x)$ for every $x$. Prove that there exist at least $2$ points in $[0, 1]$ where the derivative of $f$ equals zero.</p>
<p>I understand how to prove there is one point, by applying the mean value theor... | Gappy Hilmore | 219,783 | <p>You get </p>
<p>$$f'(x)=\frac{\cos x}{1-2\sin x+\sin^2x}$$</p>
<p>To integrate it use $u=\sin x$</p>
<p>$$\frac{1}{1-\sin x}$$</p>
<p>Now insert $x=0$ (in the original equation) to find the constant</p>
<p>$$f(x)=\frac{1}{1-\sin x}-1$$</p>
|
1,317,206 | <p>Let $f$ be a function defined on $\mathbb{R}$ which is differentiable at every point, and such that $f (x + 1) = f (x)$ for every $x$. Prove that there exist at least $2$ points in $[0, 1]$ where the derivative of $f$ equals zero.</p>
<p>I understand how to prove there is one point, by applying the mean value theor... | me_ravi_ | 221,457 | <p>You applied Newton-Leibniz rule correctly. </p>
<p>$$f'(x) = \frac{cos x}{1-2sinx+ sin^2 x}=\frac{cosx}{(sinx-1)^2}$$ and then integrate.</p>
<p>$$f(x)=\frac{-1}{sinx-1}+c$$</p>
<p>From given initial equation $f(0)=0$,
which gives $c=-1$.</p>
<p>Hence, $$f(x)=\frac{-1}{sinx-1}-1$$</p>
|
3,396,816 | <p>Let <span class="math-container">$k\in \mathbb{N}$</span>. Which of <span class="math-container">$[0]_3, [1]_3, [2]_3$</span> is <span class="math-container">$[5^k]_3$</span> equal to? Prove your answer.</p>
<p>Below is my proof so far. I figured out what it equals when <span class="math-container">$k$</span> is ev... | Sri-Amirthan Theivendran | 302,692 | <p>Write
<span class="math-container">$$
5^k=(3+2)^k=\sum_{j=0}^k\binom{k}{j}3^j2^{k-j}=2^k+\sum_{j=1}^k \binom{k}{j}3^j2^{k-j}
$$</span>
to deduce that for <span class="math-container">$k\geq 1$</span> it is the case that <span class="math-container">$5^k=2^k$</span> modulo <span class="math-container">$3$</span>.</p>... |
3,396,816 | <p>Let <span class="math-container">$k\in \mathbb{N}$</span>. Which of <span class="math-container">$[0]_3, [1]_3, [2]_3$</span> is <span class="math-container">$[5^k]_3$</span> equal to? Prove your answer.</p>
<p>Below is my proof so far. I figured out what it equals when <span class="math-container">$k$</span> is ev... | J. W. Tanner | 615,567 | <p><em>Case 1:</em> Let <span class="math-container">$k$</span> be odd. Then <span class="math-container">$k=2n+1$</span> for some <span class="math-container">$n,$</span> </p>
<p>so <span class="math-container">$[5^k]_3=[5^{2n+1}]_3=[(5^2)^n]_3[5]_3=[1^n]_3[5]_3=[2]_3.$</span></p>
<p><em>Case 2:</em> Let <span clas... |
1,348,714 | <p>Is there always a square between two consecutive cubes? </p>
<p>I just thought of this question. It seems really simple and the answer is probably yes.</p>
<p>Edit: I should have given this more than 2 seconds of thought.</p>
| individ | 128,505 | <p>We write the equation:</p>
<p>$$(x+1)^3-x^3=y^2$$</p>
<p>$$3x^2+3x+1=y^2$$</p>
<p>Let's use the formula. <a href="https://mathoverflow.net/questions/31118/integer-polynomials-taking-square-values/195614#195614">https://mathoverflow.net/questions/31118/integer-polynomials-taking-square-values/195614#195614</a></p... |
31,113 | <p>Zagier has a very short proof (<a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1041893" rel="noreferrer">MR1041893</a>, <a href="https://www.jstor.org/stable/2323918" rel="noreferrer">JSTOR</a>) for the fact that every prime number $p$ of the form $4k+1$ is the sum of two squares.
The proof defines an inv... | Zurab Silagadze | 32,389 | <p>Zagier's proof is non-constructive. Part II of this paper</p>
<ul>
<li>Aimeric Malter, Dierk Schleicher and Don Zagier, <em>New Looks at Old Number Theory</em>, AMM <strong>120</strong>, No. 3 (2013), pp. 243–264 doi:<a href="https://doi.org/10.4169/amer.math.monthly.120.03.243" rel="nofollow noreferrer">10.4169/ame... |
641,492 | <p>I'm given a continuously differentiable $f:\mathbb{R}\to\mathbb{R}$ , we define a function $\vec{F}:\mathbb{R}^2 \to \mathbb{R}^2 $ by:
$\vec{F}( \vec{x} ) =f(|| \vec{x} || ) \vec{x} $ .</p>
<p>The question is:
Prove that $\vec{F}$ is a conservative field and find its potential. </p>
<p>I guess that if I'll find a... | TZakrevskiy | 77,314 | <p>We know that a field is conservative iff it's rotor-free. Let's note $r=\|x\|$.
First, our field is continuous everywhere - it's a product of two continuous functions.</p>
<p>Now, let's prove that its gradient is also continuous everywhere. It writes</p>
<p>$$\nabla F = \begin{pmatrix}F_{1,1}& F_{1,2}\\F_{2,1}... |
641,492 | <p>I'm given a continuously differentiable $f:\mathbb{R}\to\mathbb{R}$ , we define a function $\vec{F}:\mathbb{R}^2 \to \mathbb{R}^2 $ by:
$\vec{F}( \vec{x} ) =f(|| \vec{x} || ) \vec{x} $ .</p>
<p>The question is:
Prove that $\vec{F}$ is a conservative field and find its potential. </p>
<p>I guess that if I'll find a... | Christian Blatter | 1,303 | <p>A hint:</p>
<p>The given field is rotationally symmetric. It is not unreasonable to assume that a possible potential $(x,y)\mapsto V(x,y)$ is rotationally symmetric as well. Therefore make the "Ansatz"
$$V(x,y)=v\bigl(\sqrt{x^2+y^2}\bigr)$$
with a function $v$ of one variable, and check whether this leads somewhere... |
935,358 | <p>What is an example of a six-element group that is not abelian?</p>
<p>I can't think of any. It is very possible that I am overthinking this. Thank you for any help.</p>
| hchar | 120,815 | <p>In general,</p>
<blockquote>
<p><span class="math-container">$\{e, a,b,b^2,ab,ab^2\}$</span></p>
</blockquote>
<p>is a six-element <strong>non-abelian</strong> group, isomorphic to <span class="math-container">$S_3$</span>, when <span class="math-container">$ba=ab^2$</span>, where <span class="math-container">$a... |
573,518 | <p>I have learnt very basic category theory (up to Yoneda lemma from Hungerford's Algebra text). My question is how much category theory should <em>every</em> Mathematics student who is not planning to specialize in that area learn ? </p>
<p>I am not sure which area of Mathematics I would like to specialize in, though... | Yuri Sulyma | 13,632 | <p>My personal opinion is that category theory is like set theory; it's a language, everyone should know the basics, and everything in the "basics" is essentially trivial. Here "basics" for set theory means subsets, products, power sets, and identities like $f^{-1}(\bigcap A) = \bigcap f^{-1}(A)$. For category theory, ... |
573,518 | <p>I have learnt very basic category theory (up to Yoneda lemma from Hungerford's Algebra text). My question is how much category theory should <em>every</em> Mathematics student who is not planning to specialize in that area learn ? </p>
<p>I am not sure which area of Mathematics I would like to specialize in, though... | Giorgio Mossa | 11,888 | <p>Let's start to say that I'm one of those people that believe that the more mathematics you know the better it is.</p>
<p>That said based on what it seems that yours interests are I believe that you could be interested in learning quite enough category theory.
Category theory was originally developed by Eilenberg an... |
2,410,118 | <p>Edit: Excuse me, the numerator of the given solution should've been a rising factorial. That said, I still don't understand where it comes from?</p>
<p>The question is as follows :</p>
<blockquote>
<p>n friends visit k exhibitions, each person visits only 1 exhibition. Find the number of possibilities if</p>
<p>b) O... | dxiv | 291,201 | <p>The following gives $a_0=1, a_1=k, a_2=k+1, a_3=2k, \,\dots\,$:</p>
<p>$$
a_n = \left\lfloor \frac{n+1}{2} \right\rfloor \, k + (n+1) \bmod 2
$$</p>
|
1,179,154 | <blockquote>
<p>Let <span class="math-container">$f$</span> be a twice continuously differentiable function from <span class="math-container">$[0,1]$</span> into <span class="math-container">$\mathbb R$</span>. Given that
<span class="math-container">$$f(0)+2f(\frac{1}{2})+f(1)=0,$$</span>
show that
<span class... | Kelenner | 159,886 | <ol>
<li>Let <span class="math-container">$g_1(x)=x(x-1/2)$</span>, <span class="math-container">$g_2(x)=(x-1)(x-1/2)$</span>. By two integration by parts, we have</li>
</ol>
<p><span class="math-container">$$\int_0^{1/2}f^{\prime\prime}(x)g_1(x)dx=-\frac{f(1/2)+f(0)}{2}+2\int_0^{1/2}f(x)dx$$</span>
and
<span class="ma... |
4,205,525 | <p>I have to prove by mathematical induction that <span class="math-container">$L = R$</span></p>
<p>Question:</p>
<p><a href="https://i.stack.imgur.com/3UYsv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3UYsv.png" alt="enter image description here" /></a></p>
<p>What I've tried so far:</p>
<p><a ... | José Carlos Santos | 446,262 | <p>You made a mistake: when you wrote <span class="math-container">$n+1\bigl(5(n+1)+3\bigr)$</span>, you should have written <span class="math-container">$(n+1)\bigl(5(n+1)+3\bigr)$</span>. A proof can be done as follows:<span class="math-container">\begin{multline}\frac{n(5n+3)}2+5(n+1)-1=\frac{(n+1)\bigl(5(n+1)+3\big... |
564,918 | <p>$$\sec(2x)=\frac{\sec^2x+\sec^4x}{2+\sec^2x-\sec^4x}$$</p>
<p>I have no idea how to verify this. I've tried changing it into cosine but it doesn't work. </p>
| math110 | 58,742 | <p>Note
$$\dfrac{\sec^2{x}+\sec^4{x}}{2+\sec^2{x}-\sec^4{x}}=\dfrac{\sec^2{x}(1+\sec^2{x})}{(1+\sec^2{x})(2-\sec^2{x})}=\dfrac{\sec^2{x}}{2-\sec^2{x}}=\dfrac{1}{2\cos^2{x}-1}=\sec{(2x)}$$</p>
|
1,345,503 | <p>I have no idea about the answer, but I'm viewing the question this way;</p>
<p>If the smallest score obtained from the any three throws of the die is $1$, then largest among the other two throws must be $3$, and likewise
if the smallest score obtained from any of the three throw of the die is $2$ then a $6$ must a... | Asinomás | 33,907 | <p>Classify depending on the smallest die.</p>
<p>If the smallest dice is $1$ the largest is $3$ and the third can be $1,2$ or $3$, then the possible outcomes are:</p>
<p>$1,1,3$ ($3$ ways to order them)</p>
<p>$1,2,3$ ($6$ ways to order them)</p>
<p>$1,3,3$ ($3$ ways to order them)</p>
<p>If the smallest dice is ... |
4,188,176 | <p>Consider the second order ordinary differential equation with one parameter <span class="math-container">$k$</span>
<span class="math-container">$$(x-3)^7 ((x^2-4 x+3) f''(x)+2 x f'(x))+3456 k (3-2 x)^2 (x-1) x^2 f(x)=0, \quad 0 \leq x \leq 1$$</span>
and a set of boundary conditions
<span class="math-container">$$... | uranix | 200,750 | <p>This is a linear <a href="https://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory" rel="nofollow noreferrer">Sturm–Liouville</a> eigenvalue problem (if <span class="math-container">$f'(0) = 1$</span> condition is dropped, it is only needed to fix the unique solution).
<span class="math-container">$$
-\frac{d}{d... |
204,037 | <p>Let $H$ and $K$ be two subgroups of $G$ of finite order. Can $HK$ be infinite?</p>
| Hagen von Eitzen | 39,174 | <p>Yes.
Consider in the euclidean plane the reflection along the $x$ axis and the reflection along some other axis at an angle $\alpha$ with $\frac\alpha\pi\notin\mathbb Q$. These reflection generate a group that also contains infinitely many rotations by multiples of $2\alpha$.</p>
|
204,037 | <p>Let $H$ and $K$ be two subgroups of $G$ of finite order. Can $HK$ be infinite?</p>
| Logan M | 8,473 | <p>This will depend on what you mean by $HK$. </p>
<p>If you mean just those elements in $G$ of the form $hk$ for $h \in H$, $k \in K$, then $|HK| \le |H||K|$. </p>
<p>If you mean the group generated by products of elements in $H$ and $K$, then the infinite dihedral group $\langle x,y | x^2=y^2=1 \rangle$ is an examp... |
952,922 | <p>Evaluate <span class="math-container">$\begin{align} \int^{\infty}_{0} \dfrac{x^\alpha}{(1+x^2)^2}\end{align}dx, \ -1 < \alpha<3.$</span></p>
<p>May I verify if my solution is correct? Thank you. </p>
<p>Consider <span class="math-container">$\gamma_1:=\{x:-\dfrac{1}{r}\leq x \leq-r\}, \ \gamma_2:=\{re^{it}:... | Yiorgos S. Smyrlis | 57,021 | <p>Your proof is correct. To make it more rigorous you should add the following:</p>
<p>a. Specify the direction of the contour of your integrals.</p>
<p>b. It does not hurt to explain the necessity that $a\in (-1,3)$.</p>
<p>c. Most important, in order to define $z^a$ you need to pick the branch of the logarithm
... |
43,043 | <p>I'm working through Carl Bender's <a href="http://youtu.be/LYNOGk3ZjFM">Mathematical Physics lectures</a> on YouTube (which are great fun), and I'd like <em>Mathematica</em>'s help solving terms in the perturbation series. It would be convenient if expressions like</p>
<pre><code>SeriesCoefficient[Sum[a[n] b^n, {n,... | Carl Woll | 45,431 | <p>The new in M12 function <a href="http://reference.wolfram.com/language/ref/AsymptoticSolve" rel="nofollow noreferrer"><code>AsymptoticSolve</code></a> can be used to find the perturbation expansions:</p>
<pre><code>AsymptoticSolve[x^5 + b x == 1, {x, 1}, {b, 0, 5}]
</code></pre>
<blockquote>
<p>{{x -> 1 - b/5 - ... |
1,552,292 | <p>I would like to share a point on which I was really stuck with you, in order for you to help me to see my mistake:</p>
<p>Here is the theorem:</p>
<p><strong>A set X is infinite if and only if it is equipotent (equivalent) to a proper subset of itself</strong></p>
<p>And here is my question.
The set of positive i... | Mankind | 207,432 | <p>Here's a more precise statement of the theorem:</p>
<p>A set $X$ is infinite, if and only if <em>there exists</em> a proper subset $Y\subset X$, such that $X$ and $Y$ are equipotent.</p>
<p>So, $X$ is not expected to be equipotent to any proper subset of itself, but if $X$ is infinite, then it is equipotent to <em... |
1,552,292 | <p>I would like to share a point on which I was really stuck with you, in order for you to help me to see my mistake:</p>
<p>Here is the theorem:</p>
<p><strong>A set X is infinite if and only if it is equipotent (equivalent) to a proper subset of itself</strong></p>
<p>And here is my question.
The set of positive i... | Justpassingby | 293,332 | <p>The statement should be read "if there exists a proper subset to which it is equipotent". As you have shown, that does not mean that every proper subset qualifies.</p>
|
1,469,737 | <p>I think I have a solution so this sequence in:</p>
<p>$a_n = \lfloor \frac{n-1}{2}\rfloor + n$.</p>
<p>I have been plugging numbers in and so far it works. However, there must be a better way to check if this indeed a solution.</p>
<p>My questions are:</p>
<p>1) Is this a solution, and</p>
<p>2) What is the bes... | lulu | 252,071 | <p>Well, $n$ is either even or odd.</p>
<p>If $n$ is even, $n=2k$ and $$\lfloor\frac {n-1}2\rfloor=\lfloor k-\frac 12 \rfloor=k-1$$
So $$a_{2k}=k-1+2k=3k-1$$</p>
<p>If $n$ is odd, $n=2k+1$ and $$\lfloor\frac {n-1}2\rfloor=\lfloor k \rfloor=k$$
So $$a_{2k+1}=k+2k+1=3k+1$$</p>
<p>Either way $a_n$ can't be divisible by... |
980,470 | <p>How to find the continued fraction of $\sqrt{n}$, for an integer $n$?</p>
<p>I saw a site where they explained it, but it required a calculator. Is it possible to do it without a calculator? </p>
| Bumblebee | 156,886 | <p>If your $n$ is not a perfect square, I'll show it with an example. Consider $\sqrt{23}.$ $$\sqrt{23}\approx4.8$$ Here I denote successive irrational by $x_k$ and it's integer part by $a_k.$
$$x_0=\sqrt{23}=4+(\sqrt{23}-4)\rightarrow \,\ a_0=4$$
$$x_1=\dfrac{1}{x_0-a_0}=\dfrac{1}{\sqrt{23}-4}=\dfrac{\sqrt{23}+4}{7}=1... |
1,071,981 | <p>I have the following integral:
$$\int_V ((\vec\nabla.\vec A)(\vec\nabla.\vec B)+\vec B.\vec\nabla(\vec\nabla.\vec A))dV$$</p>
<p>Using Gauss' theorem I can convert this into a surface integral. However, I need to rewrite the integrandum into something of the form $\vec\nabla.\vec V$ in order to apply Gauss.</p>
<p... | Muphrid | 45,296 | <p>The key is that you have both $B \cdot \nabla$ and $\nabla \cdot B$. This is a tell-tale sign of the product rule being applied. You can use "overdot notation" to write the expression as a total divergence. First, recognize that</p>
<p>$$(B \cdot \nabla) (\nabla \cdot A) + (\nabla \cdot B)(\nabla \cdot A)= (\dot ... |
30,339 | <p>I need to construct a vector similar to:</p>
<pre><code>v[x_]:={0, 0, x, 0, 0, 2*x, 0, 0, 3*x, ....., 0, 0, n*x};
</code></pre>
<p>where <code>n=10^9</code>. How can I make such a vector wisely?</p>
| Mr.Wizard | 121 | <p>After testing a number of different methods I have found this code to be the fastest on packable data:</p>
<pre><code>Flatten[ArrayPad[{Range[x, x n, x]}, {{2, 0}, 0}]\[Transpose]]
</code></pre>
<p>And this fastest on unpackable data:</p>
<pre><code>Module[{rs = ConstantArray[0, 3 n]}, rs[[Range[3, 3 n, 3]]] = Ra... |
340,485 | <p>If $f^{'}(3) = 2$ then find</p>
<p>$$\lim_{h \rightarrow 0}\left(\frac{f(3+h^2)-f(3-h^2)}{2h^2} \right)$$</p>
| Inceptio | 63,477 | <p><strong>Hint</strong>: $3+h^2=k$, $(k-2h^2)=3-h^2$, Now that's a differential question.</p>
<p>$$\lim_{h \rightarrow 0}\left(\frac{f(k)-f(k-2h^2)}{2h^2} \right)=f'(3)$$</p>
|
1,463,152 | <p>Let $$f(x)= \begin{cases}
e^{-1/x} & x>0 \\
0 & x\le 0
\end{cases}
$$
I want to show that $f^{(n)}(0)=0$ for every $n\in\mathbb{N}$.</p>
<p>My idea is to do this by induction on $n$. I can compute the first derivative by taking the limit of $$\frac{f(x)-f(0)}{x-0}=\frac{e^{-1/x}}{x}$$ as ... | Claude Leibovici | 82,404 | <p>As user84413 answered, the answer given in the book is not correct.</p>
<p>The general solution being $$a_n=c_1 3^n+c_2 3^n n$$ we can compute the coefficients $c_1$ and $c_2$ from $a_1$ and $a_2$. This gives at the end $$a_n=3^{n-2} \Big( (6a_1-a_2)+(a_2-3a_1)n\Big)$$ So, to get the answer from the book, it would ... |
1,358,140 | <p>I haven't been able to solve this limit, so help would be appreciated.
L'Hopital is not allowed.
Thanks in advance.</p>
| Community | -1 | <p><strong>Hint:</strong></p>
<p>Putting $f(x) = x e^{x^2}$, that limit is just $f'(1)$.</p>
|
436,071 | <blockquote>
<p>Calculate the double integral $\iint_D {(1+x^2 + y^2)ln(1+x^2+y^2)dxdy} $ where $D = \{(x,y) \in \mathbb R^2 | \frac{x}{\sqrt3} \leq y \leq x , x^2 + y^2 \leq 4\}$.</p>
</blockquote>
<p>I heard there is a way called Polar Coordinates but the more I looked and read about it the more I did not understa... | user860374 | 137,485 | <p>Looking at your sketch and remembering that $x^2 + y^2 = r^2$, it can be seen, in polar coordinates that your angle starts at $\frac{\pi}{4}$ and moves up to $\frac{\pi}{6}$, whereas your radius, goes from 0 to 2, thus gving the following limits:</p>
<p>$\frac{\pi}{6}\leq\theta\leq\frac{\pi}{4}$, $0\leq r\leq 2$</p... |
1,516,482 | <p>I need to proof that $\sqrt{3} + \sqrt{2}$ is irrational, without using the fact that an irrational number plus a rational number equals irrational. also, i can't use the rational root theorem. that's why i posted a new question..</p>
<p>thanks for help! </p>
| Arpit Kansal | 175,006 | <p>Note that $\sqrt 2=\dfrac 12\left(\sqrt 2+\sqrt 3-\dfrac 1{\sqrt 2+\sqrt 3}\right)\notin\mathbb Q$, hence $\sqrt 2+\sqrt 3\notin \mathbb Q$</p>
<p><strong>OR,</strong> </p>
<p>Suppose $\sqrt{2} + \sqrt{3}$ is rational, then so is $(\sqrt{2} + \sqrt{3})^2 = 5 + 2 \sqrt{6}$. Hence, $\sqrt{6}$ is rational which is o... |
1,225,009 | <p>Find the volume of a solid inside a sphere $x^{2} + y^{2} + z^{2} = 4$ and the cylinder $x^{2} + y^{2} + 2y = 0$</p>
<p>Any help is appreciated</p>
| David | 119,775 | <p>The claim is that <strong>there exists</strong> a cut with the specified property. You have given an example of a cut which doesn't work, but there are also many examples which do work, for instance
$$\eqalign{
A&=\{A_1,A_3,\ldots,A_{m-1},B_1,B_3,\ldots,B_{m-1}\}\cr
B&=\{A_2,A_4,\ldots,A_m,B_2,B_4,\ldot... |
1,225,009 | <p>Find the volume of a solid inside a sphere $x^{2} + y^{2} + z^{2} = 4$ and the cylinder $x^{2} + y^{2} + 2y = 0$</p>
<p>Any help is appreciated</p>
| Robert Israel | 8,508 | <p>Yes, you are. The two sets $A$ and $B$ don't have to be connected.<br>
So for your line graph, number the vertices from left to right and take the odd-numbered and even-numbered vertices for $A$ and $B$ respectively.</p>
|
1,531,488 | <p>I hope to know the singularity and pole of $\frac{z^2 e^z}{1+e^{2z}}$.</p>
<p>I try $\frac{z^2 e^z}{1+e^{2z}} = \frac{z^2}{e^{-z}+e^{z}}$ and observe that the denominator seems like cosine function. So I think the singularites are i(2n+1)$\pi$/2. But when I try to evaluate the poles, I fail and different values of ... | skyking | 265,767 | <p>You wouldn't have to do the rewrite, but it's easier for the residue calculation. The only thing you have to see is that the nominator doesn't have any singularities and neither does the denominator (except at $\infty$).</p>
<p>The only singularities are due to the denominator being zero. Then you use Euler formula... |
1,531,488 | <p>I hope to know the singularity and pole of $\frac{z^2 e^z}{1+e^{2z}}$.</p>
<p>I try $\frac{z^2 e^z}{1+e^{2z}} = \frac{z^2}{e^{-z}+e^{z}}$ and observe that the denominator seems like cosine function. So I think the singularites are i(2n+1)$\pi$/2. But when I try to evaluate the poles, I fail and different values of ... | Philippe Malot | 39,781 | <p>You can write your function $f$ as the quotient $g/h$ of the two entire functions $g\colon z\mapsto ze^z$ and $h\colon z\mapsto 1+e^{2z}$.</p>
<p>If $w$ is such that $h(w)=1+e^{2w}=0$ (you already found all such $w$'s), then since $h'(z)=2e^{2z}$ for all complex number $z$, you find that $h'(w)=2e^{2w}=-2\neq 0$, a... |
3,079,696 | <p>Let <span class="math-container">$ f : \mathbb{R} \rightarrow \mathbb{R} $</span> be a function which takes a convergent sequence and gives us a convergent sequence.</p>
<p>Show that <span class="math-container">$f$</span> is continuous.</p>
<p>So I saw a proof for this but I don't get it.</p>
<p><strong>proof</s... | J.F | 552,053 | <p>Well the first equality of the first equation holds because the limit of a sequence (assuming it exists) is equal to the limit of any of its subsequence. </p>
<p>As for the "it follows that" part, You apply the same argument with <span class="math-container">$b_{2n}$</span> instead of <span class="math-container">$... |
3,079,696 | <p>Let <span class="math-container">$ f : \mathbb{R} \rightarrow \mathbb{R} $</span> be a function which takes a convergent sequence and gives us a convergent sequence.</p>
<p>Show that <span class="math-container">$f$</span> is continuous.</p>
<p>So I saw a proof for this but I don't get it.</p>
<p><strong>proof</s... | fleablood | 280,126 | <p>This proof centers on the Theorem:</p>
<blockquote>
<p>If <span class="math-container">$ \{s_n\}\to s$</span> is a convergent sequence then for any sub sequence <span class="math-container">$\{s_{k_i}\}$</span>, <span class="math-container">$\{s_{k_i}\}$</span> is a convergent sequence that converges to <span cla... |
383,259 | <p>We know that the real ordered field can be characterized up to isomorphism as a complete ordered field. However this is a second order characterization. That raises the following question. Consider the following theory. We take as axioms the axioms for ordered fields, and then add an axiom schema that states that ev... | Ali Enayat | 9,269 | <blockquote>
<p>In my hasty first reading I did not notice the stipulation about parameter-freeness; I am leaving this answer "only for the record".</p>
</blockquote>
<p>The answer to the question is in the positive, provided "real ordered field" is interpreted as "real closed field", and ... |
383,259 | <p>We know that the real ordered field can be characterized up to isomorphism as a complete ordered field. However this is a second order characterization. That raises the following question. Consider the following theory. We take as axioms the axioms for ordered fields, and then add an axiom schema that states that ev... | Emil Jeřábek | 12,705 | <p>This is a non-answer too long for a comment.</p>
<p>If <span class="math-container">$K$</span> is an ordered field that satisfies the least upper bound property for sets definable <em>without</em> parameters, then suprema of sets definable <em>with</em> parameters can be arbitrarily closely <em>approximated</em> in ... |
152,649 | <p>How can I show the divergence of </p>
<p>$$ \int_0^x \frac{1}{1+\sqrt{t}\sin(t)^2} dt$$</p>
<p>as $x\rightarrow\infty?$</p>
| Ayman Hourieh | 4,583 | <p>For $t \gt 0$:</p>
<p>$$
1 + t \ge 1 + \sqrt{t}\sin^2t
$$</p>
<p>Or:</p>
<p>$$
\frac{1}{1 + t} \le \frac{1}{1 + \sqrt{t}\sin^2t}
$$</p>
<p>Now consider:</p>
<p>$$
\int_0^x \frac{dt}{1 + t} \le \int_0^x \frac{dt}{1 + \sqrt{t}\sin^2t}
$$</p>
<p>The LHS diverges as $x \to +\infty$, so the RHS does too.</p>
|
152,649 | <p>How can I show the divergence of </p>
<p>$$ \int_0^x \frac{1}{1+\sqrt{t}\sin(t)^2} dt$$</p>
<p>as $x\rightarrow\infty?$</p>
| Did | 6,179 | <p>Use $1+\sqrt{t}\sin^2(t)\leqslant1+\sqrt{t}$ uniformly over $t$.</p>
|
152,707 | <p>I have a simple equation (if you are interested it is originated from the field of hydrology):</p>
<pre><code>S^0.5 (1-(1-S^6)^0.166667)^2 == 1/1000
</code></pre>
<p>with <code>S</code> certainly being between 0 and 1. The LHS of the equation is strictly growing function of <code>S</code>. So, the equation can be ... | kglr | 125 | <pre><code>pp1 = ParametricPlot[Evaluate[2 {Cos[t], Sin[t]} # & /@ {1 - Cos[t], 1 + Cos[t]}],
{t, 0, 2 π}]
</code></pre>
<p>Shaded region can be obtained in two ways:</p>
<ol>
<li>Post-process a one parameter <code>ParametricPlot</code> constrained to an appropriate region: </li>
</ol>
<p>For example:</p>... |
1,880,396 | <p>According to article, author wrote "since $|b_n|\to\infty$, we may and do assume that $|b_n|\ge 1$ for every $n\ge 1$". Can you explain for me this? And what will $\sum\limits_{1\le |b_n| \le |X_1|} \dfrac{1}{|b_n|} $ become? (with $X_1$ be a random variable). Can I assum that $|b_n|\ge a$ for all $n>1$? And th... | Marcus M | 215,322 | <p>Since $|b_n| \to \infty$, we know that there exists an $N$ so that for all $n > N$, we have $|b_n| \geq 1$. It's impossible to know for sure without context, but it looks like the writer is replacing the sequence $\{b_n\}$ with the sequence $\{b_{n+N}\}$.</p>
|
2,856,121 | <p>Before I start explaining what I will say regarding the question, read the question before going any further.</p>
<p>Question: <em>Mariam and Leena both live on the bank of the same river, a distance apart. When Mariam visits Leena, she swims downstream (with the current), but then must swim upstream (against the c... | Anonymous | 399,787 | <p>While there are several other excellent answers that formally show why swimming without current is easier, a quick way to gain an intuition of it is to think what happens if the speed of the current <em>exactly</em> matches the speed of the swimmer. Then, swimming with the current in favour will take half the time. ... |
3,339,105 | <p>Can anyone explain to me how the expectation of geometric distribution can be used here?</p>
| Marco Lecci | 344,362 | <p>Let define <span class="math-container">$f(x)= \sum_{k=1}^{\infty}\frac {(-1)^k}kx^k$</span>. This function is defined in <span class="math-container">$[0,1]$</span> and here is <span class="math-container">$C^\infty$</span>.</p>
<p>Now <span class="math-container">$f'(x)= -\sum_{k=0}^{\infty} {(-1)^k}x^{k}= - \fra... |
1,609,845 | <p><strong>Problem</strong>: Let $\alpha, \beta$ be non-negative numbers.
Suppose the number of strictly increasing sequences $a_0, a_1, a_2 \cdots a_{2014}$ satisfying $0 \leq 3m$ is $2^{\alpha}(2\beta+1)$.
Find $\alpha$.</p>
<p>My attempt:</p>
<p>I tried to find an explicit form for the $nth$ of the sequence using... | Cody | 13,295 | <p>Choi and Srivastava list a rather obscure, but useful, series:</p>
<p>$$\psi(x)-\log(x)=\sum_{k=0}^{\infty}\left(\log\left(1+\frac{1}{x+k}\right)-\frac{1}{x+k}\right)\tag{1}$$</p>
<p>Let $x\to x^{2}+1$, then we have:</p>
<p>$$\log(x^{2}+1)-\psi(x^{2}+1)=-\sum_{k=0}^{\infty}\left(\log\left(1+\frac{1}{x^{2}+k+1}\ri... |
2,802,094 | <blockquote>
<p>If $y=\varphi(x) $ is a particular solution of $y''+(\sin(x))y'+2y=e^x$
and $\psi(x)$ is a particular solution of
$y''+(\sin(x))y'+2y=\cos(2x)$,then what is the particular solution of
$y''+(\sin(x))y'+2y=e^x+2\sin^2(x)$?</p>
</blockquote>
<p>Since $LHS$ of equation is same while their $RHS$ ar... | Doug M | 317,162 | <p>It is given that:</p>
<p>$\phi'' + \sin x \phi + 2\phi = e^x\\
\psi'' + \sin x \psi + 2\psi = \cos 2x$</p>
<p>You have that:</p>
<p>$\cos 2x = 1-2\sin^2 x$</p>
<p>Differentiation is linear: </p>
<p>i.e. $(u+v)' = u' + v'$ </p>
<p>$(\phi - \psi)'' + \sin x (\phi-\psi)' + 2(\phi - \psi) = e^x - \cos 2x = e^x + ... |
1,974,865 | <p>Looking for the sum of the infinite series: </p>
<p>$\sum_{n=k}^{\infty}\binom{n}{k}p^{k}(1-p)^{n-k}$ </p>
<p>You can pull a few terms out to get: </p>
<p>$\frac{p^k}{k!}\sum_{n=k}^{\infty}\frac{n!}{(n-k)!}(1-p)^{n-k}$ </p>
<p>Strangely enough, I've actually calculated it for a bunch of different values of $... | copper.hat | 27,978 | <p>The polynomials $x \mapsto 1$, $x \mapsto x$, $x \mapsto ix^2$ form a basis for the space. Now apply Gram Schmidt, but start with the $x \mapsto ix^2$ vector.</p>
<p>Then the other two will be orthogonal to this and hence span the orthogonal complement.
$f_0(x) = ix^2$</p>
<p>$g_0(x) = \sqrt{5}ix^2$ (after normali... |
1,532,263 | <p>$$\lim_{n\to \infty}\sum_{k=1}^n \frac{1}{ k (k + 1) (k + 2) \cdots (k + m)
}, m\in\Bbb N$$</p>
<p>Please help to do this</p>
| slader.com | 290,016 | <p>$$\lim_{n\to \infty}\sum_{k=1}^n \frac{1}{ k (k + 1) (k + 2) \cdots (k + m)
}=\lim_{n\to \infty}\dfrac{1}{m}\sum_{k=1}^n \frac{(k+m)-(k)}{ k (k + 1) (k + 2) \cdots (k + m)
}$$</p>
<p>$$=\lim_{n\to \infty}\dfrac{1}{m}\sum_{k=1}^n \frac{1}{ k (k + 1) (k + 2) \cdots (k + m-1)
}-\frac{1}{ (k + 1) (k + 2) \cdot... |
2,132,863 | <p>I am studying trigonometry in my school and learned that in a triangle the side opposite to angle theta should be taken as perpendicular side - hypotenuse remains the same and the third remaining side is the base.</p>
<p>The same is required for calculating the sine / cosine etc of the angle theta for below formula... | Kanwaljit Singh | 401,635 | <p>One other way to find angles -</p>
<p>$\sin \theta = \frac{\text{Opposite Side}}{\text{Hyphotenuse}}$</p>
<p>$\cos \theta = \frac{\text{Adjacent Side}}{\text{Hyphotenuse}}$</p>
<blockquote>
<p>Edit -</p>
</blockquote>
<p>In your case -</p>
<p>$\sin 90° = \frac{\text{Opposite Side}}{\text{Hyphotenuse}}$</p>
<... |
2,444,641 | <p>A student asked a very insightful question in my Calculus class this morning. I did not know how to answer him. (Admittedly, I am not an analyst by trade: once I passed my qualifiers I never looked back.) I would like to know if anyone here can give a precise answer, and if that can be massaged into an answer un... | Community | -1 | <p>The answer could depend on how exactly we <a href="https://en.wikipedia.org/wiki/Riemann_series_theorem" rel="nofollow noreferrer">sum the series</a>, because, conditionally convergent series possibly could be permutated so that terms at the same rate go to zero but one sequence converges and permutated one diverges... |
1,706,741 | <p>I am having trouble with the following exercise from Dollon and Gilet's <em>Géométrie plane</em>.</p>
<blockquote>
<p>Two parallel lines <span class="math-container">$\Delta$</span> and <span class="math-container">$\Delta'$</span> are given, as well as a point <span class="math-container">$A$</span> on <span class=... | Narasimham | 95,860 | <p>$AB$ is balance length after subtracting $OA$ from given total perimeter length. $OABP$ is a <em>parallelogram</em> but the figure between $ \Delta,\Delta^{\prime}$ is a <em>rhombus</em> as the adjacent sides have equal length as radii.It is now probably more clear what he says.It is too simple...</p>
<p><a href="... |
2,837,230 | <p>When using vector notation in coordinate systems (Cartesian coordinates) we see that the magnitude of a vector in two dimensions is equal to the square root of its Y component squared added to its X component squared (Pythagorean theorem).<br>
But the same calculation is done for a three dimensional vector that has ... | Ragnar | 230,367 | <p>You can think of it as doing the pythagorean theorem twice. Imagine you have the vector (x,y,z) denoted by the red line in the figure below. The magnitude of the green line is given by</p>
<p>$\sqrt{x^2+y^2}$ </p>
<p>and the magnitude of the blue line is $z$. So when you use the pythagorean theorem on the triangle... |
1,509,205 | <p>If $n$ is a positive Integer,then $43^{5n}-21^{5n}$ is always divisible by</p>
<p>options:</p>
<p>a)$11\quad$ b)$18\quad$ c)$25\quad$ d)$64$</p>
<p>MyApproach:</p>
<p>I am not able to think how to solve this problem but.</p>
<p>I did $43^{5n}-21^{5n}=a^n-b^n$ where a=$43^5$ and b=$21^5$</p>
<p>So, $a^n-b^... | littleO | 40,119 | <p>What about just writing a function that multiplies a vector by this matrix using a for loop? You can overwrite $x$ as the multiplication is performed, something like this:</p>
<pre><code>for i = 2,3,...
aNext = x[i]
x[i] = (1/6)a + (2/3)x[i] + (1/6)x[i+1]
a = aNext
end
</code></pre>
<p>After the loop, th... |
1,937,351 | <p>$$\sum_{n=1}^\infty (-1)^n$$
Is this mathematical expression zero or undefined?
I think it looks like zero but i can't explain the reason mathematically.
In addition, $\infty - \infty$ is undefined afaik?</p>
| Andres Mejia | 297,998 | <p>As a general rule, if $\sum a_n$ converges, it must be the case that $a_n \to 0$. In the case of this series, $(-1)^n=-1,0,-1,0$ doesn't converge to <em>anything</em>,so it certainly does not tend to $0$, and so you can deduce that the series is divergent.</p>
<p>For your last comment, referring to $\infty-\infty$ ... |
3,960,180 | <p>There is a question about convex set.</p>
<blockquote>
<p>Let <span class="math-container">$\Omega\subset\mathbb R^n$</span> be an open, connected set. If for any <span class="math-container">$x\in\overline{\Omega}$</span>, there is a neighborhood <span class="math-container">$U_x$</span> such that <span class="math... | Paul Frost | 349,785 | <p>Let <span class="math-container">$B_r(x) \subset \mathbb R^n$</span> denote the open ball with center <span class="math-container">$x \in \mathbb R^n$</span> and radius <span class="math-container">$r > 0$</span>. For <span class="math-container">$x,y \in \mathbb R^n$</span> and <span class="math-container">$t \i... |
2,324,718 | <p>To find the inverse of a function, you switch the spots of $x$ and $y$ in the function and solve for the new $y$ output. For example:</p>
<p>$y = 3 + \sin\ x$</p>
<ul>
<li>Switch positions of $x$ and $y$</li>
</ul>
<p>$x = 3 + \sin\ y$</p>
<ul>
<li>Solve for this new $y$</li>
</ul>
<p>$y = \sin^{-1}(x-3)$</p>
... | praveen kr | 453,426 | <p>suppose you have a function f:X->Y. If f is bijective, then we can identify a function g:Y->X which maps the images under f to its pre images. This is called the inverse function.</p>
|
3,100,381 | <p>I am trying to prove this, but I don't know if I'm on the right track.</p>
<p>So far I've said <span class="math-container">$d=(a,c)=1$</span>, therefore, <span class="math-container">$d|a, d|b,$</span> and <span class="math-container">$d|c$</span>. I'm not sure what to do from here or if I'm even going in the righ... | Surb | 154,545 | <p>If <span class="math-container">$(a,c)=1$</span>, by Bezout identity there are <span class="math-container">$x,y\in\mathbb Z$</span> s.t. <span class="math-container">$$ax+cy=1.$$</span>
Now <span class="math-container">$c=kb$</span> for a <span class="math-container">$k\in\mathbb Z$</span>. Therefore <span class="m... |
3,100,381 | <p>I am trying to prove this, but I don't know if I'm on the right track.</p>
<p>So far I've said <span class="math-container">$d=(a,c)=1$</span>, therefore, <span class="math-container">$d|a, d|b,$</span> and <span class="math-container">$d|c$</span>. I'm not sure what to do from here or if I'm even going in the righ... | Bermudes | 437,646 | <p>Ad absurdum: assume <span class="math-container">$(a, b) = n \gneqq 1$</span>. You know <span class="math-container">$n | a$</span> and <span class="math-container">$n | b$</span>. But since <span class="math-container">$b | c$</span>, you have <span class="math-container">$n | c$</span>. Hence <span class="math-con... |
2,389,247 | <p>I need to solve $$\int|x^n| \, dx$$ <strong>if n is an odd positive integer</strong>.</p>
<p>I think it should be $\dfrac{|x|^{n+1}}{n+1}+c$, but this is wrong. I am new to integration so I don’t know what to do when there is modulus function.</p>
<p>The answer given at back of book is $\dfrac{|x^n|x}{n+1} +c$.</p... | Masacroso | 173,262 | <p>HINT: you can rewrite the indefinite integral as</p>
<p>$$\int |x^n|\mathrm dx=\int |x|^n\mathrm dx=\int(\operatorname{sign}(x))^nx^n\mathrm dx=\int\operatorname{sign}(x^n)x^n\mathrm dx,\quad n\in\Bbb N_{>0}\tag1$$</p>
<p>where</p>
<p>$$\operatorname{sign}(x):=\begin{cases}1,&x> 0\\-1,&x<0\\0,&am... |
1,453,336 | <p>Let us define half-space as
$$ C = \{x\mid a^Tx\leq b\} $$
Intuitively (or geometrically), I understand why halfspace is not affine. But while I prove that half-space is convex, it seems to hold for affine case.</p>
<hr>
<p>Let us choose any $x_1,x_2\in C$ and $x=\theta x_1 + (1-\theta)x_2$, then
$$ a^Tx = a^T(\th... | shisui | 72,173 | <p>Geometrically you can think of it like this:<br>
Take a pair of points $x_1$ and $x_2$ that respect $\leq b$ and they form a segment parallel to the hyperplane's normal $a$. The $\theta$ moves the resulting point $x$ formed from $x_1$ and $x_2$ on this segment to the right(if $\theta < 0$) or to the left(if $\the... |
3,940,974 | <p><span class="math-container">$((A \land (A \rightarrow B)) \rightarrow B$</span></p>
<p><span class="math-container">$((A \land (A \rightarrow B))$</span> is premise
and <span class="math-container">$B$</span> is goal</p>
<p><span class="math-container">$((A \land (\neg A \lor B))$</span></p>
<p><span class="math-c... | Ongky Denny Wijaya | 556,947 | <p><span class="math-container">\begin{align}
((A \land (A \rightarrow B)) \rightarrow B &\equiv (( A \land (\neg A \lor B)) \rightarrow B\\
&\equiv (( A \land \neg A) \lor (A \land B)) \rightarrow B\\
&\equiv (F \lor (A \land B)) \rightarrow B\\
&\equiv (A \land B) \rightarrow B\\
&\equiv \neg... |
2,140,570 | <p>I'm not really sure on how to go about solving this problem...</p>
<p>Consider the initial value problem. </p>
<blockquote>
<p>$$y' (x) = \frac{1}{2}(-x + \sqrt{x^2 + 4y})$$
$$y(2) = -1$$</p>
</blockquote>
<p>$a)$ Show that $y(x) = 1-x$ and $y(x) = -\frac{x^2}{4}$ are two solutions to the above IVP.</p>
| Olod | 1,100 | <p>There are three crucial steps:</p>
<p>1) Bolzano-Weierstrass Theorem stating that</p>
<blockquote>
<p>Every bounded sequence of real numbers
has a convergent subsequence.</p>
</blockquote>
<p>The method of the proof sometimes is jokingly
referred to as "how to catch the lion in the
Sahara desert".</p>
<p>2)... |
2,140,570 | <p>I'm not really sure on how to go about solving this problem...</p>
<p>Consider the initial value problem. </p>
<blockquote>
<p>$$y' (x) = \frac{1}{2}(-x + \sqrt{x^2 + 4y})$$
$$y(2) = -1$$</p>
</blockquote>
<p>$a)$ Show that $y(x) = 1-x$ and $y(x) = -\frac{x^2}{4}$ are two solutions to the above IVP.</p>
| Paramanand Singh | 72,031 | <p>This is not exactly Bolzano Weierstrass theorem. Bolzano Weierstrass theorem has two forms:</p>
<ul>
<li>Any infinite bounded subset of real numbers has an accumulation point. </li>
<li>Any bounded sequence has a convergent subsequence. </li>
</ul>
<p>You can see that both these forms of Bolzano Weierstrass are eq... |
1,942,599 | <p>How can I calculate sums of the form:
$$\sum_{i=-1}^{5}\sum_{j = -5}^{1}(1 + 2i + 3j + 4ij)?$$</p>
<p>Shall I calculate it like first for $i = -1, j = \{-5,\dots, 1\}$, then for $i = 2, j = \{-5, \dots, 1\}$? Is there some shortcut for calculating those kinds of sums, i.e. something akin to summation of arithmetic ... | haqnatural | 247,767 | <p>One way ,you can start from $i$,and in this situation $j$ should be considered as constant $$\sum _{ i=-1 }^{ 5 } \sum _{ j=-5 }^{ 1 } (1+2i+3j+4ij)=\sum _{ i=-1 }^{ 5 } \left( \sum _{ j=-5 }^{ 1 } +2i\sum _{ j=-5 }^{ 1 } +3\sum _{ j=-5 }^{ 1 } j+4i\sum _{ j=-5 }^{ 1 } \right) =\\ =\sum _{ i=-1 }^{ 5 } \left( 7+2i... |
1,942,599 | <p>How can I calculate sums of the form:
$$\sum_{i=-1}^{5}\sum_{j = -5}^{1}(1 + 2i + 3j + 4ij)?$$</p>
<p>Shall I calculate it like first for $i = -1, j = \{-5,\dots, 1\}$, then for $i = 2, j = \{-5, \dots, 1\}$? Is there some shortcut for calculating those kinds of sums, i.e. something akin to summation of arithmetic ... | Lozenges | 219,277 | <p>If you write
$$1+2i+3j+4i j=\frac{1}{2}(4i+3)(2j+1)-\frac{1}{2}$$
you can use the sum of arithmetic sequence formula</p>
<p>$$\frac{1}{2}\sum _{i=-1}^5 (4i+3)\sum _{j=-5}^1 (2j+1)-\frac{1}{2}49$$</p>
<p>$$=\frac{1}{2}(77)(-21)-\frac{1}{2}49$$</p>
<p>$$=\text{-833}$$</p>
|
2,071,977 | <p>I have a matrix $M$ and a vector $x$. I would like to compute the vector $y= M^n x$.</p>
<p>I found the eigenvalues $λ_k$ of $M$ and the left and right eigenvectors, properly normalized. If M was symmetric, I would now project $x$ on each eigenvector, multiply by $λ_k^n$, and then recombine them. That would give me... | hardmath | 3,111 | <p>If you have a complete basis of right eigenvectors for $M$, say $\{v_1,v_2,\ldots,v_\ell\}$ such that $Mv_k = \lambda_k v_k$, then finding $M^n x$ would be easy if we knew the linear combination $x = \sum_{k=1}^\ell c_k v_k$:</p>
<p>$$ M^n x = \sum_{k=1}^\ell c_k \lambda_k^n v_k $$</p>
<p>The OP asks in an edit wh... |
4,156,262 | <p>I am working from a textbook that gives us that</p>
<blockquote>
<p>The derivative of the inverse function is <span class="math-container">$$ \frac{dx}{dy} = \large{\frac{1}{\frac{dy}{dx}}}.\qquad (*)$$</span></p>
</blockquote>
<p><span class="math-container">$$$$</span>
Now I am stuck on part <span class="math-con... | cdeamaze | 429,551 | <p>(a) Let g(.) be the inverse function of f(.)</p>
<pre><code>f'(x) = 3(x^2 + 1) > 0 for all x.
f is strictly increasing, therefore, it is one to one.
Hence its inverse g: R-->R, defined by g(x) = <span class="math-container">$f^{-1}(x)$</span>
and g'(f(x)) = 1/f'(x)
By definition g(f(x)) = x.
Differen... |
4,458,665 | <p>I am looking for a (system of) functional equation or difference equation that has a solution: <span class="math-container">$f(x)=(ax+b)^c$</span>, where <span class="math-container">$a,b,c$</span> are constants. Constant <span class="math-container">$a,b,c$</span> cannot appear in the equation.</p>
<p>For example, ... | MachineLearner | 647,466 | <p><span class="math-container">$$f’(x)=c(ax+b)^{c-1}a=ac\dfrac{f(x)}{ax+b}$$</span></p>
|
232,613 | <p>Let $A$ be a set has $n$ element, my question is how many partial order on it?</p>
<p>For $n=0,1$, $N_P(n)=1$
Case $n=2$, $N_P(n)=3$
Case $n=3$, $N_P(n)=19$</p>
<p>Is there a general formula?</p>
<p>Update:</p>
<p>It seems closed formula have not been found yet. But does any generating function be known current?... | Community | -1 | <p>The motivation is to write $\dfrac1{(2n+1)(2n-1)}$ as $\dfrac{A}{(2n-1)} + \dfrac{B}{(2n+1)}$. Hence, we need $A$ and $B$ such that
$$\dfrac1{(2n+1)(2n-1)} = \dfrac{A}{(2n-1)} + \dfrac{B}{(2n+1)}$$
Simplifying the right hand side, we get that
$$\dfrac1{(2n+1)(2n-1)} = \dfrac{A(2n+1) + B(2n-1)}{(2n+1)(2n-1)} = \dfrac... |
232,613 | <p>Let $A$ be a set has $n$ element, my question is how many partial order on it?</p>
<p>For $n=0,1$, $N_P(n)=1$
Case $n=2$, $N_P(n)=3$
Case $n=3$, $N_P(n)=19$</p>
<p>Is there a general formula?</p>
<p>Update:</p>
<p>It seems closed formula have not been found yet. But does any generating function be known current?... | Joe | 24,942 | <p>I'll shed some light on another way to do this using a slightly different method with partial fractions. Note that the way Marvis did it is generally a more effective method for more difficult expressions (say with several variables where systems of equations are nice to work with).
$$\frac{1}{{4n^2 - 1}} = \frac{1... |
958,486 | <p>I'm trying to prove that</p>
<p>$$x^2 + x + 1 = y^2$$</p>
<p>has no integer solution. So far I've tried proof by contradiction, but all of that seems to rely on me being able to factor this expression into some neat form where the solution is obvious. I've not been able to do so.</p>
| Kevin Sheng | 150,297 | <p>You could try completing the square to obtain $(x+\frac{1}{2})^2 + 1 - \frac{1}{4} = y^2 $. So if there exist integer solutions to this equation, then $(x+\frac{1}{2})^2 + 1 - \frac{1}{4}$ is a perfect square. Go from here to reach a contradiction.</p>
|
3,214,753 | <p>In class, my professor showed that the function:</p>
<p><span class="math-container">$$f(z)=\frac{1}{z}$$</span></p>
<p>does not have a primitive function in the domain <span class="math-container">$\mathbb C$</span> since:</p>
<p><span class="math-container">$$\int_{|z|=R}=2\pi i$$</span></p>
<p>She then said t... | José Carlos Santos | 446,262 | <p>For each <span class="math-container">$z\in\mathbb C\setminus(-\infty,0]$</span>, let <span class="math-container">$\gamma_z(t)=1+t(z-1)$</span> (<span class="math-container">$t\in[0,1]$</span>). Define<span class="math-container">$$F(z)=\int_{\gamma_z}\frac1w\,\mathrm dw.$$</span>Then <span class="math-container">$... |
545,978 | <p>Can it be done?
We have the real field $(\Bbb R,+,-,\times,0,1,<)$, of course $(0,1,-,<)$ are definable using the rest.
We take an elementary non-archimedean extension. Can we define the original set of reals in it?</p>
| Rob Arthan | 23,171 | <p>The first-order theory of the real numbers coincides with the first-order theory of real closed fields (<a href="http://en.wikipedia.org/wiki/Real_closed_field" rel="nofollow noreferrer">http://en.wikipedia.org/wiki/Real_closed_field</a>). This theory is O-minimal (<a href="http://en.wikipedia.org/wiki/O-minimal_the... |
2,294,354 | <p>How to check the convergence of the series
$$
\sum_{n=1}^\infty \frac{\ln\left(1+\frac{1}{n}\right)}{n^a}
$$
I tried using all the tests: D'Alembert's, ratio test, comparison test but I failed . how to find when the series converges?
Thanks in advance.</p>
| C. Falcon | 285,416 | <p>The topology on $\mathbb{C}$ is the product topology on $\mathbb{R}^2$ for which the <a href="https://math.stackexchange.com/questions/247542/projection-is-an-open-map">projections are open</a>.</p>
<p>In my opinion, you have done the <em>hard</em> part of showing that $f(A)$ is open using the inverse function theo... |
2,852,234 | <p>I am certainly new to statistics. I did some simulations and got a lot of data. From the data I ran a AWK script to calculate the average $\bar x$; minimum, $x_0$ and standard deviation, $\sigma$ (the one where you divide by $N$, not $N-1$).</p>
<p>Now I want to plot the data. I guess, I can draw the histogram $\ba... | Martin Roberts | 575,045 | <p>Consistent with the other comments, there is no universally agreed practice across all fields and disciplines.
Thus, the length of your error bars should be:</p>
<ul>
<li>the standard as typically used in your industry / field;</li>
<li>what you believe your audience ( or reviewers for journal publications!) most ... |
2,356,789 | <p><strong>Suppose you have a set of x objects.</strong> In it is an object y. Suppose you pick at random one object from this set. This set is uniformly distributed. There is a 1 / x chance of picking y. Let's say that you pick x times from the set, then eliminating it from the set. This means there is a 100% chance o... | Bram28 | 256,001 | <p>In your example of 9 white marbles and 1 black marbles, you say that if you pick 1 marble out of those 10 3 times, you have a chance of 30% of picking the black marble (at least) once. That is actually a very common reasoning mistake, and you basically realized this yourself, for when you pick a marble $10$ times, ... |
745,922 | <p>How did Leibniz know to write derivatives as $$\frac{dy}{dx}$$ so that everything would work out? For example, the chain rule: $$\frac{dy}{dz}=\frac{dy}{dx}\frac{dx}{dz}$$ Integration by Parts: $$xy=\int d(xy)=\int x\,dy+y\,dx \implies \int x\,dy =xy-\int y\,dx$$ Separable differential equations: $$\frac{dy}{dx}=\fr... | Mikhail Katz | 72,694 | <p>The received wisdom that the success of Leibniz's framework was a "lucky accident" is in error. Leibniz realized, and explicitly stated in a number of texts, that when he was working with "equality" this was a generalized relation where the right hand side and the left hand side were allowed to differ by a negligib... |
542,582 | <p>I have a question which I'm unable to provide much working for as I'm not sure how to start it. </p>
<p>A metal sphere is placed in seawater to study the corrosive effect of seawater. If the surface area decreases at a rate of $35cm^2/year$ due to corrosion, how fast is the radius changing when the radius is $7cm$?... | Daniel Robert-Nicoud | 60,713 | <p>Notice that
$$\frac{d}{dt}S=\frac{d}{dt}(4\pi r^2) = 8\pi r\frac{dr}{dt}$$
thus from $\frac{d}{dt}S=35$ you get
$$\frac{dr}{dt}=\frac{35}{8\pi r}$$
For $r=7$
$$\frac{dr}{dt}=\frac{35}{7\cdot8\pi} = \frac{5}{8\pi}$$
Through all this the units of length are $cm$, of area $cm^2$ and of time $years$.</p>
|
2,431,679 | <p>First I denote $x=\arctan{2}$ and $y=\arcsin{\frac{1}{\sqrt{10}}}$ and then use the addition formula for sine:</p>
<p>$$\sin{(x-y)}=\sin{x}\cos{y}-\cos{x}\sin{y}=\sin{x}\cos{y}-\cos{x}\cdot \frac{1}{\sqrt{10}}.$$</p>
<p>Now I use the fact that $\cos{y}=1-\sin^2{y}$ which gives</p>
<p>$$\cos{y}=1-\sin^2{\left(\arc... | GAVD | 255,061 | <p>HINT: $$\sin(\arctan x) = \frac{x}{\sqrt{1+x^2}}.$$</p>
<p><strong>EDIT 1</strong>: Prove the hint. Let $u = \arctan x$, then $x = \tan u$.</p>
<p>So, we have $$\frac{x}{\sqrt{1+x^2}} = \frac{\tan u}{\sqrt{1+\tan^2 u}} = \tan u \cos u = \sin u = \sin( \arctan x).$$</p>
|
2,431,679 | <p>First I denote $x=\arctan{2}$ and $y=\arcsin{\frac{1}{\sqrt{10}}}$ and then use the addition formula for sine:</p>
<p>$$\sin{(x-y)}=\sin{x}\cos{y}-\cos{x}\sin{y}=\sin{x}\cos{y}-\cos{x}\cdot \frac{1}{\sqrt{10}}.$$</p>
<p>Now I use the fact that $\cos{y}=1-\sin^2{y}$ which gives</p>
<p>$$\cos{y}=1-\sin^2{\left(\arc... | lab bhattacharjee | 33,337 | <p>If $t=\arcsin\dfrac1{\sqrt{10}},0<t<\dfrac\pi2$</p>
<p>$\sin t=\dfrac1{\sqrt{10}},\tan t=+\dfrac{\sin t}{\sqrt{1-\sin^2t}}=\dfrac13$</p>
<p>$\implies t=\arctan\dfrac13$</p>
<p>Now $\arctan2-\arctan\dfrac13=\arctan\dfrac{2-\dfrac13}{1+\dfrac23}=\dfrac\pi4$</p>
|
2,143,658 | <blockquote>
<p>Show that no two of the spaces $(0, 1), (0, 1], [0, 1]$ are homeomorphic</p>
</blockquote>
<p><em>My Attempted Proof</em></p>
<p>We have $(0, 1) \subset (0, 1] \subset [0, 1]$.</p>
<p>We also have $(0, 1] = (0, 1) \cup \{1\}$ and thus $|(0, 1)| \neq |(0, 1]|$ and thus no bijective mapping $f : (0,... | Ege Erdil | 326,053 | <p>Use the commonly used method of removing points from your space. Removing any point from $ (0, 1) $ disconnects it, while you can remove at most $ 1 $ point from $ (0, 1] $ without disconnecting it, and at most $ 2 $ from $ [0, 1] $. Since homeomorphism preserves connected components, the result follows.</p>
|
1,046,864 | <p>Some time ago, I came across an anecdotal story about the "Big Book of Proofs" that God always keeps up in the heaven, which records valid proofs of all theorems in the world. A noted mathematician was quoted as saying that if he was to die tomorrow, the first thing he would do is to take a look at the book, since h... | Michael Hardy | 11,667 | <p>Based on Erdős's idea of that book of proofs that God keeps in Heaven, another book, called <a href="http://rads.stackoverflow.com/amzn/click/3642008550"><em>Proofs from the Book</em></a> was written by Martin Aigner and Günter M. Ziegler.</p>
|
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