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 |
|---|---|---|---|---|
3,270,382 | <p>I'm having trouble constructing, and understanding the construction, of a metric that induces the dictionary order topology on <span class="math-container">$\mathbb R \times \mathbb R$</span>. There are example metrics posted here on Math.SE, but that is not what I'm looking for. I'd like to understand how to <stron... | Henno Brandsma | 4,280 | <p>Intuitively, the plane <span class="math-container">$\Bbb R^2$</span> in the lexicographic order, is just a set of vertical lines, all of which are topological copies of <span class="math-container">$\Bbb R$</span> and such that different vertical lines do not really interact: the horizonal line <span class="math-co... |
65,083 | <p>Rotman's book <em>An Introduction to the Theory of Groups</em> (Fourth Edition) asks, on page 22, Exercise 2.8, to show that <span class="math-container">$S(n)$</span> cannot be embedded in <span class="math-container">$A(n+1)$</span>, where <span class="math-container">$S(n)$</span> = the symmetric group on <span c... | darij grinberg | 2,530 | <p>I think this is solved on <a href="http://www.artofproblemsolving.com/Forum/viewtopic.php?f=61&t=333049">http://www.artofproblemsolving.com/Forum/viewtopic.php?f=61&t=333049</a> .</p>
|
65,083 | <p>Rotman's book <em>An Introduction to the Theory of Groups</em> (Fourth Edition) asks, on page 22, Exercise 2.8, to show that <span class="math-container">$S(n)$</span> cannot be embedded in <span class="math-container">$A(n+1)$</span>, where <span class="math-container">$S(n)$</span> = the symmetric group on <span c... | Derek Holt | 35,840 | <p>Of course this is not a research level question, and so is not appropriate for MO, but I remember being puzzled myself about what proof Rotman had in mind for this. I think we had better assume Lagrange's Theorem or it will be completely hopeless! Perhaps the proof using Bertrand's Postulate was intended, because st... |
65,083 | <p>Rotman's book <em>An Introduction to the Theory of Groups</em> (Fourth Edition) asks, on page 22, Exercise 2.8, to show that <span class="math-container">$S(n)$</span> cannot be embedded in <span class="math-container">$A(n+1)$</span>, where <span class="math-container">$S(n)$</span> = the symmetric group on <span c... | KhashF | 128,556 | <p>Here is a short proof that follows from Rotman's material: Automorphisms of <span class="math-container">$S_m$</span> are all inner unless <span class="math-container">$m=2 \text{ or } 6$</span>. It is not hard to use this to show that for <span class="math-container">$m\neq 2, 6$</span> subgroups of <span class="ma... |
1,233,700 | <p>A infinitely generated additive group G and its subgroup K, when they are isomorphic to each other? Is there any theorem on this?</p>
| P Vanchinathan | 28,915 | <p>Take the set of polynomials in a variable $x$ with integer coefficients, it is an infinitely generated group on the generating set $\{1,x,x^2,x^3,\ldots\}$.
Take the proper subgroup of polynomials involving only the even powers:
It is a proper subgroup isomorphic to the whole group (it is infact isomorphic even as ... |
606,656 | <p>So I heard this a long time ago and I recently started thinking about it again. So I was told that the complex function $f(z)=1/z$ maps everything inside a circle to points outside the circle (the remaining part of the complex plane). Why is this?</p>
<p>I realize we can write $$f(z)=\frac{1}{z}=\frac{\overline{z}}... | Robert Israel | 8,508 | <p>$0 < |z| < 1$ iff $1 < |1/z| < \infty$</p>
|
3,864,364 | <p>Let <span class="math-container">$g \colon \mathbb{N} \rightarrow \{0,1\}$</span> such that <span class="math-container">$g(n)=0$</span>, if <span class="math-container">$n \equiv 0$</span> or <span class="math-container">$n \equiv 1 ~(\bmod~4)$</span>; or <span class="math-container">$g(n)=1$</span>, if <span clas... | Angina Seng | 436,618 | <p>I think that's
<span class="math-container">$$\sum_{m=0}^\infty\left(\frac1{\sqrt{4m+1}}-\frac1{\sqrt{4m+2}}
-\frac1{\sqrt{4m+3}}+\frac1{\sqrt{4m+4}}\right).$$</span>
The mean value theorem implies that both
<span class="math-container">$$\frac1{\sqrt{4m+1}}-\frac1{\sqrt{4m+2}}$$</span>
and
<span class="math-contain... |
3,864,364 | <p>Let <span class="math-container">$g \colon \mathbb{N} \rightarrow \{0,1\}$</span> such that <span class="math-container">$g(n)=0$</span>, if <span class="math-container">$n \equiv 0$</span> or <span class="math-container">$n \equiv 1 ~(\bmod~4)$</span>; or <span class="math-container">$g(n)=1$</span>, if <span clas... | QC_QAOA | 364,346 | <p>We have</p>
<p><span class="math-container">$$\sum_{n=1}^\infty \frac{(-1)^{g(n)}}{\sqrt{n}}=\frac{1}{1}-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+\cdots$$</span></p>
<p>Now, note that for for <span class="math-container">$n\equiv 2\ (\text{mod}\ 4)$</span> we have the terms</p>
<p><span class="math-c... |
3,864,364 | <p>Let <span class="math-container">$g \colon \mathbb{N} \rightarrow \{0,1\}$</span> such that <span class="math-container">$g(n)=0$</span>, if <span class="math-container">$n \equiv 0$</span> or <span class="math-container">$n \equiv 1 ~(\bmod~4)$</span>; or <span class="math-container">$g(n)=1$</span>, if <span clas... | saulspatz | 235,128 | <p>The series converges by <a href="https://en.wikipedia.org/wiki/Dirichlet%27s_test" rel="nofollow noreferrer">Dirichlet's test</a>. In the notation of the Wikipedia article, let <span class="math-container">$a_n=\frac1{\sqrt n}$</span>, and let <span class="math-container">$b_n=(-1)^{g(n)}$</span>. Then <span class=... |
142,858 | <p>Let $f:X\to Y$ be a proper surjection of complex algebraic varieties. Let $H_i$ denote Borel-Moore homology. Then $$ \mathrm{Gr}^W_{-k} H_k(X) \to \mathrm{Gr}^W_{-k} H_k(Y) $$ is surjective.</p>
<p><strong>Question:</strong> Does anyone know a reference for this fact?</p>
<p>I have a proof, but it's not as simple ... | novice | 40,331 | <p>See <a href="http://books.google.com/books?id=uNeu1hjOh7YC&lpg=PA284&ots=gPlXNOtiXu&dq=borel-moore%20weights%20mixed%20hodge%20structure&pg=PA285#v=onepage&q=borel-moore%20weights%20mixed%20hodge%20structure&f=false" rel="nofollow">Lewis's book</a> (page 285) for the case of projective variet... |
3,551,976 | <p>Let <span class="math-container">$E$</span> be a uniformly convex banach space, <span class="math-container">$K$</span> convex and closed in <span class="math-container">$E$</span>, and <span class="math-container">$x\in B\setminus K$</span>.</p>
<p>Can someone give a concise proof that there is a unique <span clas... | Matematleta | 138,929 | <p><a href="https://books.google.com.mx/books/about/Functional_Analysis.html?id=kSJZAAAACAAJ&redir_esc=y" rel="nofollow noreferrer">Larsen's Functional Analysis, an Introduction, Thm 8.2.2</a> has an elementary proof of this, that relies on a lemma, which he leaves to the reader. Here is the statement and proof of ... |
3,243,655 | <p>Question:</p>
<blockquote>
<p>Prove the equation <span class="math-container">$2x - 6y = 3$</span> has no integer solution to <span class="math-container">$x$</span> and <span class="math-container">$y$</span>.</p>
</blockquote>
<p>I need to verify my proof I think I did it correctly, but am not fully sure since... | Parcly Taxel | 357,390 | <p>Rewind to the point where you say <span class="math-container">$x=3y+3/2$</span>. We rearrange this to <span class="math-container">$x-3y=3/2$</span>, then note that since we have taken <span class="math-container">$x$</span> and <span class="math-container">$y$</span> to be integers, <span class="math-container">$x... |
301,038 | <p>If you want to show that a sequence $(a_{n})$ in $\mathbb{R}$ is convergent, when is it sufficient to show that there is a number $b\in\mathbb{R}$ such that
$$ \liminf a_{n} \geq b \geq \limsup a_{n}$$</p>
<p>In particular, I have a situation where my sequence is bounded and I wanted to use this approach, but I'm n... | superAnnoyingUser | 34,371 | <p>Geometrically, $\lim \inf$ is the leftmost (or smallest) accumulation point of a sequence, while $\lim \sup$ is the rightmost (or biggest) one. If you can prove that the one on the left is in fact bigger than or equal to the one on the right, then they coincide. Thus, your sequence is convergent, by Weierstrass theo... |
109,307 | <p><strong>Problem.</strong> Let $d=(a,b)$, $b=\beta d$ and $n>1$. If $\beta$ is odd number, prove that $(n^a+1,n^b-1)\le 2$.</p>
<p><em>Solution (from the book).</em> Each common divisor of numbers $n^a+1$ and $n^b-1$ has to be divisor of their sum $n^a+n^b=n^b(n^{a-b}+1)$, that is, it has to be a common divisor o... | Math Gems | 75,092 | <p>The second half of the proof is more clearly presented using congruence arithmetic. Namely, working mod $\:m := (n^a+1,n^b-1)$ we have $ n^b\equiv 1$, so from $n^d\equiv -1$ we deduce</p>
<p>$$ 1 \equiv\ n^b \equiv (n^d)^{\beta} \equiv (-1)^{\beta}\equiv -1\ \ {\text by}\ \ \beta\ {\text odd}$$</p>
<p>Therefore $... |
794,032 | <p>Prove by induction that the complement of $ A1 \cup A2...An = A1^c \cap A2^c ...\cap An^c$</p>
<p>My approach: basic step is true, $\overline A1 = A1^c$,</p>
<p>then assume $ A1 \cup A2...Ak = A1^c \cap A2^c ...\cap Ak^c$, prove the case of $k+1$ is true. How should I do that?</p>
| Paul Hurst | 149,898 | <p>The basic structure is to show</p>
<p>1) $(A_1 \bigcup A_2 \ldots A_k \bigcup A_{k+1})^c \subset A_1^c \cap A_2^c \ldots A_k^c \cap A_{k+1}^c$ </p>
<p>and</p>
<p>2) $ A_1^c \cap A_2^c \ldots A_k^c \cap A_{k+1}^c \subset (A_1 \bigcup A_2 \ldots A_k \bigcup A_{k+1})^c $</p>
<p>Let $x \in (A_1 \bigcup A_2 \ldots A_... |
4,435,497 | <p>I have to prove the following:</p>
<p>Let <span class="math-container">$p$</span> be a prime and denote <span class="math-container">$\mathbb{Z}_p=\mathbb{Z}/p\mathbb{Z}$</span>. Show that <span class="math-container">$\mathbb{Z}_p[x]/(x^2+x+1)$</span> is not a field if and only if <span class="math-container">$p=3$... | Sammy Black | 6,509 | <p>Everything you write is correct, but it winds about unnecessarily, and comes up short of the conclusion: How do we know that <span class="math-container">$x^2 - 2x - 2$</span> is reducible?</p>
<p>Instead, just add/subtract polynomial multiples of <span class="math-container">$3$</span> to put the expression in a fo... |
362,944 | <blockquote>
<p>Consider a Brownian bridge <span class="math-container">$B: [0,1]\to \mathbb{R}$</span> with <span class="math-container">$B(0)=B(1)=0$</span>. Let <span class="math-container">$M[0, 1/2]=\max_{x\in[0,1/2]}B(x)$</span>. How to prove that
<span class="math-container">$$\mathbb{P}(M[0, 1/2]\geq s)\le... | Iosif Pinelis | 36,721 | <p>Let <span class="math-container">$B_t:=B(t)$</span>. We have to show that
<span class="math-container">\begin{equation}
P(M_{1/2}\ge s)\le2P(B_{1/2}\ge s/2) \tag{1}
\end{equation}</span>
for <span class="math-container">$s\ge0$</span>, where <span class="math-container">$M_T:=\max_{0\le t\le T}B_t$</span> for <s... |
3,212,488 | <p><span class="math-container">$$\int_{0}^{\infty} \ln{(1+a x)}{x^{-b-1}} dx$$</span><br>
I defined two branch cuts along the real axis: <span class="math-container">$[-\infty ,-\frac{1}{a}]$</span> & <span class="math-container">$[0,\infty]$</span> with the following contour: <a href="https://i.stack.imgur.com/4r... | I was suspended for talking | 474,690 | <p>If <span class="math-container">$P(X=\infty)=1$</span> then <span class="math-container">$E e^{-X} = E[1\{X=\infty\}e^{-X}] = 0$</span>. Conversely, suppose <span class="math-container">$P(X<\infty)>0$</span>. Then <span class="math-container">$E[e^{-X}] \geq E[1\{X<\infty\}e^{-X}] > 0$</span> since <spa... |
1,761,273 | <p>Good morning. I have a problem with this:</p>
<p>Find the maximum and minimum distances from the origin to the curve* <span class="math-container">$$g\left(x,y\right)=5x^{2}+6xy+5y^{2}$$</span></p>
<p>I have done this:</p>
<p>Function to optimize:<span class="math-container">$f\left(x,y\right)=x^{2}+y^{2}$</span></p... | boojum | 882,145 | <p>It turns out that this system of Lagrange equations doesn't really warrant the use of Mathematica. We are faced with</p>
<p><span class="math-container">$$ \ 2x \ = \ (2x \ + \ \frac{6}{5}y)\lambda \ \ , \ \ 2y \ = \ (2y \ + \ \frac{6}{5}x)\lambda \ \ . $$</span></p>
<p>Since bringing all the terms to one side in e... |
2,823,568 | <p>It is well known that the sum</p>
<p>$$
\sum _{{k=0}}^{\infty }{\frac {x^{k}}{k!}}
$$</p>
<p>converges to $e^{x}$. In particular, for $x=1$ we have $\sum _{{k=0}}^{\infty }{\frac {1}{k!}}=e$. But what about the sum over the reciprocals of primorials, i.e.,</p>
<p>$$
\sum _{{k=0}}^{\infty }{\frac {x^{k}}{k\#}},... | Pagode | 564,189 | <p>Far more precisely we use Chebyscheff $\theta$ function:</p>
<p>So </p>
<p>$$ (a_n)_{n \in \mathbb{N*}} \triangleq\dfrac{1}{n\#}= \dfrac{1}{\prod_{p\leq n} p} = \dfrac{1}{e^{\theta(n)}} $$</p>
<p>Thus ( <a href="https://rads.stackoverflow.com/amzn/click/0198531710" rel="nofollow noreferrer">Hardy proof in this ... |
1,831,250 | <p>I have come to conclusion that the most efficient and thorough way to prove whether or not a limit exists in three dimensions is to use polar coordinates. </p>
<p>$lim_{x,y \to (0,0)} \frac{x^3+y^3}{x^2+y^2}$</p>
<p>Substituting for polar coordinates:</p>
<p>$lim_{r \to 0^+}$ $\frac{r^3(cos^3 \Theta + sin^3 \Thet... | hmakholm left over Monica | 14,366 | <p>This works in the particular case where the numerator and denominator of your function are both <em>homogenous</em> polynomials in $x$ and $y$ (and the one in the denominator has no nontrivial roots) -- that is, in every term the sum of the exponents if $x$ and $y$ is the same. In that case you have
$$ \frac{p(x,y)}... |
597,610 | <p>can someone explain why when a radical equation is solved some solutions don't work? Is there a rule when a certain transformation is performed on the equation it gives extra solutions? thanks very much</p>
| 2'5 9'2 | 11,123 | <p>The steps you may be used to for solving an equation are one-to-one. If all I know about $x$ is that $2x+1=0$, then</p>
<p>$$\begin{align}
&&2x+1&=0\\
&\iff&2x&=-1\\
&\iff&x&=-\frac12
\end{align}$$</p>
<p>Each step is an implication but so too would be the steps in the reverse o... |
120,863 | <blockquote>
<p>A perfect dice is drawn $4$ times.What's the probability to the same number comes out at least $2$ times?</p>
</blockquote>
<p>At first, I applied to binomial law.I made all calculations.</p>
<p>I set up a random variable.The possible values for the variable were $0,1,2,3$ and $4$.
My thought was th... | Peđa | 15,660 | <p>"Alea iacta est !!"</p>
<p>$$P=\frac{6^4-\frac{6!}{2!}}{6^4} \approx 0.722$$</p>
|
3,270,504 | <blockquote>
<p>Prove that <span class="math-container">$\log|e^z-z|\leq |z|+1$</span> where <span class="math-container">$z\in\mathbb{C}$</span> with <span class="math-container">$|z|\geq e$</span>.</p>
</blockquote>
<p><strong>Background:</strong></p>
<p>This is from a proof that <span class="math-container">$e^z... | Lutz Lehmann | 115,115 | <p>There exist a direct quick solution to the problem behind the question, and a solution using the logarithmic instead of the exponential approach that is probably also easier to handle in using the Rouché theorem.</p>
<h3>Quick solution using the poly-log or Lambert-W functions</h3>
<p>Roots of <span class="math-cont... |
1,647,442 | <p>Prof using the binomial theorem: for all integers $n ≥0$ and for all nonnegative real numbers $x$, $1+nx ≤(1+x)^n$. </p>
<p>Don't have a idea to start this one. I don't know how to use math induction yet, so I need a answer without that.</p>
| JP McCarthy | 19,352 | <p>I define $\%=\frac{1}{100}$ so</p>
<p>$$10\%=10\cdot\frac{1}{100}=\frac{1}{10}.$$</p>
|
1,647,442 | <p>Prof using the binomial theorem: for all integers $n ≥0$ and for all nonnegative real numbers $x$, $1+nx ≤(1+x)^n$. </p>
<p>Don't have a idea to start this one. I don't know how to use math induction yet, so I need a answer without that.</p>
| Christian Blatter | 1,303 | <p>If a school has to send ${1\over10}$ of its students then strictly speaking that school cannot take part if its total number of students is not divisible by $10$. If the school has to send $10\%$ of its students then everyone accepts that the resulting number is rounded to the nearest integer.</p>
|
4,148,930 | <p>Let's consider the function <span class="math-container">$\log x$</span>; how can I prove that it is a transcendental function on the function field of rational functions, i.e. that a polynomial in two variables <span class="math-container">$p(x,y)$</span> such that <span class="math-container">$p(x,\log x)=0$</span... | orangeskid | 168,051 | <p>HINT:</p>
<p>Let us show that we cannot have an equality</p>
<p><span class="math-container">$$\sum_{k=0}^m P_k(x) \exp ( \lambda_k x) = 0$$</span>
where <span class="math-container">$P_k(x)$</span> are non-zero polynomials and <span class="math-container">$\lambda_k$</span> are distinct complex numbers. Otherwise w... |
500,446 | <p>Let $p$ be a prime and $p \geq 5$, Consider the congruence $x^3 \equiv a$ (mod p) with $\gcd(a,p)=1$. Show that The congruence has either no solution or three incongruent solutions modulo $p$ if $p \equiv 1$ (mod 6) and has unique solution modulo $p$ if $p \equiv 5$ (mod 6).</p>
<p>My attempt: By Lagrange theorem... | lab bhattacharjee | 33,337 | <p>$$x^3\equiv a\pmod p\ \ \ \ (1)$$</p>
<p>Taking <a href="http://mathworld.wolfram.com/DiscreteLogarithm.html" rel="nofollow">Discrete logarithm</a>, wrt some primitive root $g\pmod p$ as we can <a href="http://planetmath.org/node/38477" rel="nofollow">prove</a> that every prime has at least one primitive root </p>... |
2,859,411 | <p>Let $y(t)$ be a real valued function defined on the real line such that
$y'= y(1 − y)$, with $y(0) \in [0, 1]$. Then $\lim_{t\to\infty} y(t) = 1$.</p>
<p>The solution is given as false .But i have no idea about that. I try some counterexample but it won't work.</p>
<p>How can i find solution with 3 minutes?</p>
| Mohammad Riazi-Kermani | 514,496 | <p>One way is to solve this logistic equation and find the limit, but it might take more than $3$ minutes.</p>
<p>The other way is qualitative analysis.</p>
<p>Find the equilibrium states which are $y=0$ and $y=1$</p>
<p>Note that if $ y_0 \in (0,1) $ then your $y'=y(1-y)>0$ and it stays positive, so your functio... |
1,863,868 | <p>Coordinates: <span class="math-container">$(0,0), (3,3), (6,4.5), (9, 5.25)$</span></p>
<p>If this is a curve is there a formula for determining the <span class="math-container">$y$</span> value for any given <span class="math-container">$x$</span> within the range <span class="math-container">$0$</span> to <span c... | Andrei | 331,661 | <p>This particular curve can be also $f(x)=6(1-2^{-x/3})$. Here is how I guessed this function:</p>
<ol>
<li><p>you go in steps of 3, so you might have something $x/3$.</p></li>
<li><p>At the first step the function increases by 3, then at the next step by 3/2, then 3/4. From here, I would guess that the formula is $$... |
275,526 | <p>What would be an easy example of a sequence of functions defined on a compact interval so that $f_n$ goes to $f$ pointwise but $\sup f_n$ does not go to $sup f$.</p>
<p>I thought of the usual example we take to show that the limits in integration can't be interchanged when we only have pointwise convergence. Is th... | Thomas Andrews | 7,933 | <p>Let $f_n(x)=0$ if $x<n$ and $1$ if $x\geq n$. Then $f_n\to 0$ pointwise, but $\sup f_n = 1$ for all $n$.</p>
<p>You can get continuous examples easily enough.</p>
|
3,068,270 | <p>Given that <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are real constants and that the equation <span class="math-container">$x^4+ax^3+2x^2+bx+1=0$</span> has at least one real root, find the minimum possible value of <span class="math-container">$a^2+b^2$</span>.</p>
<p>I be... | dmtri | 482,116 | <p>I would write first <span class="math-container">$(x^2+cx+d) (x^2+ex+f) $</span> and then find <span class="math-container">$c, d, e, f$</span> in terms of <span class="math-container">$a, b$</span>.</p>
|
2,880,271 | <p>Let a, b, c and d be real numbers that are not all zero. Let
ax + by = p
cx + dy = q
be a pair of equations in the variables x and y with p, q ∈ R. </p>
<p>Show this system of equations has a unique solution if and only if ab − cd != 0.</p>
<hr>
<p>From Determinant of coefficient matrix, I know (ad -bc) =0 => no ... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: let $b\ne 0$ then we get from the first equation</p>
<p>$y=\frac{p}{b}-\frac{a}{b}x$ plugging this in the second equation we get</p>
<p>$$cx+d\left(\frac{p}{b}-\frac{a}{b}x\right)=q$$ and this is</p>
<p>$$x\left(c-\frac{ad}{b}\right)=q-\frac{pd}{b}$$</p>
<p>multiplying by $b$ we get
$$x(bc-ad)=qb-dp$$</p>
... |
1,549,506 | <p>To Prove $$\lim_{x \to 0}\frac{x^2+2\cos x-2}{x\sin^3 x}=\frac{1}{12}$$
I tried with L'Hospital rule but in vain.</p>
| David Holden | 79,543 | <p>from
$$
\cos x = 1 - \frac{x^2}2+\frac{x^4}{24}-\dots
$$
we have
$$
x^2+2\cos x - 2 = \frac{2x^4}{24}-\dots
$$
and
$$
\sin x = x-... \\
x\sin^3 x = x^4-...
$$</p>
|
1,282,592 | <p>I am often asked to prove properties of regular bipartite graphs, and beyond the two parts having equal size nothing seems obvious. Are these graphs more intuitive than they first seem?</p>
<p>In particular, right now I can't work out why an r-regular bipartite graph is r-edge-colourable.</p>
<p>Thanks</p>
| Salomo | 226,957 | <p>A $r$-regular bipartite graph has not only two parts of equal size, but by Hall's Marriage Theorem, it has a perfect matching. Hence, you can take out that perfect matching, and the graph will become regular bipartite. Keep doing this, we will get a partition of $r$ independent classes of the edges, that is the $r$-... |
423,793 | <p>A group of 3141 students gather together. Some of them have 13 friends in this group, some have 33 friends, and the rest has 37 friends. Prove using graph theory that this group does not exist. Assume that if A is friends with B, then B is friends with A.</p>
| Ross Millikan | 1,827 | <p>The handshaking theorem tells you that the sum of the degrees of a graph is even.</p>
|
15,093 | <p>For example, I am confident that very few students majoring in pure mathematics can write a complete proof to the <a href="https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem" rel="noreferrer">Abel–Ruffini theorem</a> (there is no algebraic solution to general polynomial equations of degree five or higher wit... | Tommi | 2,083 | <p>For the project, you would need to define what are "classical results in mathematics".
I suspect that different people would disagree on the classicalness of various results.</p>
<p>Furthermore, it is doubtful if everyone should learn the same classical results - mathematics is a big field. You can, of course, cut ... |
660,034 | <p>I wondered if all decimal expansions of $\frac{1}{n}$ could be thought of in such a way, but clearly for $n=6$,</p>
<p>$$.12+.0024+.000048+.00000096+.0000000192+...\neq.1\bar{6}$$</p>
<p>Why does it work for 7 but not 6? Is there only one such number per base, <em>i.e.</em> 7 in base 10? If so what is the general ... | Daniel Robert-Nicoud | 60,713 | <p>Your expansion comes from the fact that you can write:
$$\frac{1}{7}=\frac{7}{50}\sum_{k=0}^\infty 50^{-k}=0.14(1+2\cdot0.01+4\cdot0.0001+\ldots)$$
You can check the equality by using the standard result that for $|q|<1$ you have $\sum_{k=0}^\infty q^k=\frac{1}{1-k}$.</p>
<p>It doesn't work for $6$ because if yo... |
134,815 | <p>Assume I have a shuffled deck of cards (52 cards, all normal, no jokers) I'd like to record the order in my computer in such a way that the <em>ordering</em> requires the least bits (I'm not counting look up tables ect as part of the deal, just the ordering itself. </p>
<p>For example, I could record a set of strin... | leonbloy | 312 | <p>As the other answers point out, you can code the permutations in several ways, but you'll need at least 226 bits (in average).
To read about concrete encoding schemes, you can start <a href="http://en.wikipedia.org/wiki/Permutation#Numbering_permutations" rel="nofollow">here</a></p>
|
662,637 | <p>I have a quick question</p>
<p>If S is an orthonormal set of vectors in Rn, then S is a basis for the subspace it spans. I am not sure if this is true or not based off the definition of orthonormal basis which is a subspace w of Rn is a basis for w that is also an orthogonal set.</p>
| Community | -1 | <p>Let denote the orthonormal vectors $v_1,\ldots,v_p$ and let $a_1,\ldots, a_p\in\mathbb R$ such that
$$\sum_{k=1}^pa_k v_k=0$$
then</p>
<p>$$0=\left\langle \sum_{k=1}^pa_k v_k/v_i\right\rangle=a_j,\quad \forall j=1,\ldots,p$$
so the vectors are linearly independant.</p>
|
3,436,804 | <p>Does the following series converge? If yes, what is its value in simplest form?</p>
<p><span class="math-container">$$\left( \frac{1}{1} \right)^2+\left( \frac{1}{2}+\frac{1}{3} \right)^2+\left( \frac{1}{4}+\frac{1}{5}+\frac{1}{6} \right)^2+\left( \frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10} \right)^2+\dots$$</... | Jack D'Aurizio | 44,121 | <p>Since for moderately large values of <span class="math-container">$n$</span> we have
<span class="math-container">$$ H_n \approx \log(n)+\gamma+\frac{1}{2n}-\frac{1}{12n^2} $$</span>
we also have
<span class="math-container">$$ H_{n(n+1)/2}-H_{n(n-1)/2} \approx \frac{2}{n}-\frac{4}{3n^3}$$</span>
and
<span class="ma... |
902,407 | <p>Exercise</p>
<p>Let $f:G \to G'$ be an isomorphism and let $H\unlhd G$. If $H'=f(H)$, prove that $G/H \cong G'/H'$.</p>
<p>As I've shown that $H'\unlhd G'$, I thought of defining $$\phi(Ha)=H'f(a)$$I was trying to show that this function is well defined and that it is an isomorphism.</p>
<p>I'll work with right c... | bof | 111,012 | <p>The set $A$ is the union of the two disjoint sets $A\cap B$ and $A-B$, so $|A|=|A\cap B|+|A-B|$. Now, if only we could show that $|B|=|A\cap B|+|B-A|$ . . .</p>
|
4,024,871 | <p>I'm looking to build a function <span class="math-container">$f:S^2 \to \mathbb R^2$</span> such that <span class="math-container">$f(x)\neq f(−x)$</span> for all <span class="math-container">$x\in S^2$</span>.</p>
<p>By Borsuk-Ulam Theorem, this function must be discontinuous. I was trying to build a not too compli... | R.V.N. | 730,220 | <p>In that case you could try with: <span class="math-container">$$(x_1,x_2,x_3)\in\mathbb{S}^1\longmapsto f(x_1,x_2,x_3):=\begin{cases}(x_1,x_2) &\text{ if } (x_1,x_2,x_3)\neq (0,0,1) \\ (2,2) &\text{ if } (x_1,x_2,x_3)=(0,0,1) \end{cases}$$</span></p>
<p>Then obviously <span class="math-container">$f(x_1,x_2,... |
939,911 | <p>Question Re-phrased:</p>
<hr>
<p>I'm having a lot of trouble wrapping my head around this problem. While I've looked through similar posts, It's difficult understanding the maths because I currently have another approach stuck in my head. Is this valid?</p>
<p>Question:
Prove that an at most countable union of at... | copper.hat | 27,978 | <p>Let $i_k:\mathbb{N} \to A_k$ be a surjection.</p>
<p>Define $W_n = \cup_{k=1}^n i_k(\{ 1,...,n-k+1\})$. Note the $W_n$ are nested, finite and $\cup_{n=1}^\infty W_n = \cup_{k=1}^\infty A_k$.</p>
<p>Let $\Delta_n = W_n \setminus \cup_{k=1}^{n-1} W_k$. By renumbering if necessary, we may assume that all the $\Delta... |
532,785 | <p>Let $f:[0, 1] \rightarrow \mathbb{R}$ a continuous function. If $a>0$, show that:</p>
<p>$$ f(0)\ln(\frac{b}{a})=\lim_{\epsilon\rightarrow 0}\int_{\epsilon a}^{\epsilon b} \frac{f(x)}{x}dx$$</p>
<p>Tried using Riemann sum, but did not succeed.</p>
| Mark Bennet | 2,906 | <p>The area of a triangle is half base times height - here $\frac 12 \times 6 \times 8 = 24$</p>
<p>Now join each of the vertices of the triangle to the in centre to get three triangles of height $r$ and base each of the sides - so the area, as you say, is $\frac 12 \times r \times (6+8+10)=12r$ so that $r=2$.</p>
<p... |
1,307,460 | <p>Need to integrate this function. Need help with my assignment. Thanks</p>
| Pranav Arora | 117,767 | <p>$$I=\int_0^{\pi/2} \frac{\sin^2x}{\sin x+\cos x}\,dx=\int_0^{\pi/2}\frac{\cos^2x}{\sin x+\cos x}\,dx$$
$$\Rightarrow 2I=\int_0^{\pi/2} \frac{1}{\sin x+\cos x}\,dx=2\int_0^{\pi/4} \frac{1}{\sin x+\cos x}\,dx$$
$$\Rightarrow I=\int_0^{\pi/4} \frac{1}{\sin\left(\frac{\pi}{4}-x\right)+\cos\left(\frac{\pi}{4}-x\right)}\,... |
4,644,847 | <p>Is there a formal fallacy to describe lack of an observation is not proof that it does not exist, or lack of an occurrence is not proof that it can never happen?</p>
| ryang | 21,813 | <blockquote>
<p>Is there a formal fallacy to describe that lack of an observation is not proof that it does not exist, or that lack of an occurrence is not proof that it can never happen?</p>
</blockquote>
<p><em>Absence of evidence is not evidence of absence.</em> -Carl Sagan(<a href="https://en.wikipedia.org/wiki/Evi... |
471,710 | <p>Why do small angle approximations only hold in radians? All the books I have say this is so but don't explain why.</p>
| E.P. | 30,935 | <p>A 'small angle' is equally small whatever system you use to measure it. Thus if an angle is, say, much smaller than 0.1 rad, it will be much smaller than the equivalent in degrees. More typically, saying 'small angle approximation' typically means $\theta\ll1$, where $\theta$ is in radians; this can be rephrased in ... |
471,710 | <p>Why do small angle approximations only hold in radians? All the books I have say this is so but don't explain why.</p>
| Norman | 186,107 | <p>because the sine of an angle is dimensionless [i.e. just a number], the angle has to be also dimensionless [i.e. the radian is dimensionless] ... units on both sides of an equation have to match</p>
|
2,698,555 | <p><strong>Question</strong></p>
<p>Find all the rational values of $x$ at which $y=\sqrt{x^2+x+3}$</p>
<p><strong>My attempt</strong></p>
<p>Since we only have to find the rational values of $x$ and $y$, we can assume that
$$ x \in Q$$
$$ y \in Q$$
$$ y-x \in Q $$
Let$$ d = y-x$$
$$d=\sqrt{x^2+x+3}-x$$
$$d+x=\sqrt... | John | 543,506 | <p>Let r = the radius of the large quarter-circle = the side of the square. \
Area of the big quarter-circle = (\pi * r^2) / 4 \
Area of two smaller semi-circles = (\pi * r^2) / 4 \
These two areas are equal. \
Graphically, the area of the big quarter-circle (\pi * r^2) / 4 = area of two smaller semi-circles (\pi * r^2... |
2,710,656 | <p>I now there is a continuous surjective map from $\Bbb{R}\to\Bbb{R}^2$ thanks to Peano curve.</p>
<p>My question is simple: does there exist a $C^1$ surjective map from $\Bbb{R}\to\Bbb{R}^2$ ? I think that the answer is no and I have seen this long time ago but I was too young to understand the proof. Unfortunately ... | Angina Seng | 436,618 | <p>The image of a $C^1$ map from $\Bbb R$ to $\Bbb R^2$ has Lebesgue measure zero, so it cannot be surjective, nor have non-empty interior.</p>
|
2,983,231 | <p>I'm working out of Salamon's <em>Measure and Integration</em> in preparation for studying probability, and I came across the following exercise.</p>
<blockquote>
<p>Let <span class="math-container">$X$</span> be uncountable and let <span class="math-container">$\mathcal{A} \subset 2^X$</span> be the set of all su... | Alonso Delfín | 206,395 | <p><strong>(1) <span class="math-container">$\mu$</span> is a measure:</strong>
Since <span class="math-container">$\varnothing$</span> is countable, then <span class="math-container">$\mu(\varnothing)=0$</span>. Let's take <span class="math-container">$E_1,E_2, \cdots \in \mathcal{A}$</span> pairwise disjoint. If ever... |
1,495,885 | <p>I'm trying to integrate this but I think the function can't be integrated? Just wanted to check, and see if anyone is able to find the answer (I used integration by parts but it doesn't work). Thanks in advance; the function I need to integrate is $$\int\frac{x}{x^5+2}dx$$</p>
| Stefan Hante | 42,060 | <p>You can do a partial fraction decomposition of $\frac{x}{x^5+2}$ which results in a very ugly result of the form</p>
<p>$$
\frac{x}{x^5+2} = \frac{a_1 x + a_2}{a_3x^2 + a_4x + a_5} + \frac{a_6 x + a_7}{a_8x^2 + a_9x + a_{10}} + \frac1{b_1 + xb_2},
$$
where all the $a$ an $b$ are some really ugly constants. Now you ... |
3,080,566 | <p>please tell me how I can solve the following equation. </p>
<p><span class="math-container">$$z^3+\frac{(\sqrt2+\sqrt2i)^7}{i^{11}(-6+2\sqrt3i)^{13}}=0$$</span></p>
<p>What formula should I use? If possible, tell me how to solve this equation or write where I can find a formula for solving such an equation. I sear... | Martin Sleziak | 8,297 | <p>Noticing that <span class="math-container">$\sqrt2+\sqrt2i=2\left(\frac{\sqrt2}2+i\frac{\sqrt2}2\right)$</span> and <span class="math-container">$-6+2\sqrt3i=4\sqrt3\left(-\frac{\sqrt3}2+i\frac12\right)$</span> should help you to express the given complex numbers in the polar form. You can then use <a href="https://... |
273,580 | <p>I notice that the built-in "Coordinates Tool" has a very efficient zoom tooltip displaying an enlarged portion of the image along with coordinates and row/column indices:</p>
<p><a href="https://i.stack.imgur.com/KteX5.png" rel="noreferrer"><img src="https://i.stack.imgur.com/KteX5.png" alt="screenshot" />... | Lukas Lang | 36,508 | <p>Here's a proof-of-concept demonstration of how to build your own tooltip with reasonable performance:</p>
<pre><code>tooltip[img_] :=
EventHandler[
MouseAppearance[
img,
Dynamic@Graphics[
{
Red,
Line@{{{0.05, 0.9}, {0.05, 1}}, {{0, 0.95}, {0.1, 0.95}}},
Inset[
Graphics[
... |
2,459,651 | <p>What additional properties must an operation have besides commutativity so that commutativity along with other properties implies associativity?</p>
<p>Where can I read about such structures?</p>
| Community | -1 | <p>Here's why it's not possible, in most cases, to have that something. Commutativity deals with the inputs themselves. Associativity meanwhile, is all about parenthesis placement. The common part they have, is they are both defined for repeated operations, of the same type. Are we locked in by PEMDAS etc.? That's most... |
2,713,555 | <p>I have a question stating</p>
<blockquote>
<p>Let $A= \{ x_1 , x_2 , x_3 , \ldots , x_n \}$ be a set consisting of $n$ distinct elements. How many subsets does it have? How many proper subsets?</p>
</blockquote>
<p>My thought is that there would be subsets with $1$ element, $2$ elements, $3$ element and so on, u... | Martín-Blas Pérez Pinilla | 98,199 | <p>Is a well-known fact that the answer is $2^n$. Proof by induction:</p>
<ul>
<li>$A_1 = \{x_1\}$ has two subsets: $\emptyset,A_1$.</li>
<li>Suppose true up to $n$. The subsets of $A_{n+1} = \{x_1,\dots,x_{n+1}\}$ are the subsets of $A_n = \{x_1,\dots,x_n\}$ plus the subsets of $A_n$ union $\{x_{n+1}\}$.</li>
</ul>
|
4,419,360 | <p>In chapter 9 of Spivak's <em>Calculus</em>, on derivatives, he mentions the "Leibnizian Notation" for the derivative of a function <span class="math-container">$f$</span>, <span class="math-container">$\frac{df(x)}{dx}$</span>. In a footnote on page 155, he writes</p>
<blockquote>
<p>Leibniz was led to thi... | Joe | 623,665 | <p>The precise meaning of "infinitesimal" depends on context, but it common to define an infinitesimal number as a positive number <span class="math-container">$x$</span> such that <span class="math-container">$x<1/n$</span> for every natural number <span class="math-container">$n$</span>; similarly, an in... |
1,105,056 | <p>There's something I've never understood about polynomials.</p>
<p>Suppose $p(x) \in \mathbb{R}[x]$ is a real polynomial. Then obviously,</p>
<p>$$(x-a) \mid p(x)\, \longrightarrow\, p(a) = 0.$$</p>
<p>The converse of this statement was used throughout high school, but I never really understood why it was true. I ... | orangeskid | 168,051 | <p>You can also understand division by $(x-a)$ as follows:</p>
<p>Take a polynomial</p>
<p>$$P(x) = \sum c_k x^k$$</p>
<p>Plug in $x+a$ instead of $x$. You get a new polynomial in $x$, $P(x+a) = \sum_k c_k (x+a)^k = \sum_k d_k x^k$
that is
$$P(x+a) = \sum d_k x^k$$</p>
<p>Now in the last equality instead of $x$ plu... |
1,613,857 | <p>I'm reading <a href="https://en.wikipedia.org/wiki/The_Music_of_the_Primes" rel="nofollow">The Music of the Primes</a> by du Sautoy and I've come across a section that I'm having difficulty understanding:</p>
<blockquote>
<p>Euler fed imaginary numbers into the function $2^x$. To his surprise, out came waves whic... | pills | 394,220 | <p>The issue is that as far as I know, there is no canonical way of translating a "musical note" into a complex number - however you can translate it into a complex <em>function</em>. As SolUmbrae wrote, a note is simply a sine or cosine function defined by its amplitude, frequency and offset. To keep it simple, let's ... |
3,440,171 | <p>How does one write the Taylor expansion of <span class="math-container">$\log\big(\frac{\sin(x)}x\big)$</span>? It is not defined on <span class="math-container">$x=0$</span>.</p>
| Claude Leibovici | 82,404 | <p>If you are not looking for the infinite series, since <span class="math-container">$\forall x$</span>, <span class="math-container">$\frac{\sin(x)} x <1$</span> and that <span class="math-container">$x=0$</span> is a removable singularity, let <span class="math-container">$a=\frac{\sin(x)} x -1$</span> and write
... |
3,440,171 | <p>How does one write the Taylor expansion of <span class="math-container">$\log\big(\frac{\sin(x)}x\big)$</span>? It is not defined on <span class="math-container">$x=0$</span>.</p>
| user90369 | 332,823 | <p>Euler product formula for sine:</p>
<p><span class="math-container">$$\frac{\sin x}{x}=\prod\limits_{k=1}^\infty\left(1-\left(\frac{x}{\pi k}\right)^2\right)$$</span></p>
<p>You see: <span class="math-container">$~\ln\frac{\sin x}{x}~$</span> is defined everywhere in arbitrary proximity of <span class="math-contai... |
1,046,792 | <p>Given:
$$\sin(x+iy)=\cos\theta+i\sin\theta$$
To prove:
$$x=\arccos (\sqrt{\sin\theta})$$
How I tried:
$$\begin{align*}
\sin x \cosh y &= \cos\theta \\
\cos x \sinh y &= \sin\theta
\end{align*}$$
Then tried to use logarithm of hyperbolic complex number.</p>
<p>Also various trignometric form manipulation but ... | Ashish | 196,782 | <pre><code>given:
</code></pre>
<p>$sin(x+iy)=e^{i\theta}$</p>
<p>$sin(x)*cosh(y)+icos(x)*sinh(y)=cos(\theta)+isin(\theta)$</p>
<p><strong><em>comparing the the real and imaginary part</em></strong></p>
<p>$\implies sin(x)*cosh(y)=cos(\theta)$ </p>
<p>&</p>
<p>$\implies cos(x)*sinh(y)=sin(\theta)$</p>
<p><st... |
199,235 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/9505/xy-yx-for-integers-x-and-y">$x^y = y^x$ for integers $x$ and $y$</a> </p>
</blockquote>
<p>Determine the number of solutions of the equation
$n^m = m^n$
where both m and n are integers.</p>
| Ross Millikan | 1,827 | <p>Hint: it might be that $n$ and $m$ are powers of a prime. They would clearly need to be powers of the same prime. Define $n=p^a, m=p^b$ then apply the laws of exponents. Otherwise, they might be a product of primes. Again, they need to be products of the same primes. Again, use the laws of exponents to rule it... |
2,334,305 | <blockquote>
<p>Let $F$ be the distribution function of the continuous-type random variable $X$, and assume that $F(x)=0$ for $x \le 0$ and $0 \lt F(x) \lt1$ for $0 \lt x$. Prove that if $P(X \gt x+y |X \gt x)=P(X \gt y),$ then $F(x)=1-e^{- \lambda x}, 0 \lt x.$</p>
</blockquote>
<p>My attempt: </p>
<p>Let $F(x)=P(... | leonbloy | 312 | <p>Hint: let $y \to 0$ and try to get a differential equation for $F(x)$</p>
|
292,159 | <blockquote>
<p>Find the relation between <span class="math-container">$a, b, c, d$</span> if the roots of <span class="math-container">$ax^3+bx^2+cx+d=0$</span> are in geometric progression.</p>
<p>By considering <span class="math-container">$(\alpha+\beta)(\beta+\gamma)(\alpha+\gamma)$</span> show that the above cubi... | Hagen von Eitzen | 39,174 | <p>The monic polynomial with roots $\beta/r,\beta, r\beta$ is
$$ x^3-\beta\left(1+r+\frac1r\right) x^2+\beta^2\left(1+r+\frac1r\right)x-\beta^3$$
We find $\beta=-\frac cb$ and $\beta^3=-\frac da$, hence
$$db^3=ac^3$$
as necessary condition. Why is it also sufficient? What additional condition(s) would we need if we wan... |
292,159 | <blockquote>
<p>Find the relation between <span class="math-container">$a, b, c, d$</span> if the roots of <span class="math-container">$ax^3+bx^2+cx+d=0$</span> are in geometric progression.</p>
<p>By considering <span class="math-container">$(\alpha+\beta)(\beta+\gamma)(\alpha+\gamma)$</span> show that the above cubi... | namasikanam | 482,214 | <p>Suppose $a \not = 0$ , let $p=\frac{b}{a},q=\frac{c}{a},r=\frac{d}{a}$, then the equation is $$x^3+px^2+qx+r=0$$
Using Vieta's Formulas,$$\alpha + \beta + \gamma=-p \\ \alpha \beta + \beta \gamma + \alpha \gamma =q \\ \alpha \beta \gamma = -r$$Suppose $\beta^2=\alpha \gamma$, we get $$q^3=p^3r$$that is $$\left \{ \b... |
1,226,162 | <p>\begin{align}
x' &= -x^3 + x^5 + (x^4)(y^5)\\[.7em]
y' &= -8y^3 + y^5 - 10(y^4)(x^5)
\end{align}
$(0,0)$ is obviously a critical point of the system, and we are given that it is asymptotically stable, but have to show it. </p>
<p>I have tried to make a Lyapunov function $V(x,y) = ax^2 + cy^2$, with ... | Cesareo | 397,348 | <p>This problem can be handled with an optimization procedure, having in mind that generally is a non convex problem. The result depends on the test Lyapunov function used so we will generalize to a quadratic Lyapunov function</p>
<p><span class="math-container">$$
V(p) = p^{\dagger}\cdot M\cdot p = a x^2+b x y + c y^2... |
936,830 | <p>I made the differential equation : $$dQ = (-1/100)2Q dt$$ </p>
<p>I separate it and get: $\int_a^b x (dQ/Q) = \int_a^b x (-2/100)dt$</p>
<p>this leads me to: $\log(|Q|) = (-t/50) + C$</p>
<p>I simplify that to $Q = e^{-t/50}$</p>
<p>My TI-Nspire differential equation solver, however, gives me: $Q = Ce^{-t/50}$<... | Dilip Sarwate | 15,941 | <p>It is quite likely that you were told that $X$ and $Y$ are <em>independent</em> random variables, but neglected to pass on this information to us.</p>
<p><em>Assuming</em> that $X$ and $Y$ are independent Poisson random variables, $U = X+Y$ is
also a Poisson random variable with parameter $\theta+\lambda$. Thus, th... |
3,672,948 | <p>Consider the series
<span class="math-container">$$
S = \sum_{n=1}^\infty e^{-n^2x}
$$</span>
then I have to argue for that <span class="math-container">$S$</span> is convergent if and only if <span class="math-container">$x>0$</span>.</p>
<p>As this is if and only if I think I have to assume first that S is con... | Derpp | 596,585 | <p>If the series converges <span class="math-container">$\implies \lim_{n\to\infty} e^{-n^{2}x} = 0$</span>. Do you see why <span class="math-container">$x$</span> needs to be positive now?</p>
|
696,370 | <p>Is it possible that $R/I$ is a field when $R$ is non-commutative ring with unit and $I$ is a maximal left ideal of $R$? If it is not, can anyone give an example of such $R$ and $I$? Thanks.</p>
| J.R. | 44,389 | <p>$F$ cannot have characteristic $0$, because it is a finite field.</p>
<p>If $F$ has characteristic $p>0$, then $1$ generates an additive subgroup of order $p$. By Lagrange's theorem, $p$ divides the order of the whole group $F$, which is $2^n$.</p>
<p>Since $p$ is prime, $p=2$.</p>
|
1,330,078 | <p>Given a quadrilateral $MNPQ$ for which $MN=26$, $NP=30$, $PQ=17$, $QM=25$ and $MP=28$ how do I find the length of $NQ$?</p>
| E.H.E | 187,799 | <p>Hint:
use Heron's Formula for the area of a triangle
$$A=\sqrt{s(s-a)(s-b)(s-c)}$$
$$s=\frac{a+b+c}{2}$$
<img src="https://i.stack.imgur.com/tuTXD.png" alt="enter image description here">
Firstly find the total area by using the two triangles $MNP$ and $MQP$, then you can use same procedures for triangles $NMQ$ and ... |
18,691 | <p>Let's say I have a folowing set of data:</p>
<ul>
<li>k = 1 : list of values </li>
<li>k = 3 : list of values </li>
<li>k = 10 : list of values</li>
</ul>
<p>I know that to make a <code>BoxWhiskerChart</code> I have to give it a list of listen of these values as data and ks as labels. </p>
<p>How do I force the o... | David Park | 5,549 | <p>At the request of one user I included a BoxWhisherDraw statement in the latest release of Presentations (which I sell for $50.). This essentially draws a chart centered at zero and then you can easily Scale/Rotate/Translate it to wherever you wish. So here is a solution to your problem using <code>BoxWiskerDraw</cod... |
2,936,994 | <p>There are 7 white balls in a row and a fair die (a cube with numbers 1, 2,..., 6 in its six faces). We roll the die 7 times and paint the ith balls into black if we get either 5 or 6 in the ith roll.</p>
<p>(a) What is the expected number of black balls?</p>
<p>(b) What is the chance that there exist 6 or more con... | Satish Ramanathan | 99,745 | <p>@lulu</p>
<p>But here is my account for a string of 6 and 7 string with not 4 or more consecutive whites(I find the 4 or more consecutive and subtract it from 1)</p>
<p>It absolutely confirms your calculation. Lulu can't be wrong</p>
<p><a href="https://i.stack.imgur.com/gUqP6.png" rel="nofollow noreferrer"><img ... |
116,392 | <p>As <a href="https://mathematica.stackexchange.com/questions/110808/a-custom-function-is-too-slow-when-i-first-time-to-run-it">my this previous post</a>.I get a custom function which can help me got some function contain a certain option.And I have put it in my "init.m" file.</p>
<pre><code>LookupOptionFunc... | Edmund | 19,542 | <p>You may make use of the built-in index caching for <code>Entity</code> objects; <code>"WolframLanguageSymbol"</code>.</p>
<pre><code>lookupOptionFunction[optionName_String] :=
ToExpression /@
EntityValue[EntityList[
Entity["WolframLanguageSymbol",
{EntityProperty["WolframLanguageSymbol", "OptionName... |
839,516 | <p>What is the number of binary sequences of length $n$, with no two consecutive zeros, and if starts with $0$ has to end with $1$.</p>
<p>Would appreciate suggestions and help.</p>
<p>I tried counting the total sequences and than substractung the ones containing 2 consecutive zeros, and then substracting the ones st... | Pavan Sangha | 154,686 | <p>Let $B(n)$ be the number of strings starting with a $1$. Further more let $B_{1}(n)$ be the number of strings starting with a $1$ and ending in a $1$ and let $B_{0}(n)$ be the number of strings starting with a $1$ and ending in a $0$, it follows that $B(n)=B_{1}(n)+B_{0}(n)$. With a bit of calculation you can see th... |
1,000,938 | <p>$\lim_{X \to \infty} \int_0^Xf(x)^2 > 2(\lim_{X \to \infty}\int_0^Xf(x))^2$</p>
| Sarastro | 31,047 | <p>I assume that $X>0$. Consider
$f(x) = \left\{
\begin{array}{lr}
0 & :x &= &0\\
1 & :x &\in &(\frac{1}{n+1},\frac{1}{n}], \text{n odd},n\geq N\\
-1 & :x &\in &(\frac{1}{n+1},\frac{1}{n}], \text{n even},n\geq N\\
0&:x&>&\frac{1}{N} \end{array}
\right... |
293,371 | <p>This is part of a homework assignment for a real analysis course taught out of "Baby Rudin." Just looking for a push in the right direction, not a full-blown solution. We are to suppose that $f(x)f(y)=f(x+y)$ for all real x and y, and that f is continuous and not zero. The first part of this question let me assum... | ibnAbu | 334,224 | <p><span class="math-container">$f(x)= f(\frac{x}{2})f(\frac{x}{2}) $</span> and together with <span class="math-container">$f(x) \neq 0$</span> implies <span class="math-container">$f(x)$</span> is positive (actually if <span class="math-container">$f(x) $</span> is zero anywhere then it is zero everywhere)</p>
<p>De... |
98,317 | <p>This comes from Artin Second Edition, page 219. Artin defined $G = \langle x,y\mid x^3, y^3, yxyxy\rangle$, and uses the Todd-Coxeter Algorithm to show that the subgroup $H = \langle y\rangle$ has index 1, and therefore $G = H$ is the cyclic group of order 3.</p>
<p>That being the case, $x$ cannot be either $y$ or ... | Did | 6,179 | <p>In other words, assuming that $x^3=y^3=yxyxy=e$, the goal is to prove that $x=e$. </p>
<p>Note that $xyx=y^2(yxyxy)y^2=y^4=y$ hence $xy=(xyx)x^2=yx^2$ $(*)$.</p>
<p>Imagine one wants to carry every $y$ in $x=xy^3$ to the leftmost end of the product. Using $(*)$ twice, one first gets
$$
x=xy^3=(xy)y^2=(yx^2)y^2=yx... |
573,484 | <p>Prove that if $A \setminus B = \emptyset$, then $A \subseteq B$.</p>
<p>The Venn Diagram helped me to visualize what I'm trying to show (thanks @GA316), but the book asks for a written proof (step by step) by contradiction. Sorry if I wasn't more specific at first, is just that I've had many troubles in the past w... | Brian M. Scott | 12,042 | <p>HINT: Just follow the definitions. In order to show that $A\subseteq B$, you should let $x$ be an arbitrary element of $A$ and somehow use the hypothesis that $A\setminus B=\varnothing$ to show that $x\in B$. What if $x$ were not in $B$? Then you’d have $x\in A$ <strong>and</strong> $x\notin B$, which would tell you... |
573,484 | <p>Prove that if $A \setminus B = \emptyset$, then $A \subseteq B$.</p>
<p>The Venn Diagram helped me to visualize what I'm trying to show (thanks @GA316), but the book asks for a written proof (step by step) by contradiction. Sorry if I wasn't more specific at first, is just that I've had many troubles in the past w... | GA316 | 72,257 | <p>Draw venn diagram of $A - B = A \cap B^c$. can you see when it will be empty?</p>
|
1,756,448 | <p>Suppose the function, $f$, is differentiable at $x = 1$. $$\lim_{h\rightarrow\ 0}\frac{f(1+h)}{h} = 5$$</p>
<p>Find a) $f(1)$ and, b) $f'(1)$. </p>
<p>I know b) (well at least I think it can) can be found by the definition of the derivative, i.e. </p>
<p>$$\lim_{h\rightarrow\ 0}\frac{f(1+h)-f(1)}{h} $$</p>
<p>Th... | DonAntonio | 31,254 | <p>If $\;f\;$ differentiable at $\;x=1\;$ then it is <em>continuous</em> and, of course, defined there. But</p>
<p>$$f(1)\neq 0\implies \lim_{h\to0}\frac{f(1)}h\;\;\text{doesn't exist}$$</p>
|
1,646,460 | <p>I was thinking about the following integral if I could solve it without using trigonometric formulas. If there is no other way to solve it, could you please explain me why do we replace $x$ with $2\sqrt 2 \sin(t)$? I'm really confused about these types of integrals.</p>
<p>$$\int \sqrt{8 - x^2} dx$$</p>
| Ant | 66,711 | <p>The idea is that we want to get rid of the square root. So we use the fact that $1 - \sin^2 = \cos^2$, so that $$\sqrt{1 - \sin ^2 x } = \sqrt{\cos^2 x} = |\cos x|$$</p>
<p>which we then can integrate (of course there will be another factor coming from $dx$ but that works out)</p>
<p>Same idea when you're faced wi... |
1,646,460 | <p>I was thinking about the following integral if I could solve it without using trigonometric formulas. If there is no other way to solve it, could you please explain me why do we replace $x$ with $2\sqrt 2 \sin(t)$? I'm really confused about these types of integrals.</p>
<p>$$\int \sqrt{8 - x^2} dx$$</p>
| Community | -1 | <p>By rescaling the variable, let us replace the constant $8$ by $1$, for convenience.</p>
<p>The equation $y=\sqrt{1-x^2}$ represents the upper-half of the unit circle, and the integral</p>
<p>$$\int_{t=0}^x\sqrt{1-t^2}dt$$ is the area of a vertical "slice" between the abscissas $0$ and $x$. You can compute it as th... |
256,785 | <p>Let $T_0$ be the set theory axiomatized by $ZFC^-$ (that is $ZFC$ without powerset) + every set is countable + $\mathbb{V}=\mathbb{L}$. </p>
<p><strong>Question 1:</strong> Suppose $\phi$ is a sentence of set theory. Must there be a large cardinal axiom $A$ such that $\phi$ is decided by $T_0$ + "there are transiti... | Emil Jeřábek | 12,705 | <p>Without a definition of “large cardinal axiom”, I’m going to ignore Q1 and Q3.</p>
<p>The answers to Q2 and Q4 are negative by the following general principle. (For Q4, we take $T_0$ to be the set of $\mathcal L_{\mathrm{set}}$-consequences of $T_1$.)</p>
<blockquote>
<p>$\DeclareMathOperator\Tr{Tr}\DeclareMathO... |
2,947,218 | <p>I know that if a curve <span class="math-container">$C$</span> is parametrize as <span class="math-container">$$s(t)=(x(t),y(t)),\quad t\in [a,b]$$</span>
then <span class="math-container">$$\ell(C)=\int ds=\int_a^b \sqrt{\dot x^2(t)+\dot y^2(t)}dt.\tag{*}$$</span></p>
<p>In the <a href="https://fr.wikipedia.org/w... | md2perpe | 168,433 | <p>Formally we have
<span class="math-container">$$
\int \sqrt{\dot x^2(t)+\dot y^2(t)} \, dt
= \int \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2} \, dt \\
= \int \sqrt{\left[\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2\right]dt^2}
= \int \sqrt{dx^2+dy^2}
$$</span>
It's not really strang... |
2,947,218 | <p>I know that if a curve <span class="math-container">$C$</span> is parametrize as <span class="math-container">$$s(t)=(x(t),y(t)),\quad t\in [a,b]$$</span>
then <span class="math-container">$$\ell(C)=\int ds=\int_a^b \sqrt{\dot x^2(t)+\dot y^2(t)}dt.\tag{*}$$</span></p>
<p>In the <a href="https://fr.wikipedia.org/w... | edm | 356,114 | <p>The object <span class="math-container">$ds^2$</span> is a tensor field. It assigns to each point a tensor. The formula <span class="math-container">$ds^2=dx^2+dy^2$</span> tells you how this tensor field is defined.</p>
<p>The terminology looks fancy, but it simply does the following:</p>
<p>Write the canonical i... |
3,868,571 | <p>Let <span class="math-container">$n\ge 1$</span> and <span class="math-container">$A,B\in\mathrm M_n(\mathbb R)$</span>.</p>
<p>Let's assume that</p>
<p><span class="math-container">$$\forall Q\in\mathrm M_n(\mathbb R), \quad \det\begin{pmatrix} I_n & A \\ Q & B\end{pmatrix}=0$$</span></p>
<p>where <span cla... | user1551 | 1,551 | <p>It is true over any field, not just <span class="math-container">$\mathbb R$</span>. Note that <span class="math-container">$\det\pmatrix{I&A\\ Q&B}=\det(B-QA)$</span>. View <span class="math-container">$A$</span> and <span class="math-container">$B$</span> as two linear maps from a vector space <span class=... |
889,628 | <p>The question and answer is shown but I don't fully understand the answer for part a. Could someone please explain to me why the integral setup for the marginal density function of y1 is from y1 to 1, and not 0 to 1? And also the same thing for y2. Thank you
<img src="https://i.stack.imgur.com/ECQQ7.png" alt="Questio... | user487291 | 487,291 | <p>Infinity and Me is a children's book that literally and figuratively illustrates the concept of infinity, in a way that is accessible to all ages. </p>
<p>(<a href="https://i.stack.imgur.com/o5T6t.jpg" rel="nofollow noreferrer">https://i.stack.imgur.com/o5T6t.jpg</a>)</p>
|
1,714,053 | <p>How can I prove that if $\sum\limits_{n\in\Bbb N} a_n^2 $ converges then also $\sum\limits_{n\in\Bbb N} \frac{\lvert a_n\rvert}{n}$ converges?</p>
<p>Thanks ! </p>
| Spenser | 39,285 | <p>By <a href="https://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality">Cauchy–Schwarz inequality</a>,
$$\sum_{n=1}^N\frac{|a_n|}{n}\le \left(\sum_{n=1}^Na_n^2\right)^{1/2}\left(\sum_{n=1}^N\frac{1}{n^2}\right)^{1/2}.$$
Both sums on the right converge, so the partial sums of $\sum|a_n|/n$ are bounded and hence ... |
1,714,053 | <p>How can I prove that if $\sum\limits_{n\in\Bbb N} a_n^2 $ converges then also $\sum\limits_{n\in\Bbb N} \frac{\lvert a_n\rvert}{n}$ converges?</p>
<p>Thanks ! </p>
| David Mitra | 18,986 | <p>Expanding $(a-b)^2\ge0$ gives the inequality $|ab|\le{1\over2}(a^2+b^2)$. </p>
<p>Apply this with $a=a_n$ and $b={1\over n}$ to obtain ${|a_n|\over n}\le{1\over2}(a_n^2+{1\over n^2})$ for every $n$. Now you're set up nicely to use the Comparison Test.</p>
<p>(Alternatively: ${|a_n|\over n} \le( \max \{|a_n|, 1/n\... |
2,031,842 | <p>If 3 is the remainder when dividing $P(x)$ with $(x-3)$, and $5$ is the remainder when dividing $P(x)$ with $(x-4)$, what is the remainder when dividing $P(x)$ with $(x-3)(x-4)$?</p>
<p>I'm completely puzzled by this, I'm not sure where to start...</p>
<p>Any hint would be much appreciated. </p>
| Bernard | 202,857 | <p>Euclidean division by $(x-3)(x-4)$ can be written as
$$P(x)=(x-3)(x-4)Q(x)+R(x),\quad\deg R(x)\le1.$$
Instead of writing $R(x)$ with the standard basis $\{1,x\}$, use the basis $\{x-3,x-4\}$. Thus
$$P(x)=(x-3)(x-4)Q(x)+A(x-3)+B(x-4)$$
Setting $x=3$, you get right away $B=-P(3)=-3$. Similarly $A=P(4)=5$, and finally... |
3,996,790 | <p>I just realized that there may be a case where L'Hopital's rule fails, specifically</p>
<p><span class="math-container">$$ \lim_{x \to \infty} \frac{e^x}{e^x}$$</span></p>
<p>which evaluates to an indeterminate form, specifically <span class="math-container">$\frac{\infty}{\infty}$</span>. Sure, we can cancel the <s... | Spectree | 849,152 | <p>L'Hôpital's Rule says that if <span class="math-container">$\lim_{x\to a}{\frac{f'(x)}{g'(x)}}=L $</span>, then <span class="math-container">$\lim_{x\to a}{\frac{f(x)}{g(x)}}=L$</span>. With <span class="math-container">$f'(x)=e^x$</span> and <span class="math-container">$ g'(x)=e^x$</span> is obviously true. L'Hôpi... |
2,339,479 | <p>I am an avid board game player, especially ones with miniatures that duke it out (Risk, Axis and Allies, etc.) and I'm trying to analyze efficiency of different units in one of my more recent pickups. </p>
<p>In the game (Battlelore Second Edition for those who are curious), you get a hit on either a 5 or a 6. This... | import random | 462,373 | <p>I'm not particularly familiar with programming in Excel, but I do like Python, so here's some code based on David K's excellent answer and <a href="https://stackoverflow.com/a/26561091/7675174">this</a> binomial coefficient implementation:</p>
<pre><code>from math import factorial as fac
def binomial(x, y):
tr... |
2,153,128 | <p>I have been given an exercise to convert switch coordinated from cylindrical to rectangular ones. This task is easy but one of them is a strange looking. The point in cylindrical coordinates is $(0,45,10)$. This corresponds to $r=0$. What is this point? Is it not the origin? But why then the angle and z-coordinate. ... | Julio Maldonado Henríquez | 406,412 | <p>With binomials, so $\binom{11}{2}$ or $\binom{11}{3}$ respectively, where in general a subset of $k$ from a group of $n$ is $\binom{n}{k}=\frac{n!}{k!(n-k)!}$</p>
|
3,941,728 | <p>I know that every function from the empty set to any other set is the empty function.</p>
<p>I also know that there is no function to the empty set from any other set.</p>
<p>Now, what if both the domain and codomain are empty? Would such a function exist?</p>
| user1551 | 1,551 | <p><em>Hint.</em> Consider the second-order differences of the rows. When <span class="math-container">$n\ge3$</span>,
<span class="math-container">$$
\pmatrix{1&-2&1\\ &1&-2&1\\ &&\ddots&\ddots&\ddots\\ &&&1&-2&1\\ &&&&1&0\\ &&&&am... |
552,384 | <blockquote>
<p>Prove that</p>
<p>$$
\int_{0}^{\infty}\frac{e^{-bx}-e^{-ax}}{x}\,dx = \ln\left(\frac{a}{b}\right)
$$</p>
</blockquote>
<p><strong>My Attempt:</strong></p>
<p>Define the function $I(a,b)$ as</p>
<p>$$
I(a,b) = \int_{0}^{\infty}\frac{e^{-bx}-e^{-ax}}{x}\,dx
$$</p>
<p>Differentiate both side wit... | Sangchul Lee | 9,340 | <p>A problem-specific solution is as follows:</p>
<p>\begin{align*}
\int_{0}^{\infty} \frac{e^{-bx} - e^{-ax}}{x} \, dx
&= - \int_{0}^{\infty} \int_{a}^{b} e^{-xt} dt \, dx \\
&= - \int_{a}^{b} \int_{0}^{\infty} e^{-xt} dx \, dt \\
&= - \int_{a}^{b} \frac{dt}{t}
= - \left[ \log x \right]_{a}^{b} = \log\le... |
1,940,249 | <p>According to Wikipedia, <a href="https://en.wikipedia.org/wiki/First-order_logic#Completeness_and_undecidability" rel="nofollow">first order logic is complete</a>. What is the proof of this?</p>
<p>(Also, in the same paragraph, it says that its undecidable. Couldn't you just enumerate all possible proofs and dispro... | user2902293 | 138,157 | <p>I would recommend Lou Van den dries' lecture notes in first order logic. He does it in three chapters. Here's a link <a href="https://www.google.ca/url?sa=t&source=web&rct=j&url=http://www.math.uiuc.edu/~henson/Math570/Fall2009/Math570notes.pdf&ved=0ahUKEwiR--aI5KnPAhVS0GMKHdYADEAQFggbMAA&usg... |
3,853,896 | <p>Do I consider the probability before drawing both cards or after?</p>
<p><strong>Question more clearly</strong>:
A single card is removed at random from a deck of <span class="math-container">$52$</span> cards. From the remainder we draw <span class="math-container">$2$</span> cards at random and find that they are ... | Barry Cipra | 86,747 | <p>The trick here is to realize that the situation described is equivalent to removing two spades from the deck and then drawing a card at random from what remains. If you can calculate the probability of a spade occurring as the third card in that simpler setting, you're good to go.</p>
<p>Many probability problems ca... |
3,831,310 | <p>I try to integrate</p>
<p><span class="math-container">$$\int \frac {dv}{\frac {-c}{m}v^2 - g \sin \theta}$$</span></p>
<p>I did substituted <span class="math-container">$u = \frac{c}{m}$</span> and <span class="math-container">$w = g \sin \theta$</span> to get</p>
<p><span class="math-container">$$-\int \frac {dv}{... | 19aksh | 668,124 | <p><span class="math-container">$$I= \int\dfrac{dv}{-\frac{c}{m}v^2-g\sin\theta} = -\int\dfrac{dv}{\frac{c}{m}v^2+g\sin\theta} = -\int\dfrac{dv}{\left(\sqrt\frac{c}{m}\ v\right)^2+(\sqrt{g\sin\theta})^2} $$</span></p>
<p>(Assuming <span class="math-container">$0<\sin\theta<1$</span>)</p>
<p><span class="math-cont... |
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