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 |
|---|---|---|---|---|
788,671 | <p>What is the ratio of the area of a triangle $ABC$ to the area of the triangle whose sides are equal in length to the medians of triangle $ABC$?</p>
<p>I see an obvious method of brute-force wherein I can impose a coordinate system onto the figure. But is there a better solution?</p>
| Quanto | 686,284 | <p><a href="https://i.stack.imgur.com/FIXn3.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/FIXn3.png" alt="enter image description here"></a></p>
<p>Let <span class="math-container">$X$</span>, <span class="math-container">$Y$</span> and <span class="math-container">$Z$</span> be the side midpoints... |
75,925 | <p>I hope this question is focused enough – it's not about real problem I have, but to find out if anyone knows about a similar thing.</p>
<p>You probably know the <a href="https://en.wikipedia.org/wiki/Uncertainty_principle" rel="nofollow noreferrer">Heisenberg uncertainty principle</a>: For any function <span class="... | Antoine Levitt | 19,334 | <p>I find the neatest "standard" uncertainty principle is the one with commutators, see e.g. <a href="http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/GenUncertPrinciple.htm" rel="nofollow">http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/GenUncertPrinciple.htm</a>. I think that readily gives both your ... |
694,279 | <p>I am learning convex analysis by myself and I need help.</p>
<p>How to show that if $X=U=\mathbb{R}$
and $f\left(x\right)=\frac{|x|^{p}}{p}$
then the convex conjugate $f^{*}\left(u\right)=\frac{|u|^{q}}{q}$
when $\frac{1}{p}+\frac{1}{q}=1$?
There exists a particular technique that I have to apply in order to ... | user127096 | 127,096 | <p>For practical computation I would use the fact that $\nabla f^*$ is the inverse of $\nabla f$ (<a href="http://en.wikipedia.org/wiki/Convex_conjugate#Maximizing_argument" rel="nofollow">see here</a>). By the chain rule, $$\nabla f(x) =|x|^{p-1} \nabla |x| = |x|^{p-1} \frac{x}{|x|}$$ which means the direction of $x$ ... |
4,375,994 | <blockquote>
<p>Question:</p>
<p>Show that, <span class="math-container">$$\pi =3\arccos(\frac{5}{\sqrt{28}}) +
3\arctan(\frac{\sqrt{3}}{2}) ~~~~~~ (*)$$</span></p>
</blockquote>
<p><em>My proof method for this question has received mixed responses. Some people say it's fine, others say that it is a verification, inst... | user2661923 | 464,411 | <p>To show:</p>
<p><span class="math-container">$$\text{arctan}\left(\frac{\sqrt{3}}{5}\right) ~+~ \text{arctan}\left(\frac{\sqrt{3}}{2}\right) ~~<~~ \frac{\pi}{2}.$$</span></p>
<p>In fact this conclusion is immediate by the following analysis.</p>
<p>Let <span class="math-container">$~\displaystyle \theta ~~=~~ \te... |
4,601,727 | <p>I'm aware that the title might be a bit off, I am unsure on how to describe this.</p>
<p>For <span class="math-container">$n\in \mathbb{N}$</span>, define <span class="math-container">$n+1$</span> independent random variables <span class="math-container">$X_0, \ldots , X_n$</span> which are uniformly distributed ove... | Aishgadol | 861,510 | <p>The answer I've been able to compute, using the following formula:
<span class="math-container">$$ \mathbb{P}(X\in A)=\int_{-\infty}^{\infty}\mathbb{P}(X\in A |Y=y)f_Y(y)dy$$</span></p>
<p>We know that the boundaries are <span class="math-container">$[0,1]$</span>.</p>
<p>Therefor, We're looking to find <span class=... |
18,459 | <p>Does anyone know of any studies or have personal experience dealing with difficulties (if any) faced by students studying mathematics if they come from countries which use languages written from right-to-left or top-down? </p>
<p>I have been wondering about this recently because I have been working on supplemental ... | IrbidMath | 5,993 | <p>Am Arabian Mathematician. In my country in all school we used to write math from right to left i scored very well in 12th grade 179/200. But in the universities we use English books so we write from left to right it was hard for me to get through that in the first course I scored 65/100 by the time I used to that a... |
3,378,004 | <p>If <span class="math-container">$H$</span> and <span class="math-container">$K$</span> are abelian subgroups of a group <span class="math-container">$G$</span>, then <span class="math-container">$H\cap K$</span> is a normal subgroup of <span class="math-container">$\left\langle H\cup K\right\rangle$</span>.</p>
<p>... | José Carlos Santos | 446,262 | <p>Yes, every element of <span class="math-container">$H\cap K$</span> commutes with every element of <span class="math-container">$H$</span> and with every element of <span class="math-container">$K$</span>. In my opinion, you should add a proof of that fact that it follows from this that every element of <span class=... |
153,902 | <p>Let $A_i$ be open subsets of $\Omega$. Then $A_0 \cap A_1$ and $A_0 \cup A_1$ are open sets as well.</p>
<p>Thereby follows, that also $\bigcap_{i=1}^N A_i$ and $\bigcup_{i=1}^N A_i$ are open sets.</p>
<p>My question is, does thereby follow that $\bigcap_{i \in \mathbb{N}} A_i$ and $\bigcup_{i \in \mathbb{N}} A_i$... | Davide Giraudo | 9,849 | <p>An arbitrary union (coutable or not) of open sets is open, but even for a countable intersection it's not true in general. For example, when $\Omega$ is the real line endowed with the usual topology, and $A_i:=\left(-\frac 1i,\frac 1i\right)$, $A_i$ is open but $\bigcap_{i\in \Bbb N}A_i=\{0\}$ which is not open. </p... |
158,810 | <p>Let $ (\hat i, \hat j, \hat k) $ be unit vectors in Cartesian coordinate and $ (\hat e_\rho, \hat e_\theta, \hat e_z)$ be on spherical coordinate.
Using the relation, $$ \hat e_\rho = \frac{\frac{\partial \vec r}{\partial \rho}}{ \left | \frac{\partial \vec r}{\partial \rho} \right |}, \hat e_\theta = \frac{\frac{... | Chloé Seppälä Zetterberg | 514,885 | <p>A far more simple method would be to use the gradient.<br/>
Lets say we want to get the unit vector $\boldsymbol { \hat e_x } $. What we then do is to take $\boldsymbol { grad(x) } $ or $\boldsymbol { ∇x } $.<br/>
This; $\boldsymbol ∇ $, is the nabla-operator. It is a vector containing each partial derivative like t... |
46,631 | <p>I'm writing a program to play a game of <a href="http://en.wikipedia.org/wiki/Pente" rel="noreferrer">Pente</a>, and I'm struggling with the following question:</p>
<blockquote>
<p>What's the best way to detect patterns on a two-dimensional board?</p>
</blockquote>
<p>For example, in Pente a pair of neighboring ... | Pellesatansfant | 13,904 | <p><strong>EDIT</strong></p>
<p>Making this code runnable with Java reloader.</p>
<ol>
<li><p>Load the <a href="https://mathematica.stackexchange.com/questions/6144/looking-for-longest-common-substring-solution/6376#6376">Java reloader</a> (run the code from that post. For Mac OS X, see the comments below the post fo... |
1,158,489 | <p>Is it the case that, as $N\to\infty$, $$\binom{2N}{N+j}_q\to (-1)^j,$$ where convergence of the $q$-binomial coefficient is seen as a convergence of formal power series in the variable $q$? </p>
| Community | -1 | <p>$$A_{N,j}=\binom{2N}{N+j}_q=\frac{(1-q^{2N})(1-q^{2N-1})\ldots(1-q^{2N-N-j+1})}{(1-q)(1-q^2)\ldots(1-q^{N-j})}$$</p>
<p>Take $\log$ and expand using that $$\log(1-x)=\sum_{n=1}^{\infty}\frac{x^n}{n}.$$</p>
<p>We obtain that $$\begin{align}\log(A_{N,j})&=\sum_{k=N-j+1}^{2N}\log(1-q^{k})-\sum_{k=1}^{N-j}\log(1-... |
394,085 | <p>How is it possible to establish proof for the following statement?</p>
<p>$$n = \frac{1}{2}(5x+4),\;2<x,\;\text{isPrime}(n)\;\Rightarrow\;n=10k+7$$</p>
<p>Where $n,x,k$ are $\text{integers}$.</p>
<hr>
<p>To be more verbose:</p>
<p>I conjecture that;</p>
<p>If $\frac{1}{2}(5x+4),\;2<x$ is a prime number, ... | Berci | 41,488 | <p>Since $n=5\,x/2\ +2$ is integer, we have that $x$ is even, and that $n$ gives remainder $2$ modulo $5$ (written $n\equiv 2\pmod{5}$). So, modulo $10$ there are only two possibilities: $2$ and $7=5+2$. As all primes $>2$ are odd, but $10k+2$ is even, it can't be prime.</p>
<p>(So, instead of 'prime' and $x>2$ ... |
425,969 | <p>It seems striking that the cardinalities of <span class="math-container">$\aleph_0$</span> and <span class="math-container">$\mathfrak c = 2^{\aleph_0}$</span> each admit what I will call a "homogeneous cyclic order", via the examples of <span class="math-container">$ℚ/ℤ$</span> and <span class="math-conta... | Joseph Van Name | 22,277 | <p>Every totally ordered abelian group is automatically homogeneous as a total order, and every totally ordered set that is homogeneous as a total order is automatically homogeneous as a cyclic order. Therefore, every totally ordered abelian group is automatically homogeneous as a cyclic order.</p>
<p>Let <span class="... |
3,624,662 | <p><a href="https://i.stack.imgur.com/kwAMn.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/kwAMn.png" alt=" c"></a></p>
<p>In my mind, I can think of below example which seems to work.</p>
<p>If <span class="math-container">$(X,T) = \mathbb{R}$</span>, and <span class="math-container">$A = (0,\inf... | peek-a-boo | 568,204 | <p>Let <span class="math-container">$A = [0, \infty)$</span>, then its complement <span class="math-container">$A^c = (-\infty, 0)$</span> is an open interval, and hence <span class="math-container">$A^c$</span> is an open set. Thus, by definition, <span class="math-container">$A$</span> is a closed set.</p>
<p>To sho... |
3,624,662 | <p><a href="https://i.stack.imgur.com/kwAMn.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/kwAMn.png" alt=" c"></a></p>
<p>In my mind, I can think of below example which seems to work.</p>
<p>If <span class="math-container">$(X,T) = \mathbb{R}$</span>, and <span class="math-container">$A = (0,\inf... | Community | -1 | <p>By Heine-Borel, a compact subset of <span class="math-container">$\Bbb R^n$</span> is closed and bounded. </p>
<p>So take any closed, unbounded set.</p>
|
97,449 | <p>I am trying to compute $\chi(\mathbb{C}\mathrm{P}^2)$ using only elementary techniques from differential topology and this is proving to be trickier than I thought. I am aware of the usual proof for this result, which uses the cellular decomposition of $\mathbb{C}\mathrm{P}^2$ to get $\chi(\mathbb{C}\mathrm{P}^2) = ... | Igor Rivin | 11,142 | <p>There is a canonical way to construct <em>holomorphic</em> vector fields on $\mathbb{C}P^2,$ and that way is <a href="http://www.springer.com/us/book/9783764375355" rel="nofollow noreferrer">described in Zoladek's "Monodromy Group", page 335.</a> If you read the description, it will be pretty clear what the index is... |
97,449 | <p>I am trying to compute $\chi(\mathbb{C}\mathrm{P}^2)$ using only elementary techniques from differential topology and this is proving to be trickier than I thought. I am aware of the usual proof for this result, which uses the cellular decomposition of $\mathbb{C}\mathrm{P}^2$ to get $\chi(\mathbb{C}\mathrm{P}^2) = ... | Johannes Ebert | 9,928 | <p>Take a $3 \times 3$ complex diagonal matrix $A$ with distinct nonzero diagonal entries. The 1-parameter subgroup $exp(At)$ acts on $CP^2$; the fixed points are the lines in $C^3$ containing eigenvectors of $A$. There are $3$ of them and the derivative of the action is a vector field with $3$ zeroes. As the vector fi... |
1,572,351 | <p>Solve the differential equation;</p>
<p>$(xdx+ydy)=x(xdy-ydx)$</p>
<p>L.H.S. can be written as $\frac{d(x^2+y^2)}{2}$ but what should be done for R.H.S.?</p>
| SchrodingersCat | 278,967 | <p>As much as I can see, there is no standard solution. <br>
From <a href="http://www.wolframalpha.com/input/?i=%5B%2F%2Fmath%3Axdx%2Bydy%3Dx(xdy-ydx)%2F%2F%5D" rel="nofollow noreferrer">Wolfram Alpha</a>, I have obtained a very complicated solution. You can check the link, if you wish.</p>
<p><a href="https://i.stack... |
1,572,351 | <p>Solve the differential equation;</p>
<p>$(xdx+ydy)=x(xdy-ydx)$</p>
<p>L.H.S. can be written as $\frac{d(x^2+y^2)}{2}$ but what should be done for R.H.S.?</p>
| Empy2 | 81,790 | <p>Let $x^2=z+y$. The equation $(xdx+ydy)=x(xdy-ydx)$ becomes
$$(dz+dy)+2ydy=2(z+y)dy-y(dz+dy)\\
(1+y)dz=(2z-y-1)dy\\
y=w-1\\
wdz=(2z-w)dw\\
\frac{dz}{w^2}-\frac{2zdw}{w^3}=-\frac{dw}{w^2}\\
\frac{z}{w^2}-\frac1w=const$$</p>
|
3,110,508 | <p>I read that implication like a=>b can be proof using the following steps :
1) suppose a true.
2) Then deduce b from a.
3) Then you can conclude that a=>b is true.</p>
<p>Actually my real problem is to understand why step 1 and 2 are sufficient to prove that a=>b is true. I mean, how can you prove the truth table of... | Dr. Mathva | 588,272 | <p>Just let <span class="math-container">$k=-y$</span> to obtain <span class="math-container">$$x^3-y^3=(x-y)(x^2+xy+y^2)\iff x^3+k^3=(x+k)(x^2-xk+k^2)$$</span>
Note that the same works whenever <span class="math-container">$n$</span> is odd. Given <span class="math-container">$$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\dots + x... |
3,405,914 | <blockquote>
<p>It's known that <span class="math-container">$\lim_{n \to \infty} \left(1 + \frac{x}{n} \right)^n = e^x$</span>.</p>
<p>Using the above statement, prove <span class="math-container">$\lim_{n \to \infty} \left(\frac{3n-2}{3n+1}\right)^{2n} = \frac{1}{e^2}$</span>.</p>
</blockquote>
<h2>My attempt... | Z Ahmed | 671,540 | <p><span class="math-container">$$L=\lim_{n \rightarrow} \left(\frac{3n-2}{3n+1} \right)^{2n}= \lim_{n \rightarrow \infty} \frac{\left([1-2/(3n)]^{3n/2}\right)^{4/3}}{\left([1+1/(3n)]^{3n}\right)^{2/3}}=\frac{ e^{-4/3}}{e^{2/3}}=e^{-2}$$</span></p>
|
1,458,960 | <p>I have been working on this problem for about(over) two months now. This is <strong>not school work</strong>.<br>
The final objective of this problem is to find the <strong>area</strong> of these <strong>11 cylinders</strong> which are enclosed in a sphere.</p>
<p>In the image below, every line represents the cente... | Archis Welankar | 275,884 | <p>For this problem consider sun .The ball is centre and rays are the cylinders so the area of the 11 cylinders would be a shape like a flat tyre like cylinder whose radius is length of the small cylinders and height is the diameter of small cylinders .This my way of thinking. Soory if I have interpreted something wr... |
1,458,960 | <p>I have been working on this problem for about(over) two months now. This is <strong>not school work</strong>.<br>
The final objective of this problem is to find the <strong>area</strong> of these <strong>11 cylinders</strong> which are enclosed in a sphere.</p>
<p>In the image below, every line represents the cente... | Ian Miller | 278,461 | <p>Is this what you meant? (Sorry I can't put an image in a comment.)</p>
<p><a href="https://i.stack.imgur.com/evxeb.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/evxeb.png" alt="Your thing"></a></p>
<p>The sphere is partially transparent so you can see the cylinders inside.</p>
|
2,704,770 | <p>I need help in calculating this strange limit.</p>
<p>$$
\lim_{n \to \infty} n^2 \int_{0}^{\infty} \frac{sin(x)}{(1 + x)^n} dx
$$</p>
| Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\... |
2,772,895 | <p>I've noticed one classical way of defining certain topologies is to define them as the "weakest" (or coarsest) topology such that a certain set of functions is continuous. For example,</p>
<blockquote>
<p>The <strong>product topology</strong> on <span class="math-container">$X=\prod X_i$</span> is the weak... | Billy | 13,942 | <blockquote>
<p>"weakest"</p>
</blockquote>
<p>Fix a space X, and let Top(X) be the set of topologies on X. That is, an element T of Top(X) can be thought of as a subset of the power set P(X) (satisfying axioms). Now, Top(X) naturally forms a poset under $\subseteq$: that is, if S and T are elements of Top(X), you c... |
834,678 | <p>An object $X$ is a <em>generator</em> of a category $\mathcal{C}$ if the functor $Hom_{\mathcal{C}}(X,\_) : \mathcal{C} \rightarrow Set$ is faithful. </p>
<p>I encountered the notion in the context of Morita-equivalence of rings, but I don't understand what its use is.
Why is $X$ called a "generator"? What does it ... | Community | -1 | <p>While I don't know the <em>actual</em> etymology, I interpret the word "generate" as generating elements of the category.</p>
<p>There is a notion of a "generalized element" of an object: a generalized element of $A$ is simply an arrow with target $A$. This is the "right" notion of element: for example, a morphism ... |
3,572,842 | <p><strong>Context:</strong> 1st year BSc Mathematics, Vectors and Mechanics module, constant circular motion.</p>
<p>This may be trivial, but can someone tell me what's wrong with the following reasoning?</p>
<p><span class="math-container">$$\underline{e_r}=\underline{i}\cos\theta+\underline{j}\sin\theta=(1,\theta)... | emacs drives me nuts | 746,312 | <blockquote>
<p>Are there some available computational results much beyond <span class="math-container">$c≤100$</span> that would confirm intuition that really <span class="math-container">$a+b$</span> is almost-always a prime number or this is just some instance of the rule that this is a small sample so it is not q... |
791,535 | <blockquote>
<p>Let $f,g:\mathbb [a,b] \to \mathbb [a,b]$ be monotonically increasing functions
such that $f\circ g=g\circ f$</p>
<p>Prove that $f$ and $g$ have a common fixed point.</p>
</blockquote>
<p>I found this problem in a problem set, it's quite similar to this <a href="https://math.stackexchange.com... | Lutz Lehmann | 115,115 | <p>Let <span class="math-container">$z$</span> be a fixed point of <span class="math-container">$f$</span>. Then <span class="math-container">$$f(g(z))=g(f(z))=g(z),$$</span> so also <span class="math-container">$g(z)$</span> is a fixed point of <span class="math-container">$f$</span>. Now if they were unique, then <sp... |
1,134,145 | <p>A set S is bounded if every point in S lies inside some circle |z| = R other it is unbound. Without appealing to any limit laws, theorems, or tools from calculus, prove or disprove that the set {$\frac{z}{z^2 + 1}$; z in R} is bounded.</p>
<p>I imagine that it's simple, but I have no clue where to start due to the ... | AlexR | 86,940 | <p><strong>Hint</strong><br>
Estimate case-wise $\frac{|z|}{z^2+1}$ for $|z|<1$ and $|z|\ge 1$:
$$\frac{|z|}{z^2 + 1} \le \frac{|z|}{z^2} = \frac1{|z|} \stackrel{|z|\ge 1}\le 1$$
And for $|z| < 1$ use $z^2 \ge 0$.<br>
This will give you the bound $S\subset[-1,1]$.<br>
An optimal bound is $[-\frac12, \frac12]$, th... |
393,712 | <p>I studied elementary probability theory. For that, density functions were enough. What is a practical necessity to develop measure theory? What is a problem that cannot be solved using elementary density functions?</p>
| krishnab | 38,239 | <p>I come from a probability and statistics background, so I kinda get where the OP is coming from. The basic question is about the motivation behind learning measure theory when working with probability. And sure enough, when I look at many measure theory texts, the motivation for learning measure theory--Lebesgue mea... |
1,366,023 | <p>Here's a problem I was just working on:</p>
<blockquote>
<p>Let $f$ have an essential singularity at $0$. Show that there is a sequence of points $z_n \to 0$ such that $z_n^n f(z_n)$ tends to infinity.</p>
</blockquote>
<p>I know already that there exists a sequence $z_n \to 0$ such that $f(z_n)$ tends to any c... | Theo Bendit | 248,286 | <p>You should be able to extract such a sequence from $g_n(z_{n,k})$. To put this sequence property slightly differently, you know that for any $n \in \mathbb{N}$, any complex number $w$, and any $\varepsilon > 0$, you can find a complex number $z$ such that $|g_n(z) - w| < \varepsilon$. So, fix $n \in \mathbb{N}... |
3,011,758 | <p>So I was going through a sum for </p>
<p>Prove <span class="math-container">$ex \leq e^x$</span> , <span class="math-container">$\forall x \in \mathbb{R} $</span></p>
<p>I took <span class="math-container">$g(x) = e^x - ex$</span></p>
<p>Then <span class="math-container">$g'(x)= e^x - e$</span></p>
<p>I understo... | gb2017 | 496,656 | <p>it is easy to see <span class="math-container">$x=1$</span> is the absolute minimum, that is, <span class="math-container">$ f(1)\leq f(x)$</span> for all <span class="math-container">$x\in R$</span> so that <span class="math-container">$0\leq ex-e^x$</span>.Thus <span class="math-container">$ex\leq e^x$</span> for... |
1,220,923 | <p>Find the value of the integral
$$\int_0^\infty \frac{x^{\frac25}}{1+x^2}dx.$$
I tried the substitution $x=t^5$ to obtain
$$\int_0^\infty \frac{5t^6}{1+t^{10}}dt.$$
Now we can factor the denominator to polynomials of degree two (because we can easily find all roots of polynomial occured in the denominator of the form... | k1.M | 132,351 | <p>With some substitutions and ordinary works on the integral we have
$$\int_0^\infty \frac{x^{\frac25}}{1+x^2}dx=\frac52\int_{-\infty}^\infty\frac{t^6}{1+t^{10}}dt$$
For every $R\gt0$ we have
$$\int_{C}\frac{z^6}{1+z^{10}}dz=\int_{-R}^R\frac{t^6}{1+t^{10}}dt+\int_{C_R}\frac{z^6}{1+z^{10}}dz$$
where $C_R$ is the contou... |
4,246,048 | <p>As I understand it, Cantor defined two sets as having the same cardinality iff their members can be paired 1-to-1. He applied this to infinite sets, so ostensibly the integers (Z) and the even integers (E) have the same cardinality because we can pair each element of Z with exactly one element of E.</p>
<p>For infi... | Mark Saving | 798,694 | <p>I'm not exactly sure what the supposed problem is, but I can walk through some basic results to hopefully shed light on what's going on.</p>
<p>Two basic notions in set theory are sets and functions. I'll assume you agree that the notion of a set and the notion of a function make sense.</p>
<p>Consider a function <s... |
2,962,203 | <p>I got stuck at : <span class="math-container">$a^2/b^2 = 12+2 \sqrt 35$</span></p>
<p>I understand that <span class="math-container">$12$</span> is rational and now I need to prove that <span class="math-container">$\sqrt{35}$</span> is irrational.</p>
<p>so I defined <span class="math-container">$∀c,d∈R$</span> ... | Barry Cipra | 86,747 | <p>Here's a different approach, assuming you know that <span class="math-container">$\sqrt5$</span> and/or <span class="math-container">$\sqrt7$</span> are irrational.</p>
<p>If <span class="math-container">$\sqrt7+\sqrt5$</span> were rational, then </p>
<p><span class="math-container">$${2\over\sqrt7+\sqrt5}=\sqrt7-... |
463,139 | <p>I have this:</p>
<p>Case 1)</p>
<p><img src="https://i.stack.imgur.com/xEEFQ.png" alt="enter image description here"></p>
<p>If <em>f</em> is a pair function $f(-x)=f(x)$ then $\int_{-a}^a f(x)dx=2\int_0^af(x)dx$</p>
<p>Case 2)</p>
<p><img src="https://i.stack.imgur.com/mnHbB.png" alt="enter image description h... | Jean-Sébastien | 31,493 | <p>There are many reasons why we call those even and odd functions. One is that the taylor series of an even function only includes even powers whereas the Taylor series of an odd function only includes odd power.</p>
<p>But to recognize when and what type of symmetry there is, the most useful property is that they ar... |
2,642,144 | <p>How would I prove or disprove the following statement?
$ \forall a \in \mathbb{Z} \forall b \in \mathbb{N}$ , if $a < b$ then $a^2 < b^2$</p>
| Community | -1 | <p>If you choose $a$ as negative number and $b$ as a positive number with module less then $a$ you find a counter example.</p>
<p>In fact your statement is false $\forall a\in\mathbb{Z},a<0\forall b\in\mathbb{N}:|a|^2\geq b^2$</p>
<p>As @kevin said if you take $a=-5$ and $b=1$ the statement is false.</p>
|
3,213,464 | <p>Does 22.449 approximate to 22 or 23?
If we see it one way
<span class="math-container">$22.449≈22$</span>
But on the other hand
<span class="math-container">$22.449≈22.45≈22.5≈23$</span>
Which one is correct?</p>
| Borelian | 290,706 | <p><a href="https://en.wikipedia.org/wiki/IEEE_754" rel="nofollow noreferrer">IEEE 754</a> is a standard that defines how to round a number to the nearest integer (at least for computers). There are many conventions, and in some of them 22.5 is rounded to either 22 or 23. However, in all the cases 22.449 is rounded t... |
1,716,656 | <p>I am having trouble solving this problem</p>
<blockquote>
<p>Julie bought a house with a 100,000 mortgage for 30 years being repaid with payments at the end of each month at an interest rate of 8% compounded monthly. If Julie pays an extra 100 each month, what is the outstanding balance at the end of 10 years im... | Sava B. | 326,626 | <p>Could use the multiplication rule:</p>
<p>The probability of Die 1 landing on 1-4 is 4/6.
The probability of the Die 2 landing on the number that's Die1+2 is then 1/6. </p>
<p>(4/6) * (1/6) = 4/36</p>
<p>We multiply this by 2 to account the scenario where Die 2 is the 1-4 die, and then Die 1 is two higher than Di... |
2,426,263 | <p>Nowadays, the most widely-taught model of computation (at least in the English-speaking world) is that of Turing Machines, however, it wasn't the first Turing-Complete model out there: μ-recursive functions came a few years earlier, and λ-calculus came a year earlier. Why is it that Turing machines are so popular to... | J.-E. Pin | 89,374 | <p>There are several reasons of the popularity of Turing machines: </p>
<p>(1) Turing machines fit well in a course of automata theory: you can start by introducing simpler models, like finite automata or pushdown automata and then move to TMs.</p>
<p>(2) The statement of the most famous open problem in theoretical c... |
1,411,305 | <p>I have been trying to solve the following problem:</p>
<blockquote>
<p>What is the probability that among 3 random digits, there appear
exactly 2 different ones?</p>
</blockquote>
<p>The formula for no repititions is:</p>
<pre><code>(n*(n-1)...(n-r+1))/n^r
</code></pre>
<p>So, for the first digit there are 1... | vonbrand | 43,946 | <p>Your mistake is that you don't consider the case where the first digit repeats. You have the following exhaustive and exclusive cases:</p>
<ul>
<li><strong>The first digit repeats:</strong> There are $10$ options for the first digit, which can appear in any of the $2$ remaining positions, and the other digit is one... |
2,243,542 | <p>In a previous question here <a href="https://math.stackexchange.com/q/2240195/369757">Can we define the Cantor Set in this way?</a></p>
<p>we defined a family of sets $ \left\{ C_0,C_1,C_2,C_3,\dots \right\}$</p>
<p>We can call this set $S_1$ , where the values of these elements is</p>
<p>$C_0 = \left\{ 0.0 \ri... | Ivan Hieno | 369,757 | <p>Casting this in terms of the set of natural numbers, there seems to be no contradiction if $ \mathbb{N} \in S_1 $.</p>
<p>Let:$$ S_1 = \left\{ C_m = \left\{ n \in \mathbb{N} : n \le m \right\} : m \in \mathbb{N} \right\} $$
Then:$$C_0= \left\{ 0 \right\}$$</p>
<p>$$C_1= \left\{ 0,1 \right\}$$</p>
<p>$$C_2=... |
12,717 | <p>In the familiar case of (smooth projective) curves over an algebraically closed fields, (closed) points correspond to DVR's.</p>
<p>What if we have a non-singular projective curve over a non-algebraically closed field? The closed points will certainly induce DVR's, but would all DVR's come from closed points? Is th... | Qing Liu | 3,485 | <p>To complete partly the answer of Emerton, the picture for DVR is relatively clear. Let $X$ be an integral noetherian scheme and let $R$ be a DVR with field of fractions equal to the field of rational functions $k(X)$ on $X$. Suppose that $R$ has a center $x\in X$ (e.g. if $X$ is proper over a subring of $R$). Let $k... |
1,442,240 | <p>I have a little question, that run threw my thoughts, when i saw this exercise: $$\lim _{x\to \infty }\left(\frac{\int _{sin\left(x\right)}^xe^{\sqrt{t^2+1}}dt}{e^{\sqrt{x^2-1}}}\:\right)$$</p>
<p>Of course I want to implement here Lophital's rule, but without showing a calculation, is there intuitive and logical e... | Jack D'Aurizio | 44,121 | <p><strong>Hint:</strong> By De l'Hopital theorem,
$$ \lim_{x\to +\infty} e^{-x}\int_{0}^{x}e^{\sqrt{t^2+1}}\,dt = \lim_{x\to +\infty}\exp\left(\sqrt{x^2+1}-x\right)=\exp(0)=\color{red}{1}.$$</p>
|
954,419 | <p>I am teaching myself mathematics, my objective being a thorough understanding of game theory and probability. In particular, I want to be able to go through A Course in Game Theory by Osborne and Probability Theory by Jaynes.</p>
<p>I understand I want to cover a lot of ground so I'm not expecting to learn it in le... | Shane | 61,746 | <p>Perhaps this should be a comment, but I don't have the reputation. I TA a class that uses the Osborne and Rubinstein text you mention. I can't speak in depth regarding your goals in probability, but here's my advice on Game Theory:</p>
<p>Unless you're looking to publish research in game theory, the mathematics req... |
300,867 | <p>I am having a difficult time understanding where I went wrong with the following:
$$\begin{matrix}4x-y = 1 \\ 2x+3y = 3 \end{matrix} $$
$$\begin{matrix}4x-y = -3 \\ 2x+3y = 3 \end{matrix} $$</p>
<p>I found the inverse of the common coefficient matrix of the systems:
$$A^{-1} \begin{cases} \frac3{14}, \frac1{14} \\ ... | Mariano Suárez-Álvarez | 274 | <p>If you want to do this, you can always find an arc-connected compact subset $K\subseteq U$ such that $C\subseteq K$. Then do your argument for $K$ and, since you can restrict the conclusion to $C$, you are happy.</p>
|
3,165,781 | <p>Ok, I am a bit confused.</p>
<p>Does first countability mean that for every element <span class="math-container">$x$</span> in the space, there is a collection of open sets and each of those open sets are countable containing <span class="math-container">$x$</span> and for every open neighborhood of <span class="ma... | Kavi Rama Murthy | 142,385 | <p>The first one is wrong. Even in the real line open sets are all uncountable so the condition there is not satisfied. The second definition is correct. </p>
|
1,395,619 | <p>One of my friend asked this doubt.Even in lower class we use both as synonyms,he says that these two concepts have difference.Empty set $\{ \}$ is a set which does not contain any elements,while null set ,$\emptyset$ says about a set which does not contain any elements.</p>
<p>I could not make out that...is his arg... | Khallil | 99,916 | <p>They aren't the same although they were used interchangeable way back when.</p>
<blockquote>
<p>In mathematics, a <a href="https://en.wikipedia.org/wiki/Null_set" rel="noreferrer">null set</a> is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure the... |
320,452 | <p>For any positive integer <span class="math-container">$n\in\mathbb{N}$</span> let <span class="math-container">$S_n$</span> denote the set of all bijective maps <span class="math-container">$\pi:\{1,\ldots,n\}\to\{1,\ldots,n\}$</span>. For <span class="math-container">$n>1$</span> and <span class="math-container"... | Michael Lugo | 143 | <p>Some quick simulation leads me to conjecture that <span class="math-container">$\lim_{n \to \infty} E_n$</span> exists and is around <span class="math-container">$1.15$</span>. </p>
<p>The number of permutations in <span class="math-container">$S_n$</span> with <span class="math-container">$N_n(\pi) = 1$</span> is... |
909,741 | <blockquote>
<p><strong>ALREADY ANSWERED</strong></p>
</blockquote>
<p>I was trying to prove the result that the OP of <a href="https://math.stackexchange.com/questions/909712/evaluate-int-0-frac-pi2-ln1-cos-x-dx"><strong><em>this</em></strong></a> question is given as a hint.</p>
<p>That is to say: <em>imagine tha... | dustin | 78,317 | <p>As shown by @idm, we have that
$$
\int_0^{\pi/2}\ln(\cos(x))dx = \int_0^{\pi/2}\ln(\sin(x))dx.
$$
We can exploit this identity for another one and use Feynman's method (differentiating under the integral).
Consider
$$
\int_0^{\pi/2}x\cot(x)dx.\tag{1}
$$
By integration by parts, we have
$$
\int_0^{\pi/2}x\cot(x)dx =... |
792,924 | <p>If a quantity can be either a scalar or a vector, how would one call that property? I could think of scalarity but I don't think such a term exists.</p>
| Mikhail Katz | 72,694 | <p>No because by the Cauchy-Schwarz inequality, $\int_0^1 x^n f(x)dx\leq \sqrt{\int_0^1 x^{2n}dx}\sqrt{\int_0^1 f^2(x)dx}$,where the second factor is constant whereas the first factor tends to zero. This shows there is not even an $L^2$ function with such a property.</p>
|
347,385 | <p>Assume $f(x) \in C^1([0,1])$,and $\int_0^{\frac{1}{2}}f(x)\text{d}x=0$,show that:
$$\left(\int_0^1f(x)\text{d}x\right)^2 \leq \frac{1}{12}\int_0^1[f'(x)]^2\text{d}x$$</p>
<p>and how to find the smallest constant $C$ which satisfies
$$\left(\int_0^1f(x)\text{d}x\right)^2 \leq C\int_0^1[f'(x)]^2\text{d}x$$</p>
| math110 | 58,742 | <p>solutin 2:</p>
<p>by Schwarz,we have
$$\int_{0}^{\frac{1}{2}}[f'(x)]^2dx\int_{0}^{\frac{1}{2}}x^2dx\ge\left(\int_{0}^{\frac{1}{2}}xf'(x)dx\right)^2=\left[\dfrac{1}{2}f(\dfrac{1}{2})-\int_{0}^{\frac{1}{2}}f(x)dx\right]^2$$
so
$$\int_{0}^{\frac{1}{2}}[f'(x)]^2dx\ge 24\left[\dfrac{1}{2}f(\dfrac{1}{2})-\int_{0}^{\fra... |
706,980 | <p>If I know that $f(z)$ is differentiable at $z_0$, $z_0 = x_0 + iy_0$.
How do I prove that $g(z) = \overline{f(\overline{z})}$ is differentiable at $\overline z_0$?</p>
| BigM | 90,395 | <p>Let $f(z)=f(x+iy)=u(x,y)+iv(x,y)$, then $f(\bar{z})=f(x-iy)=u(x,-y)+iv(x,-y)\Rightarrow \overline{f(\bar{z})}=u(x,-y)-iv(x,-y)$.Put $u_{1}(x,y)=u(x,-y)$ and $v_1(x,y)=-v(x,-y)$.</p>
<p>We have $\frac{\partial}{\partial x}u_1=\frac{\partial }{\partial x}u(x,-y)=\frac{\partial }{\partial x}u(x,-y)$ and $\frac{\partia... |
3,288,815 | <p>I'm reading theorem 3.11 in Rudin's RCA what says <span class="math-container">$L^p$</span> space is a complete metric space.
At the end of the proof, Rudin says that "Then <span class="math-container">$\mu(E)=0$</span>, and on the complement of <span class="math-container">$E$</span> the sequence <span class="math... | Kavi Rama Murthy | 142,385 | <p>Since <span class="math-container">$\|f_n(x)-f_m(x)| \leq \|f_n-f_m\|_{\infty}$</span> for all <span class="math-container">$n,m$</span>, for <span class="math-container">$x$</span> not in <span class="math-container">$E$</span> and since <span class="math-container">$\|f_n-f_m\|_{\infty} \to 0$</span> the converge... |
490,802 | <p>Is $(x,3,5)$ a plane, for $x\in\mathbb{R}$?</p>
<p>I know that if two of the coordinates are "arbitrary", like $(x,y,4)$or $(3,y,z)$, then it creates a plane (for $x,y,z\in \mathbb{R}).$</p>
<p>Is there a way to tell if it would create a plane in $\mathbb{R}^3?$</p>
| imranfat | 64,546 | <p>It doesn't. It is a line in space parallel to the x-axis. Graph the points (-1,3,5) , (0,3,5) , (1,3,5) , (2,3,5) etc. What do you notice?</p>
|
4,190,301 | <p>I found <span class="math-container">$\tilde{R}$</span> in a mathematical text, and I would like to know how this is pronounced. I tried to search on the internet but was not able to find anything related.</p>
| Lee Mosher | 26,501 | <p>That symbol on top is a <a href="https://en.wikipedia.org/wiki/Tilde" rel="noreferrer">tilde</a>.</p>
<p>So you can pronounce that "arr tilde" or sometimes "tilde arr", depending on your preference. I've gotten used to "tilde arr" because that's how it's typed in LaTeX, namely \tilde R ... |
2,155,589 | <p>I'm reading a computer science book that gives several functions, in the mathematical sense. There are two that are the basis of this question.</p>
<p>These are equations used to convert a number represented in base ten to a bit representation using two's complement and back.</p>
<p>One function makes the conversi... | Veridian Dynamics | 408,632 | <p>Let $\epsilon >0$. Take $\delta = \min\{1,\frac{\epsilon}{3(1+|z_{0}|)^{2}}\}$. If $|z-z_{0}|<\delta$, then $|z|\leq|z-z_{0}|+|z_{0}|< \delta+|z_{0}|$. So,
$$|z^{3}-z_{0}^{3}|=|z-z_{0}||z^{2}+zz_{0}+z_{0}^{2}|< \delta((\delta+|z_{0}|)^{2}+(\delta+|z_{0}|)||z_{0}|+|z_{0}|^{2})\leq3\delta (1+|z_{0}|)^{2}\l... |
3,819,202 | <p>Can anyone explain to solve the identity posted by my friend <span class="math-container">$$2\cos12°= \sqrt{2+{\sqrt{2+\sqrt{2-\sqrt{2-...}}} }}$$</span> which is an infinite nested square roots of 2. <strong>(Pattern <span class="math-container">$++--$</span> repeating infinitely)</strong></p>
<p>Converging to fini... | Sivakumar Krishnamoorthi | 686,991 | <p>Somehow I got the answer from my subsequent post (after a long homework for cyclic infinite nested square roots of 2)</p>
<p>Sivakumar Krishnamoorthi (<a href="https://math.stackexchange.com/users/686991/sivakumar-krishnamoorthi">https://math.stackexchange.com/users/686991/sivakumar-krishnamoorthi</a>), Solving cycl... |
1,436,867 | <p>I don´t know an example wich $ \rho (Ax,Ay)< \rho (x,y) $ $ \forall x\neq y $ is not sufficient for the existence of a fixed point .
can anybody help me? please</p>
| eudes | 265,976 | <p>Even simpler example, although not on whole $\mathbb R$:<br>
$$f\!: \mathbb R\setminus\{0\}\to\mathbb R,\quad f(x) = \frac 12 x.$$
(This is what ThePortakal said in commment.)</p>
|
71,636 | <p>For a self-map $\varphi:X\longrightarrow X$ of a space $X$, many important notions of entropy are defined through a limit of the form $$\lim_{n\rightarrow\infty}\frac{1}{n}\log a_n,$$ where in each case $a_n$ represents some appropriate quantity (see, for example, <a href="https://mathoverflow.net/questions/69218/if... | Vaughn Climenhaga | 5,701 | <p>The limit exists for the first two examples that come to mind, namely topological entropy on the full shift and on certain simple Markov shifts.</p>
<p>If $X \subset \Sigma_d^+ = \{1,2, \dots, d\}^{\mathbb{N}}$ and $\sigma$ is the shift map, then for the topological entropy the quantity $a_n$ denotes the number of ... |
71,636 | <p>For a self-map $\varphi:X\longrightarrow X$ of a space $X$, many important notions of entropy are defined through a limit of the form $$\lim_{n\rightarrow\infty}\frac{1}{n}\log a_n,$$ where in each case $a_n$ represents some appropriate quantity (see, for example, <a href="https://mathoverflow.net/questions/69218/if... | Barbara Schapira | 30,691 | <p>In the case of the geodesic flow acting on the unit tangent bundle of a compact negatively curved manifold, if $a_n$ is the number of closed geodesics of length at most $n$, and $h$ the topological entropy of the geodesic flow, Margulis proved that $a_n$ is equivalent to $\frac{e^{hn}}{hn}$. </p>
<p>The original a... |
197,603 | <p>I'm a newcomer in topology, so I have many things chaotic in my minds, so I hope you could help me.
In order topology, an basis has structure $(a,b)$, right. This is no problem when considering a topology like R, but, what if the number of elements between a and b is finite, so we can write $$(a,b) = [a_1, b_1]$$ wh... | G. R. | 41,674 | <p>In some ordered sets like $[0,1]$, in order to get a base for the order topology, you need consider too the intervals of the form $(\leftarrow,a)$ and $(b,\rightarrow)$.</p>
|
2,038,189 | <p>(Note: I didn't learn how to solve equations the conventional way; instead I was just taught to "move numbers from side to side", inverting the sign or the operation accordingly. I am learning the conventional way though because I think it makes the process of solving equations clearer. That being said, I apologize ... | MattG88 | 159,928 | <p>You can do it by means the second equivalence principle: "multiplying or dividing both sides of an equation by a <strong>non-zero</strong> constant" we obtain an equivalent equation. This is the basis of your calculations in the example $5=\frac{2}{x}$.</p>
|
229,966 | <p>I want to put a title to the plotlegends I am using. I get a solution <a href="https://mathematica.stackexchange.com/questions/201353/title-for-plotlegends">here</a> which says to use <code>PlotLegends -> SwatchLegend[{0, 3.3, 6.7, 10, 13, 17, 20}, LegendLabel -> "mu"]</code>. But I also want to p... | kglr | 125 | <p>You can use <code>PlotLegends - > Placed[labels, Top, Labeled[#, legendlabel, Top] &]</code> .</p>
<p>Using the example input from Tim Laska's answer:</p>
<pre><code>mu = {0, 3.3, 6.7, 10, 13, 17, 20};
fns = Table[n^(1/p), {p, 7}, {n, 10}];
ListLinePlot[fns, PlotMarkers -> Automatic,
PlotLegends -> P... |
275,775 | <p>For the FrameLabel, I have:</p>
<pre><code>Style["\[NumberSign] Humans per city", FontFamily -> "Latin Modern Math"]
</code></pre>
<p>How can I make only "Humans" in the label to be italic?</p>
| lericr | 84,894 | <p>Something like this might work:</p>
<pre><code>Row[
{"\[NumberSign] ", Style["Humans", FontSlant -> Italic], " per city"},
BaseStyle -> {FontFamily -> "Latin Modern Math"}]
</code></pre>
<p>If you don't want to insert your own spaces, you can add a spacer argumen... |
275,775 | <p>For the FrameLabel, I have:</p>
<pre><code>Style["\[NumberSign] Humans per city", FontFamily -> "Latin Modern Math"]
</code></pre>
<p>How can I make only "Humans" in the label to be italic?</p>
| Chris Degnen | 363 | <p>As per <a href="https://mathematica.stackexchange.com/a/11002/363">WReach</a></p>
<pre><code>Style[
StringJoin["\[NumberSign] ",
ToString[Style["Humans", Italic], StandardForm], " per city"],
FontFamily -> "Latin Modern Math"]
</code></pre>
<blockquote>
<p># <em>Humans<... |
1,363,860 | <p>This problem is for my own exploration, not for class. The problem goes as follows:</p>
<blockquote>
<p>There are <span class="math-container">$n$</span> pairs of people with restraining orders against one another. However, all <span class="math-container">$2n$</span> people are friends with the other <span class="m... | Marcus M | 215,322 | <p>This can be done for all $n$. Order the $2n$ people so that person $k$ has a restraining order against person $k + n$ (and vice versa) for every $k$. Then put person $k$ on a circle centered at the origin; more specifically, position them at $r e^{2\pi k i/ 2n}$, with $r$ to be decided later. Then on the opposite... |
3,570,688 | <p>For example, if a ball can be any of 3 colors, then the number of configurations (with repetition of colors) of 2 balls is <span class="math-container">$(3+2-1)C_{2} = 4C_{2} = 6$</span> Why?</p>
| LHF | 744,207 | <p>A similar idea as the accepted answer after rewriting the condition as:</p>
<p><span class="math-container">$$\frac{1}{a+2}+\frac{1}{b+2}+\frac{1}{c+2}=1$$</span></p>
<p>and using Cauchy-Schwarz:</p>
<p><span class="math-container">$$
\begin{aligned}
\frac{a+1}{a+2}&=\frac{1}{b+2}+\frac{1}{c+2}\\
&=5\cdot... |
1,702,616 | <p>I was working on a programming problem to find all 10-digit perfect squares when I started wondering if I could figure out how many perfects squares have exactly N-digits. I believe that I am close to finding a formula, but I am still off by one in some cases.</p>
<p>Current formula where $n$ is the number of digit... | gnasher729 | 137,175 | <p>If you think about it, you should have a formula that says the number of squares is f (n) - f (n-1) for some function f, so that every perfect square is counted exactly once if you calculate the squares from 10^1 to 10^2, from 10^2 to 10^3 and so on. </p>
<p>In your formula, the squares 100, 10,000, 1,000,000 and s... |
115,483 | <p>Edited:</p>
<p>I guess </p>
<p>$$H^2_{(x,y)}\left(\frac{\Bbb Z[x,y]}{(5x+4y)}\right)=0$$</p>
<p>We know that if $\operatorname{Supp} H^i_I(M)\subseteq V(I)\cap \operatorname{Supp}(M)$, then
$$\operatorname{Supp} H^2_{(x,y)}\frac{\Bbb Z[x,y]}{(5x+4y)})\subseteq V((x,y))\cap V((5x+4y))=V((x,y))=\lbrace(x,y) \rbr... | Damian Rössler | 17,308 | <p>I wasn't able to use the method you suggest. Here is a different proof.
Let $M$ be any ${\bf Z}[x,y]$-module. We have (see Brodmann and Sharp, Th. 1.3.8),
$$
H^2_{(x,y)}(M)=\varinjlim_n \operatorname{Ext}^2({\bf Z}[x,y]/(x,y)^n,M)
$$
where the limit is an inductive limit. Now we have
$$
\varinjlim_n \operatorname{... |
4,487,654 | <blockquote>
<p>Demonstrate recursively that</p>
<p><span class="math-container">$$\prod_{k = 0}^\infty (1 + x^{2^k}) = \frac{1}{1-x}$$</span></p>
</blockquote>
<p><strong>My work:</strong></p>
<p>Define</p>
<p><span class="math-container">$$a_n = \prod_{k = 0}^n (1 + x^{2^k}) = (1 + x^{2^n})a_{n - 1} \iff a_n - (1 + x... | Igor Rivin | 109,865 | <p>Write your equation as <span class="math-container">$\frac{\sin(\pi x)}{x} = \frac{1}{n}.$</span> It is clear that any solution is at most <span class="math-container">$n$</span>, since if <span class="math-container">$x > n,$</span> the LHS is smaller than <span class="math-container">$1/n.$</span> Thus, the lim... |
576,379 | <p>I know how to show that $f(x)=x^2$ is uniformly continuous, but I am confused when it is $x^2 +x$</p>
| user1337 | 62,839 | <p><strong>Hint:</strong></p>
<p>The collection of all uniformly continuous functions in a given interval form a vector space.</p>
|
1,506,741 | <p>Here's an exercise: </p>
<p>Let $(X,M,\mu)$ be a measure space with $\mu(X)<\infty$. Let $N\subseteq M$ be a $\sigma$-algebra. If $f\geq 0$ is $M$-measurable and $\mu$-integrable, then there exists some $N$-measurable and $\mu$-integrable function $g\geq 0$ such that
$$
\int_E g \, d\mu=\int_E f \, d\mu,~~~~E\i... | Michael Hardy | 11,667 | <p>For $E \in N$, let
$$
\lambda(E) = \int_E f\, d\mu.
$$
Then the measure $\lambda$ on $N$ is absolutely continuous to the measure $\mu$ on $N$ (i.e. the restriction to $N$ of the measure $\mu$ on $M$). Therefore by the Radon–Nikodym theorem, there is an $N$-measurable function $g=\dfrac{d\lambda}{d\mu}$ such that fo... |
1,506,741 | <p>Here's an exercise: </p>
<p>Let $(X,M,\mu)$ be a measure space with $\mu(X)<\infty$. Let $N\subseteq M$ be a $\sigma$-algebra. If $f\geq 0$ is $M$-measurable and $\mu$-integrable, then there exists some $N$-measurable and $\mu$-integrable function $g\geq 0$ such that
$$
\int_E g \, d\mu=\int_E f \, d\mu,~~~~E\i... | David C. Ullrich | 248,223 | <p><strong>NEW:</strong> In fact the result <em>always</em> fails unless the restriction $\mu|_N$ is $\sigma$-finite. See below.</p>
<p>First the simple counterexample showing that the result <em>may</em> fail without some finiteness hypothesis:</p>
<p>Without the assumption that $\mu(X)<\infty$ the result is fals... |
134,937 | <p>Let $p \equiv q \equiv 3 \pmod 4$ for distinct odd primes $p$ and $q$. Show that $x^2 - qy^2 = p$ has no integer solutions $x,y$.</p>
<p>My solution is as follows.</p>
<p>Firstly we know that as $p \equiv q \pmod 4$ then $\big(\frac{p}{q}\big) = -\big(\frac{q}{p}\big)$</p>
<p>Assume that a solution $(x,y)$ does e... | Kerry | 7,887 | <p>You probably tried too hard on this. We know $x^{2},y^{2}\equiv 1/0\pmod{4}$. So $x^{2}$ choices are $0,1$. We have $0-1=3$, $1-2=3$. $qy^{2}$ cannot be $1\pmod{4}$. Thus $x$ must be odd and $y$ be even. But if that is so $qy^{2}$ must be $0\pmod{4}$, which contradicts with our assumption $qy^{2}\equiv 2\pmod{4}$. T... |
70,946 | <p>I'm an REU student who has just recently been thrown into a dynamical system problem without basically any background in the subject. My project advisor has told me that I should represent regions of my dynamical system by letters and look at the sequence of letters formed by the trajectory of a point under the iter... | Anthony Quas | 11,054 | <p>I think symbolic dynamics is the study of what you get <em>after</em> you've introduced your partition and coded points by their itinerary. Your question is essentially "when is the symbolic dynamics a faithful representation of the original dynamical system?".</p>
<p>As Vaughn says, the thing you're looking for is... |
2,353,190 | <p>Let $f(x)=\dfrac{1+x}{1-x}$ The nth derivative of f is equal to:</p>
<ol>
<li>$\dfrac{2n}{(1-x)^{n+1}} $</li>
<li>$\dfrac{2(n!)}{(1-x)^{2n}} $</li>
<li>$\dfrac{2(n!)}{(1-x)^{n+1}} $</li>
</ol>
<p>by Leibniz formula </p>
<p>$$ {\displaystyle \left( \dfrac{1+x}{1-x}\right)^{(n)}=\sum _{k=0}^{n}{\binom {n}{k}}\ (1+x... | egreg | 62,967 | <p>You can do it with Leibniz's formula (not that it's easier than without it); just consider that
$$
(1+x)^{(k)}=
\begin{cases}
1+x & k=0 \\
1 & k=1 \\
0 & k>1
\end{cases}
$$
so the formula gives
$$
\left( \frac{1+x}{1-x}\right)^{\!(n)}=
(1+x)\left(\dfrac{1}{1-x}\right)^{\!(n)}+
n\left(\frac{1}{1-x}\rig... |
1,392,661 | <p>For a National Board Exam Review: </p>
<blockquote>
<p>Find the equation of the perpendicular bisector of the line joining
(4,0) and (-6, -3)</p>
</blockquote>
<p>Answer is 20x + 6y + 29 = 0</p>
<p>I dont know where I went wrong. This is supposed to be very easy:</p>
<p>Find slope between two points:</p>
<p... | pancini | 252,495 | <p>lol. a lot of confusion on this thread. When you are computing the midpoints, what you actually do is find the distance between them and divide by two. This isn't the coordinate of the midpoint; this is just the distance form one end to the midpoint.</p>
<p>What you meant to do is take the average of the x and y <e... |
1,392,661 | <p>For a National Board Exam Review: </p>
<blockquote>
<p>Find the equation of the perpendicular bisector of the line joining
(4,0) and (-6, -3)</p>
</blockquote>
<p>Answer is 20x + 6y + 29 = 0</p>
<p>I dont know where I went wrong. This is supposed to be very easy:</p>
<p>Find slope between two points:</p>
<p... | bubba | 31,744 | <p>Just for variety, a different approach:</p>
<p>If a point $(x,y)$ is on the perpendicular bisector, then it is equidistant from $(4,0)$ and $(-6,-3)$, so
$$
(x-4)^2 + (y-0)^2 = (x+6)^2 + (y+3)^2
$$
Multiplying out, we get
$$
x^2 -8x +16 + y^2 = x^2 +12x +36 \; + \; y^2 +6y + 9
$$
So
$$
20x + 6y + 29 = 0
$$</p>
|
3,960,404 | <p>Let <span class="math-container">$T$</span> be a tree.</p>
<p>Suppose that <span class="math-container">$T$</span> doesn’t have a perfect matching and let <span class="math-container">$M$</span> be a matching of <span class="math-container">$T$</span>, <span class="math-container">$|M| = k$</span>. Prove
that there ... | Brian M. Scott | 12,042 | <p>HINT: If <span class="math-container">$M$</span> exposes some pendant vertex, we’re done, so suppose that <span class="math-container">$M$</span> covers every pendant vertex. Root <span class="math-container">$T$</span> at a non-pendant vertex <span class="math-container">$v$</span>, so that the pendant vertices are... |
661,026 | <p>prove or disprove this
$$\sum_{k=0}^{n}\binom{n}{k}^3\approx\dfrac{2}{\pi\sqrt{3}n}\cdot 8^n,n\to\infty?$$</p>
<p>this problem is from when Find this limit
$$\lim_{n\to\infty}\dfrac{\displaystyle\sum_{k=0}^{n}\binom{n}{k}^3}{\displaystyle\sum_{k=0}^{n+1}\binom{n+1}{k}^3}=\dfrac{1}{8}?$$</p>
<p>first,follow I c... | vonbrand | 43,946 | <p>This is a <a href="http://en.wikipedia.org/wiki/Quartic_function">quartic</a>, its discriminant tells all you need.</p>
|
661,026 | <p>prove or disprove this
$$\sum_{k=0}^{n}\binom{n}{k}^3\approx\dfrac{2}{\pi\sqrt{3}n}\cdot 8^n,n\to\infty?$$</p>
<p>this problem is from when Find this limit
$$\lim_{n\to\infty}\dfrac{\displaystyle\sum_{k=0}^{n}\binom{n}{k}^3}{\displaystyle\sum_{k=0}^{n+1}\binom{n+1}{k}^3}=\dfrac{1}{8}?$$</p>
<p>first,follow I c... | Steve Kass | 60,500 | <p>Sketch $\frac{1}{2}x^4 - x^3 -x + 100$. There are two changes in concavity between $x=-2$ and $x=3$, so the graph can't "change direction" outside those values, and the minimum must be between them. Direct calculation shows that within that interval, the function exceeds 50 at each integer-valued $x$.</p>
<p>For th... |
794,736 | <blockquote>
<p>Let $60$ students and $10$ teachers. How many arrangements are there, such that, between two teachers must be exactly $6$ students? </p>
</blockquote>
<p>I know that there are $10!$ permutations for the teachers, and there are $54$ places between them for the students. Nothing said about the edges. ... | Community | -1 | <p>The proof is essentially correct, but a bit too brief. The part "$g(x)=g(1)$ for all $x=0$" is written nicely. For $x<0$, I would write instead: $g(x)=g(x^2)=g(1)$, where the second equality holds since $x^2>0$. And as Jean Claude Arbaut added, the proof should conclude with $g(0)=\lim_{x\to 0}g(x) = g(1)$. ... |
1,115,222 | <blockquote>
<p>Suppose <span class="math-container">$f$</span> is a continuous, strictly increasing function defined on a closed interval <span class="math-container">$[a,b]$</span> such that <span class="math-container">$f^{-1}$</span> is the inverse function of <span class="math-container">$f$</span>. Prove that,
... | Tom-Tom | 116,182 | <p>For completeness, the case where <span class="math-container">$f$</span> is differentiable is handled this way
<span class="math-container">$$\begin{split} I&=\int_a^bf(x)\mathrm dx+\int_{f(a)}^{f(b)}f^{-1}(x)\mathrm dx\\
& = \int_a^bf(x)\mathrm dx+\int_a^bf^{-1}\big(f(u)\big)f'(u)\mathrm du \quad
\text{(by ... |
75,875 | <p>I am asking in the sense of isometry groups of a manifold. SU(3) is the group of isometries of CP2, and SO(5) is the group of isometries of the 4-sphere. Now, it happens that both manifolds are related by Arnold-Kuiper-Massey theorem: $\mathbb{CP}^2/conj \approx S^4$; one is a branched covering of the other, the quo... | José Figueroa-O'Farrill | 394 | <p>Why would you think that they are related?</p>
<p>The map $\mathbb{CP}^2/\text{conjugation} \to S^4$ is only $SO(3)$-equivariant, where $SO(3) \subset SU(3)$ consists of the real matrices and $SO(3) \subset SO(5)$ is the maximal subgroup acting irreducibly on the 5-dimensional vector representation of $SO(5)$.</p>
... |
2,324,850 | <p>How to find the shortest distance from line to parabola?</p>
<p>parabola: $$2x^2-4xy+2y^2-x-y=0$$and the line is: $$9x-7y+16=0$$
Already tried use this formula for distance:
$$\frac{|ax_{0}+by_{0}+c|}{\sqrt{a^2+b^2}}$$</p>
| Steven Alexis Gregory | 75,410 | <p>$$2x^2-4xy+2y^2-x-y=0$$</p>
<p>At a point $(x_0, y_0)$ on the parabola, the gradient is
$\nabla f(x_0,y_0) = (4x_0-4y_0-1, -4x_0+4y_0-1)$</p>
<p>The direction of a normal to the line $9x-7y+16=0$ is $(7,9)$. So we must have</p>
<p>\begin{align}
(4x_0-4y_0-1, -4x_0+4y_0-1) \circ (7,9) &= 0 \\
-8 x_0 + 8... |
3,979,674 | <p>Let
<span class="math-container">$$I=\int\frac{dx}{\sqrt{ax^2+bx+c}}$$</span>
I know this can be either
<span class="math-container">$$\displaystyle I=\frac{1}{\sqrt{a}}\ln\left({2\sqrt{a}\sqrt{ax^2+bx+c}+2ax+b}\right)+C$$</span>
<span class="math-container">$$\displaystyle I=-\frac{1}{\sqrt{-a}}\arcsin{\left(\frac{... | Leonard Neon | 818,617 | <p><span class="math-container">$$
\displaystyle I=\frac{1}{\sqrt{a}}\ln\left({2\sqrt{a}\sqrt{ax^2+bx+c}+2ax+b}\right)+C
\label{eq1} \tag{eq1}\\
$$</span></p>
<p><span class="math-container">$$
\displaystyle I=-\frac{1}{\sqrt{-a}}\arcsin{\left(\frac{2ax+b}{\sqrt{b^2-4ac}}\right)}+C
\label{eq2} \tag{eq2}\\
$$</span></p>... |
217,291 | <p>I am trying to recreate the following image in latex (pgfplots), but in order to do so I need to figure out the mathematical expressions for the functions</p>
<p><img src="https://i.stack.imgur.com/jYGNP.png" alt="wavepacket"></p>
<p>So far I am sure that the gray line is $\sin x$, and that
the redline is some ver... | Neal | 20,569 | <p>Here's a nice theorem illustrating how some basic algebraic concepts are useful in seemingly unrelated areas of mathematics. The <a href="http://en.wikipedia.org/wiki/Uniformization_theorem" rel="nofollow">uniformization theorem for surfaces</a> states that</p>
<blockquote>
<p>Every orientable topological surfac... |
1,077,504 | <p>Evaluate:</p>
<p>$$\int_{0}^{\infty} \frac{1}{x^6 + 1} \,\mathrm dx$$</p>
<p>Without <strong>the use of complex-analysis.</strong></p>
<p>With complex analysis it is a very simple problem, how can this be done WITHOUT complex analysis?</p>
| Aditya Hase | 190,645 | <p>Let $\displaystyle \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\mathcal I=\int_0^\infty\frac{1}{1+x^6} \,\mathrm dx
$</p>
<p>$$\begin{align}
I&=\frac{1}{2}\left[ \int_0^\infty \frac{(1-x^2+x^4)+x^2+(1-x^4)}{(1+x^2)(1-x^2+x^4)} \,\mathrm dx \right]\tag{1}\\
&=
\frac{1}{2}\left[\int_0^\infty... |
1,570,044 | <p>How many arrangements of banana such that the "b" occurs before any of the "a's"?</p>
<p>This is more an inquiry into what I did wrong in my counting. I came up with a solution of: $$\binom{3}{1} \binom{5}{3}$$ where i did C(3,1) to account for the 3 possible places the "b" could go and C(5,3) to choose the posi... | joriki | 6,622 | <p>There are only $5$ slots for the $3$ a's if the 'b' is in the beginning. If it's in second place, there are $4$, and if it's in third place, there are $3$, so the total is</p>
<p>$$
\binom53+\binom43+\binom33=10+4+1=15=\binom64\;,
$$</p>
<p>which is the number of ways to choose $4$ out of $6$ slots for the letters... |
762,651 | <p>I have to prove that "any straight line $\alpha$ contained on a surface $S$ is an asymptotic curve and geodesic (modulo parametrization) of that surface $S$". Can I have hints at tackling this problem? It seems so general that I am not sure even how to formulate it well, let alone prove it. Intuitively, I imagine ... | Community | -1 | <p>Short answer: We can't simply square both sides because that's exactly what we're trying to prove: $$0 < a < b \implies a^2 < b^2$$</p>
<p>More somewhat related details: I think it may be a common misconception that simply squaring both sides of an inequality is ok because we can do it indiscriminately w... |
410,105 | <p>I have this recurrence relation:</p>
<p>$$
R(1)=1, RE(1)=0, EE(1)=0$$</p>
<p>$$a(n)=R(n) + RE(n)$$</p>
<p>$$R(n)=EE(n-1)+RE(n-1),$$$$ RE(n)=R(n-1),$$$$ EE(n)=RE(n-1)
$$</p>
<p>How do I get $a(15)$?
What kind of method do I use?</p>
| N. S. | 9,176 | <p>$$R(n)=EE(n-1)+RE(n-1)=RE(n-2)+RE(n-1)=R(n-3)+R(n-2) \,.$$
$$RE(n)=R(n-1)=EE(n-2)+RE(n-2)=RE(n-3)+RE(n-2) \,.$$</p>
<p>Adding them you get</p>
<p>$$a(n)=a(n-3)+a(n-2) \,.$$</p>
<p>Solve it!</p>
|
4,402,262 | <p>A class of 24 pupils consists of 11 girls and 13 boys. To form the class committee, four of the pupils are chosen at random as "Chairperson", "Vice-Chairperson", "Treasurer", and "Secretary". Find the number of ways the committee can be formed if<br>
(i) the committee consists... | Acccumulation | 476,070 | <p>It wasn't clear at first that these are three different questions, rather than three conditions for the answer. Your teacher should write questions more clearly.</p>
<p>(i) There are <span class="math-container">$24P4$</span> ways to choose <span class="math-container">$4$</span> people from a set of <span class="ma... |
292,831 | <p>Usually the question whether the <a href="https://en.wikipedia.org/wiki/Diamond_principle" rel="noreferrer">diamond principle</a> $\diamondsuit(\kappa)$ holds for some large cardinal $\kappa$ only concerns large cardinal notions of very low consistency (among the weakly compacts). Partly since it <em>does</em> hold ... | Yair Hayut | 41,953 | <p>This is a partial answer. I will show that if $\delta$ is Woodin then $\diamondsuit_\delta$ holds. </p>
<p><strong>Claim:</strong> Any Woodin cardinal is subtle.</p>
<p><strong>Proof:</strong> Let $\delta$ be a Woodin cardinal. Let $\vec{A} = \langle A_\alpha \mid \alpha < \delta\rangle$ be a sequence a sets, $... |
241,612 | <blockquote>
<p>Find all eigenvalues and eigenvectors:</p>
<p>a.) $\pmatrix{i&1\\0&-1+i}$</p>
<p>b.) $\pmatrix{\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta}$</p>
</blockquote>
<p>For a I got:
$$\operatorname{det} \pmatrix{i-\lambda&1\\0&-1+i-\lambda}= \lambda^{2} - 2\lambda ... | dantopa | 206,581 | <p><strong>Basic tools</strong></p>
<p>For $2 \times 2$ matrices the characteristic polynomial is:
$$
p(\lambda) = \lambda^{2} - \lambda\, \text{tr }\mathbf{A} + \det \mathbf{A}
$$</p>
<p>The roots of this function are the eigenvalues, $\lambda_{k}$, k=1,2$.</p>
<p>The eigenvectors solve the eigenvalue equation
$$
... |
1,218,582 | <p>I was presented with the function $max (|x|,|y|)$ which should output a maximum value of given two.... I can only suppose this one creates some body in $\mathbb{R}^3$ but how do you sketch it and what does it mean in $\mathbb{R}^3$? for that matter in $\mathbb{R}^2$ I cant really imagine it also.. </p>
| Hagen von Eitzen | 39,174 | <p>It's a top-down pyramid, the faces lying in the four planes $z=x$, $z=-x$, $z=y$, $z=-y$.</p>
|
1,215,542 | <p>I am not sure why in my textbook there is a long proof for this because on the pages before this was prooven:</p>
<p>Let $S = \{ x \mid a \leq x \leq b \hbox{ and } x \in \mathbb{R} \}$. Then $\sup S = b$ must hold.</p>
<p>EDIT: also $\inf S=a$ follows. But the proof for $\sup S$ does not depend on there being a l... | Regret | 184,794 | <p>The problem is that $S+T$ is not defined as</p>
<p>$$S+T=\{x\in\Bbb R\mid\inf S + \inf T\le x\le\sup S+\sup T \}$$</p>
<p>It is defined as</p>
<p>$$S+T=\{s+t\mid s\in S,t\in T\}$$</p>
<p>So you can not use your previous result that if $X=\{x\in\Bbb R\mid a\le x \le b\}$ then $\sup X=b$. While it is true that $s+... |
1,215,542 | <p>I am not sure why in my textbook there is a long proof for this because on the pages before this was prooven:</p>
<p>Let $S = \{ x \mid a \leq x \leq b \hbox{ and } x \in \mathbb{R} \}$. Then $\sup S = b$ must hold.</p>
<p>EDIT: also $\inf S=a$ follows. But the proof for $\sup S$ does not depend on there being a l... | egreg | 62,967 | <p>If $S=\{x\in\mathbb{R}:a\le x\le b\}$, then it's of course true that $b=\sup S$: indeed, $b\le x$, for all $x\in S$ and $b\in S$, so $b$ is actually the maximum of $S$.</p>
<p>When you have two sets $S$ and $T$ (subsets of $\mathbb{R}$), then
$$
S+T=\{x+y:x\in S,y\in T\}
$$
is the set of all numbers that can be exp... |
4,195,399 | <p>Given that <span class="math-container">$a,b,c > 0$</span> are real numbers such that <span class="math-container">$$\frac{a}{b+c+1}+\frac{b}{c+a+1}+\frac{c}{a+b+1}\le 1,$$</span> prove that <span class="math-container">$$\frac{1}{b+c+1}+\frac{1}{c+a+1}+\frac{1}{a+b+1}\ge 1.$$</span></p>
<hr />
<p>I first rewrote... | achille hui | 59,379 | <p>Since the two inequalities are completely symmetric in <span class="math-container">$a,b,c$</span>. WLOG, we only need to study the case <span class="math-container">$a \ge b \ge c$</span>.</p>
<p>Let <span class="math-container">$\Lambda = a + b + c + 1$</span>. The two inequalities can be rewritten
as</p>
<p><spa... |
21,262 | <p><strong>Bug introduced in 9.0 and fixed in 11.1</strong></p>
<hr>
<p><code>NDSolve</code> in Mathematica 9.0.0 (MacOS) is behaving strangely with a piecewise right hand side. The following code (a simplified version of my real problem):</p>
<pre><code>sol = NDSolve[{x'[t] ==
Piecewise[{{2, 0 <= Mod[t... | Chris K | 6,358 | <p>Following a lead from Albert Retey, I found an <code>NDSolve</code> option that fixes this problem:</p>
<pre><code>sol = NDSolve[{x'[t] ==
Piecewise[{{2, 0 <= Mod[t, 1] < 0.5},
{-1, 0.5 <= Mod[t, 1] < 1}}
], x[0] == 0}, x, {t, 0, 1}, Method -> {"DiscontinuityProcessing" ->... |
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