qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,034,697 | <p>The exercise goes like this:</p>
<p>-Let $W= {(x,y,z)|2x+3y-z=0}$ Then $W\subseteq\mathbb{R}^3$, find the dimension of $W$.</p>
<p>-Find the dimension $[\mathbb{R}^3|W]$</p>
<p>This was a problem from my algebra exam, it was a team exam and this problem was solved by another member of the team (we...he had it rig... | Learnmore | 294,365 | <p>I think an easy way to solve this is as follows:</p>
<p>dim $(\mathbb R^3)/W$=dim $\mathbb R^3$-dim $W$</p>
<p>Now let $(a,b,c)\in W$ then $2a+3b-c=0$
so $c=2a+3b$</p>
<p>so a basis for $W$ is $\{(1,0,2)^t,(0,1,3)^t\}$ so dim$W$=2</p>
<p>So you get your answer.</p>
<p>Another way out is try the linear mapping ... |
244,433 | <p>I have a list:</p>
<pre><code>data = {{2.*10^-9, 0.0025}, {4.*10^-9, 0.0025}, {6.*10^-9, 0.0025}, {8.*10^-9, 0.0025}, {1.*10^-8, 0.0025}, {7.*10^-9, 0.0023}, {3.*10^-9, 0.0025},...}
</code></pre>
<p>And I wanted to remove every third pair and get</p>
<pre><code> newdata = {{2.*10^-9, 0.0025}, {4.*10^-9, 0.0025}, {8.... | xzczd | 1,871 | <pre><code>newdata = data;
newdata[[3 ;; ;; 3]] = Nothing;
newdata
</code></pre>
<p>If it's OK to overwrite <code>data</code>, then simply</p>
<pre><code>data[[3 ;; ;; 3]] = Nothing;
</code></pre>
|
704,917 | <p>I need your help evaluating this integral:
<span class="math-container">$$I=\int_0^\infty F(x)\,F\left(x\,\sqrt2\right)\frac{e^{-x^2}}{x^2} \, dx,\tag1$$</span>
where <span class="math-container">$F(x)$</span> represents <a href="http://mathworld.wolfram.com/DawsonsIntegral.html" rel="nofollow noreferrer">Dawson's f... | Cleo | 97,378 | <p>$$I=\frac{\pi^{3/2}}8\left(\sqrt2-4\right)+\frac{3\,\pi^{1/2}}2\arctan\sqrt2$$</p>
|
4,459,439 | <p>Suppose <span class="math-container">$G$</span> is an abelian finite group, and the number of order-2 elements in <span class="math-container">$G$</span> is denoted by <span class="math-container">$N$</span>.</p>
<p>I have found that <span class="math-container">$N= 2^n-1$</span> for some <span class="math-container... | Berci | 41,488 | <p>Yes, your argument is fine.</p>
<p>Note that the given subgroup of <span class="math-container">$N+1$</span> elements naturally carries a <span class="math-container">$\Bbb Z/2\Bbb Z$</span> vector space structure, so indeed its cardinality must be <span class="math-container">$2^n$</span>.</p>
<p>Since such a vecto... |
39,684 | <p>In order to have a good view of the whole mathematical landscape one might want to know a deep theorem from the main subjects (I think my view is too narrow so I want to extend it).</p>
<p>For example in <em>natural number theory</em> it is good to know quadratic reciprocity and in <em>linear algebra</em> it's good... | Zev Chonoles | 264 | <p>Differential Geometry: the <a href="http://en.wikipedia.org/wiki/Gauss%E2%80%93Bonnet_theorem" rel="nofollow">Gauss-Bonnet theorem</a>. </p>
<p>I took a one-semester intro course on differential geometry class and we got to this towards the end of the semester, so I feel that a couple of months is an appropriate ti... |
39,684 | <p>In order to have a good view of the whole mathematical landscape one might want to know a deep theorem from the main subjects (I think my view is too narrow so I want to extend it).</p>
<p>For example in <em>natural number theory</em> it is good to know quadratic reciprocity and in <em>linear algebra</em> it's good... | Community | -1 | <p>In Complex Analysis the <a href="http://en.wikipedia.org/wiki/Riemann_mapping_theorem" rel="nofollow">Riemann Mapping Theorem</a>.</p>
|
39,684 | <p>In order to have a good view of the whole mathematical landscape one might want to know a deep theorem from the main subjects (I think my view is too narrow so I want to extend it).</p>
<p>For example in <em>natural number theory</em> it is good to know quadratic reciprocity and in <em>linear algebra</em> it's good... | Community | -1 | <ul>
<li>Primary decomposition theorem in linear algebra.</li>
</ul>
|
2,530,458 | <p>Find Range of $$ y =\frac{x}{(x-2)(x+1)} $$</p>
<p>Why is the range all real numbers ? </p>
<p>the denominator cannot be $0$ Hence isn't range suppose to be $y$ not equals to $0$ ?</p>
| farruhota | 425,072 | <p>Alternatively: the function can be expressed as the sum:
$$y =\frac{x}{(x-2)(x+1)}=\frac13\left(\frac{2}{x-2}+\frac{1}{x+1}\right)=\frac13\left(g(x)+h(x)\right)$$
The functions $g(x)$ and $h(x)$ are hyperbolas with the range $g\ne0$ and $h\ne0$. And the function $y$ has the range in $\mathbb{R}$, in particular, $y(0... |
3,009,387 | <p>I'm asking the following: it is true that if <span class="math-container">$K$</span> is a normal subgroup of <span class="math-container">$G$</span> and <span class="math-container">$K\leq H\leq G$</span> then <span class="math-container">$K$</span> is normal in <span class="math-container">$H$</span>? I tried to pr... | Siong Thye Goh | 306,553 | <p>For your way <span class="math-container">$1$</span>, check the computation of your denominator, it should give you <span class="math-container">$0$</span> again.</p>
<p>For your way <span class="math-container">$2$</span>, check your factorization in your denominator as well.</p>
<p>Use L'hopital's rule:</p>
<p>... |
3,009,387 | <p>I'm asking the following: it is true that if <span class="math-container">$K$</span> is a normal subgroup of <span class="math-container">$G$</span> and <span class="math-container">$K\leq H\leq G$</span> then <span class="math-container">$K$</span> is normal in <span class="math-container">$H$</span>? I tried to pr... | user | 505,767 | <p>As an alternative by <span class="math-container">$y=x+5 \to 0$</span></p>
<p><span class="math-container">$$\lim_{x\rightarrow -5} \frac{2x^2-50}{2x^2+3x-35}=\lim_{y\rightarrow 0} \frac{2(y-5)^2-50}{2(y-5)^2+3(y-5)-35}=\lim_{y\rightarrow 0} \frac{2y^2-20y}{2y^2-17y}=\lim_{y\rightarrow 0} \frac{2y-20}{2y-17}$$</spa... |
1,156,874 | <p>How to show that $\mathbb{Z}[i]/I$ is a finite field whenever $I$ is a prime ideal? Is it possible to find the cardinality of $\mathbb{Z}[i]/I$ as well?</p>
<p>I know how to show that it is an integral domain, because that follows very quickly.</p>
| Lubin | 17,760 | <p>Another approach, especially if you’re only interested in counting $\Bbb Z[i]/I\,$: There’s a theorem in algebraic number theory that if $I$ is a nonzero principal ideal of the ring of integers $R$ in an algebraic number field $K$, say with $I=(\xi)$, then the cardinality of $R/I$ is $\big|\text N(\xi)\big|$, where ... |
130,564 | <p>Hi, everyone.</p>
<p>I am looking for some references for period matrix of abelian variety over arbitrary field, if you know, could you please tell me?</p>
<p>For period matrix of abelian varieties, I means that if $A$ is an abelian variety over complex number field, $A \cong V/\Gamma$. If we chose a basis of $V$,... | anon | 22,503 | <p>For an abelian variety $A$ over a subfield $k$ of the complex numbers, you have the first de Rham cohomology group of $A$ over $k$, which is a $k$-vector space, and the first singular cohomology group of $A/\mathbb{C}$, which is a $\mathbb{Q}$-vector space. When tensored up to $\mathbb{C}$ the two vector spaces are ... |
749,926 | <p>I have a group of 10 players and I want to form two groups with them.Each group must have atleast one member.In how many ways can I do it?</p>
| Jean-Sébastien | 31,493 | <p>Take one of the guy, say, Joe. We will form Joe's team. Either of the $9$ other players can be or not be in Joe's team, so there are $2^9$ choices.</p>
<p>But this include the case where all $10$ players are on the same team, so we remove $1$ from that, leading to $2^9-1$.</p>
<p><strong>Added</strong> For complet... |
1,485,310 | <blockquote>
<p>Write a formula/formulae for the following sequence:</p>
<p>b). 1,3,6,10,15,...</p>
</blockquote>
<p>I am not getting any pattern here, from which to derive a formula.
This sequence does not look like the examples I could solve: like</p>
<blockquote>
<p>a) 1,0,1,0,1...</p>
</blockquote>
<p>where I got t... | Amey Deshpande | 210,002 | <p>For (b) you can observe the difference between next and previous terms is in A.P.</p>
<p>Thus you can define it recursively as $s_1 = 1$, $s_{n+1}=s_n + (n+1)$.</p>
|
2,064,984 | <p>I am learning about the student t-test. </p>
<p>I am struggling, however, to be given a reasonable explanation why the standard deviation of the standard normal distribution curve is 1. </p>
<p>It says "The Standard Normal Variable is denoted Z and has mean 0 and S.D 1..."</p>
<p>"... this is written as Z ~ N(0,1... | Logician6 | 306,688 | <p>The "standard normal distribution" has mean 0 and standard deviation 1 because that is in fact the definition of the "standard normal distribution". To understand this, what I think you need to understand is what a "normal distribution" in general is.</p>
<p>A "normal distribution" is a distribution of "relative fr... |
644,935 | <p>I'm having trouble integrating $3^x$ using the $px + q$ rule. Can some please walk me through this?</p>
<p>Thanks</p>
| amWhy | 9,003 | <p>We can simply take as primitive $$\int a^x \,\text{d}x = \dfrac{a^x}{\log a} + C$$</p>
<p>Suggestion: Verify for yourself that $$\frac{\text{d}}{dx}\left(\frac {a^x}{\log a} + C\right) = a^x$$</p>
<p>So $$\int 3^x \,\text{d}x = \dfrac{3^x}{\log 3} + C.$$</p>
|
90,656 | <p>In the introduction to 'A convenient setting for Global Analysis', Michor & Kriegl make this claim: "The study of Banach manifolds per se is not very interesting, since they turnout to be open subsets of the modeling space for many modeling spaces."</p>
<p>But finite-dimensional manifolds are found to be intere... | Georges Elencwajg | 450 | <p>In his <a href="http://archive.numdam.org/ARCHIVE/AIF/AIF_1966__16_1/AIF_1966__16_1_1_0/AIF_1966__16_1_1_0.pdf">remarkable thesis</a> Douady proved that, given a compact complex analytic space $X$, the set $H(X)$ of analytic subspaces of $X$ has itself a natural structure of analytic space .<br>
If $X=\mathbb P^n(\... |
1,339,649 | <p>Summation convention holds. If $\frac{\partial}{\partial t}g_{ij}=\frac{2}{n}rg_{ij}-2R_{ij}$, then ,I compute:
$$
\frac{1}{2}g^{ij}\frac{\partial}{\partial t}g_{ij}=\frac{1}{2}g^{ij}(\frac{2}{n}rg_{ij}-2R_{ij})=\frac{1}{n}r(\sum\limits_i\sum\limits_jg^{ij}g_{ij})-g^{ij}R_{ij}=nr-R
$$</p>
<p>But on the Hamilton's ... | Community | -1 | <p>Note: </p>
<p>$$\sum_{i,j=1}^n g^{ij} g_{ij} = \text{tr} (g^{-1}g) = \text{tr} (I_n) = n.$$</p>
|
3,219,428 | <p>Sorry for the strange title, as I don't really know the proper terminology.</p>
<p>I need a formula that returns 1 if the supplied value is anything from 10 to 99, returns 10 if the value is anything from 100 to 999, returns 100 if the value is anything from 1000 to 9999, and so on.</p>
<p>I will be translating th... | Nilotpal Sinha | 60,930 | <p><span class="math-container">$f(x) = 10^{[\log_{10} x]-1}$</span> where <span class="math-container">$[x]$</span> denotes floor</p>
|
2,031,964 | <p>I am required to prove that the following series
$$a_1=0, a_{n+1}=(a_n+1)/3, n \in N$$
is bounded from above and is monotonously increasing through induction and calculate its limit. Proving that it's monotonously increasing was simple enough, but I don't quite understand how I can prove that it's bounded from above... | mfl | 148,513 | <p>We have </p>
<p>$$\begin{cases}a_0&=0\\a_1&=\frac13 \\a_2& =\frac 13+\frac{1}{3^2}\\a_3& =\frac 13+\frac{1}{3^2}+\frac{1}{3^3} \end{cases}$$ What does this sugest? That $$a_n=\sum_{k=1}^n\frac{1}{3^k}=\dfrac{\frac13-\frac{1}{3^{n+1}}}{\frac23}.$$ Show this by induction and get that $a_n<\frac 12,... |
3,043,598 | <p>I have seen a procedure to calculate <span class="math-container">$A^{100}B$</span> like products without actually multiplying where <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are matrices. But the procedure will work only if <span class="math-container">$A$</span> is diagona... | Gratus | 544,878 | <p>I think the fastest way in general is trying to calculate <span class="math-container">$A^{100}$</span> faster.
Have you heard of fast exponentiation algorithm?</p>
<p>When you have to calculate <span class="math-container">$A^{100}$</span>,
You can calculate <span class="math-container">$A^2, A^4, A^8....A^{64}$<... |
399,804 | <p>The Question was:</p>
<blockquote>
<p>Express $2\cos{X} = \sin{X}$ in terms of $\sin{X}$ only.</p>
</blockquote>
<p>I have had dealings with similar problems but for some reason, due to I believe a minor oversight, I am terribly vexed.</p>
| Julien | 38,053 | <p><strong>Note:</strong> too long for a comment...</p>
<p>Squaring to force the use of $\cos^2x+\sin^2x=1$ results in an equation which is not equivalent to the original equation. This actually creates another countable set of solutions. </p>
<p>First note that, $\cos \theta=0$ does not occur when the equation is fu... |
5,896 | <p>$\sum_{n=1}^{\infty} \frac{\varphi(n)}{n}$ where $\varphi(n)$ is 1 if the variable $\text n$ has the number $\text 7$ in its typical base-$\text10$ representation, and $\text0$ otherwise.</p>
<p>I am supposed to find out if this series converges or diverges. I think it diverges, and here is why.</p>
<p>We can see ... | Michael Lugo | 173 | <p>Here's another proof that your sum diverges. </p>
<p>Consider the sum $\sum_{n-1}^\infty (1-\phi(n))/n$. This is the sum of the reciprocals of integers which <i>don't</i> have a 7 in their decimal expansion.</p>
<p>The number of integers $n$ with $1-\phi(n)=1$ and $1 \le n < 10^k$ is $9^k - 1$. (We can choose ... |
1,220,790 | <p>Consider the black scholes equation, </p>
<p>$$
\frac{\partial V}{\partial t } + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2 } + ( r-q )S\frac{\partial V}{\partial S }-rV =0
$$</p>
<p>How do I show that if $V( S, t)$ is a solution, then $S(\frac{\partial V}{\partial S })$ is also a solution?</p>
<p>I... | Raskolnikov | 3,567 | <p>First note that</p>
<p>$$\frac{\partial}{\partial S}\left(S\frac{\partial V}{\partial S}\right) = \frac{\partial V}{\partial S} + S \frac{\partial^2 V}{\partial S^2}$$</p>
<p>and</p>
<p>$$\frac{\partial^2}{\partial S^2}\left(S\frac{\partial V}{\partial S}\right) = 2\frac{\partial^2 V}{\partial S^2} + S \frac{\par... |
506,720 | <p>Hi how can I find the dimension of a vector space? For example :
$V = \mathbb{C} , F = \mathbb{Q}$
what is the dimension of $V$ over $\mathbb{Q}$?</p>
| John Gowers | 26,267 | <p>For this particular example, there are two ways to show that the dimension must be infinite: </p>
<p>$\mathbb Q$ is countable, so for any countable set $\{a_1,a_2,\dots\}\subset \mathbb C$, the set of linear combinations of elements of that set over $\mathbb Q$ must be countable. But $\mathbb C$ is uncountable, so... |
11,519 | <p>Hello,</p>
<p>If $f:\mathbb{R} \to \mathbb{R}$ a differentiable function, it is very easy to find its Lipschitz constant. Is there any way to extend this to functions $f: \mathbb{R} \to \mathbb{R}^n$ (or similar)?</p>
| Pete L. Clark | 1,149 | <p>Let $f = (f_1,\ldots,f_n): [a,b] \rightarrow \mathbb{R}^n$ be a continuously differentiable function. (See the comments above for an explanation as to why the hypotheses have been strengthened.) </p>
<p>For $1 \leq i \leq n$, let </p>
<p>$L_i = \max_{x \in [a,b]} |f_i'(x)|$, </p>
<p>so that, by the Mean Value T... |
11,519 | <p>Hello,</p>
<p>If $f:\mathbb{R} \to \mathbb{R}$ a differentiable function, it is very easy to find its Lipschitz constant. Is there any way to extend this to functions $f: \mathbb{R} \to \mathbb{R}^n$ (or similar)?</p>
| Phil Isett | 7,193 | <p>Another nice way to do this computation (which generally gives more precise information) is to use the formula</p>
<p>$ f(x + h) - f(x) = [ \int_0^1 Df(x + th) dt ] h$</p>
<p>Also, the formula for $Df(x)$ in Hahn's post above is not correct.</p>
|
1,057,675 | <p>I was asked to prove that $\lim\limits_{x\to\infty}\frac{x^n}{a^x}=0$ when $n$ is some natural number and $a>1$. However, taking second and third derivatives according to L'Hôpital's rule didn't bring any fresh insights nor did it clarify anything. How can this be proven? </p>
| Ben Grossmann | 81,360 | <p>Sometimes, we want to consider $\infty$ as a value that can be attained. For example, we could define
$$
f(x) =
\begin{cases}
1/|x| & x \neq 0\\
\infty & x = 0
\end{cases}
$$
In this case, we would say that $f:\Bbb R \to (0,\infty]$ is an onto function</p>
<hr>
<p>We can consider $\Bbb R = (-\infty,\inft... |
1,057,675 | <p>I was asked to prove that $\lim\limits_{x\to\infty}\frac{x^n}{a^x}=0$ when $n$ is some natural number and $a>1$. However, taking second and third derivatives according to L'Hôpital's rule didn't bring any fresh insights nor did it clarify anything. How can this be proven? </p>
| Michael Albanese | 39,599 | <p>Note that $\infty \notin \mathbb{R}$, so it doesn't make sense to say that $[0, \infty]$ is a subset of $\mathbb{R}$. On the other hand, the set $$[0, \infty) = \{x \in \mathbb{R} \mid 0 \leq x\}$$ completely makes sense and defines a subset of $\mathbb{R}$.</p>
<p>However, one can consider the extended real number... |
1,634,325 | <blockquote>
<p><strong>Problem</strong>:
Is there sequence that sublimit are $\mathbb{N}$? If it's eqsitist prove this.</p>
</blockquote>
<p>I try to solve this problem by guessing what type of sequence need to be. <br>For example:
$a_n=(-1)^n$ has two sublimit $\{1,-1\}$.
<br>
$a_n=n
\times\sin(\frac{\pi}{2})$ h... | DanielWainfleet | 254,665 | <p>Let $p_m$ be the $m$th prime.For $n\in N,$ if $n=(p_m)^k$ for some $m,k\in N,$ let $x_n=m+1.$ If $n$ is not a power of a prime, let $x_n=1.$</p>
|
1,556,298 | <p>If we have $p\implies q$, then the only case the logical value of this implication is false is when $p$ is true, but $q$ false.</p>
<p>So suppose I have a broken soda machine - it will never give me any can of coke, no matter if I insert some coins in it or not.</p>
<p>Let $p$ be 'I insert a coin', and $q$ - 'I ge... | Dan Christensen | 3,515 | <blockquote>
<p>When is implication true?</p>
</blockquote>
<p><span class="math-container">$p\implies q$</span> is true when both <span class="math-container">$p$</span> and <span class="math-container">$q$</span> are true, or when <span class="math-container">$p$</span> is false. Assuming your problem is with the lat... |
179,230 | <p>I draw <a href="http://reference.wolfram.com/language/ref/Cos.html" rel="nofollow noreferrer"><code>Cos</code></a> function using the code line : </p>
<pre><code>GraphicsColumn[
{
Plot[Cos[0.0625*Pi x], {x, 0, 40*Pi}, Axes -> False],
Plot[Cos[0.0625*Pi x], {x, 0, 40*Pi}, Axes -> False],
Plot[Cos... | Bill | 18,890 | <p>If you can adapt this then <code>Export</code> works just fine</p>
<pre><code>g1 = GraphicsColumn[{
Plot[Cos[0.0625*Pi x], {x, 0, 40*Pi}, Axes -> False,
Epilog -> {Disk[{5 Pi, 3/4}, {8, .2}], Disk[{5 Pi, 1/4}, {8, .2}],
Arrowheads[{-.05, .05}],
Arrow[BezierCurve[{{5 Pi, 1.1}, {10 Pi, 1.3}, {15 Pi... |
2,883,023 | <p>Find the number of zeros of $f(z)=z^6-5z^4+3z^2-1$ in $|z|\leq1$. </p>
<p>My attempts have not gotten far. </p>
<p>I know we can examine the related equation $f(w)=w^3-5w^2+3w-1$ in $|w|\leq1$, letting $w=z^2$.</p>
<p>It is clear that $f(w)=0$ for $|w|=1$ if and only if $w=-1$. </p>
<p>My main problem is that th... | dxiv | 291,201 | <p>Alt. hint: show that the cubic $\,w^3-5w^2+3w-1\,$ has a unique real root which is $\,\gt 1\,$, then use that the product of all three roots is $\,1\,$.</p>
|
2,883,023 | <p>Find the number of zeros of $f(z)=z^6-5z^4+3z^2-1$ in $|z|\leq1$. </p>
<p>My attempts have not gotten far. </p>
<p>I know we can examine the related equation $f(w)=w^3-5w^2+3w-1$ in $|w|\leq1$, letting $w=z^2$.</p>
<p>It is clear that $f(w)=0$ for $|w|=1$ if and only if $w=-1$. </p>
<p>My main problem is that th... | Nosrati | 108,128 | <p><strong>Hint:</strong> On $|z|=1$ :
$$|z^6-1|\leqslant|z|^6+1=2\leqslant5|z|^4-3|z|^2\leqslant|-5z^4+3z^2|$$</p>
|
622,552 | <p>In the context of (most of the times convex) optimization problems -</p>
<p>I understand that I can build a Lagrange dual problem and assuming I know there is strong duality (no gap) I can find the optimum of the primal problem from the one of the dual. Now I want to find the primal optimum point (i.e. the point in... | jkn | 37,377 | <p>Ok, so I'm not totally sure whether this addresses all (some!) of your doubts, so let me know if it does not. As a disclaimer, what follows is just one out of several ways to view duality in optimisation (you can find others in the answers to <a href="https://math.stackexchange.com/questions/223235/please-explain-th... |
28,532 | <p><code>MapIndexed</code> is a very handy built-in function. Suppose that I have the following list, called <code>list</code>:</p>
<pre><code>list = {10, 20, 30, 40};
</code></pre>
<p>I can use <code>MapIndexed</code> to map an arbitrary function <code>f</code> across <code>list</code>:</p>
<pre><code>{f[10, {1}],... | rm -rf | 5 | <p><code>If</code> does the job and is simple enough:</p>
<pre><code>MapIndexed[If[2 ≤ First@#2 ≤ 3, f[#, #2], #] &, list]
(* {10, f[20, {2}], f[30, {3}], 40} *)
</code></pre>
|
113,797 | <p>I'm trying to extract every 21st character from this text, s (given below), to create new strings of all 1st characters, 2nd characters, etc.</p>
<p>I have already separated the long string into substrings of 21 characters each using</p>
<pre><code> splitstring[String : str_, n_] :=
StringJoin @@@ Partitio... | kglr | 125 | <pre><code>StringJoin /@ Transpose[Characters @ StringPartition[s,21]]
</code></pre>
<p><img src="https://i.stack.imgur.com/iFapi.png" alt="Mathematica graphics"></p>
|
1,192,338 | <p>How to prove that there are infinite taxicab numbers?
ok i was reading this <a href="http://en.wikipedia.org/wiki/Taxicab_number#Known_taxicab_numbers" rel="nofollow">http://en.wikipedia.org/wiki/Taxicab_number#Known_taxicab_numbers</a>
and thought of this question..any ideas?</p>
| Dietrich Burde | 83,966 | <p>It is easy to show that there are infinitely many positive integers which are representable as the sum of two cubes, e.g., see the article <a href="https://www.google.at/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CB8QFjAA&url=https%3A%2F%2Fcs.uwaterloo.ca%2Fjournals%2FJIS%2FVOL6%2FBrou... |
3,710,710 | <p>Suppose <span class="math-container">$A(t,x)$</span> is a <span class="math-container">$n\times n$</span> matrix that depends on a parameter <span class="math-container">$t$</span> and a variable <span class="math-container">$x$</span>, and let <span class="math-container">$f(t,x)$</span> be such that <span class="m... | greg | 357,854 | <p>Consider the scalar function <span class="math-container">$\alpha$</span> which matches the proposed
functional form, i.e.
<span class="math-container">$$\eqalign{
&\alpha = \alpha(t,f) \qquad &f = f(t,x) \\
&\alpha,t\in{\mathbb R}^{1} \qquad &f,x\in{\mathbb R}^{n}
}$$</span>
Everyone knows how to g... |
464,426 | <p>Find the limit of $$\lim_{x\to 1}\frac{x^{1/5}-1}{x^{1/3}-1}$$</p>
<p>How should I approach it? I tried to use L'Hopital's Rule but it's just keep giving me 0/0.</p>
| Community | -1 | <p>What do you mean by "keep giving you 0/0"?</p>
<p>After you apply the L'Hopital's Rule, you should get:</p>
<p>\begin{align*}
\lim_{x\to1}\frac{x^{1/5}-1}{x^{1/3}-1}&=\lim_{x\to1}\frac{\frac d{dx}(x^{1/5}-1)}{\frac d{dx}(x^{1/3}-1)}\\
&=\lim_{x\to1}\frac{\frac15x^{-4/5}}{\frac13x^{-2/3}}\\
&=\boxed{\df... |
915,054 | <p>I'm trying to find a closed form of this sum:
$$S=\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}.\tag{1}$$
<a href="http://www.wolframalpha.com/input/?i=Sum%5BGamma%28n%2B1%2F2%29%2F%28%282n%2B1%29%5E4+4%5En+n%21%29%2C+%7Bn%2C1%2CInfinity%7D%5D"><em>WolframAlpha</em></a> gives a large ex... | user153012 | 153,012 | <p>Another possible closed form of $S$ is the following. It containts also a generalized hypergeometric function, but just one.</p>
<p>$$S = \frac{\sqrt{\pi}}{648} {_6F_5}\left(\begin{array}c\ 1,\frac32,\frac32,\frac32,\frac32,\frac32\\2,\frac52,\frac52,\frac52,\frac52\end{array}\middle|\,\frac14\right).$$</p>
<p><a ... |
915,054 | <p>I'm trying to find a closed form of this sum:
$$S=\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}.\tag{1}$$
<a href="http://www.wolframalpha.com/input/?i=Sum%5BGamma%28n%2B1%2F2%29%2F%28%282n%2B1%29%5E4+4%5En+n%21%29%2C+%7Bn%2C1%2CInfinity%7D%5D"><em>WolframAlpha</em></a> gives a large ex... | Noam Shalev - nospoon | 219,995 | <p>First, in view of Legrende's duplication formula,
<span class="math-container">$$S=\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}=2\sqrt{\pi}\sum_{n=1}^\infty\frac{\Gamma(2n)}{\Gamma(n)\,n!\,(2n+1)^4\, 16^n}
\\=-\frac{\sqrt{\pi}}{3}\int_0^1 \ln^3(x)\sum_{n=1}^{\infty}\frac{\Gamma(2n)}{\... |
405,866 | <p>Original question:
For compact metric spaces, plenty of subtly different definitions converge to the same concept. Overtness can be viewed as a property dual to compactness. So is there a similar story for overt metric spaces?</p>
<p>Edit: Since overtness is trivially true assuming the Law of the Excluded Middle, cl... | Arno | 15,002 | <p>In computable analysis, the typical approach to metric spaces is that of a <em>computable metric space</em>. If we assume that there already is some external concept of the metric space we want to handle, we will ask for a particular dense sequence such that we can compute the distance of any two points in that sequ... |
405,866 | <p>Original question:
For compact metric spaces, plenty of subtly different definitions converge to the same concept. Overtness can be viewed as a property dual to compactness. So is there a similar story for overt metric spaces?</p>
<p>Edit: Since overtness is trivially true assuming the Law of the Excluded Middle, cl... | Andrej Bauer | 1,176 | <p>To conjure up a non-overt space we must change slightly the definition of topology, since even inuitionistically every space is overt, so long as every union of opens is open.</p>
<p>Let <span class="math-container">$\Sigma$</span> be the <a href="https://ncatlab.org/nlab/show/Sierpinski+space" rel="nofollow norefer... |
2,312,016 | <p>Prove limit of three variables using (ε, δ)-definition.</p>
<p>$$\lim_{(x, y, z)\to (0, 1, 2)} (3x+3y-z)=1$$</p>
<p>I have no idea how to do this with three variables.</p>
| 5xum | 112,884 | <p>You have to prove that for every $\epsilon > 0$, there exists some $\delta > 0$ such that if $$\sqrt{x^2+y^2+z^2} < \delta$$ then $|3x+3y-z-1| < \epsilon$.</p>
<hr>
<p>To do that, here's a <strong>hint</strong>:</p>
<ul>
<li>If $\sqrt{x^2+(y-1)^2+(z-2)^2} < \delta$, then $|x|<\delta$ and $|y-1|&... |
158,662 | <p>I know to prove a language is regular, drawing NFA/DFA that satisfies it is a decent way. But what to do in cases like</p>
<p>$$
L=\{ww \mid w \text{ belongs to } \{a,b\}*\}
$$</p>
<p>where we need to find it it is regular or not. Pumping lemma can be used for irregularity but how to justify in a case where it can... | sxd | 12,500 | <p>This language is not regular.</p>
<p>HINT Suppose that it is, then the pumping lemma should hold.</p>
<p>Let $p$ be the pumping length, and pick $w = a^pba^pb$. Can you procede now?</p>
|
578,337 | <p>For $n=1,2,3,\dots,$ and $|x| < 1$ I need to prove that $\frac{x}{1+nx^2}$ converges uniformly to zero function. How ?. For $|x| > 1$ it is easy. </p>
| ncmathsadist | 4,154 | <p>You have
$${x\over 1 + nx^2} = {1\over \sqrt{n}} {{\sqrt{n}x\over 1 + nx^2}} $$
The function $x\mapsto{x\over 1 + x^2}$ is bounded; let $M$ be the supremum of its absolute value. Then you have
$$\left|{x\over 1 + nx^2}\right| \le {M\over \sqrt{n}}.$$</p>
|
290,903 | <p>I am unable to understand the fundamental difference between a Gradient vector and a Tangent vector.
I need to understand the geometrical difference between the both. </p>
<p>By Gradient I mean a vector $\nabla F(X)$ , where $ X \in [X_1 X_2\cdots X_n]^T $</p>
<p>Note: I saw similar questions on "Difference betw... | Community | -1 | <p>I suppose the question has been answered in the comments.</p>
<p>The gradient of a function $(x_1,x_2,\ldots,x_n)\mapsto y$ is the vector $\left(\dfrac{\partial y}{\partial x_1},\dfrac{\partial y}{\partial x_2},\ldots,\dfrac{\partial y}{\partial x_n}\right)$.</p>
<p>The tangent to a curve $x\mapsto(y_1,y_2,\ldots,... |
489,109 | <p>I've been stumped on this problem for hours and cannot figure out how to do it from tons of tutorials.</p>
<p>Please note: This is an intro to calculus, so we haven't learned derivatives or anything too complex.</p>
<p>Here's the question: </p>
<p>Let $f(x) = x^5 + x + 7$. Find the value of the inverse function a... | QED | 91,884 | <p>$$1035-7=1028=1024+4=4^5+4$$
Therefore $f^{-1}(1035)=4$.</p>
|
1,855,641 | <p>$$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$$</p>
<p>We have three unknowns if they were two it were easy but I have no idea when it becomes three unknowns any hints?</p>
<p><strong>note</strong>:There isn't any information about value of $a$,$b$ and $c$</p>
| Roby5 | 243,045 | <p>Using AM-GM inequality, we have</p>
<blockquote>
<p>$$\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \geq 3 \cdot \sqrt[3]{\frac{a}{b} \cdot \frac{b}{c} \cdot \frac{c}{a}}=3$$</p>
</blockquote>
<p>The equality is indeed attained when</p>
<blockquote>
<p>$$\frac{a}{b}=\frac{b}{c}=\frac{c}{a}\tag{1}$$</p>
</blockquote>
<... |
1,855,641 | <p>$$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$$</p>
<p>We have three unknowns if they were two it were easy but I have no idea when it becomes three unknowns any hints?</p>
<p><strong>note</strong>:There isn't any information about value of $a$,$b$ and $c$</p>
| Ryan Roberson | 353,357 | <p>Well, the best I can think of, [that you stated the <em>minimum</em>, not the <em>least absolute value</em>]:</p>
<ul>
<li><p>if $A$ is some arbitrarily <em>large</em> negative, i.e. $-10^{100}$</p></li>
<li><p>and $B$ is some arbitrarily <em>small</em> positive, i.e. $10^{-100}$ (so therefore $A/B= -10^{200}$)</p>... |
1,855,641 | <p>$$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$$</p>
<p>We have three unknowns if they were two it were easy but I have no idea when it becomes three unknowns any hints?</p>
<p><strong>note</strong>:There isn't any information about value of $a$,$b$ and $c$</p>
| chaviaras michalis | 247,390 | <p>Another idea to see this is the following :
Let's supoose we have $$f(a,b,c)= \frac{a}{b} + \frac{b}{c} + \frac{c}{a}, \hspace{5mm}a,b,c\in \mathbb{R}^{*}$$
We have
$\frac{\partial f}{\partial a}= \frac{1}{b}-\frac{c}{a^2} , \hspace{2mm} \frac{\partial f}{\partial b} = \frac{1}{c} - \frac{a}{b^2}, \hspace{2mm} \fr... |
1,009,503 | <p>Theorem 15 in Chapter 15 of Peter Lax's functional analysis book says</p>
<p>$X$ is a Banach space, $Y$ and $Z$ are closed subspaces of $X$ that complement each other $X = Y \oplus Z$, in the sense that every $x\in X$ can be decomposed uniquely as $x = y+z$ where $y\in Y$, $z\in Z$. Denote the two complements of $... | Graham Kemp | 135,106 | <p>In order to count occurrence of 'heads in a row' you stop counting just before the first tail. This is a “count successes before one failure” senario; so the number of heads in a row $X$ given a biased coin, with probability of heads $p$, has a <strong>negative binomial probability distribution</s... |
1,369,990 | <p>I came across a quesion -
<a href="https://www.hackerrank.com/contests/ode-to-code-finals/challenges/pingu-and-pinglings" rel="nofollow">https://www.hackerrank.com/contests/ode-to-code-finals/challenges/pingu-and-pinglings</a></p>
<p>The question basically asks to generate all combinations of size k and sum up the ... | Rus May | 17,853 | <p>If the numbers in your set are $a_1,\dots,a_n$, then your sum is the coefficient of $x^k$ in the product $\prod_i (1+a_ix)$. If there's a relationship between the $a_i$'s, then you might be able to simplify the product and then extract the coefficient of $x^k$. Otherwise, I doubt there's much simplification.</p>
|
115,367 | <p>Let $f(z)$ be an analytic function on $D=\{z : |z|\leq 1\}$. $f(z) < 1$ if $|z|=1$. How to show that there exists $z_0 \in D$ such that $f(z_0)=z_0$. I try to define $f(z)/z$ and use Schwarz Lemma but is not successful. </p>
<p>Edit: Hypothesis is changed to $f(z) < 1$ if $|z|=1$. I try the following. If $f$ ... | user8268 | 8,268 | <p>$|f|$ has maximum on the boundary, hence it maps $D$ to $D$, and you can use Brouwer fixed point theorem.</p>
|
2,173,918 | <p>Let $f(z)=\sum\limits_{k=1}^\infty\frac{z^k}{1-z^k}$. I want to show that this series represents a holomorphic function in the unit disk. I'm, however, quite confused. For example, is $f(z)$ even a power series? It doesn't look as such. Here's what I have so far come up with.</p>
<blockquote>
<p>Proof:</p>
</bloc... | zhw. | 228,045 | <p>Suppose $0<r<1.$ Then $r^k \to 0.$ Thus $r^k\le 1/2$ for large $k.$ For these $k$ and $|z|\le r$ we have</p>
<p>$$\left| \frac{z^k}{1-z^k}\right | \le \frac{r^k}{1-r^k} \le 2r^k.$$</p>
<p>Since $\sum 2r^k<\infty,$ Weierstrass M implies our series converges uniformly on $\{|z|\le r\}.$ This is true for eve... |
1,353,498 | <p>The problem is to prove or disprove that there is a noncyclic abelian group of order $51$. </p>
<p>I don't think such a group exists. Here is a brief outline of my proof:</p>
<p>Assume for a contradiction that there exists a noncyclic abelian group of order $51$.</p>
<p>We know that every element (except the iden... | Arthur | 15,500 | <p>Any finite abelian group is isomorphic to a group of the form $$\Bbb Z_{p_1^{a_1}}\times \Bbb Z_{p_2^{a_2}}\times\cdots\times \Bbb Z_{p_n^{a_n}}$$ where $p_i$ are (not necessarily distinct) primes. The order of such a group is $p_1^{a_1}\cdots p_n^{a_n}$. How many ways can this be done for $51$?</p>
|
795,193 | <p><strong>THEOREM</strong>: Suppose $\{f_n\}$ is a sequence of continuous functions from $[a,b]$ to $\Bbb R$ that converge pointwise to a continuous function $f$ over $[a,b]$. If $f_{n+1}\leq f_n$, then convergence is uniform. </p>
<p>Then, why is the continuity of the the functions $f_i$'s important for the theorem?... | Tony Piccolo | 71,180 | <p>Compactness cannot be avoided.</p>
<p>Let $\varepsilon>0$ and $x \in [a,b]$.</p>
<p>Since $f_n(x)$ converges pointwise to $f(x)$, there exists $N_x$ such that $$|f_n(x)-f(x)|<\varepsilon/3 \quad \text{for} \quad n \ge N_x$$ Also, since $f$ and $f_{N_x}$ are continuous, there is an open neighborhood $U_x$ of ... |
1,443,812 | <p>Suppose that $X$ and $Y$ have joint p.d.f.</p>
<p>$$ f(x, y) = 3x, \; 0 < y < x < 1.$$ </p>
<p>Find $f_X(x)$,the marginal p.d.f. of $X$.</p>
<p>this is what i got</p>
<p>$$f_X(x) = \int_0^x f(x, y)dy = \int_0^x 3x dy = 3x^2$$
for $0 < x < 1$.</p>
<p>however, if want to know whether $X$ and $Y$ ar... | purugin | 271,254 | <p>f(x, y) = fX(x)fY(y) since you already get the fx(X), find fy(y) and then verify, if fx(x)fy(y) is the same withf(x,y) then they are independent.</p>
|
11,090 | <p>In <em>MMA</em> (8.0.0/Linux), I tried to to create an animation using the command</p>
<pre><code>Export["s4s5mov.mov", listOfFigures]
</code></pre>
<p>and got the output</p>
<p><img src="https://i.stack.imgur.com/bFWPP.png" alt="enter image description here"></p>
<p>Doing a little research, one can <a href="htt... | amr | 950 | <pre><code>stringToHex[str_] := ToExpression["16^^" <> str];
</code></pre>
<p>This is just a way of automating the normal notation you would use, which is 16^^6b (check <a href="http://reference.wolfram.com/mathematica/tutorial/InputSyntax.html">here</a> for the documentation).</p>
|
3,118,462 | <p>cars arrives according to a Poisson process with rate=2 per hour and trucks arrives according to a Poisson process with rate=1 per hour. They are independent. </p>
<p>What is the probability that <strong>at least</strong> 3 cars arrive before a truck arrives? </p>
<p>My thoughts:
Interarrival of cars A ~ Exp(2 p... | Paras Khosla | 478,779 | <p>Consider any one factor pair of <span class="math-container">$68$</span>, say <span class="math-container">$1 \times 68$</span>. Now go on dividing one number by <span class="math-container">$2$</span> and multiplying other by <span class="math-container">$2$</span>, this way the product remains the same but the sum... |
531,342 | <p>Prove or disprove that the greedy algorithm for making change always uses the fewest coins possible when the denominations available are pennies (1-cent coins), nickels (5-cent coins), and quarters (25-cent coins).</p>
<p>Does anyone know how to solve this?</p>
| vadim123 | 73,324 | <p>After giving out the maximal number of quarters, there will be $0-24$ cents remaining. Then there will be at most 4 nickels to give out. After giving out nickels greedily, there will be $0-4$ cents remaining, so there will be at most 4 pennies to give out. Now, can you prove that we cannot rearrange our change to... |
2,853,278 | <p><a href="https://math.stackexchange.com/a/203701/312406">This answer</a> suggests the idea, that a local ring $(R, \mathfrak{m})$ whose maximal ideal is nilpotent is in fact an Artinian ring.</p>
<p>Is this true? If so, how is it proven?</p>
| rschwieb | 29,335 | <p>There is a version of this theorem even for noncommutative rings. It is called the <a href="https://en.wikipedia.org/wiki/Hopkins%E2%80%93Levitzki_theorem" rel="nofollow noreferrer">Hopkins-Levitzki theorem</a>.</p>
<p>It says, in a nutshell, that if $R/J(R)$ is semisimple and $J(R)$ is nilpotent, then $R$ is right... |
3,921,847 | <p>I had the following question:</p>
<blockquote>
<p>Does there exist a nonzero polynomial <span class="math-container">$P(x)$</span> with integer coefficients satisfying both of the following conditions?</p>
<ul>
<li><span class="math-container">$P(x)$</span> has no rational root;</li>
<li>For every positive integer <... | Paul Sinclair | 258,282 | <p>Your definition of <span class="math-container">$b$</span> depends on <span class="math-container">$n$</span>. So it is not a constant <span class="math-container">$b$</span>, but rather each <span class="math-container">$n$</span> has its own <span class="math-container">$b_n$</span>. And it is not the case that <s... |
3,921,847 | <p>I had the following question:</p>
<blockquote>
<p>Does there exist a nonzero polynomial <span class="math-container">$P(x)$</span> with integer coefficients satisfying both of the following conditions?</p>
<ul>
<li><span class="math-container">$P(x)$</span> has no rational root;</li>
<li>For every positive integer <... | GreginGre | 447,764 | <p>Here is an elementary way to construct such a polynomial explicitely. I will need the fact that if <span class="math-container">$p\nmid a$</span>, then <span class="math-container">$a$</span> is invertible modulo <span class="math-container">$p$</span>, which can be proven using Bézout theorem.</p>
<p>The proof of F... |
1,602,312 | <p>Can we prove that $\sum_{i=1}^k x_i$ and $\prod_{i=1}^k x_i$ is unique for $x_i \in R > 0$?<br>
I stated that conjecture to solve CS task, but I do not know how to prove it (or stop using it if it is false assumption). </p>
<p>Is the pair of Sum and Product of array of k real numbers > 0 unique per array? I do ... | Eli Rose | 123,848 | <p>Let's work in $\mathbb{R}^2$. You can visualize vectors as arrows, and add them by laying the arrows head-to-tail.</p>
<p><a href="https://i.stack.imgur.com/pieZh.gif" rel="noreferrer"><img src="https://i.stack.imgur.com/pieZh.gif" alt="enter image description here"></a></p>
<p>That's what we're going to do to vis... |
801,668 | <p>Sources: <a href="https://rads.stackoverflow.com/amzn/click/0495011665" rel="nofollow noreferrer"><em>Calculus: Early Transcendentals</em> (6 edn 2007)</a>. p. 206, Section 3.4. Question 95.<br>
<a href="https://rads.stackoverflow.com/amzn/click/0470383348" rel="nofollow noreferrer"><em>Elementary Differential Equat... | Deepak | 151,732 | <p>Just chain rule in action. Let $\displaystyle\frac{dy}{dx} = z$</p>
<p>So $\displaystyle \frac{d}{dt}(\frac{dy}{dx}) = \frac{dz}{dt} = \frac{dz}{dx}.\frac{dx}{dt} = \frac{d}{dx}(\frac{dy}{dx}).\frac{dx}{dt}$, as you wanted to prove.</p>
|
2,279,120 | <p>How do we prove that for any complex number $z$ the minimum value of $|z|+|z-1|$ is $1$ ?
$$
|z|+|z-1|=|z|+|-(z-1)|\geq|z-(z-1)|=|z-z+1|=|1|=1\\\implies|z|+|z-1|\geq1
$$</p>
<p>But, when I do as follows
$$
|z|+|z-1|\geq|z+z-1|=|2z-1|\geq2|z|-|1|\geq-|1|=-1
$$
Since LHS can never be less than 0, $|z|+|z-1|\geq0$</p>... | Michael Rozenberg | 190,319 | <p>$$|z|+|1-z|\geq|z+1-z|=1$$
The equality occurs for $z=\frac{1}{2}$.</p>
|
25,485 | <p>Not sure if this is more appropriate for here or for Math.SE, but here goes:</p>
<p>How does one who is self-studying mathematics determine if a textbook is too hard for you?</p>
<p>Math is hard in general, but when does a textbook cross that line from being challenging to being nearly intractable?</p>
<p>Sometimes ... | guest troll | 20,204 | <p>When you are self-studying, it is a much harder environment because (a) you lack external "whip" of the grade. And (b) you don't have a teacher, lectures, etc. With that in mind, you need to pick the books that are most beginner friendly. Especially useful are "programmed instruction" books (e... |
4,631,463 | <p>I came cross the following equation:<br />
<span class="math-container">$$\lim_{x\to+\infty}\frac{1}{x}\int_0^x|\sin t|\mathrm{d}t=\frac{2}{\pi}$$</span>
I wonder how to prove it.</p>
<p>Using the Mathematica I got the following result:
<a href="https://i.stack.imgur.com/CVWgR.png" rel="nofollow noreferrer"><img src... | Ninad Munshi | 698,724 | <p><span class="math-container">$\textbf{Hint}$</span>: the limmand can be rewritten as</p>
<p><span class="math-container">$$ \frac{\int_0^{\left\lfloor \frac{x}{\pi}\right\rfloor \pi} |\sin t|dt + \int_{\left\lfloor \frac{x}{\pi}\right\rfloor \pi}^x |\sin t|dt}{x}$$</span></p>
<p><span class="math-container">$$=2 \fr... |
254,126 | <p>If 0 < a < b, where a, b $\in\mathbb{R}$, determine $\lim \bigg(\dfrac{a^{n+1} + b^{n+1}}{a^{n} + b^{n}}\bigg)$</p>
<p>The answer (from the back of the text) is $\lim \bigg(\dfrac{a^{n+1} + b^{n+1}}{a^{n} + b^{n}}\bigg) = b$ but I have no idea how to get there. The course is Real Analysis 1, so its a course o... | Chris Gerig | 22,295 | <p>To answer your only question: No. A 1-manifold is either a (open or closed) line segment or a circle or a disjoint union of such. But you can <em>fatten</em> the figure-$8$ (for example) out to get an honest manifold homotopy-equivalent to what you want (plane minus two points).</p>
|
1,980,510 | <p>$$ \lim_{(x,y) \to (1,0)} \frac{(x-1)^2\ln(x)}{(x-1)^2 + y^2}$$</p>
<p>I tried L' Hospitals Rule and got now where. Then I tried using $x = r\cos(\theta)$ and $y = r\sin(\theta)$, but no help. How would I approach this? T</p>
| Tsemo Aristide | 280,301 | <p>$$\frac{(x-1)^2\ln(x)}{(x-1)^2+y^2} = \frac{\ln(x)}{1+{y^2\over {(x-1)^2}}}.$$ The limit of the numerator is $-\infty$ and the limit of the denominator is $1$ so the limit is $-\infty$.</p>
|
1,393,154 | <p><span class="math-container">$4n$</span> to the power of <span class="math-container">$3$</span> over <span class="math-container">$2 = 8$</span> to the power of negative <span class="math-container">$1$</span> over <span class="math-container">$3$</span></p>
<p>Written Differently for Clarity:</p>
<p><span class="m... | Siwel | 259,274 | <p>We have that $4n^{3/2}=8^{-1/3}$. Proceed as follows, to find $n$.</p>
<ul>
<li>Simplify $8^{-1/3}=1/\sqrt[3]{8}=1/2$</li>
<li>Divide both sides by 4 to get $n^{3/2}=1/8$. This is the next step, bearing in mind the order of operations, which is often remembered with the phrase BODMAS(/BIDMAS) (Brackets, Other(/Indi... |
2,843,560 | <p>If $\sin x +\sin 2x + \sin 3x = \sin y\:$ and $\:\cos x + \cos 2x + \cos 3x =\cos y$, then $x$ is equal to</p>
<p>(a) $y$</p>
<p>(b) $y/2$</p>
<p>(c) $2y$</p>
<p>(d) $y/6$</p>
<p>I expanded the first equation to reach $2\sin x(2+\cos x-2\sin x)= \sin y$, but I doubt it leads me anywhere. A little hint would be... | Aryabhata | 1,102 | <p>If $z = e^{ix} = \cos x + i \sin x$</p>
<p>Then we have $z + z^2 + z^3 = e^{iy} = w$ (say)</p>
<p>Divide by $z^2$ to see that</p>
<p>$$z + \frac{1}{z} + 1 = \frac{w}{z^2}$$</p>
<p>The left side is real and thus</p>
<p>$$w = az^2$$</p>
<p>Since $|w| = |z| = 1$ we must have that $|a| = 1$</p>
<p>Thus $$w = \pm ... |
2,949,011 | <blockquote>
<p>If <span class="math-container">$b_n\to\infty$</span> and <span class="math-container">$\{a_n\}$</span> is such that <span class="math-container">$b_n>a_n$</span> for all <span class="math-container">$n$</span>, then <span class="math-container">$a_n\to\infty$</span>.</p>
</blockquote>
<p>We are t... | hamam_Abdallah | 369,188 | <p>Your statement is false.
as a counter example,</p>
<p>take</p>
<p><span class="math-container">$$b_n=n \text{ and } \; a_n=-n.$$</span></p>
<p>then
<span class="math-container">$$a_n\le b_n,$$</span></p>
<p><span class="math-container">$$b_n\to +\infty$$</span>
but
<span class="math-container">$$a_n\to -\infty$... |
1,013,346 | <p>Given a box which contains $3$ red balls and $7$ blue balls. A ball is drawn from the box and a ball of the other color is then put into the box. A second ball is drawn from the box, What is the probability that the second ball is blue? </p>
<p>could anyone provide me any hint? </p>
<p>Please, don't offer a comple... | Timbuc | 118,527 | <p>$$\frac{\pi^4}{90}=\sum_{n=1}^\infty\frac1{n^4}=\sum_{n=1}^\infty\frac1{(2n)^4}+\sum_{n=1}^\infty\frac1{(2n-1)^4}=\frac1{16}\sum_{n=1}^\infty\frac1{n^4}+\sum_{n=1}^\infty\frac1{(2n-1)^4}\implies$$</p>
<p>$$\implies\sum_{n=1}^\infty\frac1{(2n-1)^4}=\left(1-\frac1{16}\right)\frac{\pi^4}{90}=\frac{\pi^4}{96} $$</p>
|
1,013,346 | <p>Given a box which contains $3$ red balls and $7$ blue balls. A ball is drawn from the box and a ball of the other color is then put into the box. A second ball is drawn from the box, What is the probability that the second ball is blue? </p>
<p>could anyone provide me any hint? </p>
<p>Please, don't offer a comple... | Vivek Kaushik | 169,367 | <p>Here is another proof I came up with. Start with the quadruple integral:</p>
<p>$$J=\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1} \frac{1}{1-x^2_1x^2_2x^2_3x^2_4}dx_1dx_2dx_3dx_4.$$</p>
<p>Replace the integrand with a geometric series as such:
$$\frac{1}{1-x^2_1x^2_2x^2_3x^2_4}=\sum_{n=0}^{\infty}(x_1x_2x_3x_4)... |
353 | <p>One of the challenges of undergraduate teaching is logical implication. The case by case definition, in particular, is quite disturbing for most students, that have trouble accepting "false implies false" and "false implies true" as true sentences. </p>
<blockquote>
<p><strong>Question:</strong> What are good poi... | Henry Towsner | 62 | <p>When I've taught propositional logic I acknowledge that this is a formalism that doesn't perfectly match the English usage, and use it as an opportunity to point out </p>
<ol>
<li>The evaluation of $\rightarrow$ has to be purely a property of truth values, whereas "implies" in English involves the meaning of the st... |
353 | <p>One of the challenges of undergraduate teaching is logical implication. The case by case definition, in particular, is quite disturbing for most students, that have trouble accepting "false implies false" and "false implies true" as true sentences. </p>
<blockquote>
<p><strong>Question:</strong> What are good poi... | user578 | 578 | <ol>
<li><p>Avoid real world analogies. They confuse students because real world means natural language. Instead remind them of implication and its truth table whenever you write a mathematical proposition on the board.</p></li>
<li><p>Remind students to work with the definitions. Over and over again.</p></li>
<li><p>I... |
353 | <p>One of the challenges of undergraduate teaching is logical implication. The case by case definition, in particular, is quite disturbing for most students, that have trouble accepting "false implies false" and "false implies true" as true sentences. </p>
<blockquote>
<p><strong>Question:</strong> What are good poi... | Michael Joyce | 1,397 | <p>I think the intuition for implication is aided when you consider it in the context of universal quantification.</p>
<p>A claim of the form $(\forall x)(A(x) \Rightarrow B(x))$ can be translated roughly as every $x$ that has property $A$ must also have property $B$. Most of the use of conditional claims used in prac... |
1,897,538 | <p>Next week I will start teaching Calculus for the first time. I am preparing my notes, and, as pure mathematician, I cannot come up with a good real world example of the following.</p>
<p>Are there good examples of
\begin{equation}
\lim_{x \to c} f(x) \neq f(c),
\end{equation}
or of cases when $c$ is not in the doma... | dxiv | 291,201 | <p>If instantaneous elastic collisions count as <code>real world</code> then the speed $f(t)$ of a ball traveling at constant speed $v$ and hitting a wall at $t = t_0$ is $v$ for $t \lt t_0$, $0$ at $t = t_0$, and $-v$ for $t \gt t_0$, so both lateral limits exist at $t_0$ but are different between them and different f... |
588,214 | <p>What is the meaning of $[G:C_G(x)]$ in group theory? Is this equivalent to $\frac{|G|}{|Z_G(x)|}$, or to $|Z_G(x)|$?</p>
| Henry Swanson | 55,540 | <p>$C_G(x)$ is the <em>centralizer</em> of $x$ in $G$. That is, $\{ g \in G \mid gxg^{-1} = x \}$ or equivalently, $\{ g \in G \mid gx = xg \}$.</p>
<p>The $[G : H]$ notation means the <em>index</em> of $H$ in $G$, and it is defined as the number of cosets of $H$ in $G$. Lagrange's theorem says that this is equal to $... |
937,912 | <p>I'm looking for a closed form of this integral.</p>
<p>$$I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx ,$$</p>
<p>where $\operatorname{Li}_2$ is the <a href="http://mathworld.wolfram.com/Dilogarithm.html" rel="noreferrer">dilogarithm function</a>.</p>
<p>A numerical approximation of i... | Juan Ospina | 170,228 | <p>Using Maple I am obtaining</p>
<p>$$1+\frac{\pi }{16}{\ _4F_3(1,1,1,3/2;\,2,2,2;\,1)}+\frac{\sqrt {\pi }}{8}
G^{4, 1}_{4, 4}\left(-1\, \Big\vert\,^{1, 5/2, 5/2, 5/2}_{2, 3/2, 3/2, 1}\right)
$$</p>
<p>and a numerical approximation is</p>
<p>$$1.3913063720392030337$$</p>
|
937,912 | <p>I'm looking for a closed form of this integral.</p>
<p>$$I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx ,$$</p>
<p>where $\operatorname{Li}_2$ is the <a href="http://mathworld.wolfram.com/Dilogarithm.html" rel="noreferrer">dilogarithm function</a>.</p>
<p>A numerical approximation of i... | Vladimir Reshetnikov | 19,661 | <p>We will go through a sequence of integrals, and, remarkably, we will see that at each step an integrand will have a continuous closed-form antiderivative in terms of elementary functions, dilogarithms and trilogarithms, so evaluation of an integral is then just a matter of calculating values (or limits) at end-point... |
907,851 | <p>I am really new into math, why is $-2^2 = -4 $ and $(-2)^2 = 4 $? </p>
| GEdgar | 442 | <p>It is true that $-2^2$ is ambiguous (unless you know the convention), because $-(2^2) \ne (-2)^2$. (And, indeed, some calculators or programing languages may do it using their own convention, different from the mathematicians' convention.) The mathematicians' convention is $-2^2 = -(2^2)$. Why? Presumably because... |
907,851 | <p>I am really new into math, why is $-2^2 = -4 $ and $(-2)^2 = 4 $? </p>
| DanielV | 97,045 | <p>Universally, exponentiation is agreed to be evaluated before subtraction.</p>
<p>However, there are two opposing wishes that go into the design of a grammar:</p>
<ul>
<li>Negation is evaluated at the same time as subtraction, to avoid irregularities</li>
<li>Unary operators are all evaluated before binary operator... |
10,666 | <p>My question is about <a href="http://en.wikipedia.org/wiki/Non-standard_analysis">nonstandard analysis</a>, and the diverse possibilities for the choice of the nonstandard model R*. Although one hears talk of <em>the</em> nonstandard reals R*, there are of course many non-isomorphic possibilities for R*. My question... | Gerald Edgar | 454 | <p>See Robinson's book <em>Non-Standard Analysis</em> (North-Holland 1966)...<br>
Section 3.1 has some remarks about the order type of the non-standard natural numbers.</p>
|
2,338,123 | <p>Function $f(z)$ is an entire function such that $$|f(z)| \le |z^{n}|$$ for $z \in \mathbb{C}$ and some $n \in \mathbb{N}$.</p>
<p>Show that the singularities of the function $$\frac {f(z)}{z^{n}}$$ are removable. What can be implied about the function $f(z)$ if moreover $f(1) = i$? Draw a far-reaching conclusion.</... | Jack D'Aurizio | 44,121 | <p>Since $f(z)$ is an entire function, $g(z)=\frac{f(z)}{z^n}$ may only have a pole at the origin. If that was the case, then $\left|g(z)\right|$ would be unbounded in a neighbourhood of the origin, but that contradicts $\left|g(z)\right|\leq 1$ for any $z\neq 0$. It follows that $z=0$ is a removable singularity for $g... |
18,895 | <p>I know that the answer is $C(8,2)$, but I don't get, why.
Can anyone, please, explain it?</p>
| aioobe | 27,807 | <p>$C(8, 2)$ means "$8$ choose $2$", which in this case should be interpreted as "how many ways can you choose $2$ items out of $8$ possible". Here the $8$ items are the bits, and the $2$ comes from the bits you "select to be $1$". That is, there are just as many bytes with exactly two 1s as there are ways to select $2... |
1,776,726 | <p>I'm trying to determine whether or not </p>
<blockquote>
<p>$$\sum_{k=1}^\infty \frac{2+\cos k}{\sqrt{k+1}}$$ </p>
</blockquote>
<p>converges or not. </p>
<p>I have tried using the ratio test but this isn't getting me very far. Is this a sensible way to go about it or should I be doing something else?</p>
| ivanhu42 | 337,970 | <p>there is a high-school method.</p>
<p>since $\frac{a^2+1}{a^2+2}$ is an even function, and it's decreasing when $x \lt 0$, and increasing when $x \gt 0$, all we need to do is to find the min value of $\mid a \mid$.</p>
<p>since $x^2 + (a - 3) x - (a - 2) = 0$ has root(s) as real number(s), it follows that
$$
\Delt... |
894,909 | <p>Given </p>
<p>$(x+3)(y−4)=0 $</p>
<p>Quantity $A = xy $</p>
<p>Quantity $B = -12 $</p>
<p>A Quantity $A$ is greater.<br>
B Quantity $B$ is greater.<br>
C The two quantities are equal.<br>
D The relationship cannot be determined from the information given. </p>
<p>How is the answer D and not C ? </p>
<p>Pl... | Shabbeh | 165,678 | <p>Because the first equation says either $x=-3$ or $y=4$ or both and only in last case C is correct.</p>
<p>for example </p>
<p>$$\begin{cases}
x=-3\\y=0\end{cases}\rightarrow \begin{cases}
A=0\\B=-12\end{cases}$$</p>
|
3,755,032 | <p>I was given <span class="math-container">$(x,y)$</span> that satisfies both of this equation:</p>
<p><span class="math-container">$4|xy| - y^2 - 2 = 0\\(2x+y)^2 + 4x^2 = 2$</span></p>
<p>And was asked to find the maximum value of <span class="math-container">$4x + y$</span>.</p>
<p>Solving for <span class="math-cont... | Thulashitharan D | 786,768 | <p>No it does not, It represents a hyperbola.<br>
General equation of a straight line is <span class="math-container">$ax+by+c=$</span> where <span class="math-container">$a,b,c \in\mathbb R$</span>. But your equation when simplified looks like <span class="math-container">$2y+xy-3=0$</span> Notice that straight line e... |
3,755,032 | <p>I was given <span class="math-container">$(x,y)$</span> that satisfies both of this equation:</p>
<p><span class="math-container">$4|xy| - y^2 - 2 = 0\\(2x+y)^2 + 4x^2 = 2$</span></p>
<p>And was asked to find the maximum value of <span class="math-container">$4x + y$</span>.</p>
<p>Solving for <span class="math-cont... | Tanmay Gajapati | 634,349 | <p>There's a simple way to distinguish between a linear equation (or any polynomial) and a non-polynomial equation. A polynomial is defines as <span class="math-container">$$y=a_1x^1 + a_2x^2 + ... + a_nx^n$$</span> Observe the variable <span class="math-container">$x$</span> is on one side and <span class="math-contai... |
2,130,699 | <p>I have the coordinates of potentially any points within a 3D arc, representing the path an object will take when launched through the air. X is forward, Z is up, if that is relevant.
Using any of these points, I need to create a bezier curve that follows this arc. The curve requires the tangent values for the very s... | rschwieb | 29,335 | <p>A commutative ring with only trivial idempotents is called <em>connected</em>, and there are <a href="http://ringtheory.herokuapp.com/commsearch/commresults/?has=58&lacks=5" rel="nofollow noreferrer">lots of connected rings that aren't domains</a>. </p>
<p>Most of the ones appearing in the search query can be t... |
2,461,820 | <p>I am a bit confused with the following question, I get that P(T|D) = 0.95 and P(D) = 0.0001 however because i'm unable to work out P(T|~D) i'm struggling to apply the theorem, am i missing something? Also i'm unsure about what to do with the information relating to testing negative when you don't have the disease co... | fleablood | 280,126 | <p>$e = e^1$</p>
<p>$1 = e^0$</p>
<p>So $\ln(e) =1$ and $\ln(1) = 0$.</p>
<p>And $\ln(\ln e) = \ln (1) = 0$.</p>
|
54,541 | <p>Apparently, Mathematica has no real sprintf-equivalent (unlike any other high-level language known to man). <a href="https://mathematica.stackexchange.com/questions/970/sprintf-or-close-equivalent-or-re-implementation">This has been asked before</a>, but I'm wondering if the new <code>StringTemplate</code> function ... | a06e | 534 | <p>You can also try:</p>
<pre><code>StringTemplate["Number: `1` some other text"]
[ToString[NumberForm[N[Pi], {\[Infinity], 2}]]]
</code></pre>
<p>That is, leave the formatting to <code>ToString</code>, which does handle <code>NumberForm</code> and everything else.</p>
|
72,854 | <p>Hi everybody,</p>
<p>Does there exist an explicit formula for the Stirling Numbers of the First Kind which are given by the formula
$$
x(x-1)\cdots (x-n+1) = \sum_{k=0}^n s(n,k)x^k.
$$</p>
<p>Otherwise, what is the computationally fastest formula one knows?</p>
| Feldmann Denis | 17,164 | <p>There is an explicit formula : $s(n,m)=\frac{(2n-m)!}{(m-1)!}\sum_{k=0}^{n-m}\frac{1}{(n+k)(n-m-k)!(n-m+k)!}\sum_{j=0}^{k}\frac{(-1)^{j} j^{n-m+k} }{j!(k-j)!}.$ For once, it is not in Wikipedia (en), but in the french version of it (and I posted it there myself, if I may so brag)</p>
|
2,492,107 | <p>The question asks $\text{span}(A1,A2)$</p>
<p>$$A1 =\begin{bmatrix}1&2\\0&1\end{bmatrix}$$
$$A2 = \begin{bmatrix}0&1\\2&1\end{bmatrix}$$</p>
<p>I began by calculating $c_1[A1] + c_2[A2]$ then converting it into a matrix and row reducing. I found the restrictions where the stuff after the augment m... | Claude Leibovici | 82,404 | <p><strong>Hint</strong></p>
<p>Consider $$S_n=\sum_{i=1}^n \frac i{1+i^2+i^4}=\frac 12\left(\sum_{i=1}^n \frac 1{1-i+i^2}-\sum_{i=1}^n \frac 1{1+i+i^2}\right)$$ which beautifully telescopes.</p>
|
1,250,755 | <p>I'm sure there's something I'm missing here; probably a naive confusion of mathematics with metamathematics. Regardless, I've come up with what looks to me like a proof that (first-order) ZF+AD is inconsistent:</p>
<p>Assume otherwise. Then it cannot be shown that ZF+AD is consistent relative to ZF. However, ZFC is... | Sam | 221,113 | <p>What Asaf Karagila says in the second paragraph of his answer is basically what you are overlooking in your argument. Namely, it is implied in your argument that the statement $''\text{Con}(ZF+AD)\implies ZFC\vdash\text{Con}(ZF+AD)''$ is true in the meta-theory. But this is not necessarily true. What is true in the ... |
406,437 | <p>Calculus the extreme value of the $f(x,y)=x^{2}+y^{2}+xy+\dfrac{1}{x}+\dfrac{1}{y}$</p>
<p>pleasee help me.</p>
| Alex Wertheim | 73,817 | <p>How about the ring of quaternions? This is an example of a non-commutative ring and I don't think it's super difficult to wrap your head around. Understanding the ring of quaternions is also in fact very useful in understanding representations of rotations and groups of isometries for polyhedra. </p>
|
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