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 |
|---|---|---|---|---|
1,982,102 | <p>If I wanted to figure out for example, how many tutorial exercises I completed today.</p>
<p>And the first question I do is <strong>question $45$</strong>, </p>
<p>And the last question I do is <strong>question $55$</strong></p>
<p>If I do $55-45$ I get $10$.</p>
<p>But I have actually done $11$ questions:<br>
$... | Nitin | 131,456 | <p>You'll always need to add one in such cases. Consider - if you do problems 45 through 45, you'll have done 45-45 + 1 = 1 questions. </p>
|
1,982,102 | <p>If I wanted to figure out for example, how many tutorial exercises I completed today.</p>
<p>And the first question I do is <strong>question $45$</strong>, </p>
<p>And the last question I do is <strong>question $55$</strong></p>
<p>If I do $55-45$ I get $10$.</p>
<p>But I have actually done $11$ questions:<br>
$... | BigPanda | 343,523 | <p>What you are trying to calculate is the sum of exercises you did from question $a$ to question $b$.
You have $1$ exercise per question so the number of exercises is :
$$\underbrace{\sum_{a}^b 1}_{\text{Sum from a to b}} =\underbrace{\sum_{1}^b 1}_{\text{Sum from 1 to b}} - \underbrace{\sum_{1}^{a-1} 1}_{\text{Sum fr... |
1,888,729 | <p>It is stated in Wikipedia (and other pages too) that the spheres $S^n$ are all not contractible. </p>
<p>Take $n=1$. Would anyone explain to me why $$S^1\times [0,1]\to S^1$$$$(e^{2\pi i t},s)\mapsto e^{2\pi i ts}$$is not an homotopy between the identity and a point?</p>
| user39082 | 97,620 | <p>$$e^{2\pi i}=1,$$
so that
$$(e^{2\pi i},s)=(1,s)$$
but
$$(e^{2\pi i},s)$$
is mapped to $e^{2\pi is},$ while $$(1,s)$$ (which corresponds to $t=0$) is mapped to $1.$</p>
<p>For $0<s<1$ you have
$$ e^{2\pi is}\not=1$$
which shows that your map is not well-defined on the circle. (Not to talk about continuity.)<... |
1,888,729 | <p>It is stated in Wikipedia (and other pages too) that the spheres $S^n$ are all not contractible. </p>
<p>Take $n=1$. Would anyone explain to me why $$S^1\times [0,1]\to S^1$$$$(e^{2\pi i t},s)\mapsto e^{2\pi i ts}$$is not an homotopy between the identity and a point?</p>
| APURVA AVINASH PUJARI | 982,024 | <p>We can easily prove that S¹ in not contractible.
Let's recall some definations and results for the proof.</p>
<p>★Simply Connected Space: A path Connected space X is called as simply connected, if every closed curve in X is a null homotopy.
After studying Fundamental groups, one can define the term as: "A path ... |
2,377,816 | <p>I was solving problems based on Bayes theorem from the book "A First Course in Probability by Sheldon Ross". The problem reads as follows:</p>
<blockquote>
<p>An insurance company believes that there are two types of people: accident prone and not accident prone. Company statistics states that accident prone pers... | Graham Kemp | 135,106 | <blockquote>
<ol>
<li>can someone generalize it, so as to make my understanding more clear? Say for $n$ events? </li>
</ol>
</blockquote>
<p>If $(B_k)_n$ is a sequence of $n$ events that partition the sample space (or if at least $(B_k\cap A_1)_n$ partitions $A_1$) then, $\mathsf P(A_2\mid A_1) = \sum_{k=1}^n \m... |
73,991 | <p>I have the axiom from Peano's axioms:</p>
<p>If $A\subseteq \mathbb{N}$ and $1\in A$ and $m\in A \Rightarrow S(m)\in A$, then $A=\mathbb{N}$.</p>
<p>My book tells me that it secures that there are no more natural numbers than the numbers produced by the below 3 axioms (also from Peano's axioms):</p>
<p>$1\in \mat... | Damian Sobota | 12,690 | <p>The three axioms guarantee you that the set $A=\{1,S(1),S(S(1)),S(S(S(1))),...\}$ is infinite (mainly because of the injectivity of $S$). But they do not guarantee the equality $A=\mathbb{N}$. To set it you need the axiom of induction.</p>
<p>For example, put $S(n)=2n$. It satisfies the three axioms, but $A=\{1\}\c... |
1,038,579 | <p>The question is from Joseph .J Rotman's book - Introduction to the Theory of Groups and it goes like this:
<br/> $A,B,C$ are subgroups of $G$, so $A\leq B$, prove that if $(AC=BC\ \text{and}\ A\cap C=B\cap C)$. (we do not assume that either AB or AC is a subgroup) than A=B.
<br/><br/>
I need you guys to tell me if ... | user 59363 | 192,084 | <p>We have
$$B=\bigcup_{b\in B}b(B\cap C)$$
and
$$A=\bigcup_{a\in A}a(A\cap C).$$
Now, for $b,b'\in B$ we have $b(B\cap C)=b'(B\cap C)$ if and only if $b^{-1}b'\in B\cap C$ if an only if $b^{-1}b'\in C$ since $b^{-1}b'\in B$ is automatically true. From $B\subseteq BC$ and the assumption that $BC=AC$ it follows that f... |
2,245,408 | <blockquote>
<p>How is the following result of a parabola with focus <span class="math-container">$F(0,0)$</span> and directrix <span class="math-container">$y=-p$</span>, for <span class="math-container">$p \gt 0$</span> reached? It is said to be <span class="math-container">$$r(\theta)=\frac{p}{1-\sin \theta} $$</spa... | Brian Tung | 224,454 | <p>The equation of the parabola you want is</p>
<p>$$
y = \frac{x^2}{2p} - \frac{p}{2}
$$</p>
<p>Substituting</p>
<p>$$
x = r \cos \theta
$$
$$
y = r \sin \theta
$$</p>
<p>gives us</p>
<p>$$
\frac{\cos^2\theta}{2p} r^2 - (\sin \theta) r - \frac{p}{2} = 0
$$</p>
<p>If you solve this quadratic expression for $r$, a... |
2,245,408 | <blockquote>
<p>How is the following result of a parabola with focus <span class="math-container">$F(0,0)$</span> and directrix <span class="math-container">$y=-p$</span>, for <span class="math-container">$p \gt 0$</span> reached? It is said to be <span class="math-container">$$r(\theta)=\frac{p}{1-\sin \theta} $$</spa... | robert timmer-arends | 468,626 | <p>For a parabola with $F(0,0)$ and directrix $y=-p$, first write a “distance equation” relating any point on the parabola, $P(x,y)$, to $F$ and to the directrix, $D$.<p>
A parabola can be defined by its locus: $distanceFP = distanceDP$.<p>
$distanceFP = \sqrt{(x-0)^2+(y-0)^2} = \sqrt{x^2+y^2}$<p>
$distanceDP = y-(-p) ... |
4,251,233 | <p>Find</p>
<p><span class="math-container">$\int\frac{x+1}{x^2+x+1}dx$</span></p>
<p><span class="math-container">$\int \frac{x+1dx}{x^2+x+1}=\int \frac{x+1}{(x+\frac{1}{2})^2+\frac{3}{4}}dx$</span></p>
<p>From here I don't know what to do.Write <span class="math-container">$(x+1)$</span> = <span class="math-container... | AbdelAziz AbdelLatef | 692,431 | <p>You will have to make the numerator as a sum of two functions like this</p>
<p><span class="math-container">$\int \frac{x+\frac{1}{2}}{(x+\frac{1}{2})^2+\frac{3}{4}}dx+\int \frac{\frac{1}{2}}{(x+\frac{1}{2})^2+\frac{3}{4}}dx$</span></p>
<p>The first integral will result in <span class="math-container">$ln$</span> an... |
458 | <p>If you go to the bottom of any page in the SE network (e.g. this one!), you'll see a list of SE sites. In particular there's a link to MathOverflow, that is potentially seen by a large number of people (many of whom are outside of our target audience).</p>
<p>When you put your cursor over that link, there's a hover... | Scott Morrison | 3 | <p>Research mathematicians</p>
<p>.......</p>
|
458 | <p>If you go to the bottom of any page in the SE network (e.g. this one!), you'll see a list of SE sites. In particular there's a link to MathOverflow, that is potentially seen by a large number of people (many of whom are outside of our target audience).</p>
<p>When you put your cursor over that link, there's a hover... | Scott Morrison | 3 | <p>Professional mathematicians</p>
<p>...</p>
|
2,953,371 | <p>How can I find the derivative of this function ?
<span class="math-container">$$f(x)= (4x^2 + 2x +5)^{0.5}$$</span></p>
| Kaleb R. | 566,000 | <p>You could do it using the Laplace transform and the convolution theorem for Laplace transforms. The Laplace transform of a Dirac delta is
<span class="math-container">$$\mathcal{L}(\delta(t-a)) = e^{-as}$$</span>
and the convolution theorem states that <span class="math-container">$\mathcal{L} ((f*g)(t)) = \mathcal{... |
441,374 | <p>Let $K_{\alpha}(z)$ be the <a href="https://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions:_I.CE.B1_.2C_K.CE.B1" rel="nofollow noreferrer">modified Bessel function of the second kind of order $\alpha$</a>.</p>
<p>I need to compute the following integral:</p>
<p>$$\int_0^\infty\;\;K_0\left(\sqrt{a(... | Random Variable | 16,033 | <p>The evaluation is a bit easier if we use the <a href="http://dlmf.nist.gov/10.32#E10" rel="nofollow noreferrer">integral representation</a> $$K_{0}(x) = \frac{1}{2} \int_{0}^{\infty} \frac{1}{t} \, \exp \left(-t - \frac{x^{2}}{4t} \right) \, dt , \quad x>0, $$ which can be derived from the integral representatio... |
598,635 | <p>Prove the two Identities for
$-1 < r < 1$</p>
<p>$$\sum_{n=0}^{\infty} r^n\cos n\theta =\frac{1-r\cos\theta}{1-2r\cos\theta+r^2}$$</p>
<p>$$\sum_{n=0}^{\infty} r^n\sin{n\theta}=\frac{r \sin\theta }{1-2r\cos\theta+r^2}$$</p>
<p>Sorry could not figure out how to format equations</p>
| Mark | 24,958 | <p>HINT:Mathematical induction. For:
$$
\sum_{n=0}^{\infty} r^n\cos n\theta =\frac{1-r\cos\theta}{1-2r\cos\theta+r^2}
$$
Let us consider:
$$
\sum_{n=0}^{\infty} r^{n+1}\cos (n+1)\theta
$$
which is rewritten
$$
r\sum_{n=0}^{\infty} r^{n}\left[\cos n\theta \cos \theta-\sin n\theta \sin \theta \right]
$$
$$
=r\cos \theta\... |
675,718 | <p>I have a non-linear system of equations, $$\left\{ \begin{array}{rcl} x^2 - xy + 8 = 0 \\ x^2 - 8x + y = 0 \\ \end{array} \right.$$
I have tried equating the expressions (because both equal 0), which tells me: $$x^2 - xy + 8 = x^2 - 8x + y$$
Moving all expressions to the right yields: $$0 = xy - 8x + y - 8$$
Factori... | ir7 | 26,651 | <p>Hint: You got $x=-1$ OR $y=8$. BTW, cool idea to equate the expressions.</p>
|
2,011,236 | <p>I was reading <a href="https://web.williams.edu/Mathematics/lg5/Hindman.pdf" rel="nofollow noreferrer">this</a> discussion of Hindman's Theorem by Leo Goldmakher, and was tripped up by his introduction of a topology on $U(\mathbb N)$. (He is using $U(\mathbb N)$ to denote the space of ultrafilters on the natural num... | Brian M. Scott | 12,042 | <p>Yes, you’re correct. For each $A\subseteq\Bbb N$ let $B_A=\{\mathscr{U}\in U(\Bbb N):A\in\mathscr{U}\}$; then $\{B_A:A\subseteq\Bbb N\}$ is a base for the topology in question. Goldmakher should have put ‘for some $A\subseteq\Bbb N$’ outside the curly braces.</p>
<p>It’s not an easy topology to visualize, to put it... |
3,086,878 | <p>Recently I came across this general integral,
<span class="math-container">$$\int \frac {dx}{(x^2-2ax+b)^n}$$</span>
Putting <span class="math-container">$x^2-2ax+b=0$</span> we have,
<span class="math-container">$$x = a±\sqrt {a^2-b} = a±\sqrt {∆}$$</span>
Hence the integrand can be written as,
<span class="math-co... | jmerry | 619,637 | <p>All right, now I've got it.</p>
<p>The easiest way to get all the coefficients? Expand in a Laurent series around one of the roots. Substituting <span class="math-container">$z=x-a-\sqrt{\Delta}$</span> and later defining <span class="math-container">$D=\frac1{2\sqrt{\Delta}}$</span>, we get
<span class="math-conta... |
3,644,710 | <p>I wish to classify the Galois group of <span class="math-container">$\mathbb{Q}(e^{i\pi/4})/\mathbb{Q}$</span>. Let me denote the eighth root of unity as <span class="math-container">$\epsilon$</span>. I see that <span class="math-container">$1, \epsilon, \epsilon^2, \epsilon^3$</span> are linearly independent over ... | Martin Argerami | 22,857 | <p>Once you have <span class="math-container">$|x_{n+1}-x_n|\leq c q^n$</span> with <span class="math-container">$c>0$</span> and <span class="math-container">$0<q<1$</span>, you can telescope. This means
<span class="math-container">$$
|x_{n+k}-x_n|=|\sum_{j=1}^kx_{n+j}-x_{n+j-1}|\leq \sum_{j=1}^k|x_{n+j}-x_{... |
737,915 | <p>I'm reading Calculus: Basic Concepts for High School Students and am trying to digest the definition of 'limit of function'. There are two details that I am struggling to fully accept:</p>
<ol>
<li><p>If you are supposed to pick an interval $(a - \delta, a + \delta)$ but $a$ can be an undefined point at the end of ... | Siminore | 29,672 | <ol>
<li>Consider $f(x)=\sqrt{x}$. It is naturally defined on the set $[0,+\infty)$, and it would be wrong to consider subsets like $(-1,1)$, since such an interval is not a subset of the domain of definition.</li>
<li>What you are saying could be translated into the sentence that "every function is a continuous functi... |
3,100,831 | <p>What is the domain of <span class="math-container">$g(x)=\frac{1}{1-\tan x}$</span> </p>
<p>I tried it and got this. But I'm not really sure if it is right. Is that gonna be like this ? <span class="math-container">$(\mathbb{R}, \frac{\pi}{4})$</span></p>
| JustAnAmateur | 589,373 | <p>Any fraction is not defined if its denominator is zero. Hence you must exclude all the points where <span class="math-container">$\tan x=1$</span>,which are <span class="math-container">$x\in\{k\pi+\arctan 1 | k\in \mathbb{Z}\}=\{k\pi + \frac{\pi}{4}| k\in \mathbb{Z}\}$</span>.<br>
However, the <span class="math-con... |
2,979,315 | <p>Let <span class="math-container">$X$</span> be a continuous random variable with uniform distribution between <span class="math-container">$0$</span> and <span class="math-container">$1$</span>. Compute the distribution of <span class="math-container">$Y = \sin(2\pi X)$</span>.</p>
<p><span class="math-container">$... | Greg Johnson | 897,096 | <p>Here is a proof that there are at least 48 symmetries. (Goal: make it obvious and easy to understand.)</p>
<p>Go to a craps table and borrow one of the dice.</p>
<p>There are eight corners on your die, and so you can position the die in any of eight ways based on the corner you are looking at.</p>
<p>For each of t... |
65,886 | <p>It is clear that Sylow theorems are an essential tool for the classification of finite groups.
I recently read an article by Marcel Wild, <em>The Groups of Order Sixteen Made Easy</em>, where he gives a complete classification of the groups of order $16$ that is based on
elementary facts, in particular, he does not ... | Bruce Cooperstein | 754,986 | <p>There is a nice geometric proof: using only Sylow theorems one can show that G has 30 subgroups isomorphic to Z_2 x Z_2 which break into classes of 15 each. Denote by P one class and by L the other. Refer to elements of P as points and elements of L as lines. Say a "point", E, is on the "line", F, E and F interse... |
3,299,296 | <p>The question is </p>
<blockquote>
<p>When <span class="math-container">$~2x^3 + x^2 - 2kx + f~$</span> is divided by <span class="math-container">$~x - 1~$</span>, the remainder is
<span class="math-container">$~-4~$</span>, and when it is divided by <span class="math-container">$~x+2~$</span>, the remainder i... | José Carlos Santos | 446,262 | <p><strong>Hint:</strong> If <span class="math-container">$p(x)=2x^3+x^2-2kx+f$</span>, then <span class="math-container">$p(1)=-4$</span> and <span class="math-container">$p(-2)=11$</span>. So…</p>
|
815,065 | <p>$$\int^{\pi /2}_{0} \frac{\ln(\sin x)}{\sqrt x}dx$$</p>
<p>Use the segment integral formula? The $\sqrt x$ is zero at $x=0$ and $\ln\sin x$ is $-\infty$ </p>
| RRL | 148,510 | <p>Use a limit comparison test with the function $g(x) = x^{-2/3}$. </p>
<p>We have $g$ integrable on $[0,\pi/2]$:</p>
<p>$$\int_{0}^{\pi/2}x^{-2/3}dx = 3(\pi/2)^{1/3} < \infty.$$</p>
<p>Now consider the limit</p>
<p>$$\lim_{x \rightarrow 0} \frac{f(x)}{g(x)}=\lim_{x \rightarrow 0} \frac{x^{-1/2}\ln(\sin x)}{x^{... |
815,065 | <p>$$\int^{\pi /2}_{0} \frac{\ln(\sin x)}{\sqrt x}dx$$</p>
<p>Use the segment integral formula? The $\sqrt x$ is zero at $x=0$ and $\ln\sin x$ is $-\infty$ </p>
| Santosh Linkha | 2,199 | <p>Integrate it by parts
$$\int_{0}^{\pi/2} \frac{\log(\sin (x))}{\sqrt x} dx = \left [ \log( \sin x) (2\sqrt x)\right]_{0}^{\pi/2} - \int_{0}^{\pi/2} \frac{2 \sqrt x}{\sin(x)}\cos(x)dx$$taking limit on first you get you get zero. For the last integral, use <a href="http://en.wikipedia.org/wiki/Jordan%27s_inequality"... |
2,202,382 | <p>When $A^TA = I$, I am told it is orthogonal. What does that mean?</p>
<p>$A = \begin{bmatrix}cos\theta & & -sin\theta \\ \\ sin\theta & & cos\theta\end{bmatrix}, A^T = \begin{bmatrix}cos\theta & & sin\theta \\ \\ -sin\theta & & cos\theta\end{bmatrix}$</p>
| Peter | 82,961 | <p>It means that the row vectors (and also the column vectors) form an orthogonal basis, that means if $A$ has dimension $n\times n$, $A$ consists of $n$ linear independent and pairwise orthogonal vectors spanning $\mathbb R_n$</p>
|
1,939,937 | <p>$(172195)(572167)=985242x6565$</p>
<p>Obviously the answer is 9 if you have a calculator, but how can you find x without redoing the multiplication?</p>
<p>The book says to use congruences, but I don't see how that is very helpful. </p>
| hamam_Abdallah | 369,188 | <p>we use congruence modulo 9.
if a=b (9) and c=d (9) then
ac=bd (9)
what we call the proof by 9.</p>
|
1,753,719 | <p>Definition of rapidly decreasing function</p>
<p>$$\sup_{x\in\mathbb{R}} |x|^k |f^{(l)}(x)| < \infty$$ for every $k,l\ge 0$.</p>
<p>Given the Gaussian function $f(x) = e^{-x^2}$, I know that its derivatives will always be in form of $P(x)e^{-x^2}$ where $P(x)$ is a polynomial of degree, say, $n$. Then $|x|^k |f... | marty cohen | 13,079 | <p>Suppose
$(e^{-x^2})^{(n)}
=p_n(x) e^{-x^2}
$.
Then</p>
<p>$\begin{array}\\
(e^{-x^2})^{(n+1)}
&=(p_n(x) e^{-x^2})'\\
&=p_n'(x) e^{-x^2}-p_n(x)(2x) e^{-x^2}\\
&=e^{-x^2}(p_n'(x) -2xp_n(x))\\
\end{array}
$</p>
<p>so if we define
$p_0(x) = 1$
and
$p_{n+1}(x)
=p_n'(x) -2xp_n(x)
$,
then
$(e^{-x^2})^{(n)}
=p... |
2,725,455 | <p>Probably this is pretty simple (or even trivial), but I'm stucked.</p>
<p>If $H\leq G$ is a subgroup, does it follow that $hH=Hh$, if $h\in H$ ? I can't prove or find a counter-example. If anyone could help me, I'd be grateful!</p>
| Bernard | 202,857 | <p>If $h\in H$, $hH=Hh$ because both are equal to $H$.</p>
<p>Indeed, by definition of a subgroup, $hH\subset H$. Conversely, if $k\in H$, we can write
$\; k=h(h^{-1}k)\in hH$, so $H\subset hH$. Similarly, one checks $H=Hh$.</p>
|
745,436 | <p>I'm reading this pdf <a href="http://rutherglen.science.mq.edu.au/wchen/lndpnfolder/dpn01.pdf" rel="nofollow">http://rutherglen.science.mq.edu.au/wchen/lndpnfolder/dpn01.pdf</a> I understand some of the expression used in this but I don't understand the part $(m,n) = 1$</p>
<p>Is this a cartesian coordinate or some... | amWhy | 9,003 | <p>$(m, n)$ is used by some to denote $\gcd(m, n),$ the <em>greatest common divisor</em> function of two integers, $m, n$. </p>
<p>So $(m, n) = 1$ means that the greatest common divisor of $m, n$ is $1$; i.e., $m, n$ are relatively prime.</p>
<p>Whether $(m, n)$ is used to mean $\gcd(m, n)$, or an ordered pair, depen... |
3,479,953 | <p>Let <span class="math-container">$v={\{v_1,v_2,...,v_k}\}$</span> Linearly independent</p>
<p><span class="math-container">$\mathbb{F} = \mathbb{R}$</span> or <span class="math-container">$\mathbb{F}=\mathbb{C}$</span></p>
<blockquote>
<p>Prove that <span class="math-container">${\{v_1 + v_2 , v_2+v_3, v_3+v_4,.... | tch | 352,534 | <p>The standard numerical approaches would be computing a (rank revealing) QR decomposition or SVD. Both <a href="https://docs.scipy.org/doc/numpy-1.10.4/reference/generated/numpy.linalg.matrix_rank.html" rel="nofollow noreferrer">Numpy</a> and <a href="https://www.mathworks.com/help/matlab/ref/rank.html" rel="nofollow... |
79,658 | <blockquote>
<p>Let $U$ and $W$ be subspaces of an inner product space $V$. If $U$ is a subspace
of $W$, then $W^{\bot}$ is a subspace of $U^{\bot}$?.</p>
</blockquote>
<p>I don't find the above statement intuitively obvious. Could someone provide a proof?</p>
| Jonas Meyer | 1,424 | <p>If you're orthogonal to everything in a set, then you're also orthogonal to everything in every subset of that set.</p>
<p>Put another way: Elements of $W^\perp$ have to be orthogonal to more vectors than elements of $U^\perp$; they have to be orthogonal to the vectors in $U$ <em>and</em> the vectors in $W\setminus... |
3,461,762 | <blockquote>
<p>Is it true that, for any Pythagorean triple <span class="math-container">$4ab > c^2$</span>?</p>
</blockquote>
<p>So this came up in a proof I was working on and it seems experimentally correct from what I've tried and I would imagine the proof is similar to proving,</p>
<p><span class="math-containe... | Trevor Gunn | 437,127 | <p><span class="math-container">$15^2 + 112^2 = 113^2$</span>, but <span class="math-container">$4 \cdot 15 \cdot 112 \le 113^2$</span>.</p>
|
1,448,476 | <p>Let's assume that we are given $f_{X}(x)=0.5e^{-|x|}$, with x being in the set of all real numbers and Y=$|X|^{1/3}$. If I'm asked to find the pdf of Y, do I just follow the formula and do the following?</p>
<p>$f_{Y}(Y)$=$f_{x}(g^{-1}(y))$|$g^{-1}$'(y) to get something like:
$0.5e^{-|y^{1/3}|} |y^{-2/3}/3|$</p>
... | juantheron | 14,311 | <p>Using First derivative $$\displaystyle f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$$</p>
<p>So we get $$\displaystyle f'(x) = \lim_{h\rightarrow 0}\frac{(x+h)^2\cos \left(\frac{1}{x+h}\right)-x^2\cos \left(\frac{1}{x}\right)}{h}$$</p>
<p>So we get $$\displaystyle f'(x) = \lim_{h\rightarrow 0}x^2\cdot \frac{... |
1,448,476 | <p>Let's assume that we are given $f_{X}(x)=0.5e^{-|x|}$, with x being in the set of all real numbers and Y=$|X|^{1/3}$. If I'm asked to find the pdf of Y, do I just follow the formula and do the following?</p>
<p>$f_{Y}(Y)$=$f_{x}(g^{-1}(y))$|$g^{-1}$'(y) to get something like:
$0.5e^{-|y^{1/3}|} |y^{-2/3}/3|$</p>
... | mrf | 19,440 | <p>Assuming you put $f(0)=0$ to make $f$ continuous there, you have
$$
f'(0) = \lim_{h\to0} \frac{f(h)-f(0)}{h} = \lim_{h\to0} \frac{h^2\cos(1/h)}{h} = \lim_{h\to0} h \cos \frac1h = 0
$$
since $|\cos \frac1h| \le 1$ and $h \to 0$.</p>
|
2,031,699 | <p>Let $A,B$ be open subsets of $\mathbb{R}^n$. </p>
<p>Does the following equality hold?</p>
<p>$$\partial(A\cap B)= (\bar A \cap \partial B) \cup (\partial A \cap \bar B)$$</p>
<p>Edit: Thanks for showing me in the answers that above formula fails if $A$ and $B$ are disjoint but their boundaries still intersect. I... | Leo163 | 185,102 | <p>It does not hold. Consider for example $$A=\{x\in \mathbb{R}^n:|x|<1\}$$ and
$$B=\{x\in \mathbb{R}^n:|x-(2,0,\dots,0)|<1\}.$$
Since $A\cap B=\varnothing$, $\partial(A\cap B)=\varnothing$, but the RHS in your formula is the set $\{(1,0,\dots,0)\}$.</p>
|
3,993,727 | <p>For the fucntion <span class="math-container">$f:\mathbb{R}\rightarrow \mathbb{R}$</span> which are differentiable at <span class="math-container">$x=x_0$</span>
imply <span class="math-container">$f'$</span> is continuous at <span class="math-container">$x=x_0$</span>?</p>
<p><span class="math-container">$f$</span>... | JKL | 874,247 | <p>Differentiability does not necessarily imply the derivative is continuous. For a counterexample, see <a href="https://math.stackexchange.com/questions/1391544/differentiable-but-not-continuously-differentiable">Differentiable but not continuously differentiable.</a>.</p>
<p>The class of differentiable functions that... |
3,993,727 | <p>For the fucntion <span class="math-container">$f:\mathbb{R}\rightarrow \mathbb{R}$</span> which are differentiable at <span class="math-container">$x=x_0$</span>
imply <span class="math-container">$f'$</span> is continuous at <span class="math-container">$x=x_0$</span>?</p>
<p><span class="math-container">$f$</span>... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$f(x)=x^{2}\sin (\frac1 x)$</span> for <span class="math-container">$x \neq 0$</span>, <span class="math-container">$ f(0)=0$</span> defines a function which is differentiable at <span class="math-container">$0$</span> but its deriavtive is not continuous at <span class="math-container">... |
1,450,497 | <p>Consider the class of topological spaces $\langle X,\mathcal T\rangle$ such that the following are equivalent for $A\subseteq X$:</p>
<ul>
<li>$A$ is a $G_\delta$ set with respect to $\mathcal T$</li>
<li>$A\in\mathcal T$ or $X\smallsetminus A\in\mathcal T$</li>
</ul>
<p>Open sets, of course, are always $G_\delta$... | marty cohen | 13,079 | <p>For positive integers $x$ and $y$
if $x|y$ and $y|x$
then $x=y$.</p>
<p>Proof:</p>
<p>By prime factorization,
let
$x = \prod_p p^{x_p}$
and
$y = \prod_p p^{y_p}$
.</p>
<p>If $x|y$ then
$x_p \le y_p$.
If $y|x$ then
$y_p \le x_p$.</p>
<p>Therefore,
if $x|y$ and $y|x$
then
$x_p \le y_p$
and
$y_p \le x_p$,
so
$x_p =... |
2,842,217 | <p>im looking to understand the tangent taylor series, but im struggling to understand how to use long division to divide one series (sine) into the other (cosine). I also can't find examples of the Tangent series much beyond X^5 (wikipedia and youtube videos both stop at the second or third term), which is not enough ... | Robert Israel | 8,508 | <p>$$\tan(x) = x+{\frac{1}{3}}{x}^{3}+{\frac{2}{15}}{x}^{5}+{\frac{17}{315}}{x}^{7}+
{\frac{62}{2835}}{x}^{9}+{\frac{1382}{155925}}{x}^{11}+{\frac{21844}{
6081075}}{x}^{13}+\ldots$$</p>
<p>EDIT: Long division:</p>
<p>$$ \matrix{& & x &+ \frac{x^3}{3} &+ \frac{2 x^5}{15} &+ \frac{17 x^7}{315}&+... |
3,057,819 | <p>I giving a second try to this question. Hopefully, with better problem definition.</p>
<p>I have a circle inscribed inside a square and would like to know the point the radius touches when extended. In the figure A, we have calculated the angle(<code>θ</code>), <code>C</code>(center) , <code>D</code> and <code>E</c... | Mohammad Riazi-Kermani | 514,496 | <p>If you know the coordinates of the center then you add <span class="math-container">$r$</span> to the <span class="math-container">$x$</span> coordinate and you add <span class="math-container">$r \tan (\theta)$</span> to the <span class="math-container">$y$</span> coordinate of the center to get coordinates of <spa... |
3,057,819 | <p>I giving a second try to this question. Hopefully, with better problem definition.</p>
<p>I have a circle inscribed inside a square and would like to know the point the radius touches when extended. In the figure A, we have calculated the angle(<code>θ</code>), <code>C</code>(center) , <code>D</code> and <code>E</c... | Michael Hoppe | 93,935 | <p>Describe the circle as
<span class="math-container">$$\vec x=\vec m+\begin{pmatrix} r\cos(t)\\ r\sin(t)
\end{pmatrix}.$$</span>
Now consider the ray
<span class="math-container">$$\vec y=\vec m+\lambda\begin{pmatrix} r\cos(t)\\ r\sin(t)
\end{pmatrix}$$</span>
with <span class="math-container">$\lambda>0$</span>... |
3,581,390 | <p>The problem is as follows:</p>
<p>Mike was born on <span class="math-container">$\textrm{October 1st, 2012,}$</span> and Jack on <span class="math-container">$\textrm{December 1st, 2013}$</span>. Find the date when the triple the age of Jack is the double of Mike's age.</p>
<p>The alternatives given in my book are... | fleablood | 280,126 | <p>Just do it.</p>
<p>Mike was born Oct 1st, 2012. So <span class="math-container">$365$</span> days later is Oct 1st, 2013. And <span class="math-container">$31$</span> days after that is Nov. 1st, 2013 and Mike is <span class="math-container">$365+31 = 396$</span> days old. And <span class="math-container">$30$</s... |
1,392,340 | <p>Suppose there is a function $f:\mathbb R^n \to \mathbb R$. One way to find a stationary value is to solve the ODE $\dot x = - \nabla f(x)$, and look at $\lim_{t\to\infty} x(t)$.</p>
<p>However I want to consider a variation of this method where we solve
$$ dx = - \nabla f(x) dt + C(t,x) \cdot dW_t ,$$
where $C(x,t... | denis | 73,222 | <p>Not directly your question, but
Wales, <a href="http://books.google.com/books?isbn=0521814154" rel="nofollow">Energy landscapes</a> (2003, 681p)
might be interesting.
That leads to "basin-hopping", a <em>practical</em> stochastic algorithm for minimizing e.g.
configurations of molecules.
<a href="http://docs.scipy.o... |
1,734,680 | <p>How can I find $F'(x)$ given $F(x) = \int_0^{x^3}\sin(t) dt$ ? <br>
I think that (by the fundamental theorem of calculus) since $f = \sin(x)$ is continuous in $[0, x^3]$, then $F$ is differentiable and $F'(x) = f(x) = \sin(x)$ but I'm not sure...</p>
| Zelos Malum | 197,853 | <p>You have that $F(x)=1-\cos x^3$ which can then be derived. If you do not however have it, think like this. Let $g(x)=\int f(x)$ for some function. Then for $F(x)=\int^{h(x)}_0 f(x)dx$ we have
$$F'(x)=(g(h(x)-g(0))'=(g(h(x)))'=g'(h(x))h'(x)=f(h(x))h'(x)$$
where I used the chain rule, fundamental theorem of calculus a... |
1,785,444 | <p>The question says to 'Express the last equation of each system as a sum of multiples of the first two equations." </p>
<p>System in question being: </p>
<p>$ x_1+x_2+x_3=1 $</p>
<p>$ 2x_1-x_2+3x_3=3 $</p>
<p>$ x_1-2x_2+2x_3=2 $</p>
<p>The question gives a hint saying "Label the equations, use the gaussian algor... | alphacapture | 334,625 | <p>Hint:</p>
<p>We want to solve</p>
<p>$$a(x_1+x_2+x_3)+b(2x_1-x_2+3x_3)=x_1-2x_2+2x_3$$</p>
<p>Matching coefficients, we want to solve</p>
<p>$$a+2b=1$$</p>
<p>$$a-b=-2$$</p>
<p>$$a+3b=2.$$</p>
<p>Can you take it from here?</p>
|
1,785,444 | <p>The question says to 'Express the last equation of each system as a sum of multiples of the first two equations." </p>
<p>System in question being: </p>
<p>$ x_1+x_2+x_3=1 $</p>
<p>$ 2x_1-x_2+3x_3=3 $</p>
<p>$ x_1-2x_2+2x_3=2 $</p>
<p>The question gives a hint saying "Label the equations, use the gaussian algor... | egreg | 62,967 | <p>You can do row reduction on the <em>transpose</em>:
\begin{align}
\begin{bmatrix}
1 & 2 & 1\\
1 & -1 & -2\\
1 & 3 & 2\\
1 & 3 & 2\\
\end{bmatrix}
&\to
\begin{bmatrix}
1 & 2 & 1\\
0 & -3 & -3\\
0 & 1 & 1\\
0 & 1 & 1\\
\end{bmatrix}
&&\begin{a... |
3,054,321 | <p>I'm looking for a closed form for this sequence,</p>
<blockquote>
<p><span class="math-container">$$\sum_{n=1}^{\infty}\left(\sum_{k=1}^{n}\frac{1}{(25k^2+25k+4)(n-k+1)^3} \right)$$</span></p>
</blockquote>
<p>I applied convergence test. The series converges.I want to know if the series is expressed with any mat... | marty cohen | 13,079 | <p>Proceeding in
my usual naive way,</p>
<p><span class="math-container">$\begin{array}\\
S
&=\sum_{n=1}^{\infty}\sum_{k=1}^{n}\frac{1}{(25k^2+25k+4)(n-k+1)^3}\\
&=\sum_{n=1}^{\infty}\sum_{k=1}^{n}\frac{1}{(25k^2+25k+4)(n-k+1)^3} \\
&=\sum_{k=1}^{\infty}\sum_{n=k}^{\infty}\frac{1}{(25k^2+25k+4)(n-k+1)^3} \... |
4,004 | <p>This is related to <a href="https://math.stackexchange.com/q/133615/26306">this post</a>, please read the comments.</p>
<p>What is the usual way of dealing with that kind of problems on math.SE?
(By "that kind of problems" I mean someone posting tasks from an ongoing contest.)</p>
<p>I mean I did email the contest... | Mariano Suárez-Álvarez | 274 | <p>If anyone notices this happening, it is nice to inform the contest coordinators. </p>
<p>On the other hand, I don't think it is reasonable (nor realistic) to have a policy against this. </p>
|
4,004 | <p>This is related to <a href="https://math.stackexchange.com/q/133615/26306">this post</a>, please read the comments.</p>
<p>What is the usual way of dealing with that kind of problems on math.SE?
(By "that kind of problems" I mean someone posting tasks from an ongoing contest.)</p>
<p>I mean I did email the contest... | Gilles 'SO- stop being evil' | 1,853 | <p>If someone posts a task from an ongoing contest as a question, you may mention it in a comment. Or not, since the fact that the question is a contest task is a related bit of trivia but not particularly relevant in answering the question.</p>
<p>If the question is <em>original</em> from the contest, you should comm... |
1,657,557 | <p>For example, how would I enter y^(IV) - 16y = 0? </p>
<p>typing out fourth derivative, and putting four ' marks does not seem to work. </p>
| Enrico M. | 266,764 | <p>Input:</p>
<p>y''''[x]</p>
<p><a href="http://www.wolframalpha.com/input/?i=y''''(x)+-+16y(x)+%3D+0" rel="nofollow">http://www.wolframalpha.com/input/?i=y''''(x)+-+16y(x)+%3D+0</a>.</p>
|
3,354,566 | <p>I see integrals defined as anti-derivatives but for some reason I haven't come across the reverse. Both seem equally implied by the fundamental theorem of calculus.</p>
<p>This emerged as a sticking point in <a href="https://math.stackexchange.com/questions/3354502/are-integrals-thought-of-as-antiderivatives-to-avo... | hmakholm left over Monica | 14,366 | <p>Let <span class="math-container">$f(x)=0$</span> for all real <span class="math-container">$x$</span>.</p>
<p>Here is one anti-integral for <span class="math-container">$f$</span>:</p>
<p><span class="math-container">$$ g(x) = \begin{cases} x &\text{when }x\in\mathbb Z \\ 0 & \text{otherwise} \end{cases} $... |
2,129,086 | <p>I know that the total number of choosing without constraint is </p>
<p>$\binom{3+11−1}{11}= \binom{13}{11}= \frac{13·12}{2} =78$</p>
<p>Then with x1 ≥ 1, x2 ≥ 2, and x3 ≥ 3. </p>
<p>the textbook has the following solution </p>
<p>$\binom{3+5−1}{5}=\binom{7}{5}=21$ I can't figure out where is the 5 coming from?</... | Rodrigo de Azevedo | 339,790 | <p>The number of nonnegative integer solutions of $x_1 + x_2 + x_3 = 11$ is the coefficient of $t^{11}$ in the following generating function [JDL]</p>
<p>$$\dfrac{1}{(1-t)^3}$$</p>
<p>Suppose now that we are interested in integer solutions with $x_1 \geq 1$, $x_2 \geq 2$ and $x_3 \geq 3$. We thus introduce three new ... |
897,756 | <p>How can I solve the following trigonometric inequation?</p>
<p>$$\sin\left(x\right)\ne \sin\left(y\right)\>,\>x,y\in \mathbb{R}$$</p>
<p>Why I'm asking this question... I was doing my calculus homework, trying to plot the domain of the function $f\left(x,y\right)=\frac{x-y}{sin\left(x\right)-sin\left(y\right... | Varun Iyer | 118,690 | <p>This is the limit definition of the derivative for $\sin x$</p>
<p>So,</p>
<p>$$(\sin x)' = \cos x$$</p>
<p><strong>EDIT</strong>, given any $f(x)$, to find its derivative, through limits we can express it as:</p>
<p>$$\large \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = f'(x)$$</p>
<p>Here $f(x) = \sin x$</p>
|
3,608,625 | <blockquote>
<p>Assume that a function <span class="math-container">$f$</span> satisfies the condition <span class="math-container">$f'(1) = 2.$</span> Figure out the limit <span class="math-container">$$\lim_{h\to0}\frac{f(1+h)-f(1-3h)}{5h}.$$</span></p>
</blockquote>
<p>This seemed to be a very simple problem just... | Robert Israel | 8,508 | <p>The series (starting at <span class="math-container">$k=0$</span>) is <span class="math-container">$a^{-1} \Phi(z, 1, 1/a)$</span> where <span class="math-container">$\Phi$</span> is the Lerch Phi function.</p>
|
3,608,625 | <blockquote>
<p>Assume that a function <span class="math-container">$f$</span> satisfies the condition <span class="math-container">$f'(1) = 2.$</span> Figure out the limit <span class="math-container">$$\lim_{h\to0}\frac{f(1+h)-f(1-3h)}{5h}.$$</span></p>
</blockquote>
<p>This seemed to be a very simple problem just... | marty cohen | 13,079 | <p>If a is an integer,
this will be a multisection
of <span class="math-container">$\ln(x)$</span>.</p>
|
38,731 | <p>The <a href="http://en.wikipedia.org/wiki/Ramanujan_summation">Ramanujan Summation</a> of some infinite sums is consistent with a couple of sets of values of the Riemann zeta function. We have, for instance, $$\zeta(-2n)=\sum_{n=1}^{\infty} n^{2k} = 0 (\mathfrak{R}) $$ (for non-negative integer $k$) and $$\zeta(-(2n... | Sumit Kumar Jha | 37,260 | <p>I would urge you to do analyze the <a href="http://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29" rel="nofollow">harmonic series</a> using <a href="http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula" rel="nofollow">Euler-Maclaurin Summation</a> </p>
<p>You will be able to prove</p>
<p>\begin{eq... |
1,251,914 | <p>I do not understand how to set up the following problem:</p>
<p>"Forces of 20 lb and 32 lb make an angle of 52 degrees with each other. find the magnitude of the resultant force."</p>
<p>An actually picture would really help.</p>
| Pål GD | 55,346 | <p>David Foster Wallace in <em>everything and more, a compact history of ∞</em>:</p>
<blockquote>
<p>There is something I "know," which is that spatial dimensions beyond the Big 3 exist. I can even construct a tesseract or a hypercube out of cardboard. A weird sort of cube-within-a-cube, a tesseract is... |
1,251,914 | <p>I do not understand how to set up the following problem:</p>
<p>"Forces of 20 lb and 32 lb make an angle of 52 degrees with each other. find the magnitude of the resultant force."</p>
<p>An actually picture would really help.</p>
| user21820 | 21,820 | <p><strong>Dimension</strong> usually is just the number of 'components' of some piece of information. 3 dimensions are just nice for describing a position in (Euclidean) space, but you definitely need 4 dimensions if you want to include the time also. Now you are in the room. A while later you are not. Your position h... |
21,141 | <p>Is there a way to extract the arguments of a function? Consider the following example:</p>
<p>I have this sum</p>
<pre><code>g[1] + g[2] + g[3] + g[1]*g[3] + 3*g[1]*g[2] + 6*g[1]*g[2]*g[3]
</code></pre>
<p>Now, what I want to do is exctract the function arguments and apply them to another function <code>func</cod... | Artes | 184 | <p>Use <code>ReplaceAll</code> (<code>/.</code>)</p>
<pre><code>g[1] + g[2] + g[3] + g[1]*g[3] + 3*g[1]*g[2] + 6*g[1]*g[2]*g[3] /. g -> func
</code></pre>
<blockquote>
<pre><code>func[1] + func[2] + 3 func[1] func[2] + func[3] + func[1] func[3]
+ 6 func[1] func[2] func[3]
</code></pre>
</blockquote>
<p><... |
21,141 | <p>Is there a way to extract the arguments of a function? Consider the following example:</p>
<p>I have this sum</p>
<pre><code>g[1] + g[2] + g[3] + g[1]*g[3] + 3*g[1]*g[2] + 6*g[1]*g[2]*g[3]
</code></pre>
<p>Now, what I want to do is exctract the function arguments and apply them to another function <code>func</cod... | gpap | 1,079 | <p>Would this do, or does the ordering matter?</p>
<pre><code> Clear@func
func /: func[a_] func[b_] := func[Flatten@{a, b}]
g[1] + g[2] + g[3] + g[1]*g[3] + 3*g[1]*g[2] + 6*g[1]*g[2]*g[3]/. g[a_] -> func[{a}]
</code></pre>
|
3,173,266 | <p>In the case that <span class="math-container">$M$</span> is a <em>closed</em> orientable <span class="math-container">$3$</span>-manifold, using Wu's formula we can show <span class="math-container">$w_1(M) =0 \implies w_2(M) =0$</span>, and so <span class="math-container">$w_3 = w_1w_2 + Sq^1 w_2 = 0$</span> (or yo... | Tyrone | 258,571 | <p>According to R. Kirby in <em>The Topology of 4-Manifold</em>, section VIII, Theorem 1 on page 46</p>
<blockquote>
<p>Every orientable 3-manifold <span class="math-container">$M^3$</span> is spin and hence parallelizable.</p>
</blockquote>
<p>First he proves the case when <span class="math-container">$M$</span> i... |
3,173,266 | <p>In the case that <span class="math-container">$M$</span> is a <em>closed</em> orientable <span class="math-container">$3$</span>-manifold, using Wu's formula we can show <span class="math-container">$w_1(M) =0 \implies w_2(M) =0$</span>, and so <span class="math-container">$w_3 = w_1w_2 + Sq^1 w_2 = 0$</span> (or yo... | Mizar | 24,753 | <p>All orientable three-manifolds <span class="math-container">$M$</span> are parallelizable. If you just want to deduce the noncompact case from the closed one, this requires little machinery.</p>
<p>Basically, you find first an exhaustion of <span class="math-container">$M$</span> with connected compact manifolds wi... |
3,181,502 | <p>We have <span class="math-container">$\tan(x)=\dfrac{\sin(x)}{\cos(x)}$</span>. I was wondering why <span class="math-container">$\tan(x+{\pi/2})=\tan(x)$</span>?</p>
<p>I wanted to Show </p>
<p><span class="math-container">$$\frac{\sin(x+\pi/2)}{\cos(x+\pi/2)}=\frac{\sin(x)}{\cos(x)}\iff\frac{\sin(x+\pi/2)\cos(x)... | José Carlos Santos | 446,262 | <p>No, since <span class="math-container">$\tan\left(\frac\pi4\right)=1$</span>, whereas <span class="math-container">$\tan\left(\frac\pi2+\frac\pi4\right)=-1$</span>.</p>
|
3,181,502 | <p>We have <span class="math-container">$\tan(x)=\dfrac{\sin(x)}{\cos(x)}$</span>. I was wondering why <span class="math-container">$\tan(x+{\pi/2})=\tan(x)$</span>?</p>
<p>I wanted to Show </p>
<p><span class="math-container">$$\frac{\sin(x+\pi/2)}{\cos(x+\pi/2)}=\frac{\sin(x)}{\cos(x)}\iff\frac{\sin(x+\pi/2)\cos(x)... | Bernard | 202,857 | <p>No, we don't. The real relation is
<span class="math-container">$$\tan\bigl(x+\tfrac\pi 2\bigr)=-\cot x=-\frac 1{\tan x}$$</span>
since <span class="math-container">$\;\sin\bigl(x+\frac\pi 2)=\cos x$</span>, whereas <span class="math-container">$\;\cos\bigl(x+\frac\pi 2\bigr)=\color{red}-\sin x$</span>.</p>
<p>What... |
2,847,277 | <p>Are there primes $p=47\cdot 2^n+1$, where $n\in\mathbb Z_+$? Tested for all primes $p<100,000,000$ without equality.</p>
| 0x.dummyVar | 575,408 | <p>For a cone representing the dispersion of a substrate through a medium with the following properties:</p>
<p>Apex displacement vector   $\vec{s}=[\matrix{s_x \ s_y \ s_z}]$,
<br>
Axis direction vector     $\vec{d}=[\matrix{d_x \ d_y \ d_z}]$ (non-zero),
<br>
Internal angle    &ems... |
201,122 | <p>A little bit of <em>motivation</em> (the question starts below the line): I am studying a proper, generically finite map of varieties $X \to Y$, with $X$ and $Y$ smooth. Since the map is proper, we can use the Stein factorization $X \to \hat{X} \to Y$. Since the composition is generically finite, $X \to \hat{X}$ is ... | jmc | 21,815 | <p>There are three good answers to this question, and together they have more or less answered what I wanted to know. I find it hard to choose one of them as best, but nevertheless I think this question should have an accepted answer to move it from the <em>unanswered list</em>. Hence a CW-answer summarizing the (in my... |
2,225,834 | <blockquote>
<p>Let $f$ and $g$ be functions on $\mathbb{R}^{2}$ defined respectively by </p>
<p>$$f(x,y) = \frac{1}{3}x^3 - \frac{3}{2}y^2 + 2x$$ and</p>
<p>$$g(x,y)=x−y.$$</p>
<p>Consider the problems of maximizing and minimizing $f$ on the
constraint set $$C=\{(x,y)\in\mathbb{R}\,:\,g(x,y)=0\}.$$<... | mlc | 360,141 | <p>You found a <em>local</em> maximum and a <em>local</em> minimum. There is no <em>global</em> maximum because for $x=y \rightarrow \pm\infty$ the objective function $f(x,y) \rightarrow \pm \infty$. </p>
<p>You may also spot this by taking the limits of $\frac{1}{3}x^3-\frac{3}{2}x^2+2x$ as $x \rightarrow \pm \infty$... |
2,225,834 | <blockquote>
<p>Let $f$ and $g$ be functions on $\mathbb{R}^{2}$ defined respectively by </p>
<p>$$f(x,y) = \frac{1}{3}x^3 - \frac{3}{2}y^2 + 2x$$ and</p>
<p>$$g(x,y)=x−y.$$</p>
<p>Consider the problems of maximizing and minimizing $f$ on the
constraint set $$C=\{(x,y)\in\mathbb{R}\,:\,g(x,y)=0\}.$$<... | OnoL | 65,018 | <p>I guess you obtained the conclusion by letting
$$\left(\frac 13 x^3-\frac 32 x^2+2x\right)'=0,$$
which yields that
$$(x-2)(x-1)=0,$$
and so you concluded that the function achieves extrema at $(1,1)$ and $(2,2)$, respectively. This is, however, not true, because the first-order condition only characterizes <strong... |
2,544,864 | <p>I have been trying to prove the continuity of the function:
$f:\mathbb{R}\to \mathbb{R}, f(x) =x \sin(x) $ using the $\epsilon -\delta$ method. </p>
<p>The particular objective of posting this question is to understand <strong>the dependence of $\delta$ on $\epsilon$ and $x$</strong>. I know that $f(x) =x \sin(x) $... | Brian Tung | 224,454 | <p><strong>Basic approach.</strong> Use the fact that the slope of $\sin x$ is everywhere between $-1$ and $1$, so the slope of $x\sin x$ at any point $x$ is guaranteed to be between $-x-1$ and $x+1$. (Thanks to YvesDaoust for the catch.) Thus, if you want to get the function to within $\varepsilon$, you need to get ... |
305,166 | <p>If two undirected graphs are identical except that one has an additional loop at vertex $A$, do they actually have the same complement?</p>
| Community | -1 | <p>The complement of a graph is only defined for simple graphs.</p>
<p><strong>Source</strong>: M.N.S. Swamy and K. Thulasiraman: <em>Graphs, Networks and Algorithms</em> (1981): $\S 1.2$</p>
<p>If we extend the definition to include loopgraphs then the answer is no as well for the following reason:</p>
<p>Suppose $... |
4,475,082 | <p>Problem:</p>
<ul>
<li>Three-of-a-kind poker hand: Three cards have one rank and the remaining two cards have
two other ranks. e.g. {2♥, 2♠, 2♣, 5♣, K♦}</li>
</ul>
<p>Calculate the probability of drawing this kind of poker hand.</p>
<p>My confusion: When choosing the three ranks, the explanation used <span class="mat... | Rebecca J. Stones | 91,818 | <p>Actually, we can use <span class="math-container">$\binom{13}{3}$</span>: it counts the number of ways of choosing 3 distinct ranks. Just don't forget to also choose which of those three ranks (i.e., <span class="math-container">$\binom{3}{1}$</span>) is the special rank with 3 cards. It's another way of counting ... |
1,407,131 | <p>I need to prove the following integral is convergent and find an upper bound
$$\int_{0}^{\infty} \int_{0}^{\infty} \frac{1}{1+x^2+y^4} dx dy$$</p>
<p>I've tried integrating $\frac{1}{1+x^2+y^2} \lt \frac{1}{1+x^2+y^4}$ but it doesn't converge</p>
| Jack D'Aurizio | 44,121 | <p>Continuing from zhw.'s answer,
$$ \int_{0}^{+\infty}\frac{\sqrt{r}}{2(1+r^2)}\,dr = \int_{0}^{+\infty}\frac{u^2\,du}{1+u^4}=\frac{\pi}{2\sqrt{2}} $$
by the residue theorem, while:
$$ \int_{0}^{\pi/2}\frac{d\theta}{\sqrt{\sin\theta}}=\int_{0}^{1}\frac{du}{\sqrt{u(1-u^2)}}=\frac{1}{2\sqrt{2\pi}}\,\Gamma\left(\frac{1}{... |
1,148,720 | <blockquote>
<p>Toni and her friends are building triangular pyramids with golf balls.
Write a formula for the number of golf balls in a pyramid with n
layers, if a $1$-layer pyramid contains 1 ball, a 2-layer pyramid contains 4
balls, a 3-layer one contains 10 balls, and so on.</p>
</blockquote>
<p>What is th... | Marc van Leeuwen | 18,880 | <p>Each layer is a triangular number $\binom n2=\frac{n(n-1)}2 = \sum_{i=0}^{n-1}i$, where the side of the triangle contains $n-1$ balls. Note that the second equality is a special case of the general identity
$$
\sum_{i=0}^{n-1}\binom ik =\binom n{k+1}
$$
which can be proved by an easy induction using Pascal's recur... |
131,206 | <p>According to the wiki of <a href="http://en.wikipedia.org/wiki/Kakutani_fixed-point_theorem">Kakutani's fixed-point theorem</a>, A set-valued mapping $\varphi$ from a topological space $X$ into a powerset $\wp(Y)$ called upper semi-continuous if for every open set $W \subseteq Y$, $\lbrace x| \varphi(x) \subseteq W ... | Gerald Edgar | 454 | <p>The definition quoted is an "order" notion of upper semicontinuous, not a "topology" notion. For real-valued functions, the two coincide. But in other settings you can have one but not the other.</p>
|
131,206 | <p>According to the wiki of <a href="http://en.wikipedia.org/wiki/Kakutani_fixed-point_theorem">Kakutani's fixed-point theorem</a>, A set-valued mapping $\varphi$ from a topological space $X$ into a powerset $\wp(Y)$ called upper semi-continuous if for every open set $W \subseteq Y$, $\lbrace x| \varphi(x) \subseteq W ... | Mikhail Katz | 28,128 | <p>One sensible way of generalizing continuity to set-valued functions (from $X$ to subsets of $Y$) is to require the graph of the function to be closed in the product $X\times Y$. This would be equivalent to the continuity of the function if $Y$ is compact. Thus, the Heaviside function is not continuous because one o... |
663,363 | <p>I do not know if this is an ill-posed question but ...
is $\delta(t)e^{-\gamma t}$ equal to $\delta(t)$?</p>
<p>Thanks,
biologist</p>
| Wei Zhou | 106,010 | <p>Let $a_1,\cdots, a_n, \cdots $ is a basis for $V$. Let $S$ be the vector subspace spanned by $a_2, \cdots, a_{2k}, \cdots$, then $V/S$ is spanned by $a_1+S, \cdots, a_{2k+1}+S. \cdots, $, which is infinite dimension.</p>
|
3,736,706 | <p>Let <span class="math-container">$M$</span> be an <span class="math-container">$A$</span>-module and let <span class="math-container">$\mathfrak{a}$</span> and <span class="math-container">$\mathfrak{b}$</span> be coprime ideals of A.</p>
<p>I must show that <span class="math-container">$M/ \mathfrak{a}M \oplus M/ \... | TomGrubb | 223,701 | <p>In fact it's a bit easier to go backwards. Let <span class="math-container">$[x]\in M/(\mathfrak{a}\cap\mathfrak{b})M$</span>. Since <span class="math-container">$\mathfrak{a}$</span> contains the ideal <span class="math-container">$\mathfrak{a}\cap\mathfrak{b}$</span>, we can restrict <span class="math-container">$... |
4,478,486 | <p>I have just started to read Stein's Singular Integrals and Differentiability properties of functions.</p>
<p>The Hardy-Littlewood maximal function has just been introduced i.e. <span class="math-container">$$M(f)(x):= \sup_{r > 0} \frac{1}{m(B(x,r))}\int_{B(x,r)}|f(y)|dy$$</span></p>
<p>where <span class="math-co... | Mason | 752,243 | <p>For finite dimensional vector spaces, you can view tensors as multilinear maps. The tensor product <span class="math-container">$v_1 \otimes v_2$</span> is the bilinear map on <span class="math-container">$V_1^* \times V_2^*$</span> with <span class="math-container">$$(v_1 \otimes v_2)(\omega_1, \omega_2) = v_1(\ome... |
1,379,188 | <p>The Riemann distance function $d(p,q)$ is usually defined as the infimum of the lengths of all <strong>piecewise</strong> smooth paths between $p$ and $q$.</p>
<p><strong>Does it change if we take the infimum only over smooth paths?</strong>
(Note that if a smooth manifold is connected, <a href="https://math.stacke... | Community | -1 | <p>No, the distance stays the same. As mentioned in <a href="https://math.stackexchange.com/questions/957324/the-formula-for-a-distance-between-two-point-on-riemannian-manifold">The formula for a distance between two point on Riemannian manifold</a>, the reason to allow piecewise smooth curves is to be able to concaten... |
2,060,694 | <p>Problem:</p>
<blockquote>
<p>I have 610 friends. Each one of them will invite me to his birthday party, and I will accept every invitation. What is the probability that I will be attending at least one birthday party on every day of the year?</p>
</blockquote>
<p>My attempt has been to try counting how many ways... | Fimpellizzeri | 173,410 | <p>I know this uses inclusion-exclusion which you stated is not what you wanted. I don't know how far you got into it, or if other users know the end result, but I figured this is too long for a comment and thought someone might find it useful.</p>
<p>Let $p$ be the desired probability.
We will calculate the complime... |
4,537,489 | <p>Assume that <span class="math-container">$a>0$</span>, Suppose we have :<br />
<span class="math-container">$$X = \{x\in \mathbb{R} \ : \ x^2 < a \}$$</span><br />
We should prove that this set has a supremum, and that's <strong><span class="math-container">$\sqrt{a}$</span></strong> .<br />
I saw <a href="htt... | Lorenzo | 365,199 | <p>Given a topological space <span class="math-container">$(X,\tau)$</span> and an uncountable closed discrete set <span class="math-container">$C\subseteq X$</span>, then we can consider the following open cover of <span class="math-container">$X$</span>: <span class="math-container">$$\{\{x\} \mid x \in C\} \cup \{X\... |
2,737,869 | <p>Determine the value of real parameter $p$ </p>
<p>in such a way that the equation</p>
<p>$$\sqrt{x^2+2p} = p+x $$ </p>
<p>has just one real solution</p>
<p>a. $p \ne 0$</p>
<p>b. There is no such value of parameter$p$</p>
<p>c. None of the remaining possibilities is correct.</p>
<p>d. $p\in [−2,\infty)$</p>
... | Misha Lavrov | 383,078 | <p>It follows from <a href="https://en.wikipedia.org/wiki/Roth%27s_theorem" rel="nofollow noreferrer">Roth's theorem</a> that for every <em>algebraic number</em> $\alpha \in [0,1]$ (that is, every $\alpha$ that is the root of some polynomial with rational coefficients) there is a sufficiently small $\epsilon > 0$ fo... |
258,205 | <p>I want to know if $\displaystyle{\int_{0}^{+\infty}\frac{e^{-x} - e^{-2x}}{x}dx}$ is finite, or in the other words, if the function $\displaystyle{\frac{e^{-x} - e^{-2x}}{x}}$ is integrable in the neighborhood of zero.</p>
| Martin Argerami | 22,857 | <p>$$
\frac1x\,(e^{-x}-e^{-2x})=\frac1x\,(1-x+O(x^2)-(1-2x+O(x^2)))=\frac1x\,(x+O(x^2))=1+O(x).
$$
So the function can be extended to $x=0$ in a continuous way, and it thus integrable on any interval $[0,k]$. </p>
|
2,216,778 | <p>My question is if there exists a way to evaluate the sum</p>
<p>$$
{{s}\choose{s}}^{\!2} + {{s + 1}\choose{s}}^{\!2} + \ldots {{s+r}\choose{s}}^{\!2}.
$$</p>
<p>In other words, it's the sum of the squares of the first r binomial coefficients on the s-th right-to-left diagonal of Pascal's triangle. Moreover, is it... | subdiver | 505,246 | <p>The zero element <span class="math-container">$P_n$</span> is not in the set, but I feel some clarification to the above statements is necessary:
The zero of this space is the polynomial with all zero coefficients - not the polynomial that equals zero when you set <span class="math-container">$t=0$</span>. Thus, mul... |
7,871 | <p>I'm trying to make a demonstration of how rounding to different numbers of digits affects things but I can't find a way to round numbers to a specified number of digits. </p>
<p>The <code>Round</code>function only round to the nearest whole integer, and that is not what I always want. Other ways seems to only chang... | Chris Degnen | 363 | <p>Just specify the nearest multiple in the second argument.</p>
<pre><code>Round[123.456, 0.01]
</code></pre>
<blockquote>
<p>123.46</p>
</blockquote>
|
7,871 | <p>I'm trying to make a demonstration of how rounding to different numbers of digits affects things but I can't find a way to round numbers to a specified number of digits. </p>
<p>The <code>Round</code>function only round to the nearest whole integer, and that is not what I always want. Other ways seems to only chang... | Mr.Wizard | 121 | <p>Suppose <code>Round</code> did not take a second argument as it does. What to do?</p>
<pre><code>myround[n_, a_] := Round[n/a] a
myround[π, 0.001]
myround[π, 1/7]
</code></pre>
<blockquote>
<pre><code>3.142
22/7
</code></pre>
</blockquote>
|
2,653,829 | <blockquote>
<p>How can I show $(x^2+1, y^2+1)$ is not maximal in $\mathbb R[x,y]$?</p>
</blockquote>
<p>I know I can mod out the ideal one piece at a time and show $\mathbb C[x]/(x^2+1)$ is not a field since $(x^2+1)$ is not maximal in $\mathbb C[x]$, <strong>but is there another way of showing this?</strong></p>
| quasi | 400,434 | <p>Let $I = (x^2+1,y^2+1)$, let $J=(x^2+1,x-y)$, and let $K=(x^2+1,x+y)$.
<p>
If $1 \in J$, then $a(x^2+1)+b(x-y)=1$, for some $a,b \in \mathbb{R}[x,y]$. But then, letting $y=x$, we get $a(x,x)(x^2+1)=1$, contradiction, since a nonzero multiple of $x^2+1$ must have degree at least $2$.
<p>
If $1 \in K$, then $c(x^2+1)+... |
4,019,119 | <p>im struggeling to find <span class="math-container">$$\lim _{x\to 0}\left(2-e^{\arcsin^{2}\left(\sqrt{x}\right)}\right)^{\frac{3}{x}}$$</span></p>
<p>Ive tried the following:
<span class="math-container">$$\lim_{x \to x_0} ax^{bx} = \lim_{x \to x_0} e^{ax^{bx}} = \lim_{x \to x_0} e^{bx \ln(ax)} = e^{\lim_{x \to x_... | Community | -1 | <p>Apologies in advance for shooting sparrows with cannons, but I can't give away such a funny opportunity to see "pure" math being applied to "simple" geometry problems.</p>
<p>I'll be using Gödel's compactness theorem, which is well-explained <a href="https://math.stackexchange.com/a/663187/632577... |
3,087,570 | <p>The "school identities with derivatives", like
<span class="math-container">$$
(x^2)'=2x
$$</span>
are not identities in the normal sense, since they do not admint substitutions. For example if we insert <span class="math-container">$1$</span> instead of <span class="math-container">$x$</span> into the identity abov... | Derek Elkins left SE | 305,738 | <p>There are at least a few approaches to this.</p>
<p>At a conceptual level the key is that differentiation acts on functions, not numbers. This leads to the first approach. Often notation like <span class="math-container">$\text{D}f$</span> is used where <span class="math-container">$f$</span> is a function. From th... |
4,118,297 | <p>I have been trying to find the closed form for integral below <span class="math-container">$$\int_1^{\infty}\frac{x^2\tan^{-1}(ax)}{x^4+x^2+1}dx ,\; \; a>0 $$</span>
My progress to this integral <span class="math-container">$$\cong\frac{\pi^2}{8\sqrt 3}+\frac{\pi}{8}\log(3)-\frac{\pi}{6a\sqrt 2}+\frac{1}{2a^3}\l... | TheSimpliFire | 471,884 | <p>We have <span class="math-container">\begin{align}\int_1^{\infty}\frac{x^2\arctan ax}{x^4+x^2+1}\,dx&=\int_0^1\frac{\arctan a/u}{u^2(1/u^4+1/u^2+1)}\frac{du}{u^2}\\&=\int_0^1\frac{\pi/2-\arctan u/a}{u^4+u^2+1}\,dx\\&=\frac\pi2\left(\frac14\log3+\frac{\pi\sqrt3}{12}\right)-\int_0^1\frac{\arctan bu}{u^4+u^... |
4,118,297 | <p>I have been trying to find the closed form for integral below <span class="math-container">$$\int_1^{\infty}\frac{x^2\tan^{-1}(ax)}{x^4+x^2+1}dx ,\; \; a>0 $$</span>
My progress to this integral <span class="math-container">$$\cong\frac{\pi^2}{8\sqrt 3}+\frac{\pi}{8}\log(3)-\frac{\pi}{6a\sqrt 2}+\frac{1}{2a^3}\l... | Claude Leibovici | 82,404 | <p>You could simplify the problem using
<span class="math-container">$$\frac{x^2}{x^4+x^2+1}=\frac {x^2}{(x^2-r)(x^2-s)}=\frac 1{r-s} \left(\frac{r}{x^2-r}-\frac{s}{x^2-s}\right)$$</span> where
<span class="math-container">$$r=-\frac{1+i \sqrt{3}}{2} \qquad \text{and} \qquad s=-\frac{1-i \sqrt{3}}{2}$$</span> So, the p... |
3,787,167 | <p>Let <span class="math-container">$\{a_{jk}\}$</span> be an infinite matrix such that corresponding mapping <span class="math-container">$$A:(x_i) \mapsto (\sum_{j=1}^\infty a_{ij}x_j)$$</span> is well defined linear operator <span class="math-container">$A:l^2\to l^2$</span>.
I need help with showing that this ope... | Bart Michels | 43,288 | <p>It is helpful to consider the 'finite rank' case first:</p>
<p><strong>Lemma.</strong> Let <span class="math-container">$(a_n)$</span> be a sequence of real numbers such that for every <span class="math-container">$x = (x_n) \in \ell^2$</span> we have <span class="math-container">$Ax := \sum_{n = 1}^\infty a_n x_n$<... |
1,578,783 | <p>A friend of mine found a book in which the author said that the dual space of $L^\infty$ is $L^1$, of course not with the norm topology but with the weak-* topology. Does anyone know where I can find this result? Thanks.</p>
| cruiser0223 | 684,172 | <p>This is a consequence of the Mackey-Arens Theorem, see for instance <a href="https://www.sciencedirect.com/book/9780125850018/methods-of-modern-mathematical-physics" rel="nofollow noreferrer">Reed, Simon: Methods of modern mathematical physics</a>, Theorem V.22. It says:</p>
<blockquote>
<p>For a dual pair <span cla... |
888,101 | <p>Suppose I am asked to show that some topology is not metrizable. What I have to prove exactly for that ?</p>
| Thomas Andrews | 7,933 | <p>Certainly, if the topology is not Hausdorff, it is not metrizable.</p>
<p>From <a href="http://en.wikipedia.org/wiki/Metrization_theorems" rel="nofollow">Wikipedia</a>:</p>
<blockquote>
<p>A compact Hausdorff space is metrizable if and only if it is <a href="http://en.wikipedia.org/wiki/Second-countable" rel="no... |
888,101 | <p>Suppose I am asked to show that some topology is not metrizable. What I have to prove exactly for that ?</p>
| Asaf Karagila | 622 | <p>Recall that if $(X,d)$ is a metric space, then the metric induces a topology. But different metrics can induce the same topology, for example $(X,d')$ where $d'(x,y)=2d(x,y)$, is a different metric, but induces the same topology.</p>
<p>We say that a topological space is <em>metrizable</em>, if there is a metric wh... |
127,108 | <p>If you take a subtraction-free rational identity like $(xxx+yyy)/(x+y)+xy=xx+yy$ and replace $\times$,$/$,$+$,$1$ by $+$,$-$,min,$0$, do you always get a valid min,plus,minus identity like min(min($x+x+x,y+y+y$)$-$min($x,y$),$\:x+y$)$\ =\ $min($x+x,y+y$)?</p>
| Will Sawin | 18,060 | <p>Yes. Replace $x$ with $e^{-Na}$, $y$ with $e^{-Nb}$, etc. Then take the log, then divide by $N$. One gets a new identity where $\times$ is replaced by $+$, $/$ by $-$, $1$ by $0$, and $u+v$ by $-\ln (e^{-N u} + e^{-N v}) / N= \min(u,v) - \ln\left( 1+ e^{-N |u-v|}\right)/N = \min(u,v) + O(1/N)$. Then take the limit ... |
1,907,743 | <p>I'm having trouble with a step in a paper which I believe boils down to the following inequality:
$$
\left\| \sum_{k\in\mathbb{Z}} f(\cdot+k) \right\|_{L^2(0,1)}
\leq c \|f\|_{L^2(\mathbb{R})}.
$$
I haven't come up with many ideas. Hitting the left-hand side with Minkowski, for example, produces something which... | dls | 1,761 | <p>The counterexample from Artic Char suggested that the statement is true with weight added:
\begin{align}
\left\| \sum_{k\in\mathbb{Z}} f(\cdot+k) \right\|_{L^2(0,1)}
&\leq \sum_{k\in\mathbb{Z}} \|f\|_{L^2(k,k+1)} \\\\
&= \sum_{k\in\mathbb{Z}} \left(\int_k^{k+1} \left|\frac{1+x^2}{1+x^2} \cdot f(x)\... |
48,626 | <p>In <code>ListPointPlot3D</code>, it seems the only point style available is the default, because there is no <code>PlotMarkers</code> option for this function. Is there a way to change the point style? For example, what if I want to draw the points as small cubes?</p>
| anderstood | 18,767 | <p>Another possibility is to use <code>Graphics3D</code>. For example (one with <code>Cuboid</code>, one with <code>Sphere</code>):</p>
<pre><code>pts = RandomReal[10, {20, 3}];
{Graphics3D[{Blue, Cuboid[#, # + .3 {1, 1, 1}] & /@ pts}],
Graphics3D[{Red, Sphere[#, 0.2] & /@ pts}]}
</code></pre>
<p><a href="... |
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