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 |
|---|---|---|---|---|
314,416 | <p>Part of what it means to be a functor between two categories is to have a map of morphisms e.g. $F$ sends $f: A \to B$ to $Ff: FA \to FB$.</p>
<p>Suppose $F$ is a functor from a category to itself, and that category has (some or all) exponential objects. $B^A$ intuitively corresponds to an object of morphisms $A \t... | Manos | 11,921 | <p>Let $M = U \Sigma V^*$ be an SVD of $M$. Suppose that $V=U$. Then $M$ is symmetric positive-definite. Conversely, if $M$ is symmetric positive semi-definite, then any unitary eigendecomposition $M=U \Lambda U^*$ in which the diagonal entries of $\Lambda$ are ordered from maximal to minimal, is an SVD of $M$.</p>
<p... |
13,922 | <p>How to sum up this series :</p>
<p>$$2C_o + \frac{2^2}{2}C_1 + \frac{2^3}{3}C_2 + \cdots + \frac{2^{n+1}}{n+1}C_n$$
</p>
<p>Any hint that will lead me to the correct solution will be highly appreciated.</p>
<p>EDIT: Here $C_i = ^nC_i $</p>
| Timothy Wagner | 3,431 | <p>Hint: Use binomial expansion for $(1+x)^n$ and integrate once. Then choose an appropriate value for $x$.</p>
|
602,973 | <p>I have a problem:</p>
<blockquote>
<p>For $\Omega$ be a domain in $\Bbb R^n$. Show that the function $u(x)=1 \in W^{m,\ 2}(\Omega)$, but not in $W_0^{m,\ 2}(\Omega)$, for all $m \ge 1$.</p>
</blockquote>
<p>=============================</p>
<p>Any help will be appreciated! Thanks!</p>
| Marek | 3,223 | <p>HINT: any smooth function $u_n$ with compact support approximating $1$ well will be close to $1$ on most of $\Omega$ and close to $0$ elsewhere. This implies that there will always be a region where the approximating function has a big gradient. Try to derive a contradiction by showing that the norm of $1 - u_n$ sta... |
175,775 | <p>To clarify the terms in the question above:</p>
<p>The symmetric group Sym($\Omega$) on a set $\Omega$ consists of all bijections from $\Omega$ to $\Omega$ under composition of functions. A generating set $X \subseteq \Omega$ is minimal if no proper subset of $X$ generates Sym($\Omega$).</p>
<p>This might be a dif... | Jeremy Rickard | 22,989 | <p>I think it follows from Theorem 1.1 of "Subgroups of Infinite Symmetric Groups" by Macpherson and Neumann (J. London Math. Soc. (1990) s2-42 (1): 64-84) that there is no minimal generating set of $S(\Omega)$for infinite $\Omega$.</p>
<p>The theorem states that any chain of proper subgroups of $S(\Omega)$ whose unio... |
4,093,406 | <p>Which of the equations have at least two real roots?
<span class="math-container">\begin{aligned}
x^4-5x^2-36 & = 0 & (1) \\
x^4-13x^2+36 & = 0 & (2) \\
4x^4-10x^2+25 & = 0 & (3)
\end{aligned}</span>
I wasn't able to notice something clever, so I solved each of the equations. The first one h... | boojum | 882,145 | <p>There is a test that can be made for biquadratics without the need for square-roots that is akin to a test for the existence of real zeroes of a quadratic polynomial (there is an evident connection with the Rule of Signs that <strong>Gregory</strong> discusses). There, "completing the square" produces
<sp... |
2,078,264 | <p>I've been see the following question on group theory:</p>
<p>Let $p$ be a prime, and let $G = SL2(p)$ be the group of $2 \times 2$ matrices of determinant $1$ with entries in the field $F(p)$ of integers $\mod p$.</p>
<p>(i) Define the action of $G$ on $X = F(p) \cup \{ \infty \}$ by Mobius transformations. [You nee... | AdLibitum | 210,743 | <p>Let $F$ be any field. The group ${\rm GL}_2(F)$ (and thus any of each subgroups) has a natural linear action on the $F$-vector space $F^2$.
By linearity, this action descends to an action on the projective line
${\Bbb P}^1(F)$ which is explicitely given by
$$
\begin{pmatrix} a & b\\ c & d\end{pmatrix}\cdot z... |
3,318,735 | <p>In a solution to a certain Rolle's theorem problem, I have <span class="math-container">$x_k=a+\frac{k}{n}(b-a)$</span> and
<span class="math-container">$$
0=f(b)-f(a)=\sum_{k=1}^nf(x_k)-f(x_{k-1})=\frac{b-a}n\sum_{k=1}^n\frac{f(x_k)-f(x_{k-1})}{x_k-x_{k-1}}
$$</span></p>
<p>How exactly does this work? I mean, let'... | John Omielan | 602,049 | <p>You're given that</p>
<p><span class="math-container">$$x_k=a+\frac{k}{n}(b-a) \tag{1}\label{eq1}$$</span></p>
<p>Also, it's stated that</p>
<p><span class="math-container">$$0=f(b)-f(a)=\sum_{k=1}^nf(x_k)-f(x_{k-1})=\frac{b-a}n\sum_{k=1}^n\frac{f(x_k)-f(x_{k-1})}{x_k-x_{k-1}} \tag{2}\label{eq2}$$</span></p>
<p>... |
3,318,735 | <p>In a solution to a certain Rolle's theorem problem, I have <span class="math-container">$x_k=a+\frac{k}{n}(b-a)$</span> and
<span class="math-container">$$
0=f(b)-f(a)=\sum_{k=1}^nf(x_k)-f(x_{k-1})=\frac{b-a}n\sum_{k=1}^n\frac{f(x_k)-f(x_{k-1})}{x_k-x_{k-1}}
$$</span></p>
<p>How exactly does this work? I mean, let'... | GEdgar | 442 | <p>For those inexperienced in <span class="math-container">$\Sigma$</span> notation, try writing a particular example. Say <span class="math-container">$n=4$</span>. Then <span class="math-container">$x_0 = a$</span> and <span class="math-container">$x_4 = b$</span>.<br>
So
<span class="math-container">$$
\sum_{k=1}^... |
2,230,417 | <p>I need to formally prove that $f(x)=x^2$ is a contraction on each interval on $[0,a],0<a<0.5$. From intuition, we know that its derivative is in the range $(-1,1)$ implies that the distance between $f(x)$ and $f(y)$ is less then the distance between $x$ and $y$. </p>
<p>But now I need an explicit $\lambda$ su... | Siminore | 29,672 | <p>Remark that
$$
|f(x)-f(y)| = |(x+y)(x-y)| \leq (|x|+|y|)|x-y| < 2a |x-y|
$$
if $x$, $y \in [0,a]$. Now $2a<1$ by assumption.</p>
|
1,920,776 | <p>As I know there is a theorem, which says the following: </p>
<blockquote>
<p>The prime ideals of $B[y]$, where $B$ is a PID, are $(0), (f)$, for irreducible $f \in B[y]$, and all maximal ideals. Moreover, each maximal ideal is of the form $m = (p,q)$, where $p$ is an irreducible element in $B$ and $q$ is an irred... | Dylan Sommer | 1,141,434 | <p>With the help of the other answers on this post (especially Ben Grossmann's), I managed to figure out Two's Complement and why it works for myself, but I wanted to add another complete barebones answer for anyone who still can't understand. This is my first post, so thank you for reading in advance. Also, much of my... |
2,470,062 | <blockquote>
<p><span class="math-container">$$\sqrt{k-\sqrt{k+x}}-x = 0$$</span></p>
<p>Solve for <span class="math-container">$k$</span> in terms of <span class="math-container">$x$</span></p>
</blockquote>
<p>I got all the way to
<span class="math-container">$$x^{4}-2kx^{2}-x+k^{2}-x^{2}$$</span>
but could not facto... | Michael Rozenberg | 190,319 | <p>Let $\sqrt{k+x}=y$, where $y\geq0$.</p>
<p>Thus, $\sqrt{k-y}=x$, where $x\geq0$ and we got the following system.
$$k+x=y^2$$ and
$$k-y=x^2,$$
which gives
$$x+y=y^2-x^2$$ or
$$(x+y)(1+x-y)=0.$$
If $x+y=0$, then $x=y=k=0$ otherwise, $y=1+x$ and $k=x^2+x+1$.</p>
<p>Id est, we got the following answer.</p>
<p>If $x&l... |
202,043 | <p>A <a href="http://en.wikipedia.org/wiki/Square-free_word">square-free word</a>
is a string of symbols (a "word") that avoids the pattern $XX$, where $X$ is any
consecutive sequence of symbols in the string.
For alphabets of two symbols, the longest square-free word has length $3$.
But for alphabets of three symbols,... | JRN | 12,357 | <p>This is not an answer, but it may help.</p>
<p>According to <a href="http://www.sciencedirect.com/science/article/pii/S0304397505004019">A generalization of repetition threshold</a> by Lucian Iliea, Pascal Ochemb, and Jeffrey Shallit (<em>Theoretical Computer Science</em>, Volume 345, Issues 2–3, 22 November 2005, ... |
202,043 | <p>A <a href="http://en.wikipedia.org/wiki/Square-free_word">square-free word</a>
is a string of symbols (a "word") that avoids the pattern $XX$, where $X$ is any
consecutive sequence of symbols in the string.
For alphabets of two symbols, the longest square-free word has length $3$.
But for alphabets of three symbols,... | Tony Huynh | 2,233 | <p>A stronger property is true for four-letter alphabets. That is, in <a href="http://cms.math.ca/cmb/v35/cmb1992v35.0161-0166.pdf">Words without near-repetitions</a>, Currie and Bendor-Samuel prove that there is an infinite word $\omega$ over a four-letter alphabet such that whenever $XYX$ occurs as a subword of $\om... |
2,263,431 | <p>Let $$v=\sqrt{a^2\cos^2(x)+b^2\sin^2(x)}+\sqrt{b^2\cos^2(x)+a^2\sin^2(x)}$$</p>
<p>Then find difference between maximum and minimum of $v^2$.</p>
<p>I understand both of them are distance of a point on ellipse from origin, but how do we find maximum and minimum?</p>
<p>I tried guessing, and got maximum $v$ when $... | Maverick | 171,392 | <p>Let $v=\sqrt{a^2\cos^2(x)+b^2\sin^2(x)}+\sqrt{b^2\cos^2(x)+a^2\sin^2(x)}$</p>
<p>$v^2=a^2+b^2+2\sqrt{\left(a^2\cos^2(x)+b^2\sin^2(x)\right)\left(b^2\cos^2(x)+a^2\sin^2(x)\right)}$</p>
<p>$v^2=a^2+b^2+2\sqrt{a^2b^2\left(\cos^4(x)+\sin^4(x)\right)+\left(a^4+b^4\right)\cos^2(x)\sin^2(x)}$</p>
<p>$v^2=a^2+b^2+\sqrt{4... |
535,948 | <p>How do I prove that $(1+\frac{1}{2})^{n} \ge 1 + \frac{n}{2}$ for every $n \ge 1$</p>
<p>My base case is $n=1$ <br/>
Inductive step is $n=k$ <br/>
Assume $n=k+1$<br/></p>
<p>$(\frac{3}{2})^{k} \times \frac{3}{2} \ge (1 + \frac{k+1}{2})$</p>
<p>I'm not sure how to proceed.</p>
| lab bhattacharjee | 33,337 | <p>Let $P(n):(1+x)^n\ge 1+nx$</p>
<p>Clearly $P(n)$ is true for $n=1$</p>
<p>Let $P(n)$ is true for $n=m\implies (1+x)^m\ge 1+mx$</p>
<p>$\implies (1+x)^{m+1}\ge (1+mx)(1+x)=1+(m+1)x+mx^2\ge 1+(m+1)x$ if $m\ge0$ and $x$ is real</p>
|
1,358,574 | <p>When I searched for the derivative of the Gamma function I got something of the form:</p>
<p>$$\Gamma'(x)=\Gamma(x) \psi(x)$$</p>
<p>But from the definition of the Digamma function to me it's like writing:</p>
<p>$$\Gamma'(x)=\Gamma'(x)$$</p>
<p>And this doesn't seem very useful to me (if I'm wrong feel free to ... | lab bhattacharjee | 33,337 | <p>$\dfrac{(n!)^24^n}{(2n)!}=\dfrac{2^n\prod_{r=1}^n(2r)}{2^n\prod_{r=1}^n(2r-1)}=\dfrac{\prod_{r=1}^n(2r)}{\prod_{r=1}^n(2r-1)}$</p>
<p>Now $\dfrac{2r}{2r-1}>1$ for $2r-1>0$</p>
|
1,358,574 | <p>When I searched for the derivative of the Gamma function I got something of the form:</p>
<p>$$\Gamma'(x)=\Gamma(x) \psi(x)$$</p>
<p>But from the definition of the Digamma function to me it's like writing:</p>
<p>$$\Gamma'(x)=\Gamma'(x)$$</p>
<p>And this doesn't seem very useful to me (if I'm wrong feel free to ... | Mark Viola | 218,419 | <p>Recall Stirling's Formula </p>
<p>$$n!=\sqrt{2\pi n}\left(\frac{n}{e}\right)^n\left(1+O\left(\frac1n\right)\right)$$</p>
<p>Then, we have</p>
<p>$$\begin{align}
\frac{(n!)^2\,4^n}{(2n)!}&=\frac{(2\pi n)\left(\frac{n}{e}\right)^{2n}4^n}{\sqrt{4\pi n}\left(\frac{2n}{e}\right)^{2n}}\left(1+O\left(\frac1n\right)\... |
3,812,224 | <p>I have a circle sector for which I know:</p>
<p><strong>The coordinates of the three points <span class="math-container">$A, B$</span>, and <span class="math-container">$S$</span>, and the radius of circle <span class="math-container">$S$</span>.</strong></p>
<p><img src="https://i.stack.imgur.com/RWzHJ.png" alt="en... | John Omielan | 602,049 | <p>First, determine the angle of the entire arc, i.e., <span class="math-container">$\alpha = \measuredangle ASB$</span>. For this, calculate the perpendicular, i.e., shortest, distance from <span class="math-container">$A$</span> to <span class="math-container">$SB$</span>. This is given by the formula for the distanc... |
3,812,224 | <p>I have a circle sector for which I know:</p>
<p><strong>The coordinates of the three points <span class="math-container">$A, B$</span>, and <span class="math-container">$S$</span>, and the radius of circle <span class="math-container">$S$</span>.</strong></p>
<p><img src="https://i.stack.imgur.com/RWzHJ.png" alt="en... | mjw | 655,367 | <p>Write <span class="math-container">$s=x_3+i y_3$</span>, compute the radius <span class="math-container">$r=((x_2-x_3)^2+(y_2-y_3)^2)^{1/2}$</span>, compute <span class="math-container">$\theta_0 = \text{atan} \frac{y_2-y_3}{x_2-x_3}$</span>, and <span class="math-container">$\theta_1=\text{atan} \frac{y_1-y_3}{x_1-... |
264,450 | <p>We know that the class of open intervals $(a,b)$, where $a,b$ are rational numbers is a countable base for $\mathbb R$. </p>
<p>But, $[a,b]$ where $a,b$ are rational numbers does not produce a base for $\mathbb R$.</p>
<p>Can we say that any $(a,b)$ or $[a,b]$ where <strong>$a$ is rational number and $b$ is an ir... | Ittay Weiss | 30,953 | <p>I assume that when you say "a base for $\mathbb R$" you mean "a base for the standard topology on $\mathbb R$. With that, the answer to your question is no since $[a,b]$ is never an open set in the standard topology on $\mathbb R$. </p>
<p>If you also meant to ask whether the collection of all $(a,b)$ where $a$ is ... |
1,526,014 | <p>I have no idea how to compute this infinite sum. It seems to pass the convergence test. It even seems to be equal to $\frac{p}{(1-p)^2}$, but I cannot prove it. Any insightful piece of advice will be appreciated.</p>
| Surb | 154,545 | <p><strong>Hint</strong></p>
<p>$$\sum_{k=1}^\infty kp^k=p\sum_{k=1}^\infty kp^{k-1}.$$</p>
|
1,526,014 | <p>I have no idea how to compute this infinite sum. It seems to pass the convergence test. It even seems to be equal to $\frac{p}{(1-p)^2}$, but I cannot prove it. Any insightful piece of advice will be appreciated.</p>
| Jack D'Aurizio | 44,121 | <p>$$\begin{eqnarray*}(1-p)^2\sum_{k\geq 1}kp^k &=& \sum_{k\geq 1}kp^k -2\sum_{k\geq 1}kp^{k+1} +\sum_{k\geq 1} kp^{k+2}\\&=&(p+2p^2)-(2p^2)+\sum_{k\geq 3}\left(k-2(k-1)+(k-2)\right)p^{k}\\&=&\color{red}{p}.\end{eqnarray*}$$</p>
|
2,590,537 | <p>Let $A$ be a proper subset of of $X$, $B$ be a proper subset of $Y$. If $X$ and $Y$ are connected, show that $(X\times Y)-(A\times B)$ is connected.</p>
<p>I have already seen the solution of this problem here on math.stackexchange, but my confusion is this.
Suppose $X$=(a,b) & $Y$=(c,d) and let $A$={m} &... | Sri-Amirthan Theivendran | 302,692 | <p>A binary $n+1$ vector $(x_1,x_2,\dotsc, x_n, y)$ can be written as $(x,y)$ where $x$ is a binary n-vector. There are $2^n$ choices for $x$ by the inductive hypothesis and $2$ choices for $y$. Hence there are
$$
2^n\times 2
$$
binary $n+1$ vectors.</p>
|
2,309,900 | <p>EDIT: How many ways are there to distribute $0$ and $1$ in a vector of length $N$ such that the number of zeros and 1 is the same (Suppose $N$ is even). I'm guessing it's $\binom{N}{2}$, but I'm not sure.</p>
| Zubin Mukerjee | 111,946 | <p>There are $N/2$ zeroes and $N/2$ ones. It suffices to choose the position of all of the zeroes, so we are choosing $N/2$ slots out of $N$ possibilities:</p>
<p>$$\binom{N}{N/2} = \frac{N!}{\left((N/2)!\right)^2}$$</p>
<p>This is a <a href="https://en.wikipedia.org/wiki/Central_binomial_coefficient" rel="nofollow n... |
2,811,908 | <p>I have this monster as the part of a longer calculation. My goal would be to somehow make it... nicer.</p>
<p>Intuitively, I would try to somehow utilize the derivate and the integral against each other, but I have no idea, exactly how.</p>
<p>I suspect, this might be a relative common problem, i.e. if we want to ... | Disintegrating By Parts | 112,478 | <p>I looked for a way to simplify such things, and found out that trading differentiation for integration was an efficient way to simplify the task. In your case,
\begin{align}
\int_{0}^{1}e^{bx}f(x)dx &= \int_{0}^{1}\left(\int_{0}^{b}xe^{cx}dc+1\right) f(x)dx\\
& = \int_{0}^{b}\int_{0}^{1}e^{cx}xf(x)dx... |
4,480,276 | <p>[This question is rather a very easy one which I found to be a little bit tough for me to grasp. If there is any other question that has been asked earlier which addresses the same topic then kindly link this here as I am unable to find such questions by far.]</p>
<p>let <span class="math-container">$f(x)=x.$</span... | Connor Gordon | 1,049,415 | <p>The textbook is not <em>proving</em> that <span class="math-container">$\delta=\epsilon$</span>. What it means for a limit to exist is that for every such <span class="math-container">$\epsilon$</span> you can choose <em>a</em> <span class="math-container">$\delta$</span> that works. There is no requirement for uniq... |
1,509,502 | <p>I am trying to prove the following statement:</p>
<blockquote>
<p>Suppose <span class="math-container">$\mu$</span> and <span class="math-container">$\nu$</span> are finite measures on the measurable space <span class="math-container">$(X,\mathcal A)$</span> which have the same null sets. Show that there exists a me... | Umberto P. | 67,536 | <p>You know from Radon-Nikodym that there exists a nonnegative measurable function $g$ with the property that $$\mu(A) = \int_A g \, d\nu$$ for every measurable set $A$. Consequently $$\int \phi \, d\mu = \int \phi\, g \, d\nu$$ for every nonnegative simple function $\phi$, and by taking monotone limits you have $$\nu(... |
679,712 | <p>Find the volume bounded by the cylinder $x^2 + y^2=1$ and the planes $y=z , x=0 ,z=0$ in the first octant.</p>
<p>How do I go about doing this?</p>
| G Tony Jacobs | 92,129 | <p>This might be easier in cylindrical coordinates. You just want the region in which $r$ runs from $0$ to $1$, $\theta$ runs from $0$ to $\frac{\pi}2$, and $z$ runs from $0$ to $y=r\sin\theta$. That gives:</p>
<p>$$\int_0^{\frac\pi2}\int_0^1\int_0^{r\sin\theta}r\,dz\,dr\,d\theta$$,</p>
<p>Which I believe comes out t... |
2,253,676 | <p>$f(x)$ continuous in $[0,\infty)$. Both $\int^\infty_0f^4(x)dx, \int^\infty_0|f(x)|dx$ converge. </p>
<p>I need to prove that $\int^\infty_0f^2(x)dx$ converges.</p>
<p>Knowing that $\int^\infty_0|f(x)|dx$ converges, I can tell that $\int^\infty_0f(x)dx$ converges.</p>
<p>Other than that, I don't know anything. I ... | user296113 | 296,113 | <p><strong>Hint</strong> Notice that if $a\ge1$ then $a^4\ge a^2$ and if $ \vert a\vert\le1$ then $a^2\le \vert a\vert$ so in all cases
$$a^2\le \max(\vert a\vert,a^4)$$</p>
|
3,497,919 | <p><span class="math-container">$$\sum_{k=0}^{\Big\lfloor \frac{(n-1)}{2} \Big\rfloor} (-1)^k {n+1 \choose k} {2n-2k-1 \choose n} = \frac{n(n+1)}{2} $$</span></p>
<p>So I feel like <span class="math-container">$(-1)^k$</span> is almost designed for the inclusion-exclusion principle. And the left-hand side looks like s... | Marko Riedel | 44,883 | <p>We seek to show that</p>
<p><span class="math-container">$$\sum_{k=0}^{\lfloor (n-1)/2 \rfloor}
(-1)^k {n+1\choose k} {2n-2k-1\choose n}
= \frac{1}{2} n (n+1).$$</span></p>
<p>The LHS is</p>
<p><span class="math-container">$$\sum_{k=0}^{\lfloor (n-1)/2 \rfloor}
(-1)^k {n+1\choose k} {2n-2k-1\choose n-1-2k}
\\ = [... |
1,703,332 | <p>Let $f \in L^2 \backslash L^\infty$ on some bounded domain. Let $h$ be a function such that $h(p) \to \frac 32$ as $p \to \infty$.</p>
<p>Consider $$I_p = \left(\int |f|^{h(p)}\right)^{\frac{p}{h(p)}}.$$</p>
<p>Is it true that $\lim_{p \to \infty} I_p \leq C$ exist for some finite constant $C$?</p>
<p>I don't th... | MikeP | 323,373 | <p>I think that you can do it in n deletes. Delete the first column (below the useful file). Note that the useful files in 2nd - nth rows now are one element closer to the left. Delete the first column below the (now two!) useful files. Continue this a total of n-1 times, at which point all useful files are along t... |
805,596 | <p>Evaluate $$\lim_{x\to\infty}\dfrac{5^{x+1}+7^{x+2}}{5^x + 7^{x+1}}$$</p>
<p>I'm getting a different result but not the exact one.
I got $$\dfrac{5\cdot\dfrac{5^n}{7^n} +49}{\dfrac{5^n}{7^n} + 7}.$$
I know the result is $7$ but I cannot figure out the steps.</p>
| Michael Hardy | 11,667 | <p>The powers of $5$ are negligible compared to the powers of $7$ when $x$ is big, so you have in effect $7^{x+2}/7^{x+1}$, so the limit is $7$.</p>
<p>One can write
$$
\lim_{x\to\infty} \frac{5^{x+1}+7^{x+2}}{5^x + 7^{x+1}} = \lim_{x\to\infty}\frac{\left(\frac 5 7\right)^{x+1} + 7}{\frac 1 7\left(\frac 5 7 \right)^x ... |
1,363,074 | <p>Q- If roots of quad. Equation $x^2-2ax+a^2+a-3=0$ are real and less than $3$ then,</p>
<p>a) $a<2$ </p>
<p>b)$2<a<3$ </p>
<p>c)$a>4$</p>
<p>In this ques., i used $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ and then if $a$ will be $1,2$ only then the root will be defined but if we use $3$ then there will be only... | Bhaskar Vashishth | 101,661 | <p>Here roots are $a \pm\sqrt{3-a}$. </p>
<p>So roots to be real, $3-a>0 \implies a<3$.</p>
<p>And to satisfy second condition that roots be less than $3$, we see that </p>
<p>$a -\sqrt{3-a}<3$ iff $a<3$ and $a +\sqrt{3-a}<3$ iff $a<2$ , </p>
<p>hence combining them all we get $a<2$.</p>
|
91,766 | <p>Hello,
I am looking for a proof for the Chern-Gauss-Bonnet theorem. All I have found so far that I find satisfactory is a proof that the euler class defined via Chern-Weil theory is equal to the pullback of the Thom class by the zero section, but I would like a proof of the fact that this class gives the Euler char... | drbobmeister | 8,472 | <p>Chern's original paper may be found at: Chern, Shiing-Shen (1945), "On the curvatura integra in Riemannian manifold", Annals of Mathematics 46 (4): 674–684; this citing quoted from the Wikipedia entry, though I have a copy of the
original paper somewhere in the piles in my office.</p>
|
31,790 | <p>See following three snippet code:</p>
<pre><code>(*can compile*)
range = Range[-2, 2, 0.005];
Compile[{},
With[{r = range}, Table[ArcTan[x, y], {x, r}, {y, r}]]][];
(*can compile*)
Compile[{},
With[{r = Range[-2, 2, 0.005 - 10^-8]}, Table[ArcTan[x, y], {x, r}, {y, r}]]][];
(*can't compile*)
Compile[{},
... | Mr.Wizard | 121 | <p>The warning messages tell you:</p>
<blockquote>
<p>CompiledFunction::cfn: Numerical error encountered at instruction 33; proceeding with uncompiled evaluation. >></p>
<p>ArcTan::indet: Indeterminate expression ArcTan[0.,0.] encountered. >></p>
</blockquote>
<p>By using an offset of <code>10^-8</code> you... |
2,411,081 | <p>How can I show (with one of the tests for convergence , <strong>not by solving</strong>) that the integral $$\int _{1}^\infty\frac{\ln^5(x)}{x^2}dx$$ converges?</p>
| Fred | 380,717 | <p>Show that $ \lim_{x \to \infty} \frac{\ln^5(x)}{\sqrt{x}}=0$. Hence there is $a>1$ sucht that </p>
<p>$0 < \frac{\ln^5(x)}{\sqrt{x}} \le 1$ for $x>a$. It follows:</p>
<p>$0 < \frac{\ln^5(x)}{x^2} \le \frac{1}{x^{3/2}}$ for $x>a$. </p>
<p>Can you proceed ?</p>
|
2,078,943 | <p>My book says this but doesn't explain it:</p>
<blockquote>
<p>Row operations do not change the dependency relationships among columns.</p>
</blockquote>
<p>Can someone explain this to me? Also what is a dependency relationship? Are they referring to linear dependence? </p>
| hardmath | 3,111 | <p>Consider an $m\times n$ matrix with columns:</p>
<p>$$\begin{pmatrix} v_1 & v_2 & \ldots & v_n \end{pmatrix}$$</p>
<p>Now a dependence relation on the columns would express zero as a nontrivial linear combination of them:</p>
<p>$$ \sum_{i=1}^n c_i v_i = 0 $$</p>
<p>where $c_1,c_2,\ldots,c_n$ are sca... |
1,397,646 | <blockquote>
<p>Given $f_n:= e^{-n(nx-1)^2} $, any suggested approaches for showing that $\displaystyle \lim_{n \to \infty}\int^1_0 f_n(x)dx = 0$? </p>
</blockquote>
<p>I have already shown that $f_n \to 0$ pointwise (and not uniformly) on $[0,1]$ and have been trying to show that $$ \lim_{n \to \infty}\int^1_0 f_n(... | Cameron Williams | 22,551 | <p>Here is a somewhat.. cheap way of doing the problem that you might feel a bit unsatisfied with but consider the following instead:</p>
<p>$$\int_{-\infty}^{\infty} e^{-n(nx-1)^2}\,dx.$$</p>
<p>Clearly</p>
<p>$$\int_0^1 e^{-n(nx-1)^2}\,dx \le \int_{-\infty}^{\infty} e^{-n(nx-1)^2}\,dx$$</p>
<p>since the integrand... |
1,634 | <p>I am not too certain what these two properties mean geometrically. It sounds very vaguely to me that finite type corresponds to some sort of "finite dimensionality", while finite corresponds to "ramified cover". Is there any way to make this precise? Or can anyone elaborate on the geometric meaning of it?</p>
| Peter Arndt | 733 | <p>If you have a morphism X-->Y of schemes, finite type means that the fibers are finite dimensional and finite, that the fibers are zero-dimensional. </p>
<p>Take for a finite type example K[x]-->K[x,y]. This corresponds to the projection A^2-->A^1, the fibers are 1-dimensional, which is reflected by K[x,y] being a K... |
20,683 | <p>The most elementary construction I know of quantum groups associated to a finite dimensional simple Hopf algebra is to construct an algebra with generators $E_i$ and $F_i$ corresponding to the simple positive roots, and invertible $K_j$'s generating a copy of the weight lattice. Then one has a flurry of relations b... | Simon Lentner | 22,709 | <p><strong>Yes!</strong> It is indeed, to get the <strong>quantum group</strong> $\mathcal{U}_q(\mathfrak{g})$, you have to...</p>
<ul>
<li><p>start with a groupring $\mathcal{U}^0(\mathfrak{g})=\mathbb{k}[\mathbb{Z}^n]=\langle K_1,\ldots K_n\rangle_{Alg}$ representing the <strong>Cartan algebra</strong> $\mathcal{U}^... |
2,409,377 | <p>I'm trying to prove that if $F \simeq h_C(X)$ or "$X$ represents the functor $F$", then $X$ is unique up to unique isomorphism. I already know that if $h_C(X) \simeq F \simeq h_C(Y)$ that $s: X \simeq Y$ since Yoneda says that $h_C(X)$ is fully faithful, so reflects isomorphisms (in either direction). If $h_C(X) ... | Derek Elkins left SE | 305,738 | <p>There are as many isomorphisms between $h_C(X)$ and $h_C(Y)$ as there are isomorphisms between $X$ and $Y$.</p>
<p>If $\varphi: h_C(X)\cong F$ and $\psi : h_C(Y)\cong F$, then there is a unique isomorphism $\alpha : h_C(X) \cong h_C(Y)$ <em>such that</em> $\psi \circ \alpha = \varphi$ (just post-compose both sides ... |
308,426 | <p>Assume that a sequence $(x_n)_{n\in\omega}$ of points of a locally convex topological vector space converges to zero. Is it always possible to find increasing number sequences $(n_k)_{k\in\omega}$ and $(m_k)_{k\in\omega}$ such that the sequence $(m_kx_{n_k})_{k\in\omega}$ still converges to zero? </p>
<p><strong>Ad... | Taras Banakh | 61,536 | <p>Oh, sorry! I posed this question too quickly. </p>
<p>A simple example is the dual space $\ell_1=c_0^*$ to the Banach space $c_0$, endpowed with the weak$^*$ topology. The sequence $(e^*_n)_{n\in\omega}$ of coordinate functionals tends to zero, but for any increasing number sequences $(m_k)_{n\in\omega}$ and $(n_k)... |
4,070,810 | <p>I'm trying to find the number of subgroups of <span class="math-container">$\mathbb{Z}^n$</span> such that the quotient is a <span class="math-container">$\bf{cyclic}$</span> group of order <span class="math-container">$p^k$</span> (<span class="math-container">$p$</span> a prime).</p>
<p>I guess I don't know enough... | Futurologist | 357,211 | <p><span class="math-container">$AB = BD$</span> implies that their corresponding circular arcs are equal too, so <span class="math-container">$EB$</span> is the angle bisector of <span class="math-container">$\angle \, AED$</span> and thus <span class="math-container">$\angle \, AEB = \angle \, DEB$</span>. Let <span ... |
2,662,792 | <p>I'm looking for a Galois extension $F$ of $\mathbb{Q}$ whose associated Galois group $\mbox{Gal}(F, \mathbb{Q})$ is isomorphic to $\mathbb{Z}_3 \oplus \mathbb{Z}_3$. I'm wondering if I should just consider some $u \notin Q$ whose minimum polynomial in $Q[x]$ has degree 9. In that case we'd have $[\mathbb{Q}(u) : \ma... | hunter | 108,129 | <p>It turns out that if you pick a polynomial at random, it's unlikely that the extension will be Galois (even if you could guarantee what it's Galois group would be). This can be quantified in various ways.</p>
<p>if $L/\mathbb{Q}$ and $K/\mathbb{Q}$ are linearly disjoint, Galois extensions, then the Galois group of ... |
2,439,278 | <p>I wanted to solve the following problem.</p>
<blockquote>
<p>In $\triangle ABC$ we have $$\sin^2 A + \sin^2 B = \sin^2 C + \sin A \sin B \sin C.$$ Compute $\sin C$.</p>
</blockquote>
<p>Since it's an equation for a triangle, I assumed that $\pi = A + B + C$ would be important to consider.</p>
<p>I've tried solv... | lab bhattacharjee | 33,337 | <p>Using <a href="https://math.stackexchange.com/questions/175143/prove-sinab-sina-b-sin2a-sin2b">Prove $ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $</a>,</p>
<p>$$\sin^2A+\sin(B-C)\sin(B+C)=\sin A\sin B\sin C$$</p>
<p>As $\sin(B+C)=\sin(\pi-A)=\sin A$ and $\sin A>0$ cancelling it in both sides</p>
<p>$$\sin A+\sin(B-C)... |
2,229,244 | <p>combinatorial proof of $2^{n+1}\nmid (n+1)(n+2)\dots (2n)$.</p>
<p>I don't have any ideas about proving $sth \nmid sthx$ using combinatorics for showing $sth \mid sthx$ we show that there is a problem that gives $\frac{sthx}{sth*k}$ but what about proving sth doesn't divide sth using combinatorics?</p>
| the_firehawk | 416,286 | <p>by definition $\phi$ is surjective if $\forall z\in \mathbb{C} \;\exists \mathcal{P}\in \mathbb{R}[X]$ such that $\phi(\mathcal{P}) = z = a+ib$
and here the $\mathcal{P}$ is simply $a+bx$</p>
|
3,162,294 | <p>I've been trying to prove this statement by opening up things on the left hand side using the chain rule but am really getting nowhere. Any tips/hints would be very helpful and appreciated!</p>
| btilly | 6,708 | <p>This is the same as the answer I gave to the now closed <a href="https://stats.stackexchange.com/questions/399396/probability-that-rolling-n-dice-that-the-two-larger-dice-sum-to-x/399409#399409">https://stats.stackexchange.com/questions/399396/probability-that-rolling-n-dice-that-the-two-larger-dice-sum-to-x/399409#... |
1,063,774 | <blockquote>
<p>Prove that the <span class="math-container">$\lim_{n\to \infty} r^n = 0$</span> for <span class="math-container">$|r|\lt 1$</span>.</p>
</blockquote>
<p>I can't think of a sequence to compare this to that'll work. L'Hopital's rule doesn't apply. I know there's some simple way of doing this, but it ... | Richard | 173,949 | <p>The case where $r=0$ is trivial. WLOG, suppose that $0<r<1$. </p>
<p>We
let $M= \frac{1}{r}-1$. </p>
<p>Now take any $\epsilon >0$. There exists $N \in \mathbb{N}$ such that $N > \frac{1}{\epsilon M}$.</p>
<p>By Binomial expansion, for $n \geq N$,</p>
<p>\begin{eqnarray}
r^n &= & \frac{1}{(1+... |
1,659,224 | <p>Find a sequence $a_n$ such that $\lim_{n \to \infty}|a_{n+1} - a_n | = 0$ while the sequence does not converge.</p>
<p>I am thrown for a loop. For a sequence not to converge it means that for all $\epsilon \geq 0$ and for all $N$, when $n \geq N$, $|a_n - L| \geq \epsilon$ as $n \longrightarrow \infty$</p>
<p>or I... | Daniel R. Collins | 266,243 | <p>How about $a_n = \sum_1^n \frac{1}{m}$? The sequential differences are of course equal to $\frac{1}{n}$, but the limit of the sequence is the sum of all such fractions. </p>
<p>Of course, that's just one concrete example of what @Jake said.</p>
|
4,075,667 | <p>[this is question. I want to know about iv.] I want to know that without being defined everywhere ,can a mapping be onto and one-to-one?
In iv - D has four elements and B has three elements, while in question only three elements are used. So it cannot be function as per definition. Then how it will onto or one to o... | DanielWainfleet | 254,665 | <p>This is a matter of terminology and notation. In Set Theory a function <span class="math-container">$is$</span> its graph, and when we say "<span class="math-container">$f$</span> is a function from <span class="math-container">$A$</span> to <span class="math-container">$B\,$</span>", or write <span class... |
1,108,787 | <p><img src="https://i.stack.imgur.com/YfmKh.png" alt="enter image description here"></p>
<p>I know we have to prove these 10 properties to prove a set is a vector space. However, I don't understand how to prove numbers 4 and 5 on the list.</p>
| DeepSea | 101,504 | <p>Its better to give an example of a set with a given binary operation on it and we can find the zero vector, and the additive inverse of any given vector. I would take the set $V = \{M: M \in R^{2\times 2}\}$, and the operation is addition of matrices $\text{+}$ , then you can find the zero vector and the additive in... |
868,384 | <p>Let's assume I have a set like $S = \{2,1,3,4,8,10\}$. </p>
<p>What's the math notation for the smallest number in the set? </p>
| Mr.Fry | 68,477 | <p>In general for a given set $S$ which is nonempty and a subset of an ordered field we define the smallest element in the set to be the element $x \in S$ such that $x\leq y, \ \forall y \in S$. Since you said in a set, I will not introduce the notion of inf. I hope this helps.</p>
|
1,167,438 | <p>I am currently solving a differential equation but I am having a little trouble figuring out my integrative factor. </p>
<p>I have </p>
<p>$$\exp\bigg({∫\frac{3}{t}dt}\bigg)$$</p>
<p>so to integrate I made it $\exp(\ln(t^3))$ what is the final solution? $t^3$?</p>
| Tad | 85,024 | <p>$$\int_0^1 da \int_0^1 db \int_0^1 dc \int_0^{{\rm min}(1,a+b+c)} dd \int_0^{{\rm min}(1,a+b+c-d)} de = \frac{31}{40}=0.775$$</p>
|
1,535,914 | <p>When I plot the following function, the graph behaves strangely:</p>
<p><span class="math-container">$$f(x) = \left(1+\frac{1}{x^{16}}\right)^{x^{16}}$$</span></p>
<p>While <span class="math-container">$\lim_{x\to +\infty} f(x) = e$</span> the graph starts to fade at <span class="math-container">$x \approx 6$</span>... | Hagen von Eitzen | 39,174 | <p>32bit vs. 64bit affects which integer types are used by default, which is of no interest here. Rather, the floting point computations are made (by default) with IEEE <code>double</code> type.
With this <code>double</code> precision (53 bit mantissa), the relative error of $(1+\frac1{x^{16}})$ is approximately $2^{-5... |
40,463 | <p>If a rectangle is formed from rigid bars for edges and joints
at vertices, then it is flexible in the plane: it can flex
to a parallelogram.
On any smooth surface with a metric, one can define a linkage
(e.g., a rectangle) whose edges are geodesics of fixed length,
and whose vertices are joints, and again ask if it ... | maproom | 9,547 | <p>There is an obvious metric on the sphere, which you are assuming applies. There is more than one way to put a nice metric on a torus - but I think you mean a torus embedded in three-space, which does not admit such a nice metric. You need to specify what metric you are assuming on the surfaces you are interested in.... |
2,565,763 | <p>I was playing a standard solitaire game on my mobile app and I came across a round where I couldn't perform a single move thus resulting in a loss. I was then thinking as to what the probability of this event to happen in a single game. Would anybody know this probability and show the calculations given a standard d... | symplectomorphic | 23,611 | <p>I think neither of the other two answers succinctly answers your question "why $Z$?"</p>
<p>The point is that $Z$ only appears in this example <em>because of the distributional assumption made in the problem statement</em>. If you make different distributional assumtions, the likelihood ratio inequality may not yie... |
4,036,975 | <p><span class="math-container">$\require{begingroup} \begingroup$</span>
<span class="math-container">$\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}\def\Catalan{\mathsf{Catalan}}$</span></p>
<p><a href="https://math.stackexchange.com/q/3590905/122782">Related quest... | Svyatoslav | 869,237 | <p>Just to put in more closed form what @Varu Vejalla has evaluated</p>
<p><span class="math-container">$I=\int_0^1\frac{\ln(1+x^{2n})}{1+x^2} dx$</span></p>
<p><span class="math-container">$\int_0^\infty\frac{\ln(1+x^{2n})}{1+x^2} dx=\frac{1}{2}\int_{-\infty}^\infty\frac{\ln(1+x^{2n})}{1+x^2} dx=\int_0^1\frac{\ln(1+x^... |
603,291 | <p>Suppose $f:(a,b) \to \mathbb{R} $ satisfy $|f(x) - f(y) | \le M |x-y|^\alpha$ for some $\alpha >1$
and all $x,y \in (a,b) $. Prove that $f$ is constant on $(a,b)$. </p>
<p>I'm not sure which theorem should I look to prove this question. Can you guys give me a bit of hint? First of all how to prove some function... | Did | 6,179 | <blockquote>
<p>Derivatives are not needed here (which is a good thing since proofs based on them can lead to erroneous arguments, see the accepted answer for an example)...</p>
</blockquote>
<p>Choose $y\gt x$ in the interval $(a,b)$. For every $n\geqslant1$, divide the interval $(x,y)$ into $n$ subintervals $(x_i,... |
2,463,052 | <p>I admit $\frac{1}{1+b}<1$ is trivial for $b>0.$ However, the above claim boils down to suggesting that, any $y\in\{x\in\mathbb{R}:x>1\}$ can be uniquely represented by $1+b$ for some $b>0.$ Can you give me some hints how to prove it?</p>
| Piquito | 219,998 | <p>HINT.-Because if $f(x)=\dfrac{1}{1+x}$ then $f([0,+\infty[)=]0,1]$</p>
|
597,986 | <p>It's a classical theorem of Lie group theory that any compact connected abelian Lie group must be a torus. So it's natural to ask what if we delete the connectedness, i.e. the problem of classification of the compact abelian Lie groups.</p>
| Cronus | 229,396 | <p>Let <span class="math-container">$G$</span> be an abelian Lie group. The connected component <span class="math-container">$G_0$</span> is a connected abelian Lie group, so it's isomorphic to <span class="math-container">$\Bbb{R}^n\times\Bbb{T}^m$</span> for some <span class="math-container">$n,m\in\omega$</span> (th... |
169,919 | <blockquote>
<p>If $p$ is a prime, show that the product of the $\phi(p-1)$ primitive roots of $p$ is congruent modulo $p$ to $(-1)^{\phi(p-1)}$.</p>
</blockquote>
<p>I know that if $a^k$ is a primitive root of $p$ if gcd$(k,p-1)=1$.And sum of all those $k's$ is $\frac{1}{2}p\phi(p-1)$,but then I don't know how use... | lab bhattacharjee | 33,337 | <p>We know $ ord_m(a^k) =\frac{d}{(d, k)} $ where d=$ord_ma$ => $ord_pa=ord_p(a^{-1})$</p>
<p>There are $\phi(p-1)$ primitive roots for prime $p$.</p>
<p>As $\phi(n)$ is even for $n>2$ ,i.e. for $p-1>2$.</p>
<p>So,for $p>3$, we can always find $a^{-1}$ for each primitive root $a$ of $p$.</p>
<p>Now if $a... |
268,482 | <p>One has written a paper, the main contribution of which is a few conjectures. Several known theorems turned out to be special cases of the conjectures, however no new case of the conjectures was proven in the paper. In fact, no new theorem was proven in the paper. </p>
<p>The work was reported on a few seminars, an... | T. Amdeberhan | 66,131 | <p>This is interesting to me too. In my case, I decided for a simple approach: by posting on the arXiv, here is <a href="https://arxiv.org/abs/1207.4045" rel="nofollow noreferrer">one such paper</a>. I also created a separate route on my own website. Since then the conjectures have received several attention as you can... |
268,482 | <p>One has written a paper, the main contribution of which is a few conjectures. Several known theorems turned out to be special cases of the conjectures, however no new case of the conjectures was proven in the paper. In fact, no new theorem was proven in the paper. </p>
<p>The work was reported on a few seminars, an... | Brian Hopkins | 14,807 | <p>For open problems/conjectures in combinatorial and discrete geometry, there's the journal <a href="http://geombina.uccs.edu/" rel="nofollow noreferrer"><em>Geombinatorics</em></a> focusing on "live mathematics," i.e., research in progress. </p>
<p>One big title is Aubrey de Grey's 2018 "The chromatic number of the... |
1,265,987 | <p>I am working on some Automata practice problems. I am working a 2 part question. Here it is:</p>
<blockquote>
<p>Let $\Sigma = \{a,b\}$ be an alphabet.
Let $L = \left\{w \in \Sigma^* \mid n_a(w) \le 4\right\}$</p>
<p>a) Create a transition graph for $L$</p>
<p>b) Find a regular expression for $L$</p>... | Brian M. Scott | 12,042 | <p>Your work for (a) is fine, but your regular expression is wrong: its last term matches $aaaaa$, for instance, which is not in $L$, and none of it matches $bab$, which is in $L$. It looks as if you tried to generate it from the transition graph and made quite a few mistakes; in particular, it looks as if you forgot t... |
4,097,262 | <p><a href="https://i.stack.imgur.com/E1UoR.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/E1UoR.png" alt="Cubic equation with a unit radius circle" /></a>
A cubic equation and circle (unit radius) has intersection at A,B,C,D. ABCD is a square. <em>Find the angle <span class="math-container">$\theta... | RicardoCruz | 36,340 | <p>Let
<span class="math-container">$$f(x)=ax^3+bx. \quad \quad (1)$$</span>
Let's find out the values of <span class="math-container">$a$</span> and <span class="math-container">$b$</span> for a specific value of <span class="math-container">$R$</span>.</p>
<p>Let the points <span class="math-container">$ A=(R \cos \t... |
545,728 | <p>So I have $X \sim \text{Geom}(p)$ and the probability mass function is:</p>
<p>$$p(1-p)^{x-1}$$</p>
<p>From the definition that:</p>
<p>$$\sum_{n=1}^\infty ns^{n-1} = \frac {1}{(1-s)^2}$$</p>
<p>How would I show that the $E(X)=\frac 1p$</p>
| thepiercingarrow | 274,737 | <p>A simpler way would be to plug in $q=1-p$ and solve it that way using formula for geometric sequences:</p>
<p>\begin{align*}
E(X) &= \sum\limits_{k=1}^\infty kpq^{k-1}\\
&= \frac{p}{q} \sum\limits_{k=1}^\infty kq^{k}\\
&= \frac{p}{q} \frac{q}{(1-q)^2}\\
&= \frac{p}{q} \frac{q}{p^2}\\
&= \frac{p}... |
357,138 | <blockquote>
<p>If $a_1,a_2,\dotsc,a_n $ are positive real numbers, then prove that</p>
</blockquote>
<p>$$\lim_{x \to \infty} \left[\frac {a_1^{1/x}+a_2^{1/x}+.....+a_n^{1/x}}{n}\right]^{nx}=a_1 a_2 \dotsb a_n.$$ </p>
<p>My Attempt: </p>
<p>Let $P=\lim_{x \to \infty} \left[\dfrac {a_1^{\frac{1}{x}}+a_2^{\frac {1... | Xiaolang | 71,857 | <p>I will just prove this limit
<span class="math-container">$$
\lim_{x\to\infty} \left(\frac {a_1^{\frac{1}{x}}+a_2^{\frac{1}{x}}+.....+a_n^{\frac{1}{x}}}{n}\right)^{x}=(a_1.a_2.....a_n)^{\frac{1}{n}}
$$</span>
We have
<span class="math-container">$$\begin{align}
\lim_{x\to\infty} \left(\frac {a_1^{\frac{1}{x}}+a_2^{\... |
3,752,948 | <p>Find <span class="math-container">$\displaystyle \lim_{x \to\frac{\pi} {2}} \{1^{\sec^2 x} + 2^{\sec^2 x} + \cdots + n^{\sec^2 x}\}^{\cos^2 x} $</span><br />
I tried to do like this :
Let <span class="math-container">$A=\displaystyle \lim_{x \to\frac{\pi}{2}} \{1^{\sec^2 x} + 2^{\sec^2 x} + \cdots + n^{\sec^2 x}\}^{... | Mark Viola | 218,419 | <p>Let <span class="math-container">$f_n(x)$</span> be the sequence given by</p>
<p><span class="math-container">$$f_n(x)=\left(\sum_{k=1}^n k^{\sec^2(x)}\right)^{\cos^2(x)}$$</span></p>
<p>Then, we have</p>
<p><span class="math-container">$$\begin{align}
f_n(x)&=\left(n^{\sec^2(x)}\sum_{k=1}^n \left(\frac kn\right... |
478,523 | <p>I'm trying to reason through whether $\int_{x=2}^\infty \frac{1}{xe^x} dx$ converges.</p>
<p>Intuitively, It would seem that since $\int_{x=2}^\infty \frac{1}{x} dx$ diverges, then multiplying the denominator by something to make it smaller would make it converge.</p>
<p>If I apply a Taylor expansion to the $\fra... | André Nicolas | 6,312 | <p>The Taylor expansion works. Note from $e^x=1+x+\frac{x^2}{2!}+\cdots$, we have $e^x\gt x$ for positive $x$.</p>
<p>It follows that
$$\frac{1}{xe^x}\lt \frac{1}{x^2}.$$
But we know that $\displaystyle\int_2^\infty \frac{1}{x^2}\,dx$ converges, and therefore by Comparison so does our integral.</p>
<p>As has been po... |
1,197,875 | <p>I'm just starting partials and don't understand this at all. I'm told to hold $y$ "constant", so I treat $y$ like just some number and take the derivative of $\frac{1}{x}$, which I hope I'm correct in saying is $-\frac{1}{x^2}$, then multiply by $y$, getting $-\frac{y}{x^2}$.</p>
<p>But apparently the correct answe... | ASB | 111,607 | <p>By first principles,</p>
<p>$ \dfrac{\partial}{\partial y}\left( \dfrac{y}{x}\right)=\lim\limits_{\delta y\to 0}\left( \dfrac{\dfrac{y+\delta y}{x}-\dfrac{y}{x}}{\delta y}\right)=\lim\limits_{\delta y\to 0}\dfrac{\delta y}{x\delta y}=\lim\limits_{\delta y\to 0}\dfrac{1}{x}=\dfrac{1}{x} $</p>
|
84,982 | <p>I am a new professor in Mathematics and I am running an independent study on Diophantine equations with a student of mine. Online I have found a wealth of very helpful expository notes written by other professors, and I would like to use them for guided reading. <strong>I am wondering whether it is customary to as... | anon | 18,030 | <p>Rule of thumb: it's OK to print out one copy of online notes for your own private use without requesting permission (that's why they are posted). So if you and your students will each be printing your own copies, there should be no need to request permission. If you are going to print a stack of copies to hand out, ... |
815,418 | <p>Ok, so I've been playing around with radical graphs and such lately, and I discovered that if the </p>
<pre><code>nth x = √(1st x √ 2nd x ... √nth x);
</code></pre>
<p>Then</p>
<p>$$\text{the "infinith" } x = x$$</p>
<p>Example: </p>
<p>$$\sqrt{4\sqrt{4\sqrt{4\sqrt{4\ldots}}}}=4$$<br>
Try it yourself, type calc... | AnalysisStudent0414 | 97,327 | <p>You're basically doing this:</p>
<p>$x_0 = \sqrt x$</p>
<p>$x_1 = \sqrt{x x_0}$</p>
<p>$\displaystyle x_n = \sqrt{x x_{n-1}} = x^{\sum_{k=1}^n \frac{1}{2^k}} $</p>
<p>So it's pretty obvious that converges to $x$</p>
|
815,418 | <p>Ok, so I've been playing around with radical graphs and such lately, and I discovered that if the </p>
<pre><code>nth x = √(1st x √ 2nd x ... √nth x);
</code></pre>
<p>Then</p>
<p>$$\text{the "infinith" } x = x$$</p>
<p>Example: </p>
<p>$$\sqrt{4\sqrt{4\sqrt{4\sqrt{4\ldots}}}}=4$$<br>
Try it yourself, type calc... | robjohn | 13,854 | <p>It is important to show that the limit exists. Let define the sequence
$$
a_k=\sqrt{\vphantom{A}na_{k-1}}
$$
Since $\dfrac{a_k}{a_{k-1}}=\sqrt{\dfrac{n}{a_{k-1}}}$ and $\dfrac{a_k}{n}=\sqrt{\dfrac{a_{k-1}}{n}}$, we have</p>
<ol>
<li><p>if $a_{k-1}\le n$, then $a_{k-1}\le a_k\le n$; that is, $a_k$ is increasing and ... |
815,418 | <p>Ok, so I've been playing around with radical graphs and such lately, and I discovered that if the </p>
<pre><code>nth x = √(1st x √ 2nd x ... √nth x);
</code></pre>
<p>Then</p>
<p>$$\text{the "infinith" } x = x$$</p>
<p>Example: </p>
<p>$$\sqrt{4\sqrt{4\sqrt{4\sqrt{4\ldots}}}}=4$$<br>
Try it yourself, type calc... | Community | -1 | <p>Assume that you iterated infinitely many times (can take a while), and observed a convergence to $n$.</p>
<p>One more iteration yields $\sqrt{n.n}=n$.</p>
|
3,967,118 | <p>How is it possible to define <span class="math-container">$\lim_{x \rightarrow a} f(x)$</span> if <span class="math-container">$f(x)$</span> is undefined at <span class="math-container">$a$</span>? There is an infinite amount of <span class="math-container">$x$</span>-values on either side of <span class="math-conta... | Community | -1 | <p>A limit uses the values of <span class="math-container">$f(x)$</span> everywhere <em>except</em> at <span class="math-container">$x=a$</span> ! Whether <span class="math-container">$f(a)$</span> is defined or not and whether <span class="math-container">$f(a)$</span> coincides with the limit or not is irrelevant. Th... |
3,967,118 | <p>How is it possible to define <span class="math-container">$\lim_{x \rightarrow a} f(x)$</span> if <span class="math-container">$f(x)$</span> is undefined at <span class="math-container">$a$</span>? There is an infinite amount of <span class="math-container">$x$</span>-values on either side of <span class="math-conta... | DonAntonio | 31,254 | <p>I present you with the grap of the function <span class="math-container">$\;f(x)=\cfrac{\sin x}x\;$</span> :</p>
<p><a href="https://i.stack.imgur.com/gFnXl.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/gFnXl.png" alt="enter image description here" /></a></p>
<p>Clearly the functions isn't defin... |
261,156 | <p>Well, I am trying to write a code that makes the number:</p>
<p><span class="math-container">$$123456\dots n\tag1$$</span></p>
<p>So, when <span class="math-container">$n=10$</span> we get:</p>
<p><span class="math-container">$$12345678910$$</span></p>
<p>And when <span class="math-container">$n=15$</span> we get:</... | Syed | 81,355 | <p>Identical to the solution by @murray but written as a composition:</p>
<pre><code>f = FromDigits@*
Flatten@*
IntegerDigits@*
Range@
# &
f /@ Range[8, 15]
</code></pre>
<hr />
<p>Using Reap/Sow:</p>
<pre><code>g = FromDigits@
Flatten@Last@Reap@Scan[Sow[IntegerDigits[#]] & ]@Range[#] &
... |
2,801,294 | <p>I cam across an interesting claim that:</p>
<p>$\mathcal N (\mu_f, \sigma_f^2) \; \mathcal N (\mu_g, \sigma_g^2) = \mathcal N \left(\frac{\mu_f\sigma_g^2+\mu_g\sigma_f^2}{\sigma_f^2+\sigma_g^2}, \frac {\sigma_f^2\sigma_g^2}{\sigma_f^2+\sigma_g^2}\right)$</p>
<p>In trying to understand it I consulted Bromiley:</p>
... | Nadiels | 394,719 | <p>My (subjective) opinion:</p>
<h3>Do write</h3>
<p>$$
\mathcal{N}(x \mid \mu_1,\sigma_1^2)\cdot\mathcal{N}(x \mid \mu_2,\sigma_2^2)
$$</p>
<h3>Don't write</h3>
<p>$$
\mathcal{N}(\mu_1,\sigma_1^2)\mathcal{N}(\mu_2, \sigma^2)
$$</p>
<hr>
<p>The notation $\mathcal{N}(x|\mu_1,\sigma_1^2)$ makes it clear that this i... |
1,110,652 | <p>Would anyone happen to know any introductory video lectures / courses on partial differential equations? I have tried to find it without success (I found, however, on ODEs). </p>
<p>It does not have to be free material, but something not to expensive would be nice.</p>
| Community | -1 | <p>Khan Academy is very good with teaching Differential Equations. You should check it out:</p>
<p><a href="https://www.khanacademy.org/math/differential-equations" rel="nofollow">Khan Academy</a></p>
|
1,110,652 | <p>Would anyone happen to know any introductory video lectures / courses on partial differential equations? I have tried to find it without success (I found, however, on ODEs). </p>
<p>It does not have to be free material, but something not to expensive would be nice.</p>
| DilipSahu | 95,033 | <p>May Be these Lectures will be useful to you </p>
<p><a href="http://www.math.lamar.edu/faculty/maesumi/PDE1.html" rel="nofollow">http://www.math.lamar.edu/faculty/maesumi/PDE1.html</a></p>
|
3,657,751 | <p>Consider the series <span class="math-container">$$\sum_{n=1}^{\infty}\frac{(-1)^{\frac{n(n+1)}{2}+1}}{n}=1+\dfrac12-\dfrac13-\dfrac14+\dfrac15+\dfrac16-\cdots.$$</span> This is clearly not absolutely convergent. On the other hand, obvious choice, alternating series does not work here. Seems like the partial sum seq... | user780985 | 780,985 | <p>The triangular numbers alternate odd, odd, even, even, odd, odd, even, even, etc. The reason is that to move from <span class="math-container">$T_n$</span> to <span class="math-container">$T_{n+2}$</span>, we add <span class="math-container">$n + (n + 1) = 2n + 1$</span>, an odd number, so we obtain an alternating p... |
3,657,751 | <p>Consider the series <span class="math-container">$$\sum_{n=1}^{\infty}\frac{(-1)^{\frac{n(n+1)}{2}+1}}{n}=1+\dfrac12-\dfrac13-\dfrac14+\dfrac15+\dfrac16-\cdots.$$</span> This is clearly not absolutely convergent. On the other hand, obvious choice, alternating series does not work here. Seems like the partial sum seq... | wjmccann | 426,335 | <p>The triangle numbers themselves follow the pattern of </p>
<p><span class="math-container">$$odd, odd, even, even, odd, odd, even, even, \ldots$$</span></p>
<p>This can be shown as this pattern alternates whether or not the even number in <span class="math-container">$\frac{n(n+1)}{2}$</span> is <span class="math-... |
1,341,896 | <p>given three lines $\ell_1,\ell_2, \ell_3 $ which intersect in one point $P$. How can one construct a triangle such that the given lines become its angle bisectors?</p>
<p>So far I tried to find conditions on how the three given lines have to intersect such that the desired triangle exists. There are altogether six ... | Narasimham | 95,860 | <p>I was attempting to address the general question you posed earlier <a href="http://%20[in%20Mathematica][1]" rel="nofollow noreferrer">(by Mathematica)</a> with an assumption of existence of common angular bisectors of given the three lines for variable triangles schematically hand sketched here:</p>
<p><a href="ht... |
4,228,535 | <p>First the geometric inversion map <span class="math-container">$f:\mathbb{R}^n\setminus \{0\}\rightarrow \mathbb{R}^n$</span> is defined by
<span class="math-container">$$f(x)=\frac{x}{|x|^2}=:X.$$</span>
The one of its properties is following:</p>
<p>For any <span class="math-container">$c\in\mathbb{R}^n, c\not=0$<... | paul garrett | 12,291 | <p>Your argument is correct. Indeed, characteristic functions <span class="math-container">$\chi_\varepsilon$</span> of smaller and smaller intervals <span class="math-container">$[-\varepsilon,\varepsilon]$</span> do not converge to <span class="math-container">$\delta$</span> (in a distributional sense), but, rather,... |
3,954,066 | <p>How do you show that for some <span class="math-container">$A\subset \mathbb R$</span>, <span class="math-container">$A\not \in \mathbb R$</span>.</p>
<p>Intuitively it makes sense, but how do you actually prove it?</p>
| saulspatz | 235,128 | <p>It depends upon how the real numbers are defined. For example, if a real number is an equivalence class of Cauchy sequences of rational numbers, then a set of real numbers is not a real number, because its elements are not equivalence classes of Cauchy sequences, but sets of such equivalence classes.</p>
|
4,557,576 | <p>Working on a 3U CubeSat as part of a project for a Space Engineering club. To calculate the maximum solar disturbance force, we are trying to calculate the largest shadow a 0.1 * 0.1 * 0.3 rectangular prism can cast.</p>
<p>If the satellite was oriented with the largest side facing the sun directly, the shadow cast ... | Andrew Draganov | 432,295 | <p>I can work out the answer more thoroughly if you'd like but here's a write-up of the steps you want to take to find a solution.</p>
<p>First, the size of the shadow is a continuous function of (I think) seven variables. We have already defined the length, width and height, though, so I'll just treat those as constan... |
3,331,865 | <p>Find the Orthonormal basis of vector space <span class="math-container">$V$</span> of the linear polynomials of the form <span class="math-container">$ax+b$</span> such that <span class="math-container">$\:$</span>, <span class="math-container">$p:[0,1] \to \mathbb{R}$</span>. with inner product</p>
<p><span clas... | projectilemotion | 323,432 | <p>A standard approach for these types of problems is to use the Gram-Schmidt procedure:</p>
<hr>
<p>You already found a basis of <span class="math-container">$V$</span>, so let <span class="math-container">$\mathbf{v}_1=1$</span> and <span class="math-container">$\mathbf{v}_2=x$</span>. Then an orthogonal basis <spa... |
14,541 | <p>I know this is a quite general question , but I've always heard about creating mathematical models in economics, social sciences, engineering,... And I would want to know what are the starting points and roadmap to understand what a mathematical model is and how to create them. Thanks</p>
| Joseph Malkevitch | 1,369 | <p>COMAP, the Consortium for Mathematics and Its Applications has developed lots of materials at different levels that address your question:</p>
<p><a href="http://www.comap.com/" rel="nofollow">http://www.comap.com/</a></p>
|
14,541 | <p>I know this is a quite general question , but I've always heard about creating mathematical models in economics, social sciences, engineering,... And I would want to know what are the starting points and roadmap to understand what a mathematical model is and how to create them. Thanks</p>
| Mike Spivey | 2,370 | <p>The last time I taught mathematical modeling we used <em><a href="http://books.google.com/books?id=XPxrItVsXpQC&printsec=frontcover&dq=a+first+course+in+mathematical+modeling&source=bl&ots=4EcHhIII_M&sig=K1KZ1lq9OmZWj-aMKMdJiv5wg64&hl=en&ei=pVQKTeTAI4nUtQPioODDCg&sa=X&oi=book_resu... |
2,214,231 | <blockquote>
<p>Find the derivative of $\tan^3[\sin(2x^2-17)]$. </p>
</blockquote>
<p>Sorry if my question is a little too specific but I am confused on this trig equation. After completing the derivative I was wondering why does the $3tan^2$ not distribute to $sec^2$? Is there a rule for this? How come the exponent... | PiE | 110,208 | <p>What you have to do is to solve the following equation:</p>
<p>$$F(x) = \ln(x) - 1 = 0.$$</p>
<p>When you do, you'll find out that $x = e$.</p>
<p>This is where the function crosses the $x$-axis. Thus, the point is $(e,0)$.</p>
|
20,802 | <p>Look at the following example:</p>
<p>Which picture has four apples?</p>
<p>A<a href="https://i.stack.imgur.com/Tpm46.png" rel="noreferrer"><img src="https://i.stack.imgur.com/Tpm46.png" alt="enter image description here" /></a></p>
<hr />
<p>B <a href="https://i.stack.imgur.com/AOv29.png" rel="noreferrer"><img src=... | Thierry | 12,671 | <p>I don't think there's anything wrong with the wording; it's clear what is being asked. Your example with the three dollars is also not always the way we speak in everyday language. If you ask someone with three children if they have two children, they're unlikely to say "yes" and leave it at that.</p>
<p>G... |
20,802 | <p>Look at the following example:</p>
<p>Which picture has four apples?</p>
<p>A<a href="https://i.stack.imgur.com/Tpm46.png" rel="noreferrer"><img src="https://i.stack.imgur.com/Tpm46.png" alt="enter image description here" /></a></p>
<hr />
<p>B <a href="https://i.stack.imgur.com/AOv29.png" rel="noreferrer"><img src=... | Jason E | 15,949 | <p>Nearly every test like this includes instructions to choose the "best answer" to cover exactly this scenario. This looks like it's part of a test of basic counting skills, and in that context, the best answer is B. While one could make an argument for either C or D, I can't imagine an argument for either o... |
2,530,298 | <p>I tried putting y alone and got y=(-6x-5)/5. Which I then put into the distance formula sqrt((x-1)^2+(y+5) and substitute the number above in for y but my answer never comes out correct.. Wondering if I could get some help.</p>
| lab bhattacharjee | 33,337 | <p><strong>Method</strong>$\#1:$</p>
<p>As the perpendicular distance is the shortest find the equation of the perpendicular of $$6x+5y+5=0$$ passing through $(1,-5)$</p>
<p>Find the intersection of the two lines.</p>
<p><strong>Method</strong>$\#2:$</p>
<p>$$6x+5y+5=0\iff\dfrac x5=\dfrac{y+1}{-6}$$ $=k$(say)</p>... |
2,530,298 | <p>I tried putting y alone and got y=(-6x-5)/5. Which I then put into the distance formula sqrt((x-1)^2+(y+5) and substitute the number above in for y but my answer never comes out correct.. Wondering if I could get some help.</p>
| Michael Rozenberg | 190,319 | <p>By C-S
$$\sqrt{(6^2+5^2)\left((x-1)^2+(y+5)^2\right)}\geq6(x-1)+5(y+5)=14.$$
Thus, $$\sqrt{(x-1)^2+(y+5)^2}\geq\frac{14}{\sqrt{61}}.$$
The equality occurs for $(x-1,y+5)||(6,5),$ which gives the needed point.</p>
<p>I got $\left(\frac{145}{61},-\frac{235}{61}\right).$</p>
|
28,389 | <p>I've been studying the axiomatic definition of the real numbers, and there's one thing I'm not entirely sure about.</p>
<p>I think I've understood that the Archimedean axiom is added in order to discard ordered complete fields containing infinitesimals like the hyperreal numbers. Additionally, this property clearly... | Pete L. Clark | 299 | <p>The answer to your question depends critically on what you mean by a "complete ordered field" <span class="math-container">$(F,<)$</span>. Here are two rival definitions:</p>
<ol>
<li><p>[<b>added</b>: <em>sequentially</em>] Cauchy complete: every Cauchy sequence in <span class="math-container">$F$</sp... |
326,600 | <p>The intervals are: </p>
<p>$I_1=\{ x\in\mathbb{R}:a\leq x\leq b\}$ with $-1<a<b<1$.</p>
<p>$I_2=\{ x\in\mathbb{R}:-1< x< 1\}$.</p>
<p>I've shown $f_n$ is pointwise convergent to $0$ on $I_1$ and $I_2$. But what is the difference for showing uniform convergence on these two intervals? I have a few o... | Cameron Buie | 28,900 | <p>To show uniform convergence to a function $f$ on an interval $I,$ you must show that $$\lim_{n\to\infty}\left[\sup_{x\in I}|f_n(x)-f(x)|\right]=0.$$ In this particular case, you must show that $$\lim_{n\to\infty}\left[\sup_{x\in I}|f_n(x)|\right]=0.$$</p>
<p>Since $|f_n(x)|\leq |x^n|$ for all $x$, then to see unifo... |
79,782 | <p>Calculate $17^{14} \pmod{71}$</p>
<p>By Fermat's little theorem:<br>
$17^{70} \equiv 1 \pmod{71}$<br>
$17^{14} \equiv 17^{(70\cdot\frac{14}{70})}\pmod{71}$</p>
<p>And then I don't really know what to do from this point on. In another example, the terms were small enough that I could just simplify down to an answer... | Arthur | 15,500 | <p>We have that</p>
<p>$$
17^{14}\equiv 17^{70} \cdot 17^{14} \equiv 17^{84}
$$</p>
<p>or</p>
<p>$$
17^{14}\equiv 17^{-70} \cdot 17^{14} \equiv 17^{-56}
$$</p>
<p>I don't see any of these as especially easy to calculate, as $17^{14} = 289^{7} \equiv 5^{7}$. If you don't have to use Fermat's, I'd suggest going w... |
2,439,863 | <p>I was working on the series </p>
<p>$\sum_{n=1}^{\infty}{\frac{(-1)^n}{n}z^{n(n + 1)}}$ and I was to consider when $z = i$. I have that $$\sum_{n=1}^{\infty}{\frac{(-1)^n}{n}i^{n(n + 1)}} = \sum_{n=1}^{\infty}{\frac{(-1)^{\frac{3}{2}n+\frac{1}{2}n^2}}{n}} = 1 - \frac{1}{2} - \frac{1}{3} + \frac{1}{4} + \frac{1}{5} ... | Community | -1 | <p>Your sum is
\begin{align}
s&=\sum^\infty_{n=0}\left(\frac1{4n+1}-\frac1{4n+2}-\frac1{4n+3}+\frac1{4n+4}\right)\\&=\frac14\sum^\infty_{n=0}\left(\frac1{n+1/4}-\frac1{n+1/2}-\frac1{n+3/4}+\frac1{n+1}\right)
\\&=\frac14\left(-\psi(1/4)+\psi(1/2)+\psi(3/4)-\psi(1)\right)
\end{align}
using the series represe... |
2,438,236 | <p>Suppose a finite group $G$ has order smaller than the dimension of a vector space $V$ over any field, how to prove that the representation of $G$ on $V$ is reducible？</p>
| Long | 41,744 | <p>The following works over the complex numbers -- I'm not sure if there is an analogous result for arbitrary fields:</p>
<p>Let $Irr(G)$ be the set of all irreducible representations of $G$, then
$$|G| = \sum_{\rho \in Irr(G)} \deg(\rho)^2$$</p>
<p>(See for instance Corollary 4.4.5 of Benjamin Steinberg's <a href="h... |
3,327,094 | <p>Give an example of a non abelian group of order <span class="math-container">$55$</span>.</p>
<p>To find non abelian group the simplest way is to find one non abelian group whose order divides the order of given group and then we take the group which is the external direct product of the non abelian group and some ... | Community | -1 | <p>Notince that <span class="math-container">$\operatorname{aut}\Bbb Z_{11}\cong \Bbb Z_{11}^*\cong \Bbb Z_{10}$</span>. Therefore you can consider an element <span class="math-container">$g\in\operatorname{aut}\Bbb Z_{11}$</span> of order <span class="math-container">$5$</span> and the homomorphism <span class="math-c... |
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