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 |
|---|---|---|---|---|
2,252,256 | <p>The Chernoff bound, being a sharp quantitative version of the law of large numbers, is incredibly useful in many contexts. Some general applications that come to mind (which I guess are really the same idea) are:</p>
<ul>
<li>bounding the sample complexity of PAC algorithms;</li>
<li>estimating confidence intervals... | amakelov | 286,977 | <p>To answer my question a bit, I've recently learned that Kyng and Sachdeva have used some matrix concentration inequalities to give a very simple Laplacian solver algorithm; paper <a href="https://arxiv.org/pdf/1605.02353.pdf" rel="nofollow noreferrer">here</a>.</p>
|
87,091 | <p>Find $C$ such that</p>
<p>$$Ce^{(-4x^2-xy-4y^2)/2}$$</p>
<p>is a joint probability density of a $2$-variable Gaussian.</p>
<p>If someone could give me a jumping off point, or a process as to how to go about this, I'd really appreciate it.</p>
| DonAntonio | 31,254 | <p>For some reason the OP won't post, or can't post, an answer, so summarizing the comments:</p>
<p>$(1)\,\,\forall\,g,x,y\in G\,$ , and putting $\,a^b:=b^{-1}ab\,\,,\,a,b\in G\,$:
$$[x,y]^g:=g^{-1}[x,y]g:=g^{-1}x^{-1}y^{-1}xy g=\left(x^{-1}\right)^g\left(y^{-1}\right)^gx^gy^g=[x^g,y^g]\in G'\Longrightarrow G'\triangl... |
2,407,183 | <blockquote>
<p>Calculate $\int_{\lambda} dz/(z^2-1)^2$, where $\lambda$ is the path
in $\mathbb{R^2}-\{1,-1\}$ plotted below:</p>
<p><a href="https://i.stack.imgur.com/T1uFH.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/T1uFH.png" alt="enter image description here"></a></p>
</blockquote>
... | robjohn | 13,854 | <p>Using the contour</p>
<p><a href="https://i.stack.imgur.com/sEDKB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sEDKB.png" alt="enter image description here"></a></p>
<p>contour integration gives
$$
\begin{align}
\overbrace{\color{#C00}{\text{PV}\int_0^\infty\frac{e^{i\pi x}}{\log(x)}\,\mathrm... |
4,315,844 | <p>Find the greatest number <span class="math-container">$k$</span> such that there exists a perfect square that is not a multiple of 10, with its last <span class="math-container">$k$</span> digits the same</p>
<p>I could find <span class="math-container">$12^2 = 144$</span>, <span class="math-container">$38^2 = 1444$... | Callum | 416,266 | <p>This is definitely not true. An important example of this is the Cartan decomposition of a semisimple Lie Algebra. These are decompositions of the form <span class="math-container">$\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$</span> where <span class="math-container">$\mathfrak{k}$</span> is a maximal compact s... |
117,836 | <p>I'm not the best at math(but eager to learn) so please excuse me if I'm not explaining this problem correctly, I will try to add as much info to make it clear. I basically receive 2 pieces of data, one is a list of integers and the other is a target_sum, and I want to figure out all the ways I can use the list to ... | Henry | 6,460 | <p>I assume that you are trying to generate partitions rather than the simpler task of counting how many there are. </p>
<p>What you need to do is generate smaller partitions with part of the data and then add the next piece of data up to the target. There is a cost here as you are also building all partitions from t... |
887,282 | <p>I was given a question to find a greatest possible length to measure $495$, $900$,$1665$ (in centimetres). </p>
<p>The solution is finding GCF or GCD or HCF (highest common factor) of these numbers which is $45$.</p>
<p>How does this works in real time, please explain in layman words</p>
<p>Thank you all</p>
| DSinghvi | 148,018 | <p>Obviously question has imprecise wording . It assumes whatever you get must be a whole number if you take HCF it will have some whole number into HCF so you can find the length.</p>
|
887,282 | <p>I was given a question to find a greatest possible length to measure $495$, $900$,$1665$ (in centimetres). </p>
<p>The solution is finding GCF or GCD or HCF (highest common factor) of these numbers which is $45$.</p>
<p>How does this works in real time, please explain in layman words</p>
<p>Thank you all</p>
| Michael Albanese | 39,599 | <p>The way that we measure lengths is to have a standard unit of measurement and then count how many of these units a given length is. Common standard units include millimetre, centimetre, metre, kilometre, inch, foot, mile, lightyear, astronomical unit, etc. </p>
<p><strong>Example:</strong> An Olympic size swimming ... |
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}\}^{... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
1,761,775 | <p>Suppose we have angular momentum operators $L_1,L_2,L_3$ which satisfy $[L_1,L_2]=iL_3$, $[L_2,L_3]=iL_1$ and $[L_3,L_1]=iL_2$. We can show that the operator $L^2:=L_1^2+L_2^2+L_3^2$ commutes with $L_1,L_2$ and $L_3$. Now define $L_{\pm}=L_1\pm iL_2$ and then we can also show that $$L_+L_-=L^2-L_3(L_3-I)$$ and $$L_-... | Dac0 | 291,786 | <p>You have to work out the commutators, i.e.
$$[L_+,L_-]=-2iL_3$$
Then consider that
$$L_-L_+=L_+L_--[L_+,L_-]=L_+L_--2iL_3$$
so that
$$(L_-L_+)v=(L_+L_-)v-2iL_3v$$
I don't know where you should plug this one in but I'm pretty sure that if you didn't use it already you should use it now.</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... | Gerhard Paseman | 3,568 | <p>Yes, asking permission is the right thing to do. If you don't get it, the right thing to do is to mention the resource to your students, along with the proviso that you do not have permission to copy it, and suggest they do their best not to make or distribute paper or electronic copies beyond viewing it in a brows... |
2,390,215 | <p>There exist a bijection from $\mathbb N$ to $\mathbb Q$, so $\mathbb Q$ is countable. And by well ordering principle $\mathbb N$ has a least member say $n_1$ which is mapped to something in $\mathbb Q$, In this way $\mathbb Q$ can be well arranged? Is my argument good? </p>
| TRUSKI | 91,784 | <p>$\mathbb{Q}$ is not well ordered. That can be seen very easily for example $(\sqrt{2},3)$ intersection $\mathbb{Q}$ has no least element in $\mathbb{Q}$. Comparing with $\mathbb{N}$ is not a good idea because there does not exist an order preserving bijective map form $\mathbb{N} \to \mathbb{Q}$.</p>
|
234,063 | <p>There are some answers on how to get a smooth squarewave function. But I would like to have a smooth boxcar function or rectangle function with 2 different widths.: <code>wup</code>, and <code>wdown</code></p>
<p>One solution is the Fourier Transform, but I prefer having an approximation with a smoothness factor.</p... | Ted Ersek | 460 | <p>Suppose you want the square wave high 20% of the time. The following helps.</p>
<pre><code>DutyCycle = 0.2; Plot[Piecewise[{{x/(2 DutyCycle),
x < DutyCycle}, {(1 - 2 DutyCycle + x)/(2 - 2 DutyCycle),
DutyCycle < 1}}], {x, 0, 1}]
</code></pre>
<p><a href="https://i.stack.imgur.com/7Ht2V.png" rel="nofollow nor... |
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$$<br>
Try it yourself, type calc... | DanielV | 97,045 | <p>$$\begin{align}
X & = \sqrt{n \cdot \sqrt {n \cdot \sqrt{n \dots} } } \\
& = \sqrt{n} \cdot \sqrt{\sqrt{n}} \cdot \sqrt{\sqrt{\sqrt{n}}}
\cdot \dots \\
&= n^{1/2} \cdot n^{1/4} \cdot n^{1/8} \dots \\
&= n^{1/2 + 1/4 + 1/8 \dots} \\
&= n^1 \\
&= n \\
\end{align}$$</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>
| Tomás | 42,394 | <p>You can find a lecture on Hyperbolic Conservation Laws, given by Constantine Dafermos <a href="http://video.impa.br/index.php?page=Curso-Hyperbolic-Conservation-Laws" rel="nofollow">here</a>.</p>
<p>And <a href="http://video.impa.br/index.php?page=escola-de-altos-estudos-unique-continuation-and-nonlinear-dispersive... |
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>
| krishnab | 38,239 | <p>The best PDE introduction that I have found is by Dr. Pavel Grinfeld on youtube. The link is:</p>
<p><a href="https://www.youtube.com/playlist?list=PLlXfTHzgMRUK56vbQgzCVM9vxjKxc8DCr" rel="nofollow noreferrer">https://www.youtube.com/playlist?list=PLlXfTHzgMRUK56vbQgzCVM9vxjKxc8DCr</a></p>
<p>These are really really... |
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>
| Stinking Bishop | 700,480 | <p>To prove that <em>none</em> of the subsets <span class="math-container">$A\subset\mathbb R$</span> are contained in <span class="math-container">$\mathbb R$</span> as elements, one needs to craft the proof tailored to the particular definition of <span class="math-container">$\mathbb R$</span>. For some definitions ... |
3,355,215 | <blockquote>
<p>For <span class="math-container">$x$</span> and <span class="math-container">$k$</span> real numbers, for what values of <span class="math-container">$k$</span> will the graphs of <span class="math-container">$f(x)=-2\sqrt{x+1}$</span> and <span class="math-container">$g(x)=\sqrt{x-2}+k$</span> inters... | Toby Mak | 285,313 | <p>Rearranging, we have:</p>
<p><span class="math-container">$$k = -2 \sqrt{x+1} - \sqrt{x-2}$$</span></p>
<p>The domain of the RHS is <span class="math-container">$[2, \infty)$</span>. Therefore, the graphs will intersect when</p>
<p><span class="math-container">$$k ≤ -2 \sqrt{2+1} - \sqrt{2-2} \Rightarrow k ≤ -2 \... |
3,355,215 | <blockquote>
<p>For <span class="math-container">$x$</span> and <span class="math-container">$k$</span> real numbers, for what values of <span class="math-container">$k$</span> will the graphs of <span class="math-container">$f(x)=-2\sqrt{x+1}$</span> and <span class="math-container">$g(x)=\sqrt{x-2}+k$</span> inters... | Olivier Roche | 649,615 | <p>This problem is not about solving an equation in the variables <span class="math-container">$x$</span> and <span class="math-container">$k$</span>. Keep in mind that you're looking for values of <span class="math-container">$k$</span> such that there exist <span class="math-container">$x$</span> such that <span clas... |
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 ... | cbob | 1,109,675 | <p>Thank you for everyone's help! I was able to use a similar method by defining a matrix of the vertices of the prism and multiplying by the X rotation matrix and Y rotation matrix.</p>
<p>We then have a function of thetaX and thetaY, which can be maximized to describe the optimal angles. I also calculated an answer o... |
231,317 | <p>The task I'm faced with is to implement a poly-time algorithm that finds a nontrivial factor of a Carmichael number. Many resources on the web state that this is easy, however without further explanation why.</p>
<p>Furthermore, since Miller-Rabin exits when a nontrivial square root of 1 is found, this can be used ... | Ravindra HV | 262,480 | <p>Try this (a description of the comment above - also blogged the same!) :</p>
<p>For each prime base $(2,3,5,7,11...)$ try checking the remainder for the exponents under $\frac{n-1}{2},\frac{n-1}{4},\frac{n-1}{8}\dots$ and so on. Once a number other than $1$ is found then try : </p>
<p>$\gcd (x-1,n)$ </p>
<p>or</... |
101,974 | <p>We are familiar with <strong><em>Hurwitz’s theorem</em></strong> which implies there is only the Fibonacci 2-Square, Euler 4-Square, Degen 8-Square, and no more. However, if we relax conditions and allow for <em>rational expressions</em>, then <strong><em>Pfister's theorem</em></strong> states that similar identitie... | Tito Piezas III | 4,781 | <p>Persistence pays off. I really wanted to see <strong><em>Pfister’s 8-Square Identity</em></strong> (distinct from Degen's version) and, since no one was answering my question, I took another look at K. Conrad’s paper and managed, with some heuristics, to find the identity myself. Without further ado,</p>
<p>$\begin... |
3,331,865 | <p>Find the Orthonormal basis of vector space <span class="math-container">$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}$</span>. with inner product</p>
<p><span clas... | ZAF | 609,023 | <p>With Gram-Schmidt</p>
<p><span class="math-container">$v_{1} = 1$</span>, <span class="math-container">$v_{2} = \frac{x - <x,v_{1}>v_{1}}{ \| x - <x,v_{1}> v_{1}\|} $</span></p>
<p>So <span class="math-container">$<x,1> = \int_{[0,1]}xdx = \frac{1}{2}$</span></p>
<p>And <span class="math-contain... |
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=... | JVC | 13,160 | <p>I think a lot of it has to do with the age of the child, and what the goal of the question is. If this is for children just learning their numbers, like say 4 or 5 years old, then I think B is the correct answer as thy are not being asked to stretch their logical capabilities, but to simply recognize and call out t... |
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>
| Community | -1 | <p><strong>Hint:-</strong><br>
$(1,-5)$ projected on the point $(x, \frac{-5-6x}{5})$. Line passes through $(0,-1)$. Vector joining points $(0,-1)$and $(x, \frac{-5-6x}{5})$ is perpendicular to the Vector joining $(x, \frac{-5-6x}{5})$ to $(1,-5)$. Find $x$ from the condition. </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... | Scott Carter | 722 | <p>There are non-archemedian completions of the rationals, called <a href="http://en.wikipedia.org/wiki/P-adic_number"> p-adic </a> completions. The book Gouvêa, Fernando Q. (2000). p-adic Numbers : An Introduction (2nd ed.). Springer is an excellent introduction to these.</p>
|
619,526 | <p>I'm studying up for my algebra exam, and I'm not exactly sure how to solve a problem like the following</p>
<blockquote>
<p>Let $f = X^2 + 1 \in \mathbb{F}_5[X]$, $R = \mathbb{F}_5[X]/\langle f \rangle$ and $\alpha = X + \langle f \rangle \in R$. Show that $\alpha \in R^*$ and that $\vert \alpha \vert = 4$ in $R^... | Andreas Caranti | 58,401 | <p>Working in $R$ means working with polynomials in $\mathbb{F}_5[X]$ modulo the polynomial $f = X^{2} + 1$.</p>
<p>So you have $\alpha^{2} = X^{2} \equiv -1 \pmod{f}$, and $\alpha^{4} \equiv (-1)^{2} = 1 \pmod{f}$. This shows that $\alpha$ is invertible, with inverse $\alpha^{3} \equiv -X \pmod{f}$. (This is because ... |
363,881 | <p>The problem gives the curl of a vector field, and tells us to calculate the line integral over $C$ where $C$ is the intersection of $x^2 + y^2 = 1$ and $z= y^2$. I know I should use Stokes Theorem, but how do I find $dS$? </p>
<p>I did $z = \frac{1}{2}(y^2 + 1 - x^2)$ and calculated $dS$ as $\langle-dz/dx,-dz/dy,1\... | Glen O | 67,842 | <p>So long as there isn't any sort of problem point (such as if the vector field you're integrating is undefined at the origin), you can integrate over any surface that is closed within the definition. So let $z=y^2$ be the surface, and let $x^2+y^2=1$ describe the limits of the surface (because $x^2+y^2=1$ doesn't for... |
128,695 | <p>Is there any good guide on covering space for idiots? Like a really dumped down approach to it . As I have an exam on this, but don't understand it and it's like 1/6th of the exam. </p>
<p>So I'm doing Hatcher problem and stuck on 4.</p>
<ol>
<li>Construct a simply-connected covering space of the space $X \subset ... | Eric O. Korman | 9 | <p>I think it will help if you "pull the diameter out of the sphere using the 4th dimension" (think about the analogous situation of a diameter in a circle) to see that space is homeomorphic to <img src="https://i.stack.imgur.com/eiMLK.jpg" alt="enter image description here"></p>
<p>Now this is similar to the wedge su... |
4,523,363 | <p>A set <a href="https://en.m.wikipedia.org/wiki/Lattice_(order)#:%7E:text=A%20lattice%20is%20an%20abstract,greatest%20lower%20bound%20or%20meet" rel="nofollow noreferrer">lattice</a> is a lattice where the meet is the intersection, the join is the union, and the partial order is <span class="math-container">$\subsete... | Dumper D Garb | 1,076,376 | <p>Total orders with <span class="math-container">$\land = \min$</span> and <span class="math-container">$\lor = \max$</span> are lattices and any of them, call it <span class="math-container">$(X,\le)$</span> and call <span class="math-container">$(-\infty,x)=\{y\in X\,:\, y<x\}$</span>, is isomorphic to the set la... |
4,135,449 | <p>A few years ago, I got interested in an apparently hard integration problem which had me fascinated. This was the integral of a Sophomore Dream like integral except with the bounds over the real positive numbers denoted by <span class="math-container">$\Bbb R^+$</span> and not from 0 to 1 over the unit square, or un... | Nikos Bagis | 223,191 | <p><strong>This is a note</strong>. I will try to find more in the near future.
<span class="math-container">$$
I=\int^{\infty}_{0}x^{-x}dx=\int^{\infty}_{1}e^{-x\log x}dx+\int^{1}_{0}x^{-x}dx.
$$</span>
Using <span class="math-container">$\sum_{n\geq 1}n^{-n}=\int^{1}_{0}x^{-x}dx$</span> and making the change of varia... |
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 $$\sum_{n=1}^{\infty}{\frac{(-1)^n}{n}i^{n(n + 1)}} = \sum_{n=1}^{\infty}{\frac{(-1)^{\frac{3}{2}n+\frac{1}{2}n^2}}{n}} = 1 - \frac{1}{2} - \frac{1}{3} + \frac{1}{4} + \frac{1}{5} ... | Claude Leibovici | 82,404 | <p>The number you give is very close to $$\frac{\pi }{4}-\frac{\log (2)}{2}$$ They differ by $5 \times 10^{-12}$.</p>
<p>I bet that you obtained this value summing up to $n=10^6$.</p>
<p>Now, prove it !</p>
|
4,356,938 | <p>I'm reading through <a href="https://press.princeton.edu/books/hardcover/9780691151199/elliptic-tales" rel="nofollow noreferrer">Elliptic Tales</a>.</p>
<p>Addition of 2 points on an elliptic curve is described as follows:</p>
<p><a href="https://i.stack.imgur.com/Hifi9.png" rel="nofollow noreferrer"><img src="https... | brainjam | 1,257 | <p>The construction and diagrams you're quoting are in Section 8.1 of the book.</p>
<p>In Section 8.2, the author changes the definition of <span class="math-container">$\mathcal O$</span> to <span class="math-container">$(0:1:0)$</span>, i.e. the point at infinity in the vertical direction. In this new context, <span... |
2,203,770 | <p>If I am asked to find all values of $z$ such that $z^3=-8i$, what is the best method to go about that?</p>
<p>I have the following formula:
$$z^{\frac{1}{n}}=r^\frac{1}{n}\left[\cos\left(\frac{\theta}{n}+\frac{2\pi k}{n}\right)+i\sin\left(\frac{\theta}{n}+\frac{2\pi k}{n}\right)\right]$$</p>
<p>for $k=0,\pm1, \pm2... | Warren Hill | 86,986 | <p>There are several ways to solve this but one of the simplest is to write $ z $ of the form $ r \angle \theta $.</p>
<p>In this case $ z = 8 \angle \frac{-\pi}{2} $ We want cube roots and the cube root of 8 is 2. We get the first one by dividing $ \theta$ by 3</p>
<p>$ 2 \angle \frac{-\pi}{6}$ is thus one solution.... |
675,695 | <p>[<b>Added by PLC</b>: This question is a followup to <a href="https://math.stackexchange.com/questions/298870/inner-product-change-of-axioms/298891">this already answered question</a>.]</p>
<p>Keep the axioms for a real inner product (symmetry, linearity, and homogeneity).
But make the fourth be
$$\langle x,x \ran... | Berci | 41,488 | <p>We can assume by scaling that $\langle x,x\rangle =1$ and $\langle y,y\rangle=-1$.</p>
<p>The method should work: let $z=x+\lambda y$. Then we have
$$\langle z,z\rangle\ =\ 1+2\lambda\langle x,y\rangle-\lambda^2$$
which has a real root in $\lambda$, as $\langle x,y\rangle$ is considered constant (given).</p>
|
1,329,539 | <p>I'm trying to figure out how to solve this word problem. I'm pretty sure it involves calculus or something even harder, but I don't know how to solve <strong><em>the general form</em></strong>.</p>
<p>Let me start with the concrete form, however:</p>
<p><strong>Concrete Form:</strong></p>
<p>You start with 5 scie... | tomi | 215,986 | <p>I will introduce the following notation:</p>
<p>Let $s_0$ be the initial number of scientists.</p>
<p>Let $s$ be the number of scientists at a particular point.</p>
<p>Let $X_i$ be the different types of projects.</p>
<p>Let $W_i$ be the work required for project $X_i$ - the units will be "scientist years".</p>
... |
106,775 | <p>I don't get this, need some help, examples and information</p>
<blockquote>
<p>The linear function $f$ is given by
$$f(x) = 3x - 2 ,\quad -2 \leq x \leq 4.$$</p>
<ol>
<li><p>Enter the independent variable and the dependent variable.</p></li>
<li><p>Determine the function values $f (-2)$, $f (-1)$, $f... | André Nicolas | 6,312 | <p>The independent variable is $x$. The dependent "variable" is probably intended to be $f(x)$. This is a somewhat unusual use of language. It is used more often when write the relationship as $y=3x-2$. Then $y$ is called the dependent variable. In modern mathematics, the terms "independent variable" and "dependent var... |
1,773,303 | <p>Let $k$ be a field and $L=k(\sqrt{a_1},\sqrt{a_2},\cdots,\sqrt{a_n})$ with $a_i\in k$. Suppose $[L:k]=2^n$. Let $a=\sqrt{a_1}+\sqrt{a_2}+\cdots\sqrt{a_n}$. Show that $L=k(a)$.</p>
| egreg | 62,967 | <p>Let $x_0\in X$; for each $x\in X$, $x\ne x_0$, there is an open neighborhood $V_x$ of $x$ that doesn't contain $x$. Since $X$ is finite,
$$
\bigcap_{\substack{x\in X\\x\ne x_0}}V_x
$$
is an open set containing $x_0$ and no other element of $X$. So $\{x_0\}$ is open. As $x_0$ is arbitrary, you're done.</p>
<p>Note t... |
199,199 | <p>Suppose a box contains 5 white balls and 5 black balls.</p>
<p>If you want to extract a ball and then another:</p>
<p>What is the probability of getting a black ball and then a black one?</p>
<p>I think that this is the answer:</p>
<p>Let $A:$ get a black ball in the first extraction, $B:$ get a black ball in th... | Cameron Buie | 28,900 | <p>Not quite. Now, if the first ball <strong>isn't</strong> replaced before drawing the second, then $P(B|A)=\frac49$, from which $$P(A\cap B)=P(A)\cdot P(B|A)=\frac12\cdot\frac49=\frac29.$$ If we <em>are</em> replacing the first ball, $P(B|A)=P(B)=\frac12$, and that changes things.</p>
<p>As for the second question, ... |
3,506,340 | <p>I have no idea how to calculate <span class="math-container">$z_x+z_y$</span> at a point <span class="math-container">$\left( \frac{\pi +3}{3}, \frac{\pi+1}{2}\right)$</span>, if <span class="math-container">$z=uv^2$</span> and <span class="math-container">$x=u+sinv$</span>, <span class="math-container">$y=v+cosu$</... | Keen-ameteur | 421,273 | <p><strong>Hint:</strong></p>
<p>Define <span class="math-container">$Z(u,v)=z\big( x(u,v),y(u,v) \big)$</span>. By the chain rule (assuming the conditions hold) you can write:</p>
<p><span class="math-container">$$ \frac{\partial Z}{\partial u}(u,v)= \frac{\partial z}{\partial x}\big( x(u,v),y(u,v) \big) \cdot \fra... |
3,019,882 | <p>Í already have found a proof About the statment which I did not understand, I will write it down, so maybe someone can explain me the part that I did not understand. But if there are easier ways to prove the statement I would be delighted to know. </p>
<p><span class="math-container">$\alpha:=\sqrt{2}+\sqrt{3}\noti... | Sujit Bhattacharyya | 524,692 | <p>Instead of that just try to check is this <span class="math-container">$U:=\mathbb{Q}(\sqrt 2)\cup \mathbb{Q}(\sqrt 3)$</span> a Group?</p>
<p>Let us take <span class="math-container">$\sqrt2,\sqrt3\in U$</span> [Both exists] but <span class="math-container">$\sqrt2+\sqrt3\notin U$</span>.</p>
<p>So definitely it ... |
2,682,531 | <p>The definition of a convex set is the following:</p>
<blockquote>
<p>A set $\Omega \subset \mathbb R^n$ is convex if $\alpha x + (1 − \alpha) y \in \Omega, \forall x, y \in \Omega$ and $\forall \alpha \in [0, 1]$.</p>
</blockquote>
<p>With this it should be easy enough to prove that a set is not convex: just fin... | Rodrigo de Azevedo | 339,790 | <p>If one can prove that a given set is a <a href="https://en.wikipedia.org/wiki/Spectrahedron" rel="nofollow noreferrer">spectrahedron</a>, then one can conclude that given set is <em>convex</em>.</p>
<p>For example, the unit disk can be represented by the following <a href="https://en.wikipedia.org/wiki/Linear_matrix... |
2,606,999 | <p>$f: X \to Y$ prove that if $A \subseteq X$, then A is a subset of the pre-image of the image of $A$, which is shown by these symbols respectively: $f^{-1}[f[A]]$</p>
<p>This is my proof:</p>
<p>If $A \nsubseteq f^{-1}[f[A]]$, then there exists "$x$" as an element of $A$, such that $f^{-1}[f[A]] \neq x$
which is a ... | The Phenotype | 514,183 | <p>$f^{-1}[f[A]]$ is a set, and $x$ is an element. They cannot be equal.</p>
<p>The correct way of proving this is: let $x\in A$, then $$f(x)\in \{f(x)\ |\ x\in A\}=f[A]$$ by the definition of image. Now because $f(x)\in f[A]$ (and obviously $x\in X$), we have that $$x\in\{x\in X\ |\ f(x)\in f[A]\}=f^{-1}[f[A]]$$ by t... |
4,489,898 | <p>After 18 months of studying an advanced junior high school mathematics course, I'm doing a review of the previous 6 months, starting with solving difficult quadratics that are not easily factored, for example:
<span class="math-container">$$x^2+6x+2=0$$</span>
This could be processed via the quadratic equation but t... | emacs drives me nuts | 746,312 | <p>One example is multiplying an expression by some term and dividing it out again, like:</p>
<p><span class="math-container">$$\begin{align}
\frac1{\sqrt{n+1}+\sqrt n}
&= \frac{\sqrt{n+1}-\sqrt n}{(\sqrt{n+1}+\sqrt n)(\sqrt{n+1}-\sqrt n)} \\
&= \frac{\sqrt{n+1}-\sqrt n}{n+1 - n} \\
&= \sqrt{n+1}-\sqrt n\\
... |
4,489,898 | <p>After 18 months of studying an advanced junior high school mathematics course, I'm doing a review of the previous 6 months, starting with solving difficult quadratics that are not easily factored, for example:
<span class="math-container">$$x^2+6x+2=0$$</span>
This could be processed via the quadratic equation but t... | Community | -1 | <blockquote>
<p>Evaluating recurrence relations using generating functions.</p>
</blockquote>
<p>To evaluate <span class="math-container">$F_n = F_{n-1} + F_{n-2}$</span> (<span class="math-container">$F_0 = 0, F_1 = 1$</span>), we can construct the generating function <span class="math-container">$F(x) = \sum_{n=0}^\i... |
4,489,898 | <p>After 18 months of studying an advanced junior high school mathematics course, I'm doing a review of the previous 6 months, starting with solving difficult quadratics that are not easily factored, for example:
<span class="math-container">$$x^2+6x+2=0$$</span>
This could be processed via the quadratic equation but t... | CR Drost | 42,154 | <p>With tensors/matrices, we often do this to provide various identities. For example, every matrix is the sum of a symmetric matrix <span class="math-container">$A_{ij} = A_{ji}$</span> and a skew-symmetric matrix <span class="math-container">$A_{ij} = -A_{ji}.$</span> The easiest proof is,
<span class="math-container... |
2,550,655 | <p>When we're integrating a one variable function, there is only one path to follow between points $a$ and $b$.</p>
<p>$F(b)=F(a)+h(f(a)+f(a+h)+f(a+2h)+........)$ where $h$ is very small. So, we approach from $a$ to $b$ in steps of $h$ along the line joining unique path joining $a$ and $b$.</p>
<p>I was wondering wha... | user | 505,767 | <p>Only for some special functions the path-integral is path independent.</p>
<p>In general it gives different results.</p>
<p>Take a look here:</p>
<p><a href="https://math.stackexchange.com/questions/1734106/what-does-it-mean-for-an-integral-to-be-independent-of-a-path">What does it mean for an integral to be inde... |
3,901,613 | <p>Determine <span class="math-container">$\int_Axy\space\space d(x,y)$</span>, when <span class="math-container">$A$</span> is a closed set, bounded by the curve <span class="math-container">$y=x-1$</span>, and the parabel <span class="math-container">$y^2 = 2x+6.$</span></p>
<p>I believe we only need to know the rang... | Taras | 583,783 | <p>See that <span class="math-container">$2^{16^x} = 16^{2^x}$</span> is <span class="math-container">$2^{2^{4x}} = 2^{2^{x+2}}$</span></p>
<p>then <span class="math-container">$4x = x + 2$</span> and get <span class="math-container">$x = 2/3$</span>.</p>
|
353,920 | <p>I am studying a problem where a quadratic matrix equation emerges. The equation is as follow (all capital letters are n by n matrices)</p>
<p><span class="math-container">$(I-X^{\prime}L)X=D$</span></p>
<p>where <span class="math-container">$L$</span> and <span class="math-container">$D$</span> are both symmetric ... | yc0000 | 153,034 | <p>I figure out something and would like to share and check whether it is correct.</p>
<p>It is perfectly analogous to real quadratic equations.</p>
<p>Since <span class="math-container">$L$</span> is symmetric and positive definite, they can be decomposed:</p>
<p><span class="math-container">$L=U_L^{\prime} \Lambda... |
4,570,396 | <p>Is the set <span class="math-container">$\{\frac 1n :n\in\mathbb{N}\}$</span> countable?</p>
<p>This was part of the Exercises in a book on Real Analysis. The answer given is that this set is uncountable.</p>
<p><strong>My work:</strong></p>
<p>I'm just starting on Real Analysis and , to my knowledge, this set is co... | Nicolás Atanes Santos | 1,116,223 | <p>Yes. As a subset of a countable set is countable, and rationals are (Countability of Rational Numbers), the set <span class="math-container">$\{\frac 1n :n\in\mathbb{N}\}$</span> is countable.
Another way to see it is that for all <span class="math-container">$n$</span>, <span class="math-container">$n+1$</span> exi... |
719,056 | <p>I'm writing a Maple procedure and I have a line that is "if ... then ..." and I would like it to be if k is an integer (or if k^2 is a square) - how would onw say that in Maple? Thanks!</p>
| DanielV | 97,045 | <p>The relation
$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$</p>
<p>requires that $u$ has a nonzero relationship with $x$ and $y$. This doesn't have to be explicitly stated when the chain rule is written as
$$D_x f(g(x)) = f'g(x) \cdot g'(x)$$</p>
<p>Example:</p>
<p>Consider using $g = \text{gravitation... |
758,950 | <p>I have a pretty straightforward combinatorical problem which is an exercise to one paper about generating functions.</p>
<ol>
<li>How many ways are there to get a sum of 14 when 4 distinguishable dice are rolled? </li>
</ol>
<p>So, one die has numbers 1..6 and as dice are distinguishable then we should use exponen... | Stefan Gruenwald | 149,416 | <p>You can do it quite easily with Python:</p>
<pre><code> from sympy.abc import x
from sympy import expand
from math import factorial
d1 = expand((1+x+x**2/2+x**3/factorial(3)+x**4/factorial(4)+x**5/factorial(5)+x**6/factorial(6))**4).as_coefficients_dict()
print d1[x**14]
</code></pre>
<p>It retu... |
2,439,202 | <p>i want to solve this question but i don't know what way i use :
$$\lim_{n\to \infty}a_n= \lim_{n\to\infty} \frac{\binom n 1} {n} +\frac {\binom n 2} {n^2} + \cdots + \frac {\binom n n}{n^n} $$</p>
| user577215664 | 475,762 | <p><strong><em>hint</em></strong></p>
<p>Newton's binomial $(x+y)^n=\sum_{k=0}^n \binom n k x^ky^{n-k}$</p>
<p>With $y=1$ and $x=\frac 1 n$</p>
<p>and the fact that $\lim_{ n\to \infty} (1+ \frac 1 n)^n =e$</p>
|
2,941,311 | <p>Given diophantine equation <span class="math-container">$11x+17y +19z =2561$</span> , which <span class="math-container">$x,y,z \geq 1$</span></p>
<p>Find minimum and maximum value of <span class="math-container">$x+y+z$</span></p>
<p>I'm start with reduces equation to <span class="math-container">$11x+17y +1... | davidlowryduda | 9,754 | <p>My pocket example of such a limit is
<span class="math-container">$$ \lim_{x \to \infty} \frac{x + \sin x}{x + \cos x}.$$</span>
This limit is very clearly <span class="math-container">$1$</span>. But an application of l'Hopital's rule would lead to the consideration of
<span class="math-container">$$ \lim_{x \to \i... |
4,465,528 | <p>Solving the integral of <span class="math-container">$\cos^2x\sin^2 x$</span>:</p>
<p>My steps are: <span class="math-container">$(\cos x\sin x)^2=\left(\frac{\sin(2x)}{2}\right)^2$</span>. Now we know that</p>
<p><span class="math-container">$$\sin^2(\alpha)=\frac{1-\cos (2x)}{2}\iff\left(\frac{\sin(2x)}{2}\right)^... | Gregory J. Puleo | 183,812 | <p>I disagree with your reasoning: you are assuming that the <span class="math-container">$300$</span> students who answered wrong to the third question and the <span class="math-container">$400$</span> students who answered wrong to the fourth question are totally separate from each other, but there's no guarantee of ... |
3,207,453 | <p>studying the series <span class="math-container">$\sum_\limits{n=2}^\infty \frac{1}{n(\log n)^ {2}}$</span>.</p>
<p>I've tried with the root criterion</p>
<p><span class="math-container">$\lim_{n \to \infty} \sqrt[n]{\frac{1}{n(\log n)^ {2}}}>1$</span>
and the series should diverge.</p>
<p>But I'm not sure
Ca... | Bernard | 202,857 | <p><strong>Hint</strong>:</p>
<p>Use the integral test to show that all series <span class="math-container">$\;\displaystyle\sum_\limits{n=2}^\infty \frac{1}{n(\log n)^{\alpha}}$</span> converge if and only if <span class="math-container">$\alpha >1$</span>.</p>
|
3,784,889 | <p><span class="math-container">$$\begin{array}{ll} \text{minimize} & f(x,y,z) := x^2 + y^2 + z^2\\ \text{subject to} & g(x,y,z) := xy - z + 1 = 0\end{array}$$</span></p>
<hr />
<p>I tried the Lagrange multipliers method and the system resulted from has no solution. So I posted it to see if the question is wron... | Rodrigo de Azevedo | 339,790 | <p>Since <span class="math-container">$f$</span> and <span class="math-container">$g$</span> are polynomial, using <a href="http://www.sympy.org" rel="nofollow noreferrer">SymPy</a>'s <code>solve_poly_system</code>:</p>
<pre><code>>>> from sympy import *
>>> x, y, z, mu = symbols('x y z mu', real=True... |
3,784,889 | <p><span class="math-container">$$\begin{array}{ll} \text{minimize} & f(x,y,z) := x^2 + y^2 + z^2\\ \text{subject to} & g(x,y,z) := xy - z + 1 = 0\end{array}$$</span></p>
<hr />
<p>I tried the Lagrange multipliers method and the system resulted from has no solution. So I posted it to see if the question is wron... | Moko19 | 618,171 | <p>This can be solved in at least two methods.
First, let's solve without Lagrange, using convenient changes of variables. Let <span class="math-container">$u=x+y, v=xy$</span>. This results in <span class="math-container">$u^2=x^2+y^2+2xy=x^2+y^2+2v$</span>.</p>
<p>We now need to minimize <span class="math-container... |
1,364,417 | <p>Find a real number k such that the limit $$\lim_{n\to\infty}\ \left(\frac{1^4 + 2^4 + 3^4 +....+ n^4}{n^k}\right)$$ has as positive value.
If I am not mistaken every even $k$ can be the answer. But the answer is 5.</p>
| MadSax | 253,866 | <p>Observe that
$$
1^4+2^4+...+n^4 = \sum_{i=1}^n i^4 = \frac{1}{30}(6n^5+15n^4+10n^3-n)
$$ so if $k<5$ the limit is $+\infty$ (does not exist) but if $k>5$ the limit is $0$.</p>
|
3,102,218 | <p>Given a fraction:</p>
<p><span class="math-container">$$\frac{a}{b}$$</span></p>
<p>I now add a number <span class="math-container">$n$</span> to both numerator and denominator in the following fashion:</p>
<p><span class="math-container">$$\frac{a+n}{b+n}$$</span></p>
<p>The basic property is that the second fr... | Claude Leibovici | 82,404 | <p><em>Just for the fun of it, since you already received very good answers.</em></p>
<p>Perform the long division to get
<span class="math-container">$$\frac{a+n}{b+n}=1+\frac{a-b}n\left(1-\frac{b}{n}+\frac{b^2}{n^2}-\frac{b^3}{n^3} +\cdots\right)=1+\frac{a-b}n\sum_{k=0}^\infty (-1)^k \left(\frac bn\right)^k$$</span>... |
299,795 | <p>I seem to have completely lost my bearing with implicit differentiation. Just a quick question:</p>
<p>Given $y = y(x)$ what is $$\frac{d}{dx} (e^x(x^2 + y^2))$$</p>
<p>I think its the $\frac d{dx}$ confusing me, I don't what effect it has compared to $\frac{dy}{dx}$. Any help will be greatly appreciated.</p>
| Community | -1 | <p>You could write the equation as
\begin{equation*}
x^7(x^2+x+1)=0.
\end{equation*}
Now, solve the equation $x^7=0$ and $x^2+x+1=0$.</p>
|
73,039 | <p>I have a huge data file which I can't ListPlot.</p>
<p>This code generates similar kind of data:</p>
<pre><code>datatest =RandomSample[Join[RandomReal[{0.5, 15}, 20], RandomReal[.1, 10000]]];
datatest2 = 5 + Riffle[datatest, -datatest];
</code></pre>
<p>I want to filter (delete) the part of the data that is not n... | yohbs | 367 | <p>I'm not sure I understand what you need, but here's my try: If you want to keep only <code>2d+1</code> data points around each peak, you can use</p>
<pre><code>toKeep = Map[# + Range[-d, d] &, peaks[[All, 1]]];
choppedData = Map[Part[datatest2, #] &, toKeep];
</code></pre>
<p><code>choppedData</code> is a ... |
5,927 | <p>I have a problem with the binomial coefficient $\binom{5}{7}$. I know that the solution is zero, but I have problems to reproduce that:</p>
<p>${\displaystyle \binom{5}{7}=\frac{5!}{7!\times(5-7)!}=\frac{5!}{7!\times(-2)!}=\frac{120}{5040\times-2}=\frac{120}{-10080}=-\frac{1}{84}}$</p>
<p>Where is my mistake?</p>
| Bill Dubuque | 242 | <p>By definition $\rm\binom{n}k$ is the coefficient of $\rm x^k$ in $\rm (1+x)^n$ so it is $0$ for $\rm k > n\:$.</p>
|
6,929 | <p>For example, I have posted quite a few questions regarding mathematics in general, because I want to learn more about the field and I just want to make sure I get an answer or an opinion from someone who has good knowledge about my question. I tag it with the advice tag and I have seen, more than once, that people w... | Mariano Suárez-Álvarez | 274 | <p>There are currently 106 (not deleted) questions tagged with the «advice» tag, and only a few of them are closed: exactly 12.</p>
<p>There is the possibility that your questions are being closed for reasons other than the tag!</p>
<p>Moreover, as far as I can tell, none of your three advice questions is closed (alt... |
592,963 | <p>Find $x^4+y^4$ if $x+y=2$ and $x^2+y^2=8$</p>
<p>So i started the problem by nothing that $x^2+y^2=(x+y)^2 - 2xy$ but that doesn't help!</p>
<p>I also seen that $x+y=2^1$ and $x^2+y^2=2^3$ so maybe $x^3+y^3=2^5$ and $x^4+y^4=2^7$ but i think this is just coincidence</p>
<p>So how can i solve this problem?</p>
<p... | Mercy King | 23,304 | <p>\begin{eqnarray}
x^4+y^4&=&(x^2+y^2)^2-2x^2y^2=(x^2+y^2)^2-\frac12(2xy)^2=(x^2+y^2)^2-\frac12\left[(x+y)^2-(x^2+y^2)\right]^2\\
&=&8^2-\frac12(2^2-8)^2=56.
\end{eqnarray}</p>
|
2,913,156 | <p>I'm working through the definitions given in <strong>2.18</strong> in Baby Rudin, and I was wondering how to prove that <span class="math-container">$E = [0,1] \cap \Bbb{Q}$</span> is closed in <span class="math-container">$\Bbb{Q}$</span>.</p>
<p>Clearly, the set <span class="math-container">$[0,1] \cap \Bbb{R}$</... | José Carlos Santos | 446,262 | <p>The density of $\mathbb Q$ plays no role here. If $A$ and $B$ are subsets of $\mathbb R$ and $A$ is a closed subset of $\mathbb R$, then $A\cap B$ is a closed subset of $B$. That's so because the complement of $A\cap B$ in $B$ is equal to $A^\complement\cap B$. Now, since $A$ is a closed subset of $\mathbb R$, $A^\c... |
54,856 | <p>I'm trying to simplify an expression with sums of <code>ArcTan</code>, for which I've written a transformation rule that works for simple sums, so that <code>ArcTan[a] + ArcTan[b] -> ArcTan[(a+b)/(1-ab)]</code>. However, it fails for terms of the type <code>5 ArcTan[a] + ArcTan[b]</code>. One way to simplify this... | Michael E2 | 4,999 | <p>How about:</p>
<pre><code>5 ArcTan[a] + ArcTan[b] /.
{(m_Integer: 1) ArcTan[a_] + (n_Integer: 1) ArcTan[b_] /; Positive[m] && Positive[n] :>
With[{min = Min[m, n]},
(m - min) ArcTan[a] + (n - min) ArcTan[b] +
min ArcTan[(a + b)/(1 - ab)]]}
(* 4 ArcTan[a] + ArcTan[(a + b)/(1 - ab)] *)
</cod... |
1,109,706 | <p>I have just started going over abstract algebra.</p>
<p>One of the question is </p>
<p>$*$ is defined on $\mathbb C$ such that $a*b=|ab|$</p>
<p>I tried to check three axioms : 1) Associativity 2) identity 3) inverse</p>
<p>I found out with long computation that Associativity works.</p>
<p>For identity, $a*e=e*... | user207950 | 207,950 | <p>It is not group. Problem is with a=-s s$\in R^+$.</p>
|
85,288 | <p>For some reason, I'd like to use <code>ParallelTable</code> with a variable number of iterators.</p>
<ul>
<li><code>Table[a[1], {a[1], 0, 10}]</code> works fine:</li>
</ul>
<blockquote>
<p><em>Output:</em> <code>{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}</code></p>
</blockquote>
<ul>
<li><code>ParallelTable[a[1], {a[1], 0, ... | Albert Retey | 169 | <p>This seems to just be a limitation of <code>ParallelTable</code>, can't comment whether that has a deeper reason due to parallelism or is just a simple oversight. I think it was not possible to use expressions like <code>a[i]</code> as e.g. iterators in older versions but in newer version that has been added as a fe... |
2,900,014 | <p>How would solve for $a$ in this equation without using an approximation ?
is it possible?</p>
<p>where $x>0$ and $0<a<\infty$</p>
<p>$x=\Sigma _{i=1}^{n} i^a$</p>
<p>for example $120=\Sigma _{i=1}^{6} i^a$ what is $a$ in this equation?</p>
| Community | -1 | <p>If you set $a=\log t$, you can write the question as</p>
<p>$$x=\sum_{k=1}^n k^a=\sum_{k=1}^n t^{\log k}$$</p>
<p>where the RHS is a generalized polynomial (with irrational powers). So this is even less solvable than a polynomial and you can't avoid numerical methods.</p>
<p>The function (of $a$) is strictly grow... |
2,588,408 | <p>A question from <em>Introduction to Analysis</em> by Arthur Mattuck:</p>
<p>Suppose $f(x)$ is continuous for all $x$ and $f(a+b)=f(a)+f(b)$ for all $a$ and $b$. Prove that $f(x)=Cx$, where $C=f(1)$, as follows:</p>
<p>(a)prove, in order, that it is true when $x=n, {1\over n}$ and $m\over n$, where $m, n$ are integ... | QuIcKmAtHs | 515,633 | <p>This is true for all: $f(x)=Cx$. For instance, for $f({1\over n})$, you would need $n$ of that term to make C. Hence, it is ${C\over n}$, which is also $C*{1\over n}$. For $f({m\over n})$, it is ${C*m\over n}$. Firstly, ${m\over n}$ is actually just $m*{1\over n}$. Since $f({1\over n})$ = ${C\over n}$, $f({m\ov... |
690,569 | <p>Suppose a function is given by:
$$
f(x)=
\begin{cases}
\cos\left(\dfrac{1}{x}\right) & x\neq 0 \\
0 & x=0
\end{cases}
$$</p>
<p>Show that this function is not continuous. Please help - I don't know how to proceed with formally using the limits.</p>
| BoZenKhaa | 60,953 | <p>To show that it is not continous, show that the function is not continous at 0. To do that, think about what hapens to cos(1/x) as x goes to zero, i.e. What happens to cos(x) as x goes to infinity. Does it have zero as a limit? Nah, it oscilates. So because function f is continuous at a point c iff f(c) equals to th... |
2,349,476 | <p>Suppose $n$ and $k$ are positive integers. Are there any known conditions on $n$ and $k$ such that the polynomial $x^{k+1}-x^k-n$ has rational or integer roots?</p>
| Dietrich Burde | 83,966 | <p>Because of the <a href="https://en.wikipedia.org/wiki/Rational_root_theorem" rel="nofollow noreferrer">Rational Root Theorem</a>, every rational root $x=\frac{p}{q}$ of $x^{k+1}-x^k-n$ must satisfy $p\mid n$ and $q\mid 1$ in $\mathbb{Z}$. For most $k$, in practice, there will be no integer roots for given $n$ of the... |
244 | <p>I know that Hilbert schemes can be very singular. But are there any interesting and nontrivial Hilbert schemes that are smooth? Are there any necessary conditions or sufficient conditions for a Hilbert scheme to be smooth?</p>
| G.G. | 118,706 | <p>For some more examples of smooth HS see <a href="https://arxiv.org/abs/1702.00080" rel="nofollow noreferrer">A.P.Staal: The ubiquity of smooth Hilbert schemes, arxiv AG 31.Jan. 2017</a>.</p>
|
496,882 | <p>If I have an experiment that has $1000$ trials, and $10$% of the time there is an error, what is the approximate probability that I will have $125$ failures? I figured out that $\mu =100$ and $\sigma =9.4868$. I'm trying to approximate with a normal distribution, since with this many trials, and the fact that each e... | Dilip Sarwate | 15,941 | <p>The DeMoivre-Laplace theorem says that if $X$ is a binomial random variable with parameters $(n,p)$, then
$$P\{a < X < b\} \approx \Phi\left(\frac{b-np}{\sqrt{np(1-p)}}\right) - \Phi\left(\frac{a-np}{\sqrt{np(1-p)}}\right)$$
and so, for your problem,
$$P\{X \leq 125\} \approx \Phi\left(\frac{125-100}{\sqrt{10... |
3,118,238 | <p>I know there is a pretty simply way to check if a 2D point is inside a circle, I wanted to know how to do there same but in the 3rd dimension. The variables are the point's X, Y, Z and the sphere's 3D point and radius.</p>
<p>Let's say we know that X and Y is inside the circle, can we rotate it 90 degrees and prete... | Parcly Taxel | 357,390 | <p>Rather than using two 2D tests to check (which is incorrect – points passing both tests are only guaranteed to lie inside a <a href="https://en.wikipedia.org/wiki/Steinmetz_solid" rel="noreferrer">mouhefanggai/Steinmetz solid</a> which strictly encloses the sphere), a simple extension of the 2D test will work.... |
16,740 | <p>Excuse me for my unclear question title, math terms have never been my forté. However being new here and knowing I shall be staying here for a while, I thought it's best to do things right with the community and site.</p>
<p>I just asked my first <a href="https://math.stackexchange.com/questions/920192/differentiat... | user642796 | 8,348 | <p>In similarity to posters of link-only answers, users who post questions which rely on familiarity with or access to a <em>specific</em> external source to be understood and answered should be urged to include as much relevant information from the source so as to make the question self-contained (roughly meaning inte... |
2,424,722 | <blockquote>
<p>Let a,b,c be real numbers such that $a+2b+c=4$. What is the value of $\max(ab+bc+ac)$</p>
</blockquote>
<p>My attempt:</p>
<p>Squaring both the sides: </p>
<p>$a^2 +4b^2+c^2+2ac+4bc+4ab=16$</p>
<p>Then I tried factoring after bringing 16 to LHS but couldn't. It's not even a quadratic in one variab... | GAVD | 255,061 | <p>Note that $4xy \leq (x+y)^2$ for all $x$, $y$.</p>
<p>You have $$ab+bc+ca = b(a+c) + ac \leq b(a+c) + \frac{1}{4}(a+c)^2 = (a+c) \frac{a+c+4b}{4} = \frac{1}{4}(a+c)(8-(a+c)) \leq \frac{1}{4}\frac{1}{4}8^2 = 4.$$</p>
<p>Equality is happen when $a=c$ and $a+c = 8-(a+c)$, or $a+c=4$.</p>
|
4,005,738 | <p>I am interested in a method of how to compute the expected number of steps for absorpion in a Markov Chain with only one absorption node and given the starting and pre-final node (before absorption).</p>
<p>So, if the transition probability matrix is given, for example by:</p>
<p><span class="math-container">$P = \b... | Dieter Kadelka | 882,293 | <p>Let <span class="math-container">$S := \{1,2,3,4,5\}$</span> be the state space of this homogeneous Markov chain (MC) and <span class="math-container">$\xi_n$</span> be the random state at time <span class="math-container">$n \in \mathbb{N}_0$</span>. Let <span class="math-container">$\mathbb{P}_s$</span> be the joi... |
855,088 | <p>Consider two vectors $V_1$ and $V_2$ in $\mathbb{R}^3$.
When we take their dot product we get a real number.
How is that number related to the vectors?
Is there any way we can visualize it? </p>
| blue | 34,139 | <p>The dot product ${\bf a}\cdot{\bf b}$ measures the length of ${\bf a}$'s orthogonal projection onto $\bf b$ (the $1$-dimensional subspace it is a part of), scaled by the length of $\bf b$ itself. And conversely. The scaling is nice to have because it means the dot product is bilinear in its two arguments. Physically... |
2,485,447 | <p>My attempt:</p>
<p>3x≡1 mod 7 (1)</p>
<p>4x≡1 mod 9 (2)</p>
<p>Multiply (1) by 5</p>
<p>Multiply (2) by 7</p>
<p>x≡5 mod 7</p>
<p>x≡7 mod 9</p>
<p>So x≡9k+7</p>
<p>9k+7=5(mod7)</p>
<p>k=5(mod7)</p>
<p>k=7j+5</p>
<p>x=9(7j+5)+7</p>
<p>=63j+52</p>
<p>x≡52(mod63)</p>
| Deepak | 151,732 | <p>From the first equation (note that $5$ is the multplicative inverse of $3 \pmod 5$),</p>
<p>$3(5)x \equiv 5 \pmod 7 \implies 15x \equiv 5 \pmod 7 \implies x \equiv 5 \pmod 7$. So $x = 7k + 5$</p>
<p>Substitute into the second to give $28k + 20 \equiv 1 \pmod 9 \implies k \equiv -19\pmod 9 \implies k \equiv -1 \pmo... |
1,531,171 | <p>if $fg$ and $f$ are differentiable at $a$, must $g$ be differentiable at $a$? If not, what condition is needed to imply that $g$ be differentiable at $a$?</p>
<p>I realized that people asked similar question before, like
"If $f+g$ and $f$ are differentiable at $a$, must $g$ be differentiable at $a?$" It can be easi... | Tsemo Aristide | 280,301 | <p>f=x and g=$\sqrt x$ at 0 fg=x$\sqrt x$ g is not derivable at 0</p>
|
2,233,185 | <p>I'm finding how many integers under a limit, $L$, only have prime factors from a given set of prime numbers, $P$. The numbers that meet these conditions are called n-smooth numbers. (I've never used sets before so feel free to correct any mistakes I make). Take, for example, $P = \left\{2, 3 \right\}$, and $L = 25$,... | Paramanand Singh | 72,031 | <p>We can directly prove that $M_{a}$ is maximal without using any theorems.</p>
<p>Let us take any ideal $N \supset M_{a}$ and we show that $N = R$. Clearly since $N \supset M_{a}$ there is an $f \in N$ such that $f(a) \neq 0$. Now let $g$ be any arbitrary member of $R$. Note that the function $h(x)$ given $$h(x) = g... |
90,548 | <p>Suppose we are handed an algebra $A$ over a field $k$. What should we look at if we want to determine whether $A$ can or cannot be equipped with structure maps to make it a Hopf algebra?</p>
<p>I guess in order to narrow it down a bit, I'll phrase it like this: what are some necessary conditions on an algebra for ... | Qiaochu Yuan | 290 | <p>I'm a bit late, but here's a simple observation. Consider a topological version of the question: given a topological space $X$, how can we recognize when $X$ can be given the structure of a topological group? A simple necessary condition is that that $X$ must be homogeneous, and in particular each point should have ... |
1,760,148 | <p>If I have a connected metric space $X$, is any ball around a point $x\in X$ also connected?</p>
| Moishe Kohan | 84,907 | <p>On the positive side, if $(X,d)$ is a <a href="https://en.wikipedia.org/wiki/Intrinsic_metric" rel="nofollow noreferrer">length metric space</a> then every open ball $B(a, r)\subset (X,d)$ is path connected. (Every closed ball is connected as well, since, in this situation, each closed ball is either a singleton or ... |
4,164,069 | <p>I guess it's true for functions that are Lipshitz or uniformly continuous since we can limit the length of the intervals after the transformation.<br />
However, I don't know if it's true or not, and since <span class="math-container">$1/x$</span> is not one of those, I don't know how to solve this problem.</p>
| Oliver Díaz | 121,671 | <p>Here is an interesting result that helps with your problem:</p>
<blockquote>
<p><strong>Theorem:</strong> If <span class="math-container">$f:(a,b)\rightarrow\mathbb{R}$</span> is measurable, and <span class="math-container">$f$</span> is differentiable on a measurable set <span class="math-container">$C\subset(a,b)$... |
1,764,729 | <blockquote>
<p>Let $x(t) : [0,T] \rightarrow \mathbb{R}^n$ be a solution of a differential equation
$$
\frac{d}{dt} x(t) = f(x(t),t).
$$
In addition we have functions $E :\mathbb{R}^n \rightarrow \mathbb{R}$ and $h:\mathbb{R}^{n+1}\rightarrow \mathbb{R}$ such that
$$
\frac{d}{dt}{E(x(t))} = h(x(t),t) E(x(t))... | Salsifis | 334,805 | <p>Winther was right. You can use Gronwall's Lemma to understand it. Define the function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that: $$f(t)=E(x(t))^2\times 1_{E(x(t))\leqslant 0}$$
For sure, this is a non-negative function. Take the derivative:
\begin{align}
\partial_t f(t)&\leqslant h(x(t),t)E(x(t))^2 \times 1... |
782,427 | <p><strong>I - First doubt</strong> : free variables on open formulas . </p>
<p>I'm having a hard time discovering what different kinds of variables in an open formula of fol are refering to., For example, lets take some open formulas : </p>
<p>1: $"x+2=5"$<br>
2: $"x=x"$<br>
3: $"x+y = y+x"$<br>
4: $"x+2 = ... | Mauro ALLEGRANZA | 108,274 | <p><em>Question n°1</em> :</p>
<blockquote>
<p>let's suppose we have a domain <span class="math-container">$D$</span> of a structure which is interpreting that fol language. [...] the I think the variables refer to different kind of things depending on the kind of formula... to what exactly, in each case ?</p>
</bl... |
3,428,546 | <p>Consider whether the following series converges or diverges <span class="math-container">$$\sum_{n=1}^\infty\frac{(-1)^n\sin\frac{n\pi}{3n+1}}{\sqrt{n+3}}$$</span></p>
<p>I have tried Leibniz's test but failed to show that <span class="math-container">$\sin\frac{n\pi}{3n+1} / \sqrt{n+3}$</span> is monotone, which I... | eternalGoldenBraid | 225,419 | <p>Terence Tao has some interesting related commentary in his blog post "<a href="https://terrytao.wordpress.com/2010/01/01/254a-notes-0-a-review-of-probability-theory/" rel="noreferrer">A review of probability theory</a>":</p>
<blockquote>
<p>Elements of the sample space <span class="math-container">$\Omega$</span>... |
989,740 | <p>How do I prove the following statement?</p>
<blockquote>
<p>If $x^2$ is irrational, then $x$ is irrational. The number $y = π^2$ is irrational. Therefore, the number $x = π$ is irrational</p>
</blockquote>
| mfl | 148,513 | <p>The statement </p>
<p>"$x^2$ irrational $\implies$ $x$ irrational" </p>
<p>is logically equivalent to </p>
<p>"$x$ not irrational $\implies x^2$ not irrational".</p>
|
3,655,215 | <p>There are many contradictions in literature on tensors and differential forms. Authors use the words coordinate-free and geometric. For example, the book Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers say differential forms are coordinate free while tensors are dependendent on coor... | Lee Mosher | 26,501 | <p>I cannot suggest any particular textbook because mathematical physics is quite far from my area of expertise, although I will confess a fondness for Misner, Thorne and Wheeler.</p>
<p>But I will say that there does not exist any "authoritative textbook" in which all ambiguities of terminology and notation are erase... |
1,517,086 | <p>I spent a long time trying to find a natural deduction derivation for the formula $\exists x(\exists y A(y) \rightarrow A(x))$, but I always got stuck at some point with free variables in the leaves. Could someone please help me or give me some hints to find a proof. </p>
<p>Thanks.</p>
| oerpli | 184,354 | <p>If you prefer the tree style notation this is a valid proof:</p>
<p>$$\dfrac{\dfrac{\lower{1.5ex}{[\exists y~A(y)]^1}~~\dfrac{[A(s)]^2}{A(s)}{}}{\dfrac{A(s)}{\exists y~A(y)~\to~A(s)}{(\to i,1)}}{(\exists e,1)}}{\exists x~(\exists y~A(y)~\to~A(x))}{(\exists i)}$$</p>
|
121,791 | <p>Now I'm trying to use </p>
<pre><code>Show[plota, Epilog->Inset[inset, Scaled->[{0.2, 0.7}]]]
</code></pre>
<p>operation to insert an inset of a figure, where plota is an Overlay of two plots</p>
<pre><code>plota = Overlay[{plotb, plotc}];
</code></pre>
<p>and then I cannot be successfully inserted in beca... | Mr.Wizard | 121 | <p>As JasonB temporarily commented you can <a href="http://reference.wolfram.com/language/ref/Rasterize.html" rel="nofollow noreferrer"><code>Rasterize</code></a> an <a href="http://reference.wolfram.com/language/ref/Overlay.html" rel="nofollow noreferrer"><code>Overlay</code></a> to convert it into a (rasterized) <cod... |
267,753 | <p>A number of seemingly unrelated elementary notions can be defined uniformly with help of (iterated) Quillen lifting property
(a category-theoretic construction I define below) "starting" to a single (counter)example or a simple class of morphisms,
for example a finite group being nilpotent, solvable, p-group, a top... | Tim Campion | 2,362 | <p>A word on the bigger picture. Examples are absolutely ubiquitous in category theory. First note that if <span class="math-container">$\mathcal{C} \perp \mathcal{D}$</span>, then <span class="math-container">$\mathcal{C}$</span> and <span class="math-container">$\mathcal{D}$</span> determine each other uniquely up to... |
1,304,948 | <p>If I solve $Tx=0$ where $T$ is some square matrix
then if I multiply both sides by $T$ and solve for $T^2x=0$, will my x be the same?
In other words if I were to multiply to both sides of the equation $Tx=0$ to any order of $T$, will my x be the same? or at least satisfy the first equation?</p>
| Kevin | 61,342 | <p>As Prahlad Vaidyanathan stated, this is, in general, not true. However, it is true that $W\subseteq (W^\perp)^\perp$. </p>
|
1,304,948 | <p>If I solve $Tx=0$ where $T$ is some square matrix
then if I multiply both sides by $T$ and solve for $T^2x=0$, will my x be the same?
In other words if I were to multiply to both sides of the equation $Tx=0$ to any order of $T$, will my x be the same? or at least satisfy the first equation?</p>
| copper.hat | 27,978 | <p>In general you have $W^{\bot \bot} = \overline{\operatorname{sp} W}$.</p>
<p>For a simple example, take $l_2$ and let $W =\operatorname{sp} \{e_k \}$, a proper dense subspace of $l_2$.
Then $W^\bot = \{0\}$ and so $W^{\bot \bot} = l_2$.</p>
|
1,109,918 | <p>Is it always possible to add terms into limits, like in the following example? (Or must certain conditions be fulfilled first, such as for example the numerator by itself must converge etc)</p>
<p>$\lim_{h \to 0} {f(x)} = \lim_{h \to 0} \frac{e^xf(x)}{e^x}$</p>
| Pp.. | 203,995 | <p>$$\frac{\partial f(x+h)}{\partial h}=f'(x+h)$$</p>
<p>$$\frac{\partial f(x-h)}{\partial h}=-f'(x-h)$$</p>
<p>$$\frac{\partial f(x)}{\partial h}=0$$</p>
<p>$$(h^2)'=2h$$</p>
|
1,423,456 | <p>I am completely stuck on this problem: $C[0,1] = \{f: f\text{ is continuous function on } [0,1] \}$ with metric $d_1$ defined as follows:</p>
<p>$d_1(f,g) = \int_{0}^{1} |f(x) - g(x)|dx $.</p>
<p>Let the sequence $\{f_n\}_{n =1}^{\infty}\subseteq C[0,1]$ be defined as follows:</p>
<p>$
f_n(x) = \left\{
\beg... | DanielWainfleet | 254,665 | <p>Estimate $d_1(f_m,f_n)$ for $0<m<n$ as follows: For any $n>0$ let $p(n)=(1-1/n)/2$ and $q(n)=(1+1/n)/2$. For $0<m<n$ we have $0\le p(m)<p(n)<q(n)<q(m) \le 1$ .We have $$f_m(x)=f_n(x)=1$$ for $x \in [0,p(m)]$ and $$f_m(x)=f_n(x)=0$$ for $y \in [q(m),1]$.Therefore $$d_1(f_m,f_n)=\int_{p(m)}^{q... |
2,538,553 | <p>For example in statistics we learn that mean = E(x) of a function which is defined as</p>
<p>$$\mu = \int_a^b xf(x) \,dx$$</p>
<p>however in calculus we learn that</p>
<p>$$\mu = \frac {1}{b-a}\int_a^b f(x) \,dx $$ </p>
<p>What is the difference between the means in statistics and calculus and why don't they gi... | Michael Hardy | 11,667 | <p>This seems to be based on confusion resulting from resemblance between the notations used in the two situations.</p>
<p>In probability and statistics, one learns that $\displaystyle\int_{-\infty}^\infty x f(x)\,dx$ is the mean, <b>NOT</b> of the function $f$, but of a random variable denoted (capital) $X$ (whereas ... |
4,333,062 | <p>I stumbled upon the following identity
<span class="math-container">$$\sum_{k=0}^n(-1)^kC_k\binom{k+2}{n-k}=0\qquad n\ge2$$</span>
where <span class="math-container">$C_n$</span> is the <span class="math-container">$n$</span>th Catalan number. Any suggestions on how to prove it are welcome!</p>
<p>This came up as a ... | Clyde Kertzer | 1,026,367 | <p>I'll sum up the concepts it seems a few have addressed in the comments.</p>
<p>Setting the square root of something equal to a negative will only result in an imaginary solution, as I'm sure you are aware. Thus, squaring both sides will ALWAYS result in an extraneous solution. So, if you can show that an equation is... |
167,221 | <p>I'm wondering if this is possible for the general case. In other words, I'd like to take $$\int_a^b{g(x)dx} + \int_c^d{h(x)dx} = \int_e^f{j(x)dx}$$ and determine $e$, $f$, and $j(x)$ from the other (known) formulas and integrals. I'm wondering what restrictions, limitations, and problems arise.</p>
<p>If this is ... | Mercy King | 23,304 | <p>Assume $a< b$ and $c< d$. Let<br>
$$
e \le \min\{a,c\}, \ f \ge \max\{b,d\},
$$
and set $j=\tilde{g}+\tilde{h}$ where
$$
\tilde{g}:=g.1_{[a,b]},\tilde{h}:=h.1_{[c,d]}: [e,f] \to \mathbb{R}.
$$
Then
$$
\int_a^b g+\int_c^d h=\int_e^f(\tilde{g}+\tilde{h})=\int_e^f j.
$$</p>
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.