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 |
|---|---|---|---|---|
2,601,851 | <p>I have a binary variable $y_{t}$ that is equal to $1$ iff the job is scheduled at slot $t$. I need to write constraints that guarantee that if the job is scheduled somewhere, then it must be scheduled for a period of $A$ consecutive slots. I tried to write it this way:</p>
<p>$\sum_{t'=t}^{t+A-1}y_{t'}\geqslant A y... | user160110 | 160,110 | <p>Imagine this is the real number line</p>
<p>$--------------$</p>
<p>Let's put $U=U(f,P)$ and$L=L(f,P)$ on there</p>
<p>$---L----------U---$</p>
<p>Now we know that the distance between these two is less than $\epsilon$ </p>
<p>$---L----------U---$
$---|<---(\epsilon< )--->|--$</p>
<p>Given that the sm... |
3,511,660 | <p>Can you help me please I could not figure this out.</p>
<p>Given: </p>
<p><span class="math-container">$f:\mathbb{R}\to\mathbb{R}$</span>, <span class="math-container">$f'(0)$</span> exists, <span class="math-container">$f(x)\neq0$</span> and for all <span class="math-container">$a, b\in\mathbb{R}$</span>, <span ... | Dominik Kutek | 601,852 | <p>Since <span class="math-container">$f'(0)$</span> exists, there is <span class="math-container">$\delta >0$</span> such that <span class="math-container">$f$</span> is continuous on <span class="math-container">$(-\delta,\delta)$</span>. Since <span class="math-container">$f > 0$</span> (why), we can take <spa... |
4,624,058 | <p>Godsil&Royle <a href="https://doi.org/10.1007/978-1-4613-0163-9" rel="nofollow noreferrer">Algebraic Graph Theory</a> section 2.5 states (slightly paraphrased):</p>
<blockquote>
<p>Let <span class="math-container">$G$</span> be a transitive group acting on a set <span class="math-container">$V$</span>. A nonempt... | Devo | 1,092,170 | <p>If <span class="math-container">$g(S)=f(S)$</span>, you are done. So, let's assume <span class="math-container">$g(S)\ne f(S)$</span>.</p>
<ul>
<li>Case 1: <span class="math-container">$g(S)=S$</span> and <span class="math-container">$f(S)\ne S$</span>; then, <span class="math-container">$g(S)\cap f(S)=S\cap f(S)=\... |
864,212 | <p>While trying to look up examples of PIDs that are not Euclidean domains, I found a statement (without reference) on the <a href="http://en.wikipedia.org/wiki/Euclidean_domain">Euclidean domain</a> page of Wikipedia that</p>
<p>$$\mathbb{R}[X,Y]/(X^2+Y^2+1)$$</p>
<p>is such a ring. After a good deal of searching, I... | Jessica B | 81,247 | <p>Here are some more details for those who, like me, are not so familiar with this material.</p>
<p>$1$) Take a non-zero prime ideal. We wish to show that it is maximal. It contains a prime element (since we are in a UFD, choose an element with a shortest factorisation). Take this to be $ax+by+c$. We will show that t... |
3,858,517 | <p>Is it possible to count exactly the number of binary strings of length <span class="math-container">$n$</span> that contain no two adjacent blocks of 1s of the same length? More precisely, if we represent the string as <span class="math-container">$0^{x_1}1^{y_1}0^{x_2}1^{y_2}\cdots 0^{x_{k-1}}1^{y_{k-1}}0^{x_k}$</s... | Phicar | 78,870 | <p>I am gonna attempt to complement <strong>RobPratt</strong>'s proposed approach involving inclusion exclusion and stars and bars and be that person who posts a horribly long formula.<br><br>
Consider <span class="math-container">$$A_{n,k,r}=\left |\left \{0^{l_1}1^{k_1}\cdots 0^{l_r}1^{k_r}0^{l_{r+1}}\in \{0,1\}^n: k... |
286,798 | <blockquote>
<p>Find the limit $$\lim_{n \to \infty}\left[\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\cdots\left(1-\frac{1}{n^2}\right)\right]$$</p>
</blockquote>
<p>I take log and get $$\lim_{n \to \infty}\sum_{k=2}^{n} \log\left(1-\frac{1}{k^2}\right)$$</p>
| Did | 6,179 | <p>The identity $k^2-1=(k+1)(k-1)$ shows that
$$
\prod_{k=2}^n\left(1-\frac{1}{k^2}\right)=\prod_{k=2}^n\frac{k^2-1}{k^2}=\prod_{k=2}^n\frac{k-1}k\cdot\prod_{k=2}^n\frac{k+1}k=\frac1n\cdot\frac{n+1}2,
$$
and the value of limit should follow.</p>
|
660,899 | <p>find the unit normal $\bf \hat{N}$ of</p>
<p>$${\bf r}=6 \mathrm{e}^{-14 t}\cos(t){\bf i}+6 \mathrm{e}^{-14 t}\sin(t){\bf j}$$</p>
<p>The answer should be in vector form. Use t as parameter. Write $e^x$ for exponentials.</p>
<p>Have been working with this a long time now but cant get the right answer.
My answer i... | Did | 6,179 | <p>Two different questions here:</p>
<blockquote>
<p>How do you derive that properly?</p>
</blockquote>
<p>For example, by noting that the definition of conditional distribution says exactly that $Y$ is distributed as $aX+b+\sigma_YZ$ where $Z$ is standard normal (and independent of $X$) hence $E(Y)=aE(X)+b+\sigma_... |
2,966,010 | <p>how to show that <span class="math-container">$$\sum_{n=1}^{\infty}(-1)^{n}\dfrac{3n-1}{n^2 + n} = \log\left({32}\right) - 4$$</span>? Can I use the Alternating Series test and how? </p>
| user | 505,767 | <p><strong>HINT</strong></p>
<p>We have that</p>
<p><span class="math-container">$$(-1)^{n}\dfrac{3n-1}{n^2 + n} =(-1)^{n}\frac1n\dfrac{3n+3-4}{n + 1} =3\frac{(-1)^{n}}n-4(-1)^{n}\dfrac{1}{n(n + 1)}=\ldots$$</span></p>
<p>and since by telescoping</p>
<p><span class="math-container">$$\dfrac{1}{n(n + 1)}=\frac1{n}-\... |
2,966,010 | <p>how to show that <span class="math-container">$$\sum_{n=1}^{\infty}(-1)^{n}\dfrac{3n-1}{n^2 + n} = \log\left({32}\right) - 4$$</span>? Can I use the Alternating Series test and how? </p>
| lab bhattacharjee | 33,337 | <p><span class="math-container">$$\dfrac{3n-1}{n(n+1)}=\dfrac{3(n+1)-4}{n(n+1)}=\dfrac3n-4\left(\dfrac1n-\dfrac1{n+1}\right)=\dfrac4{n+1}-\dfrac1n$$</span></p>
<p>This can also be achieved by <a href="http://mathworld.wolfram.com/PartialFractionDecomposition.html" rel="noreferrer">Partial Fraction Decomposition</a> <... |
2,745,884 | <p>If random variables $X$ and $Y$ are independent and $X$ and $Z$ are independent, are $X$ and $Y \cup Z$ independent?</p>
| lulu | 252,071 | <p>No. </p>
<p>suppose you are tossing two coins, a penny and a dime. $X$ is the event that the penny comes up $H$, $Y$ is the event that the dime comes up $H$, and $Z$ is the event that the coins match. Then the events that comprise $Y\cup Z$ are $$HH,TH,TT\implies P(Y\cup Z)=\frac 34$$
Where $TH$, for example, d... |
919,699 | <p>I am very bad at problems involving expected return and was hoping some one could help me out.</p>
<p>You are offered a chance to play a game for $48 against 99 other players(100 including you) the game consists of 16 rounds and in each round you have 4 chances to win. The first winner picked in each round gets 30 ... | Durin | 144,406 | <p>Lets consider $X_i$ to be the random variable which represents our earning in a given round.
We play 16 rounds so our total earning will be be represented by a random variable $Y=\sum_{n=1}^{16} X_i$.
The expected value of total earning is $E(Y) = E(X_1) + E(X_2) + ...E(X_{16})$
Since all random variables represent ... |
34,049 | <p>A person has a sheets of metal of a fixed size.</p>
<p>They are required to cut parts from the sheets of metal. </p>
<p>It's desireable to waste as little metal as possible. </p>
<p>Assume they have sufficient requirements before making the first cut to more than use one sheet of metal</p>
<p>What is the name of... | phv3773 | 1,312 | <p>Depending on context and specifics, it could be considered an problem in analysis, optimization, combinatorics, or simply applied math. If I wanted a text to help me out, I'd look for terms like Operations Research or Management Science in the title. </p>
|
397,040 | <p>What is the domain for $$\dfrac{1}{x}\leq\dfrac{1}{2}$$</p>
<p>according to the rules of taking the reciprocals, $A\leq B \Leftrightarrow \dfrac{1}{A}\geq \dfrac{1}{B}$, then the domain should be simply $$x\geq2$$</p>
<p>however negative numbers less than $-2$ also satisfy the original inequality. When am I missin... | Mark Bennet | 2,906 | <p>When you convert $A \le B$ to $\cfrac 1 A \ge \cfrac 1 B$, you are in fact dividing by $AB$. This works if $AB$ is positive, but if $AB \lt 0$ you have to reverse the inequality. </p>
<p>The inequality in the question is true whenever $x \lt 0$ because the left hand side is negative and the right-hand side is posit... |
1,097,658 | <p>I read in a notes: A semi-function is a relation (not a function) with of the form $y^2=f(x)$. </p>
<p>It seems that we can get more that one values for $f(x)$ for a single value of $x$. </p>
<p>Could any-one please help me to understand this notion.</p>
<p>The link of the note is <a href="http://www.google.co.in... | Fizz | 173,347 | <p>"Semi-function" is a rather seldom-encountered term, I think. <em><a href="https://en.wikipedia.org/wiki/Multivalued_function" rel="nofollow">Multivalued function</a></em> or <em>set-valued function</em> are the more common ones. If there are some specific issues you don't understand from those presentations, you sh... |
160,818 | <p>Could someone help me with an simple example of a profinite group that is not the p-adics integers or a finite group? It's my first course on groups and the examples that I've found of profinite groups are very complex and to understand them requires advanced theory on groups, rings, field and Galois Theory. Know a ... | Akhil Mathew | 536 | <p>One way to get a profinite group is to start with any torsion abelian group $A$, and take $\hom(A, \mathbb{Q}/\mathbb{Z})$. This acquires a topology as the inverse limit topology: it is the inverse limit of $\hom(A_0, \mathbb{Q}/\mathbb{Z})$ for $A_0 \subset A$ a finitely generated (and necessarily torsion) subgroup... |
48,629 | <p>Recently I began to consider algebraic surfaces, that is, the zero set of a polynomial in 3 (or more variables). My algebraic geometry background is poor, and I'm more used to differential and Riemannian geometry. Therefore, I'm looking for the relations between the two areas. I should also mention, that I'm interes... | Ariyan Javanpeykar | 4,333 | <p>Let $X$ be a connected normal projective C-scheme of dimension 2, i.e., an algebraic surface. The topology on an algebraic surface is the Zariski topology. But you can associate to $X$ its analytification. (See Hartshorne's appendix B or the wonderful SGA1 Exposé XII available on Arxiv.) Let $X^a$ be the analytifica... |
536,187 | <p>Let $\mathbb{R}^{\omega}$ be the countable product of $\mathbb{R}$. Make t a topological space using the box topology. Let $\pi_{n}$ denote the usual projection maps. </p>
<p>Fix $N \in \mathbb{Z}_+$ and define $A_N = \{x \in \mathbb{R}^\omega$ $|$ $\pi_{k}(x) = 0$ $\forall k>N\}$. Show that $A_N$ is closed in t... | Brian M. Scott | 12,042 | <p>HINT: Just show that each $x\in\Bbb R^\omega\setminus A_N$ has an open nbhd that is disjoint from $A_N$. This is very straightforward: if $x\notin A_N$, there is a $k>N$ such that ... what?</p>
|
516,544 | <p>The following is an <a href="http://placement.freshersworld.com/placement-papers/IBM/Placement-Paper-Whole-Testpaper-37851" rel="nofollow">aptitude problem (question no: 29-32)</a>, I am trying to solve:- </p>
<blockquote>
<p>Questions 29 - 32:</p>
<p>A, B, C, D, E and F are six positive integers such that</... | Ramchandra Apte | 38,626 | <p>Keep substituting and work with the equations to end up having equations for the other variables in terms of <code>A</code>, i.e. something like <code>B = 10A+3</code>. There are only certain values of <code>A</code> that are a prime number between 12 and 20. For those values of <code>A</code>, using your equation ... |
2,540,992 | <blockquote>
<p>An infinite sequence of increasing positive integers is given with bounded first differences.</p>
<p>Prove that there are elements <span class="math-container">$a$</span> and <span class="math-container">$b$</span> in the sequence such that <span class="math-container">$\dfrac{a}{b}$</span> is a positiv... | Abr001am | 223,829 | <p>I have doubts about the effeciency of this proof attempt, but it is just an extension of @MooS 's answer that is based upon the fact that <a href="https://oeis.org/A005250" rel="nofollow noreferrer">increasing prime gaps</a> are diverging to infinity, and <a href="https://primes.utm.edu/notes/gaps.html" rel="nofollo... |
19,585 | <p>Since I'm fairly new to Mathematica, I'm trying to learn better ways to improve my coding skills so I've turned to Project Euler and this site to speed up my learning pace. Anyways, I was trying to solve problem 32 on the project Euler forum and came up with the following code</p>
<pre><code>PanDigital[n_, m_] := S... | Mr.Wizard | 121 | <p>I'm really uncomfortable with hosting solutions to Project Euler problems here, but <a href="https://mathematica.meta.stackexchange.com/questions/634/should-we-allow-project-euler-questions">apparently the community feels otherwise</a>.</p>
<p>I'll remark that it is often best to find the smallest set that encompas... |
19,585 | <p>Since I'm fairly new to Mathematica, I'm trying to learn better ways to improve my coding skills so I've turned to Project Euler and this site to speed up my learning pace. Anyways, I was trying to solve problem 32 on the project Euler forum and came up with the following code</p>
<pre><code>PanDigital[n_, m_] := S... | chyanog | 2,090 | <pre><code>Union @@ Table[
If[a*b <= 9876 &&
Union[IntegerDigits[a], IntegerDigits[b], IntegerDigits[a*b]] ==
Range[9], a*b, 0], {a, 123, 1987}, {b, 2, 98}] // Tr // Timing
(*v7*)(*{0.686, 45228}*)
(*v8*)(*{0.078, 45228}*)
</code></pre>
|
631,163 | <p>As a student in high school, I never bothered to memorize equations or methods of solving, rather I would try to identify the logic behind the operations and apply them. However, now that I've begun to teach Algebra in high school, I find it rather frustrating when students either a) memorize methods of solving the ... | David | 651,991 | <p>I think that "lazy memorization" will always be the default strategy taken by your students, since that is what works in most of the subjects they are tought. Their minds are wired that way and that is hard to change.</p>
<p>If you really want students to think by themselves, you will have no choice but to force th... |
1,386,682 | <p>How do you calculate $\lim_{z\to0} \frac{\bar{z}^2}{z}$?</p>
<p>I tried $$\lim_{z\to0} \frac{\bar{z}^2}{z}=\lim_{\overset{x\to0}{y\to0}}\frac{(x-iy)^2}{x+iy}=\lim_{\overset{x\to0}{y\to0}}\frac{x^2-2xyi-y^2}{x+iy}=\lim_{\overset{x\to0}{y\to0}}\frac{x^2-2xyi-y^2}{x+iy}\cdot\frac{x-iy}{x-iy} \\ \\ =\lim_{\overset{x\to... | mvw | 86,776 | <p>How about
$$
\lim_{z\to 0} \frac{\bar{z}^2}{z} =
\lim_{z\to 0} \frac{\bar{z}^2z^2}{z^3} =
\lim_{z\to 0} \frac{\lvert z\rvert^4}{z^3} =
\lim_{z\to 0} \frac{\lvert z\rvert^4}{\lvert z \rvert^3 e^{3i\phi(z)}}
= 0
$$</p>
|
2,317,625 | <p>How do you compare $6-2\sqrt{3}$ and $3\sqrt{2}-2$? (no calculator)</p>
<p>Look simple but I have tried many ways and fail miserably.
Both are positive, so we cannot find which one is bigger than $0$ and the other smaller than $0$.
Taking the first minus the second in order to see the result positive or negative ge... | TomGrubb | 223,701 | <p>We have $\sqrt{3}\leq 1.8$ so $6-2\sqrt{3}\geq 2.4$, whereas $\sqrt{2}\leq 1.42$ so $3\sqrt{2}-2\leq 2.26$.</p>
|
192,125 | <p>Solve: $$\sqrt{x-4} + 10 = \sqrt{x+4}$$
Little help here? >.<</p>
| Zarrax | 3,035 | <p>Square both sides, and you get
$$x - 4 + 20\sqrt{x - 4} + 100 = x + 4$$
This simplifies to
$$20\sqrt{x - 4} = -92$$
or just
$$\sqrt{x - 4} = -\frac{92}{20}$$
Since square roots of numbers are always nonnegative, this cannot have a solution.</p>
|
192,125 | <p>Solve: $$\sqrt{x-4} + 10 = \sqrt{x+4}$$
Little help here? >.<</p>
| Ross Millikan | 1,827 | <p>As others have said, there are no solutions within the usual rules. However, once we get to $\sqrt {x-4}=-4.6$ we can remember that square roots can be negative (despite the convention that $\sqrt x \ge 0$). Then we can square and add $4$to find $x=25.16$. Checking, we find $\sqrt {x+4}=5.4, \sqrt{x-4}=-4.6$ and ... |
111,183 | <p><img src="https://i.stack.imgur.com/1MOuo.jpg" alt="Problem">
<img src="https://i.stack.imgur.com/bdRXi.png" alt="New Solution"></p>
<p>I believe I have gotten all of the ways now - thanks for the hints below Yun, Andre Nicolas, and Gerry Myerson. If anyone could confirm my answer (I feel there should be more poss... | davidlowryduda | 9,754 | <p>I can verify your solution. I have one more than you. I've also put up a list (everything is divided by 2). I also put it up in the order that mirrors the smaller example's list (at least, as I interpreted it - perhaps, putting the list in strictly number-increasing order would have been a good idea too). </p>
<p>1... |
351,846 | <p>The following problem was on a math competition that I participated in at my school about a month ago: </p>
<blockquote>
<p>Prove that the equation $\cos(\sin x)=\sin(\cos x)$ has no real solutions.</p>
</blockquote>
<p>I will outline my proof below. I think it has some holes. My approach to the problem was to... | Aryabhata | 1,102 | <p>Let $a = \cos x$ and $b = \sin x$, and so $a,b \in [-1,1]$.</p>
<p>We have to solve $\sin a = \cos b$. </p>
<p>We can assume that $0 \le b \le 1$, because, if $x$ is a root, so is $-x$.</p>
<p>Since $\sin a = \cos b$ and $b \ge 0$, we must have that $a \ge 0$ (remember, $a,b \in [-1,1]$)</p>
<p>Thus if the equat... |
3,077,629 | <p>Assume <span class="math-container">$(f_i)_{i\in I}$</span> is an orthonormal/orthogonal system in an (complex) inner product space. Does <span class="math-container">$$\sum_{i\in I}\langle f_i,f\rangle f_i$$</span> always converges for any <span class="math-container">$f$</span> (may not to <span class="math-contai... | lonza leggiera | 632,373 | <p>The constraint <span class="math-container">$\ x^2 - y^2 - z^2 + 16 \le 0\ $</span> is not satisfied by <span class="math-container">$\ \left(\pm 4, 0, 0\right)\ $</span>, so it's not even a feasible point, let alone an extremum.</p>
<p>All points of the form <span class="math-container">$\ \left(\,0, y, z\,\right)... |
210,658 | <p>$$\sum_{i=1}^n t^{i-1}$$
I am stuck with the proof of this equality. </p>
| Alex | 38,873 | <p>Use perturbation method from Concrete Mathematics:
$$
S_n=\sum_{k=1}^{n}t^{i-1}\\
S_{n+1}=S_n + t^{n}=\sum_{k=1}^{n}t^{k} + 1=t\sum_{k=1}^{n}t^{k-1}+1=t S_n + 1
$$
After the algebra you get
$$
S_n=\frac{1-t^{n}}{1-t}
$$</p>
|
2,336,535 | <p>I have a limit:</p>
<p>$$\lim_{(x,y)\rightarrow(0,0)} \frac{x^3+y^3}{x^4+y^2}$$</p>
<p>I need to show that it doesn't equal 0.</p>
<p>Since the power of $x$ is 3 and 4 down it seems like that part could go to $0$ but the power of $y$ is 3 and 2 down so that seems like it's going to $\infty$.</p>
<p>I wonder if t... | Glorfindel | 228,959 | <p>The limit just <strong>doesn't exist</strong> ('is undefined'), for exactly the reasons you describe. Therefore, it is not equal to zero.</p>
|
279,808 | <p>I was working on a way of calculating the square root of a number by the method of x/y → (x+4y)/(x+y) as shown by bobbym at <a href="https://math.stackexchange.com/questions/861509/">https://math.stackexchange.com/questions/861509/</a></p>
<p>I tried to do it via functions on mathematica, everything seems correct. W... | Nasser | 70 | <p>Another option is is to use basic Do loop or a Table.</p>
<pre><code>update[a_Integer, b_Integer] := (a + 2*b)/(a + b)
a = 1;
b = 2;
lis = Last@
Reap@Do[ t = update[a, b]; a = Numerator[t]; b = Denominator[t];
Sow[a/b],
{n, 1, 10}
]
</code></pre>
<p><img src="https://i.stack.imgur.com/TL2KI.png" alt=... |
279,808 | <p>I was working on a way of calculating the square root of a number by the method of x/y → (x+4y)/(x+y) as shown by bobbym at <a href="https://math.stackexchange.com/questions/861509/">https://math.stackexchange.com/questions/861509/</a></p>
<p>I tried to do it via functions on mathematica, everything seems correct. W... | E. Chan-López | 53,427 | <p>Following Syed's idea, but using <code>FoldList</code>:</p>
<pre><code>FoldList[(Numerator@# + 2 Denominator@#)/(Numerator@# + Denominator@#) &, 1/2, Range[5]]
(*{1/2, 5/3, 11/8, 27/19, 65/46, 157/111}*)
</code></pre>
|
2,558,267 | <p>Let $M$ be a finite dimensional von-Neumann algebra. We know this algebra is generated by its projections. My question maybe simple. Can one computing these projections? What about its minimal or central projections? </p>
<p>If possible please give me a reference for this. </p>
<p>Thanks </p>
| Martin Argerami | 22,857 | <p>You would have to define "compute". To "compute" an element of an algebra (or any other mathematical object, for that matter) you need to have some kind of presentation of the object. </p>
<p>In the case of a finite-dimensional von Neumann/C$^*$-algebra, one can prove that they are always (isomorphic to) a direct s... |
3,274,172 | <p>Let <span class="math-container">$X$</span> a compact set. Prove that if every connected component is open then the number of components is finite.</p>
<p>Ok, <span class="math-container">$X = \bigcup C(x)$</span> where <span class="math-container">$C(x)$</span> is the connected component of <span class="math-conta... | Henno Brandsma | 4,280 | <p>Under the assumption that components are open, they form an <em>open</em> cover of the compact space <span class="math-container">$X$</span>. They form a partition of non-empty sets (always), so we cannot omit any one of them and still have a cover. </p>
<p>So <span class="math-container">$X$</span> has a finite su... |
2,601,088 | <p>I'm new to the group theory and want to get familar with the theorems in it, so I choose a number $52$
to try making some obseveration on all group that has this rank. Below are my thoughts. I don't know if there is any better way to think of these (i.e., an experienced group theorist would think), and I still have... | ajotatxe | 132,456 | <p>Consider the group $D_{26}$. That is, take a regular polygon with $26$ sides. Let $V_1,\ldots,V_{26}$ be its vertices (in order). Define these elements:</p>
<ul>
<li><p>$\tau$ is the rotation that maps each vertex to the next one. That is, $\tau(V_n)=V_{n+1}$ if $1\le n\le 25$ and $\tau(V_{26})=V_1$. The order of t... |
2,393,872 | <p>I was doing the following problem</p>
<blockquote>
<p>An isoceles triangle is a triangle in which two sides are equal. Prove that the angles opposite to the equal sides are equal.</p>
</blockquote>
<p>I drew this diagram (sorry for the large picture):</p>
<p><a href="https://i.stack.imgur.com/uxLlm.jpg" rel="no... | Jay Zha | 379,853 | <p>Yes, you are right that "there exists a way to place triangle $ABD$ onto triangle $BDC$ so that they overlap perfectly", and you might argue: "well there are more than one way to place, how do I know which way they overlap perfectly" - and that's exactly your question. </p>
<p>However, do not forget that you get th... |
2,393,872 | <p>I was doing the following problem</p>
<blockquote>
<p>An isoceles triangle is a triangle in which two sides are equal. Prove that the angles opposite to the equal sides are equal.</p>
</blockquote>
<p>I drew this diagram (sorry for the large picture):</p>
<p><a href="https://i.stack.imgur.com/uxLlm.jpg" rel="no... | Intelligenti pauca | 255,730 | <p>Even if your proof is the preferred one in high-school textbooks, I think a simpler proof is worth mentioning.</p>
<p>Compare triangles $ABC$ and $CBA$: they are congruent by $SAS$ and thus $\angle A\cong\angle C$.</p>
|
2,736,426 | <p>Let's imagine a point in 3D coordinate such that its distance to the origin is <span class="math-container">$1 \text{ unit}$</span>.</p>
<p>The coordinates of that point have been given as <span class="math-container">$x = a$</span>, <span class="math-container">$y = b$</span>, and <span class="math-container">$z = ... | The Integrator | 538,397 | <p>Suppose you have a vector $\vec v = xi+yj+zk$ where $i,j,k $ are the basis unit vectors then the angles $\alpha,\beta, \gamma$ of the vector to the $x,y,z $ axes respectively is given by ;</p>
<p>$\alpha = \frac{x}{\sqrt{x^2+y^2+z^2}} = \cos(a)\\\beta = \frac{y}{\sqrt{x^2+y^2+z^2}}=\cos(b)\\\gamma = \frac{z}{\sqr... |
3,757,213 | <blockquote>
<p>Prove that the maximum area of a rectangle inscribed in an ellipse <span class="math-container">$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$</span> is <span class="math-container">$2ab$</span>.</p>
</blockquote>
<p><strong>My attempt:</strong></p>
<p>Equation of ellipse: <span class="math-container">$\dfrac{x^... | Harish Chandra Rajpoot | 210,295 | <p>Here is an easier method to prove (without using Calculus).</p>
<p>The sides (i.e. length and width) of inscribed rectangle have to be parallel to the axes of ellipse to make all four vertices of inscribed rectangle lie on the ellipse.</p>
<p>Consider <span class="math-container">$(\pm a\cos\theta, \pm b\sin\theta)$... |
1,040,932 | <p>I have a system of congruence equations</p>
<p>$$
\begin{cases}
x \equiv 17 \pmod{15} \\
x \equiv 14 \pmod{33}
\end{cases}
$$</p>
<p>I need to investigate the system and see if they've got any solutions.</p>
<p>I know that I should use the Chinese remainder theorem "in a reverse order" so I think I should split e... | lab bhattacharjee | 33,337 | <p>$$x\equiv17\pmod{15}\equiv2$$</p>
<p>$$\implies x\equiv2\pmod3\ \ \ \ (1),$$</p>
<p>$$x\equiv2\pmod5\ \ \ \ (2)$$</p>
<p>$$x\equiv14\pmod{33}\implies x\equiv14\pmod3\equiv2,$$</p>
<p>$$x\equiv14\pmod{11}\equiv3\ \ \ \ (3)$$</p>
<p>Now apply <a href="http://mathworld.wolfram.com/ChineseRemainderTheorem.html" rel... |
1,146,824 | <p>The Russel's Paradox, showing $X=\{x|x\notin x\}$ can't exist is not very hard.
If $X \in X$, then $X \notin X$ by definiition, in the other case, $X \notin X$, then $X \in X$ by definition. Both cases are impossible.</p>
<p>But how about whole things $X=\{x|x=x\}$? $X \in X$ probably cause the problem, but I don't... | Mike Earnest | 177,399 | <p>Cantor's Theorem says that for any set $Y$, the power set of $Y$ is strictly bigger than $Y$. But $X$ contains all the elements of any set, and is therefore at least as big as any set, including its own power set, giving the contradiction
$$
\mathcal P(X)\le X<\mathcal P(X)
$$</p>
|
3,013,355 | <p>I have been asked to prove that </p>
<p>(<span class="math-container">$a \to $</span>b) <span class="math-container">$\vee$</span> (<span class="math-container">$a \to $</span>c) = <span class="math-container">$a \to ($</span>b <span class="math-container">$\vee$</span> c).</p>
<p>I believe it is just the simple c... | Trevor Gunn | 437,127 | <p>I'm not sure if this is exactly what you're looking for, but the equivalence of <span class="math-container">$y^2$</span> and <span class="math-container">$(1 - x)(1 + x)$</span> for <span class="math-container">$\mathbf{S}^1$</span> says that you can look at the divisor <span class="math-container">$D = 2((1,0) + (... |
3,382,464 | <p>Let <span class="math-container">$g$</span> be a <strong>smooth</strong> Riemannian metric on the closed <span class="math-container">$n$</span>-dimensional unit disk <span class="math-container">$\mathbb{D}^n$</span>. Let <span class="math-container">$f$</span> be a harmonic function w.r.t <span class="math-contain... | Allawonder | 145,126 | <p><em>Hint.</em> No, multiply through by <span class="math-container">$a$</span> instead, and with a little rearrangement, you now have <span class="math-container">$$a^6=a^4-a^2+2a,$$</span> whose right hand side is a biquadratic in <span class="math-container">$a.$</span> You may still attempt to complete squares on... |
1,939,382 | <p>I've read about integration, and i believe i understood concept correctly. But, unfortunately, the simplest exercise already got my stumbled. I need to find an integral of $x{\sqrt {x+x^2}}$. So i proceed as follows,</p>
<p>By the fundamental theorem of calculus:</p>
<p>$f(x)=\int[f'(x)]=\int[x\sqrt{x+x^2}]$,</p>
... | Jack D'Aurizio | 44,121 | <p>I hope you do not mind if I prefer to start from scratch. We have
$$ \int (2x+1)\sqrt{x^2+x}\,dx = C+\frac{2}{3}(x^2+x)^{3/2} \tag{1}$$
and the problem boils down to computing $\int\sqrt{x^2+x}\,dx$. Integration by parts gives
$$ \int \sqrt{x^2+x}\,dx = x\sqrt{x^2+x}-\int\frac{x+2x^2}{2\sqrt{x+x^2}}\,dx \tag{2}$$
he... |
256,666 | <p>Let $X$ be a set. Suppose $\beta$ is a basis for the topology $\tau_\beta$ of $X$. Since each base element is open (with respect to $\tau_\beta$) we have that $$B\in \beta\Rightarrow B\in \tau_\beta.$$ Thus, $\beta\subset \tau_\beta$. </p>
<p>However, since $\beta$ is a union of base elements (I assume a set can al... | Brian M. Scott | 12,042 | <p>Let $S$ be any set. Then $$S=\bigcup_{x\in S}\{x\}\;,$$ but in general $$S\ne\bigcup_{x\in S}x\;.$$ That is, $S$ is the union of the <em>singletons</em> of its elements, but it is not in general the union of its elements. It can only be the union of subsets, and in general $x\in S$ does not imply that $x\subseteq S$... |
355,740 | <p>Today in class we learned that for exponential functions $f(x) = b^x$ and their derivatives $f'(x)$, the ratio is always constant for any $x$. For example for $f(x) = 2^x$ and its derivative $f'(x) = 2^x \cdot \ln 2$</p>
<p>$$\begin{array}{c | c | c | c}
x & f(x) & f'(x) & \frac{f'(x)}{f(x)}\\ \hline
-1... | spin | 12,623 | <p>In the ring $\mathbb{Z}_4 = \mathbb{Z}/4\mathbb{Z}$ the characteristic is $4$, but $2 + 2 = 0$. You could call this the order of an element in the additive group, perhaps "additive order" would be a good term.</p>
|
1,431,464 | <p>Does anyone know a good reference where it is shown that the Schwartz class $\mathcal{S}(\mathbb R)$ is a dense subset of $L^2(\mathbb R)$?</p>
<p>Many thanks</p>
| Luigi Nocera | 687,083 | <p>A proof that <span class="math-container">$C_{0}$</span> is dense in <span class="math-container">$L^{p}$</span> can be found in Naylor and Sell's "Linear Operator Theory in Engineering and Science", Appendix D, paragraph 12 "Dense Subspaces in <span class="math-container">$L^{p}$</span>, <span class="math-container... |
183,077 | <p>A complex Lie group may have several real forms.
Are there any duality/trinity... between them?
Maybe a trivial question to ask, is $SL(3,\mathbb{C})$ a real form of $SL(3,\mathbb{C})\times SL(3,\mathbb{C})$ ?</p>
| Geoff Robinson | 14,450 | <p>As Amritanshu Prasad points out, the $6$-dimensional irreducible complex representation of $S_{5}$ is indeed monomial (with respect to a suitable basis), and thinking about how to prove this directly led me to a general observation: let $G$ be a finite group, and $\chi$ be a non-linear complex irreducible character ... |
20,314 | <p>Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.</p>
| Harun Šiljak | 4,925 | <p>Just my 2c: Being a student with a limited mathematical education, I used V.S. Vladimirov's <em>Generalized Functions in Mathematical Physics</em> (Mir Moscow 1979) and it was not as hard as I expected it to be - Vladimirov was rigorous and pedantic, as a book in mathematics should be, but not too complicated in exp... |
4,291,864 | <p>I have the following equation:</p>
<p><span class="math-container">$y=\frac{3x}{x^{2}+1}$</span></p>
<p>and I want to obtain x in terms of y, so far what I have done is the following:</p>
<p><span class="math-container">$3x=y(x^{2}+1)$</span></p>
<p><span class="math-container">$3x=x^{2}y+y$</span></p>
<p><span clas... | José Carlos Santos | 446,262 | <p>Yes, <span class="math-container">$\Bbb R$</span> is an open set. Nonetheless, it is also a closed set (the whole space is <em>always</em> a closed set for <em>any</em> topology) and, with respect to the co-finite topology, it is the smallest closed set that contains <span class="math-container">$(a,b)$</span>. Ther... |
4,411,096 | <p>I know closure of connected set in a topological space must be connected as well. However, I can't understand why this counterexample fails.
Take <span class="math-container">$X=[0,2)\cup\{3\}, B_2(1)=(0,2)$</span> which is connected. Now take the closed ball <span class="math-container">$C_2(1)=[0,2)\cup \{3\}$</sp... | Ken Hung | 626,360 | <p>In this case, the closure of the open ball <span class="math-container">$ B_2(1) $</span> is not the closed ball <span class="math-container">$ C_2(1)$</span>. This is because we have <span class="math-container">$ (B_{1/2}(3) \cap X) \cap B_2(1) = \varnothing $</span> and this shows that <span class="math-containe... |
3,369,069 | <p>Let <span class="math-container">$l_1$</span> and <span class="math-container">$l_2$</span> be two distributions in disjoint variables <span class="math-container">$x_1, ..., x_n$</span> and <span class="math-container">$y_1, ..., y_m$</span>. Then it is said to be possible to define a product distribution.</p>
<p>... | quarague | 169,704 | <p>If you want to multiply <span class="math-container">$\delta(x_1)$</span> and <span class="math-container">$\delta(x_2)$</span> you first need to make them into functions acting on the space space, so you multiply <span class="math-container">$\delta(x_1)Id(x_2)$</span> and <span class="math-container">$Id(x_1)\delt... |
1,125,842 | <p>In $\sf ZFC$ we have the axiom of infinity and thus can define the natural numbers $$\mathbb N \equiv \bigcap\{X:\emptyset\in X\land \forall n(n\in X\implies n\cup\{n\}\in X)\}.$$ From this it's not particularly hard (exercises 1.6 and 1.7 in Jech - <em>Set Theory</em>) to prove that, firstly, every $n\in\mathbb N$... | Rene Schipperus | 149,912 | <p>The von Neumann herirarchy is defined as
$$V_0=\emptyset$$
$$V_{\alpha+1}=P(V_{\alpha})$$
$$V_{\lambda}=\cup_{\alpha < \lambda} V_{\alpha}$$</p>
<p>Now all elements of $V$ satisfy foundation. What foundation really means is that all sets belong to $V$. Since in practice all the sets we naturally deal with are al... |
3,752,455 | <blockquote>
<p><strong>Problem.</strong> Show that for <span class="math-container">$n\ge 2$</span> there are no solution <span class="math-container">$$x^n+y^n=z^n$$</span> such that <span class="math-container">$x$</span>, <span class="math-container">$y$</span>, <span class="math-container">$z$</span> are prime num... | Devansh Kamra | 625,028 | <p>It can be observed that if <span class="math-container">$k$</span> is prime, then either <span class="math-container">$k\equiv 1 \space(\text {mod 6})$</span> or <span class="math-container">$k\equiv 5 \space(\text {mod 6})$</span>.</p>
<p>It is also observable that if <span class="math-container">$k\equiv 1 \space(... |
2,647,000 | <p>Consider a function $ϕ$ such that $$\lim_{h→0} ϕ(h) = L$$ and $$L − ϕ(h) ≈ ce^{−1/h}$$ for some constant $c$. By combining $ϕ(h)$, $ϕ(h/2)$, and $ϕ(h/3)$, find an accurate estimate of $L$.</p>
<p>Isn't $ϕ(h)=-ce^{−1/h}+L$? I think I am over-simplfying this...</p>
| videlity | 70,729 | <p>It would probably be best to talk to someone at a university who researches in number theory. There's many different area and aspects of research which would largely depend on a possible supervisor that you would have. They would also have possible projects that you could look into.</p>
|
753,881 | <p>I want to know some typical forms of system of equations generating from practical problems in engineering/economics/physics,etc.</p>
<p>Some examples or research articles would be good.</p>
<p>Specifically, I am looking for some examples of nonlinear system of equations generated from practical problems.</p>
<p>... | celtschk | 34,930 | <p>If no odd digits can be placed on odd places, you must fill all odd places with even digits. There are four odd places, and you've got four even digits ($2$, $2$, $4$, $4$), so they are just enough to fill the four odd slots. The number of ways to do so is
$$n_{\text{odd places}} = \frac{4!}{2!\cdot 2!} = 6.$$
Now y... |
3,837,548 | <p>The triple integral
<span class="math-container">$$\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{dx dy dz}{1-xyz}=\zeta(3) \dots (1)$$</span>
is not separable in <span class="math-container">$x,y,z$</span> and the integral representation of reciprocal: <span class="math-container">$\frac{1}{1-xyz}=\int_{0}^{1} t^{-xy... | FDP | 186,817 | <p><span class="math-container">\begin{align}\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{dx dy dz}{1-xyz}&=-\int_0^1\int_0^1 \frac{\ln u}{1-uz}dudz\\
&=-\frac{1}{2}\int_0^1\int_0^1 \frac{\ln (uz)}{1-uz}dudz\\
&=\frac{1}{2}\int_0^1 \frac{\ln^2 v}{1-v}dv\\
&=\frac{1}{2}\times 2\zeta(3)\\
&=\boxed{\ze... |
1,762,268 | <p>Let $X$ be a Hausdorff space and let $f:X\to \mathbb{R}$. If grapph of $f$ is compact we have to show that $f$ is continuous. </p>
<p>Since every closed subset of a Hausdorff space is closed, therefore grapph of $f$ is closed. WE know that if $f:X\to Y$ and $Y$ is compact, then graph of $f$ is clsed implies $f$ is... | Rick Sanchez | 332,412 | <p>Fix $x_0 \in X$ and let $x_a$ be a net converging to $x_0$ in $X$. You need to show $f(x_a)\rightarrow f(x_0)$ in $\mathbb{R}$. Let $G_f$ denote the graph of $f$, which is a subset of $X \times \mathbb{R}$. Then $(x_a,f(x_a))$ is a net in $G_f$, which is compact, so we can extract a convergent subnet $(x_b,f(x_b)) \... |
1,724,554 | <p>Say, $A$ is an $ n\times n $ matrix over $\Bbb R$, with</p>
<p>$$ A_{ij} = \begin{cases} a \qquad \text{if } i=j\\
b \qquad \text{otherwise.}
\end{cases} $$</p>
<p>How do we compute the determinant of this symmetrix matrix $A$?</p>
| mathreadler | 213,607 | <p>If the first row is the vector $\bf v$ and it's DFT (discrete fourier-transform) is $\bf w$, then the determinant is $\prod_i {\bf w}_i$. This is true for any <a href="https://en.wikipedia.org/wiki/Circulant_matrix" rel="nofollow">circulant</a> matrix as it's eigenvalues coincide with those fourier coefficients $$\... |
3,077,084 | <blockquote>
<p>If <span class="math-container">$a,b,c>0.$</span> Then minimum value of</p>
<p><span class="math-container">$(8a^2+b^2+c^2)\cdot (a^{-1}+b^{-1}+c^{-1})^2$</span></p>
</blockquote>
<p>Try: Arithmetic geometric inequality</p>
<p><span class="math-container">$8a^2+b^2+c^2\geq 3\cdot 2\sqrt{2}(abc)^{1/3}... | Martin R | 42,969 | <p><em>Hint:</em> Apply <span class="math-container">$AM \ge GM$</span> not to <span class="math-container">$8a^2 + b^2 + c^2$</span>, but to
<span class="math-container">$$
(2a)^2 + (2a)^2 + b^2 + c^2
$$</span>
and <span class="math-container">$HM \le GM$</span> not to <span class="math-container">$a^{-1}+b^{-1}+c^{-... |
44,562 | <p>The question is motivated from the definition of $C^r(\Omega)$ I learned from S.S.Chern's <em>Lectures on Differential Geometry</em>:</p>
<p>Suppose $f$ is a real-valued function defined on an open set $\Omega\subset{\bf R}^m$. If all the $k$-th order partial derivatives of $f$ exist and are continuous for $k\leq r... | Condor | 11,915 | <p>A sketch proof...</p>
<p>Consider $\mathbb{R}$ as the set of equivalence classes of Cauchy sequences of rationals. We define "the normal" partial order on $\mathbb{R}$ by $x \leq y$ iff $(x = y)$ OR $(\forall \langle x_{i} \rangle)(\forall \langle y_{i} \rangle)(\langle x_{i} \rangle \in x$ AND $\langle y_{i} \rang... |
3,491,816 | <p>Find the min and max values of the function <span class="math-container">$$f(x,y)=10y^2-4x^2$$</span> with the constraint <span class="math-container">$$g(x,y)=x^4+y^4=1$$</span>
I have done the following working;
<span class="math-container">$$\frac{\partial f}{\partial x} = \lambda \frac{\partial g}{\partial x... | José Carlos Santos | 446,262 | <p>Your system of equations is<span class="math-container">$$\left\{\begin{array}{l}-8x=4\lambda x^3\\20y=4\lambda y^3\\x^4+y^4=1.\end{array}\right.$$</span>If <span class="math-container">$x=0$</span>, then you can obviously take <span class="math-container">$y=\pm1$</span> and if <span class="math-container">$y=0$</s... |
3,491,816 | <p>Find the min and max values of the function <span class="math-container">$$f(x,y)=10y^2-4x^2$$</span> with the constraint <span class="math-container">$$g(x,y)=x^4+y^4=1$$</span>
I have done the following working;
<span class="math-container">$$\frac{\partial f}{\partial x} = \lambda \frac{\partial g}{\partial x... | Michael Rozenberg | 190,319 | <p><span class="math-container">$$-4=-4\sqrt{x^4+y^4}\leq10y^2-4x^2\leq10\sqrt{x^4+y^4}=10.$$</span>
The equality occurs for <span class="math-container">$(x,y)=(1,0)$</span> for the left inequality and for <span class="math-container">$(x,y)=(1,0)$</span> for the right inequality, which says that we got the minimal va... |
3,491,816 | <p>Find the min and max values of the function <span class="math-container">$$f(x,y)=10y^2-4x^2$$</span> with the constraint <span class="math-container">$$g(x,y)=x^4+y^4=1$$</span>
I have done the following working;
<span class="math-container">$$\frac{\partial f}{\partial x} = \lambda \frac{\partial g}{\partial x... | DeepSea | 101,504 | <p>The solutions by Lagrange Multipliers are shown above. But if you like a traditional method, here is one. One way is to lower the "power" of the constraint, namely: <span class="math-container">$x^4+y^4 =1$</span>. So let's put <span class="math-container">$a = x^2, b = y^2$</span>, thus <span class="math-container"... |
1,392,209 | <blockquote>
<p>Evaluate the limit $$\lim_{x \to 0}\left( \frac{1}{x^{2}}-\frac{1}{\tan^{2}x}\right)$$</p>
</blockquote>
<p>My attempt </p>
<p>So we have $$\frac{1}{x^{2}}-\frac{\cos^{2}x}{\sin^{2}x}$$</p>
<p>$$=\frac{\sin^2 x-x^2\cos^2 x}{x^2\sin^2 x}$$
$$=\frac{x^2}{\sin^2 x}\cdot\frac{\sin x+x\cos x}{x}\cdot\fr... | CivilSigma | 229,877 | <p>Using L'Hospital's rule (since direct evaluation gives $\bigl(\frac{0}{0}\bigr)$ ), we have the following:</p>
<p>$$\lim_{x \to 0} \frac{\cos x-\cos x +x\sin x}{3x^2}= \lim_{x \to 0} \frac{\sin x}{3x}.$$</p>
<p>We take the derivative of the numerator and denominator again:</p>
<p>$$\lim_{x \to 0} \frac{\cos x}{3}... |
1,392,209 | <blockquote>
<p>Evaluate the limit $$\lim_{x \to 0}\left( \frac{1}{x^{2}}-\frac{1}{\tan^{2}x}\right)$$</p>
</blockquote>
<p>My attempt </p>
<p>So we have $$\frac{1}{x^{2}}-\frac{\cos^{2}x}{\sin^{2}x}$$</p>
<p>$$=\frac{\sin^2 x-x^2\cos^2 x}{x^2\sin^2 x}$$
$$=\frac{x^2}{\sin^2 x}\cdot\frac{\sin x+x\cos x}{x}\cdot\fr... | Rio Alvarado | 253,991 | <p>For your final problem I'd use L'Hospital's Rule to obtain:</p>
<p>$$\lim_{x \to 0} \frac{x \sin(x)}{3x^2} \implies \lim_{x \to 0} \frac{\sin(x)}{3x} \\ \hspace{.1cm} \text{using L'Hospital's again}, \hspace{.1cm} \\ \lim_{x \to 0}\frac{\cos(x)}{3} = \frac{1}{3}.$$</p>
|
1,680,862 | <p>I am attempting to find the expected value and variance of the random variable $X$ analytically (in addition to a decimal answer). $X$ is the random variable <code>expression(100)[-1]</code> where <code>expression</code> is defined by:</p>
<pre><code>def meander(n):
x = [0]
for t in range(n):
... | Clement C. | 75,808 | <p>So, your random variable is $$
X = 3X_1+\dots+3X_{100} = \sum_{k=1}^n 3X_k
$$
with $n=100$, where $X_1,\dots, X_n$ are independent, identically distributed random variables that are uniform in $[0,1)$. <a href="https://en.wikipedia.org/wiki/Uniform_distribution_(continuous)" rel="nofollow">In particular</a>, $\mathb... |
685,681 | <p>I want to prove that $\dim V/(X \cap Y)$ in finite, if $V$ be a vector space and $X$, $Y$ two sub spaces of $V$ such that $\dim V/Y$ and $\dim V/X$ are finite.</p>
| ajd | 90,897 | <p>Let $\{a_1 + Y,\ldots,a_r +Y\}$ be a basis of $V/Y$, where the $a_i$s are elements of $V$. Since $V/X$ is finite-dimensional, we have that $Y/(Y\cap X)$ is finite-dimensional (since we have maps $Y\to V\to V/X$ and the composition has kernel $Y\cap X$, so we have an injection $Y/(Y\cap X)\to V/X$, so $\dim Y/(Y\cap ... |
2,544,261 | <p>The question:</p>
<blockquote>
<p>Find values of $x$ such that $2^x+3^x-4^x+6^x-9^x=1$, $\forall x \in \mathbb R$.</p>
</blockquote>
<p>Notice the numbers $4$, $6$ and $9$ can be expressed as powers of $2$ and/or $3$. Hence let $a = 2^x$ and $b=3^x$.</p>
<p>\begin{align}
1 & = 2^x+3^x-4^x+6^x-9^x \\
& =... | stressed out | 436,477 | <p>If the sum of a finite number of non-negative expressions is $0$, each of them has to be zero. </p>
<p>In other words, when $a,b,c\geq0$</p>
<p>$a+b+c=0 \implies a=b=c=0$</p>
<p>You have done the hard part by showing that it can be written as the sum of three squares.
This means that $a=b$, $a=1$ and $b=1$. What ... |
2,073,410 | <p>If $$ a-(a \bmod x)<b$$ how do I prove that $$c-(c\bmod x)<b \;\forall c<a?$$ </p>
| Sergei Golovan | 400,926 | <p>The expression $c - (c\mathrel{\mathrm{mod}} x)$ represents the greatest integer $kx$ which is not greater than $c$. So, the larger $c$, the larger $c - (c\mathrel{\mathrm{mod}} x)$.</p>
|
1,240,212 | <blockquote>
<p>How to find the degree of an extension field ?</p>
</blockquote>
<p>Let $f:=T^3-T^2+2T+8\in\mathbb Z[T]$ and $\alpha$ be the real root of $f$. Why is then $\mathbb Q(\alpha)$ is a number field of degree $3$ ?</p>
<p>I've seen somewhere that $[\mathbb Q(r):\mathbb Q]\le n$ if $r$ is a root of an irre... | A.P. | 65,389 | <blockquote>
<p><strong>Fact</strong>: Consider two polynomials $f$ and $p$ over $\Bbb{Q}$, with $p$ irreducible. It can be proved that if $f$ and $p$ share a root, then $p$ divides $f$.</p>
</blockquote>
<p>How does this help? Suppose that $\alpha$ is a root of an irreducible polynomial $f \in \Bbb{Q}[X]$ of degree... |
17,134 | <p>On a very regular basis we see new users that are not accustomed with the use of MathJaX on MSE. Sometimes even some users that aren't that new to the site. Most of us, when this happens, kindly bring to this users attention that there is a <a href="http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tut... | Paradox 101 | 177,844 | <p>That's a really good suggestion. I'm relatively new here and had no idea that MathJax even existed. Your suggestion made me aware of it so thank you. The link should be given with the email.</p>
|
1,178,361 | <p>The surface with equation $z = x^{3} + xy^{2} $ intersects the plane with equation $2x-2y = 1$ in a curve. What is the slope of that curve at $x=1$ and $ y = \frac{1}{2} $</p>
<p>So I put $ x^{3} + xy^{2} = 2x - 2y - 1 $</p>
<p>We have $ x^{3} + xy^{2} - 2x + 2y + 1 $</p>
<p>Do I then differentiate wrt x and y si... | John Brevik | 210,492 | <p>Sorry to add yet another answer, but for a learner I think that a straightforward approach is best. How do you prove two sets equal? Prove that each is contained in the other. So let $y\in C$. Since $f$ is onto, there exists $x\in X$ such that $f(x)=y.$ Now $x\in f^{-1}(C)$. But then $x\in f^{-1}(D)$, so $f(x)\in D$... |
4,058,884 | <p>I have an orthonormal basis <span class="math-container">${\bf{b}}_1$</span> and <span class="math-container">${\bf{b}}_2$</span> in <span class="math-container">$\mathbb{R}^2$</span>. I want to find out the angle of rotation. I added a little picture here. I essentially want to find <span class="math-container">$\t... | Glärbo | 892,839 | <p>Let <span class="math-container">$\mathbf{b}_1 = (x_1, y_1)$</span>. Then, <span class="math-container">$\theta = \operatorname{atan2}(y_1, x_1)$</span>.</p>
<p><span class="math-container">$\operatorname{atan2}(y_1, x_1)$</span> is the two-argument form of arcus tangent, equivalent to <span class="math-container">... |
4,058,884 | <p>I have an orthonormal basis <span class="math-container">${\bf{b}}_1$</span> and <span class="math-container">${\bf{b}}_2$</span> in <span class="math-container">$\mathbb{R}^2$</span>. I want to find out the angle of rotation. I added a little picture here. I essentially want to find <span class="math-container">$\t... | Widawensen | 334,463 | <p>The simplest way is to construct orthogonal matrix with column vectors <span class="math-container">$B=[b_1 \ \ b_2 \ \ b_1 \times b_2]$</span>.<br />
(I assume here that <span class="math-container">$b_1$</span> and <span class="math-container">$b_2$</span> are normalized to unit length)
and to use trace of such ma... |
2,646,363 | <p>Let $A_1, A_2, \ldots , A_{63}$ be the 63 nonempty subsets of $\{ 1,2,3,4,5,6 \}$. For each of these sets $A_i$, let $\pi(A_i)$ denote the product of all the elements in $A_i$. Then what is the value of $\pi(A_1)+\pi(A_2)+\cdots+\pi(A_{63})$?</p>
<p>Here is the solution </p>
<p>For size 1: sum of the elements, whi... | Clive Newstead | 19,542 | <p>Let $X$ be a set containing $6$ geese a-laying, $5$ gold rings, $4$ calling birds, $3$ French hens, $2$ turtle doves and $1$ partidge in a pear tree.</p>
<p>For a fixed $A \subseteq \{ 1, 2, 3, 4, 5, 6 \}$, the value $\pi(A)$ is the number of ways of picking one of each of the animals (or rings, I guess) from $X$ a... |
3,085,842 | <p>What can be said about the uniform Convergence of <span class="math-container">$\sum_{n=1}^{\infty}\frac{x}{[(n-1)x+1][nx+1]}$</span> in the interval <span class="math-container">$[0,1]$</span>?</p>
<p>The sequence inside the summation bracket doesn't seem to yield to root or ratio tests. The pointwise convergence ... | vidyarthi | 349,094 | <p>The series can be computed by using telescoping. We have <span class="math-container">$\frac{x}{[(n-1)x+1][nx+1]}=\frac1{(n-1)x+1}-\frac1{nx+1}$</span>. Thus, the sequence of partial sums would be <span class="math-container">$s_n=\frac{nx}{nx+1}$</span>. Hence, sum is equal to <span class="math-container">$$\begin{... |
3,101,098 | <p>From 11, 12 in the book Logic in Computer Science by M. Ryan and M. Huth:</p>
<p>**</p>
<blockquote>
<p>"What we are saying is: let’s make the assumption of ¬q. To do this,
we open a box and put ¬q at the top. Then we continue applying other
rules as normal, for example to obtain ¬p. But this still depends o... | Mees de Vries | 75,429 | <blockquote>
<p>once you finish what you want to prove it is no more part of the assumption box and is accessible universally in the proof?</p>
</blockquote>
<p>This is not what happens. You open a proof box with <span class="math-container">$\neg q$</span>, and within the proof box <span class="math-container">$\ne... |
4,247,888 | <p>I'm having a lot of trouble about an apparently simple task. I have the following trigonometric equation:</p>
<p><span class="math-container">$A\cos(\omega_1t+\phi_1)=B\cos(\omega_2t+\phi_2)$</span></p>
<p>which holds for every <span class="math-container">$t \in [0,+\infty)$</span>, where <span class="math-containe... | vonbrand | 43,946 | <p>You know that <span class="math-container">$\cos \theta = 0$</span> if and only if <span class="math-container">$\theta = (2 k + 1) \pi / 2$</span>, use that to find a relation between <span class="math-container">$\omega_1 t + \phi_1$</span> and <span class="math-container">$\omega_2 t + \phi_2$</span>. Then the va... |
1,222,064 | <p>Given is an ellipse with $x=a\cos(t),~~y=b\sin(t)$</p>
<p>I do this by using $S=|\int_c^d x(t)y'(t) dt|$, so calculating the area regarding the vertical axis.
Since $t$ runs from $0$ to $2\pi$ I figured I only had to calculate it from $c=\pi/2$ to $d=3\pi/2$ and then this times $2$. But when I integrate over those... | MvG | 35,416 | <h1>Projective transformations in general</h1>
<p>Projective transformation matrices work on <a href="http://en.wikipedia.org/wiki/Homogeneous_coordinates" rel="nofollow noreferrer">homogeneous coordinates</a>. So the transformation</p>
<p>$$\begin{bmatrix} a & b & c & d \\ e & f & g & h \\ i ... |
2,776,089 | <p>Let $T = \mathbb{S}^1 \times \mathbb{S}^1$ be a torus and $x \in T$. Prove or disprove: There exists a continuous
surjective map $f : T \rightarrow T$ such that the induced homomorphism $f^* : H_1(T,x) \rightarrow
H_1(T,x)$is the zero-map. </p>
<p>I have no idea how to solve this kind of problems. All I know is th... | Jason DeVito | 331 | <p>Here is an outline on constructing such a map.</p>
<p>I'm thinking of $S^1 = \{z\in \mathbb{C}: |z|=1\}$. Then define $g:S^1\rightarrow S^1$ by $g(z) = \begin{cases} z^2 & \operatorname{Im}(z)\geq 0\\ \overline{z}^2 & \operatorname{Im}(z)\leq 0\end{cases}$.</p>
<p>Note that if $\operatorname{Im}(z) = 0$, ... |
2,776,089 | <p>Let $T = \mathbb{S}^1 \times \mathbb{S}^1$ be a torus and $x \in T$. Prove or disprove: There exists a continuous
surjective map $f : T \rightarrow T$ such that the induced homomorphism $f^* : H_1(T,x) \rightarrow
H_1(T,x)$is the zero-map. </p>
<p>I have no idea how to solve this kind of problems. All I know is th... | Igor Sikora | 464,503 | <p>You can try also the following solution: Take any surjective continuous map from $T$ to $I$ - unit interval, for example a height function. Then using Peano curve you can find surjective continuous map from $I$ to $I^2$, and then the quotient map from $I^2$ to $T$. This map will be continuous and surjective, and ind... |
2,414,965 |
<p>I am following along and reading this notes: <a href="https://www.maths.tcd.ie/~levene/221/pdf/cantor.pdf" rel="nofollow noreferrer">https://www.maths.tcd.ie/~levene/221/pdf/cantor.pdf</a></p>
<p>I am having trouble understanding why we necessarily have $e_n=d_n+1$,
$d_{n+1}= d_{n+2} =···= 2$
and $e_{n+1} = e_{... | kishlaya | 369,027 | <p>You'll have to play around with the inequalities a little bit to establish that. </p>
<p>To make life easy, assume without loss of generality $d_1 \neq e_1$. Next, notice,</p>
<p>$$x = \sum_{n \geq 1} \frac{e_n}{3^n} = \frac{e_1}{3} + \sum_{n \geq 2} \frac{e_n}{3^n} \geq \frac{e_1}{3}$$</p>
<p>And also, </p>
<p>... |
172,131 | <p>Given <span class="math-container">$P$</span>, a polynomial of degree <span class="math-container">$n$</span>, such that <span class="math-container">$P(x) = r^x$</span> for <span class="math-container">$x = 0,1, \ldots, n$</span> and some real number <span class="math-container">$r$</span>, I need to calculate <spa... | Théophile | 26,091 | <p>$P(n+1) = r^{n+1}-(r-1)^{n+1}$.</p>
<p>Construct the successive differences between terms, thus:</p>
<pre><code>1 r r^2 r^3 ... r^n
r-1 r(r-1) r^2(r-1) ... r^(n-1)*(r-1)
(r-1)^2 r(r-1)^2 ...
...
... |
172,131 | <p>Given <span class="math-container">$P$</span>, a polynomial of degree <span class="math-container">$n$</span>, such that <span class="math-container">$P(x) = r^x$</span> for <span class="math-container">$x = 0,1, \ldots, n$</span> and some real number <span class="math-container">$r$</span>, I need to calculate <spa... | David E Speyer | 448 | <p>By the binomial theorem,
$$r^x = \sum_{k=0}^{\infty} \binom{x}{k} (r-1)^k$$
for any $x$. Now, if $x$ is a integer from $0$ to $n$, then $\binom{x}{k}=0$ for $k>n$. So
$$r^x = \sum_{k=0}^n \binom{x}{k} (r-1)^k \quad \mbox{for} \ x \in \{ 0,1,2,\ldots, n \}.$$</p>
<p>Notice that the right hand side is a degree $... |
3,225,784 | <p>Solve for x:</p>
<blockquote>
<p><span class="math-container">$$2\sin(x) + 3\sin(2x) = 0 $$</span></p>
<p><span class="math-container">$$2\sin(x)(1 + 3\cos(x)) = 0$$</span></p>
</blockquote>
<p>Stuck here. The solution mentions some arccos function, but I need a detailed explanation on this one.</p>
| Peter Foreman | 631,494 | <p>If you have that
<span class="math-container">$$2\sin{(x)}(1+3\cos{(x)})=0$$</span>
then one or both of the factors must be equal to zero, hence either
<span class="math-container">$$2\sin{(x)}=0$$</span>
<span class="math-container">$$\sin{(x)}=0$$</span>
<span class="math-container">$$x=\pi k $$</span>
or
<span cl... |
573,964 | <blockquote>
<p>Let set $S$ be the set of all functions $f:\mathbb{Z_+} \rightarrow \mathbb{Z_+}$. Define a realtion $R$ on $S$ by $(f,g)\in R$ iff there is a constant $M$ such that $\forall n (\frac{1}{M} < \frac{f(n)}{g(n)}<M). $ Prove that $R$ is an equivalence relation and that there are infinitely mane equ... | user107952 | 107,952 | <p>It is not a subspace of <span class="math-container">$\mathbb{ R}^2$</span> because <span class="math-container">$kx$</span> has to be in it for every real <span class="math-container">$k$</span>, not just integer <span class="math-container">$k$</span>.</p>
|
783,502 | <p>Here in my exercise I have to study the function and draw its graph. Can you please tell me what's the best method to do this, because I don't think that's reasonable to use the input output method, it's quite imprecise.
$$f(x)={|x+1|\over x}$$</p>
<p>Thank you!!!</p>
| evil999man | 102,285 | <p>First you have to break the mod into cases when $x>-1$ and...</p>
<p>Let me talk of case : $\frac{x+1}{x}=1+\frac 1 x$</p>
<p>$1/x $ is odd function. It tends to infinity at $0^+$ and tends to $0$ at infinity.</p>
<p>Make graph of $\frac 1 x $ and shift it one unit upward. Erase all part left of $x=-1$. Can yo... |
2,404,176 | <p>From the days I started to learn Maths, I've have been taught that </p>
<blockquote>
<p>Adding Odd times Odd numbers the Answer always would be Odd; e.g.,
<span class="math-container">$$3 + 5 + 1 = 9$$</span></p>
</blockquote>
<p>OK, but look at this question </p>
<p><a href="https://i.stack.imgur.com/TmYsJ.... | Caleb Stanford | 68,107 | <p>You have answered your own question: the sum of an odd number of odd numbers must be odd. Therefore it cannot equal 30.</p>
<p>You should not believe everything you read in a photo on the internet.</p>
|
251,705 | <p>I would like to find the residue of $$f(z)=\frac{e^{iz}}{z\,(z^2+1)^2}$$ at $z=i$. One way to do it is simply to take the derivative of $\frac{e^{iz}}{z\,(z^2+1)^2}$. Another is to find the Laurent expansion of the function.</p>
<p>I managed to do it using the first way, and the answer is $-3/(4e)$. However, I'm ou... | Community | -1 | <p>$$e^{iz} = e^{i(z-i) + i^2} = \dfrac{e^{i(z-i)}}{e} = \dfrac1e \sum_{k=0}^{\infty} \dfrac{i^k(z-i)^k}{k!}$$
$$\dfrac1z = \dfrac1{z-i+i} = \dfrac1i \dfrac1{1 + \dfrac{z-i}i} = \dfrac1i \sum_{k=0}^{\infty} (-1)^k\left(\dfrac{z-i}i \right)^k = \sum_{k=0}^{\infty} i^{k-1} (z-i)^k$$
$$\dfrac1{(z+i)^2} = \dfrac1{(z-i+2i)^... |
2,741,686 | <p>If I have the following vector space $ V, \text{{$e_0, e_1, e_2$}} \text{ where } e_0(x) = 1, e_1(x) = x \text{ and } e_2(x) = x^2$.I want to know the linear dependency of it how can I proceed? I thought of following the definition of linearly independent $$c_0e_0 + c_1e_1 + c_2e_2 = c_0+ c_1x + c_2x^2=0\iff c_0 = c... | Theo Bendit | 248,286 | <p>One way to show this is repeated differentiation. If
$$c_0 + c_1 x + c_2 x^2 \equiv 0,$$
then
\begin{align*}
c_1 + 2c_2 x &\equiv 0, \\
2c_2 &\equiv 0.
\end{align*}
From evaluating all these polynomials at $x = 0$, we obtain $c_0 = c_1 = c_2 = 0$.</p>
|
1,112,081 | <p>Does $\int_0^\infty e^{-x}\sqrt{x}dx$ converge? Thanks in advance.</p>
| Angelo | 208,573 | <p>yes, $\sqrt{\pi}/2$ should be your answer. let $\sqrt{x}=t$ make the substitution, then integrate by parts. you'll get an integral involving $\int_{0}^\infty e^{-t^2}\,dt$ which is equal to the answer stated above.</p>
<p>you should integrate the resulting integral, i.e., $2\int_{0}^\infty t^2\,e^{-t^2}\,dt$ by par... |
1,921,302 | <p>I can't believe I am asking such a silly question. So I have the function
$$\ln\tan^{-1}x$$
I am asked to find the range of this function. I know that the range of $\ln x$ is all real numbers and that the range of $\tan^{-1}(x)$ is $(-\frac\pi2$, $\frac\pi2)$. Wouldn't the range of $\ln\tan^{-1}x$ also be $(-\frac\p... | Hagen von Eitzen | 39,174 | <p>We are given that $\beta I\subseteq I$.
If $I$ is principal, say $I=(\iota)$ with $\iota\ne 0$, then $\beta \iota= r\iota $ for some $r\in R$ and hence $\beta=r\in R$.</p>
<p>For any other non-zero ideal $J$, we can replace $I$ with $IJ$: If $\beta I \subseteq I$ then also $\beta IJ\subseteq IJ$.
Using Theorem 8.1... |
992,068 | <p>I am having a little trouble understanding this question.</p>
<p>For a DFA M = (Q, Σ, δ, q0, F), we say that a state q ∈ Q is reachable if there
exists some string w ∈ Σ∗ such that q = δ∗(q0, w).</p>
<p>Give an algorithm that, given as input a DFA expressed as a five-tuple M =
(Q, Σ, δ, q0, F), returns the set of ... | Compiii | 633,845 | <p>I hope that this algorithm will be useful for you:</p>
<pre><code>algorithm: determine the set if reachable states of an DFA;
intput: an DFA M = (Q, Σ, δ, q0, F);
output: an list of reachable state;
begin
create a list result and insert q0;
create a stack s and push q0;
while s is not emp... |
7,237 | <p>this came up in class yesterday and I feel like my explanation could have been more clear/rigorous. The students were given the task of finding the zeros of the following equation $$6x^2 = 12x$$ and one of the students did $$\frac{6x^2}{6x}=\frac{12x}{6x}$$ $$x = 2$$ which is a valid solution but this method elimin... | Gerhard Paseman | 3,468 | <p>I think it is important to emphasize Mark Fantini's remark: factoring is different from dividing, and factoring is the way to get a complete solution. I would also suggest that the
problem be rearranged and then factored to give 6x(x-2)=0, or even 2*3*x*(x-2)=0. Then
one can see that one of 2 , 3, x, or x-2 must ... |
32,137 | <p>I have an equation that I evaluate at some point (let's say $x=1$) that have terms of the form</p>
<pre><code>f[1,y] D[g[1,y],{y,2}]
</code></pre>
<p>Is there an easy way to replace [1,y] by [x,y] with a simple replacement rule? The thing is that </p>
<pre><code>g[1,y] /. g[1,y] -> g[x,y]
</code></pre>
<p>wil... | Szabolcs | 12 | <p>Taking the question as "how to replace the first argument in $g(a,b)$ as well as all of its derivatives", you can do this:</p>
<p>Check the InputForm of the derivative:</p>
<pre><code>D[g[1, y], {y, 2}] // InputForm
(* ==> Derivative[0, 2][g][1, y] *)
</code></pre>
<p>Add a corresponding pattern to the replac... |
1,323,845 | <p>For a nonnegative integer $n$, a composition of $n$ means a partition in which the order of the parts matters.</p>
<p>Consider the generating function
$$C(x) = \sum_{n=0}^{\infty} c_nx^n,$$
where $c_n$ is the number of distinct compositions of $n$ (note that $c_0=1$ by convention).</p>
<p>What is the value of $C\l... | Lee Mosher | 26,501 | <p>Let $X$ be a path connected space with more than one point, and let $B \subset X$ be an open ball. If $B$ is just a single point then by an easy argument $B=X$, a contradiction.</p>
<p>So there are two points $x \ne y \in B$, and using them I'll prove that $B$ has contains a subset of the cardinality of the reals.<... |
4,128,046 | <p>I am working through a pure maths book as a hobby. This question puzzles me.</p>
<p>The line y=mx intersects the curve <span class="math-container">$y=x^2-1$</span> at the points A and B. Find the equation of the locus of the mid point of AB as m varies.</p>
<p>I have said at intersection:</p>
<p><span class="math-c... | Math Lover | 801,574 | <p>If <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> are x-coordinates of intersection points, x-coordinate of midpoint,</p>
<p><span class="math-container">$x_m = \frac{x_1 + x_2}{2} = \frac{m}{2}$</span></p>
<p><span class="math-container">$y = mx \implies y_m = 2 x_m^2$</span... |
2,638,679 | <p><a href="https://i.stack.imgur.com/S4p0Y.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/S4p0Y.jpg" alt="enter image description here"></a></p>
<p>Due apologies for this rustic image. But while drawing this lattice arrangement about the "square numbers" , I discovered a pattern here wherein if I ... | Michael Hardy | 11,667 | <p>\begin{align}
\lambda_1 + \lambda_2 = 3 \\
\lambda_1\lambda_2 = 3
\end{align}
So you get $\lambda_2 = \dfrac 3 {\lambda_1},$ so the first equation above becomes
$$
\lambda_1 + \frac 3 \lambda_1 = 3.
$$
Multiply both sides by $\lambda_1$ and you have an ordinary quadratic equation.</p>
<p>The answer posted by "N.S."... |
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