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 |
|---|---|---|---|---|
176,055 | <p>I heard teachers say [cosh x] instead of saying "hyperbolic cosine of x".</p>
<p>I also heard [sinch x] for "hyperboic sine of x". Is this correct?</p>
<p>How would you pronounce tanh x? Instead of saying "hyperbolic tangent of x"?</p>
<p>Thank you very much in advance.</p>
| Argon | 27,624 | <p>Here are some pronunciations that I use with alternate pronunciations given by others.</p>
<ul>
<li>$\sinh$ - Sinch (sɪntʃ) (Others say "shine" (ʃaɪn) according to Olivier Bégassat et al.)</li>
<li>$\cosh$ - Kosh (kɒʃ or koʊʃ)</li>
<li>$\tanh$ - Tanch (tæntʃ) (Others say "tsan" (tsæn) or "tank" (teɪnk) according t... |
2,847,301 | <p>Find elements $a,b,$ and $c$ in the ring $\mathbb{Z}×\mathbb{Z}×\mathbb{Z}$ such that $ab, ac,$ and $bc$ are zero divisors but $abc$ is not a zero divisor.</p>
<p>Work:</p>
<ul>
<li><p>$a=(1,1,0)$</p></li>
<li><p>$b=(1,0,1)$</p></li>
<li><p>$c=(0,1,1)$</p></li>
</ul>
<p>Why this works: because $ab=(1,0,0)\neq(0,0... | Aaron | 9,863 | <p>It is worth pointing out that in a commutative ring, if $x$ is a zero divisor, then so is $xy$, unless $xy=0$. The reason is that, if there is some $z$ such that $xz=0$, then $(xy)z=(xz)y=0$. This means that the condition "$abc$ is not a zero divisor" means that $abc=0$. Further, since $ab, bc, ac$ are all nonzer... |
3,096,351 | <blockquote>
<p><span class="math-container">$$x_1 + y_1 = 3$$</span>
<span class="math-container">$$y_1 - x_3 = -1$$</span>
<span class="math-container">$$x_3 + y_3 = 7$$</span>
<span class="math-container">$$x_1 - y_3 = -3$$</span></p>
<p>Find the values of <span class="math-container">$x_1$</span>, <spa... | Bernard | 202,857 | <p><strong>Hint</strong>:</p>
<p>This has nothing to do with <span class="math-container">$A$</span> and <span class="math-container">$B$</span> being finite or not.</p>
<p>What can you say of a subset <span class="math-container">$X$</span> of <span class="math-container">$A$</span> which is not in <span class="math... |
4,173,559 | <p>The integral is</p>
<p><span class="math-container">$$\int\frac{1}{x\sqrt{1-x^2}}dx\tag{1}$$</span></p>
<p>I tried solving it by parts, but that didn't work out. I couldn't integrate the result of substituting <span class="math-container">$t=1-x^2$</span> either.</p>
<p>The answer is</p>
<p><span class="math-contain... | user71207 | 814,679 | <p>Let <span class="math-container">$\sqrt{-x^{2}+1}=xt+1$</span> to transform the integral into an easier rational function. Rearrange for <span class="math-container">$x$</span> and we have <span class="math-container">$$x = \frac{2t}{-1-t^{2}}$$</span></p>
<p>Find the derivative <span class="math-container">$\frac{d... |
4,173,559 | <p>The integral is</p>
<p><span class="math-container">$$\int\frac{1}{x\sqrt{1-x^2}}dx\tag{1}$$</span></p>
<p>I tried solving it by parts, but that didn't work out. I couldn't integrate the result of substituting <span class="math-container">$t=1-x^2$</span> either.</p>
<p>The answer is</p>
<p><span class="math-contain... | Joe | 623,665 | <p>Edit: I just realised that you wanted to evaluate this integral without trigonometric substitution. I'm sorry if this answer is not of any use to you. I will keep it up for the benefit of other readers.</p>
<hr />
<p>Because of the identity <span class="math-container">$\sin^2\theta+\cos^2\theta=1$</span>, a good ca... |
2,685,822 | <p>How can we prove that $L = \lim_{n \to \infty}\frac{\log\left(\frac{n^n}{n!}\right)}{n} = 1$</p>
<p>This is part of a much bigger question however I have reduced my answer to this, I have to determine the limit of $\log(n^{n}/n!)/n$ when $n$ goes to infinity.</p>
<p>Apparently the answer is 1 by wolfram alpha but ... | Ethan Splaver | 50,290 | <p>Using an integral approximation of the sum of the logarithms of the first $n$ natural numbers gives:</p>
<p>$$n \log n - n=\int_{1}^n\log(t)dt\leq\underbrace{\sum_{k=1}^n\text{log}(n)}_{\log(n!)}\leq \int_{1}^{n+1}\text{log}(t)dt=\underbrace{(n+1)\log(n+1)-(n+1)}_{n\log(n)-n-1+\log(n)+(n+1)\log(1+1/n)}\\\implies n\... |
718,590 | <p>Is it true that in general complete metric space $(M,d)$, a closed ball of radius $r$ centered at $p\in M$ is always compact? That is, the ball is the set of all points $\left\{x:d(x,p)\leq r\right\}$.</p>
| Hamid Shafie Asl | 97,252 | <p>A subset $E$ of a metric space $X$ is said to be totally bounded if $E$ is contained in the union of finitely many open balls of radius $\varepsilon$, for every $\varepsilon>0$. </p>
<p>Let $X$ be a complete metric space.Then $E\subset X$ is compact if and only if it is closed and totally bounded.(<strong>Functi... |
1,231,781 | <p>Given matrix an n-by-n matrix $A$ and its $n$ eigenvalues.
How do I determine the coefficient of the term $x^2$ of the polynomial given by</p>
<p>$q(x) = \det(I_n + xA)$ </p>
| abel | 9,252 | <p>$$\det(I+ x A) = x^n \det (A + \frac 1 x I) = x^n\left(\lambda_1 + \frac 1 x\right)\left(\lambda_2 + \frac 1 x\right)\cdots \left(\lambda_n + \frac 1 x\right) $$</p>
|
1,231,781 | <p>Given matrix an n-by-n matrix $A$ and its $n$ eigenvalues.
How do I determine the coefficient of the term $x^2$ of the polynomial given by</p>
<p>$q(x) = \det(I_n + xA)$ </p>
| achille hui | 59,379 | <p>Translating result appeared on the wiki page of <a href="http://en.wikipedia.org/wiki/Characteristic_polynomial" rel="nofollow">Characteristic polynomial</a>,</p>
<p>$$\det(I_n + x A ) = \sum_{k=0}^n x^k \text{tr}(\wedge^k A)$$</p>
<p>where $\text{tr}(\wedge^k A)$ is the trace of the $k^{th}$ exterior power of $A$... |
470,506 | <p>When is this true?
$$\lim_{r\to 0}\int_{-K}^K f(rx)dx=\int_{-K}^K \lim_{r\to 0} f(rx)dx$$
Is it true without the hypothesis of continuity of
$f$? </p>
<p>Thank you.</p>
| Brian M. Scott | 12,042 | <p>Let your space be $X$. Your argument fails because for any $a>0$ we can choose $J$ to contain all of the points of $X$ in the square $[a,1]\times[a,1]$: there are only finitely many points in that square. The remaining points of $X$ just don’t cover much space, and it’s not too hard to see how to cover them with ... |
2,887,858 | <p>I'm learning how to take surface integrals on the surface of spheres in $\mathbb{R}^n$. This question is related to <a href="https://math.stackexchange.com/questions/2887371/calculating-the-surface-integral-int-s-10y-j-d-sigmay">Calculating the surface integral $\int_{S_1(0)}y_j \ d\sigma(y)$</a> where I try to comp... | zhw. | 228,045 | <p>You have made some errors in setting up the integral, both here and in your previous post. The parameterization of the upper hemisphere we're using is</p>
<p>$$\Sigma (y) = (y,(1-|y|^2)^{1/2},$$</p>
<p>for $y$ in the open unit ball $B_{n-1}$ in $\mathbb R^{n-1}.$ We calculate</p>
<p>$$\tag 1\frac{\partial \Sigma(... |
1,046,961 | <p>Find all continuous functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x \in \mathbb{R}$, $f(x) + f(2x) = 0$ <br/>
I'm thinking; <br/>
Let $f(x)=-f(2x)$ <br/>
Use a substitution $x=y/2$ for $y \in \mathbb{R}$. <br/>
That way $f(y)=-f(y/2)=-f(y/4)=-f(y/8)=....$ <br/>
Im just not sure if this is a good approac... | Community | -1 | <p>By induction we prove that</p>
<p>$$f(x)=(-1)^nf\left(\frac x{2^n}\right)$$
but by the sequential characterization of the continuity we have</p>
<p>$$f\left(\frac x{2^n}\right)\xrightarrow{n\to\infty}f(0)$$
so since for $x=0$ we get $f(0)=0$ so $f(x)$ is constant
$$f(x)=f(0)=0\quad\forall x\in\Bbb R$$</p>
|
35,736 | <p>I have heard that the canonical divisor can be defined on a normal variety X since the smooth locus has codimension 2. Then, I have heard as well that for ANY algebraic variety such that the canonical bundle is defined:</p>
<p>$$\mathcal{K}=\mathcal{O}_{X,-\sum D_i}$$</p>
<p>where the $D_i$ are representatives of ... | Sándor Kovács | 10,076 | <blockquote>
<p><strong>Edit</strong> (11/12/12): I added an explanation of the phrase "this is essentially equivalent to $X$ being $S_2$" at the end to answer aglearner's question in the comments.
[See also <a href="https://mathoverflow.net/questions/45347/why-does-the-s2-property-of-a-ring-correspond-to-the-harto... |
4,240,794 | <p>I was trying to obtain the square root of a matrix through the eigenvalues and eigenvectors, and there is something that doesn't add up in some of the demonstrations that I observed after getting stuck.</p>
<p>So, being <span class="math-container">$Q$</span> the eigenvector column matrix of <span class="math-contai... | Azlif | 513,870 | <p>Your function is continuous. A continuous function is one-to-one if and only if it is strictly decreasing or strictly increasing.</p>
<p>In your case, the function is differentiable so we only need to see its first derivative which by simple computation turns out to be <span class="math-container">$a_2$</span> times... |
2,298,359 | <p>Which of the following has an element that is less than any another element in that set?</p>
<p>I. The set of positive rational numbers</p>
<p>II. The set of positive rational numbers $r$ such that $r^2$ is greater than or equal to $2$</p>
<p>III. The set of positive rational numbers such that $r^2 > 4$. </p>
... | fleablood | 280,126 | <p>" 'there is no such thing as the smallest positive rational number' which is why None of the choices has a smallest element." Be a <em>little</em> careful! The is no smallest rational number but a <em>set</em> of rational numbers could have a smallest element <em>in the set</em>.</p>
<p>Example: The set of ratio... |
2,347,892 | <p>Let $f$ be a continuous differentiable function in $\mathbb{R}$. If $f(0)=0$ and $|f'(x)|\leq|f(x)|,\;\forall x \in \mathbb{R}$, then $f$ is the null function.</p>
<p>Following @DougM's idea, I think I figured it out:
Consider $f$ restricted to the interval $[0,1]$. Then, by Weirstrass theorem there exists $x_1,x_2... | Philip Roe | 430,997 | <p>The general rule for hyperbolic PDEs in two variables is that on any boundary you are allowed as many boundary conditions as there are characteristics entering the domain. This question is about the special case where the domain boundary coincides with a characteristic. On such a boundary we may only prescribe bound... |
1,780,352 | <p>Suppose we have a series</p>
<p>$$\sum^{\infty}_{n = 0} \frac{\sqrt{n}}{(n+1)^2 + |z|^2}$$</p>
<p>As mentioned in the title, how to find the radius of convergence?</p>
<p>In my opinion, $|z| \in \mathbb{C}$, $|z|^2$ is giving a real number, hence we can fix $|z|^2$ regardlessly. But then again, this is not a powe... | Mark Viola | 218,419 | <p>Note that for $|a|=1$, we can write $a=e^{i\psi}$. Then, exploiting the $2\pi$-periodicity of the integrand, we have</p>
<p>$$\begin{align}
\int_{-\pi}^\pi \log|1-ae^{i\theta}|\,d\theta&=\int_{-\pi}^\pi \log|1-e^{i(\theta+\psi)}|\,d\theta\\\\
&=\int_{-\pi+\psi}^{\pi+\psi} \log|1-e^{i\theta}|\,d\theta\\\\
&... |
1,409,932 | <p>I am trying to make a random number generator that is sorta special. Basically it generates a number between 5, and 17. The twist that I need math help on is that I want to have a variable "P" that works like the following.</p>
<p>The higher p is the more likely numbers closer to 17 are to appear, the lower it is... | Brian Tung | 224,454 | <p>To answer the question of what kind of distribution to use (as opposed to David K's good answer regarding how to "implement" a distribution), you might consider the <a href="https://en.wikipedia.org/wiki/Beta_distribution" rel="nofollow">beta distribution</a>, scaled and translated appropriately (multiply by $12$ an... |
1,499,822 | <p>So we were shown this problem from our first functional analysis lecture and I was wondering if someone could help or give a hint:</p>
<blockquote>
<p>Show that the unit sphere $\mathbb{S}^{n-1} := \{ x \in \mathbb{R}^n: \| x \| = 1 \}$ is a complete metric space equipped with $d(x,y):= \arccos \langle x,y \rangl... | PhoemueX | 151,552 | <p>Let us first show the triangle inequality for $d$, i.e. $d\left(x,z\right)\leq d\left(x,y\right)+d\left(y,z\right)$.
Since we have $\left\langle Ux,Uy\right\rangle =\left\langle x,y\right\rangle $
for every orthogonal matrix, by choosing $U$ with $Uy=e_{1}=\left(1,0,\dots,0\right)$,
we can assume $y=e_{1}$ (in short... |
41,450 | <p>I want to perform a Pearson's $\chi^2$ test to analyse contingency tables; but because I have small numbers, it is recommended to perform instead what is called a Fisher's Exact Test.</p>
<p>This requires generating all integer matrices with the same column and row totals as the one given, and compute and sum all p... | halirutan | 187 | <p>As you have found out yourself, the hard part is generating all possible matrices that have a specific row- and column-sum. I don't think that using <code>Reduce</code> for this is a good way to go if you are targeting an algorithm that always works. Consider this simple example with your approach</p>
<blockquote>
<... |
2,897,904 | <p><a href="https://i.stack.imgur.com/ZwR5o.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ZwR5o.png" alt="enter image description here" /></a></p>
<h2>Attempt:</h2>
<p>Let <span class="math-container">$x_{INi}$</span> be the number of people who travel from Ithaca to Newark in class <span class="ma... | Community | -1 | <p>The following constraints must be respected.</p>
<blockquote>
<p>The plane cannot seat more than 30 passengers.</p>
</blockquote>
<p>What you know is that:</p>
<ol>
<li>At first, the plane contains passengers that travel from Ithaca to Boston and Ithaca to Newark</li>
<li>After landing in Newark, the place will... |
2,586,501 | <p>I studied the Ebbinghaus's logic textbook, which is done under ZF (sometimes is ZFC) set theory, so the Gödel Completeness Theorem is valid in ZF(C) (in my opinion).</p>
<p>When I studied Jech's set theory, it is going to show that</p>
<blockquote>
<p>If ZF is consistent, then so is ZFC+GCH.</p>
</blockquote>
... | Alex Kruckman | 7,062 | <p>You wrote:</p>
<blockquote>
<p>Nonetheless, can we really assume that ZF is consistnet? By the Gödel Completeness Theorem, if ZF is consistent, then it is satisfiable and so <strong>there is a set V collecting all sets in the universe of ZF</strong>, which seems a contradiction.</p>
</blockquote>
<p>I added the ... |
455,695 | <p>Let $V$ be the vector space of all real valued continuous functions. Prove that the linear operator $\displaystyle\int_{0}^{x}f(t)dt$ has no eigenvalues.</p>
| Alex Wertheim | 73,817 | <p>Hint: Suppose you have $\int_{0}^{x} f(t)dt = \lambda \cdot f(t)$. Differentiate both sides - you should easily be able to solve the resultant differential equation. Is your solution truly an eigenvector if it is nontrivial? </p>
|
2,618,813 | <p>What is the image of the set {$ { {z ∈ C : z = x + iy, x ≥ 0, y ≥ 0} } $} under
the mapping $ z \to z^2$</p>
<p>my answer : $f(z) = z^2 =(x+iy)^2 = x^2-y^2 +2ixy$,</p>
<p>here I get $u=x^2-y^2$ and $v=2xy$. After that I am confused that how can I find the image of the set.</p>
<p>Please help me, as any Hints can... | Konstantin | 509,087 | <p><strong>Hint</strong>: At first I would suggest you sketch your set, call it $S$. That should be quite easy after the definition.</p>
<blockquote class="spoiler">
<p> Well it is just the first quadrant.</p>
</blockquote>
<p>Then consider points in the complex plain, represented by their polar-coordinate represen... |
265,801 | <p>Consider a 12-sided fair die. What is the distribution of the number T of rolls required to roll a 1, a 2, a 3, and a 4?</p>
<p>Taking inspiration from the Coupon Collector's Problem, I believe that the expected number of rolls to achieve the goal would be</p>
<p>$$E[T] = \sum\limits_{i=0}^3 \frac{12}{4-i} = 25$$... | Giles Adams | 54,383 | <p>I think it's the sum of four different geometric distributions.</p>
<p>For example, you start off and you haven't rolled any of the numbers. The probability of rolling one of them is 4/12 so you start a geometric distribution with that parameter. Then you suddenly roll one of them. Now you need to roll one of the o... |
1,481,627 | <p>I have been given homework in which the $2\times 2$ matrices of determinant $1$ are equipped with the subspace topology of $\mathbb{R}^4$. However, $\mathbb{R}^4$ is a space of 4-tuples, while 2x2 matrices are not n-tuples. How do I get the open sets of this topology?</p>
<p>How is this set of matrices a subset of ... | Thomas Andrews | 7,933 | <p>If $A=(a_{ij})$ and $B=(b_{ij})$ then the inherited metric will be: $$d(A,B)=\sqrt{\sum_{i,j} (a_{ij}-b_{ij})^2}$$</p>
<p>no matter whether you associate $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ with $(a,b,c,d)$, $(a,c,b,d)$ or any other re-ordering of the values.</p>
|
357,847 | <p>Let <span class="math-container">$n\ge 3$</span> be an integer. I would like to know if the following property <span class="math-container">$(P_n)$</span> holds: for all real numbers <span class="math-container">$a_i$</span> such that <span class="math-container">$\sum\limits_{i=1}^na_i\geq0 $</span> and <span clas... | Conrad | 133,811 | <p>A sort of (partial) explanation for what happens:</p>
<p>Let <span class="math-container">$N \ge 3$</span> the degree and let <span class="math-container">$A=\sum{a_k}, B=\sum_{j<k}a_ja_k, C=\sum_{j\ne k\ne m \ne j}a_ja_ka_m $</span>. We are given that <span class="math-container">$A \ge 0, C \ge 0$</span> and w... |
2,278,087 | <p>The polynomial is $p(z)=\sum^n_{k=0} a_kz^k$. And I want to prove the following inequality on the unit disk$$\max_{B_1(0)}|p(z)|\geq |a_n|+|a_0|$$</p>
<p>By the maximum modulus principle, the maximum must be on the unit circle and greater than $|a_0|$ by considering $p(0)$. However, I cannot make further conclusion... | mrf | 19,440 | <p>This was trickier than I first thought. Perhaps there is a simpler solution, but here is one approach: Let $\omega = \exp(2\pi i/n)$. Then
$$
\sum_{j=0}^{n-1} {(\omega^k)}^j = \begin{cases}
n, & k \in \{ 0, n \} \\
0, & 1 \le k \le n-1.
\end{cases}
$$
(This is a well-known property of roots of unity, or ... |
406,966 | <p>Define a graph with vertex set <span class="math-container">$\mathbb{R}^2$</span> and connect two vertices if they are unit distance apart. The famous <a href="http://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem" rel="nofollow noreferrer">Hadwiger-Nelson problem</a> is to determine the <a href="http://en.wi... | Blue | 409 | <p>This isn't "famous", but it's long been of interest to me:</p>
<blockquote>
<p>What is the Pythagorean Theorem for right-corner simplices in hyperbolic $d$-space?</p>
</blockquote>
<ul>
<li>For $d=2$, we have ... $$\cosh a \cosh b = \cosh c$$ for a triangle with legs of length $a$, $b$, and hypotenuse of length ... |
3,424,686 | <p>I have a problem, I am uncertain of how to solve:</p>
<p><a href="https://i.stack.imgur.com/DALSn.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DALSn.png" alt="enter image description here"></a></p>
<p>Is it enough to show:</p>
<p><span class="math-container">$L_1$</span>: <span class="math-c... | Community | -1 | <p>Are you happy that <span class="math-container">$f(v_1)=4v_1-2v_3$</span> and <span class="math-container">$f(v_2)=-2v_1+v_3$</span> and <span class="math-container">$f(v_3)=2v_1+v_2-1v_3$</span>?</p>
<p>Good. So now consider <span class="math-container">$f(av_1+bv_2+cv_3$</span>).
By linearity, this will be:-
<spa... |
4,167,602 | <p>Does the probability that <span class="math-container">$\Pr \left[ {A < {R_3} \cup B < {R_2}} \right]$</span> with the information that <span class="math-container">${R_3} < {R_2}$</span> logically equivalent to <span class="math-container">$\Pr \left[ {\min \left[ {A,B} \right] < {R_3}} \right]$</span> ... | quasi | 400,434 | <p>As an example, if <span class="math-container">$R_3=1,R_2=2$</span>, and <span class="math-container">$A,B$</span> are random variables with the constraints <span class="math-container">$A\ge 1$</span> and <span class="math-container">$1\le B < 2$</span>, then
<span class="math-container">$$
\Pr \left[ {A < {R... |
177,643 | <p>Let $I=\{0, 1, \ldots \}$ be the multiplicative semigroup of non-negative integers. It is possible to find a ring $R$ such that the multiplicative semigroup of $R$ is isomorphic (as a semigroup) to $I$?</p>
| Clive Newstead | 19,542 | <p>Replacing $B$ by $B+Y$ in your expression for $\tan(A+B)$ gives</p>
<p>$$\dfrac{\tan A + \tan(B+Y)}{1 - \tan A \tan(B+Y)}$$</p>
<p>Expanding gives</p>
<p>$$\dfrac{\tan A + \frac{\tan B + \tan Y}{1 - \tan B \tan Y}}{1 - \tan A \frac{\tan B + \tan Y}{1 - \tan B \tan Y}}$$</p>
<p>Multiplying through by $1 - \tan B ... |
177,643 | <p>Let $I=\{0, 1, \ldots \}$ be the multiplicative semigroup of non-negative integers. It is possible to find a ring $R$ such that the multiplicative semigroup of $R$ is isomorphic (as a semigroup) to $I$?</p>
| user29999 | 29,999 | <p>Do $a=\tan A, b= \tan B$ and $y = \tan Y$. Then,
\begin{eqnarray}
\tan (A+B+Y) &=& \dfrac{\tan(A+B) + \tan(Y)}{1-\tan(A+B)y}\\
&=& \dfrac{\dfrac{a+b}{1-ab}+y}{1 - \Big(\dfrac{a+b}{1-ab}\Bigr)y}\\
&=& \dfrac{\dfrac{a+b+(1-ab)y}{1-ab}}{\dfrac{1-ab - (a+b)y}{1-ab}}\\
&=& \dfrac{a+b+y -ab... |
6,155 | <p>I have a category $C$, which is equipped with a symmetric monoidal structure (tensor product $\otimes$, unit object $1$). My category also has finite coproducts (I'll write them using $\oplus$, and write $0$ for the initial object), and $\otimes$ distributes over $\oplus$.</p>
<p>By an <em>exponential monad</em>, ... | Mattia Coloma | 125,244 | <p>I can give you my interpretation of the Chern character in terms of morphism of formal group laws. Denote with $\gamma$ the universal line bundle over $\mathbb{P}^\infty$ and denote its first Chern class with $c_1(\gamma) = x_H \in H^2(\mathbb{P}^\infty)$. Also denote with $k = \gamma-1 \in K(\mathbb{P}^\infty)$ and... |
2,466,805 | <p>Hi maths peoples I have question how you show that function is greater or equal to zero because I want show that function is dense function and this is one of two condition for show it is dense function.</p>
<p>Example we have function </p>
<p>$$f(x) = \frac{1}{2} \cdot \sin(x) \text{ where } x \in \left[0, \pi\ri... | Thomas Andrews | 7,933 | <p>We have: $$1=\mathrm{Var}(X_i)=E(X^2)-(E(X))^2 = E(X^2)$$</p>
<p>You need to know:</p>
<blockquote>
<p><strong>Lemma:</strong> If $Y$ is a non-negative random variable, then $$E(Y)\geq kP(Y\geq k).$$</p>
</blockquote>
<p>So if $Y=X_i^2$ and $k=4$ then:</p>
<p>$$1=E(X_i^2)\geq 4P(X_i^2\geq 4)=4 P(|X_i|\geq 2)$$... |
105,583 | <p>this is a simple question, and excuse me if it's already been answered; I searched around and couldn't find anything.</p>
<p>I have two listplots, both along the same number of x data points, but with different y values. I want to find the difference between the two y values, while keeping the x values the same. I ... | Community | -1 | <p>ok, you have two lists,</p>
<pre><code>list1 = {{1, 2}, {2, 3}, {3, 5}, {4, 7}, {5, 11}, {6, 13}, {7, 17}}
list2 = {{1, 3.87}, {2, 3.53}, {3, 3.40}, {4, 3.33}, {5, 3.25}, {6, 4.25}, {7, 5.24}}
</code></pre>
<p>and you know how to plot them,</p>
<pre><code>ListPlot[{list1, list2}]
</code></pre>
<p><a href="https... |
181,327 | <p>If $q$ and $w$ are the roots of the equation $$2x^2-px+7=0$$ </p>
<p>Then $q/w$ is a root of ? </p>
<p>P.s:- It is an another question of <a href="https://math.stackexchange.com/questions/181305/how-do-i-transform-the-equation-based-on-this-condition">How do I transform the equation based on this condition?</a></p... | Kartik Audhkhasi | 37,312 | <p>I would not be worried about the equality constraint since it can be handled in almost all optimization packages. This is because $g(x) = a$ can always be written as two inequality constraints: $g(x) \leq a$ and $g(x) \geq a$. You can also remove one optimization variable using this constraint, as mentioned by Ed Go... |
1,015,137 | <p>I have an exercise book from my university which doesn't specify a quantifier.</p>
<p>It uses expressions like "here $A$,$B$,$C$ are sets", or "if $x \notin A$ then ..." (it uses $x$ before it is even defined, just out of nowhere).</p>
<p>I'm going to need an answer from my university, of course, but I want to ask... | vadim123 | 73,324 | <p>$$\sum\limits_{n=1} ^\infty |x_n||y_n|=|x_1||y_1|+\sum_{n=2}^\infty|x_n||y_n|\le |x_1|\sum_{n=1}^\infty|y_n|+\sum_{n=2}^\infty |x_n||y_n|\le$$</p>
<p>$$|x_1|\sum_{n=1}^\infty|y_n|+|x_2||y_2|+\sum_{n=3}^\infty |x_n||y_n|\le |x_1|\sum_{n=1}^\infty|y_n|+|x_2|\sum_{n=1}^\infty|y_n|+\sum_{n=3}^\infty |x_n||y_n|\le$$</p>... |
1,015,137 | <p>I have an exercise book from my university which doesn't specify a quantifier.</p>
<p>It uses expressions like "here $A$,$B$,$C$ are sets", or "if $x \notin A$ then ..." (it uses $x$ before it is even defined, just out of nowhere).</p>
<p>I'm going to need an answer from my university, of course, but I want to ask... | Winther | 147,873 | <p>Apply Cauchy-Schwarz</p>
<p>$$(x\cdot y)^2 \leq ||x ||^2~||y||^2$$</p>
<p>to the vectors $x =(|x_1|,|x_2|,\ldots,|x_n|)$ and $y = (|y_1|,|y_2|,\ldots,|y_n|)$ to get</p>
<p>$$\left(\sum x_i y_i\right)^2 \leq \sum x_i^2\sum y_i^2$$</p>
<p>Now use that</p>
<p>$$\sum x_i^2 \sum y_i^2 \leq \left(\sum |x_i|\right)^2... |
187,486 | <p>Let $SL(2,{\mathbb R})$ act on ${\mathbb R}^2$ by matrix multiplication.</p>
<p>What is known about group cohomology $H^*(SL(2,{\mathbb R}),{\mathbb R}^2)$?</p>
<p>And about $$H^*(\Gamma,{\mathbb R}^2)$$ for a cocompact lattice $$\Gamma\subset SL(2,{\mathbb R})?$$</p>
| Pierre | 61,960 | <p>I will answer only for cohomology in degree 1, as I am not sure for higher degrees. </p>
<p>On the other hand I think that what I am saying remains true when R^2 is replaced by any finite dimensional representation of SL_2(R). </p>
<p>For the first cohomology of a lattice with values in a finite dimensional repres... |
80,432 | <p>I know Mathematica's if format is </p>
<pre><code>If[test, then result, else alternative]
</code></pre>
<p>For example, this</p>
<pre><code>y:=If[RandomReal[]<0.2, 1, 3.14]
</code></pre>
<p>would take a random real number between $0$ and $1$, and evaluate it. If it's less than $0.2$, it'll map <code>y</code> ... | rcollyer | 52 | <p>This is not an answer to your question, but I wanted to point out a better way of coding</p>
<pre><code>y:=If[RandomReal[]<0.2,1,3.14]
</code></pre>
<p>Use <a href="http://reference.wolfram.com/language/ref/RandomChoice.html" rel="nofollow"><code>RandomChoice</code></a>, instead, as you can specify the exact pr... |
925,005 | <p>Show that $$\|a+b+c\|^2\leq 3\|a\|^2+3\|b\|^2+3\|c\|^2$$
where $a,b,c$ are in some Hilbert space $(H,\langle\cdot,\cdot \rangle)$?</p>
<p>I see that we have $$\|a+b\|^2\leq2\|a\|^2 +2 \|b\|^2$$ due to the parallelogram law. What about the first inequality?</p>
| PenasRaul | 156,384 | <p>You can prove that $2 | \langle a, b \rangle | \leq \| a \|^2 + \| b \|^2$ as I believe you have proven to attain $\| a + b \|^2 \leq 2\| a \|^2 + 2\| b \|^2$.</p>
<p>Then you just expand:</p>
<p>$\| a + b + c \|^2 = \langle a, a \rangle + \langle b, b \rangle + \langle c, c \rangle + 2\langle a, b \rangle + 2\lan... |
1,782,855 | <p>These were a couple examples in class before learning the $u$-substitution method for integrals. I'm not sure what is going on:</p>
<p>$$\int1\, d(2x) = 2x+ C$$
$$\int \sin x \,d(\sin x) = \frac{(\sin x)^2}{2}+C$$
$$\int(2x+1) \,d(2x+1) = \frac{(2x+1)^2}{2}+C$$ </p>
<p>How exactly do we reach these answers? I gue... | Hagen von Eitzen | 39,174 | <p>Have a look at the explicit bijective map $\Bbb N\times\Bbb N\to\Bbb N$, $(a,b)\mapsto \frac{(a+b)(a+b+1)}{2}+b$.</p>
<p>The case of countable family of countable set is almost the same - except that it may involve a bit more need for Choice (we need an enumeration for each $A_i$).</p>
|
1,782,855 | <p>These were a couple examples in class before learning the $u$-substitution method for integrals. I'm not sure what is going on:</p>
<p>$$\int1\, d(2x) = 2x+ C$$
$$\int \sin x \,d(\sin x) = \frac{(\sin x)^2}{2}+C$$
$$\int(2x+1) \,d(2x+1) = \frac{(2x+1)^2}{2}+C$$ </p>
<p>How exactly do we reach these answers? I gue... | Jxt921 | 229,776 | <p>I want to suggest one more way to prove that $|\mathbb{N} \times \mathbb{N}| = |\mathbb{N}|$.
We use two injections:</p>
<p>1) $f: \mathbb{N} \times \mathbb{N} \to \mathbb{N}, f(m,n) = 2^m3^n$. Fundamental theorem of arithmetic tells us $f$ is injective.</p>
<p>2) $g: \mathbb{N} \to \mathbb{N} \times \mathbb{N}, f... |
2,556,255 | <p>For work I have been doing a lot of calculations which look sort of like summing terms similar to $\frac{A}{1+x}$ and $\frac{B}{1+y}$ for some $A, B$ and small values of $0 \leq x, y \leq 0.1$. In my experience I have found that this is approximately equal to $\frac{A + B}{1 + \frac{A}{A+B}x + \frac{B}{A+B}y }$. The... | Karn Watcharasupat | 501,685 | <p>Hint: Try Taylor/Maclaurin series:
$$
\frac{A}{1+x} = A( 1 - x + x^2 - x^3 \cdots)
$$
for $|x| < 1$. Use the start of that series to start your approximations.</p>
|
1,368,180 | <p>Here is a tricky compass and straightedge construction problem. </p>
<blockquote>
<p>Given triangle $\triangle ABC$ and point $D$ on segment $\overline{AB}$, construct point $P$ on line $\overleftrightarrow{CD}$ such that $\angle APB = \angle BAC$. </p>
</blockquote>
<p>This configuration appears frequently in O... | Mick | 42,351 | <p>There are 2 possible cases – $P$ can be outside or inside of $\triangle ABC$. Maybe that is why the OP claims that this is trick problem.</p>
<p>Case-1 ($P$ is outside, easier to start with for my $ABC$)</p>
<p>1) Draw line $g$, the perpendicular bisector of $AB$.</p>
<p>2) Draw line $h$, through $A$ and perpendi... |
760,739 | <p>Find the limit of the sequence given by
$$\frac{10+12n+20n^4}{7n^4 + 5n^3 - 20}$$</p>
<p>I think the answer is $\frac{20}{7}$ after dividing, but is that right? </p>
| Community | -1 | <p>Yes right!</p>
<p>$$\frac{10+12n+20n^4}{7n^4+5n^3-20}=\frac{\frac{10}{n^4}+\frac{12}{n^3}+20}{7+\frac5n-\frac{20}{n^4}}$$
and pass to the limit.</p>
<p><strong>Remark:</strong> There's a simple way to find the limit at $\infty$ for a fraction of two polynomials: it suffices to take the monomial of highest degree f... |
4,578,898 | <blockquote>
<p><strong>Question:</strong> Let <span class="math-container">$f$</span> be the function defined on <span class="math-container">$[0,1]$</span> by
<span class="math-container">$$
f(x)=
\begin{cases}
n(-1)^n & \textrm{if }\frac{1}{n+1}<x\leq \frac{1}{n} \\
0 & \textrm{otherwise}
\en... | Bonnaduck | 92,329 | <p>Obviously, this is not the standard definition of the <span class="math-container">$n$</span>th root in mathematics, so I will not use the <span class="math-container">$\sqrt[n]{\cdot}\ $</span> notation.</p>
<p>Lets say we want to compute the <span class="math-container">$3$</span>rd root of <span class="math-conta... |
461,268 | <p>A square hole of depth $h$ whose base is of length $a$ is given.
A dog is tied to the center of the square at the bottom of the hole by a rope of length
$L>\sqrt{2a^2+h^2}$ ,and walks on the ground around the hole.</p>
<p>The edges of the hole are smooth, so that the rope can freely slide along it. Find the sha... | Harish Kayarohanam | 30,423 | <p>It can move any where inside the pit </p>
<p>now for L > $ \sqrt{2*a^2 +h^2}$</p>
<p>so it can very well peep out of the pit (from being inside ) because $ \sqrt{2*a^2 +h^2}$ is the length of hypotenuse with one leg as diagonal of square and other being height .</p>
<p>so shape will be circle on top , but shape o... |
461,268 | <p>A square hole of depth $h$ whose base is of length $a$ is given.
A dog is tied to the center of the square at the bottom of the hole by a rope of length
$L>\sqrt{2a^2+h^2}$ ,and walks on the ground around the hole.</p>
<p>The edges of the hole are smooth, so that the rope can freely slide along it. Find the sha... | Jonas Kgomo | 45,379 | <p>Let ABCD be the upper square of the hole. We will determine the locus along a side at one time. For this, let O be the centre of the bottom square. Let T be the variable point on the edge AB through which the string passes. And let P be the point where the goat is standing, such that the string remains taut.</p>
<p... |
4,589,828 | <p>I'm stuck in proving the function <span class="math-container">$$f(x) = \vert x + e^x\vert $$</span> has a minimum.</p>
<p>This is what I did:</p>
<p><span class="math-container">$$f'(x) = (1+e^x)\text{sgn}(1+e^x)$$</span></p>
<p>But this function is never zero, because the exponential is always positive. So when I ... | Hirofumi Ryo | 300,531 | <p>I approached this integral by trigonometric substitution and partial fraction.</p>
<p>Assume the followings:</p>
<p><span class="math-container">$x=\sin\theta\ \ldots(1)\\
\text{Hence, }dx=\cos\theta\ d\theta\ \ldots(2)\\
\text{Also, }\cos\theta=\sqrt{1-x^2}\ \ldots(3)\\$</span></p>
<p>Then,</p>
<p><span class="math... |
2,760,605 | <p>$||x-1|-2|=|x-3|$.
Find the value of $x$.<br>
In my attempt I got the critical values of the expression as $1$ and $3$.
But I’m not sure is we can just not consider the $2$ in the LHS.</p>
<p>My steps are Case 1: when $x>3$, $x-1-2=x-3$, $0=0$</p>
<p>Case 2: $1<x<3$, $X-1-2=-x+3 \implies 2x=6 \implies x... | MSDG | 447,520 | <p><strong>EDIT</strong>: As pointed out in the comments, even though the final inequality is correct, it is insufficient since $(n+1)^{1/n} \to 1$ as $n \to \infty$. The lower bound can be obtained as shown in @adfriedman's answer.</p>
<p>Here's my take on it:
Whenever $n \geq 3$, we have
$$ n^n \geq (n+1)!, $$
and t... |
2,105,238 | <p>I need to prove that the function $f: \mathbb R_{>0}\times\mathbb R \to \mathbb R^2$ , $f(x,y)=(xy, x^2-y^2)$ is injective. I know I have to show that $f(x,y)=f(a,b)$ implies $x=a$ and $y=b$ but I have no idea how to prove it. Could you give me a hint?</p>
| Rob Arthan | 23,171 | <p>If you are given $xy$ and $x^2 - y^2$ and you know that $x > 0$, you can determine all of the following:
$$
\begin{align*}
(x^2 + y^2)^2 &= x^4 + y^4 + 2(xy)^2 = (x^2 - y^2)^2 +4 (xy)^2\\
x^2 + y^2 &= \sqrt{(x^2 + y^2)^2} \\
x^2 &= \frac{(x^2 - y^2) + (x^2 + y^2)}{2} \\
x &= \sqrt{x^2} \\
y &=... |
31,544 | <p>For example, how to make Mathematica display only the lines beginning with <code>In[some number]</code>?</p>
<pre><code>In[8]:= m1 = {{1, 1, 1}, {8, 4, 2}, {64, 16, 4}}
</code></pre>
<blockquote>
<pre><code>Out[8]= {{1, 1, 1}, {8, 4, 2}, {64, 16, 4}}
</code></pre>
</blockquote>
<pre><code>In[9]:= m1.{b1, b2, b3} ... | Kuba | 5,478 | <p>4)</p>
<pre><code>SetOptions[#, CellOpen -> False] & /@ Cells[EvaluationNotebook[], GeneratedCell -> True]
</code></pre>
<p>But I would choose Szabolcs's 3rd solution.</p>
|
780,588 | <p>I need help with this proof:</p>
<p>Let $V, W$ be K-vectorspaces.
Let $T_1, T_2$ be subspaces of V with $T_1 \subseteq T_2$.
Let $f \in hom_K(V,W)$.</p>
<p>Show the following: If $ f(T_1) = f(T_2)$ then exists a subspace $U$ of $ker(f)$ so that $ U \bigoplus T_1 = T_2$</p>
<p>I'm sorry, but I really have no clue ... | Christian Blatter | 1,303 | <p>Put $U_1:=T_1\cap {\rm ker}(f)$, $\>U_2:=T_2\cap {\rm ker}(f)$. It follows that $U_1\subset U_2\subset{\rm ker}(f)$. </p>
<p>Let $U$ be a complement of $U_1$ in $U_2$, so that we have
$$U_2=U_1\oplus U\ .\tag{1}$$
I claim that
$$T_2=T_1\oplus U\ .\tag{2}$$
<em>Proof.</em> As $T_1\subset T_2$ and $U\subset U_2\s... |
1,733,923 | <p>I'm working with the following sum and trying to determine what it converges to: $$\sum_{k=1}^{\infty}\ln\left(\frac{k(k+2)}{(k+1)^2}\right)$$</p>
<p>Numerically I see that it seems to be converging to $-\ln(2)$, however I can't see why that is the case. I have expanded the logarithm and expressed the sum as $\sum_... | DonAntonio | 31,254 | <p>$$\frac{k^2+2k}{k^2+2k+1}=1-\frac1{(k+1)^2}\implies$$</p>
<p>$$\sum_{k=1}^n\log\left(1-\frac1{(k+1)^2}\right)=\sum_{k=1}^n\log\left[\left(1-\frac1{k+1}\right)\left(1+\frac1{k+1}\right)\right]=$$</p>
<p>$$=\sum_{k=1}^n\left[\log\left(1-\frac1{k+1}\right)+\log\left(1+\frac1{k+1}\right)\right]=$$</p>
<p>$$=\log\frac... |
26,420 | <p>So the cup product is not well defined over co-chain groups, but all the books claim it is well defined over co-homology groups. The only thing I am not clear on is invariance under ordering/re-ordering of simplices when we go to the co-homology level. Every book seems to gloss over this, and after doing a few examp... | Reader Manifold | 376,599 | <p>At the cochain level, for <span class="math-container">$\varphi \in C^k(X)$</span> and <span class="math-container">$\psi \in C^l(X)$</span>, the cup product is defined as, <span class="math-container">$$(\varphi \smile \psi)(\sigma) = \varphi (\sigma \big|_{\leq k}) \psi (\sigma \big|_{\geq k})$$</span> and extende... |
705,662 | <p>So my function is $f(z) = \sqrt(r)e^{i\theta/2}$.</p>
<p>I want to find its contour integral over the upper half-disk of $z = e^{i\theta}$, where $0\leq\theta\leq\pi$.</p>
<p>I know that:</p>
<p>$\int f(z) dz$ = $\int_a^b f(z(t)) * z'(t) dt$</p>
<p>But in the above problem, the function I want to find the contou... | Robert Israel | 8,508 | <p>This branch of $z^{1/2}$ is analytic on the upper half plane. What is its antiderivative? You should have a theorem relating contour integrals to antiderivatives...</p>
|
1,920,624 | <p>Prove that if 1 is a linear combination of two integers $a, b$, then $a$ and $b$
are relatively prime. Use only basic divisibility properties. (Do not use GCDLC).</p>
<p>I can only figure out how to prove this using the GCDLC.</p>
| Arthur | 15,500 | <p>Hint: any common divisor of $a$ and $b$ will necessary divide any linear combination of $a$ and $b$.</p>
|
356,594 | <p>The roots of the quadratic equation $ax^ 2-bx+c=0,$ $a>0$, both lie within the interval $[2,\frac{12}{5}]$. Prove that:
(a) $a \leq b \leq c <a+b$.
(b) $\frac{a}{a+c}+\frac{b}{b+a}>\frac{c}{c+b}$</p>
<p>So we can use the quadractic formula and obtain
$2 \leq \frac{b \pm \sqrt{b^2-4ac}}{2a} \leq \frac{12}{5... | DonAntonio | 31,254 | <p>Hints: let $\,\alpha\,,\,\beta\,$ be the quadratic's roots, then Viete's formulas give</p>
<p>$$\alpha+\beta=\frac{b}{a}\;,\;\;\alpha\beta=\frac{c}{a}$$</p>
<p>Suppose $\,2\le \alpha\le\beta\le 2.4\,$ , then for example</p>
<p>$$a=\frac{b}{\alpha+\beta}\le b\ldots$$</p>
|
4,188,147 | <p>I'm trying to show the momentum operator <span class="math-container">$P : \mathcal D(P)\ \dot{=}\ C^\infty_0(\Bbb R)\to L^2(\Bbb R)$</span> with <span class="math-container">$P=-i\hbar\partial_x$</span> is essentially self-adjoint, which is equivalent to saying that it is symmetric (densely defined, which is true b... | Jose27 | 4,829 | <p>A bootstrapping argument shows that we can worry only about classical solutions: Consider the equation
<span class="math-container">\begin{equation}\tag{1}
\text{w-}\partial_x \psi= \dfrac{\psi}{h},
\end{equation}</span>
given that <span class="math-container">$\psi \in W^{1,2}$</span> (the domain doesn't matter her... |
4,224,303 | <p><strong>Question</strong></p>
<p>Let <span class="math-container">$X_0, X_1, X_2, \dots$</span> be independent and identically distributed continuous random variables with density <span class="math-container">$f(x)$</span>. Let <span class="math-container">$N$</span> be the first index <span class="math-container">$... | StubbornAtom | 321,264 | <p>When the <span class="math-container">$X_i$</span>'s are i.i.d continuous, the events <span class="math-container">$A_j=\{X_j=\max\{X_0,X_1,\ldots,X_k\}\}$</span> are equally likely for every <span class="math-container">$j=0,1,\ldots,k$</span> by symmetry.</p>
<p>Further note that <span class="math-container">$P(A_... |
557,680 | <blockquote>
<p>Let $M_{k,n}$ be the set of all $k\times n$ matrices, $S_k$ be the set of all symmetric $k\times k$ matrices, and $I_k$ the identity $k\times k$ matrix. Suppose $A\in M_{k,n}$ is such that $AA^t=I_k$. Let $f:M_{k,n}\rightarrow S_k$ be the map $f(B)=BA^t+AB^t$. Prove that $f$ is onto (surjective). (Not... | Henry | 106,067 | <p>First observation: $C=BA^t+AB^t=BA^t+(BA^t)^t$. Also, since $C$ is a symmetric matrix, there is an upper-triangular matrix $U$ such that $C=U+U^t$ (just take the $C$'s entries for the off-diagonal entries, and take a half of $C$'s diagonal entries). </p>
<p>Therefore it is enough to prove that for any $k\times k$ u... |
2,687,375 | <p>Let <span class="math-container">$x_1, x_2, \ldots, x_n$</span> be a random sample from the Bernoulli (<span class="math-container">$\theta$</span>).</p>
<p>The question is to find the UMVUE of <span class="math-container">$\theta^k$</span>.</p>
<p>I know the <span class="math-container">$\sum_1^nx_i$</span> is the ... | DanielWainfleet | 254,665 | <p>(1). Compactnes: In a $T_2$ (Hausdorff) space (such as $\Bbb R^2$) a compact subset must be closed. But $T$ is not closed. Also $S\setminus \{p\}$ is not closed for any $p\in S,$ because $p\in \overline {S\setminus \{p\}}$. </p>
<p>Let $T$ be a non-closed subset of a Hausdorff space $X$ and let $p\in \overline T ... |
2,529,990 | <p>Find the area of the shape surrounded by $y = \sin(x)$, $y = -\cos(x)$, $x = 0$, $x = π/4$.
Do I subtract $S_2$ from the $S_1$? How do I find the shape's area?
<a href="https://i.stack.imgur.com/67G7B.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/67G7B.png" alt="Picture of the graph: "></a></p>
| zoli | 203,663 | <p>The area equals</p>
<p>$$\int_0^{\frac{\pi}4}\int_{-\cos(x)}^{\sin(x)}\ dy\ dx=\int_0^{\frac{\pi}4}\sin(x)+\cos(x)\ dx=$$
$$=\left[-\cos(x)\right]_{0}^{\frac{\pi}4}+\left[\sin(x)\right]_{0}^{\frac{\pi}4}=$$
$$=-\frac1{\sqrt 2}+1+\frac1{\sqrt 2}-0=1.$$</p>
|
101,828 | <p>In what sense is the limit of discrete series representation of $SL(2, \mathbb{R})$ a limit of discrete series representations? Where does the name origin from?</p>
| paul garrett | 15,629 | <p>These repns are not actually "discrete series", in that they do not appear in $L^2(G)$. Yet their construction/description is completely parallel to that of the discrete family of repns called "discrete series". Since the relevant parameter is discrete, it is hard to conjure up any "limit-taking process", indeed, in... |
452,889 | <p>My question relates to the conditions under which the spectral decomposition of a nonnegative definite symmetric matrix can be performed. That is if $A$ is a real $n\times n$ symmetric matrix with eigenvalues $\lambda_{1},...,\lambda_{n}$, $X=(x_{1},...,x_{n})$ where $x_{1},...,x_{n}$ are a set of orthonormal eigenv... | fuglede | 10,816 | <p>For your first question: that you can diagonalize real symmetric matrices is the so-called <a href="http://en.wikipedia.org/wiki/Spectral_theorem" rel="nofollow">spectral theorem</a>.</p>
<p>For the second: your argument is general; you did not use that the eigenvalues were distinct.</p>
|
3,794,418 | <p><strong>Background:</strong> I'm a math rookie, yet to enrol in university. I randomly started reading Mendelson's <em>Introduction to Mathematical Logic</em>, when I stumbled upon this paradox in the introductory section:</p>
<blockquote>
<p><strong>Grelling's Paradox:</strong> An adjective is called <em>autologica... | Dan Christensen | 3,515 | <p>Logically speaking, Grelling's Paradox is just Russell's Paradox--only the names have been changed.</p>
<p>For better or worse, natural language allows us to define things that cannot logically exist. Logically, just like there can exist no <strong>set</strong> of all those sets that are not <strong>elements of</str... |
3,120,468 | <p>I'm wondering if it's correct to say that the absolute value squared means that the values above <span class="math-container">$1$</span> are magnified, whereas the values below <span class="math-container">$1$</span> are damped (I'm considering only positive values because the codomain of the absolute value is <span... | G Cab | 317,234 | <p>We can approach the problem geometrically.</p>
<p>Each triple toss <span class="math-container">$\left( {x_{\,1} ,x_{\,2} ,x_{\,3} } \right)$</span> corresponds to an integral point
in a cube with side <span class="math-container">$[1,6]$</span>: wlog we can take the die to be numbered <span class="math-container">... |
3,120,468 | <p>I'm wondering if it's correct to say that the absolute value squared means that the values above <span class="math-container">$1$</span> are magnified, whereas the values below <span class="math-container">$1$</span> are damped (I'm considering only positive values because the codomain of the absolute value is <span... | user | 293,846 | <p>The probability that a round ends without desicion is
<span class="math-container">$$
\underbrace{3\left(\frac16\right)^2\left(\frac56+\frac46+\frac36+\frac26+\frac16\right)}_{\frac{15}{72}}+
\underbrace{6\left(\frac16\right)^3}_{\frac{2}{72}}=\frac{17}{72}.
$$</span>
Here the first summand stands for two equal sco... |
448,227 | <p>Is it possible to define the completion of a metric space using categorical terms?</p>
| Martin Brandenburg | 1,650 | <p>At least for Lawvere metric spaces it is a special case of the <a href="http://ncatlab.org/nlab/show/Cauchy+complete+category" rel="nofollow">Cauchy completion</a>.</p>
|
1,510,124 | <p>What is the method that would be used to find the unit normal vector to the surface given by the equation $x+y^{2}+2z=4$?</p>
| Surb | 154,545 | <p>$$\{(x,y,z)\mid x+y^2+2z=4\}=\{u(y,z)=(y^2+4-2z, y,z)\mid y,z\in \mathbb R\}$$</p>
<p>I denote $u_y=\frac{\partial u}{\partial y}$ and $u_z=\frac{\partial u}{\partial z}$. A normal is given by $u_y\times u_z$.</p>
|
1,510,124 | <p>What is the method that would be used to find the unit normal vector to the surface given by the equation $x+y^{2}+2z=4$?</p>
| abel | 9,252 | <p>if you difference $x + y^2 + 2z = 4, $ and use the fact that $(dx, dy, dz)$ is tangent vector ay $(x, y, z),$ then you get $$0=dx + 2ydy + 2dz = (1,2y, 2)^\top \cdot (dx, dy, dz)^\top.$$ that is the normal vector at the point $(x,y,z)$ to the surface $x + y^2 + 2z = 4 $ is $(1,2y,z)^\top.$</p>
|
2,055,546 | <p>So I stumbled across this math problem:</p>
<p>$1015$ USD is to be shared between $3$ people: Person A, Person B and Person C.
Person A gets double of what Person B gets and Person B gets $100$ USD more than Person C</p>
<p>Basically:</p>
<p>$A = 2B$</p>
<p>$B = C + 100$</p>
<p>$A + B + C = 1015$</p>
<p>Can so... | egreg | 62,967 | <p>If we lend them $\$100$, then we can give $B$ and $C$ the same amount of money, after which $C$ will give back the $\$100$ note.</p>
<p>Now, if we give $\$1$ to $B$ and $C$, $A$ will receive $\$2$. The total would be $\$4$. </p>
<p>OK, but we have $\$1115$ to distribute.</p>
<blockquote class="spoiler">
<p> Jus... |
636,289 | <p>Here's what I've gathered so far: </p>
<p>First I calculate the number of combinations of $5$ cards from the whole deck, which is $2 598 960.$</p>
<p>I'll use the complement, so I want to the combinations of everything but the aces and kings, so it's again combinations of $5$ from $44$ cards, which is $1086008$.</... | Edward ffitch | 26,243 | <p>We need to subtract from $C^{52}_{5}$ (the total number of ways of choosing $5$ cards from $52$) the number of ways of selecting no aces and the number of ways of selecting no kings, but we must add the number of ways of selecting no aces <strong>and</strong> no kings (else we'll have counted these twice). The numbe... |
3,997,321 | <p>Let <span class="math-container">$X$</span> be a set, <span class="math-container">$\tau_1,\tau_2$</span> two topologies on <span class="math-container">$X$</span>, and consider the following statements</p>
<ol>
<li><span class="math-container">$\tau_1\subseteq \tau_2$</span> (i.e <span class="math-container">$\tau_... | Saúl RM | 807,670 | <p>The converse is true: let´s prove <span class="math-container">$\neg1\implies\neg2$</span>. If you have a set <span class="math-container">$U$</span> in <span class="math-container">$\tau_1$</span> but not in <span class="math-container">$\tau_2$</span>, you can pick a point <span class="math-container">$x\in U$</sp... |
4,010,408 | <p>good day</p>
<p>Can anybody help me? thanks</p>
<p>Let <span class="math-container">${X_i}$</span> Independent and Identically Distributed Random Variables as normals with mean <span class="math-container">$μ$</span> and standard deviation 2, What is the minimum value of n such that <span class="math-container">$P(|... | Vons | 274,987 | <p>It gives you <span class="math-container">$X\sim N(\mu, 4)$</span>, then <span class="math-container">$S\sim N(n\mu, 4n)$</span>.</p>
<p>So <span class="math-container">$$\begin{split}P(|\frac {S-n\mu}{n}|<.01)&=P(|\frac{S-n\mu}{2\sqrt n}|<.005\sqrt n)\\
&=P(|Z|<.005\sqrt n)\ge .99\end{split}$$</spa... |
406,514 | <blockquote>
<p>Find the Galois group $\operatorname{Gal}(f/\mathbb{Q})$ of the polynomial $f(x)=(x^2+3)(x^2-2)$.</p>
</blockquote>
<p>Any explanations during the demonstration, will be appreciated. Thanks!</p>
| Kortlek | 77,402 | <p>The splitting field of $f$ is $\mathbb{Q}(\sqrt{2}, i\sqrt{3})$ and since
$$[\mathbb{Q}(\sqrt{2}, i\sqrt{3}):\mathbb{Q}]=[\mathbb{Q}(\sqrt{2}, i\sqrt{3}):\mathbb{Q}(\sqrt{2})][\mathbb{Q}(\sqrt{2}):\mathbb{Q}]=2\cdot 2=4$$
the Galois group has four element.
Since every $\sigma\in$ Gal$(\mathbb{Q}(\sqrt{2}, i\sqrt{3})... |
1,182,644 | <p>The tittle says it all. I think it's true, and I tried to prove it by showing that the derivative of this function: $-2Bxe^{-Bx^2}$ is bounded from above with a bound less than 1, in order to do that, I tried to use Taylor series of $e^{-Bx^2}$, but it seems that leads nowhere. Any suggestion?</p>
<p>Here $B>0$ ... | Michael Hardy | 11,667 | <p>One can write
$$
\int x\,d(x^2) = \int x\Big( 2x\,dx\Big)=\cdots
$$
etc.</p>
<p>However, the notation
$$
\int_a^b f(x)\,dg(x) \tag 1
$$
is sometimes also taken to mean the <a href="http://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral" rel="nofollow">Riemann–Stieltjes integral</a> of $f$ with respect to $... |
2,775,443 | <p>The statement $\displaystyle \sum_{n=1}^{\infty}a_n$ converges $\implies$ $\displaystyle \sum_{n=1}^{\infty}\cfrac{1}{a_n}$ looks natural but do we have this implication? I am checking alternating series as an counter-example but could not find one yet. What can we say about the implication?</p>
| hamam_Abdallah | 369,188 | <p>If $\sum a_n $ and $\sum b_n $ are both convergent then
$$\lim_{n\to+\infty}a_nb_n=0$$</p>
<p>but here $$a_nb_n=a_n.\frac {1}{a_n}=1.$$</p>
|
502,994 | <p>If the image is f(x), why does f(|x|) look like two triangles above the x axis (basically the right side duplicated on the left)?</p>
<p><img src="https://i.imgur.com/pGIdNYZ.jpg" alt="function image"></p>
| Sudarsan | 88,391 | <p>Ok I think I should have put it in a better way like the other answers. Let $g(x)=f(|x|)$ and now $g(x)$ will look like the positive part of $f(x)$ flipped if $x<0$ since $$g(x)=\left\{ \begin{array}{ll} f(x) & x\geq0 \\ f(-x) & x<0\end{array}\right.$$
Now, $g(-2)=f(2)$ and so forth. </p>
|
1,457,478 | <p>I'm trying to teach myself some number theory. In my textbook, this proof is given:</p>
<blockquote>
<p><strong>Example (2.3.1)</strong> Show that an integer is divisible by 3 if and only if the sum of its digits is a multiple of 3.</p>
<p>Let <span class="math-container">$n=a_0a_1\ldots a_k$</span> be the decimal r... | Alexander Belopolsky | 185,688 | <blockquote>
<p>The assertion is an immediate consequence of this congruence.</p>
</blockquote>
<p>Since any power of $10$ is congruent to $1$ (mod $3$),</p>
<p>$$n=a_k+a_{k-1}10+a_{k-2}10^2+\cdots+a_010^k\equiv a_k+a_{k-1}+a_{k-2}+\cdots+a_0\pmod3.$$</p>
<p>In other words, the residual of dividing $n$ by $3$ is t... |
1,231,082 | <p>Let $\text{tr}A$ be the trace of the matrix $A \in M_n(\mathbb{R})$.</p>
<ul>
<li>I realize that $\text{tr}A: M_n(\mathbb{R}) \to \mathbb{R}$ is obviously linear (but how can I write down a <em>formal</em> proof?). However, I am confused about how I should calculate $\text{dim}(\text{Im(tr)})$ and $\text{dim}(\text... | Bernard | 202,857 | <p>Using Little Fermat, we have:
$$2^{105} +3^{105}\equiv 2^{105\bmod 6} +3^{105\bmod 6}\equiv 2^{3}+ 3^{3}\equiv 1 + 6\equiv 0 \mod 7.$$</p>
|
1,231,082 | <p>Let $\text{tr}A$ be the trace of the matrix $A \in M_n(\mathbb{R})$.</p>
<ul>
<li>I realize that $\text{tr}A: M_n(\mathbb{R}) \to \mathbb{R}$ is obviously linear (but how can I write down a <em>formal</em> proof?). However, I am confused about how I should calculate $\text{dim}(\text{Im(tr)})$ and $\text{dim}(\text... | user223741 | 223,741 | <p>$$\begin{align}
2^3 &\equiv 1 (\mod 7)\\
(2^3)^{35} &\equiv 1^{35} (\mod 7)\\
2^{105} &\equiv 1 (\mod 7)\tag1
\end{align}$$
Again,
$$\begin{align}
3^3 &\equiv -1 (\mod 7)\\
(3^3)^{35} &\equiv (-1)^{35} (\mod 7)\\
3^{105} &\equiv -1 (\mod 7)\tag2
\end{align}$$
Adding (1) and (2) we get,
$$\beg... |
344,994 | <p>Are there any decimals that times to make 1? I know that 0.25 x 4 = 1 but it's a decimal and a number. I need two decimals. Any ideas? I'm not an expert btw so... thanks</p>
| Cameron Buie | 28,900 | <p>By decimal, you seem to mean a $0$, then a decimal point, then some numbers, yes? Any such creature is positive and less than $1$. The product of two positive numbers less than $1$ will again be less than $1$.</p>
<p>Now, if you just want two non-whole numbers that multiply to $1$, then $2.5$ and $0.4$ do the job.<... |
1,549,920 | <p>Prove that
$$(1+x)^\frac{1}{x}=e-\frac{e}{2}x+\frac{11e}{24}x^2-\frac{7e}{16}x^3....$$
where e is exponenial ,
can any one give a proof...I tried with series expansion i could not get it.</p>
| Clement C. | 75,808 | <p><strong>Hint:</strong> you can use Taylor expansions, indeed.
$$\ln(1+x) = x-\frac{x^2}{2} + \frac{x^3}{3} + o(x^3)$$
when $x \to 0$; and $e^x = 1+x+\frac{x^2}{2} + \frac{x^3}{6} + o(x^3)$. Now, $$(1+x)^{\frac{1}{x}} = e^{\frac{1}{x}\ln(1+x)},$$
so composing the above two series will give you what you seek.</p>
|
1,305,661 | <p>How can I solve this Question ? </p>
<ul>
<li>Find all solutions exactly for $2\sin^2 x - \sin x = 0$</li>
</ul>
<p>my answer was $( 2k\pi,\pi+2k\pi, 11\pi/6, 7\pi/6)$
but my teacher answer was $( 2k\pi,\pi+2k\pi, \pi/6 +2k\pi, 5\pi/6 +2k\pi )$</p>
| Batman | 127,428 | <p>You have $2 \sin^2 x - \sin x =0$ or $\sin x ( 2 \sin x -1) =0$, so $\sin x = 0$ or $2 \sin x -1 = 0$.</p>
<p>$\sin x =0$ implies $x = \pi k$ for integer $k$. </p>
<p>$2 \sin x - 1 =0 $ means $\sin x = \frac{1}{2}$ which has solutions between $0, 2pi$ as $\pi/6$ and $5 \pi/6$. Since $\sin$ is $2 \pi$ periodic, yo... |
1,305,661 | <p>How can I solve this Question ? </p>
<ul>
<li>Find all solutions exactly for $2\sin^2 x - \sin x = 0$</li>
</ul>
<p>my answer was $( 2k\pi,\pi+2k\pi, 11\pi/6, 7\pi/6)$
but my teacher answer was $( 2k\pi,\pi+2k\pi, \pi/6 +2k\pi, 5\pi/6 +2k\pi )$</p>
| Community | -1 | <p>Check your solution:</p>
<p>$$\sin\left(\frac{11\pi}6\right)=\sin\left(\frac{7\pi}6\right)=-\frac12\to2\frac14+\frac12\color{red}{=0}.$$</p>
<p>And the Teacher's solution:</p>
<p>$$\sin\left(\frac{\pi}6\right)=\sin\left(\frac{5\pi}6\right)=\frac12\to2\frac14-\frac12\color{green}{=0}.$$</p>
|
587,162 | <p>I want to prove that the following holds, where the $+$ means Minkowski sum:</p>
<p>$$
conv(A+B)=conv(A)+conv(B)
$$</p>
<p>Let the convex hull of $A+B$ be
$$
conv(A+B)=\sum_{j,k}\lambda_j\mu_k(a_j+b_k)
$$</p>
<p>I don't know how to continue from here. </p>
| Community | -1 | <p>$\newcommand{\co}{\operatorname{co}}$ As already noted we have $\co (A+B) \subset \co A +\co B$. Now let $v+w \in \co A + \co B$. Write $v = \sum_i \alpha_i a_i$ and $w = \sum_j \beta_j b_j$. First we have
$$
v + b_j = \sum_i \alpha_i (a_i + b_j)
$$ so $v + b_j \in \co ( A +B)$. Now we have
$$
v+w = \sum_j \beta_j ... |
376,772 | <blockquote>
<p>Find a polynomial $q(a)$ of degree less than equal to $2$ that saitsifies the condition $q(a_0)=b_0, q'(a_0)=b'_0, \ \text{and} \ q'(a_1)=b'_1,$ where $a_0,a_1,b_0,b'_0,b'_1\in \mathbb{R}$, where $a_0\ne a_1$. And give a formula of the form $q(a)=b_0k_0(a)+b'_0k_1(a)+b'_1k_2(a).$</p>
</blockquote>
<p... | Emily | 31,475 | <p>Let $q(x) = k_0 + k_1 x + k_2 x^2 + \cdots + k_n x^n$.</p>
<p>Since the polynomial is of degree less than or equal to 2, $n \le 2$, as $n$ represents the degree of the polynomial.</p>
<p>Thus, $q(x) = k_0 + k_1x + k_2x^2$.</p>
<p>If $q(a_0) = b_0$, then $k_0+k_1 a_0 +l_2a_0^2 = b_0$. Proceed from here.</p>
|
4,033,453 | <p>How many roots does the equation <span class="math-container">$$\left\lfloor\frac x3\right\rfloor=\frac x2$$</span> have?</p>
<ol>
<li><span class="math-container">$1$</span></li>
<li><span class="math-container">$2$</span></li>
<li><span class="math-container">$3$</span></li>
<li>infinitely many</li>
</ol>
<p>I che... | hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>Put <span class="math-container">$ x=3X$</span>.
the equation becomes
<span class="math-container">$$\lfloor X\rfloor =\frac 32 X$$</span></p>
<p>which gives</p>
<p><span class="math-container">$$X-\lfloor X\rfloor =-\frac X2\in[0,1)$$</span></p>
<p>So, there are three cases
<span class=... |
3,057,200 | <blockquote>
<p>Does there exist an analytic function <span class="math-container">$f:D\rightarrow\mathbb C$</span>, where <span class="math-container">$D$</span> is a domain containing the unit disk, such that <span class="math-container">$f(z) = e^{i \text{Im} z}$</span> on the unit circle <span class="math-contain... | Community | -1 | <p>Here is an answer which only uses your hint in spirit. Ideally you should think about Greg Martin's hints before reading this, since they are of a similar spirit.</p>
<p>You know that <span class="math-container">$f(z)$</span> satisfies the relation <span class="math-container">$f(z)f(1/z) = 1$</span> on the unit c... |
2,917,024 | <p>Let $H$ be a Hilbert space and let ${e_n} ,\ n=1,2,3,\ldots$ be an orthonormal basis of $H$. Suppose $T$ is a bounded linear oprator on $H$. Then which of the following can not be true?
$$(a)\quad T(e_n)=e_1, n=1,2,3,\ldots$$
$$(b)\quad T(e_n)=e_{n+1}, n=1,2,3,\ldots$$
$$(c)\quad T(e_n)=e_{n-1} , n=2,3,4,\ldots ... | Eric Wofsey | 86,856 | <p>A bounded operator is completely determined by where it sends each $e_n$. Specifically, every element of $H$ has the form $\sum a_ne_n$ for some scalars $a_n$ (with $\sum |a_n|^2<\infty$), and then we must have $T(\sum a_ne_n)=\sum a_nT(e_n)$.</p>
<p>So, your task is to determine, in each of the three cases, wh... |
2,917,024 | <p>Let $H$ be a Hilbert space and let ${e_n} ,\ n=1,2,3,\ldots$ be an orthonormal basis of $H$. Suppose $T$ is a bounded linear oprator on $H$. Then which of the following can not be true?
$$(a)\quad T(e_n)=e_1, n=1,2,3,\ldots$$
$$(b)\quad T(e_n)=e_{n+1}, n=1,2,3,\ldots$$
$$(c)\quad T(e_n)=e_{n-1} , n=2,3,4,\ldots ... | Dionel Jaime | 462,370 | <p>You're questions been answered, but if you're curious about examples of bounded linear operators that satisfy the last two resp. Look no further than $l^2(\mathbb{N})$ and consider the left and right shift operators on the standard basis. That is </p>
<p>$$T_L( (a_1, a_2, ... a_n , ... )) = (a_2, a_3, ... a_n,\ ... |
3,873,438 | <blockquote>
<p>A particle of unit mass moves under the action of <span class="math-container">$n$</span> forces directed towards <span class="math-container">$n$</span> fixed points <span class="math-container">$A_1,A_2,...,A_n$</span>. The force towards <span class="math-container">$A_i$</span> is of magnitude <span ... | Quimey | 10,443 | <p>The first condition gives you
<span class="math-container">$$
\sum_{i=1}^n (A_i - B)k_i=0
$$</span>
and you want to compute <span class="math-container">$|\sum_{i=1}^n (A_i - C)k_i|$</span> but
<span class="math-container">$$
\sum_{i=1}^n (A_i - C)k_i = \sum_{i=1}^n (A_i - B)k_i + \sum_{i=1}^n (B - C)k_i = \sum_{i=1... |
3,877,319 | <p>I have been thinking about finding an explicit formula for the tribonacci numbers, where, namely:</p>
<p><span class="math-container">$$a_n = a_{n-1}+a_{n-2}+a_{n-3}$$</span></p>
<p>and <span class="math-container">$a_1 = 0, a_1 = 1, a_2 = 1.$</span> Obviously, these beginning terms can be shifted, but we'll leave t... | DavidW | 838,791 | <p>Please see <a href="https://www.fq.math.ca/Scanned/36-2/wolfram.pdf" rel="nofollow noreferrer">https://www.fq.math.ca/Scanned/36-2/wolfram.pdf</a> in <em>The Fibonacci Quarterly</em>, sub-section 6.2.2. That approach also considers having initial functions instead of any given initial values.</p>
<p>Changing the con... |
3,193,107 | <p>As the title says, I need to find the explicit form of the recursive sequence defined above, and I am very stuck on this.</p>
| Ross Millikan | 1,827 | <p>Hint: if you make <span class="math-container">$b_1=5$</span> and divide by <span class="math-container">$2$</span> you have a geometric series.</p>
|
737,152 | <p>Give an algebraic proof that $\binom{n+1}{m+1} = \sum_{k=m}^{n} \binom{k}{m}$.</p>
<p>I've tried using Pascal's rule and looking for a telescopic sum, but I can't find one.</p>
<p>Any help is appreciated.</p>
| Debajyoti Mondal | 434,846 | <p><a href="https://i.stack.imgur.com/ZmLnv.jpg" rel="nofollow noreferrer">Here is a geometric intuition.</a>
We started with a width of $\sqrt{n}$. At each step, the width reduced by (1/2). Hence within roughly $2\sqrt{n}$ steps we reach the base case.</p>
|
3,007,044 | <p>Sequence <span class="math-container">$a_n$</span> is defined as <span class="math-container">$a_n = \ln (1 + a_{n-1})$</span>, where <span class="math-container">$a_0 > -1$</span>. Find all <span class="math-container">$a_0$</span> for which the sequence converges.</p>
<p>Using arithmetic properties of limits, ... | hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>for <span class="math-container">$x>-1$</span>
<span class="math-container">$$\ln(1+x)\le x$$</span>
the sequence is decreasing.</p>
<p>if <span class="math-container">$-1<a_0<0$</span> the sequence is not well defined</p>
<p>if <span class="math-container">$a_0=0$</span> it... |
3,007,044 | <p>Sequence <span class="math-container">$a_n$</span> is defined as <span class="math-container">$a_n = \ln (1 + a_{n-1})$</span>, where <span class="math-container">$a_0 > -1$</span>. Find all <span class="math-container">$a_0$</span> for which the sequence converges.</p>
<p>Using arithmetic properties of limits, ... | user3482749 | 226,174 | <p>Note that we must have <span class="math-container">$a_n > 1$</span> for all <span class="math-container">$n$</span>, else <span class="math-container">$a_{n+1}$</span> is not even well defined (I'll assume this silently throughout).</p>
<p>We note that <span class="math-container">$a_n \geq a_{n+1}$</span> if a... |
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