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 |
|---|---|---|---|---|
540,830 | <p>Find the value of <span class="math-container">$$\int_0^\infty t^{x-1}e^{-\lambda t \cos(\theta)} \cos(\lambda t \sin (\theta)) dt$$</span> where <span class="math-container">$\lambda >0$</span>, <span class="math-container">$x>0$</span>, and <span class="math-container">${-1\over 2}\pi < \theta < {1\ove... | Ron Gordon | 53,268 | <p>Generally speaking, when $\Re{(a)}>0$,</p>
<p>$$\int_0^{\infty} dt \, t^{x-1} \, e^{-a t} = \frac{\Gamma(x)}{a^x}$$</p>
<p>Note that the above integral may be written as</p>
<p>$$\Re{\left [\int_0^{\infty} dt \, t^{x-1} \, e^{-\lambda t e^{i \theta}} \right ]} $$</p>
<p>Therefore, as $\cos{\theta}>0$, the ... |
2,023,222 | <p>I am facing difficulty with the following limit.</p>
<p><span class="math-container">$$ \lim_{n\to\infty}\left(\binom{n}{0}\binom{n}{1}\dots\binom{n}{n}\right)^{\frac{1}{n(n+1)}} $$</span></p>
<p>I tried to take log both sides but I could not simplify the resulting expression.</p>
<p>Please help in this regard. Than... | AlgorithmsX | 355,874 | <p>This should help simplify this problem, but I don't know how to get an exact answer and this is too long for a comment.</p>
<p>$$\prod_{k=0}^n {n\choose k}=\prod_{k=0}^n\frac{n!}{k!(n-k)!}$$
Using $\prod_{k=0}^nk!(n-k)!=(\prod_{k=0}^nk!)*(\prod_{k=0}^n(n-k)!)$ and $\prod_{k=0}^n(n-k)!=\prod_{k=0}^nk!$ and $\prod_{k... |
2,023,222 | <p>I am facing difficulty with the following limit.</p>
<p><span class="math-container">$$ \lim_{n\to\infty}\left(\binom{n}{0}\binom{n}{1}\dots\binom{n}{n}\right)^{\frac{1}{n(n+1)}} $$</span></p>
<p>I tried to take log both sides but I could not simplify the resulting expression.</p>
<p>Please help in this regard. Than... | Naren | 696,689 | <p>Let limit be denoted as <span class="math-container">$L$</span>,then <span class="math-container">$$\prod_{r=0}^{n}\binom{n}{r}= (n!)^{n+1}\left(\prod_{k=0}^n k!\right)^{-2} =(n!)^{n+1}\left(\prod_{k=0}^{n} k^{k-n-1}\right)^2= \dfrac{1}{(n!)^{n+1}}\left(\prod_{r=1}^n r^r\right)^2$$</span> Also we have the approximat... |
1,808,206 | <p>how can I find the splitting field of polynomial $x^{13}+1$ over $GF(2)$?</p>
| m.idaya | 337,282 | <p>Let $ P (X) = X ^ n-1 $, $ \Phi_n$ the n-cyclotomy polynomial over $ \Bbb{Q} $.
Let $ p $ a prime number and $F_p$ the prime field of
characteristic $ p $ and $ \Phi_ {n , F_p}$ the polynomial $
\Phi_n $ modulo $p$.</p>
<p>Link between $ \Phi_n $ and $ \Phi_ {n,F_p} $ :</p>
<p>Let $ d $ the order of $ p $ in the... |
1,417,404 | <p>The following came up in my solution to <a href="https://math.stackexchange.com/questions/1410565/can-this-congruence-be-simplified/1410579#1410579">this question</a>, but buried in the comments, so maybe it's worth a question of its own. Consider the Diophantine equation
$$ (x+y)(x+y+1) - kxy = 0$$
For $k=5$ and $... | individ | 128,505 | <p>For the equation:</p>
<p>$$(x+y)(x+y+1)=kxy$$</p>
<p>Use the standard approach of using Pell equations. If you use solutions of this equation.</p>
<p>$$p^2-k(k-4)s^2=1$$</p>
<p>Decisions can be recorded.</p>
<p>$$x=-(p^2+2kps+k(k-4)s^2)(p^2+2(k-2)ps+k(k-4)s^2)$$</p>
<p>$$y=-4ps(p^2+2kps+k(k-4)s^2)$$</p>
<p>I... |
2,578,870 | <p>$A$ is real skew symmetric matrix</p>
<p>$S$ is a positive-definite symmetric matrix</p>
<p>Prove that $\det(S) \le \det(A+S)$</p>
<p>As $S$ is diagonalizable, we can reduce the problem to :
for any real skew symmetric matrix $A$ and any diagonal matrix D with positive entries, prove that $\det(S) \le \det(D+S)... | Batominovski | 72,152 | <p><strong>Hint:</strong> You made a mistake and you should try to prove $\det(D)\leq \det(D+A)$. You may further assume that $D$ is the identity matrix. Using the fact that the eigenvalues of $A$ are $0$ or purely imaginary complex numbers that come in conjugate pairs, the claim should be now trivial.</p>
|
2,760,994 | <p><strong>I have 3 trees.</strong></p>
<ul>
<li>These particular trees are dioecious (male or female).</li>
<li>I don't know the gender of any of the trees. </li>
<li>The chance of a tree being male or female is 50/50. </li>
<li>I need at least one male and one female for successful pollination to occur.</li>
</ul>
... | User1974 | 389,017 | <p>Here is my layman's interpretation of @PrzemysławScherwentke's answer:</p>
<p><code>1</code> - <code>(1/2 * 1/2 * 1/2)</code> - <code>(1/2 * 1/2 * 1/2)</code> = <code>.75</code></p>
|
878,020 | <p>It's easy to prove that if $I$, $J$ are two-sided ideals and $R/I\cong R/J$ as modules over $R$, then $I=J$. What about left ideals? Is there a simple counterexample?</p>
<p>I believe I've found an answer, but since answering own questions is encouraged, I thought I might post it here. Other examples are obviously ... | Martin Brandenburg | 1,650 | <p>When I try to find (counter)examples, I don't consider random examples first, but rather try to simplify the general case so that, in the end, examples pop out automatically without any effort. This also works here:</p>
<p>By the universal properties of $R$ and quotient modules, a homomorphism of left $R$-modules $... |
3,624,230 | <p>An algebra, as far as I know, is closely related to a group with a family of functions being closed under addition, scalar multiplication and then the product of any two functions in the family.</p>
<p>Then there is this separate term I came across on Wikipedia called a <a href="https://en.wikipedia.org/wiki/Lie_al... | janmarqz | 74,166 | <p>Let me add that a pretty useful example is the vector space <span class="math-container">$\mathbb R^3$</span> over the field <span class="math-container">$\mathbb R$</span> where the extra operation is the cross product (also know as the vector product) which usefulness lays in the great number of geometrical constr... |
1,658,279 | <p>I'm new here so apologies if I am not clear enough. I am trying to find the zero divisors of the form $ax + b$ in $\mathbb Z_{10}$. Specifically, I need to find the values of $b$. I know that $2,4,5,6$ and $8$ are zero divisors in $\mathbb Z_{10}$ but I am not sure how to translate these into linear divisors.</p>
<... | Henry | 6,460 | <p>The model answers seem to assume that Ashmit and Amisha's abilities to answer particular questions are independent, so </p>
<ul>
<li>probability both can solve $= 0.8 \times 0.7= 0.56$</li>
<li>probability Ashmit can solve but Amisha cannot $= 0.8 \times 0.3 = 0.24$</li>
<li>probability Ashmit cannot solve but Ami... |
220,196 | <p><a href="https://i.stack.imgur.com/b2N1E.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/b2N1E.png" alt="enter image description here"></a></p>
<p>I was working on some geometric manipulation and hoping to further process this graphic's isolines however I was stumped as how best to do that when I... | J. M.'s persistent exhaustion | 50 | <p>To add to halmir's answer: if you don't need the surface, just use <code>PlotStyle -> None</code>:</p>
<pre><code>RevolutionPlot3D[{x - 0.2, -2 x}, {x, 0.7, 1}, Axes -> None, Boxed -> False,
Mesh -> {8, 4}, PlotStyle -> None]
</code></pre>
<p>which should give the same picture.</p>
... |
220,196 | <p><a href="https://i.stack.imgur.com/b2N1E.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/b2N1E.png" alt="enter image description here"></a></p>
<p>I was working on some geometric manipulation and hoping to further process this graphic's isolines however I was stumped as how best to do that when I... | Michael E2 | 4,999 | <p>Another way:</p>
<pre><code>plot /. GraphicsComplex[p_, g_, o___] :>
GraphicsComplex[p, Cases[g, _Line, Infinity]]
</code></pre>
<p><a href="https://i.stack.imgur.com/7mMSU.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7mMSU.png" alt="enter image description here"></a></p>
|
2,054,176 | <blockquote>
<p>Given $n!+(n+1)!+(n+2)!+(n+3)!$ is divisible by $181$ but $n!$ is not divisible by $181$, find three possible values for $n$.</p>
</blockquote>
| Dr. Sonnhard Graubner | 175,066 | <p>we have only two cases:
a) $$x\geq -3$$ and $$x<4$$
or
b) $$x\le -3$$ and $$x>4$$ and this is impossible.
Thus we have $$-3\le x<4$$</p>
|
4,068,216 | <p>I know that the sequence <span class="math-container">$\sqrt[n]{n}$</span> converges to 1 and that <span class="math-container">$\text{log}(\sqrt[n]{n})$</span> thus converges to 0 as <span class="math-container">$n\to\infty$</span> since the logarithmic function is continuous. But how can I calculate the limits as ... | Futurologist | 357,211 | <p><a href="https://i.stack.imgur.com/aXnfT.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/aXnfT.png" alt="enter image description here" /></a></p>
<p>If you construct <span class="math-container">$P^*$</span> and <span class="math-container">$A^*$</span> so that <span class="math-container">$$\,\,\... |
260,512 | <p>I am stuck with the following problem:</p>
<p>Let $A$ be a $3\times 3$ matrix over real numbers satisfying $A^{-1}=I-2A.$<br>
Then find the value of det$(A).$</p>
<p>I do not know how to proceed. Can someone point me in the right direction? Thanks in advance for your time.</p>
| pritam | 33,736 | <p>As pointed out in other answers, no such real matrix exist. For complex matrix $A$, note that $$I=2AA^{-1}=A(I-2A)=A-2A^2$$ Now let us assume $A$ is diagonal with all the diagonal entries equal to $a$, then each diagonal entry of $A-2A^2$ equals $a-2a^2$. Now from the equation $a-2a^2=1$, we have $a=(1\pm\sqrt{7}i)... |
995,551 | <p>How would one calculate the log of a number where the base isn't an integer (in particular, an irrational number)? For example:</p>
<p>$$0.5^x = 8 \textrm{ (where } x = -3\textrm{)}$$</p>
<p>$$\log_{0.5}8 = -3$$</p>
<p>How would you solve this, and how would this work for an irrational base (like $\sqrt{2}$)?</p>... | Dr. Sonnhard Graubner | 175,066 | <p>rewrite your equation in the form $2^{-x}=2^{3}$</p>
|
2,332,078 | <blockquote>
<p>Why $kerT=Span\{\underline 0\}$ if linear map $T: \mathbb{F}_3[x]\to \mathbb{F}_3[x]$ is defined as $T(p(x))=p(-x)$?</p>
</blockquote>
<p>$\mathbb{F}_3[x]$ is a space of polynomials of form $ax^2+bx+c$.</p>
<p>The solution I saw finds $\text{ker}T$ as follows:</p>
<p>first $p(x)=ax^2+bx+c \Rightarr... | Fred | 380,717 | <p>If $p \in ker(T)$ and $q$ defined py $q(x)=p(-x)$, then $q(x)=0$ for all $x$. Hence $p(-x)=0$ for all $x$. Thus $p=\underline 0.$</p>
|
2,332,078 | <blockquote>
<p>Why $kerT=Span\{\underline 0\}$ if linear map $T: \mathbb{F}_3[x]\to \mathbb{F}_3[x]$ is defined as $T(p(x))=p(-x)$?</p>
</blockquote>
<p>$\mathbb{F}_3[x]$ is a space of polynomials of form $ax^2+bx+c$.</p>
<p>The solution I saw finds $\text{ker}T$ as follows:</p>
<p>first $p(x)=ax^2+bx+c \Rightarr... | Martin Argerami | 22,857 | <p>The other answers mix the concept of polynomial with <em>polynomial function</em>. These concepts agree for many fields (in particular the usual ones, $\mathbb R$, etc.) but not for all.</p>
<p>What happens in the proof you were given is simply that the zero polynomial is the polynomial with all coefficients equal ... |
1,036,684 | <p>There is an inequality:</p>
<p>$$\sqrt[n]{\prod_{i = 1}^{n}{(a_i+b_i)}} \geq \sqrt[n]{\prod_{i = 1}^{n}{a_i}} + \sqrt[n]{\prod_{i = 1}^{n}{b_i}}$$</p>
<p>which I even don't know its name. I'd like to have an ask of its name and usage.</p>
| Community | -1 | <p>This is <a href="http://en.wikipedia.org/wiki/Mahler%27s_inequality" rel="nofollow noreferrer">Mahler's inequality</a>. We had it here in <a href="https://math.stackexchange.com/q/29357">this question</a> (see also the linked questions), and it was question A2 on the 2003 Putnam, so you can find some proofs in <a h... |
3,356,468 | <p>Let <span class="math-container">$\pi:E\rightarrow M$</span> be a smooth vector bundle. Let <span class="math-container">$S:M\rightarrow E$</span> be it's zero section. Let <span class="math-container">$M'=E-S(M)$</span>.</p>
<p><strong>Is <span class="math-container">$M'$</span> a smooth submanifold of <span clas... | Noah Riggenbach | 482,732 | <p>An open subset of a manifold is a manifold, just take the intersections with the charts.</p>
|
2,360,117 | <p>Let $\xi:\mathbb{R}^3\rightarrow \mathbb{R}$ be a function $\xi(x,y,z)$. I have the following PDE:
$$
\cos(\xi)\partial_y\xi = \sin(\xi)\partial_x\xi
$$
which is equivalent to $\tan(\xi)=\xi_y/\xi_x$.</p>
<p>Clearly, $\xi_x = \xi_y = 0$ (where $\partial_i\xi=\xi_i$) is a solution, which would imply $\xi(x,y,z) = \e... | Lutz Lehmann | 115,115 | <p>Characteristic curves of $0=\sin(ξ)ξ_x-\cos(ξ)ξ_y$ follow the direction field
$$
\frac{dx}{\sin(ξ)}=\frac{dy}{-\cos(ξ)}=\frac{dz}{0}=\frac{dξ}{0}
$$
which means that $\cos(ξ)x+\sin(ξ)y=c_1$, $z=c_2$, $ξ=c_3$ are constant along these curves and the curves are completely characterized by these first integrals. As ther... |
2,360,117 | <p>Let $\xi:\mathbb{R}^3\rightarrow \mathbb{R}$ be a function $\xi(x,y,z)$. I have the following PDE:
$$
\cos(\xi)\partial_y\xi = \sin(\xi)\partial_x\xi
$$
which is equivalent to $\tan(\xi)=\xi_y/\xi_x$.</p>
<p>Clearly, $\xi_x = \xi_y = 0$ (where $\partial_i\xi=\xi_i$) is a solution, which would imply $\xi(x,y,z) = \e... | JJacquelin | 108,514 | <p>$$
\cos(ξ)\frac{\partial ξ}{\partial y}- \sin(ξ)\frac{\partial ξ}{\partial x}=0 \tag 1
$$</p>
<p>With the method of characteristics, the set of ODEs for the characteristic curves is :
$$\frac{dy}{\cos(ξ)}=\frac{dx}{-\sin(ξ)}=\frac{dz}{0}=\frac{dξ}{0} \quad\implies\quad dξ=0\quad\text{and}\quad dz=0$$
A first family... |
2,220,032 | <p>Quoting:</p>
<p>"Let H be the subgroup of S3 defined by the permutations
{(1); (123); (132)}. The left cosets of H are</p>
<p>(1)H = (123)H = (132)H = {(1); (123); (132)}</p>
<p>(12)H = (13)H = (23)H = {(12); (13); (23)} "</p>
<p>I am a bit stock here, I am not understanding the meaning of the first permutation ... | Derek Elkins left SE | 305,738 | <p>The first result is covered (in dual and enriched form) in <a href="http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html" rel="nofollow noreferrer">Basic Concepts of Enriched Category Theory</a> as Theorem 4.81 which is a corollary to Theorem 4.80 which I prove below (in the unenriched case), the easily prove... |
93,325 | <p>Is there a formula for solving problems such as: If there are n coin denominations x1,x2...xn that total p cents, what is the possible number of combinations of coins that total less than or equal to p.? Where n and p are positive real numbers, of course.</p>
<p>On a side note, wikipedia says: "There is an explicit... | Boris Novikov | 18,814 | <p>You can find them in the book:</p>
<p>J.L. Ramirez Alfonsin.
The Diophantine Frobenius Problem.
Oxford Univ. Press 2005</p>
|
93,325 | <p>Is there a formula for solving problems such as: If there are n coin denominations x1,x2...xn that total p cents, what is the possible number of combinations of coins that total less than or equal to p.? Where n and p are positive real numbers, of course.</p>
<p>On a side note, wikipedia says: "There is an explicit... | Igor Rivin | 11,142 | <p>Presumably the algorithm to compute the number of ways to change exactly $p$ cents (call it $n_p$) is to write down the generating function:</p>
<p>$G(z) = \prod_{i=1}^\infty \dfrac{1}{1-z^{x_i}},$ then $n_p$ is then simply the coefficient of $z^p.$ To get the answer to the question you are asking, just sum the coe... |
522,702 | <p>Let $f(x)\colon \Bbb R\to\Bbb R$ be a bounded function satisfying the following condition:</p>
<p>$$f(x+\frac{13}{42})+f(x)=f(x+\frac16)+f(x+\frac17), \forall x\in\Bbb R$$</p>
<p>Is the function $f(x)$ periodic?</p>
| Ewan Delanoy | 15,381 | <p>The answer is YES.
Put $g(x)=f(\frac{x}{42})$. Since
$f(\frac{x}{42}+\frac{13}{42})+f(\frac{x}{42})=
f(\frac{x}{42}+\frac16)+f(\frac{x}{42}+\frac17) $, we see that
for any $x$,</p>
<p>$$
g(x+13)=g(x+6)+g(x+7)-g(x) \tag{1}
$$</p>
<p>Let $h(x)=g(x+7)-g(x)$. Then (1) says that $h(x+6)=h(x)$. Now for
$x\in{\mathbb R}... |
271,078 | <p>Please guys i need help with this limit:</p>
<p>$$\lim_{n \to \infty} \left(\frac {1}{\sqrt{n^2+1}}+ \frac{1}{\sqrt{n^2+2}}+\dots +\frac{1}{\sqrt{n^2+2n}}\right)$$</p>
<p>I don't know what to do?</p>
| Community | -1 | <p><strong>HINT</strong></p>
<p>Note that $$\underbrace{\dfrac{2n}{n+1} < \dfrac{2n}{\sqrt{n^2+2n}}}_{n^2 +2n < (n+1)^2} \leq \sum_{k=1}^{2n} \dfrac1{\sqrt{n^2+k}} \leq \underbrace{\dfrac{2n}{\sqrt{n^2+1}} < \dfrac{2n}{\sqrt{n^2}}}_{n^2 < n^2+1} = 2$$</p>
<p><strong>EDIT</strong></p>
<p>Make use of the <... |
283,926 | <p>Let $A$ be a given symmetric positive definite $N\times N$ matrix. I need to find a symmetric positive semi-definite matrix $S$ which is the solution to the following optimization problem
\begin{align}
\max_{S}~&\det(A+S) \\s.t.~&\sum_{i}^{N}\sigma_i(S)\,=\,c \\&S\geq0
\end{align}where $\sigma_i(S)$ are ... | mikitov | 11,825 | <p>See the discussion of Schur-convex/concave of the determinant function in </p>
<p>D. P. Palomar, J. M. Cioffi and M. A. Lagunas, "Joint Tx-Rx beamforming design for multicarrier MIMO channels: a unified framework for convex optimization," in IEEE Transactions on Signal Processing, vol. 51, no. 9, pp. 2381-2401, Sep... |
283,926 | <p>Let $A$ be a given symmetric positive definite $N\times N$ matrix. I need to find a symmetric positive semi-definite matrix $S$ which is the solution to the following optimization problem
\begin{align}
\max_{S}~&\det(A+S) \\s.t.~&\sum_{i}^{N}\sigma_i(S)\,=\,c \\&S\geq0
\end{align}where $\sigma_i(S)$ are ... | Igor Rivin | 11,142 | <p>Your constraints are linear on the entries of $S,$ since in your case the singular values are equal to the eigenvalues (since $S$ is symmetric positive definite) and so their sum is the trace. I assume the last inequality $S\geq 0$ is elementwise.
Log determinant is a convex function, and so your problem is a box-s... |
2,600,283 | <p>I think $f(x) = x^2$. Then $f'(0)$ should be $0$.</p>
<p>But when I try to calculate the derivative of $f(x) = |x|^2$, then I get:</p>
<p>$f'(x) = 2|x| \cdot \frac{x}{|x|}$, which is not defined for $x = 0$. Does $f'(0)$ still exist?</p>
| Ángel Mario Gallegos | 67,622 | <p><strong>Hint:</strong></p>
<p>Since $|x|=\begin{cases}x,&x\ge 0\\-x,&x<0\end{cases}$ it follows that</p>
<p>$$f(x)=\begin{cases}x^2,&x\ge 0\\(-x)^2,&x<0\end{cases}$$</p>
<p>and then, $f(x)=x^2$, for $x\in \mathbb R$.</p>
|
1,103,217 | <p>I know what does set of generators mean and that subset of $G$ generates some subgroup. But I have no idea how to prove the statement above in title. It's like comes from definition and there is nothing to prove.</p>
| Mihail | 201,204 | <p>I'm answering to my own question and I'd like to know if it's acceptable. By definition generator set $H$ generates a group $G$ if $H\subset G$ and every element in $G$ equal to some finite product of elements of $H$ and inverses of these elements. So it is only necessary to show that $x^{-1}$ can be expressed as pr... |
726,650 | <p>a) How does $x^2y''-3xy'+3y=0$ can be solved? I know how to solve for constant coefficients, but in this case they are functions...</p>
<p>b) In which maximum interval there is a solution that confirms $y'(1)=6, y(1)=4$?</p>
| Mariano Suárez-Álvarez | 274 | <p>An equation of the form $ax^2u''+bxu'+cu=0$ can be rewritten in terms of the operator $D=x\frac{d}{dx}$: indeed, we have $$ax^2u''+bxu'+cu=aD^2u+(b-a)Du+cu.$$</p>
<p>The right hand side is the result of applying the operator $aD^2+(b-a)D+c$ to $u$. If we are able to factor $aD^2+(b-a)D+c=\alpha(D-\beta)(D-\gamma)$,... |
1,648,460 | <p>Suppose that $A$ is an m by n matrix and is right invertible, such that there exists and an n by m matrix $B$ such that $AB = I_m.$ Prove that $m\leq n.$ </p>
<p>I'm not really sure how go about this problem; any help would be appreciated.</p>
| Steve Kass | 60,500 | <p>One way to see your error is to note that the circumference of a circle is measured in (say) inches, and the surface area of a sphere is measured in square inches. It can’t work out that adding up inches gives you square inches. Your approach is not useless, however.</p>
<p>Each circle you’re using contributes an i... |
642,863 | <p>Imagine there's a quiz on the internet intended for a wide audience. It contains a (unlimited) number of questions, all of them with yes/no answers. A person gets one random question and must answer it, after that he can get another one. He can continue answering any number of questions he wants. So the only data yo... | Jack M | 30,481 | <p>Here's a more naive, elementary answer that might be useful. Note that I admit to being bad at both probability and combinatorics, so you probably shouldn't accept any of this without double checking the calculations.</p>
<p>The first issue is to define what you mean by "winner". Perhaps we can assume that, as a pe... |
2,032,923 | <p>I want to rearrange the formula for angular velocity $\omega = \dfrac{2\pi}{T}$, to make $T$ the subject as I wish to find the period.</p>
<p>Would the correct answer be $T = \frac{\omega}{2\pi}$ or would it be $T = \frac{2\pi}{\omega}$? </p>
<p>And is there a certain rule you should follow when rearranging ?</p>
| Olivier Oloa | 118,798 | <p>From
$$
\frac{a}{b}=\frac{c}{d} \tag1
$$ by multiplying out $(1)$ by $bd$ one gets
$$
ad=bc\tag2
$$ by dividing $(2)$ by $cd$ one gets
$$
\frac{a}{c}=\frac{b}{d}. \tag3
$$ Applying it to
$$
w=\frac{2\pi}{T}
$$ one gets
$$
T=\frac{2\pi}{w}.
$$</p>
|
2,032,923 | <p>I want to rearrange the formula for angular velocity $\omega = \dfrac{2\pi}{T}$, to make $T$ the subject as I wish to find the period.</p>
<p>Would the correct answer be $T = \frac{\omega}{2\pi}$ or would it be $T = \frac{2\pi}{\omega}$? </p>
<p>And is there a certain rule you should follow when rearranging ?</p>
| Community | -1 | <p>Starting with $$\omega = \dfrac{2\pi}{T}$$ First multiply both sides by $T$:</p>
<p>$$\omega\color{red}{\cdot T} = \dfrac{2\pi}{T}\color{red}{\cdot T} = 2\pi$$</p>
<p>Divide both sides by $\omega$ to isolate $T$ on the left:</p>
<p>$$\color{red}{\frac{\color{black}{\omega T}}{\omega}} = \color{red}{\frac{\color{b... |
1,329,374 | <p>I'm looking for an explicit example of a BVP for a second order ODE: </p>
<blockquote>
<p>$y''+p(x)y'+q(x)y=f(x)$ (where $\,0\leq x\leq L\,$ and $\,y(0)=\alpha\,$ $\,y(L)=\beta$).</p>
</blockquote>
<p>If you also have the exact solution, the better. The reason is for test purposes, I've just finished a Mathemati... | Malcolm | 247,665 | <p>Like I said in my comment, what we would like to do is use the equality</p>
<p><em>Area of Big Triangle = Area of Little Triangle + Area of Trapezium</em></p>
<p><em>Area of Big Triangle</em> = $\frac {\sqrt 3} 4 a^2$</p>
<p><em>Area of Little Triangle</em> = $\frac {\sqrt 3} 4 (a-\alpha).b $</p>
<p><em>Area of ... |
1,329,374 | <p>I'm looking for an explicit example of a BVP for a second order ODE: </p>
<blockquote>
<p>$y''+p(x)y'+q(x)y=f(x)$ (where $\,0\leq x\leq L\,$ and $\,y(0)=\alpha\,$ $\,y(L)=\beta$).</p>
</blockquote>
<p>If you also have the exact solution, the better. The reason is for test purposes, I've just finished a Mathemati... | Harish Chandra Rajpoot | 210,295 | <p>Problem 1. Let $x$ be the base of the small triangle & the side of trapezium then using simple geometry, we have $$x=a-a\left(\frac{\frac{\alpha\sqrt{3}}{2}}{\frac{a\sqrt{3}}{2}}\right)\implies \color{blue}{x=a-\alpha}$$ Now we have $$\text{area of trapezuim}=\frac{1}{2}(\text{sum of lengths of parallel sides})\... |
4,195,841 | <p>V is a finite-dimensional <span class="math-container">$\mathbb{Q}$</span>- vector space with <span class="math-container">$\phi: V \rightarrow V$</span></p>
<p>Why does it follow that <span class="math-container">$\phi$</span> is diagonalizable if <span class="math-container">$\phi \circ \phi = id_{V}$</span>?</p>
... | user0 | 389,981 | <p>We have<span class="math-container">\begin{align}
\phi\circ\phi & = id_V\\
\phi\circ\phi - id_V & = 0\\
(\phi + id_V)(\phi - id_V) & = 0,\\
\end{align}</span>
so the minimal polynomial for <span class="math-container">$\phi$</span> divides <span class="math-container">$(x + 1)(x - 1)$</span>. Have you le... |
19,876 | <p>Very important in integrating things like $\int \cos^{2}(\theta) d\theta$ but it is hard for me to remember them. So how do you deduce this type of formulae? If I can remember right, there was some $e^{\theta i}=\cos(\theta)+i \sin(\theta)$ trick where you took $e^{2 i \theta}$ and $e^{-2 i \theta}$. While I am draf... | Community | -1 | <p>For Proving $\sin(\alpha+\beta)=\sin\alpha\cdot \cos\beta + \cos\alpha \cdot \sin\beta$ you can see this link:</p>
<ul>
<li><a href="http://www.math.wisc.edu/~leili/teaching/math222s11/problems/quizzes/trig.pdf" rel="nofollow">http://www.math.wisc.edu/~leili/teaching/math222s11/problems/quizzes/trig.pdf</a></li>
</... |
3,506 | <p>I am wondering what is the correct function in Mathematica to plot the true impulse function, better known as the <code>DiracDelta[]</code> function. When using this inside of a function or just the function itself when plotting, it renders output = zero. Quick example:</p>
<pre><code>Plot[DiracDelta[x], {x,-1,1}]
... | Francisco Rodríguez Fortuño | 30,096 | <p>Building on Sjoerd C. de Vries's solution, I wrote this little function to make the task easy for anyone interested in plotting Dirac's Delta as arrows in a Plot command in Mathematica. The function receives an equation and a variable, and returns a list of {Arrow[]}'s in the correct location of the Diract Deltas, t... |
6,695 | <p>The standard approach for showing <span class="math-container">$\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$</span> is to multiply by <span class="math-container">$\dfrac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$</span> and then do a substitution with <span class="math-cont... | omegadot | 128,913 | <p>Here is another variation on a theme. It relies on the following two double angle formulae for sine and cosine, namely
<span class="math-container">$$\sin 2\theta = 2 \sin \theta \cos \theta \qquad \text{and} \qquad \cos 2 \theta = \cos^2 \theta - \sin^2 \theta,$$</span>
two obvious substitutions, and a simple parti... |
1,575,253 | <p>I need to prove that if $A$ is an $n \times n$ matrix, then $\lambda $ is an eigenvalue of $A$ if and only if $\lambda^k $ is an eigenvalue of $A^k$ for any positive integer $k \geq 1$. I am assuming $\lambda \in \mathbb R$. Otherwise I think the propositions is false.</p>
<p>The first part ($\implies$) is very eas... | StephenG - Help Ukraine | 298,172 | <p>The eigenvalues and vectors of <span class="math-container">$A$</span> are related by :</p>
<p><span class="math-container">$$Au_k=\lambda_k u_k$$</span></p>
<p>For any <span class="math-container">$n$</span> a positive integer greater than one, we can say :</p>
<p><span class="math-container">$$A^nu_k=A^{n-1}Au_k=A... |
835,376 | <p>Is it possible for two vector functions of, for the moment's simplicity, one variable be both independent <strong>and</strong> dependent?</p>
<p>The reason I'm asking this is because on a problem from a book of mine (not homework), they put the following exercise:</p>
<p><em>Let $x^{(1)}(t)=\left (\begin{array}{cc... | Git Gud | 55,235 | <p>Bad notation is bad....</p>
<blockquote>
<p>Show that $x^{(1)}(t)$ and $x^{(2)}(t)$ are linearly dependent at each point in the interval $0 ≤ t ≤ 1$.</p>
</blockquote>
<p>What you're being asked to prove here is that given $t\in [0,1]$, the vectors $x^{(1)}(t)$ and $x^{(2)}(t)$ are linearly dependent. There is n... |
280,436 | <p>For example, if we evaluate this:</p>
<pre><code>BSplineFunction@{{0,100},{200,50},{200,0}}
</code></pre>
<p>we'll get</p>
<pre><code>BSplineFunction[1,
{{0., 1.}},
{2}, {False}, {{{0., 100.}, {200., 50.}, {200., 0.}}, Automatic},
{{0., 0., 0., 1., 1., 1.}},
{0}, MachinePrecision, "Unevaluated&q... | Domen | 75,628 | <p>This is the full internal representation of <code>BSplineFunction</code> with all relevant parameters. You can fiddle around with the options, then open the information box and compare the values to find the correspondence.</p>
<pre><code>pts = {{0, 100}, {200, 50}, {200, 0}, {300, 50}};
BSplineFunction[pts, SplineC... |
1,221,487 | <p>Problem: Let V be the subspace of all 2x2 matrices over R, and W the subspace spanned by:</p>
<p>\begin{bmatrix}
1 & -5 \\
-4 & 2 \\
\end{bmatrix}
\begin{bmatrix}
1 & 1 \\
-1 & 5 \\
\end{bmatrix}
\begin{bmatrix}
2 & -4 \\
-5 & 7 \\
\end{bmatrix}
\begin{bmatrix}
1 & -7 \\
-5 & 1 \\
\e... | Rellek | 228,621 | <p>The only reason I can see immediately is that, in base 10, of course, every number can simply be represented as $$\sum a_{n}10^{n-1}$$ where $a_{n}$ is the nth digit. Since $10^n \equiv 1 \bmod(9)$ for any positive integer n, taking the number modulo 9 is the same exact thing as adding its digits. If one of the digi... |
18,844 | <p>To elaborate a bit, I should say that the question of the existence of a complete metric is only of interest in the case of manifolds of infinite topological type; if a manifold is compact, any metric is complete, and if a noncompact manifold has finite topological type(ie is diffeomorphic to the interior of a compa... | AlexE | 13,356 | <p>We can even prove more: every smooth manifold can be equipped with a Riemannian metric of bounded geometry (positive injectivity radius, bounded curvature tensor and all derivatives of the curvature tensor are also bounded).</p>
<p>This is Theorem 2' in "R. E. Greene, Complete metrics of bounded curvature on noncom... |
56,942 | <p>If $ x $ and $ y $ have $ n $ significant places, how many significant places do $ x + y $, $ x - y $, $ x \times y $, $ x / y $, $ \sqrt{x} $ have?</p>
<p>I want to evaluate expressions like $ \frac{ \sqrt{ \left( a - b \right) + c } - \sqrt{ c } }{ a - b } $ to $ n $ significant places, where $ a $, $ b $, $ c $ ... | André Nicolas | 6,312 | <p>Let us look at your specific example, because it is very interesting. We want to evaluate
$$\frac{\sqrt{(a-b)+c}-\sqrt{c}}{a-b},$$
where $a$, $b$, and $c$ are integers. There can be serious loss of precision if $c$ is huge and $a$ and $b$ are very close to each other.</p>
<p>However, there is a straightforward wo... |
2,752,646 | <p>$\lim_{n \rightarrow \infty}$ $ n^{b}/a^{n} $ </p>
<p>I have tried to approach it using L Hopital but it is not working. Maybe using sandwich could work but i cant think of the function to enclose it </p>
| farruhota | 425,072 | <p>The cases:
$$\lim_\limits{n\to+\infty} \frac{n^b}{a^n}=\begin{cases} 0,\ \ \ \ \ \ \ a>1 \\
+\infty, \ \ a=1,b>0 \\
1, \ \ \ \ \ \ \ a=1,b=0 \\
0, \ \ \ \ \ \ \ a=1,b<0 \\
+\infty, \ \ 0<a<1 \\
+\infty, \ \ a=0,b\ne0 \\
\emptyset, \ \ \ \ \ \ \ a=0,b=0
\end{cases}$$</p>
|
89,675 | <p>Given a geometric series, how would you do this?</p>
<p>For example, how would this be done if the geometric series in question as is as follows?:</p>
<p>$$ \frac{1}{(1 - (-x^2))}$$</p>
| Bardia | 14,872 | <p>suppose that $a_n$ is a geometric series i.e. the sequence is:
$$a_0 ,\quad a_1=a_0\times q,\quad a_2=a_1\times q = a_0 \times q^2,\quad \cdots , \quad a_n=a_0 \times q^n$$</p>
<p>The summation of $n$ elements of this sequence is:
$$\sum_{i=0}^{n}a_i=a_0+a_1+a_2+\cdots+a_n=$$
$$a_0+a_0.q+a_0.q^2+\cdots+a_0.q^n=$$
$... |
17,429 | <p>I want to define my own little 'Inner Product' function satisfying properties of linearity and commutativity, and I'd like to use the "$\langle$" and "$\rangle$" symbols to output my results. For this I am using <code>AngleBracket</code> which has no built-in meaning.</p>
<p>I was able to use <code>SetAttributes[A... | Michael E2 | 4,999 | <p>If I set</p>
<pre><code>SetAttributes[AngleBracket, Orderless];
AngleBracket[a_?NumericQ u_, v_] := a AngleBracket[u, v];
</code></pre>
<p>then I get</p>
<pre><code>2 AngleBracket[u, 6 v] == AngleBracket[3 u, 4 v]
(* -> True *)
</code></pre>
|
4,341,404 | <p>Choose a point randomly on the interval <span class="math-container">$[0, 1]$</span> and label it <span class="math-container">$X_1$</span>.</p>
<p>Then
choose a point randomly on the interval <span class="math-container">$[0, X_1]$</span> and label it <span class="math-container">$X_2$</span>.</p>
<p>Finally,
choos... | Edil | 1,137,414 | <p><span class="math-container">$\newcommand{\abstraction}[2]{\lambda #1. #2}$</span>
<span class="math-container">$\newcommand{\application}[2]{\left(#1 #2\right)}$</span> <span class="math-container">$\newcommand{\substitution}[3]{#1 \left[#2 := #3\right]}$</span>
<span class="math-container">$\newcommand{\freevars}[... |
1,859,741 | <p>How do I prove that</p>
<p>$$\sqrt{20+\sqrt{20+\sqrt{20}}}-\sqrt{20-\sqrt{20-\sqrt{20}}} \approx 1$$</p>
<p>without using the calculator?</p>
| mweiss | 124,095 | <p>Use the classical approximation:
$$\sqrt{a^2 + b} \approx a + \frac{b}{2a}$$
With $a = \sqrt{20}$ and $b = \sqrt{20 + \sqrt{20}}$ we have
$$\sqrt{20 + \sqrt{20 + \sqrt{20}}} \approx \sqrt{20} + \frac{\sqrt{20 + \sqrt{20}}}{2\sqrt{20}} = \sqrt{20} + \frac{\sqrt{400 + 20\sqrt{20}}}{40} $$
Now use the same classical ap... |
1,859,741 | <p>How do I prove that</p>
<p>$$\sqrt{20+\sqrt{20+\sqrt{20}}}-\sqrt{20-\sqrt{20-\sqrt{20}}} \approx 1$$</p>
<p>without using the calculator?</p>
| fleablood | 280,126 | <p>Since Alex Meiburg didn't write an answer I'm going to steal his idea:</p>
<p>(Oh, I see he did after all... oh, well, I'm still going to steal his answer.)</p>
<p>$\sqrt{20 + \sqrt{20 + \sqrt{20}}} =$</p>
<p>$\sqrt{20 + \sqrt{20 + \sqrt{25*4/5}}}=$</p>
<p>$\sqrt{20 + \sqrt{20 + 5\sqrt{4/5}}}=$</p>
<p>$\sqrt{20... |
1,083,801 | <p>Let's define a stuttering sequence the following way :</p>
<p>Let $q\in\mathbb{N}^*,E_q=\{1,2,\dots,q\}$ and $(u_n)\in (E_q)^\mathbb{N}$.</p>
<p><strong>$(u_n)$ is a stuttering sequence of order $k$ with spacing $w$ iff $$\exists n,w\in\mathbb{N},\exists k \in \mathbb{N}^*,\forall i\in\{0,1,\dots,k-1\},u_{n+i}=u_{... | Asinomás | 33,907 | <p>Question $1$: The answer is no since the sequence $21212121\dots$ is not $1$ stuttering with $k=1$.</p>
<p>Question $2$: The answer is no, given any $k$ we can find a sequence using only $1$ and $2$ that is not $k$ stuttering with spacing $0$.</p>
<p>Proof: Take the sequence $\underbrace{00\dots0}_\text{k zeroes}1... |
1,017,026 | <blockquote>
<p><strong>Cauchy-Schwarz Inequality:</strong></p>
<p>If <span class="math-container">$\textbf{u}$</span> and <span class="math-container">$\textbf{v}$</span> are vectors in a real inner product space <span class="math-container">$V$</span>, then <span class="math-container">$$|\left\langle\textbf{u},\text... | peterwhy | 89,922 | <p>$\arctan y = atan2(y,1)$, using the definition at <a href="http://en.wikipedia.org/wiki/Atan2#Definition_and_computation" rel="nofollow">Wikipedia</a>.</p>
|
3,550,293 | <blockquote>
<p>The following are given:
<p><span class="math-container">$X$</span> is a discrete random variable
<p> The probability mass function given is <span class="math-container">$P(X=k)=Clnk$</span>
<p><span class="math-container">$k=e$</span>,<span class="math-container">$e^2$</span>,<span class="math-... | J.G. | 56,861 | <p>The formula for <span class="math-container">$\tan(A\pm B)$</span> is obtained by dividing the formula for <span class="math-container">$\sin(A\pm B)$</span> by the one for <span class="math-container">$\cos(A\pm B)$</span>, cancelling a factor of <span class="math-container">$\cos A\cos B$</span> from each to obtai... |
3,378,104 | <blockquote>
<p>Let <span class="math-container">$\mathcal{F} \subseteq \mathcal{G}$</span>.</p>
<p>Show that <span class="math-container">$$E((E(X\mid \mathcal{G})-E(X\mid \mathcal{F}))^2=E(E(X\mid \mathcal{G}))^{2}-E(E(X\mid \mathcal{F}))^{2}$$</span></p>
</blockquote>
<p>My idea:</p>
<p><span class="math-container">... | Dr. Sonnhard Graubner | 175,066 | <p>Write your term in the form
<span class="math-container">$$\frac{xy}{x^2+y^2}\times(x^2+y^2)\ln(x^2+y^2)$$</span> Now use that
<span class="math-container">$$\lim_{x\to 0^+ }x\ln(x)=...$$</span>=</p>
|
3,378,104 | <blockquote>
<p>Let <span class="math-container">$\mathcal{F} \subseteq \mathcal{G}$</span>.</p>
<p>Show that <span class="math-container">$$E((E(X\mid \mathcal{G})-E(X\mid \mathcal{F}))^2=E(E(X\mid \mathcal{G}))^{2}-E(E(X\mid \mathcal{F}))^{2}$$</span></p>
</blockquote>
<p>My idea:</p>
<p><span class="math-container">... | Who am I | 687,026 | <p>You can say when x and y, both goes to zero or very near to 0 then X is approximately equal to y.
So replace y with x then take limit as x tends to 0</p>
|
2,324,692 | <p>The Dual Group of $\mathbb{R}$ is isomorphic to $\Bbb{R}$ itself in the following way:
The map
$$\Bbb{R} \to \hat{\Bbb{R}}, \quad y \mapsto \exp(ixy) $$
is an isomorphism. Further it is stated in the literature that this map is also an homeomorphism. See for exmaple Conway, A course in functional analysis Theorem 9.... | lab bhattacharjee | 33,337 | <p>Another way:</p>
<p>$$\cos(3\cdot2x)=4\cos^3(2x)-3\cos2x$$</p>
<p>Now use $\cos2x=2\cos^2x-1$</p>
|
763,295 | <p>Show that there is no subgroup of $\mathbb{Z}_4$ containing only 3 elements.</p>
<p>I couldn't solve why 3 elements cannot exist.</p>
<p>0 and 2 are the only subgroup of $\mathbb{Z}_4$ with 2 elements. But 3 elements??</p>
| Andreas Blass | 48,510 | <p>If Lagrange's theorem is available, then Evan W's answer is exactly what you want, but if you don't (yet) have Lagrange's theorem, then you can do this small example by hand; after all, $\mathbb Z/4$ has only four subsets of 3 elements each. Whichever of the four elements of $\mathbb Z/4$ you omit in order to form a... |
4,066,653 | <p>We are given two non-intersecting circles centered at <span class="math-container">$O_{1}$</span> and <span class="math-container">$O_{2}$</span>. For simplicity sake, let <span class="math-container">$O_{1} = (0,0)$</span> and <span class="math-container">$r_{1} = r_{2} = 10$</span>. Let <span class="math-containe... | Cesareo | 397,348 | <p>Calling</p>
<p><span class="math-container">$$
\cases{
p_1 = (0,0)\\
p_2 = (27,23)\\
r_1 = 10\\
r_2 = 10\\
p=(x_1,y_1)\\
q=(x_2,y_2)\\
p_c=(y_1,x_2)
}
$$</span></p>
<p>we have the conditions</p>
<p><span class="math-container">$$
\mathcal{R(x_1,y_1,x_2,y_2)}=\cases{
\|p-p_1\|^2=r_1^2\\
\|q-p_2\|^2=r_2^2\\
(p-p_c)\cd... |
4,536,813 | <p>In my textbook, a historical motivation for the development of differentiation is given, starting with Fermat trying to find the maxima and minima of functions. What I wanted to ask is why Fermat was interested in this problem in the first place, or to ask a more mathematical and less historical question, what some ... | David | 651,991 | <p>Finding the extrema of a function can be useful in pretty much any context where you can model a real-life phenomenon through functions. Sticking to functions of one single variable:</p>
<p>Many magnitudes evolve through time, not just position. If you graph temperature versus time, you can find the range of tempera... |
727,664 | <p>I need to evaluate $I = \int^\pi_{-\pi} \cos^3(x) \cos(ax)~dx$ where $a$ is some integer. </p>
<p>I get:
$\dfrac{2a(a^2-7)\sin(\pi a)}{a^4 - 10a^2 + 9}$. However $\sin(\pi a)$ is $0$ for all $a$ so $I=0$. But as noted by an answer, there are answers for $a = 1,3,-1,-3$. </p>
| Jeff Faraci | 115,030 | <p>$$
I=\int_{-\pi}^\pi \cos^3 x \cos(ax) dx=\frac{2a(7-a^2)\sin(a\pi)}{a^4-10a^2+9}
$$
It doesn't matter that $\sin(a\pi)$ will give zero for all a, since the denominator also gives zero for some values of a! For ex: a=1, the denominator is also zero. Thus the 0/0 works in your favor and you get a result when you t... |
1,654,649 | <p>Exercise: <em>Find the fifth and tenth roots of unity in algebraic form.</em></p>
<p>This is an early exercise in Ahlfors Complex Analysis. </p>
<p>What I have tried so far:</p>
<p>For the fifth roots I have tried reducing the problem to the fact that $\Re (1+z+z^2+z^3+z^4)=1+\cos\theta+\cos2\theta+\cos3\theta+\c... | Ng Chung Tak | 299,599 | <p>By $\sin 5\theta=5\sin \theta-20\sin^{3} \theta+16\sin^{5} \theta$, put $\theta =\frac{\pi}{5}$ and $y=\sin \frac{\pi}{5}$, you have</p>
<p>$0=5y-20y^{3}+16y^{5}$</p>
<p>$0=5-20y^{2}+16y^{4}$</p>
<p>$\displaystyle y^{2}=\frac{5 \pm \sqrt{5}}{8}$</p>
<p>Reject the upper case since $y^{2}<\sin^{2} \frac{\pi}{4}... |
405,610 | <p>Consider two diffusions given by
<span class="math-container">$$X_j(t)=\int_0^t a_j(s,X_j(s))\,dW_s$$</span>
for <span class="math-container">$j=1,2$</span> and <span class="math-container">$t\ge 0$</span>, where <span class="math-container">$W_\cdot$</span> is a standard Wiener process/Brownian motion and the <span... | Mateusz Kwaśnicki | 108,637 | <p>The inequality is not true in general — additional assumptions are needed. I think some kind of monotonicity of <span class="math-container">$a_1$</span> and <span class="math-container">$a_2$</span> should help, but this is merely a guess.</p>
<p>Here is a counterexample. Consider <span class="math-container">$a_2 ... |
2,911,049 | <blockquote>
<p><strong>Question:</strong>
Can we find the gradient and Hessian of $x x^T$ w.r.t. $x$, where $x \in \mathbb{R}^{n \times 1}$ ?</p>
</blockquote>
<p>EDIT:
If we can, may I know how to compute that? Thank you.</p>
| P. Quinton | 586,757 | <p>This is standard <a href="https://en.wikipedia.org/wiki/Matrix_calculus" rel="nofollow noreferrer">Matrix differentiation</a></p>
<p>The gradient is $2x$ and the Hessian is $2 I$.</p>
<p>By the way I'm assuming you meant $x^T x$ since $x x^T$ is a matrix and don't have a gradient or Hessian.</p>
|
423,334 | <h3>Set up</h3>
<p>Suppose <span class="math-container">$\gamma$</span> a simple closed curve, oriented in a counterclockwise direction. <span class="math-container">$f(z)$</span> is a complex polynomial
<span class="math-container">$$
f(z)=a_nz^{n}+a_{n-1}z^{n-1}+\cdots+a_0.
$$</span>
We already know that the integral... | Noam D. Elkies | 14,830 | <p>Yes: change variables to <span class="math-container">$w = 1/z$</span>. Then <span class="math-container">$f(z) = f(1/w)$</span> has an
a pole of multiplicity <span class="math-container">$n$</span> at <span class="math-container">$w=0$</span>, and a zero at <span class="math-container">$1/z$</span> for each zero
<... |
2,075,238 | <p>Am I correct in assuming, when dealing with Lebesgue integrals on the Cartesian space, that we adopt the notation $\int_{a}^{b} f(x)dx$ where the notation $dx$ is used to denote the Lebesgue measure? However, this notation is identical to the notation of a standard Riemann integral. So, when faced with this sort of ... | Henricus V. | 239,207 | <p>The notation
$$ \int_a^b f(x)\;\mathrm dx
$$
is not ambiguous. When it can be interpreted both as a Riemann integral and a Lebesgue integral, the integrals coincide.</p>
<p>Integrals such as $\int_0^\infty (\sin x)/x\;\mathrm dx$ are not ordinary Riemann integrals, they are improper Riemann integrals. Although the ... |
104,210 | <p>I have this code to plot contours:</p>
<pre><code>ContourPlot[(Cos[θ] Cos[ϕ])^(1/4), {θ, -π/2, π/2}, {ϕ, -π/2, π/2}, AxesLabel -> Automatic]
</code></pre>
<p>How would I map those contours on a unit sphere (if it is even possible) where <code>θ</code> and <code>ϕ</code> are the spherical angles for the sphere (... | Michael E2 | 4,999 | <p>Well, here's a way that works when the number of seconds since Jan. 1, 1970 is odd (that is, it crashes the kernel every other time I execute it):</p>
<pre><code>reg = MeshRegion[{{0, 1}, {1, 1.001}}, Line[{1, 2}]];
points = MeshCoordinates@ DiscretizeRegion[reg, MaxCellMeasure -> {"Length" -> 0.1}];
vf = Tab... |
3,279,492 | <p>It is known that for a surface <span class="math-container">$S \subset \mathbb{R^3}$</span> it can be found the first and the second fundamental form. </p>
<p>I would like to find out if this "first and second fundamental form" can be extended for a surface <span class="math-container">$S \subset \mathbb{R^{n}}$</s... | skyking | 265,767 | <p>Let <span class="math-container">$a=\lfloor\sqrt N\rfloor$</span> then we'll have <span class="math-container">$a^2\le N< (a+1)^2 = a^2+2a+1$</span></p>
<p>So we have <span class="math-container">$N/\lfloor\sqrt N\rfloor^2 \le (a^2+2a)/a^2 = 1+2/a$</span>. So if <span class="math-container">$a>2$</span> (ie <... |
358,075 | <p>Suppose $ \lim_m \sum_n f(n,m) = c $ and $ 0 \leq c< \infty $. Is it true that $ \lim_m \sum_n f(n,m)^k =0 $ if k >1?</p>
<p>Thank you</p>
| Paul | 17,980 | <p>A picture may be helpful for you. Let the radius of the ball smaller than both $x$ and $y$. That is OK.</p>
<p><strong>PS:
You would want to know $A$ is open iff for any $x\in A$, then there exists an open set $U$ such that $x \in U \subseteq A$.</strong></p>
<p><img src="https://i.stack.imgur.com/uAhn5.jpg" alt="... |
1,752,848 | <p>A strictly increasing sequence of positive integers $a_1, a_2, a_3,...$ has the property that for every positive integer $k$, the subsequence $a_{2k-1}, a_{2k}, a_{2k+1}$ is geometric and the subsequence $a_{2k}, a_{2k+1}, a_{2k+2}$ arithmetic. If $a_{13}=539$. Then how can I find $a_5.$</p>
<p>Any help will be app... | almagest | 172,006 | <p>Hint: Let $a_1=a,a_2=ka$. Then it is not hard to show that $a_{2n+1}=a(nk-n+1)^2$. We have $539=11\cdot7^2$, so evidently we take $k=2,a=11$. That gives $a_5=11(2\cdot 2-1)^2=99$.</p>
|
3,465,963 | <p>A question from the CodeChef December Long Challenge:</p>
<blockquote>
<p>Given two binary numbers <span class="math-container">$A$</span> and <span class="math-container">$B$</span>, each with <span class="math-container">$N$</span> bits. We may reorder the bits of <span class="math-container">$A$</span> in an arbi... | Marko Riedel | 44,883 | <p>The following <a href="https://math.stackexchange.com/questions/2894653/">MSE
link</a> asks about
the probability that a set of size <span class="math-container">$n$</span> drawn from <span class="math-container">$[kn]$</span> has sum
divisible by <span cl... |
4,472,705 | <p>Say that <span class="math-container">$x \notin A \lor x \in B$</span>. Is <span class="math-container">$x\in(A\setminus B)^c$</span> ?</p>
<p>The answer is apparently yes, but I can't think of what logical intermediary steps I must take in order to reach this conclusion. Would kindly appreciate your assistance.</p>... | user2661923 | 464,411 | <blockquote>
<p>Say that <span class="math-container">$x \notin A \lor x \in B$</span>. Is <span class="math-container">$x\in(A\setminus B)^c$</span> ?</p>
</blockquote>
<blockquote>
<p>I am not trying to use De Morgan's laws, I am trying to get to the answer using the basic building blocks.</p>
</blockquote>
<p>Basica... |
76,006 | <p>In Chriss and Ginzburg's "Representation Theory and Complex Geometry", they describe a geometric construction of representations of the affine Hecke algebra, using the Borel-Moore homology of generalized Springer fibers. </p>
<p>Briefly, let $G$ be a sufficiently nice algebraic group, and choose a semisimple $s \in... | Alexander Braverman | 3,891 | <p>The statement follows from the following general claim: let $G$ be a connected group acting
on a variety $X$ and let $F$ be an $G$-equivariant constructible sheaf on $X$ (or a complex
of sheaves). Then $G$ acts trivially on $H^*(X,F)$. Borel-Moore homology is (by definition)
the cohomology of the dualizing sheaf, so... |
1,413,022 | <p>In Guillemin and Pollack's <em>Differential Topology</em>, they give as an exercise (#1.8.14) to prove the following generalization of the Inverse Function Theorem:</p>
<blockquote>
<p>Use a partition-of-unity technique to prove a noncompact version of
[the Inverse Function Theorem]. Suppose that the derivative... | Maison Margiela | 1,030,471 | <p>Eric is correct--the answer by user149792 is insufficient. Instead, we can modify his/her arguments easily by taking the <span class="math-container">$V_i$</span>'s that contain the point <span class="math-container">$y$</span>. Then, the local finiteness ensures there are only a finite number of <span class="math-c... |
255,416 | <p><em>(<strong>Update</strong>)</em>:
Courtesy of Myerson's and Elkies' answers, we find a second <em>simple</em> cyclic quintic for $\cos\frac{\pi}{p}$ with $p=10m+1$ as,
$$F(z)=z^5 - 10 p z^3 + 20 n^2 p z^2 - 5 p (3 n^4 - 25 n^2 - 625) z + 4 n^2 p(n^4 - 25 n^2 - 125)=0$$
where $p=n^4 + 25 n^2 + 125$. Its discriminan... | Gerry Myerson | 3,684 | <p>I think the depressed quintic in the question is what Emma Lehmer called the reduced quintic in her <a href="https://projecteuclid.org/euclid.dmj/1077476385" rel="nofollow noreferrer">paper</a>, The quintic character of 2 and 3, Duke Math. J. Volume 18, Number 1 (1951), 11-18, MR0040338. In the proof of Theorem 4, s... |
205,871 | <p><strong>Question.</strong> Do you know a specific example which demonstrates that the tensor product of monoids (as defined below) is not associative?</p>
<hr>
<p>Let $C$ be the category of algebraic structures of a fixed type, and let us denote by $|~|$ the underlying functor $C \to \mathsf{Set}$. For $M,N \in C$... | Ronnie Brown | 28,586 | <p>This does not exactly answer your question, but it should be pointed out that in some situations such as groups, Lie algebras, ... one wants to consider other kinds of tensor products in which the key notion is that of a biderivation. An example of this is the commutator map $[\; ,\; ]: M \times N \to G$ where $M,N$... |
1,209,546 | <p>$$F(x) = \int{f(x)}\,dx$$</p>
<p>$$G(x) = \int_0^x{g(z)}\,dz$$</p>
<p>I am confused about the exact meaning about these functions. The second function is clear to me, $G(x)$ is just the area under the graph of $g(x)$ from $0$ to some $x$. But the first function is not so clear.</p>
<p>Also, why is the following... | GEdgar | 442 | <p>$$
H(x) = \int_0^x{g(x)}\,dx
$$
is not incorrect (just confusing for unexperienced students). This function $H$ is an antiderivative of $g$; that is $H'=g$. But usually we write $G'=g$ and $F'=f$ since it is easier to remember what is what. </p>
<p>Formula
$$
F(x) = \int{f(x)}\,dx
$$
is known as an <em>indefinit... |
44,746 | <p>Okay, I'm not much of a mathematician (I'm an 8th grader in Algebra I), but I have a question about something that's been bugging me.</p>
<p>I know that $0.999 \cdots$ (repeating) = $1$. So wouldn't $1 - \frac{1}{\infty} = 1$ as well? Because $\frac{1}{\infty} $ would be infinitely close to $0$, perhaps as $1^{-\in... | GEdgar | 442 | <p>As pointed out in the other answers, in the real number system there is no item "$\infty$". Nor is there in the complex number system. There are some other number systems that DO have such an item. One is called the "Riemann sphere" ... consisting of the complex numbers with an extra point $\infty$. Legitimate c... |
44,746 | <p>Okay, I'm not much of a mathematician (I'm an 8th grader in Algebra I), but I have a question about something that's been bugging me.</p>
<p>I know that $0.999 \cdots$ (repeating) = $1$. So wouldn't $1 - \frac{1}{\infty} = 1$ as well? Because $\frac{1}{\infty} $ would be infinitely close to $0$, perhaps as $1^{-\in... | Geoff Pointer | 96,501 | <p>I believe it should be pointed out that there is a very significant and important similarity between $0.\dot 9$ and $\frac1\infty$. It may seem easy to accept that $0.\dot9 = 1$ because calculations like $0.\dot9-0.0\dot9$ <em>seem</em> obvious, but you are none the less manipulating two expressions, each of which r... |
3,214,662 | <p><strong>Q1</strong> Prove that every simple subgroup of <span class="math-container">$S_4$</span> is abelian.</p>
<p><strong>Q2</strong> Using the above result, show that if <span class="math-container">$G$</span> is a nonabelian simple group then every proper subgroup of <span class="math-container">$G$</span> has... | CY Aries | 268,334 | <p><span class="math-container">$\angle ABC<90^\circ$</span> if and only if <span class="math-container">$\vec{BC}\cdot\vec{BA}>0$</span>.</p>
<p>That is, <span class="math-container">$(x_c-x_b)(x_a-x_b)+(y_c-y_b)(y_a-y_b)>0$</span></p>
|
67,171 | <p>I am sure <a href="http://en.wikipedia.org/wiki/Modular_multiplicative_inverse">all those symbols</a> are really easy for you guys to understand, but I would appreciate it if someone could bring it down to earth for me.</p>
<p>How could I do this on a basic calculator? or with a few lines of programmer's code which... | CopyPasteIt | 432,081 | <p>This Python prints out a table of all the inverses; the output for both <span class="math-container">$\text{mod } 41$</span> and <span class="math-container">$\text{mod } 210$</span> is given<br>
(c.f. this <a href="https://stackoverflow.com/a/9758173/8929814">stackoverflow link</a>).</p>
<p>Python Program</p>
<pre>... |
2,983,553 | <p>I've been stuck trying to solve this problem for the whole day. Also, I'm trying to translate the problem as good as I can, as my English skills aren't the greatest; sorry for that.</p>
<p>Problem is as follows: <strong>Points OBDE form a quadrilateral. Points B and D are on the line x=1. Find the value of x that m... | Seth | 610,132 | <p>If you take <span class="math-container">$\theta$</span> as being measured from the y- axis, then the area of the quadrilateral should be <span class="math-container">$sin(\theta)*1+\cfrac{1}{2}sin(\theta)cos(\theta)$</span>, since the width of the rectangular part is <span class="math-container">$1$</span>, and the... |
3,828,590 | <p>I am really confused behind the mathematical meaning of a <strong>100th percentile.</strong></p>
<p>What does that mean mathematically? Does it mean that a data point in a sample space is greater in some metric and that is also greater than itself?</p>
<p><strong>That makes no sense. AFAIK, there can be no such thin... | Victor Hugo | 322,450 | <p>The function <span class="math-container">$f(x)=1-x^2$</span> is not bounded at infinity, that is, for all <span class="math-container">$A>0$</span> and <span class="math-container">$k>0$</span> there exists <span class="math-container">$x_0 \in \mathbb{R}$</span> such that <span class="math-container">$x_0&g... |
2,929,182 | <p>Let <span class="math-container">$p,q>1$</span>, <span class="math-container">$\frac{1}{p}+\frac{1}{q}=1$</span> and <span class="math-container">$x\in[0,1]$</span>. I would like to show that <span class="math-container">$x^{\tfrac{1}{p}}\leq \frac{1}{p}x+\frac{1}{q}$</span>.</p>
<p>I tried to rewrite both sides... | Daniel Schepler | 337,888 | <p>By the weighted AM-GM inequality with <span class="math-container">$\lambda_1 = \frac{1}{p}$</span>, <span class="math-container">$x_1 = x$</span>, <span class="math-container">$\lambda_2 = \frac{1}{q}$</span>, <span class="math-container">$x_2 = 1$</span>, we have
<span class="math-container">$$x^{1/p} \cdot 1^{1/q... |
160,779 | <p>I am having a bit of difficulty trying to answer the following question:</p>
<blockquote>
<p>What is the Galois group of $X^8-1$ over $\mathbb{F}_{11}$?</p>
</blockquote>
<p>So far I have factored $X^8-1$ as </p>
<p>$$X^8-1=(X+10)(X+1)(X^2+1)(X^4+1).$$ </p>
<p>I know $X^2+1$ is irreducible over $\mathbb{F}_{11... | Community | -1 | <p>Your mistake was in assuming that $x^4 + 1$ was irreducible over $\Bbb{F}_{11}$. In fact there is something incredible about this polynomial; it is irreducible over $\Bbb{Z}$ but reducible over $\Bbb{F}_p$ for <em>every</em> prime $p$!</p>
<p>The following is the proof given in <em>Dummit and Foote</em>:</p>
<p>If... |
26,636 | <p>I have a function </p>
<pre><code>f[x_, y_, z_] := Exp[-x^2]*Exp[-z^2]
</code></pre>
<p>which traces out a tube along $y$. Is there a way to plot this 4D function, where the plot color is the 4th dimension?</p>
| cormullion | 61 | <p>You can vary the color of the tube as it goes along - if you provide a list of the colours that it passes through to <code>VertexColors</code>:</p>
<pre><code>Graphics3D[
{CapForm["Round"],
Tube[
Accumulate[
RandomReal[{-3, 3}, {50, 3}]], .3,
VertexColors ->
Table[RGBColor[x, 1 - x, 0.5 - x/2]... |
26,636 | <p>I have a function </p>
<pre><code>f[x_, y_, z_] := Exp[-x^2]*Exp[-z^2]
</code></pre>
<p>which traces out a tube along $y$. Is there a way to plot this 4D function, where the plot color is the 4th dimension?</p>
| Michael E2 | 4,999 | <p>I think what you're after is <code>ContourPlot3D</code>. There are two ways to color the contours.</p>
<p><code>ColorFunction</code> has a certain attractive automatic feature to it, but it doesn't color as nicely as <code>ContourStyle</code>, especially if you are coloring the surfaces by the function value. <co... |
499,218 | <p>I've been reading the books "An introduction to knot theory" by Lickorish and "Knots, Links, Braids and 3-Manifolds" by Prosolov and Sossinsky, and while both seem to me as good books, sometimes I'd like to get a different perspective on certain topics.</p>
<p>I would be glad to get some recommendations on books de... | Sammy Black | 6,509 | <p>I really like <a href="http://books.google.com/books/about/Quantum_Invariants.html?id=_62ePy9Tv2MC" rel="nofollow"><em>Quantum Invariants</em></a> by Tomotada Ohtsuki. Unlike Kassel (mentioned in another answer), whose book is largely self-contained, Ohtsuki reads more like an encyclopedia of ideas in the intersect... |
1,190,320 | <p>I need to evaluate $$\lim_{x \to \infty}x\int_0^{1/x}e^t \cos(t) \space \text{d}t$$ and I'm not really sure how to start. Do I have to find the integral or is there another way to figure it out?</p>
| Tad | 85,024 | <p>The problem is asking for the average value of $f(t)=e^t\cos t$ over the interval $[0,1/x]$. Since $f(t)$ is continuous everywhere, its average value on a small interval near zero approaches $f(0)$, namely $1$.</p>
|
2,043,132 | <p>I am having difficulty factorising the equation. </p>
| mathreadler | 213,607 | <p><strong>Hint:</strong></p>
<p>Let us assume it holds for $n=k$, that is: $|U_k| \leq \left(\frac{\sqrt{3}}{2}\right)^k$</p>
<p>Now what we want to show for the product for $n=k+1$ reduces to $\leq \left(\frac{\sqrt{3}}{2}\right)^{k-1} \sin(2^{k}x)\sin(2^{k+1}x)$, right? If we multiply it with $\cos(2^{k}x)$, what ... |
1,059,270 | <p>I was send here from stackoverflow because they thought maybe you can help me.</p>
<p>Here my original post: <a href="https://stackoverflow.com/questions/26799476/a-faster-way-then-doing-14-for-loops">https://stackoverflow.com/questions/26799476/a-faster-way-then-doing-14-for-loops</a></p>
<p>What I want:</p>
<p>... | Ross Millikan | 1,827 | <p>It sounds like you have far too many choices to list all the possible decks. In cases like that one usually resorts to generating random decks as a sample of all possible decks. It sounds like you are drawing your deck with replacement. If there are $200$ cards, you throw $10$ random numbers from $1$ through $200... |
50,864 | <p>I have a column vector of the form $\{1, 1, 2, 3, 2, 3\}^t$ and an associated matrix that has the same number of rows as the column vector, $6$, and a number of columns that is equal to the maximum of the values in the vector, $3$.</p>
<p>I would like to construct a matrix that has a specific value at the positions... | kglr | 125 | <p>Another variation using <code>SparseArray</code>:</p>
<pre><code>ClearAll[saF];
saF= SparseArray[MapIndexed[{First@#2, #} &, #1] -> #2, ##3] &
cols = {1, 1, 2, 3, 2, 3};
saF[cols, a] //Normal
(* {{a, 0, 0}, {a, 0, 0}, {0, a, 0}, {0, 0, a}, {0, a, 0}, {0, 0, a}} *)
saF[cols, {a, b, c, d, e, f}] // No... |
4,068,830 | <p>My logic is since <span class="math-container">$3$</span> out of <span class="math-container">$4$</span> elements are chosen, each element would appear once.
So a sequence would look like: <span class="math-container">$a\,b\,c\,x\,x\,x\,x\,x\,x\,x$</span></p>
<p>We have <span class="math-container">$7$</span> spots ... | Joffan | 206,402 | <p>If you choose <span class="math-container">$10$</span> items from a pool of <span class="math-container">$3$</span> elements, you of course have <span class="math-container">$3^{10}$</span> possibilities, although that will include some where not all are used.</p>
<p>If you choose <span class="math-container">$10$</... |
4,068,830 | <p>My logic is since <span class="math-container">$3$</span> out of <span class="math-container">$4$</span> elements are chosen, each element would appear once.
So a sequence would look like: <span class="math-container">$a\,b\,c\,x\,x\,x\,x\,x\,x\,x$</span></p>
<p>We have <span class="math-container">$7$</span> spots ... | vonbrand | 43,946 | <p>First, select the <span class="math-container">$3$</span> elements to show up, in <span class="math-container">$\binom{4}{3}$</span> ways.</p>
<p>In all, there are <span class="math-container">$3^{10}$</span> sequences of <span class="math-container">$3$</span> elements, of which <span class="math-container">$\binom... |
251,118 | <p>If I have a matrix that I know it can be written as (xi.xi)*KroneckerProduct(H,xi), where xi and H are vectors. Is there a way to obtain this expression from a given matrix?</p>
| Bob Hanlon | 9,362 | <pre><code>Clear["Global`*"]
</code></pre>
<p>Generate a test matrix <code>mat</code></p>
<pre><code>n = 5;
SeedRandom[1234];
HH = RandomReal[10, n];
X = RandomReal[1, n];
mat = (X . X)*KroneckerProduct[HH, X];
</code></pre>
<p>For a given <code>mat</code>, solve for <code>xi</code> and <code>H</code></p>... |
185,527 | <p>I would love to understand the famous formula <span class="math-container">$g_{ij}(x) = \delta_{ij} + \frac{1}{3}R_{kijl}x^kx^l +O(\|x\|^3)$</span>, which is valid in Riemannian normal coordinates and possibly more general situations.</p>
<p>I'm aware of 2 proofs: One using Jacobi fields [cf. e.g. S.Sternberg's &quo... | Mohammad Ghomi | 68,969 | <p>I came to this post many years later, since I too was concerned about the absence of Riemann's formula in most texts, lengthy treatment in others, or reliance on more advanced techniques like Jacobi fields. I include here a direct concise proof which I think would be well suited for beginning students. We want to sh... |
892,580 | <p>An object moving 12m/s passes north and hits an object. Due to the wind from a west direction, it is pushed sideways at 5m/s. Find the resultant velocity.</p>
<p>I don't know where to start with this one, I can do the other ones just fine. It involves vector addition and subtraction. Would anyone know? I have the a... | JimmyK4542 | 155,509 | <p>When you have $\ln |x| = 4\ln t + C$ and exponentiate, you should get: </p>
<p>$|x| = e^{4\ln t + C} = e^{C}e^{4\ln t} = e^Ct^4$ instead of $t^4 + e^C$.</p>
<p>In general, after you solve a differential equation, its a good idea to check to make sure the solution you found does indeed satisfy the differential equa... |
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