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 |
|---|---|---|---|---|
4,364,686 | <p>I have a point in a 3D coordinate system 1 (CS1). There can be two situations: the point is constant or the point is moving along a straight line from one known position to another at constant speed.</p>
<p>The CS1 is rotating in another (static) 3D coordinate system (CS2). The rotations of CS1 are known, i.e. the s... | Community | -1 | <p><strong>Hint:</strong></p>
<p>WLOG one of the points describes a unit circle in the plane <span class="math-container">$XY$</span> and the second point a circle in a parallel plane, with some phase difference but at the same unit angular speed.</p>
<p><span class="math-container">$$P(t)=(\cos t,\sin t,0),$$</span>
<... |
2,514,236 | <p>For example, the matrix could have finitely many rows and columns, but each row/column has uncountably many elements and you can do the standard matrix multiplication by taking care to match up the entries with corresponding pairs of real number indices. </p>
<p>Do such objects exist and has there been any work on ... | fred goodman | 124,085 | <p>Two such generalizations come to mind: integral operators defined by a "kernel" $T(f)(x) = \int K(x, y)\ f(y) dy$. Such operators compose by convolving kernels $\int K_1(x, y) K_2(y, z) dy$, which is evidently a continuous generalization of matrix multiplication. The second generalization is less obviously a di... |
39,466 | <p>I could not solve this problem:</p>
<blockquote>
<p>Prove that for a non-Archimedian field $K$ with completion $L$, $$\left\{|x|\in\mathbb R \mid x\in K\right\} =\left\{|x|\in\mathbb R \mid x\in L\right\}$$</p>
</blockquote>
<p>I considered a Cauchy sequence in $K$ with norms having limit $l$, but I could not co... | t.b. | 5,363 | <p>If $|x| \lt 1$ then $\frac{1}{1-|x|} = \sum_{n = 0}^{\infty} |x|^{n}$ by the summation formula for the geometric series. Now use this, the triangle inequality and the assumption that $|a_{n}| \leq C$ for all $n$ to estimate your given series from above, hence it series converges absolutely for $|x| \lt 1$.</p>
<p>A... |
2,795,777 | <p>I encountered this problem in one of my linear algebra homeworks (Linear Algebra with Applications 5th Ed 1.3.44):</p>
<p>Consider a $n \times m$ matrix $A$, such that $n > m$. Show there is a vector $b$ in $\mathbb{R}^{n}$ such that the system $Ax=b$ is inconsistent.</p>
<p>I have a strong intuition as to why ... | quasi | 400,434 | <p>The matrix
$$
A-xI=
\begin{bmatrix}-x & 1 &0 &\ldots & 0 &0\\ 0 & -x &1 &\ldots &0 &0 \\ \vdots & \vdots & \vdots & & \vdots &\vdots \\ 0 & 0 &0 &\ldots &-x &1 \\ 10^{10} & 0 &0 &\ldots &0 & -x\end{bmatrix} _{10... |
649,239 | <p>By <a href="http://en.wikipedia.org/wiki/Post%27s_theorem" rel="nofollow">Post's Theorem</a> we know that a set $A\subseteq\mathbf{N}$ is recursively enumerable iff it is definable by a $\Sigma_1$-formula, i.e. there exists a $\Sigma_1$-formula $\varphi(x)$ with $x$ free such that for every number $n$:
\[
n\in A\lon... | Xoff | 36,246 | <p>The definition of a recursive enumerable set is that it is the domain of some partial recursive function. </p>
<p>There is a recursive primitive function $\psi$, such that $\psi(n,t,x)=0$ if and only if $\phi_n(x)$ (the recursive function $n$ on entry $x$) halts in less than $t$ steps, else $\psi(n,t,x)=1$. Any rec... |
3,913,032 | <blockquote>
<p><strong>Problem.</strong> Let <span class="math-container">$A$</span> be a non-singular <span class="math-container">$n\times n$</span> matrix and let <span class="math-container">$\Gamma=[\Gamma_1\quad\Gamma_2]$</span> be an <span class="math-container">$n\times n$</span> orthogonal matrix where <span ... | skh76 | 850,993 | <p>The radius of the sphere should be equal to the radius of the cylinder face. Though you are correct about the height of the cylinder being twice the radius of the sphere.</p>
|
4,610,394 | <p>Clearly, none of the roots are in <span class="math-container">$\mathbb{Q}$</span> so <span class="math-container">$f(x) = x^4 + 1$</span> does not have any linear factors. Thus, the only thing left to check is to show that <span class="math-container">$f(x)$</span> cannot reduce to two quadratic factors.</p>
<p>My ... | Patrick Stevens | 259,262 | <p>I'd say it's easiest to use <a href="https://en.wikipedia.org/wiki/Eisenstein%27s_criterion" rel="nofollow noreferrer">Eisenstein's criterion</a> after a <a href="https://en.wikipedia.org/wiki/Eisenstein%27s_criterion#Indirect_(after_transformation)" rel="nofollow noreferrer">shift</a>, although this might be a sled... |
4,610,394 | <p>Clearly, none of the roots are in <span class="math-container">$\mathbb{Q}$</span> so <span class="math-container">$f(x) = x^4 + 1$</span> does not have any linear factors. Thus, the only thing left to check is to show that <span class="math-container">$f(x)$</span> cannot reduce to two quadratic factors.</p>
<p>My ... | Oscar Lanzi | 248,217 | <p>You can indeed try a factorization of the form</p>
<p><span class="math-container">$x^4+1=(x^2+ax+b)(x^2+cx+d).$</span></p>
<p>Expanding the right side and matching terms with like powers gives</p>
<p><span class="math-container">$x^3$</span> terms: <span class="math-container">$a+c=0,c=-a$</span></p>
<p><span class... |
65,810 | <p>Recently, I have been learning about nef line bundles. I know that when $X$ is projective or Moishezon, a line bundle $L$ over $X$ is said to be nef iff $$L.C=\int_{C}c_{1}(L)\ge 0$$ for every curve $C$ in $X$.</p>
<p>Demailly gave a definition of nefness that works on an arbitrary compact complex manifold, i.e., ... | mrw | 15,465 | <p>what is $\omega$ here? if it is a Kahler form, then Moishezon + Kahler implies projective, and as you said they are equivalent. </p>
|
55,435 | <p>I've recently become interested in the elementary theory of groups due to Sela and Myasnikov-Kharlampovich's work with free groups. I'd like a good introduction to the field of the elementary theory of groups, and in particular I'd like a reference to contain examples of group properties that cannot be read from a ... | HJRW | 1,463 | <p>I learned a lot from reading Bestvina and Feighn's article <a href="http://arxiv.org/abs/0809.0467" rel="nofollow">Notes on Sela's work: Limit groups and Makanin-Razborov diagrams</a>. It's not a broad introduction to elementary theory, but it does express some of Sela's ideas quite succinctly. You may need a back... |
2,966,392 | <p>Suppose <span class="math-container">$\lim_{n\rightarrow\infty }z_n=z$</span>.<br>
Prove <span class="math-container">$\lim_{n\rightarrow\infty}\operatorname{Re}(z_n)=\operatorname{Re}(z)$</span></p>
<p>Where, <span class="math-container">$z\in\mathbb{C}$</span> and <span class="math-container">$z_n$</span> is a c... | T_M | 562,248 | <p>If <span class="math-container">$|Re(w)| \leq |w|$</span> is always true.... then apply it to <span class="math-container">$z_n - z$</span></p>
|
3,623,432 | <p>Say I have two independent normal distributions (both with <span class="math-container">$\mu=0$</span>, <span class="math-container">$\sigma=\sigma$</span>) one for only positive values and one for only negatives so their pdfs look like:</p>
<p><span class="math-container">$p(x, \sigma) = \frac{\sqrt{2}}{\sqrt{\pi}... | Erik Cristian Seulean | 615,501 | <p>I think you can use moment generating functions to be able to get <span class="math-container">$E\big(\big(\frac{X+Y}{2}\big)^2\big)$</span> and <span class="math-container">$E\big(\frac{X+Y}{2}\big)$</span> to calculate the variance.</p>
<p><span class="math-container">$$M_\frac{X+Y}{2}(t)= E(e^{t\frac{X+Y}{2}}) ... |
3,174,339 | <p>Let <span class="math-container">$M$</span> be a <span class="math-container">$C^{\infty}$</span> manifold. Let <span class="math-container">$U$</span> be an open subset of <span class="math-container">$M$</span>. Now take a closed subset (with respect to the subspace topology on <span class="math-container">$U$</sp... | wjmolina | 25,134 | <p>Whether <span class="math-container">$f:X\to Y$</span>, and in particular an injection, is an open map depends on the topologies, e.g., if <span class="math-container">$Y$</span> has the discrete topology, then any <span class="math-container">$f$</span> is an open map. You can check that if <span class="math-contai... |
3,174,339 | <p>Let <span class="math-container">$M$</span> be a <span class="math-container">$C^{\infty}$</span> manifold. Let <span class="math-container">$U$</span> be an open subset of <span class="math-container">$M$</span>. Now take a closed subset (with respect to the subspace topology on <span class="math-container">$U$</sp... | Bijco | 626,084 | <p>Another example : take a set <span class="math-container">$X$</span> with at least two elements. Consider <span class="math-container">$X_1$</span> to be <span class="math-container">$X$</span> equiped with the discrete topology and <span class="math-container">$X_2$</span> to be <span class="math-container">$X$</sp... |
244,333 | <p>Consider this equation : </p>
<p><span class="math-container">$$\sqrt{\left( \frac{dy\cdot u\,dt}{L}\right)^2+(dy)^2}=v\,dt,$$</span></p>
<p>where <span class="math-container">$t$</span> varies from <span class="math-container">$0$</span> to <span class="math-container">$T$</span> , and <span class="math-container... | pokrate | 50,639 | <p>Why this approach is right or wrong ? </p>
<p><img src="https://i.stack.imgur.com/I9nPW.gif" alt="enter image description here"></p>
|
244,333 | <p>Consider this equation : </p>
<p><span class="math-container">$$\sqrt{\left( \frac{dy\cdot u\,dt}{L}\right)^2+(dy)^2}=v\,dt,$$</span></p>
<p>where <span class="math-container">$t$</span> varies from <span class="math-container">$0$</span> to <span class="math-container">$T$</span> , and <span class="math-container... | Maxim | 491,644 | <p>Suppose the cat is running along the <span class="math-container">$y$</span> axis. If <span class="math-container">$\rho$</span> is the distance between the dog and the cat, <span class="math-container">$\rho_y$</span> is the distance between them along the <span class="math-container">$y$</span> axis and <span clas... |
3,327,435 | <p>I have no clue for the following problem: </p>
<blockquote>
<p>Let <span class="math-container">$G$</span> be a finite group, <span class="math-container">$p$</span> a prime number, <span class="math-container">$S$</span> a Sylow <span class="math-container">$p$</span> subgroup of <span class="math-container">$G$... | jgon | 90,543 | <p>I upvoted Arturo's answer, but I'm going to write out a more complete answer because I was also struggling with this question.</p>
<p>We'll follow Arturo's advice, and make the changes Derek Holt suggests in the comments.</p>
<p><span class="math-container">$S$</span> centralizes <span class="math-container">$X$</... |
954,130 | <p>I have to prove that the function $f(n)=3n^2-n+4$ is $O(n^2)$. So I use the definition of big oh:</p>
<blockquote>
<p>$f(n)$ is big oh $g(n)$ if there exist an integer $n_0$ and a constant $c>0$ such that for all integers $n\geq n_0$, $f(n)\leq cg(n)$.</p>
</blockquote>
<p>And it doesn't matter what those con... | Leonardo Castro | 845,603 | <p>You can think of the equation of motion</p>
<p><span class="math-container">$s(t) = s(0) + v(0) t + \frac{a}{2} t^2$</span></p>
<p>as a Taylor expansion of the function <span class="math-container">$s(t)$</span> around <span class="math-container">$t=0$</span>, exact for constant acceleration. The coefficients of th... |
947,191 | <p>Show that $\sum _{n=1 } ^{\infty } (n \pi + \pi/2)^{-1 } $ diverges.</p>
<p>Both the root test and the ratio test is inconclusive. Can you suggest a series for the series comparison test?</p>
<p>Thanks in advance!</p>
| Umberto P. | 67,536 | <p>Use the <em>limit comparison test</em> with $\sum_{n=1}^\infty n^{-1}$.</p>
|
947,191 | <p>Show that $\sum _{n=1 } ^{\infty } (n \pi + \pi/2)^{-1 } $ diverges.</p>
<p>Both the root test and the ratio test is inconclusive. Can you suggest a series for the series comparison test?</p>
<p>Thanks in advance!</p>
| Ishfaaq | 109,161 | <p>It is never too pretty to go for the Root and Ratio Tests on the onset. A much more elegant method would be to notice that:</p>
<p>$$ \dfrac{1}{n \pi + \frac{\pi }{2}} = \dfrac{2 }{ 2n \pi + \pi } = \dfrac{2}{\pi } \cdot \dfrac{1}{2n + 1} \ge \dfrac{2}{\pi } \cdot \dfrac{1}{2n + n} = \dfrac{2}{3 \pi } \cdot \d... |
653,319 | <p>I understand that $\lim_{\theta\to0}(\sin(θ)/θ) = 1$ but what is $x$ when,
$\lim_{\theta\to0}(\tan(θ)/θ) = x$ where $x$ is a real constant value. </p>
<p>Please help me, I will be eternally great full :D</p>
| user124200 | 124,200 | <p>$x=1$</p>
<p>Let $\theta=y$. Then $$\tan y=\frac{\sin y}{\cos y}$$
and $\lim_{y \to 0} \frac{\sin y}{y}=1$ and $\cos 0=1$ so $\lim_{y\to 0}\frac{\tan y}{y}=0$.</p>
|
1,529,827 | <p>In order to make it clear, I ask three questions:</p>
<ol>
<li>Does $|2^m - 3^n|<10^6$ have any integers solution for $m>20$ ?</li>
<li>Is $ \liminf |2^m - 3^n|$ infinite ?</li>
<li>Is $ \liminf |2^m - 3^n|/m$ finite ?</li>
</ol>
| Gottfried Helms | 1,714 | <p>The problem can empirically examined with the use of the continued fraction of <span class="math-container">$ \beta = \log_2(3)$</span> For each <span class="math-container">$n$</span> we can find the optimal <span class="math-container">$m_n$</span> by <span class="math-container">$m_{n,\text{lo}}=\lfloor n \cdot \... |
239,202 | <p>Let $\Gamma$ be a $C^2$ compact submanifold of $\mathbb{R}^n$. Consider the distance function $\delta$ from $\Gamma$. It is well known that, for sufficiently small $\varepsilon>0$, $\delta$ is $C^2$ on $\{ 0<\delta < \varepsilon\}$, and that it satisfies the eikonal equation</p>
<p>$$ \| \nabla \delta \| =... | Raziel | 13,915 | <p>Since this question seems to have attracted some interest, I will post my own answer (which is a proof of the statement in the comments by Anton).</p>
<p><strong>If $u : M \to \mathbb{R}$ is a $C^2$ solution of the Eikonal equation
$$ \| \nabla u\| = 1, $$
then $\mathrm{Hess}(u)$, which a priori is only continuous,... |
3,252,765 | <p>We are trying to codify in terms of modern algorithm the works of the ancient Indian mathematician <em>Udayadivakara</em> (CE 1073). In his work <em>Sundari</em>, he quotes one <em>Acarya Jayadeva</em> who has given methods to solve Pell's equations. In these methods, one can find the the cyclic <em>Chakravala</em> ... | wendy.krieger | 78,024 | <p>In the case of <span class="math-container">$X^2-DY^2=C$</span>, this applies.</p>
<ol>
<li><span class="math-container">$C$</span> is the product of short and special primes, and some <span class="math-container">$Z^2$</span> relative to base <span class="math-container">$D$</span>.</li>
<li><span class="math-cont... |
2,438,111 | <p>I hope my title somehow encapsulates my problem.</p>
<p>Let's say we have a 1-D Grid with the values 2,1,5,8,1,1. Imagine those values are of some physical quantity $\alpha$.
The mean of this would be $(2+1+5+8+1+1)/6 = 3$</p>
<p>Now let's say we have some function $f(x) = x^2$, which computes another quantity $\b... | eyeballfrog | 395,748 | <p>Remember your trig identities.</p>
<p>$$
A\cos(x - C) = A\cos(x)\cos(C) + A\sin(x)\sin(C)
$$</p>
<p>So if $f(x) = \sqrt{3}\cos(x) + \sin(x) = A\cos(x-C)$, then $A\cos(C) = \sqrt{3}$ and $A\sin(C) = 1$. If we square both sides and add them together, we get
$$
[A\cos(C)]^2 + [A\sin(C)]^2 = A^2[\cos(C)^2 +\sin(C)^2] ... |
2,438,111 | <p>I hope my title somehow encapsulates my problem.</p>
<p>Let's say we have a 1-D Grid with the values 2,1,5,8,1,1. Imagine those values are of some physical quantity $\alpha$.
The mean of this would be $(2+1+5+8+1+1)/6 = 3$</p>
<p>Now let's say we have some function $f(x) = x^2$, which computes another quantity $\b... | Saj_Eda | 168,169 | <p>Use trigonometric identities, you will see that the amplitude is 2. Why? follow these lines</p>
<p>$f(x)=tan(\pi/3)cos(x)+sin(x)=\frac{sin(\pi/3)cos(x)+cos(\pi/3)sin(x)}{cos(\pi/3)}=2sin(x+\pi/3)$</p>
|
2,512,137 | <blockquote>
<p>A social worker has 77 days to make his visits. He wants to make at least one visit a day, and has 133 visits to make. Is there a period of consecutive days in which he makes a.) 21 b.) 23 visits? Why?</p>
</blockquote>
<p>a.) Set $a_i$ to be the number of visits up to and including day $i$, for $i =... | anonymous | 375,166 | <p>In a) you should also add 21 to the relevant set; otherwise you're not accounting for the possibility that the period of consecutive days starts on the first day (i.e. currently you're merely looking for an equality of the form $a_i=a_j+21$ but not $a_i=21$). Adding 21 to the set changes the answer to the affirmativ... |
3,506,316 | <p>I am trying to evaluate this limit:</p>
<p><span class="math-container">$$\lim_{x\to0^{+}}(x-\sin x)^{\frac{1}{\log x}}$$</span></p>
<p>It's a <span class="math-container">$0^0$</span> intedeterminate form, and I am unsure how to deal with it. I have a feeling that if I could turn it to a form where L'Hopital's ru... | Clement Yung | 620,517 | <p>The general trick is to apply exponential/logarithm to the expression:
<span class="math-container">$$
\lim_{x \to 0^+} (x - \sin{x})^{\frac{1}{\log{x}}} = \lim_{x \to 0^+} e^{\frac{\log(x - \sin{x})}{\log{x}}} = e^{\lim_{x \to 0^+} \frac{\log(x - \sin{x})}{\log{x}}}
$$</span>
Now you can apply L'Hopital Rule to the... |
3,506,316 | <p>I am trying to evaluate this limit:</p>
<p><span class="math-container">$$\lim_{x\to0^{+}}(x-\sin x)^{\frac{1}{\log x}}$$</span></p>
<p>It's a <span class="math-container">$0^0$</span> intedeterminate form, and I am unsure how to deal with it. I have a feeling that if I could turn it to a form where L'Hopital's ru... | user3290550 | 278,972 | <p><span class="math-container">$$\lim_{x\to0^{+}}(x-\sin x)^{\frac{1}{\ln x}}$$</span>
<span class="math-container">$$^\lim_{x\to0^{+}}e^{\ln(x-\sin x)^{\frac{1}{\ln x}}}$$</span></p>
<p><span class="math-container">$$e^{\lim_{x\to0^{+}}\dfrac{\ln(x-\sin x)}{\ln(x)}}$$</span></p>
<p>Applying L Hospital as we have <s... |
2,346,804 | <p>Please help me finish this problem.</p>
<p>$xy''+(3x-1)y'-(4x+9)y=0$ where $y(0)=0$</p>
<p>$L[xy'']+L[(3x-1)y']-L[(4x+9)y]=L[0]$</p>
<p>$L[xy'']=\frac{d}{dp}(p^2Y)$</p>
<p>$L[(3x-1)y']=-3\frac{d}{dp}(pY)$</p>
<p>$L[(4x+9)y]=-4\frac{dY}{dp}$</p>
<p>$-\frac{d}{dp}(p^2Y)-3\frac{d}{dp}(pY)+4\frac{dY}{dp}=0$</p>
<... | J.G | 293,121 | <p>You can think of it by the symmetry of the problem. More formally, let $X_i$ be the indicator function of the $i$th student getting picked. Then $\mathbb{E}[X_i]=\mathbb{P}(\text{child $i$ got picked})$. We also know that $\mathbb{E}[\sum_{i=1}^{10} X_i]=3$, as in every possibility, we pick 3 students. By linearity ... |
2,346,804 | <p>Please help me finish this problem.</p>
<p>$xy''+(3x-1)y'-(4x+9)y=0$ where $y(0)=0$</p>
<p>$L[xy'']+L[(3x-1)y']-L[(4x+9)y]=L[0]$</p>
<p>$L[xy'']=\frac{d}{dp}(p^2Y)$</p>
<p>$L[(3x-1)y']=-3\frac{d}{dp}(pY)$</p>
<p>$L[(4x+9)y]=-4\frac{dY}{dp}$</p>
<p>$-\frac{d}{dp}(p^2Y)-3\frac{d}{dp}(pY)+4\frac{dY}{dp}=0$</p>
<... | G Tony Jacobs | 92,129 | <p>Suppose you are the particular student. You are all ten placed into ten positions, three of which are the chosen ones. What is your probability of being in one of those three positions? Three out of ten.</p>
|
2,346,804 | <p>Please help me finish this problem.</p>
<p>$xy''+(3x-1)y'-(4x+9)y=0$ where $y(0)=0$</p>
<p>$L[xy'']+L[(3x-1)y']-L[(4x+9)y]=L[0]$</p>
<p>$L[xy'']=\frac{d}{dp}(p^2Y)$</p>
<p>$L[(3x-1)y']=-3\frac{d}{dp}(pY)$</p>
<p>$L[(4x+9)y]=-4\frac{dY}{dp}$</p>
<p>$-\frac{d}{dp}(p^2Y)-3\frac{d}{dp}(pY)+4\frac{dY}{dp}=0$</p>
<... | CopyPasteIt | 432,081 | <p>In order to apply Sample Space Theory to word problems concerning odds, you have to do two things:</p>
<p>Describe the problem as a random experiment<br>
Select a probability model </p>
<p>There may be more than one valid way to do this, so you can't just plug into a formula or apply a simple technique - you have ... |
2,322,646 | <p>Let $f$ and $\varphi$ be continuous real valued functions on $\mathbb{R}$. Suppose $\varphi(x)=0$ for $|x|>5$ and that $\int_{\mathbb{R}}\varphi(x)\mathbb{d}x=1$. Show that
$$\lim_{h\to 0}\left[\frac{1}{h}\int_{\mathbb{R}}f(x-y)\varphi\left(\frac{y}{h}\right)\mathbb{d}y\right]=f(x).$$
I don't know how to proceed... | Tucker | 256,305 | <p>Note that $\varphi(y/h) = 0$ for $|y/h|>5$, so we don't need to integrate over the entire real line.</p>
<p>$$
\lim_{h\to 0}\frac{1}{h}\int_{-5h}^{5h}f(x-y)\varphi(y/h)\mathbb{d}y
$$</p>
<p>define $\xi=y/h$.</p>
<p>$$
\lim_{h\to 0}\frac{1}{h}\int_{-5}^{5}f(x-h\xi)\varphi(\xi)\mathbb{d}(\xi h)
$$</p>
<p>$$
\ma... |
738,083 | <blockquote>
<p>Show that if two random variables X and Y are equal almost surely, then they
have the same distribution. Show that the reverse direction is not correct.</p>
</blockquote>
<p>If $2$ r.v are equal a.s. can we write $\mathbb P((X\in B)\triangle (Y\in B))=0$ (How to write this better ?)</p>
<p>then </... | 5xum | 112,884 | <p>Take $X$ and $Y$ with probabilities $P(X=1)=P(X=2)=P(Y=1)=P(Y=2)=0.5$ and which are independent. Then $$P(X=Y) = P(X=1, Y=1) + P(X=2,Y=2) =\\= P(X=1)P(Y=1)+P(Y=2)P(Y=2)=0.5,$$ meaning that $X=Y$ holds with probability $0.5$, not $1$.</p>
|
432,811 | <p>I'm trying to solve $$\operatorname{Arg}(z-2) - \operatorname{Arg}(z+2) = \frac{\pi}{6}$$ for $z \in \mathbb{C}$.</p>
<p>I know that
$$\operatorname{Arg} z_1 - \operatorname{Arg} z_2 = \operatorname{Arg} \frac{z_1}{z_2},$$
but that's only valid when $\operatorname{Arg} z_1 - \operatorname{Arg} z_2 \in (-\pi,\pi]$, ... | Christian Blatter | 1,303 | <p>I suggest you draw a figure.</p>
<p>When $z$ lies in the lower half plane ${\rm Im}(z)<0$ then
$$-\pi<\arg(z-2)-\arg(z+2)<0\ .$$
It follows that there are no points in the lower half plane fulfilling your condition.</p>
<p>Consider now a point $z$ in the upper half plane $H:\ {\rm Im}(z)>0$. Then
$$0&l... |
29,766 | <p>I'm looking for a news site for Mathematics which particularly covers recently solved mathematical problems together with the unsolved ones. Is there a good site MO users can suggest me or is my only bet just to google for them?</p>
| Péter Komjáth | 6,647 | <p>The Wikipedia page <a href="http://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics" rel="nofollow">List of unsolved problems in mathematics</a> has a specific (and long) sublist for recently solved problems. </p>
|
93,621 | <p>As we know, most of the spectral sequences are doubly graded. However, this "doubly graded" condition is not a part of the formal definition of spectral sequence. Is there any useful triply (quadruply, quintuply, etc.) graded spectral sequences? If not, is there a hope that some meaningful work can be done with this... | Igor Rivin | 11,142 | <p>Of course this has been studied. You need but google "trigraded complex", and much wisdom will be found. OK, some wisdom, notably Ravenel's Complex Cobordism and Stable Homotopy Groups of Spheres, and a very cute presentation by Noah Forman on</p>
<p>Graham Denham. The combinatorial laplacian of the tutte complex. ... |
3,491,867 | <p>I'm working on an integral used to illustrate <span class="math-container">$\pi > \frac{22}{7}$</span> and I'm stuck on finding the name of a theorem for the following:</p>
<p>Let <span class="math-container">$f(x)$</span> be a continuous Real Valued function on the interval <span class="math-container">$[a,b]$<... | mathcounterexamples.net | 187,663 | <p>Not sure this theorem has a name. It is however a property named <strong>positivity of the integral</strong>.</p>
|
542,391 | <p>I understand the processes of putting a matrix into Jordan normal form and forming the transformation matrix associated to "diagonalizing" the matrix. So here's my question:</p>
<p>Why is it that when you have an eigenvalue x=0 with algebraic multiplicity greater than 1, that you don't put a 1 in the superdiagonal ... | Amzoti | 38,839 | <p>We have a single eigenvalue of $\lambda_1 = 1$ and a triple eigenvalue of $\lambda_{2,3,4} = 0$.</p>
<p>For $\lambda=0$, we need to find three linearly independent eigenvectors and can just use the null space of $A$ for this. We have:</p>
<p>$$NS(A) = NS \left(\begin{bmatrix} 1 & 0 & 0 & 0 \\ 1 & 0... |
363,911 | <p>If a function $f:[-2,3]\to \mathbb{R}$ is defined by </p>
<p>$f(x)=\begin{cases} 2|x|+1 \; ;\; \text{ if } x \in \Bbb Q \\ 0 \; ;\; \text{ if } x \notin \Bbb Q \end{cases}$ </p>
<p>Prove that $f$ is not Riemann integrable.</p>
<p>What I came up with:<br>
$m_k=0$,$M_k=7$ </p>
<p>Which implies $U(P,f)=35$ and ... | Pedro | 23,350 | <p><strong>ADD</strong> It seems you're being given Darboux's approach to integration. I guess that for each partition $P=\{a=x_0,\dots,x_n=b\}$ of the interval $[a,b]$ you're defining
$$M_k=\sup_{x\in[x_{k-1},x_{k}]}f(x)$$
$$m_k=\inf_{x\in[x_{k-1},x_{k}]}f(x)$$</p>
<p>and then the lower and upper sums of $f$ over $P... |
387,295 | <p>I need to find $$\underset{n \to \infty}{\lim} \underset{x\in [0,1]}{\sup} \left| \frac{x+x^{2}}{1+n+x} \right|.$$ How to show that supremum will be at the point $x=1$?</p>
| davidlowryduda | 9,754 | <p>It's positive, so you don't need absolute values.</p>
<p>Then you could take a derivative, sett it equal to zero to find your critical points, etc. This answers your question of deciding for which $x$ it attains its max.</p>
<p>On the other hand, you could be generous and say the numerator is at most $2$, and the ... |
3,578,191 | <p>Without tables or a calculator, find the value of <span class="math-container">$\displaystyle\frac{(\sqrt5 +2)^6 - (\sqrt5 - 2)^6}{8\sqrt5}$</span>.</p>
<p>I do not understand how the positive/negative signs are obtained as shown in the book; is there a formula for expanding these kind of things (what kind of expre... | Quanto | 686,284 | <p>Alternatively, let <span class="math-container">$a=9+4\sqrt5$</span> and <span class="math-container">$b=9-4\sqrt5$</span>. Then, <span class="math-container">$a+b=18$</span>, <span class="math-container">$ab=1$</span> and,</p>
<p><span class="math-container">$$\displaystyle\frac{(\sqrt5 +2)^6 - (\sqrt5 - 2)^6}{8\s... |
3,631,903 | <p>A Calculus A level trigonometry problem:</p>
<blockquote>
<p>Solve <span class="math-container">$\tan x = \dfrac{p}{q}$</span> where <span class="math-container">$p,q\in\mathbb{Z}$</span> such that <span class="math-container">$$3\cos x\ - 4\sin x = -5$$</span></p>
</blockquote>
<p>I tried moving terms to one s... | 19aksh | 668,124 | <p>We have <span class="math-container">$|x-a| = \begin{cases}x-a,& \text{if } x\ge a \\ -(x-a), & \text{if } x< a\end{cases}$</span></p>
<p>So, <span class="math-container">$f(x) = \begin{cases}-(x+1) +x-3(x-1)+2(x-2)-(x+2) & \text{for } x <-1 \\
(x+1) +x-3(x-1)+2(x-2)-(x+2) &\text{for }-1\le x... |
1,715,265 | <p>I've tried a method similar to showing that $\mathbb{Q}(\sqrt2, \sqrt3)$ is a primitive field extension, but the cube root of 2 just makes it a nightmare.</p>
<p>Thanks in advance </p>
| Paramanand Singh | 72,031 | <p>Try to express both $\sqrt{2}$ and $\sqrt[3]{2}$ as rational functions of $a = \sqrt{2}+\sqrt[3]{2}$. The job is simple and easily done via equation $$(a -\sqrt{2})^{3}=2\tag{1}$$ so that $$a^{3}-3\sqrt{2}a^{2}+6a-2\sqrt{2}=2$$ or $$\sqrt{2}=\frac{a^{3}+6a-2}{3a^{2}+2}\tag{2}$$ and we have $$\sqrt[3]{2}=a-\sqrt{2}$$... |
3,360,396 | <p>In trying to answer <a href="https://math.stackexchange.com/q/1854193/104041">this question</a> on MSE, I got stuck. This taunts me because I think I should be able to do it.</p>
<h2>The Question:</h2>
<blockquote>
<p>Let <span class="math-container">$\phi : G\twoheadrightarrow H$</span> be an epimorphism of gro... | diracdeltafunk | 19,006 | <p>Small note: in your attempted proof you have <span class="math-container">$g \in G$</span> and <span class="math-container">$W \subseteq H$</span>, so <span class="math-container">$gWg^{-1}$</span> isn't (a priori) a well-defined thing. Perhaps trying to work with this object made things seem more confusing than the... |
1,329,078 | <p>I am having problems in classifying the differential equation $y''=y(x^2)$ in categories like homogeneous, exact, bernoulli, separable and non-exact so I could have the general solution. </p>
<p>Or would someone help me find the solution </p>
| André Nicolas | 6,312 | <p>We count the non-empty subsets of $\{1,2,3,\dots,n\}$. There are $2^n-1$ of them.</p>
<p>There are $2^0$ subsets with biggest element $1$, $2^1$ with biggest element $2$, $2^2$ with biggest element $3$, and so on up to $2^{n-1}$ with biggest element $n$. Add up.</p>
|
1,821,582 | <blockquote>
<p>Find all solutions of $$\{x^3\}+[x^4]=1$$
where $[x]=\lfloor x\rfloor$</p>
</blockquote>
<p>$$$$</p>
<p>I know that $0\le\{x^3\}<1\Rightarrow 0<[x^4]\le 1$. Thus $[x^4]=1$. I couldn't get any further though since I'm having trouble with $x^4$ in the term $[x^4]$. $$$$As an example, in anoth... | JasonM | 343,478 | <p>$\{x^3\}=0$ and $\lfloor x^4 \rfloor=1$ implies $$x \in \{\sqrt[3]{n} | n \in \mathbb{Z} \} \cap \left([1, \sqrt[4]{2}) \cup (-\sqrt[4]{2}, -1]\right)=\{\pm 1\}$$, so $x=\pm 1$</p>
|
167,946 | <p>I seek to replace derivatives like <code>D[u[x, y], x, x]</code> which are evaluated as $u^{(2,0)}[x,y]$ by variables with names like <code>uxx</code>. Derivatives that I work with are denoted by their "order-vector", for example <code>{2,0}</code> is the vector for this particular derivative and <code>{1,1}</code> ... | Vsevolod A. | 51,665 | <p>Turned out to be:</p>
<pre><code>dsT = {1, 0, 0, 1, 0};
vars = Table[ToExpression["x" <> ToString[i]], {i, 1, Length[dsT]}];
\!\(\*SuperscriptBox[\(u\),
TagBox[
RowBox[{"(",
RowBox[{"Sequence", "@@", "dsT"}], ")"}],
Derivative],
MultilineFunction->None]\)[Sequence @@ vars]
</code></pre>
|
3,597,301 | <p>We know that formula of finding mode of grouped data is</p>
<p>Mode = <span class="math-container">$l+\frac{(f_1-f_0)}{(2f_1-f_0-f_2)}\cdot h$</span></p>
<p>Where, <span class="math-container">$f_0$</span> is frequency of the class preceding the modal class and <span class="math-container">$f_2$</span> is frequenc... | ANANT | 905,810 | <p>we can take their value as 0. The frequency of the succeeding model class is taken as 0 if model class is the last observation.</p>
<p>You can also check it from the equation as-</p>
<p><span class="math-container">$l =$</span> lower limit of the modal class,</p>
<p><span class="math-container">$h =$</span> size of ... |
1,902,455 | <p>$x=e^t$ $y=te^(-t)$</p>
<p>$\frac{dy}{dx}= \frac{e^(-t)(1-t)}{e^(t)}$</p>
<p>$\frac{d^2y}{dx^2}= \frac{\frac{dy}{dx}}{\frac{dx}{dt}}= \frac{e^(-t)(1-t)}{e^t}$</p>
<p>any t's without proper enclosement are meant to be to the power...I don't know why its giving me this trouble. I entered these answers into my homew... | Claude Leibovici | 82,404 | <p><em>A small trick without any integration.</em></p>
<p>Because of the square in denominator, you can assume that $$\int \frac{1-y^2}{(1+y^2)^2} dy = \frac{P_n(y)}{1+y^2}$$ where $P_n(y)$ is a polynomial of degree $n$.</p>
<p>Differentiate both sides to get $$\frac{1-y^2}{(1+y^2)^2}=\frac{\left(y^2+1\right) P_n'(y)... |
2,913,974 | <p>In an additive category, we say that an object $A$ is compact if the functor $\text{Hom}(A, -)$ respects coproducts. That is, if the canonical morphism
$$
\coprod_{i} \text{Hom} \left( A, X_{i} \right) \longrightarrow \text{Hom} \left( A, \coprod_{i} X_{i} \right)
$$
is a bijection. Suppose $A \oplus B$ is compact. ... | Fabio Lucchini | 54,738 | <p>You have a split exact sequence $0\to A\to A\oplus B\to B\to 0$ which produces the commutative diagram with split exact rows
$$\require{AMScd}
\begin{CD}
0 @>>>
\coprod_{i} \text{Hom} \left( A, X_{i} \right) @>>>
\coprod_{i} \text{Hom} \left( A\oplus B, X_{i} \right) @>>>
\coprod_{i} \tex... |
2,435 | <p>I'm not sure we already have something similar, but I'm working on more code inspections for the IntelliJ plugin and it's always a good idea to ask the community. Since it doesn't really fit on main, I'm posting it here on Meta.</p>
<p>Linting is an excellent way to point the developer to probable errors that he mi... | Szabolcs | 12 | <h2>Non-ASCII characters</h2>
<p>Add an inspection to warn about non-ASCII characters that appear anywhere else except in comments.</p>
<p>The encoding that Mathematica will assume when loading a file using <code>Get</code> is not predictable and typically differs between operating systems. E.g., on macOS/Linux it mi... |
2,435 | <p>I'm not sure we already have something similar, but I'm working on more code inspections for the IntelliJ plugin and it's always a good idea to ask the community. Since it doesn't really fit on main, I'm posting it here on Meta.</p>
<p>Linting is an excellent way to point the developer to probable errors that he mi... | Szabolcs | 12 | <blockquote>
<h1>Status Completed</h1>
</blockquote>
<p>I'll convert my comment as this came up again:</p>
<p>This is already handled:</p>
<pre><code>_?foo[#]&
</code></pre>
<p>But with the many operator forms, now this is a common mistake:</p>
<pre><code>_?foo[12]
</code></pre>
<p>For example,</p>
<pre><c... |
2,191,360 | <blockquote>
<p>Show that
$$ f(x,y)= \begin{cases} \dfrac{xy^2}{x^2+y^4} & (x,y) ≠ (0,0) \\ 0 & (x,y) = (0,0) \end{cases}$$
is bounded.</p>
</blockquote>
<p>I thought about splitting it up into different cases like $x<y$ but it turned out to be too many and I could not cover all of them.
As a hint I... | farruhota | 425,072 | <p>It can be estimated as $-1\leq\frac{xy^2}{x^2+y^4}\leq1:$</p>
<p>LHS: $-1\leq\frac{xy^2}{x^2+y^4} \Rightarrow y^4+xy^2+x^2\geq0 \Rightarrow D=x^2-4x^2<0$ and $y=0 \Rightarrow x^2\geq0.$</p>
<p>RHS: $\frac{xy^2}{x^2+y^4}\leq1 \Rightarrow y^4-xy^2+x^2\geq0 \Rightarrow D=x^2-4x^2<0$ and $y=0 \Rightarrow x^2\geq... |
1,336,506 | <p>We know that the usual $\leq$ is a partial order relation on the group of integers $\mathbb Z$ and $\mathbb Z$ is a totally ordered with this partial order relation. Is there any other partially order relation exist in $\mathbb Z$ which makes $\mathbb Z$ a partially ordered group (or totally ordered group)? </p>
| Rajesh | 54,310 | <p>Take $P = \{x\in\mathbb Z: x = 0 \text{ or }x\geq 2 \}.$ Then $G = (\mathbb Z, P)$ is a partially ordered group with positive cone $P.$ In this case, the element $3\wedge4$ does not exist in $G.$ If $3\wedge 4 = 3,$ then $3\leq 4, 4 - 3\in P$ but $4 - 3 = 1\notin P.$ This implies $G$ is not orderisomorphic to usual ... |
4,385,908 | <p>For an ideal <span class="math-container">$I$</span> in <span class="math-container">$A = \mathbb{C}[x, y, z]$</span> set <span class="math-container">$$Z_{xy}(I) = \{(a, b) \in \mathbb{C}^2: f(a, b, z) = 0\text{ for all }f \in I\}.$$</span></p>
<p>Let
<span class="math-container">$$J = \{f(x, y): f(a, b) = 0\text{ ... | John Dawkins | 189,130 | <p>(1) <span class="math-container">$X_n$</span> needs to be <span class="math-container">$\mathcal F_n$</span>-measurable, meaning only that it must be constant on <span class="math-container">$\{n,n+1,\ldots\}$</span>.</p>
<p>(2) you need to have <span class="math-container">$E[X_n\cdot 1_B]=E[Y\cdot 1_B]$</span> for... |
90,070 | <h2>Question:</h2>
<p>Let <span class="math-container">$A\in\mathbb{R}^{n \times n}$</span> be an orthogonal matrix and let <span class="math-container">$\varepsilon>0$</span>. Then does there exist a rational orthogonal matrix <span class="math-container">$B\in\mathbb{R}^{n \times n}$</span> such that <span class="... | Qiaochu Yuan | 290 | <p>Sure. Consider matrices which fix $n-2$ of the standard basis vectors and describe a rotation in the plane spanned by the last two about an angle $\theta$ such that $\sin \theta, \cos \theta$ are both rational; these are dense in all such rotations, and all such rotations generate the orthogonal group, so the corres... |
90,070 | <h2>Question:</h2>
<p>Let <span class="math-container">$A\in\mathbb{R}^{n \times n}$</span> be an orthogonal matrix and let <span class="math-container">$\varepsilon>0$</span>. Then does there exist a rational orthogonal matrix <span class="math-container">$B\in\mathbb{R}^{n \times n}$</span> such that <span class="... | Denis Serre | 8,799 | <p>I should say <strong>yes</strong>. For this, I shall use the fact that in the unit sphere $\mathbb S^{d-1}$, the set of rational vectors is dense. I shall proceed by induction over $n$.</p>
<p>So let $A\in {\bf O}_n(\mathbb R)$ be given. Let $\vec v_1$ be its first column, an element of ${\mathbb S}^{n-1}$. We can ... |
457,977 | <p>I am trying to use residues to compute $$\int_0^\infty\frac{\log x}{(1+x)^3}\,\operatorname d\!x.$$My first attempt involved trying to take a circular contour with the branch cut being the positive real axis, but this ended up cancelling off the term I wanted. I wasn't sure if there was another contour I should use.... | Felix Marin | 85,343 | <p><span class="math-container">\begin{align}
&\bbox[10px,#ffd]{\int_{0}^{\infty}{\ln\left(x\right) \over \left(1 + x\right)^{3}}\,{\rm d}x} =
\int_{0}^{\pi/2}
{\ln\left(\tan^{2}\left(x\right)\right)
\over
\left\lbrack 1 + \tan^{2}\left(x\right)\right\rbrack^{3}
}
\,2\tan\left(x\right)\sec^{2}\left(x\right)\,{\rm... |
3,395,098 | <p>I am trying to work out for what <span class="math-container">$\lambda_1, \lambda_2 > 0$</span> is it true that <span class="math-container">$f(y) = \lambda_1 e^{y-\lambda_1 e^y} + \lambda_2 e^{y-\lambda_2 e^y}$</span> is unimodal?</p>
<p>Experimentally it seems it is unimodal when <span class="math-container">$... | Cesareo | 397,348 | <p>Consider </p>
<p><span class="math-container">$$
f(z) = \lambda _1 z e^{-\lambda _1 z}+\lambda _2 z e^{-\lambda _2 z}
$$</span></p>
<p>with <span class="math-container">$z = e^y$</span></p>
<p>and now</p>
<p><span class="math-container">$$
f'(z) = -\lambda _1 e^{-\lambda _1 z} \left(\lambda _1 z-1\right)-\lambda... |
996,052 | <p>A disk of radius <span class="math-container">$5$</span> cm has density <span class="math-container">$10$</span> g/cm<span class="math-container">$^2$</span> at its center, density <span class="math-container">$0$</span> at its edge, and its density is a linear function of the distance from the center. Find the mass... | Fahd Siddiqui | 187,108 | <p>Dimensional analysis tells me that is incorrect. $D$ has dimensions of $g/cm^2$ integrating with $dx$ gives it dimensions of $g/cm$ and not $g$ as required by the answer.</p>
<p><strong>Solution</strong></p>
<p>Consider the elemental area of the disc $dA=2\pi xdx$</p>
<p>Now mass=density*Area</p>
<p>$m=DdA$</p>
... |
1,559,485 | <p>Suppose $\sup_{x \in \mathbb{R}} f'(x) \le M$.</p>
<p>I am trying to show that this is true if and only if $$\frac{f(x) - f(y)}{x - y} \le M$$</p>
<p>for all $x, y \in \mathbb{R}$.</p>
<p><strong>Proof</strong></p>
<p>$\text{sup}_{x \in \mathbb{R}} f'(x) \le M$</p>
<p>$f'(x) \le M$ for all $x \in \mathbb{R}$</p... | fleablood | 280,126 | <p>well, first of all, we have to presume f is continuous and differentiable on R. This statement isn't true otherwise.</p>
<p>1) Suppose $\frac{f(x) - f(y)}{x - y} > M$ for some $x, y \in \mathbb R$.</p>
<p>By the mean value theorem, there exist a $c; x <c < y$ where $f'(c) = \frac{f(x) - f(y)}{x - y}$.</... |
434,290 | <p>According to the <a href="http://arxiv.org/abs/0910.5922" rel="nofollow">equation 4</a>,
$$\phi(0,t)= \frac{A_0}{(1+\frac{2t^2}{R^4})^{3/4}}\cos \left(\sqrt2 t+ \frac{3}{2}\tan^{-1}\left[\frac{\sqrt2 t}{R^2}\right]\right)\tag{1}$$
what conditions makes, $$\cos \left(\sqrt2 t+ \frac{3}{2}\tan^{-1}\left[\frac{\sqrt2 ... | DonAntonio | 31,254 | <p>And yet another way for you to enjoy. Define</p>
<p>$$f(z):=\frac{\text{Log}\,z}{z^4+1}\;,\;\;C_R:=[-R,R]\cup\gamma_R:=\{z\in\Bbb C\;;\;z=Re^{it}\,,\,\,0<t<\pi\}\;,\;\;1<R\in\Bbb R$$</p>
<p>Now, the only poles within the region determined by $\,C_R\,$ are the simple (why? And note that $\,z=0\,$ is a pole... |
4,609,833 | <p>As far as I can tell using Mathematica, the following identity seems to hold:
<span class="math-container">$$(n+2)^n=(n+1)\sum_{k=0}^n\binom{n}{k}\frac{(k+1)^{k-1}(n-k)^{n-k}}{n+1-k},$$</span>
where we define <span class="math-container">$0^0=1$</span>. However, I am having trouble proving it. I thought that this lo... | zbrads2 | 655,480 | <p>I think I have found a solution using the principal branch of the Lambert W function. The Lagrange inversion theorem can be used to show that
<span class="math-container">$$W_0^p(x)=\sum_{n=p}^\infty\frac{-p(-n)^{n-p-1}}{(n-p)!}x^n,$$</span>
so that by shifting the index, we have
<span class="math-container">$$\left... |
4,609,833 | <p>As far as I can tell using Mathematica, the following identity seems to hold:
<span class="math-container">$$(n+2)^n=(n+1)\sum_{k=0}^n\binom{n}{k}\frac{(k+1)^{k-1}(n-k)^{n-k}}{n+1-k},$$</span>
where we define <span class="math-container">$0^0=1$</span>. However, I am having trouble proving it. I thought that this lo... | joriki | 6,622 | <p>You can actually do this by applying the first formula that you quoted from <em>Concrete Mathematics</em> three times with <span class="math-container">$r=t=1$</span>, that is,</p>
<p><span class="math-container">$$
\sum_{k=0}^n\binom{n}{k}(k+1)^{k-1}(n-k+s)^{n-k}=(n+1+s)^n\;.\tag1
$$</span></p>
<p>Split your sum li... |
1,383,781 | <p>Given $\mathbb{X}$ = $\mathbb{R^2}$, consider $\| \cdot \|_2$ and $\| \cdot \|_\infty$ </p>
<p>We can show that </p>
<p>$\| x \|_\infty \leq \| x \|_2 \leq \sqrt2 \| x \|_\infty$ </p>
<p>Hence $\| \cdot \|_2$ and $\| \cdot \|_\infty$ are equivalent norms</p>
<p>Is there some deeper implication regarding this pa... | mathcounterexamples.net | 187,663 | <p>Some application of this result and more generally that all norms are equivalent on finite dimensional spaces:</p>
<ul>
<li>The compact subspaces are the closed bounded spaces.</li>
<li>All linear maps are continuous. More generally all multilinear maps are continuous.</li>
<li>All linear maps are bounded on the un... |
646,109 | <p>For function $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfies $f\left(x+y\right)=f\left(x\right)f\left(y\right)$
and is not the zero-function I can prove that $f\left(1\right)>0$
and $f\left(x\right)=f\left(1\right)^{x}$ for each $x\in\mathbb{Q}$.
Is there a way to prove that for $x\in\mathbb{R}$?</p>
<p>This ... | Martín-Blas Pérez Pinilla | 98,199 | <p>Continuity (or continuity in some point or measurability) is required. See <a href="http://en.wikipedia.org/wiki/Cauchy%27s_functional_equation" rel="nofollow">Cauchy's functional equation</a>. Your problem is reducible to this.</p>
|
139,817 | <p>Studying stability of certain non-autonomous dynamical systems on Lie groups I have come across the following question: Exactly which finite-dimensional, real Lie groups have adjoint representations that are bounded away from zero?</p>
<p>Edit: by "bounded away from zero" I mean that the image of the adjoint repres... | Community | -1 | <p>I am not sure what you mean by "bounded away from zero", but if you mean that the closure of the image of the adjoint representation does not contain zero, that is correct. A proof might proceed by showing that the image lies in the adjoint group of the lie algebra and the latter lies in the group of elements of det... |
3,156,643 | <blockquote>
<p>Prove that <span class="math-container">$\sin(x) < x$</span> when <span class="math-container">$0<x<2\pi.$</span></p>
</blockquote>
<p>I have been struggling on this problem for quite some time and I do not understand some parts of the problem. I am supposed to use rolles theorem and Mean v... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$x-\sin\, x=\int_0^{x}[1-\cos\, t] dt \geq 0$</span> and equality can hold only of the non-negative continuous function <span class="math-container">$1-\cos\, t$</span> is identically <span class="math-container">$0$</span> from <span class="math-container">$0$</span> to <span class="mat... |
555,239 | <p>Since the polynomial has three irrational roots, I don't know how to solve the equation with familiar ways to solve the similar question. Could anyone answer the question?</p>
| André Nicolas | 6,312 | <p><strong>Added:</strong> The approach below is ugly: It would be most comfortable to delete. </p>
<p>We look at your second equation. Look first at the case $x\ge 0$, $y\ge 0$. We have $x^2+y^2-xy=(x-y)^2+xy$. Thus $x^2+y^2-xy\ge xy$. So if the equation is to hold, we need $xy\le x+y$. </p>
<p>Note that $xy-x-y=(x-... |
555,239 | <p>Since the polynomial has three irrational roots, I don't know how to solve the equation with familiar ways to solve the similar question. Could anyone answer the question?</p>
| individ | 128,505 | <p>This equation can be rewritten just in another form: $\frac{m+1}{n}+\frac{n+1}{m}=a$
Can be solved using the equation Pell:</p>
<p>$p^2-(a^2-4)s^2=1$</p>
<p>Solutions have the form:</p>
<p>$n=2(p-(a+2)s)s$</p>
<p>$m=-2(p+(a+2)s)s$</p>
<p>And more:</p>
<p>$n=\frac{2p(p+(a-2)s)}{a-2}$</p>
<p>$m=\frac{2p(p-(a-... |
944,948 | <p>$\textbf{QUESTION-}$
Let $P$ be a p-group with $|P:Z(P)|\leq p^n$. Show that $|P'| \leq p^{n(n-1)/2}$.</p>
<p>If $P=Z(P)$ it is true. Now let $n > 1$, then</p>
<p>If I see $P$ as a nilpotent group and construct its upper central series, it will end , so let it be,</p>
<p>$e=Z_0<Z_1<Z_2<......<Z_r=P... | southsinger | 80,958 | <p>First of all notice that (insert $-x$ instead of $y$)</p>
<p>$$
f' (x) = f' (0) + x^2
$$</p>
<p>On the other hand, we know that </p>
<p>$$
\lim_{x \to 0} f(x)/x = f'(0) =1
$$</p>
<p>so</p>
<p>$$
f'(x) = 1+x^2
$$</p>
|
3,151,662 | <p>Consider <span class="math-container">$a_1,\dots,a_n\in\mathbb{R}^n$</span> and identify <span class="math-container">$a_j\in\mathcal{L}(\mathbb{R},\mathbb{R}^n)$</span> via <span class="math-container">$\varphi\mapsto \varphi1$</span>.</p>
<p>Also, consider <span class="math-container">$A\in\mathcal{L}(\mathbb{R}^... | poetasis | 546,655 | <p>If the determinant of a matrix is zero, then there are no solutions to a set of equations represented by an nXn matrix set equal to a 1Xn matrix. If it is non-zero, then there are solutions and they can all be found using <a href="https://www.purplemath.com/modules/cramers.htm" rel="nofollow noreferrer">Cramer's Rul... |
3,151,662 | <p>Consider <span class="math-container">$a_1,\dots,a_n\in\mathbb{R}^n$</span> and identify <span class="math-container">$a_j\in\mathcal{L}(\mathbb{R},\mathbb{R}^n)$</span> via <span class="math-container">$\varphi\mapsto \varphi1$</span>.</p>
<p>Also, consider <span class="math-container">$A\in\mathcal{L}(\mathbb{R}^... | Raaja_is_at_topanswers.xyz | 286,483 | <p>In system-theory, </p>
<ol>
<li>systems can be represented by matrices and each column represent the internal-state of the system. </li>
<li>If a determinant of one such matrix is zero, then we can say that one of the state associated with certain dynamics is being duplicated.</li>
<li>Based on some special matrix ... |
422,118 | <p>I'm a CS major working on social network analysis and its friends.</p>
<p>In page 15 of <a href="http://open.umich.edu/sites/default/files/1446/SI508-F08-Week3.pdf" rel="nofollow">this lecture note</a>, two very interesting questions have been asked. Given a social network graph, in which cases would we find nodes ... | Gill | 86,878 | <p>So I think that $\triangledown(\top,\top)$ does follow for all normal logics in the sense you have defined. The following would be a proof in any extension of the logic you have given.</p>
<p>A $\vdash\top$ by 1</p>
<p>B $\vdash\triangledown(\top,\bot)$ by 6</p>
<p>C $\vdash\triangledown(\top,\bot)\to(\triangledo... |
4,414,843 | <p>For reference: Show that the area of triangle <span class="math-container">$ABC = R\times MN(R=BO)$</span></p>
<p>I can't demonstrate this relationship</p>
<p><a href="https://i.stack.imgur.com/A3ci3.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/A3ci3.jpg" alt="enter image description here" />... | peta arantes | 419,513 | <p>The orthocenter and the circumcenter of a triangle are isogonal conjugates, therefore
<span class="math-container">$\angle ABH=\angle NBO=α\\
HMBN(cyclic)\implies \angle BHM=\angle MNB=90−α\\
\therefore \angle BFN = 90^0 \implies BO\perp MN \\
[BMON]=[BMN]−[MNO]=\frac{BO⋅MN}{2}=\frac{R⋅MN}{2}(I)\\
\triangle MBN \si... |
4,414,843 | <p>For reference: Show that the area of triangle <span class="math-container">$ABC = R\times MN(R=BO)$</span></p>
<p>I can't demonstrate this relationship</p>
<p><a href="https://i.stack.imgur.com/A3ci3.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/A3ci3.jpg" alt="enter image description here" />... | JMP | 210,189 | <p>First establish that <span class="math-container">$MN = BH \times \sin \angle ABC$</span></p>
<p>(<a href="https://en.wikipedia.org/wiki/Law_of_sines" rel="nofollow noreferrer">law of sines</a> - <span class="math-container">$\sin (\angle HMB):BH = \sin (\angle ABC):MN$</span>).</p>
<p>Then,</p>
<p><span class="math... |
3,227,215 | <p><a href="https://i.stack.imgur.com/7pJ4t.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7pJ4t.png" alt="enter image description here" /></a></p>
<blockquote>
<p><span class="math-container">$(O, R)$</span> is the circumscribed circle of <span class="math-container">$\triangle ABC$</span>. <span c... | Michael Rozenberg | 190,319 | <p>It's wrong.</p>
<p>Indeed, let <span class="math-container">$\Delta A'B'C'\sim\Delta ABC$</span> such that <span class="math-container">$\frac{A'B'}{AB}=\epsilon.$</span></p>
<p>Thus, if your inequality is true, so we have <span class="math-container">$$\sum_{cyc}\frac{1}{A'M'\cdot B'N'}\leq\frac{4}{3(R'-O'I')}$$<... |
120,067 | <p>The <em>theta function</em> is the analytic function $\theta:U\to\mathbb{C}$ defined on the (open) right half-plane $U\subset\mathbb{C}$ by $\theta(\tau)=\sum_{n\in\mathbb{Z}}e^{-\pi n^2 \tau}$. It has the following important transformation property.</p>
<blockquote>
<p><strong>Theta reciprocity</strong>: $\theta... | Alexey Ustinov | 5,712 | <p>This is not an answer but a comment concerning the Landsberg-Schaar relation (LS). It admits not only analytic proof. The article <a href="https://arxiv.org/abs/1810.06172" rel="nofollow noreferrer">A proof of the Landsberg-Schaar relation by finite methods</a> by Ben Moore gives an elementary proof of the LS. But t... |
3,950,463 | <blockquote>
<p>What is <span class="math-container">$100$</span>th derivative of <span class="math-container">$y=\ln(2x-x^2)$</span> at <span class="math-container">$x=1$</span>?</p>
<p><span class="math-container">$a)2\times99!$</span></p>
<p><span class="math-container">$b)-2\times99!$</span></p>
<p><span class="mat... | Ninad Munshi | 698,724 | <p>We can be a bit clever instead. Notice that</p>
<p><span class="math-container">$$\ln(2x-x^2) = \ln(1-1+2x-x^2) = \ln(1-(x-1)^2)$$</span></p>
<p><span class="math-container">$\ln(1-x)$</span> has a known Taylor series</p>
<p><span class="math-container">$$-\sum_{n=1}^\infty \frac{x^n}{n}$$</span></p>
<p>which means ... |
10,505 | <p>I've only done a few questions here but already it's grinding on me. Why can't we have the writing-answer panel and the preview panel side by side, rather than below, this means for big answers I can't make use of the preview! It'd be great if side by side, two scroll-bars, or even a pop out (I have a window manager... | robjohn | 13,854 | <p>Whether you use ChatJax or not in chat, look at the bookmarks described in <a href="https://math.meta.stackexchange.com/a/3297">this answer</a>. The <code>rendering off</code> and <code>rendering on</code> bookmarks will turn off and on the MathJax rendering. While off, this prevents the rendering from affecting you... |
1,841,958 | <p>This is a claim on Wikipedia <a href="https://en.wikipedia.org/wiki/Partially_ordered_set">https://en.wikipedia.org/wiki/Partially_ordered_set</a></p>
<p>I am not sure how to make sense of the claim</p>
<p>What does it mean by ordered by inclusion? Inclusion as in $\subseteq$? </p>
<p>Can someone provide a small ... | André Nicolas | 6,312 | <p>Yes, inclusion as in $\subseteq$. A small example may clarify things. Take the usual three-dimensional vector space $\mathbb{R}^3$. Geometrically speaking, there are four types of subspaces of $\mathbb{R}^3$.</p>
<p>(i) The space consisting of the zero vector only.</p>
<p>(ii) One-dimensional subspaces, which can... |
434,061 | <p>I am given a matrix $A\in M(n\times n, \mathbb{C})$ normal (in matrix form $AA^*=A^*A$) and $A^2=A$. The task is to prove that the matrix is Hermitian.</p>
<p>But when I try something like $A^*=\,\,...$ , then I can't reach $A$, because I can't "get rid of star" in expression. Also it is not enough to show $BA=BA^*... | Sungjin Kim | 67,070 | <p><strong>Hint</strong> By spectral theorem, a normal matrix is diagonalizable by a unitary matrix. </p>
<p>Then what are the eigenvalues of a diagonalizable matrix $A$ which satisfies
$$A^2=A?$$</p>
|
3,985,177 | <p>When it comes to proving that two sets are equal, say <span class="math-container">$A = B$</span>, we're usually told that we have to prove that <span class="math-container">$A \subset B$</span> and <span class="math-container">$B \subset A$</span>. However, I'm under the impression that this strategy isn't unique. ... | Yuval Peres | 360,408 | <p>No holomorphic function can map the complex plane to a bounded open set. See
<a href="https://artofproblemsolving.com/wiki/index.php/Liouville%27s_Theorem_(complex_analysis)#:%7E:text=In%20complex%20analysis%2C%20Liouville%27s%20Theorem,Theorem%20is%20a%20stronger%20result" rel="nofollow noreferrer">Liouville's theo... |
1,039,141 | <blockquote>
<p>Let <span class="math-container">$X = \mathbb{R}$</span> and <span class="math-container">$Y = \{x \in \mathbb{R} :x ≥ 1\}$</span>, and define <span class="math-container">$G : X → Y$</span> by <span class="math-container">$$G(x) = e^{x^2}.$$</span>
Prove that <span class="math-container">$G$</span> is ... | Mimo | 195,551 | <p>For any y $\in$ Y we have to show that there exists an x in X such that G(x) = y.</p>
<p>Now,
$$G(x)=y$$
$$\implies e^{x^2} = y$$
$$\implies x^2 = \ln y $$
$$\implies x = \pm \sqrt {\ln y}$$</p>
<p>Since, y $\in$ Y, y $\ge 1$ and hence $\ln y\ge 0$ and $\pm \sqrt {\ln y}$ is well defined and is in X.Thus for any r... |
2,337,332 | <p>Tried a lot. Though unable to find starting point.</p>
| Sri-Amirthan Theivendran | 302,692 | <p>Write
$$
9^{16}-5^{16}=(9^8-5^8)(9^8+5^8)\tag{1}.
$$
By Euler's theorem,
$$
9^6\equiv 1\mod 14;\quad 5^6\equiv 1\mod 14
$$
since $5$ and $9$ are coprime to $14$. Then
$$
9^8-5^8\equiv 9^2-5^2\equiv4(14)\equiv0\mod{14}
$$
and the result follows from (1).</p>
|
3,064,501 | <p>So I was trying to find the <strong>time complexity</strong> of an algorithm to find the <span class="math-container">$N$</span>th prime number (where <span class="math-container">$N$</span> could be any positive integer).</p>
<p>So is there any way to exactly determine how far <span class="math-container">$(N+1)$<... | Community | -1 | <p>It depends on what you call large, and what theory you use. Using the fact all primes greater than 3 are 1 or -1 mod 6 and a <a href="https://en.m.wikipedia.org/wiki/Sieve_of_Sundaram" rel="nofollow noreferrer">Sieve of Sundaram</a> style argument, you can show that any natural number n, of certain forms will create... |
692,998 | <p>The inner product in a $L^2$ space can be defined as:</p>
<p>$$\langle f,g\rangle =\int_a^b \bar{f}(x)g(x)w(x)dx$$</p>
<p>For Legendre polynomials, we define it as:</p>
<p>$$\langle P_m,P_n\rangle =\int_0^1 \bar{P}_m(x)P_n(x)dx$$
so $w(x)=1$.</p>
<p>But there are case in which $w(x)\neq 1$. For example, Laguerre... | froggie | 23,685 | <p>I'm not sure this is a great answer, but in the case of the Legendre polynomials, you are working on a compact interval, so the given inner product with weight $\equiv 1$ makes sense.</p>
<p>On the other hand, for the Laguerre and Hermite polynomials, you work on the intervals $[0,\infty)$ and $(-\infty, \infty)$, ... |
5,711 | <p>I am teaching abroad to non-native English speakers with a large variance of language skills. </p>
<p>I teach both pre-calculus and AP calculus (AB & BC). For both of those classes I define the new terms. I use word problems, have them read from the textbook, take notes in English during class, have the stude... | J W | 376 | <p>I teach in an ESL environment where students come from a variety of countries. I try to be particularly sensitive to differences in notation/convention in mathematics, which seem to come up most often in the notation for the decimal point, what sign to use for multiplication and how logarithms are written.</p>
<p>I... |
5,711 | <p>I am teaching abroad to non-native English speakers with a large variance of language skills. </p>
<p>I teach both pre-calculus and AP calculus (AB & BC). For both of those classes I define the new terms. I use word problems, have them read from the textbook, take notes in English during class, have the stude... | Tom Au | 1,333 | <p>One way to teach English along with calculus is to assign so-called "word problems." (E.g. if g is the acceleration due to gravity, how do you integrate/differentiate to get Newton's Law.)</p>
<p>It's a situation where students have to "translate" from English to math. Then they have to do the math.</p>
|
2,130,397 | <p>If I want to find the power series representation of the following function:</p>
<p>$$ \ln \frac{1+x}{1-x} $$</p>
<p>I understand that it can be written as </p>
<p>$$ \ln (1+x) - \ln(1-x) $$</p>
<p>And I understand that if I now write in the power series representations for $ln(1+x)$ and $ln(1-x)$:</p>
<p>$$\su... | Jan Eerland | 226,665 | <p>Well, when you substitute $t=\cos\left(x\right)$ then we know that:</p>
<p>$$\cos^2\left(x\right)+\sin^2\left(x\right)=t^2+\sin^2\left(x\right)=1\tag1$$</p>
<p>Another way, substitute $\text{u}=\tan\left(x\right)$:</p>
<p>$$\int_0^\frac{\pi}{2}\frac{1}{1+\cos^2\left(x\right)}\space\text{d}x=\int_0^\infty\frac{1}{... |
2,435,596 | <p>Suppose we have an unsigned $8$ bit number (min=$0$, max=$255$).</p>
<p>the result of "$200 + 200$" overflows to $144$</p>
<p>the result of "$100 - 200$" (under?)overflows to $156$</p>
<p>Is there are mathematical symbol to represent this?</p>
| Community | -1 | <p>"Overflow" is the wrong term here:</p>
<blockquote>
<p>to flow over the edge or brim of (a receptacle, container, etc.).</p>
</blockquote>
<p><sub>ref: <a href="http://www.dictionary.com/browse/overflow" rel="nofollow noreferrer">http://www.dictionary.com/browse/overflow</a></sub></p>
<p>In the cited example, y... |
3,125,263 | <p>I can't solve the last exercises in a worksheet of Pre-Calculus problems. It says:</p>
<p>Quadratic function <span class="math-container">$f(x)=ax^2+bx+c$</span> determines a parabola that passes through points <span class="math-container">$(0, 2)$</span> and <span class="math-container">$(4, 2)$</span>, and its ve... | Vinyl_cape_jawa | 151,763 | <p>HINTS:</p>
<p>A graph is a collection of points where the <span class="math-container">$x$</span> and <span class="math-container">$y$</span> coordinates of these points are in a relationship. We sometimes write <span class="math-container">$y$</span> instead of <span class="math-container">$f(x)$</span> to stress ... |
1,299,474 | <p><img src="https://i.stack.imgur.com/EyYdm.jpg" alt="enter image description here"></p>
<p>Here is an attempt at a solution:</p>
<p><img src="https://i.stack.imgur.com/IWSah.jpg" alt="enter image description here"></p>
<p>Since $f(x)>0$, $f(x)>\delta$ for all x between $1$ and $2$ </p>
<p>Is this correct? <... | xavierm02 | 10,385 | <p>"By the algebra of continuous functions" doesn't look like a proper justification to me. I'd prefer something like "because it's the composition of two continuous functions".</p>
<p>But anyway, taking the inverse is unnecessary.</p>
<hr>
<p>You theorem states that if $f$ is continuous on $[1,2]$, then:</p>
<ul>
... |
1,299,474 | <p><img src="https://i.stack.imgur.com/EyYdm.jpg" alt="enter image description here"></p>
<p>Here is an attempt at a solution:</p>
<p><img src="https://i.stack.imgur.com/IWSah.jpg" alt="enter image description here"></p>
<p>Since $f(x)>0$, $f(x)>\delta$ for all x between $1$ and $2$ </p>
<p>Is this correct? <... | Project Book | 234,125 | <p>I would've said something along the line $f(x)$ attains its lower bound say at $a \in [1,2]$, $f(a) > 0$ by definition so let $\delta = f(a)/2$.</p>
|
806,532 | <p>This question takes place in a general metric space $X$. </p>
<p>Let $x$ be an interior* point of $E \subset X$ iff there exists a deleted neighborhood of $x$ that is contained in $E$. </p>
<p>This is like the normal definition of "interior point", except it uses "deleted neighborhood" instead of "neighborhood",... | user642796 | 8,348 | <p>I can't speak much for categorical reasons, but here are my opinions from the topological point of view.</p>
<p>I think a large reason is that the closure of a set is a much more fundamental concept than the derived set $A^\prime$ (<em>i.e.</em>, the set of all accumulation (limit) points of $A$). This is perhaps e... |
3,717,932 | <p>How can this identity convolution be shown?</p>
<p><span class="math-container">$$\int^\infty_{-\infty} f(\tau)\delta(t-\tau)d\tau=f(t)$$</span></p>
<p>I keep getting stuck in traps when trying to show this and need a bit of assistance</p>
| Enrico M. | 266,764 | <p>There is no <em>proof</em> of the formula you're asking, this is a definition. However, here is the way why the expression is used in so many contexts.</p>
<p>Consider a sequence of function
<span class="math-container">$$
\delta_\epsilon(x)=\begin{cases}\frac{1}{2\epsilon},&-\epsilon<x<\epsilon,\\
0,&... |
765,404 | <p>Can anyone explain the partial derivative below:</p>
<p>$\frac{\partial a^tX^{-1}b}{\partial X} = -X^{-t}ab^tX^{-t}$</p>
<p>I was trying to derive this equation using the below formula, but failed.</p>
<p><img src="https://i.stack.imgur.com/apR2q.png" alt="enter image description here"></p>
| Community | -1 | <p>Let $Y = X^{-1}$, since it's easier to type.</p>
<p>Taking the differential of $I=Y\cdot X$ you'll find that
$$dY = -Y\cdot dX\cdot Y$$</p>
<p>Now rearrange $a'\cdot Y\cdot b$ into $ab':Y$ and take the differential
$$\eqalign{
d(ab':Y) &= ab':dY \cr
&= -ab':(Y\cdot dX\cdot Y) \cr
&= ... |
1,618,042 | <p>Is there any operation that makes a set of primes i.e. {2,3,5,7.... .} a group with identity 2?</p>
| Tsemo Aristide | 280,301 | <p>Hint: Take any bijection between the set of prime and Z which takes 2 to 0 and transport the structure with this bijection.</p>
|
2,425,157 | <p>How do I show that
$$ \frac 12 \left(\frac 1 {3^2}+\frac 1{4^2}+ \frac 1{5^2}+\dots\right) < \frac 1 {3^2} + \frac 1{5^2} + \frac1{7^2} +\dots \quad ?$$</p>
| farruhota | 425,072 | <p>Alternatively: Note that the RHS:
$$\frac{\pi ^2}{6}=1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+\frac{1}{7^2}+\cdots =\\
\left(1+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\cdots \right)+\frac{1}{2^2}\left(\underbrace{1+\frac{1}{2^2}+\frac{1}{3^2}\cdots}_{\frac{\pi^2}{6}} \right) \Rightar... |
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