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 |
|---|---|---|---|---|
1,086,048 | <p>I've performed a change of variable:
$$X = \sqrt{y}$$
$$X'=\frac{1}{2}Y^{-\frac{1}{2}}$$
Thus:
$$f(\sqrt{y})*X'=f(y)=\frac{1}{2\sqrt{2\pi y}}e^{-\frac{y}{2}}$$
However the book gives:
$$f(y)=\frac{1}{\sqrt{2\pi y}}e^{-\frac{y}{2}}$$
Where did I go wrong?</p>
| Przemysław Scherwentke | 72,361 | <p>HINT: You should consider $P(-\sqrt{x}<y<\sqrt{x})$.</p>
|
1,086,048 | <p>I've performed a change of variable:
$$X = \sqrt{y}$$
$$X'=\frac{1}{2}Y^{-\frac{1}{2}}$$
Thus:
$$f(\sqrt{y})*X'=f(y)=\frac{1}{2\sqrt{2\pi y}}e^{-\frac{y}{2}}$$
However the book gives:
$$f(y)=\frac{1}{\sqrt{2\pi y}}e^{-\frac{y}{2}}$$
Where did I go wrong?</p>
| Chris | 103,950 | <p>Thanks for the hint. I computed only half the result. To complete the solution:</p>
<p>$$f(y)=\frac{1}{2\sqrt{2\pi y}}e^{-\frac{y}{2}}+\frac{1}{2\sqrt{2\pi y}}e^{-\frac{y}{2}}=\frac{1}{\sqrt{2\pi y}}e^{-\frac{y}{2}},0<y<\infty$$</p>
|
2,211,075 | <p>I don't understand the following example from Math book.</p>
<p>Solve for the equation <code>sin(theta) = -0.428</code> for <code>theta</code> in <code>radians</code> to 2 decimal places. where <code>0<= theta<= 2PI</code>.</p>
<p>And this is the answer:</p>
<p><code>theta=-0.44 + 2PI = 5.84rad and theta = ... | Clement C. | 75,808 | <p>In short: you <strong>cannot</strong> say
$$
\lim_{x\to 0} \frac{\frac{f(x)-f(0)}{x-0}}{g(x)} =
\lim_{x\to 0} \frac{f'(x)}{g(x)}
$$
which is what you used in your third step. This is simply not true in general.</p>
<hr>
<p>To see this on a simpler example, take e.g. $f$ defined by $f(x)=x^2$ and $g(x)=x$.</p>
<p... |
3,905,629 | <p>I need to compute a limit:</p>
<p><span class="math-container">$$\lim_{x \to 0+}(2\sin \sqrt x + \sqrt x \sin \frac{1}{x})^x$$</span></p>
<p>I tried to apply the L'Hôpital rule, but the emerging terms become too complicated and doesn't seem to simplify.</p>
<p><span class="math-container">$$
\lim_{x \to 0+}(2\sin \s... | Kavi Rama Murthy | 142,385 | <p>Hint: Using the fact that <span class="math-container">$\frac {\sin x} x \to 1$</span> as <span class="math-container">$x \to 0$</span> verify that <span class="math-container">$ x\ln (\frac 1 2 \sqrt x) <x\ln (2\sin\sqrt x+\sqrt x \sin (\frac 1 x)) <x \ln (3\sqrt x)$</span>. Conclude that the limit is <span... |
3,905,629 | <p>I need to compute a limit:</p>
<p><span class="math-container">$$\lim_{x \to 0+}(2\sin \sqrt x + \sqrt x \sin \frac{1}{x})^x$$</span></p>
<p>I tried to apply the L'Hôpital rule, but the emerging terms become too complicated and doesn't seem to simplify.</p>
<p><span class="math-container">$$
\lim_{x \to 0+}(2\sin \s... | Adam Rubinson | 29,156 | <p><span class="math-container">$$ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - ... $$</span></p>
<p><span class="math-container">$$\therefore \ (2\sin \sqrt x + \sqrt x \sin \frac{1}{x})^x$$</span></p>
<p><span class="math-container">$$ = \left(2 \left(x^{1/2} - \frac{x^{3/2}}{3!} + \frac{x^{5/2}}{5!} -...\right) + ... |
2,747,578 | <p>Let $S,T$ be sets with $|S|>|T|$ and $R$ some relations on $T$.<br>
Why is then $\langle S|-\rangle$ not isomorphic to $\langle T|R\rangle$ </p>
<p>This came up when I wanted to solve a different problem, <a href="https://math.stackexchange.com/q/2747383/506844">which I also asked on this site</a>. Unfortunately... | Community | -1 | <p>The group is a quotient group. The images of the elements of T generate this quotient group. If $<T|R>$ is free then any set of free generators has less than or equal to |T| generators. Thus it can't be isomorphic to $<S|->$.</p>
|
2,747,578 | <p>Let $S,T$ be sets with $|S|>|T|$ and $R$ some relations on $T$.<br>
Why is then $\langle S|-\rangle$ not isomorphic to $\langle T|R\rangle$ </p>
<p>This came up when I wanted to solve a different problem, <a href="https://math.stackexchange.com/q/2747383/506844">which I also asked on this site</a>. Unfortunately... | Giorgio Mossa | 11,888 | <p>I feel a little like cheating here, because that is basically an adaptation of the idea provided from an answer to the linked question... but again they say that mathematics is the art of finding good analogies.</p>
<p>Before I start allow me to introduce the commutator subgroup.
For every group $G$ by $G'$ I denot... |
4,349,582 | <p>Its rather easy to show that <span class="math-container">$a_n=\frac{n^{1/n}}{n}$</span> ist monotonic (which means <span class="math-container">$a_{n+1}<a_n$</span> for each <span class="math-container">$n$</span>) using derivations. But how can I do it without them? Thanks.</p>
| fleablood | 280,126 | <p>First of all base 10 and base 2 are notations and not the integers themselves. And integer will have a value no matter how it is written. So if you want a <em>function</em> that converts base 10 to base two (a function being a mapping between two sets) then this is <em>not</em> a function on the integers but a fun... |
205,671 | <p>How would one go about showing the polar version of the Cauchy Riemann Equations are sufficient to get differentiability of a complex valued function which has continuous partial derivatives? </p>
<p>I haven't found any proof of this online.</p>
<p>One of my ideas was writing out $r$ and $\theta$ in terms of $x$ a... | Shabir | 239,381 | <p><img src="https://i.stack.imgur.com/Ct6la.png" alt="Here is an easy approach">
Here is an easy approach to derive the Cauchy-Riemann equations in polar form.</p>
|
2,218,914 | <p>What is a boundary point when solving for a max/min using Lagrange Multipliers?
After you solve the required system of equation and get the critical maxima and minima, when do you have to check for boundary points and how do you identify them?</p>
<p>e.g. Optimise (1+a)(1+b)(1+c) given constraint a+b+c=1, with a,b,... | Christian Blatter | 1,303 | <p>Your example serves perfectly to explain the necessary procedure.</p>
<p>You are given a function $f(x,y,z):=(1+x)(1+y)(1+z)$ in ${\mathbb R}^3$, as well as a compact set $S\subset{\mathbb R}^3$, and you are told to determine $\max f(S)$ and $\min f(S)$.</p>
<p>Differential calculus is a help in this task insofar ... |
109,734 | <p>I am trying to do this homework problem and I have no idea how to approach it. I have tried many methods, all resulting in failure. I went to the books website and it offers no help. I am trying to find the derivative of the function
$$y=\cot^2(\sin \theta)$$</p>
<p>I could be incorrect but a trig function squared ... | Mike | 17,976 | <p>The way I was taught allows me to use the chain rule without really needing to think about it. This would be the way I'd write it:</p>
<p>$dy=d(\cot^2(\sin\theta))=2\cot(\sin\theta)d(\cot(\sin\theta))=2\cot(\sin\theta)(-\csc^2(\sin\theta))d(\sin\theta)=$</p>
<p>$-2\cot(\sin\theta)\csc^2(\sin\theta)\cos\theta d\th... |
2,486,590 | <p>Let $(X,\mathcal{B},\mu,T)$ be a dynamical system and let $A \in \mathcal{B}$ such that $\mu(A)>0$ and $\forall x \in X$ we define $$L_x=\{n \in \Bbb{N}|T^nx \in A\}$$</p>
<p>Prove that $\mu(\{x:\bar{d}(L_x)>0\})>0$ where $$\bar{d}(L_x)=\limsup_n \frac{|L_x \cap \{1,2...n\}|}{n}$$.</p>
<p>My first thought... | mathworker21 | 366,088 | <p>For almost every $x$, the limit $$\lim_{N \to \infty} \frac{1}{N}\sum_{n=1}^{N} \chi_A(T^nx) = \lim_{N \to \infty} \frac{|L_x \cap [1,N]|}{N}$$ exists. Let $f^*(x)$ be the limit. It is a fact (see Birkhoff ergodic theorem - note this does apply to non-ergodic measure preserving systems) that $\int f^*(x)d\mu = \int ... |
2,486,590 | <p>Let $(X,\mathcal{B},\mu,T)$ be a dynamical system and let $A \in \mathcal{B}$ such that $\mu(A)>0$ and $\forall x \in X$ we define $$L_x=\{n \in \Bbb{N}|T^nx \in A\}$$</p>
<p>Prove that $\mu(\{x:\bar{d}(L_x)>0\})>0$ where $$\bar{d}(L_x)=\limsup_n \frac{|L_x \cap \{1,2...n\}|}{n}$$.</p>
<p>My first thought... | D. J. Obata | 418,537 | <p>I am also assuming $\mu$ is a probability measure. For a measurable set $A$ with positive $\mu$-measure, define $A_0$ to be the set of points $x\in A$ such that $\overline{d}(L_x)=0$. Let $\chi_{A_0}(.)$ be the characteristic function over the set $A_0$ and for any $x\in X$, consider
$$
F^+(x) = \limsup_{n\to +\inft... |
4,552,723 | <p>Assume the following angles are known:
<span class="math-container">$ABD$</span>,<span class="math-container">$DBC$</span>,<span class="math-container">$BAC$</span>,<span class="math-container">$ACD$</span>.</p>
<p>Is it possible to compute <span class="math-container">$CDA$</span>?</p>
<p><a href="https://i.stack.i... | 冥王 Hades | 1,092,912 | <p>Generally speaking, it is possible to use the sine rule in <span class="math-container">$\triangle ABD$</span>, <span class="math-container">$\triangle DBC$</span> and <span class="math-container">$\triangle BAC$</span> to develop a system of trigonometric equations which also develops some sort of relation between ... |
2,483,231 | <p>If $F(x)=f(g(x))$, where $f(5) = 8$, $f'(5) = 2$, $f'(−2) = 5$, $g(−2) = 5$, and
$g'(−2) = 9$, find $F'(−2)$. I'm totally lost on this problem, I'm assuming to incorporate the Chain Rule. I get $5(5) * 9 = 225$ but I am incorrect.</p>
<p>Update: Thanks guys, I see where I messed up thanks!</p>
| Donald Splutterwit | 404,247 | <p>The chain rule gives $F'(x)=g'(x)f'(g(x))$. Now just substitute the values
\begin{eqnarray*}
F'(-2)=\underbrace{g'(-2)}_{9}f'(\underbrace{g(-2)}_{5})=9\underbrace{f'(5)}_{2}=9 \times 2 =\color{red}{18}.
\end{eqnarray*}</p>
|
2,490,128 | <p>Over the domain of integers, if $(a-c)|(ab+cd)$ then $(a-c)|(ad+bc)$.</p>
<p>Note: $x|y$ means "$x$ divides $y$," i.e. $\exists k\in \mathbb{Z}. y=x\cdot k$</p>
<p>This is part of an assignment on GCD, Euclidean algorithm, and modular arithmetic.</p>
<p>My approach:</p>
<p>If $a-c$, divides a linear combination ... | Cornman | 439,383 | <p>Hint:</p>
<p>$\int \frac{1}{x^2\sqrt{x^2+x+1}}\, dx=\int \frac{1}{x^2\sqrt{(x+\frac12)^2+\frac34}}\, dx$</p>
<p>Now substitute $u=x+\frac12$</p>
|
2,490,128 | <p>Over the domain of integers, if $(a-c)|(ab+cd)$ then $(a-c)|(ad+bc)$.</p>
<p>Note: $x|y$ means "$x$ divides $y$," i.e. $\exists k\in \mathbb{Z}. y=x\cdot k$</p>
<p>This is part of an assignment on GCD, Euclidean algorithm, and modular arithmetic.</p>
<p>My approach:</p>
<p>If $a-c$, divides a linear combination ... | Michael Rozenberg | 190,319 | <p>$x^2+x+1=\left(x+\frac{1}{2}\right)^2+\frac{3}{4}$.</p>
<p>Thus, the substitution $x+\frac{1}{2}=\frac{\sqrt3}{2}\tan{t}$ must help.</p>
|
4,487,380 | <p>I was reading my calculus book wherein I came across a note, being worth of attention. It says:</p>
<blockquote>
<p>Integrals in the form of <span class="math-container">$\int P(x)e^{ax}dx$</span> have a special property. After calculating the integral, we obtain a function in the form of <span class="math-container... | legionwhale | 685,267 | <p>The best method for this question is to use induction and integration by parts (as suggested by W. Fan), as well as the linearity of the integral. Note that if <span class="math-container">$P(x) = c_n x^n + \cdots + c_0$</span>,</p>
<p><span class="math-container">$$\int P(x) e^{ax} dx = \int (c_n x^n + \cdots + c_0... |
733,553 | <p>It's been a long time since high school, and I guess I forgot my rules of exponents. I did a web search for this rule but I could not find a rule that helps me explain this case:</p>
<p>$ 2^n + 2^n = 2^{n+1} $</p>
<p>Which rule of exponents is this?</p>
| Community | -1 | <p>$2^n + 2^n = 2^n(1+1) = 2^n(2) = 2^{n+1}$</p>
|
827,740 | <p>This is a new integral that I propose to evaluate in closed form:
$$ {\mathfrak{R}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$$
where $\Re$ denotes the real part and $\log (z)$ denotes the principal value of the logarithm defined for $z \neq 0$ by
$$ \log (z) = \ln |z| + i \mathrm{Arg}z, \qu... | pisco | 257,943 | <p>$$\color{blue}{\Re\int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)}dx = \frac{\pi }{8}\left( {1 - \ln 2 - \gamma } \right)}$$</p>
<hr>
<p>For a proof, I will continue from $$\tag{4} - 8\Re I + 2\pi J = \pi \ln 2 - \pi$$ in <a href="https://math.stackexchange.com/questions/863496">this question</a>.
where $$J=\int... |
936,138 | <p>I need help approaching a proof which deals with inequalities:</p>
<p>If p and r are the precision and recall of a test, then the F1 measure of the test is
defined to be
$$F(p, r) = \frac{2pr}{p+r}$$</p>
<p>Prove that, for all positive reals p, r, and t, if t ≥ r then F(p, t) ≥ F(p, r)</p>
<p>What's the first ste... | JimmyK4542 | 155,509 | <p><strong>Hint</strong>: $F(p,r) = \dfrac{2pr}{p+r} = \dfrac{2p^2+2pr-2p^2}{p+r} = \dfrac{2p(p+r)-2p^2}{p+r} = 2p - \dfrac{2p^2}{p+r}$. </p>
<p>Can you show that this is an increasing function in $r$?</p>
|
2,572,302 | <p>I want to calculate the limit: $$ \lim_{x\to +\infty}(1+e^{-x})^{2^x \log x}$$
The limit shows itself in an $1^\infty$ Indeterminate Form. I tried to elevate $e$ at the logarithm of the function:</p>
<p>$$\lim_{x\to +\infty} \log(e^{(1+e^{-x})^{2^x \log x}}) = e^{\lim_{x\to +\infty} \log((1+e^{-x})^{2^x \log x})} =... | Angina Seng | 436,618 | <p>If we call the expression $f(x)$ then
$$\ln f(x)=2^x\log x\,\ln(1+e^{-x})=O\left(\frac{2^x\log x}{e^{x}}\right)\to0$$
as $x\to\infty$ as $2/e<1$.</p>
|
2,572,302 | <p>I want to calculate the limit: $$ \lim_{x\to +\infty}(1+e^{-x})^{2^x \log x}$$
The limit shows itself in an $1^\infty$ Indeterminate Form. I tried to elevate $e$ at the logarithm of the function:</p>
<p>$$\lim_{x\to +\infty} \log(e^{(1+e^{-x})^{2^x \log x}}) = e^{\lim_{x\to +\infty} \log((1+e^{-x})^{2^x \log x})} =... | Claude Leibovici | 82,404 | <p>Consider the more general case of $$y=(1+e^{-x})^{a^x\, \log(x)}$$ Tak logarithms of both sides $$\log(y)={a^x \log(x)}\log(1+e^{-x})$$ When $x$ is large $$\log(1+e^{-x})\sim e^{-x}$$ making $$\log(y)\sim \left(\frac a e\right)^x \log(x)$$ Now, consider the cases where $a<e$ and $a >e$.</p>
|
1,618,373 | <p>Prove that $S_4$ cannot be generated by $(1 3),(1234)$</p>
<p>I have checked some combinations between $(13),(1234)$ and found out that those combinations cannot generated 3-cycles.</p>
<p>Updated idea:<br>
Let $A=\{\{1,3\},\{2,4\}\}$<br>
Note that $(13)A=A,(1234)A=A$<br>
Hence, $\sigma A=A,\forall\sigma\in \langl... | Marc van Leeuwen | 18,880 | <p>The partition $\{\{1,3\},\{2,4\}\}$ is invariant under the action of the two proposed generators, but not under all of $S_4$, so they cannot generate all of $S_4$.</p>
|
1,925,867 | <p>I can't find any. For saying $H$ is a subgroup of $G$ we have notation but it seems none exists for subrings.</p>
| Daniel Buck | 293,319 | <p>As you asked for the notation, I read this as how subrings are identified in the literature so I looked up the definitions for subrings in a few well known books. I checked the books on Algebra by Artin, Dummit & Foote, Hungerford, Jacobson, Lang, van der Waerden, where notation for a subring seems to be nonexis... |
111,795 | <p>I need to add a small graphics on top of a larger one, and the small graphics should stick very close to the large one, with their axis aligned. Here's a minimal code to work with, using some elements from this question/answer :</p>
<p><a href="https://mathematica.stackexchange.com/questions/22521/how-to-make-a-pl... | ubpdqn | 1,997 | <p>I think it may perhaps be easier just to combine plots and modify (e.g. suppress unnecessary frame ticks). I post this as a motivating answer rather than definitive answer. <code>li</code> is a modified version of OP function:</p>
<pre><code>li[p_, q_, phi_, {l_, u_}] :=
DensityPlot[(If[p > 0, Sin[2 Pi p^2 x]/... |
111,795 | <p>I need to add a small graphics on top of a larger one, and the small graphics should stick very close to the large one, with their axis aligned. Here's a minimal code to work with, using some elements from this question/answer :</p>
<p><a href="https://mathematica.stackexchange.com/questions/22521/how-to-make-a-pl... | Jason B. | 9,490 | <p>Just going to throw this in to the mix. When I saw this question, it immediately seemed perfect for Jens's <a href="https://mathematica.stackexchange.com/a/6882/9490">function</a>, which I modified and used previously, and in fact I have it defined in my init.m because I use it with such regularity.</p>
<p>I have ... |
111,795 | <p>I need to add a small graphics on top of a larger one, and the small graphics should stick very close to the large one, with their axis aligned. Here's a minimal code to work with, using some elements from this question/answer :</p>
<p><a href="https://mathematica.stackexchange.com/questions/22521/how-to-make-a-pl... | Edmund | 19,542 | <p>This can be achieved by matching a few layout options between the two charts. Namely, <code>ImageSize</code>, <code>ImageMargins</code>, <code>ImagePadding</code>, <code>PlotRange</code>, and <code>PlotRangePadding</code>. <code>Scaled</code> and <code>Automatic</code> can be used to simplify selection.</p>
<p>Wit... |
833,376 | <p>I know very little in the way of math history, but I question that was bothering me recently is where the terms open and closed came from in topology. I know that it's easy to ascribe a sense of openness/closedness to said sets, but I feel like there are a lot of other, more appropriate words that could have been us... | Alexander Gruber | 12,952 | <p>This answer addresses your question on who first actually used the words "open" and "closed" to describe open and closed sets. It seems, according to <a href="http://www.sciencedirect.com/science/article/pii/S0315086008000050" rel="nofollow">this article</a>, like the first mention of this language was in René Bair... |
2,789,002 | <p>How can I calculate the height of the tree? I am with geometric proportionality.</p>
<p><img src="https://i.stack.imgur.com/m4zMD.png"></p>
| Mr Pie | 477,343 | <p>I will denote by $a\cdot b$ the product of $a$ and $b$ (referred to as <em>dot multiplication</em>).</p>
<hr>
<p>You want to find the gradient of the dotted line. Since it is straight, it is in <em>linear form</em>, namely, $$y=mx+b\quad\text{ or }\quad y=mx+c.\tag*{$\bigg(\begin{align}&\text{depending on how}... |
213,338 | <p>Suppose that $f$ is analytic in the unit disc D = {$z \in \mathbb{C}$ : |$z$| < 1} and $|$f($z$)$| \le 1/(1-|$z$|)$ for all $z\in D$.</p>
<p>Let $f($z$)= \sum _{n=0}^{\infty } a_nz^n$ be the power series expansion of f about $0$.</p>
<p>Prove that $$|a_n| \le (n+1)(1+1/n)^{n} < e(n+1)$$</p>
| Max Morin | 44,026 | <p>$x^x = e^{x\ln x}$, so $$\lim_{x\to 0}x^x=\lim_{x\to 0}e^{x\ln x}=e^0=1$$.</p>
|
826,011 | <p>I want to express variable from equation in WolframAlpha web. I tried several keywords but it didn't work. For example I have equation</p>
<p>$$y=x+z+k,$$</p>
<p>and I want Wolfram to rewrite it for variable $x$, $x=y-z-k$. Is it possible and how?</p>
| ronald | 155,944 | <p>You can use the <a href="http://goo.gl/NwzZDQ" rel="noreferrer">following query</a>:</p>
<pre><code>solve(y=x+z+k,x)
</code></pre>
<p><img src="https://i.stack.imgur.com/UOrXe.png" alt="enter image description here"></p>
|
826,011 | <p>I want to express variable from equation in WolframAlpha web. I tried several keywords but it didn't work. For example I have equation</p>
<p>$$y=x+z+k,$$</p>
<p>and I want Wolfram to rewrite it for variable $x$, $x=y-z-k$. Is it possible and how?</p>
| MattAllegro | 142,842 | <p>To complete my previous comment, <a href="https://www.wolframalpha.com/input/?i=solution+for+variable+x+of%3A+y%3Dx%2Bz%2Bk" rel="nofollow noreferrer">this</a> is what you may type to achieve what you asked.</p>
<p>I just copied and pasted your equation in WolphramAlpha, the remaining "syntax" was suggested to me b... |
3,320,193 | <blockquote>
<p>If given <span class="math-container">$P(B\mid A) =4/5$</span>, <span class="math-container">$P(B\mid A^\complement)= 2/5$</span> and <span class="math-container">$P(B)= 1/2$</span>, what is the probability of <span class="math-container">$A$</span>?</p>
</blockquote>
<p>I know I need to apply Bayes ... | Robert Z | 299,698 | <p>All you need is the definition of conditional probability:
<span class="math-container">$$P(X|Y)=P(X\cap Y)/P(Y).$$</span>
We have that
<span class="math-container">$$P(A^c\cap B)=P(B|A^c)P(A^c)=\frac{2}{5}(1-P(A))$$</span>
and
<span class="math-container">$$P(A \cap B)=P(B|A)P(A)=\frac{4}{5}P(A).$$</span>
Hence
<... |
3,320,193 | <blockquote>
<p>If given <span class="math-container">$P(B\mid A) =4/5$</span>, <span class="math-container">$P(B\mid A^\complement)= 2/5$</span> and <span class="math-container">$P(B)= 1/2$</span>, what is the probability of <span class="math-container">$A$</span>?</p>
</blockquote>
<p>I know I need to apply Bayes ... | MR_BD | 195,683 | <p><a href="https://i.stack.imgur.com/hJgeo.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/hJgeo.png" alt="enter image description here" /></a></p>
<p>Use the Venn diagram. Let the blue side be the even that <span class="math-container">$A$</span> occurs. Also inside the ellipse be the event that <s... |
3,029,208 | <p>Hi I have been trying to find a way to find a combinatorial proof for <span class="math-container">${kn \choose 2}= k{n \choose 2}+n^2{k \choose 2}$</span>. </p>
| Santana Afton | 274,352 | <p>As an addendum to Daniel Robert-Nicoud’s excellent hint, I find it helpful to rewrite the equality in the following way:</p>
<p><span class="math-container">$$\binom{kn}{2} = \binom{k}{1}\binom{n}{2} + \binom{k}{2}\binom{n}{1}\binom{n}{1}.$$</span></p>
<p>By reading multiplication as “and then,” and addition as “o... |
2,895,284 | <blockquote>
<p>Find $\frac{d}{dx}\frac{x^3}{{(x-1)}^2}$</p>
</blockquote>
<p>I start by finding the derivative of the denominator, since I have to use the chain rule. </p>
<p>Thus, I make $u=x-1$ and $g=u^{-2}$. I find that $u'=1$ and $g'=-2u^{-3}$. I then multiply the two together and substitute $u$ in to get:</p... | Rhys Hughes | 487,658 | <p>It's better to use the quotient rule:
$$\frac{d(\frac fg)}{dx}=\frac{f'g-g'f}{g^2}$$
$$f=x^3\to f'=3x^2$$
$$g=(x-1)^2\to g'=2(x-1)$$
$$\to\frac{d(\frac {x^3}{(x-1)^2})}{dx}=\frac{3x^2(x-1)^2-2x^3(x-1)}{(x-1)^4}=\frac{x^2(x-3)}{(x-1)^3}$$</p>
|
4,243,344 | <blockquote>
<p><span class="math-container">${43}$</span> equally strong sportsmen take part in a ski race; 18 of
them belong to club <span class="math-container">${A}$</span>, 10 to club and 15 to club <span class="math-container">${C}$</span>. What is the
average place for (a) the best participant from club <span c... | user2661923 | 464,411 | <p>First see the answer of Mike Earnest, and the comments following his answer. Apparently, my computation numerically agrees with his answer.</p>
<hr />
<p>Alternative approach:</p>
<p><span class="math-container">$\underline{\text{Problem 1:}}$</span></p>
<p>For <span class="math-container">$k \in \{1,2,\cdots,34\},... |
244,433 | <p>I have a list:</p>
<pre><code>data = {{2.*10^-9, 0.0025}, {4.*10^-9, 0.0025}, {6.*10^-9, 0.0025}, {8.*10^-9, 0.0025}, {1.*10^-8, 0.0025}, {7.*10^-9, 0.0023}, {3.*10^-9, 0.0025},...}
</code></pre>
<p>And I wanted to remove every third pair and get</p>
<pre><code> newdata = {{2.*10^-9, 0.0025}, {4.*10^-9, 0.0025}, {8.... | Chris Degnen | 363 | <p>Using <code>Partition</code>, padded with <code>Nothing</code></p>
<pre><code>Flatten[Take[#, UpTo[2]] & /@ Partition[data, 3, 3, {1, 1}, Nothing], 1]
</code></pre>
|
1,400,352 | <p>Confusion with the eccentricity of ellipse. On <a href="https://en.wikipedia.org/wiki/Ellipse#Directrix" rel="nofollow noreferrer">wikipedia</a> I got the following in the directrix section of ellipse.</p>
<blockquote>
<p>Each focus F of the ellipse is associated with a line parallel to the minor axis called a di... | Harish Chandra Rajpoot | 210,295 | <p><strong>Hint</strong>: If the eccentricity $e$ & the major axis $2a$ of an ellipse are known then we have the following </p>
<ol>
<li>Distance of each focus from the center of ellipse
$$=\text{(semi-major axis)}\times \text{(eccentricity of ellipse)}=\color{red}{ae}$$</li>
<li>Distance of each directrix from th... |
1,400,352 | <p>Confusion with the eccentricity of ellipse. On <a href="https://en.wikipedia.org/wiki/Ellipse#Directrix" rel="nofollow noreferrer">wikipedia</a> I got the following in the directrix section of ellipse.</p>
<blockquote>
<p>Each focus F of the ellipse is associated with a line parallel to the minor axis called a di... | wltrup | 232,040 | <p>If I understood your question correctly, you're essentially asking how one can find the equation for the directrix if one only has the equation for an ellipse with a given eccentricity.</p>
<p>You start with the equation below
$$
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\qquad(1)
$$
where $a$ and $b$ are positive real... |
1,876,133 | <p>Let x be a object that is not a set</p>
<p>Let S be a set</p>
<p>Would the following statement:</p>
<p>x ⊆ S</p>
<p>evaluate to False, or considered not a well formed statement (as x is not even a set).</p>
| Ted Shifrin | 71,348 | <p><strong>HINT</strong>: Mean Value Theorem. (Consider $x\le x_0$ and $x\ge x_0$ separately.)</p>
|
1,876,133 | <p>Let x be a object that is not a set</p>
<p>Let S be a set</p>
<p>Would the following statement:</p>
<p>x ⊆ S</p>
<p>evaluate to False, or considered not a well formed statement (as x is not even a set).</p>
| Asinomás | 33,907 | <p>Prove the following with the mean value theorem, given that $x<y$:</p>
<ul>
<li>$f(x)\leq f(y)$</li>
<li>if $f(x)=f(y)$ then $x_0\in(x,y)$</li>
<li>$f(x)<f(\frac{x+y}{2})\leq f(y)$ or $f(x)\leq f(\frac{x+y}{2})<f(y)$</li>
</ul>
|
39,684 | <p>In order to have a good view of the whole mathematical landscape one might want to know a deep theorem from the main subjects (I think my view is too narrow so I want to extend it).</p>
<p>For example in <em>natural number theory</em> it is good to know quadratic reciprocity and in <em>linear algebra</em> it's good... | Eric Naslund | 6,075 | <p>In Analytic Number Theory, the <a href="http://en.wikipedia.org/wiki/Prime_number_theorem" rel="nofollow">Prime Number Theorem</a> is a great result. </p>
<p>Riemann's 1859 paper, which outlined a possible approach to proving the PNT, is credited with motivating a large amount of the research done in Complex Analy... |
39,684 | <p>In order to have a good view of the whole mathematical landscape one might want to know a deep theorem from the main subjects (I think my view is too narrow so I want to extend it).</p>
<p>For example in <em>natural number theory</em> it is good to know quadratic reciprocity and in <em>linear algebra</em> it's good... | Community | -1 | <ul>
<li>In Group theory i would say "Structure theorem for finitely generated abelian groups"</li>
</ul>
|
145,303 | <p>Another question about the convergence notes by Dr. Pete Clark:</p>
<p><a href="http://alpha.math.uga.edu/%7Epete/convergence.pdf" rel="nofollow noreferrer">http://alpha.math.uga.edu/~pete/convergence.pdf</a></p>
<p>(I'm almost at the filters chapter! Getting very excited now!)</p>
<p>On page 15, Proposition 4.6 st... | Carl | 18,702 | <p>Note that a separable metric space has a countable basis. Specifically, we take a countable dense subset $S$ and take the set of balls centered at $s$ with radius $1/n$ for each $n \in N$, $s \in S$. This can be checked to be a basis. So then $(ii) \Rightarrow (i)$ is proven, which is the missing link.</p>
|
3,583,778 | <p>This problem came to me when I was solving another binomial coefficient summation problem in this site.</p>
<p>I want to prove that <span class="math-container">$\sum_{i=0}^{p}{\sum_{j=0}^{q+i}{\sum_{k=0}^{r+j}{\binom{p}{i}\binom{q+i}{j}\binom{r+j}{k}}}}=4^{p}3^{q}2^{r}$</span></p>
| Rezha Adrian Tanuharja | 751,970 | <p><span class="math-container">$$
\begin{aligned}
(3+x)^{p}(2+x)^{q}(1+x)^{r}&=(1+(2+x))^{p}(2+x)^{q}(1+x)^{r}\\
&=\sum_{i=0}^{p}{\binom{p}{i}(2+x)^{i}(2+x)^{q}(1+x)^{r}}\\
&=\sum_{i=0}^{p}{\binom{p}{i}(2+x)^{q+i}(1+x)^{r}}\\
&=\sum_{i=0}^{p}{\binom{p}{i}(1+(1+x))^{q+i}(1+x)^{r}}\\
&=\sum_{i=0}^{p}... |
1,840,352 | <blockquote>
<p>For every $A \subset \mathbb R^3$ we define $\mathcal{P}(A)\subset \mathbb R^2$ by
$$ \mathcal{P}(A) := \{ (x,y) \mid \exists_{z \in \mathbb R}:(x,y,z) \in A \} \,. $$
Prove or disprove that $A$ closed $\implies$ $\mathcal{P}(A)$ closed and $A$ compact $\implies$ $\mathcal{P}(A)$ compact.</p>
</bl... | Shashi | 349,501 | <p><strong>Hint:</strong></p>
<p>The first one is not true.</p>
<p>For the second one try this:</p>
<p>I assume you know Heine Borel's theorem about compact sets.</p>
<p>Heine Borel says $A$ is bounded. So for all components of $(x_1,x_2,x_3)\in A $ you can find an $M>0$: $|x_i|<M$. That means $|x_i|^2<M^2... |
1,485,310 | <blockquote>
<p>Write a formula/formulae for the following sequence:</p>
<p>b). 1,3,6,10,15,...</p>
</blockquote>
<p>I am not getting any pattern here, from which to derive a formula.
This sequence does not look like the examples I could solve: like</p>
<blockquote>
<p>a) 1,0,1,0,1...</p>
</blockquote>
<p>where I got t... | PTDS | 277,299 | <p>Mike Pierce has already given the answer. </p>
<p>Still let me mention that the $n$-th term of the sequence is the sum of the first $n$ natural numbers. </p>
<p>Hence $$s_n = \frac{n(n+1)}{2}$$</p>
<p>It is also possible to derive the above by solving the recurrence relation given by Amey Deshpande.</p>
|
4,345,671 | <p>I have a series of cubic polynomials that are being used to create a trajectory. Where some constraints can be applied to each polynomial, such that these 4 parameters are satisfied.
-Initial Position
-final Position
-Initial Velocity
-final Velocity</p>
<p>The polynomials are pieced together such that the ends of o... | Andrew D. Hwang | 86,418 | <p>As noted in the comments and Karl's (+1) answer, the issue is in flattening the "<a href="https://en.wikipedia.org/wiki/Gore_(fabrics)" rel="nofollow noreferrer">gores</a>," which does not preserve area.</p>
<p>One correct reckoning is instead to use a quasi-triangle with two right angles at the base, angl... |
764,905 | <p>Calculate $$\int_{D}(x-2y)^2\sin(x+2y)\,dx\,dy$$ where $D$ is a triangle with vertices in $(0,0), (2\pi,0),(0,\pi)$.</p>
<p>I've tried using the substitution $g(u,v)=(2\pi u, \pi v)$ to make it a BIT simpler but honestly, it doesn't help much.</p>
<p>What are the patterns I need to look for in these problems so I ... | Santosh Linkha | 2,199 | <p>Here is how I calculate those types of integral on plane.
$$\int_D 1\,dy dx = \mathrm{Area \;of \; triangle}$$</p>
<p>Now, area of triangle is just area under the straight line.
$$\int_0^{2\pi}\left ( \pi - \frac x2\right ) dx$$
Now I change single integral into double integral,
$$\int_0^{2\pi}\left(\int_0^{ \pi - ... |
90,656 | <p>In the introduction to 'A convenient setting for Global Analysis', Michor & Kriegl make this claim: "The study of Banach manifolds per se is not very interesting, since they turnout to be open subsets of the modeling space for many modeling spaces."</p>
<p>But finite-dimensional manifolds are found to be intere... | Liviu Nicolaescu | 20,302 | <p>I am of two minds on this topic. It is much easier to work on Banach manifolds because the implicit function theorem on such spaces has a simple formulation. On the other hand, as the examples of gauge theory or the theory of pseudo-holomorphic curves show, in these contexts one works not with one Banach... |
679,544 | <p>How to prove this for positive real numbers?
$$a+b+c\leq\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}$$</p>
<p>I tried AM-GM, CS inequality but all failed.</p>
| r9m | 129,017 | <p>Using Cauchy-Schwarz Inequality twice:</p>
<p>$a^4 + b^4 +c^4 \geq a^2b^2 +b^2c^2 +c^2a^2 \geq ab^2c +ba^2c +ac^2b = abc(a+b+c)$</p>
|
679,544 | <p>How to prove this for positive real numbers?
$$a+b+c\leq\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}$$</p>
<p>I tried AM-GM, CS inequality but all failed.</p>
| zhangwfjh | 115,670 | <p>I have come up with an answer with myself. Using CS inequality
$$(a^4+b^4+c^4)(1+1+1)\geq(a^2+b^2+c^2)^2$$
$$(a^2+b^2+c^2)(1+1+1)\geq(a+b+c)^2$$
Hence we have
$$a^4+b^4+c^4\geq\frac{(a+b+c)^4}{27}=(a+b+c)\left(\frac{a+b+c}{3}\right)^3\geq abc(a+b+c)$$</p>
|
525,611 | <p>Let $f: [0,1] \to \mathbb{R}$ be defined by letting $f = 0 $ on $\mathcal{C}$, the Cantor set and $f(x) = k $ for every $x$ in each interval of lenght $\frac{1}{3^k}$ which has been removed from $[0,1]$. We want to calculate $\int\limits_{[0,1]} f dm $.</p>
<p>How CAn I express $f$ as a simple function?</p>
<p>So,... | Dietrich Burde | 83,966 | <p>Euclid's formula generates all primitive triples, and this can be modified to give all triples: set $a = k\cdot(m^2 - n^2) ,\ \, b = k\cdot(2mn) ,\ \, c = k\cdot(m^2 + n^2)$, where $m, n$, and $k$ are positive integers with $m > n, m − n$ odd, and with $m$ and $n$ coprime. Then $a^2+b^2=c^2$, and $c$ is the given... |
525,611 | <p>Let $f: [0,1] \to \mathbb{R}$ be defined by letting $f = 0 $ on $\mathcal{C}$, the Cantor set and $f(x) = k $ for every $x$ in each interval of lenght $\frac{1}{3^k}$ which has been removed from $[0,1]$. We want to calculate $\int\limits_{[0,1]} f dm $.</p>
<p>How CAn I express $f$ as a simple function?</p>
<p>So,... | poetasis | 546,655 | <p>You can use equations I created for a paper I am writing. To find a triplet with a matching <span class="math-container">$sideC$</span> for any odd number (<span class="math-container">$C_1$</span>), set <span class="math-container">$$k_c=\frac{-(2n-1)+\sqrt{2C_1-(2n-1)^2}}{2}$$</span></p>
<p>Now try values of <spa... |
3,200,330 | <p>Suppose <span class="math-container">$\Phi: A\to A$</span> is a transformation of the set <span class="math-container">$A$</span>. I want to understand what it means for a subset <span class="math-container">$B\subseteq A$</span> to be invariant under <span class="math-container">$\Phi$</span>. </p>
<p><a href="htt... | José Carlos Santos | 446,262 | <p>Yes, they are equivalent. Asserting that <span class="math-container">$A\subset B$</span> is equivalent to asserting that <span class="math-container">$(\forall a\in A):a\in B$</span>. And asserting that <span class="math-container">$\Phi(B)\subset B$</span>, in particular, is equivalent to <span class="math-contain... |
2,315,647 | <p>Compute the gravitational attraction on a unit mass at the origin due to the mass (of constant density) occupying the volume inside the sphere $r = 2a$ and above the plane $z=a$. Use spherical coordinates.</p>
<p>So I know the function should be
$$(G/r^2) dM$$
What are the limits of integration? What should the in... | Rafa Budría | 362,604 | <p>$0\leq\theta\leq\arccos(a/2a)$ or $0\leq\theta\leq\pi/3$; $0\leq\phi\leq 2\pi$ and $a/\cos\theta\leq r\leq 2a$</p>
<p>$$F=\int_0^{\pi/3}\int_0^{2\pi}\int_{a/\cos\theta}^{2a}\dfrac{\hat r\rho}{r^2}r^2\sin\theta d\phi d\theta dr$$</p>
<p>With $\hat r=\cos\phi\sin\theta\hat x+\sin\phi\sin\theta\hat y+\cos\theta\hat z... |
11,519 | <p>Hello,</p>
<p>If $f:\mathbb{R} \to \mathbb{R}$ a differentiable function, it is very easy to find its Lipschitz constant. Is there any way to extend this to functions $f: \mathbb{R} \to \mathbb{R}^n$ (or similar)?</p>
| Johannes Hahn | 3,041 | <p>In fact a statement similar to what was described by Pete Clark is true for all normed vector spaces:
Let $X$ and $Y$ be normed vector spaces. A (total) differentiable function $f:X\to Y$ is Lipschitz iff its derivative is bounded. Every upper bound for the differential is a Lipschitz constant.</p>
<p>One direction... |
2,208,113 | <p>Let $x$ and $y \in \mathbb{R}^{n}$ be non-zero column vectors, from the matrix $A=xy^{T}$, where $y^{T}$ is the transpose of $y$. Then the rank of $A$ is ?</p>
<hr>
<p>I am getting $1$, but need confirmation .</p>
| dxiv | 291,201 | <p>Hint: $\;x,y,z \ge \frac{1}{4}$ for the square roots to be defined. Assume WLOG that $x \ge y \ge z\,$, then $\sqrt{4z-1}=x+y \ge z+z = 2z \implies 4z-1 \ge 4z^2\iff (2z-1)^2 \le 0 \,$.</p>
|
752,045 | <p>Write the area $D$ as the union of regions. Then, calculate $$\int\int_Rxy\textrm{d}A.$$</p>
<p>First of all I do not get a lot of parameters because they are not defined explicitly (like what is $A$? what is $R$?).</p>
<p>Here is what I did for the first question:</p>
<p>The area $D$ can be written as:</p>
<p>$... | fgp | 42,986 | <p>The easiest way to do this is to write $A$ as $$
A = Q \setminus \bigl(
\underbrace{\{(x,y) \in Q \mid y > 1+x^2\}}_{:=B_1} \cup
\underbrace{\{(x,y) \in Q \mid x > y^2\}}_{:=B_2}
\big)
$$
where $Q = [-1,1]\times[-1,2]$ (i.e. a rectangle), $B_1$ is the missing part at the top and $B_2$ the missin... |
752,045 | <p>Write the area $D$ as the union of regions. Then, calculate $$\int\int_Rxy\textrm{d}A.$$</p>
<p>First of all I do not get a lot of parameters because they are not defined explicitly (like what is $A$? what is $R$?).</p>
<p>Here is what I did for the first question:</p>
<p>The area $D$ can be written as:</p>
<p>$... | robjohn | 13,854 | <p><strong>Hint:</strong> the area inside each of the parabolic indentations is
$$
\int_{-1}^1(1-x^2)\,\mathrm{d}x
$$</p>
|
1,556,298 | <p>If we have $p\implies q$, then the only case the logical value of this implication is false is when $p$ is true, but $q$ false.</p>
<p>So suppose I have a broken soda machine - it will never give me any can of coke, no matter if I insert some coins in it or not.</p>
<p>Let $p$ be 'I insert a coin', and $q$ - 'I ge... | kviiri | 187,461 | <p>Implication is just like any other logical connective, and in fact, $a \to b$ can also be written using negation and disjunction as $\neg a \lor b$. Just as with logical disjunction and conjunction, it is very possible to know the value of the entire operation by observing just one operand - if $a = F$ we know $a \t... |
166,925 | <p>I have a function <code>u[y]</code> and I want to find the limit of integration that integration is equal zero.</p>
<pre><code>Λ = -30;
u[η_] := (2*η - 2*η^3 + η^4) + Λ/6*(η - 3*η^2 + 3*η^3 - η^4);
θ = Integrate[u[η]*(1 - u[η]), {, 0, 1}] // N;
δ = 1/θ;
u[y_] := Piecewise[{{1,y > δ}}, (2*y/δ - 2*(y/δ)^3 + (y/δ)^... | chuy | 237 | <p>You can do something like this (assuming of course that the <code>InterpolatingFunction</code> <code>f1</code> and <code>f2</code> are generating using the same mesh):</p>
<pre><code>f12 = ElementMeshInterpolation[{mesh}, f1["ValuesOnGrid"] + f2["ValuesOnGrid"]]
</code></pre>
<p>Where <code>ElementMeshInterpolatio... |
887,200 | <p>So I have the permutations:
$$\pi=\left( \begin{array}{ccc}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
2 & 3 & 7 & 1 & 6 & 5 & 4 & 9 & 8
\end{array} \right)$$
$$\sigma=\left( \begin{array}{ccc}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9... | evinda | 75,843 | <p>You mixed them up.If you look at the permutation matrix $\pi \sigma $ you have to look ,firstly at $\sigma$ and then at $\pi$.</p>
<p>For example,for the first column of $\pi \sigma$,it is:</p>
<p>$$1 \to 9, 9 \to 8$$</p>
|
539,448 | <p>$a,b,c,d,e>o$. Show that</p>
<p>$$ a^{b+c+d+e}+ b^{c+d+e+a}+c^{ d+e+a+b}+ d^{e+a+b+c}+e^{a+b+c+d}>1$$ </p>
| KeithS | 23,565 | <p>Consider cases in which $a=b=c=d=e$. In such cases, the inequality (as I think you wanted to write it) simplifies to $5x^{4x} > 1$. To figure out if this is true, we need to consider the left side as a function of $x$, and find the vertex of this function. To do that, you need to take the derivative, $f'(x)$, and... |
2,536,553 | <p>I know that is possible to apply the spectral decomposition (diagonalization) to a matrix when the sum of the dimensions of its eigenspaces is equal to the size of the matrix.</p>
<p>The spectral decomposition is:</p>
<p>$$
F=P\Lambda P^{-1}
$$</p>
<p>where $\Lambda$ is the diagonal matrix of eigenvalues and $P$ ... | Nightgap | 506,645 | <p>According to JayTuma you get (w.l.o.g. $i=1$):</p>
<p>$q_1\ldots q_m=p_1\ldots p_n=u_1q_1p_2\ldots p_n$</p>
<p>so $q_2\ldots q_m=u_1p_2\ldots p_n$ since every PID is an integral domain. Now substitute $p_2$ by (w.l.o.g.) $u_2q_2$ to get $q_3\ldots q_m=u_1u_2p_3\ldots p_n$ and so on. From this you easily deduce tha... |
2,337,583 | <p>I cannot understand the inductive dimension properly. I read something on Google but mostly there only are conditions or properties. Not a definition. I got to know about it from the book “ The fractal geometry of nature”. ( I am a 12 grader.)</p>
| Henno Brandsma | 4,280 | <p>The usual definition of the small inductive dimension <span class="math-container">$\operatorname{ind}(X)$</span> is as follows and defines <span class="math-container">$\operatorname{ind}(X) \le n$</span> by recursion: <span class="math-container">$\operatorname{ind}(X)$</span> is a function from topological spaces... |
1,767,880 | <p>Let $A$ be a square matrix in the form $A=B+O(h^2)$, where $B$ is a fixed matrix, and $O(h^2)$ is a matrix with very small elements. Assume that: $$(I-A)^{-1}=I+A+A^2+A^3+...$$
How can I esimate the right hand side of the equality above ? Is that right $(I-B)^{-1}+O(h^2) ?$</p>
| Josh Hunt | 282,747 | <p><strong>Edit:</strong> <em>the below answer is incorrect - the binomial theorem doesn't apply because $O(h^2)$ doesn't necessarily commute past $B$.</em></p>
<p>A rather slapdash (<em>i.e.</em> non-rigourous) way of doing it is just to substitute $A = B + O(h^2)$ in the right-hand side, and notice that by the binom... |
1,767,880 | <p>Let $A$ be a square matrix in the form $A=B+O(h^2)$, where $B$ is a fixed matrix, and $O(h^2)$ is a matrix with very small elements. Assume that: $$(I-A)^{-1}=I+A+A^2+A^3+...$$
How can I esimate the right hand side of the equality above ? Is that right $(I-B)^{-1}+O(h^2) ?$</p>
| Community | -1 | <p>Let $A=B-K$, where $K$ is a small variable matrix. Then, by the Taylor's formula $(I-A)^{-1}=(I-B+K)^{-1}=(I-B)^{-1}-(I-B)^{-1}K(I-B)^{-1}+O(||K||^2)$. If $||K||=O(h^2)$, then $||(I-B)^{-1}K(I-B)^{-1}||=O(h^2)$ and we are done.</p>
|
43,172 | <p>I am trying to solve $\frac{dx}{dt} + \alpha x = 1$, $x(0) = 2$, $\alpha > 0$ where $\alpha$ is a constant. </p>
<p>[some very badly done mathematics deleted]</p>
<p>Continuing with Gerry's suggestion:</p>
<p>$\log|1-\alpha x | = -t\alpha + \log|1-2\alpha|$</p>
<p>$1-\alpha x = e^{-t\alpha}(1-2\alpha)$</p>
<... | joriki | 6,622 | <p>This is an inhomogeneous first-order linear ordinary differential equation. The standard way to solve such an equation is to find all solutions of the corresponding homogeneous equation and then add any particular solution of the inhomogeneous equation.</p>
<p>The inhomogeneous equation is solved by a constant:</p>... |
3,710,710 | <p>Suppose <span class="math-container">$A(t,x)$</span> is a <span class="math-container">$n\times n$</span> matrix that depends on a parameter <span class="math-container">$t$</span> and a variable <span class="math-container">$x$</span>, and let <span class="math-container">$f(t,x)$</span> be such that <span class="m... | Jacky Chong | 369,395 | <p>Let <span class="math-container">$y=(y_1, \ldots, y_n) = (f_1(t, x), \ldots, f_n(t, x))$</span> Consider the <span class="math-container">$a_{ij}$</span> entry, then we see that
<span class="math-container">\begin{align}
\frac{d}{dt}a_{ij}(t, f(t,x)) = \partial_ta_{ij}(t, f(t, x))+\sum^n_{k=1} \underbrace{a_{ij, k}... |
172,894 | <p>Suppose $A$ is an integral domain with integral closure $\overline{A}$ (inside its fraction field), $\mathfrak{q}$ is a prime ideal of $A$, and $\mathfrak{P}_1,\ldots,\mathfrak{P}_k$ are the prime ideals of $\overline{A}$ lying over $\mathfrak{q}$. Show that $\overline{A_\mathfrak{q}} = \bigcap\overline{A}_\mathfrak... | Community | -1 | <p>Consider $\dim A=1$. One can assume that $A$ is a local Noetherian domain with maximal ideal $m$ and have to prove that $\overline{A}=\bigcap_{i=1}^n\overline{A}_{P_i}$, where $P_1,\dots,P_n$ are all the prime ideals of $\overline{A}$ lying over $m$. (There are only finitely many primes in $\overline{A}$ lying over ... |
3,425,415 | <p>I need to define a bijection <span class="math-container">$f:\mathbb Q\to\mathbb Q$</span> such that <span class="math-container">$f(0) = 0$</span> and <span class="math-container">$f(1) = 1$</span> while also preserving order (i.e. if <span class="math-container">$a < b$</span>, then <span class="math-container"... | Rushabh Mehta | 537,349 | <p>There are plenty of examples. The following function <span class="math-container">$f:\mathbb Q\to\mathbb Q$</span> is one example: <span class="math-container">$$f(x) = \begin{cases}2x&x<0\\x&x\geq0\end{cases}$$</span>Now, if you also require that the bijection preserves addition/multiplication, then such... |
899,230 | <p>It seems that both isometric and unitary operators on a Hilbert space have the following property:</p>
<p><span class="math-container">$U^*U = I$</span> (<span class="math-container">$U$</span> is an operator and <span class="math-container">$I$</span> is an identity operator, <span class="math-container">$^*$</spa... | glS | 173,147 | <p>In finite dimensions, there is a straightforward characterisation of isometries and unitaries in terms of their matrix representations.</p>
<p>The basic observation is that <span class="math-container">$U^*U=I$</span> means that the columns of (the matrix representation of) <span class="math-container">$U$</span> ar... |
405,866 | <p>Original question:
For compact metric spaces, plenty of subtly different definitions converge to the same concept. Overtness can be viewed as a property dual to compactness. So is there a similar story for overt metric spaces?</p>
<p>Edit: Since overtness is trivially true assuming the Law of the Excluded Middle, cl... | Ingo Blechschmidt | 31,233 | <p>The spectrum of a commutative ring, defined as the classifying locale of its prime filters, is overt if and only if any element is nilpotent or not nilpotent (Proposition 12.51 in <a href="https://rawgit.com/iblech/internal-methods/master/notes.pdf#page=130" rel="noreferrer">these notes of mine</a>).</p>
<p>Hence fo... |
4,463,373 | <p>If <span class="math-container">$\frac{\partial f}{\partial x}(0,0) = \frac{\partial f}{\partial y}(0,0) = 0$</span>, then <span class="math-container">$f'((0,0);d)=0$</span> (directional derivative) for every direction <span class="math-container">$d \in \mathbb{R}^n$</span>.</p>
<p>Is this true? I'm trying to find... | Fred | 380,717 | <p>For <span class="math-container">$x>0$</span> we have <span class="math-container">$f(x)=x^4$</span>, hence <span class="math-container">$f'(x)=4x^3$</span>.</p>
<p>For <span class="math-container">$x<0$</span> we have <span class="math-container">$f(x)=-x^4$</span>, hence <span class="math-container">$f'(x)=-... |
55,076 | <p>Let $T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 $ be the linear operator </p>
<p>$$
A = \begin{pmatrix}
1 & 0 & 0 \\
1 & 2 & 0 \\
0 & 0 & 3 \end{pmatrix}
$$</p>
<p>The problem I am trying to understand is the following. </p>
<p>True or False? If $W$ is a $T$-invariant subspace of $\ma... | Geoff Robinson | 13,147 | <p>I am not used to your notation, but I think I understand what you mean. If a linear transformation $T$ on an $n$-dimensional vectors space $V$ has $n$ distinct eigenvalues,
say $\{\lambda_1, \lambda_2,\ldots, \lambda_n \}$, then $V$ has a basis consisting of eigenvectors of $T$. Most proofs of this fact make (at lea... |
1,275,461 | <p>I am trying to proof that $-1$ is a square in $\mathbb{F}_p$ for $p = 1 \mod{4}$. Of course, this is really easy if one uses the Legendre Symbol and Euler's criterion. However, I do not want to use those. In fact, I want to prove this using as little assumption as possible.</p>
<p>What I tried so far is not really... | Daniel Fischer | 83,702 | <p>On $\mathbb{F}_p^\ast$, define the equivalence relation</p>
<p>$$x\sim y :\!\!\iff (x = y) \lor (x = -y) \lor (xy = 1) \lor (xy = -1).$$</p>
<p>Generally, the equivalence class of $x$ has four elements, $\{ x, -x, x^{-1}, -x^{-1}\}$. Since we always have $x \neq -x$, every class has at least two elements, and if $... |
2,312,016 | <p>Prove limit of three variables using (ε, δ)-definition.</p>
<p>$$\lim_{(x, y, z)\to (0, 1, 2)} (3x+3y-z)=1$$</p>
<p>I have no idea how to do this with three variables.</p>
| SBM | 422,110 | <p>We need to show that for every $\varepsilon > 0$, there is a value $\delta \gt 0$ such that $\left|x - 0\right| \lt \delta, \left|y-1\right| \lt \delta, \text{and } \left|z-2\right| \lt \delta$ imply that $\left|3x + 3y - z - 1 \right| \lt \varepsilon$</p>
|
158,662 | <p>I know to prove a language is regular, drawing NFA/DFA that satisfies it is a decent way. But what to do in cases like</p>
<p>$$
L=\{ww \mid w \text{ belongs to } \{a,b\}*\}
$$</p>
<p>where we need to find it it is regular or not. Pumping lemma can be used for irregularity but how to justify in a case where it can... | hmakholm left over Monica | 14,366 | <p>Suppose the language was regular and had a DFA.</p>
<p>After reading, for example, "$\underbrace{aa\ldots a}_nb$" the DFA is in some state, and the identity of this state determines <em>completly</em> what the rest of an input that the machine accepts can be.</p>
<p>But if $n\ne m$, then the possible tails that ca... |
2,764,381 | <p>one thing I don't understand is what is sin(0) and sin(1) exactly? I am alright with the concept of radian (pi) but don't understand 0 and 1. What does it mean?</p>
| INDIAN | 556,781 | <p>If $1$ is considered as $1$ radian then $1$ less than $\pi /2$ so $sin x$ is a strictly increasing function on that interval.</p>
|
3,505,397 | <p>well, I have to find the Taylor polynomial of <span class="math-container">$f(x,y)=\sin(x)\sin(y)$</span> at <span class="math-container">$(0,\pi/4)$</span>. I found:</p>
<p>Is <span class="math-container">$T_3(x,y)=-\frac{1}{12}\sqrt{2}x(16x^2+48y^2-24\pi+3\pi^2)$</span> correct?</p>
| Ryan Shesler | 585,375 | <p><span class="math-container">$\sin(x) = \cos(\frac{\pi}{2} - x)$</span>.</p>
<p>Using this you get <span class="math-container">$\cos^{-1} (\cos(\frac{\pi}{2} - \frac{16\pi}{7}))$</span>. Then you can finish it from here (but keep in mind the domain of <span class="math-container">$\cos^{-1}(x)$</span>)</p>
|
3,505,397 | <p>well, I have to find the Taylor polynomial of <span class="math-container">$f(x,y)=\sin(x)\sin(y)$</span> at <span class="math-container">$(0,\pi/4)$</span>. I found:</p>
<p>Is <span class="math-container">$T_3(x,y)=-\frac{1}{12}\sqrt{2}x(16x^2+48y^2-24\pi+3\pi^2)$</span> correct?</p>
| lab bhattacharjee | 33,337 | <p><span class="math-container">$$f(x)=\sin\dfrac{16\pi}7=\sin\dfrac{2\pi}7=\cos(\dfrac\pi2-\dfrac{2\pi}7)=\cos\dfrac{\pi(7-4)}{14}$$</span></p>
<p>Now <span class="math-container">$\cos^{-1}f(x)=2n\pi\pm\dfrac{3\pi}{14}$</span> where <span class="math-container">$n$</span> is an integer such that <span class="math-co... |
578,337 | <p>For $n=1,2,3,\dots,$ and $|x| < 1$ I need to prove that $\frac{x}{1+nx^2}$ converges uniformly to zero function. How ?. For $|x| > 1$ it is easy. </p>
| Community | -1 | <p>Let $f_n(x)=\frac{x}{1+nx^2}$ then we have
$$f'_n(x)=\frac{1-nx^2}{(1+nx^2)^2}=0\iff x=n^{-1/2}:=x_n$$
hence
$$||f_n||_\infty=f_n(x_n)=\frac{1}{2}x_n\to0$$
so we have the uniform convergence to $0$.</p>
|
136,086 | <p>I've been given the following problem as homework:</p>
<blockquote>
<p>Q: <strong>Compute the number of subgraphs of <span class="math-container">$K_{15}$</span> isomorphic to <span class="math-container">$C_{15}$</span></strong>.</p>
<p><span class="math-container">$K_{15}$</span> means complete graph with 15 verti... | Yuval Filmus | 1,277 | <p>Let's start with the case $n = 3$. A triangle in $K_{15}$ is given by any three vertices. So the number of triangles is $\binom{15}{3}$.</p>
<p>Let's proceed to the case $n = 4$. A rectangle in $K_{15}$ is given by any four vertices. But the vertices $a,b,c,d$ define three different rectangles. Indeed, suppose that... |
3,168,119 | <p>How do I solve for n?</p>
<p><span class="math-container">$125 = x * 2^n$</span></p>
<p>This is what I have so far:</p>
<p><span class="math-container">$5^3 = x * 2^n$</span></p>
<p>I do remember that according to the exponential rules,
that the powers should be the same if the equation is like this:</p>
<p><sp... | YiFan | 496,634 | <p>You cannot solve for <span class="math-container">$n$</span>, unless you want your answer to be in terms of <span class="math-container">$x$</span>. In that case, <span class="math-container">$125/x=2^n$</span> tells you <span class="math-container">$n=\log_2(125/x)=3\log_2(5)-\log_2(x)$</span>.</p>
<p>The reason i... |
2,240,405 | <p>The question asks me to find the Laurent series of $$f(z) = {5z+2e^{3z}\over(z-i)^6}\,\,at\,\,z=i$$I know the following $$e^z=\sum_{n=0}^\infty {z^n\over n!}$$ What I want to know, is if I can do this: $$={1\over (z-i)^6}\sum_{n=0}^\infty ({2(3z)^n\over n!}+5z)$$ $$=\sum_{n=0}^{n=6}(z-i)^{n-6}({2(3z)^n\over n!}+5z)... | Angina Seng | 436,618 | <p>For this kind of problem I recommend changing the variable to push the pole to zero. Here I'd set $w=z-i$. The problem becomes:
find the Laurent series of
$$g(w)=\frac{5(w+i)+2e^{3w+3i}}{w^6}$$
at $w=0$. The numerator is
$$5i+5w+2e^{3i}\left(1+3w+\frac{3^2w^2}{2}+\frac{3^3w^3}{3!}+\cdots\right)$$
etc.</p>
|
1,009,503 | <p>Theorem 15 in Chapter 15 of Peter Lax's functional analysis book says</p>
<p>$X$ is a Banach space, $Y$ and $Z$ are closed subspaces of $X$ that complement each other $X = Y \oplus Z$, in the sense that every $x\in X$ can be decomposed uniquely as $x = y+z$ where $y\in Y$, $z\in Z$. Denote the two complements of $... | MJD | 25,554 | <p>If the probability of getting tails is $q = 1-p$, then you expect to get, on average, $q$ tails per throw, because that is exactly what a probability is: the average number of tails per throw.</p>
<p>Expectations are additive, so if you get $q$ tails per throw it requires $\frac nq$ throws in order to expect $n$ ta... |
1,369,990 | <p>I came across a quesion -
<a href="https://www.hackerrank.com/contests/ode-to-code-finals/challenges/pingu-and-pinglings" rel="nofollow">https://www.hackerrank.com/contests/ode-to-code-finals/challenges/pingu-and-pinglings</a></p>
<p>The question basically asks to generate all combinations of size k and sum up the ... | Shuaib Lateef | 880,979 | <p>I guess this is a general case for size 2:
<span class="math-container">$$\sum_{i=1}^{n-1}\sum_{j=i+1}^na_ia_j=\frac{(\sum_{i=1}^na_i)^2-\sum_{i=1}^na_i^2}{2}$$</span></p>
|
1,353,498 | <p>The problem is to prove or disprove that there is a noncyclic abelian group of order $51$. </p>
<p>I don't think such a group exists. Here is a brief outline of my proof:</p>
<p>Assume for a contradiction that there exists a noncyclic abelian group of order $51$.</p>
<p>We know that every element (except the iden... | coldnumber | 251,386 | <p>Using the Sylow theorems would shorten your proof considerably, because from the first Sylow theorem it follows that if a prime divides the order of a group, then the group contains an element of that order. (This eliminates the need to check cases where all elements have order 3 or all have order 17.)</p>
<p>Hence... |
3,118,462 | <p>cars arrives according to a Poisson process with rate=2 per hour and trucks arrives according to a Poisson process with rate=1 per hour. They are independent. </p>
<p>What is the probability that <strong>at least</strong> 3 cars arrive before a truck arrives? </p>
<p>My thoughts:
Interarrival of cars A ~ Exp(2 p... | Ross Millikan | 1,827 | <p>You can use the <a href="https://en.wikipedia.org/wiki/Quadratic_formula" rel="nofollow noreferrer">quadratic formula</a>. Any quadratic <span class="math-container">$ax^2+bx+c$</span> factors as <span class="math-container">$a\left(x+ \frac{b+\sqrt {b^2-4ac}}{2a}\right)\left(x+ \frac{b-\sqrt {b^2-4ac}}{2a}\right)$... |
9,462 | <p>In my question I ask for practical tips for the mathematical research practice, if not personal,I look for some articles/websites/books/guides/faq related, or if was already asked on Math.SE, the link to the question.</p>
<p><a href="https://math.stackexchange.com/questions/386520/practical-tips-research-and-discov... | Douglas S. Stones | 139 | <p>I cast the fifth re-open vote (and thus has been re-opened). Seems like a reasonable enough question to me. There seems to be one answer in the comments already.</p>
<p>Note that, in order to close a question, merely 5 close votes are required. From this, we cannot deduce that there is a consensus for closing. ... |
1,383,380 | <p>On page 12 of Stein, Shakarchi textbook 'Complex analysis', the authors state that the <em>Cauchy-Riemann equations link complex and real analysis</em>. I have completed courses on real and complex analysis, but I feel that this is somewhat of an over-statement. But perhaps it is just me which doesnt have a good eno... | Brian Tung | 224,454 | <p>There does not <em>have</em> to be a fixed point $x$ such that $f(x) = x$. Here's a counterexample: the constant function $f(x) = 2$. Obviously, there is no $x \in [0, 1]$ for which $x = 2$.</p>
<p>Given a function $f(x)$ as described in the problem, consider the function $g(x) = f(x)-2x$. We have $g(0) \in [0, ... |
464,489 | <p>We are given $H = \{(1),(13),(24),(12)(34),(13)(24),(14)(23),(1234),(1432)\}$ is a subgroup of $S_4$. Also assume $K = \{(1),(13)(24)\}$ is a normal subgroup of $H$. Show $H/K$ isomorphic to $Z_2\oplus Z_2$. </p>
<p>This is just a practice question (not assignment). So I have tried finding $H/K$ explicitly.</p>
<p... | user66733 | 66,733 | <p>I think your doubt about these two being isomorphic has been solved but just in case you still don't see how to define a precise isomorphism between them note that $\mathbb{Z}_2 \bigoplus \mathbb{Z}_2 $ has two generators, namley $(1,0)$ and $(0,1)$ also $(1,0) + (0,1) = (1,1)$. Therefore I think it's very easy for... |
1,017,965 | <p>Just as the title, my question is what is the matrix representation of Radon transform (Radon projection matrix)? I want to have an exact matrix for the Radon transformation. </p>
<p>(I want to implement some electron tomography algorithms by myself so I need to use the matrix representation of the Radon transforma... | Yi-Chun | 642,317 | <p>Currently I am solving the same problem as you guys.
I found a way to find the matrix.</p>
<p>First, we may view image(MxN) as a vector by certain reshape.
Second, set the ONLY one entry to 1, and others to 0.
Third, input this image to the function 'radon' then you way get the transformed image(LxP)</p>
<p>Loop o... |
3,345,329 | <p>In Bourbaki Lie Groups and Lie Algebras chapter 4-6 the term displacement is used a lot. For example groups generated by displacements. But I can not find a definition of the term displacement given anywhere. I also looked at Humphreys Reflection Groups and Coxeter groups book but I could not find it. Can someone pr... | Jake Levinson | 43,565 | <p>This is not an answer, but it's too long for a comment.</p>
<p>Clearly, the answer must scale proportionally to <span class="math-container">$\sigma^2$</span>, so let's just assume <span class="math-container">$\sigma=1$</span>.</p>
<p>Sampling <span class="math-container">$x_1, \ldots, x_n$</span> i.i.d. from a s... |
2,307,785 | <p>Things I understand:</p>
<p>Shannon entropy- </p>
<ul>
<li>is the expected amount of information in an event from that
distribution. </li>
<li>In game of 20 questions to guess an item, it is the
lower bound on the number questions one could ask.</li>
</ul>
<p>Doubt:</p>
<p>It gives the lower bound on the number ... | spaceisdarkgreen | 397,125 | <p>To see why there needs to be a notion of lower bound, consider that you need to specify a code. You could choose an arbitrarily inefficient code that takes a ton of bits to encode anything. So this is a statement about how well you <em>can</em> do.</p>
<p>To see why it must be an average, note that in general the l... |
3,725,007 | <p>Consider the rings, <span class="math-container">$Z_2[x]/(1 + x^2)$</span> and <span class="math-container">$ Z_2[x]/(1 + x + x^2)$</span>, despite having different polynomial as divisor, I have been told that -</p>
<p><span class="math-container">$$Z_2[x]/(1 + x^2) = \{0, 1, x, 1 + x\}$$</span></p>
<p>and</p>
<p><s... | Oliver Díaz | 121,671 | <p>Let <span class="math-container">$G(x)=\int^x_0\frac{\sin t}{t}\,dt$</span>. If One proves that <span class="math-container">$|G(x)|\leq M$</span> for some <span class="math-container">$M>0$</span> and all <span class="math-container">$x\geq0$</span>, then</p>
<p><span class="math-container">$$|G(b)-G(a)|\leq |G(... |
2,853,278 | <p><a href="https://math.stackexchange.com/a/203701/312406">This answer</a> suggests the idea, that a local ring $(R, \mathfrak{m})$ whose maximal ideal is nilpotent is in fact an Artinian ring.</p>
<p>Is this true? If so, how is it proven?</p>
| Louis | 75,278 | <p>You need that $R$ is noetherian, else there are counterexamples. </p>
<p>E.g., take $R = K[x_i]_{i \in \mathbb{N}}/(x_i | i \in \mathbb{N})^2$.</p>
<p>If $R$ is Noetherian, this is a special case of <a href="https://stacks.math.columbia.edu/tag/00KD" rel="noreferrer">Lemma 10.59.4 here</a> which states that a Noet... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.