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 |
|---|---|---|---|---|
3,386,069 | <p>Let <span class="math-container">$$A=\{\phi \in \mathcal C ([0,1], \mathbb R) \mid \forall k \in\mathbb N^+:\int^1_0 x^k\phi(x) =0\}$$</span></p>
<p>Clearly, the function <span class="math-container">$[0,1] \to \mathbb R, \, x \mapsto 0$</span> belongs to <span class="math-container">$A$</span>. I would like to ask... | Kavi Rama Murthy | 142,385 | <p>Let <span class="math-container">$g(x)=x\phi(x)$</span>. Then <span class="math-container">$\int_0^{1} x^{k} g(x)dx=0$</span> for <span class="math-container">$k=0,1,2...$</span>. Using Wierstrass approximation it follows that <span class="math-container">$\int_0^{1} g(x)^{2} dx=0$</span> so <span class="math-contai... |
125,201 | <p>I would like to show that every power series expansion for an entire function converges everywhere.</p>
| Keaton | 27,765 | <p>Maybe something is wrong with this answer, but it seems to be pretty simple.</p>
<p>First, we know that the power series of an analytic function is unique. So if a function is entire (analytic in the whole complex plane), then its power series is unique on the whole plane, and by definition is convergent.</p>
|
125,201 | <p>I would like to show that every power series expansion for an entire function converges everywhere.</p>
| robjohn | 13,854 | <p>Using <a href="http://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem" rel="noreferrer">Cauchy's Theorem</a> and integration by parts yields
$$
\begin{align}
\left|\frac{f^{(n)}(w)}{n!}\right|
&=\left|\frac{1}{2\pi i}\oint\frac{f^{(n)}(w+z)}{n!\,z}\mathrm{d}z\right|\\
&=\left|\frac{1}{2\pi i}\oint\frac... |
476,070 | <p>We consider $T:\mathbb{C}\to\mathbb{C}$ defined by $T(z)=\lambda z+\mu\overline{z}$, where $\lambda ,\mu\in\mathbb{C}$. I want to prove that if $T$ is injective, then $\lambda\cdot\overline{\lambda}\neq\mu\cdot\overline{\mu}$.</p>
<p>Clearly, if $\mu =0$, then $\lambda\neq 0$ because $T$ is injective.</p>
<p>Now, ... | Jonas Meyer | 1,424 | <p>To "generalize" your reformulation to an equivalent statement: Given $a\in\mathbb C$ with $|a|=1$, can we find $z\in\mathbb C\setminus\{0\}$ such that $z=a\overline z$?</p>
<p>The answer is yes. E.g., restricting to $|z|=1$, we can reformulate this as finding a solution $x\in\mathbb R$ to the equation $e^{ix}=e^{i... |
2,549,087 | <p>Hi was wondering if there's anyway you can figure out the value of two terms when you're giving the average of the set they're in, and the values of the other numbers in the set.
Based on a revision question I'm trying to figure out that goes like:
"The integers from 1 to 9 are written on a white board. When m 8s an... | SK19 | 509,159 | <p>I think you can't be certain. Given $s $ being the sum of the rest, $n $ the size of the set (I assume that's given too) an $m $ the mean of the set, we want to know $a $ and $b $ but only know the equality</p>
<p>$$\frac {s+a+b}{n}=m $$</p>
<p>One equality and two variables means there is no unique solution.</p>
|
153,578 | <p>I currently have $$\lim_{h\to 0}\frac{\sqrt[3]{h+1}-1}{h}$$</p>
<p>Now, I know that when you have square root instead of a cubic root it's easy. You just multiply by the conjugate and simplify afterwards. If it were a sqrt I know that the limit is 1/2. I know that the result here is 1/3 but I can't seem to get ther... | DonAntonio | 31,254 | <p>Another way without knowing before hand the derivative:$$A^3-1=(A-1)(A^2+A+1)\Longrightarrow A-1=\frac{A^3-1}{A^2+A+1}$$and putting $\,A=\sqrt[3]{h+1}\,$ one gets:$$\frac{\sqrt[3]{h+1}-1}{h}=\frac{\rlap{/}{h}+1-1}{\rlap{/}{h}\left(\sqrt[3]{(h+1)^2}+\sqrt[3]{h+1}+1\right)}\to\frac{1}{1+1+1}=\frac{1}{3}$$</p>
|
1,547,977 | <p>I have this set equation</p>
<p>$$([1,2]\times(2,5))\cap\left((\Bbb R\times[3,4])\cup\left(\left[\frac12,\frac32\right]\times\Bbb R\right)\right)$$</p>
<p>which is quite hard for me, I know that it's supposed to be a triangle but I really don't know how to draw it correct. </p>
<p>How could I write this in set th... | Ethan Alwaise | 221,420 | <p>One way is to use the class equation for $D_{10}$. Any element in the center must be in its own conjugacy class, and a normal subgroup must be a union of conjugacy classes. However, I think producing the class equation is actually more laborious in this case than just directly inspecting the group.</p>
<p>$D_{10}$ ... |
2,693,266 | <p>My teacher told me to prove this using the proof by contradiction. So, I am going to assume that there's an infinite group with finitely many subgroups. Then this will lead to finding at least one element with an infinite order, from which I can derive a contradiction. </p>
<p>But I am lost how I can prove this... ... | D_S | 28,556 | <p>If $G$ contains an element $x$ of infinite order, then you're done.</p>
<p>If all the elements of $G$ have finite order, then pick one, say $x$. Then $H = \{1_G, x, x^2, ... \}$ is a finite set as well as a subgroup of $G$. Since $G$ is infinite, you can find a $y\in G$ which is not in $H$. Then $H' = \{1_G, y, y... |
2,693,266 | <p>My teacher told me to prove this using the proof by contradiction. So, I am going to assume that there's an infinite group with finitely many subgroups. Then this will lead to finding at least one element with an infinite order, from which I can derive a contradiction. </p>
<p>But I am lost how I can prove this... ... | DanielWainfleet | 254,665 | <p>Let $G$ be an infinite group. For $x\in G$ let $G(x)=\{x^j:j\in \Bbb Z\}.$ </p>
<p>(I). If $G(x)$ is an infinite set for some $x\in G$ then for $n\in \Bbb N$ let $H(n)=\{x^{nj}:j\in \Bbb Z\}.$ Then $H(n)$ is a sub-group of $G$ and $n\ne n'\implies H(n)\ne H(n').$</p>
<p>(II). If $G(x)$ is finite for every $x\in G... |
430,126 | <p>I've asked two years ago a post on Mathematics Stack Exchange, were provided two excellent answers. I'm asking on MathOverflow in the hope that some professor can to expand/improve (if it is possible) these results answering my question. The post has the same title and identifier <a href="https://math.stackexchange.... | joro | 12,481 | <p><strong>EDIT</strong> This answer doesn't satisfy OP's assumptions.
Currently keeping it because of existence of infinitely many solutions over extensions of the integers and the relation with N-conjecture.</p>
<hr />
<p>If you set <span class="math-container">$m=n$</span> you have infinitely many solutions <span cl... |
3,063,097 | <p>On the German Wikipedia page of the Hurwitz Zeta Function I have come across the <a href="https://de.wikipedia.org/wiki/Hurwitzsche_Zeta-Funktion#Weitere" rel="nofollow noreferrer">following formula</a></p>
<blockquote>
<p><span class="math-container">$$\zeta\left(2,\frac14\right)~=~\pi^2+8G\tag1$$</span></p>
</bloc... | Ali Shadhar | 432,085 | <p>We can use the generalization identity </p>
<p><span class="math-container">$$\psi^{(n)}(a)=-\int_0^1\frac{x^{a-1}\ln^n(x)}{1-x}\ dx$$</span></p>
<p>set <span class="math-container">$n=1$</span> and <span class="math-container">$a=1/4$</span> we get</p>
<p><span class="math-container">$$\psi^{(1)}\left(\frac14\ri... |
1,223,398 | <p>Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function satisfying $g'(x) > 0$ for all $x \neq 0$. Prove that $g$ is one-to-one.</p>
<p>Proof: Case 1: Consider $x_{2} > x_{1}>0$. Then by the Mean Value Theorem, there exists a point $x \in \mathbb{R}$ such that $g'(x) = \frac{g(x_{2}) - g(... | Daniel | 150,142 | <p>You need the function to be differentiable over the interval $(x_1,x_2)$ and that's why you're considering the case $x_1\leq 0 < x_2$ apart. However, as I've said, the only thing you need to complete the proof is considering continuity of $g$ at $0$. This is a necessary condition (and you have it from differentia... |
2,288,535 | <p>Three fair coins are flipped. Let x be the number of heads. Let y equal to 1 if all coins land the same outcome I.e hhh or ttt, and 0 if otherwise. Calculate Cov(X,Y)</p>
| Doug M | 317,162 | <p>$Cov_{X,Y} = E[XY] - E[X]E[Y]$</p>
<p>We have 8 outcomes.</p>
<p>HHH $\implies XY = 3$
for all other combinations $XY = 0$</p>
<p>$E[XY] = \frac 38$</p>
<p>$E[X] = \frac 32, E[Y] = \frac 14$</p>
<p>$E[XY] - E[X]E[Y] = 0$</p>
<p>Which isn't really a surprise.</p>
|
260,991 | <p>(a) Does there exist a function $f$ defined on the <em>open</em> interval $(a,b)$ such that $f'(b^-)$ exists, and $\lim_{x\to b-}f'(x)\neq f'(b^-)$, or (b) where $f'(b^-)$ exists and $\lim_{x\to b-}f'(x)$ does not exist?</p>
<p>Since $f(b)$ is undefined, define $$f'(b^-)=\lim_{h\to0+}\frac{f(b-h)-f(b-2h)}h.$$ Are t... | Robert Israel | 8,508 | <p>Consider e.g. a function of the form $f(x) = g(\log_2(b-x))$ where $g$ is periodic with period $1$. This satisfies $f(b-2x) = f(b-x)$, and thus according to your definition $f'(b-) = 0$, while $f'(x) = - \dfrac{g'(\log_2(b-x))}{(b-x) \ln 2}$, so $\lim_{x \to b-} f'(x)$ will not exist unless $g$ is constant. </p>
... |
145,171 | <p>Can the Bourbaki series be used profitably by undergraduates and high school students?Are we the target audience? I came across the N.Bourbaki texts while surfing the internet(I have not had the opportunity to access them though) and was a bit amazed by the Wikipedia article suggesting they did mathematics very rigo... | Georges Elencwajg | 3,217 | <p>1) No, you are not the target audience. </p>
<p>2) No undergraduate is. </p>
<p>3) For every subject an undergraduate might want to learn there are books better suited than Bourbaki: ask about them here.</p>
<p>4) In a strictly historic sense the target audience was Henri Cartan and the aim of Bourbaki was to... |
2,199,650 | <p>I'm utterly stuck with no where to go. The assignment is to complete the indirect proof. I'm stuck on the following step, and have no clue how to proceed. Where do I go? Also, pardon the poor formatting. I have no clue how properly format here. I can't space it like I'd like, or use the symbol I'd like.</p>
<p>$ p ... | Bram28 | 256,001 | <p>You really need to add some parentheses to disambiguate! For example, is the goal $\neg p \land (q \lor r)$, or is it $(\neg p \land q) \lor r$? Those two statements are not the same! (use a truth table to verify that)</p>
<p>Likewise, is the second premise $s \rightarrow (r \lor \neg t)$, or is it $(s \rightarrow... |
4,787 | <pre><code>a := az + 1
b := bz + 5
list := {a, b}
</code></pre>
<p>I'd like mathematica to print</p>
<pre><code>a = az + 1
b = bz + 5
</code></pre>
<p>so basically it needs to first print the name of the variable in the list, followed by "=", and followed by the actual content of the variable.</p>
<p><strong>update... | Andy Ross | 43 | <p>This seems to work. I'm doubtful that it is very robust though.</p>
<pre><code>SetAttributes[printVar, {HoldAll, Listable}]
printVar[a_] := Row[{Defer[a], " = ", OwnValues[a][[1, 2]]}]
</code></pre>
<p>For example...</p>
<pre><code>Column[printVar[{a,b}]]
==> a = 1 + az
b = 5 + bz
</code></pre>
<p><st... |
23,029 | <p>I used computer program to generate numbers of this form, then verify if they are amicable. However, it took too long to finish, is there any special technique to verify if a pair is amicable? </p>
<p>One pair that I found is $m = 284 = 2^{2}.71, n = 220 = 2^{2}.5.11$ </p>
<p>Thanks,<br>
Chan</p>
| deinst | 943 | <p>This problem was solved by Euler who generalized a rule of Thabit ibn Kurrah, see <a href="http://www.ams.org/journals/mcom/1986-46-173/S0025-5718-1986-0815849-1/S0025-5718-1986-0815849-1.pdf" rel="nofollow">Breeding Amicable Pairs in Abundance</a> by Borho and Hoffmann for some history. To compute pairs using Eule... |
23,029 | <p>I used computer program to generate numbers of this form, then verify if they are amicable. However, it took too long to finish, is there any special technique to verify if a pair is amicable? </p>
<p>One pair that I found is $m = 284 = 2^{2}.71, n = 220 = 2^{2}.5.11$ </p>
<p>Thanks,<br>
Chan</p>
| Michel Marcus | 18,268 | <p>See also Amicable pairs, a survey
<a href="http://oai.cwi.nl/oai/asset/4143/04143D.pdf" rel="nofollow">http://oai.cwi.nl/oai/asset/4143/04143D.pdf</a>
Cheers.</p>
|
124 | <p>Harold Williams, Pablo Solis, and I were chatting and the following question came up.</p>
<p>In Lie group land (where you're doing differential geometry), given a finite-dimensional Lie algebra <em>g</em>, you can find a faithful representation <em>g</em> → End(V) by Ado's theorem. Then you can take the group ... | Ben Webster | 66 | <p>My suspicion is yes, at least over C, and that the thing to do is take the Zariski closure in GL(V) of the exponentials of the Lie algebra elements. Of course, over random fields, one doesn't have this trick. </p>
<p>Might a trick like looking at the subgroup of GL(V) fixing all invariant polynomials for the Lie ... |
1,826,866 | <p>I can't understand why the projection of those two objects on plane $xy$ gives that the angle's range from -90 degress to 90 degrees. why not from 0 to 360?</p>
<p><a href="https://i.stack.imgur.com/rFDpf.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/rFDpf.png" alt="enter image description here... | Olivier Oloa | 118,798 | <p><em>But how did they show that $x > 0$?</em></p>
<blockquote>
<p>Observe that $x^2+y^2+z^2=2x$ implies $x\geq0.$</p>
</blockquote>
|
682,170 | <p>Determine the largest 3-digit prime factor of ${2000 \choose 1000}$. </p>
<p>I could not approach the problem at all. I have no idea how to try the problem. Please help.</p>
| Hawk | 105,577 | <p>To find the solution, I need to find a number that comes exactly thrice in $2000!$ but exactly once in $1000!$ as, $3-2=1$. We get the $2$ as $1000!$ occurs twice. So, such a number is the prime just below $666$ which is $661$.</p>
|
236,235 | <p>Prove that among any 100 integers there are always two whose difference is divisible by 99. How can I prove this?</p>
| Bill Dubuque | 242 | <p>Here's a proof requiring no knowledge of division or remainders. Suppose a counterexample exists, i.e. there are $100$ integers $\rm\:a_1,\ldots, a_{100}\:$ such that $\rm\:i\ne j\:\Rightarrow\:99\nmid a_i-a_j.\:$ Then they are distinct, so we may index them so that they're strictly ordered $\rm\:a_1 < a_2 < \... |
3,227,020 | <p>let <span class="math-container">$f[0,1]\to [0,1]$</span>,and <span class="math-container">$f(x)$</span>is increasing funcion(does not mean strictly increasing) such following two condtion
<span class="math-container">$$f(x)+f(1-x)=1$$</span>
and
<span class="math-container">$$\dfrac{f(x)}{2}=f\left(\dfrac{x}{3}... | José Carlos Santos | 446,262 | <p>The paths <span class="math-container">$C_1$</span> and <span class="math-container">$C_2$</span> are homotopic in the domain of <span class="math-container">$\frac{f(z)}{z-w}$</span>. Therefore, by the homotopic version of Cauchy's integral theorem,<span class="math-container">$$\int_{C_1-C_2}\frac{f(z)}{w-z}\,\mat... |
258,806 | <p>It was pretty hard finding a short and precise title. Here is my problem:</p>
<p>The equation $$\bigg|\int_\gamma f(z)\text{d}z\bigg|\le\int_\gamma\big|f(z)||\text{d}z|$$ holds if $f$ is integrable (where $\gamma$ is a differentiable path $[0,1]\rightarrow\mathbb{C}$).
However, in what cases does "$=$" apply? Some ... | robjohn | 13,854 | <p>For simplicity's sake define
$$
g(t)=f(\gamma(t))\gamma'(t)\tag{1}
$$
Note that
$$
\int_0^1|g(t)|\,\mathrm{d}t
=\sup_{|u(z)|=1}\mathrm{Re}\left(\int_0^1g(t)u(t)\,\mathrm{d}t\right)\tag{2}
$$
and
$$
\left|\,\int_0^1g(t)\,\mathrm{d}t\,\right|
=\sup_{\substack{|u(z)|=1\\u\text{ constant}}}\mathrm{Re}\left(\int_0^1g(t)u... |
1,634,519 | <blockquote>
<p>Let $a \in \mathbb{R}$ and
$$
A_a =
\begin{pmatrix}
1 & a & 1 \\
a & 2 & 3 \\
2 & 3 & 4
\end{pmatrix}.
$$</p>
<ol>
<li>Say for each values of $a \in \mathbb{R}$ the system:
$$
A_a
\begin{pmatrix}
x \\ y \\ z
\end{pmatrix}
=
\begin{pmatrix}
1 \\ 0 \\... | mvw | 86,776 | <p>Writing $A_a$ short as $A$, and the inhomogenous system $A x = b$ as augmented matrix $[A|b]$ we get these transformations:
$$
\left[
\begin{array}{rrr|r}
1 & a & 1 & 1 \\
a & 2 & 3 & 0 \\
2 & 3 & 4 & 1
\end{array}
\right]
\to
\left[
\begin{array}{rrr|r}
1 & a & 1 &a... |
2,462,448 | <blockquote>
<p>Find all integers $m,n$ such that $$(562-5(m+n))(82-5(m-n))=98^2.$$</p>
</blockquote>
<p>What is a simpler method to tackle this? I know the results but my approach is kind of lengthy. I considered every possible factor of $98^2$ to get the result. Thanks.</p>
| Piquito | 219,998 | <p>$$(562-5S)(82-5D)=98^2=2^2\cdot7^4$$ The number $98^2$ has $15$ divisors,
$$d\in\{1,2,4,7,14,28,49,98,196,343,686,1372,2401,4802,9604\}$$
Because of $82-5D=d\iff82-d=5D$, it follows that the only divisors to be considered are $d\in\{2,7,1372,4802\}$.</p>
<p>It follows the possibilities
$$(D,S)\in\{(16,-848),(15,-... |
3,232,057 | <p><span class="math-container">$$\lim_{n \to \infty }n\int_{0}^{1}\frac{x^{n}}{1+x+x^{n}}dx$$</span></p>
<p>I factorize <span class="math-container">$x^n$</span> then I tried with <span class="math-container">$u=1+x^{1-n}$</span> but I didn't get too far. Also I tried to get something from <span class="math-container... | weirdo | 91,524 | <p>In my proof, I shall use Lebesgue's Dominated Convergence Theorem.</p>
<p>We have (I ignore the <span class="math-container">$dx$</span> for brevity)
<span class="math-container">$$
\int_0^1\frac{nx^n}{1+x+x^n}
=\int_0^1\frac{x(nx^{n-1}+1)-x}{1+x+x^n}
=\int_0^1\frac{x(nx^{n-1}+1)}{1+x+x^n}-\int_0^1\frac{x}{1+x+x^n}... |
104,698 | <p>I have a 36 inch piece of wood (Length) I will placing 6 coat hanger hooks along the length of the wood. The first hook will be placed 1 inch down the length from one end and the last hook will be placed 1 inch from the other end of this 36 inch piece of wood. Question how to find the equal distance between the ot... | CheapskateWoodworker.info | 316,203 | <p>Here's a simple formula for figuring this out.</p>
<p><strong>(A-(Bx2))/(C-1)</strong> </p>
<p>where:</p>
<p><strong>A</strong> = Length of your board </p>
<p><strong>B</strong> = Distance of First Hook From Edge of Board </p>
<p><strong>C</strong> = Total Number of Hooks</p>
<p><a href="https://www.dropbox.co... |
1,929,319 | <p>We know that a polynomial in single variable/indeterminate is defined algebraically as-</p>
<blockquote>
<p>An expression of the form $a_{n}x^n+a_{n-1}x^{n-1}+\dots+a_{2}x^2+a_{1}x+a_{0}$, <p> where $a_n,\dots,a_0$ are constants/parameters and $x$ is the indeterminate/variable.</p>
</blockquote>
<p>But I was won... | Community | -1 | <p>The graph of a zero$^{th}$ degree polynomial (a constant) is an horizontal. That of a first degree polynomial as straight line and that of a quadratic is a parabola. The cubic has a center of symmetry which is an inflection point. The higher order polynomials have no particularly remarkable shapes/geometric characte... |
1,386,750 | <p>The title pretty much says it all. I am trying to figure out how to take the $n$-th root of a number $m$ where $n$ is a fractional number. Unfortunately all I could turn up on Google was how to take a square root of a fraction, which is quite different. Is such an operation even possible?</p>
| Augustin | 241,520 | <p>No, pointwise convergence does not suffice, you need a stronger form of convergence for that to be true (e.g. uniform, monotone, dominated).</p>
|
4,432,171 | <p>The lim is about "when x approachs a, then y approachs L". Then, shouldn't the epsilon and delta be like "For all delta, no matter how small the delta is, you can always find an epsilon that makes ε < f(x)-L < ε"? But, the conventional explanation says like "for all epsilon, you find d... | pancini | 252,495 | <p>This is a common misunderstanding, and the only response I can ever think of is to just examine what the definition is really getting at.</p>
<p>The statement <span class="math-container">$\lim\limits_{x\to a}f(x)=L$</span> means that, when <span class="math-container">$x$</span> is close to <span class="math-contai... |
1,540,667 | <blockquote>
<p>Consider a triangle having integer sides such that so side is greater than $4$ units.How many such triangles are possible?</p>
</blockquote>
<p>I could not solve it by trying to use combinatorics.</p>
<p>So, how to do it?</p>
<p>Thanks fro any help.</p>
<p>The original question came from here: <a ... | Ian Miller | 278,461 | <p>The solutions in the PDF linked are incorrect. They have calculated $4\times3\times2$ which is only counting scalene triangles including rotations and reflections and then it excludes 7 without even listing them all. I would strong recommend you get official solutions from the Indian Association of Physics Teachers ... |
973,037 | <p>I was looking at sample questions from my textbook and I came across something interesting that I need a little help understanding</p>
<p>The question was to find the derivative of:</p>
<p>$\log_{10} (\frac{7}{\sqrt {x+3}})$</p>
<p>My solution was:</p>
<p>$\frac{1}{2(x+3)}$</p>
<p>My problem is that when I late... | taninamdar | 66,212 | <p>Assume your function was $\dfrac{d}{dx}\log_{10}(f(x)) = \dfrac{d}{dx} \dfrac{\ln(f(x))}{\ln 10} = \dfrac{f'(x)}{f(x)\cdot \ln10}$.</p>
<p>This is because $\log_{10}(a) = \dfrac{\ln a}{\ln 10}$, where $\ln$ is natural logarithm. Also $\dfrac{d}{dx}\ln(f(x)) = \dfrac{f'(x)}{f(x)}$ holds only when base is $e$, i.e. $... |
2,426,726 | <p>So I know that </p>
<p>$$\bar{x}y + x\bar{y} = x \oplus y$$ </p>
<p>Can this be applied to something like</p>
<p>$$ \bar{a}\bar{b}cd + ab\bar{c}\bar{d} $$</p>
<p>to get
$$ab \oplus cd$$</p>
| Especially Lime | 341,019 | <p>(Assuming I'm interpreting your notation correctly: $\overline{x}$ is "not $x$", etc.)</p>
<p>No, because the negation of $ab$ isn't $\overline{a}\overline{b}$, it's $\overline{a}+\overline{b}$. So you could write $$(\overline a+\overline b)cd+ab(\overline c+\overline d)=ab\oplus cd.$$</p>
|
3,491,523 | <p>What is the formula to calculate</p>
<p><span class="math-container">$$n^0 + n^1 + n^2 + n^3 + \cdots + n^k$$</span></p>
<p>Calculate example</p>
<p><span class="math-container">$$185^0 + 185^1 + 185^2 + \cdots + 185^{13}$$</span></p>
<p>Any hints appreciated!</p>
| Con | 682,304 | <p>This is the geometric sum formula:</p>
<p>If <span class="math-container">$n \neq 1$</span>, we have</p>
<p><span class="math-container">$$\sum_{i = 0}^k n^i = \frac{1-n^{k+1}}{1-n}.$$</span></p>
|
1,155,854 | <p>Investigate convergence of the following series: $$\sum_{n=0}^\infty\left( \frac{2+(-1)^n}{\pi} \right)^n$$
Which convergence criterion shoul be applied?</p>
| StackTD | 159,845 | <p>If $n$ is odd, then:</p>
<p>$$\left( \frac{2+(-1)^n}{\pi} \right)^n = \left( \frac{2+(-1)}{\pi} \right)^n
= \left( \frac{1}{\pi} \right)^n$$</p>
<p>If $n$ is even, then:</p>
<p>$$\left( \frac{2+(-1)^n}{\pi} \right)^n = \left( \frac{2+(+1)}{\pi} \right)^n
= \left( \frac{3}{\pi} \right)^n$$</p>
<p>Does that help?<... |
1,590,039 | <p>In programming, often the value of True is also 1, and False is 0.</p>
<p>This means that:</p>
<pre><code>(x>5)*4
</code></pre>
<p>will return 4 if x is greater than 5 (because <code>(x>5)==1</code>), else 0.</p>
<p>I need to accomplish a similar thing using mathematical operators (no piecewise functions, ... | Michael Hardy | 11,667 | <p>$$
x\mapsto \frac{|2n+1-2x| + (2n+1-2x)}{2|2n+1-2x|} = \begin{cases} 1 & \text{if } x\le n, \\ 0 & \text{if } x>n. \end{cases}
$$</p>
|
399,865 | <p>Let $A$ be a linear map from the vector space $X$ to $Y$ and $T$ be a subspace of $X$ . I want to understand what is the meaning of saying that the restriction $A_{|T}: \to A(T)$ is invertible. Could anybody explain me what is this restriction map?</p>
<p>Thanks for the help.</p>
| Asaf Karagila | 622 | <p>When we restrict a function to a subset of the domain it means we are interested now in what the function is doing to that subset. We don't care what $A(x)$ is when $x\notin T$.</p>
<p>To say that it is invertible is to say that even if $A$ itself is not invertible as a function, when we only consider what $A$ does... |
4,387,701 | <p>I have been trying to find a proof I can follow (I am not familiar with metric spaces, for instance) for the fact that any given function only has a countable number of jump discontinuities.</p>
<p>I have been looking at <a href="https://math.stackexchange.com/questions/263606/prove-that-the-number-of-jump-discontin... | B. S. Thomson | 281,004 | <p>There is now a very nice, short answer explaining the proof.</p>
<p>This is a follow-up, a suggestion for further study, a push for a different perspective, a plea for greater understanding. Also an excuse to promote a very nice survey article by a friend (see [1]).</p>
<p>Instead of thinking of this as a theorem ... |
299,984 | <p>first, I have a general question. In perturbation theory, I saw perturbations in eigenvalues and eigenvectors of square, non-symmetric matrices and the calculations were all right but no one ever proved the differentiability of the eigenvalues/eigenvectors in the perturbation-parameter! They just assumed it. So, why... | Robert Israel | 13,650 | <p>The eigenvalues of a square matrix $A$ are the roots of the characteristic polynomial, and are analytic except where their multiplicities change.
Thus if (in a certain open region of parameter space) there is one eigenvalue,
of algebraic multiplicity $m$, inside a positively oriented closed contour $\Gamma$ in $\m... |
491,851 | <p>Given that <span class="math-container">$719$</span> is prime, find the least positive residue of <span class="math-container">$11^{721} \pmod {719}$</span>, without using modular exponentiation.</p>
<p>So, I know how to use modular exponentiation and have done it to get the answer <span class="math-container">$612... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>As $719$ is prime, $(11,719)=1$ and $11^{719-1}\equiv1\pmod {719}$ using Fermat's Little Theorem</p>
<p>So, $11^{721}=11^3\cdot11^{718}\equiv 11^3\pmod{719}\equiv\cdots$</p>
|
821,747 | <p>Ok so far what i understand is this lets say...Having to draw a card from 52card-deck its probability is of course 1/52.Now the probability to say that i will keep drawing this same card 10 times of course it is 1/(52)^10 because now i have alot more possibilities.Now my question is this.I also know that drawing a ... | Marc | 132,141 | <p>You're right, given that you have drawn $9$ times the same card, it is unlikely to draw the same card again, just as it is if you had drawn that card only once or one hundred times, namely $\frac{1}{52}$. I think where your intuition goes wrong is that you fix your attention on one specific card if it appears a lot.... |
98,820 | <p>In the course of exploring <a href="https://mathematica.stackexchange.com/a/77812/531">@kglr's answer</a> to <a href="https://mathematica.stackexchange.com/q/77784/531">Edmund's question</a> gain some insight into how I might answer <a href="https://mathematica.stackexchange.com/questions/98817/how-do-i-overlay-thin... | kglr | 125 | <p>Some <code>ChartElementDataFunction</code>s have options and some plot themes (namely, <code>"Business"</code>, <code>"Marketing"</code> and <code>"Monochrome"</code>) use these options. So, the pattern we need to use in the second argument of <code>Trace</code> is <code>ChartElementDataFunction[__]</code>. With th... |
2,539,855 | <p>I'm not convinced this is possible, as soon as you have $2$ distinct elements mapping to the same number, the function is no longer $1$-$1$ and therefore not a bijection. </p>
| User8128 | 307,205 | <p>Let $f(x) = x$ for rational $x$ and $f(x) = 1-x$ for irrational $x$. This function is not only non-monotone, it is nowhere monotone (meaning that it is non-monotone on any subinterval of $[0,1]$).</p>
|
2,539,855 | <p>I'm not convinced this is possible, as soon as you have $2$ distinct elements mapping to the same number, the function is no longer $1$-$1$ and therefore not a bijection. </p>
| Cornman | 439,383 | <p>How about the bijection $f:[0,1]\to [0,1]$ with $f(x)=\begin{cases} x, \text{if}~~ 0<x<1\\ 1,\text{if}~~ x=0\\ 0,\text{if}~~ x=1\end{cases}$</p>
<p>This is not strictly monoton.</p>
|
185,591 | <p>Let $b \in (0,1)$, $m\in \mathbb{N}$ and $a>0$. I want to bound
$$\sum_{k=m+1}^\infty b^{k^a} \leq c \; b^{m^a}, $$
where $c>0$ is independent from $m$.</p>
<p>Is there a simple way of proving this inequality with $c$ not beeing too large?</p>
<p>Edit: $c$ can only be independent from $m$ if $a>1$. Otherw... | Igor Rivin | 11,142 | <p>The sum converges for positive $a$ by the integral test, and since the summand is decreasing, the sum is closely approximated by the integral. The integral can be evaluated easily. Since I am lazy, I used Mathematica:</p>
<pre><code>Assuming[0 < b < 1 && a > 0 && y > 0, Integrate[b^(x^a)... |
3,354,815 | <p>Let <span class="math-container">$ f(x)$</span> has the convergent power series representation of the form <span class="math-container">$$\sum_{n=0}^{\infty}a_nx^n$$</span>. Then What is the power series representation of the function <span class="math-container">$\left[f(x)\right]^m$</span> for some integer <span c... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$f$</span> is differentiable and <span class="math-container">$a$</span> is a point of its domain, then the limit<span class="math-container">$$\lim_{x\to a}\frac{f(x)}{x-a}$$</span>exists (in <span class="math-container">$\mathbb R$</span>) if and only if <span class="math-container"... |
306,487 | <p>$$y' = y^2 - 1$$</p>
<p>I think this cannot be solved using the integration factor method because of the square power of the y.</p>
<p>Attempted:
$$ \frac{y'}{y^2} = -1 $$
$$ y^{-1} = t + C $$
$$ \frac{1}{y} = t + C$$</p>
<p>It's not the right answer.</p>
<p>How do I go about solving this?
Thanks.</p>
| Will Jagy | 10,400 | <p>I quite like @GHL's idea of going straight to the hyperbolic trigonometric functions.
I cannot say I ever really used those in a beginning ODE course, but it is a good idea.
So I thought of putting that information here as community wiki. </p>
<p>We have $$ \frac{d}{dx} \cosh x = \sinh x,$$
$$ \frac{d}{dx} \sinh x... |
3,004,854 | <blockquote>
<p>Is it true that a natural number <span class="math-container">$n>1$</span> is prime if and only if <span class="math-container">$n|\left ( \frac{1+\sqrt{5}}{2} \right )^n+\left ( \frac{1-\sqrt{5}}{2} \right )^n-1$</span>?</p>
</blockquote>
<p>We know that <span class="math-container">$11$</span> i... | Robert Z | 299,698 | <p>Note that <span class="math-container">$\left ( \frac{1+\sqrt{5}}{2} \right )^n+\left ( \frac{1-\sqrt{5}}{2} \right )^n$</span> is the <span class="math-container">$n$</span>-th <a href="https://en.wikipedia.org/wiki/Lucas_number" rel="noreferrer">Lucas number</a>.</p>
<p>It is true that if <span class="math-contai... |
782,812 | <p>I'm trying to solve this recurrence relation: </p>
<p>$$a_m = 8 \cdot a_{m-1} + 10^{m-1}, a_1=1$$</p>
<p>By the change of variable $\displaystyle b_m = \frac{a_m}{10^m}$ I obtained this linear non-homogenous relation:</p>
<p>$$b_m = \frac{8}{10}b_{m-1} + \frac{1}{10}, b_1 = \frac{1}{10}$$</p>
<p>Now I'm trying t... | Sandeep Silwal | 138,892 | <p>From $b_m = \frac{8}{10}b_{m-1}+\frac{1}{10}$, we have $10b_m = 8b_{m-1}+1$. Shift the index to get $10b_{m+1} = 8b_m+1$. Subtract to get $10b_{m+1}-10b_{m} = 8b_m-8b_{m-1}$ or $10b_{m+1} = 18b_m-8b_{m-1}$ or $5b_{m+1} = 9b_m-4b_{m-1}$. Then use the characteristic polynomial and the initial condition to get a closed... |
1,920,504 | <p>I know, for example, that the series:</p>
<p>$$\sum\limits_{n= 1}^{\infty} \frac{1}{e^n}$$</p>
<p>is a geometric series of the form $\sum\limits_{n = 1}^{\infty} k x^n$, where $k = 1$ and $x = \frac{1}{e}$ and it is convergent<sup>1</sup>. But when I have, as in my case:</p>
<p>$$\sum\limits_{n = 1}^{\infty} \fra... | Math1000 | 38,584 | <p>If $\{a_n\}$ is a nonincreasing sequence of nonnegative numbers, then $\sum_n a_n$ converges iff $\sum_n 2^n a_{2^n} $ converges. (This is known as the Cauchy condensation test, and can be used to show that $\sum_n \frac1{n^p}$ converges iff $p>1$). Applying this test, we have
$$2^n\left(\frac{2^n}{e^{2^n}}\right... |
2,194,461 | <p><a href="https://i.stack.imgur.com/Q73yr.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Q73yr.jpg" alt="enter image description here"></a></p>
<p>in this I tries to substitute x by $\pi -t$ </p>
<p>and thought about it </p>
<p>but after that got stuck.</p>
| imranfat | 64,546 | <p>There are different ways to handle this problem. One way is to use $sin(A+B)=sinAcosB+cosAsinB$ and so applying that on the sine term in your integral, we get $sin(\frac{\pi}{4}+x)=sin\frac{\pi}{4}cosx+cos\frac{\pi}{4}sinx=\frac{\sqrt{2}}{2}(cosx+sinx)$. So actually, your integral now becomes of the form $\frac{\sqr... |
265,101 | <p>Let</p>
<pre><code>L1={1,2,4,6,8,9,10,14,15,17,21,22,72,76,80,96,106,116,117}
</code></pre>
<p><strong>Question1)</strong> Group from 10-th to 14-th elements in <code>L1</code>.</p>
<p>So that the output becomes</p>
<pre><code>L2={1,2,4,6,8,9,10,14,15,{17,21,22,72,76},80,96,106,116,117}
</code></pre>
<p>A possible s... | lericr | 84,894 | <p>TakeDrop might be interesting to you:</p>
<pre><code>TakeDrop[L1, {10, 14}]
(*gives {{17, 21, 22, 72, 76}, {1, 2, 4, 6, 8, 9, 10, 14, 15, 80, 96, 106, 116, 117}}*)
</code></pre>
<p>From there you can use Insert:</p>
<pre><code>Insert[#2, #1, 10] & @@ TakeDrop[L1, {10, 14}]
(*gives result for question 1*)
</code>... |
265,101 | <p>Let</p>
<pre><code>L1={1,2,4,6,8,9,10,14,15,17,21,22,72,76,80,96,106,116,117}
</code></pre>
<p><strong>Question1)</strong> Group from 10-th to 14-th elements in <code>L1</code>.</p>
<p>So that the output becomes</p>
<pre><code>L2={1,2,4,6,8,9,10,14,15,{17,21,22,72,76},80,96,106,116,117}
</code></pre>
<p>A possible s... | Anxon Pués | 51,310 | <p>I try to write a function to obtain list fractured with another list of points of fracture...</p>
<pre><code> Clear["Global`*"]
miLista[lista_List, filtro_List] :=
Module[{final = {}, i, j, k, provisional = {}},
(* if max{filtro] > Length[lista] error SALIR *)
If[Max[filtro] > Length[li... |
2,626,630 | <p>$$|\{n\in N:n\mid p^2q\}|$$
$p$ is an odd prime number, $q$ is a prime number and $p$ does not equal $q$.</p>
<p>I dont understand what they're asking me to find, is it to find a value of $n$ that divides into $p^2q$?</p>
<p>I need all the help i can get here, very new to this<br>
thanks</p>
| Dr. Sonnhard Graubner | 175,066 | <p>HINT: write your term $$101^{6n}-1$$ as $$\left(101^n-1\right) \left(101^n+1\right)
\left(-101^n+101^{2 n}+1\right) \left(101^n+101^{2
n}+1\right)$$</p>
|
118 | <p>The tag <a href="https://matheducators.stackexchange.com/questions/tagged/pedagogy" class="post-tag" title="show questions tagged 'pedagogy'" rel="tag">pedagogy</a> gets a lot of usage, possibly too much. I propose a discussion to sharpen the meaning of this tag (or to replace it by more specific ones). </p>... | Avi Steiner | 344 | <p>I vote for removing the tag altogether in favor of more specific tags. Pretty much every question about education could easily be construed as having to do with pedagogy. </p>
|
1,468,028 | <p>I was trying to find the volume of concentrical sphere shells but to my surprise i come up with equal volume when evidently the outer shell should have more volume than the inner shell. I can't figure out what I'm doing wrong.
This is what I did:
<a href="https://i.stack.imgur.com/sP5Eq.png" rel="nofollow noreferre... | Paul Sinclair | 258,282 | <ul>
<li>The word is "radius", not "radio". I have corrected this.</li>
<li>Not a single one of your volumes is correct. I cannot figure out the mistake you made to get them wrong. The first one is off by a factor of 10, but the other two bear no similar relation to their correct values:</li>
</ul>
<p>$$v_1 = 4188.79.... |
48,366 | <p>Does the Laplace equation on a rectangle with Dirichlet boundary conditions at two opposing sides and Neumann boundary conditions at the other two, always have a solution? If it does, is it unique? Is the same true for the discrete Laplace equation (with the standard five-point laplacian)?</p>
| Dan Lee | 1,179 | <p>I believe you can just use separation of variables.</p>
|
1,811,303 | <p>F-test statistic is the ratio of two variance and if the value of F-test statistic equals to 1 that means that those two variance are equal and we accept the null hypothesis.But when we want to check the equality of several mean we perform F-test.Here F-test statistic is the ratio of between mean sum square and with... | saz | 36,150 | <p>You already know that $[Z,Z]_t = \int_0^t \sigma^2(s) \,ds$ for any Itô process $$dZ_t = b(t) \, dt + \sigma(t) \, dW_t.$$ Moreover, the quadratic covariation is defined via the polariation formula, i.e.</p>
<p>$$[X,Y]_t = \frac{1}{2} [X+Y,X+Y]_t - [X,X]_t-[Y,Y]_t. \tag{1}$$</p>
<p>Since both $(X_t)_{t \geq 0}$ an... |
594,724 | <p>my question is:
Consider the Taylor polynomial of $e^x$ of order $n$ at $x=0$, or $P_n(x)=e^0 + e'^{(0)}x +...+\frac{e^{(n)(0)}x^n}{n!}$.
Prove that the sequence {$P_n(x)$} converges uniformly to $e^x$ on any bounded interval.
I tackled the first part of the question by proving, through Taylor's theorem, that $e^x =... | robjohn | 13,854 | <p>We have to show that for any $\epsilon\gt0$ and $M\gt0$ there is an $N$ so that if $n\ge N$ and $|x|\le M$, then
$$
\left|e^x-\sum_{k=0}^n\frac{x^k}{k!}\right|\le\epsilon
$$
We can achieve this by showing
$$
\sum_{k=n+1}^\infty\frac{|x|^k}{k!}\le\epsilon
$$
For $k\ge2M$, $\frac{M^k}{k!}=\frac{M^{2M}}{(2M)!}\overbrac... |
1,887,807 | <p>I would like to evaluate the following indefinite integral: </p>
<blockquote>
<p>$$\int\frac{1}{\sin^4\theta+\cos^4\theta}\, d\theta$$</p>
</blockquote>
| juantheron | 14,311 | <p>Let $$I = \int\frac{1}{\sin^4 \theta+\cos^4 \theta}d\theta = \int\frac{\sin^2 \theta +\cos^2 \theta}{\sin^2 \theta \cos^2 \theta (\tan^2 \theta +\cot^2 \theta)}d\theta$$</p>
<p>So $$I = \int\frac{\sec^2 \theta +\csc^2 \theta}{\tan^2 \theta +\cot^2 \theta }d\theta = \int\frac{\sec^2 \theta+\csc^2 \theta}{(\tan \thet... |
2,963,894 | <p>Suppose we would like to find a formula for some interesting real-valued function on the positive integers <span class="math-container">$f(n):\mathbb{Z}^+\rightarrow\mathbb{R}$</span>, and we have the inductive equation</p>
<p><span class="math-container">$$\forall n\in\mathbb{Z}^+,\quad f(n+1)=f(n)+p(n)$$</span></... | Noble Mushtak | 307,483 | <p>Since <span class="math-container">$f(n+1)-f(n)=p(n)$</span>, we have:</p>
<p><span class="math-container">$$f(n)=f(n-1)+p(n-1)=f(n-2)+p(n-2)+p(n-1)=...=f(1)+\sum_{i=1}^{n-1} p(i)$$</span></p>
<p>Thus, since <span class="math-container">$f(1)$</span> is a constant, we need to show <span class="math-container">$\su... |
2,983,345 | <p><span class="math-container">$3^{10^{100}} \pmod {13}$</span></p>
<p>It was originally <span class="math-container">$(10^{100})^{(10^{100})}$</span> (i.e. googol^googol), but I simplified it down to what I have there. However, I can't seem to reduce the exponent without changing the value of the entire expression.<... | Aaron | 9,863 | <p>One approach is to use Euler’s theorem/Fermat’s little theorem. Since 3 is relatively prime to 13, <span class="math-container">$3^{12} \equiv 1 \pmod{13}$</span>, and so you can reduce the exponent modulo 12 without changing the result. More explicitly, if <span class="math-container">$10^100=12a+b$</span>, then <s... |
1,549,904 | <p>If you have an investment fund which has four options $x_1,x_2,x_3,x_4$</p>
<p>More should be invested in the combination of funds $x_2$ and $x_3$ than funds $x_1$ and $x_4$ by a ratio of at least $1.5:1$.</p>
<p>The constraint I have formed is as follows:</p>
<p>$$1.5(x_2 + x_3) - (x_1 + x_4) \geq 0.$$</p>
<p>W... | Community | -1 | <p>Let $f (x) = \log (1 + e^x)$. Then the McLaurin series of $f (x)$ at $x = 0$ is $$f (x) = \sum_{n = 0}^{\infty} \frac {f^{(n)} (0)} {n!} x^n = \log 2+\frac{1}{2}x+\frac{1}{8}x^2 + \cdots$$ as desired.</p>
|
12 | <p>All introductory calculus books that I have seen spend most of their chapters on differential calculus talking about derivatives, with at most a short section defining differentials as <span class="math-container">$dy = f'(x) \, dx$</span>. However, differentials are useful for understanding a lot of things, like l... | johnnyb | 9,594 | <p>My book <a href="https://rads.stackoverflow.com/amzn/click/com/1944918027" rel="noreferrer" rel="nofollow noreferrer">Calculus from the Ground Up</a> focuses on differentials, and uses it to provide a unification of process and simplification of understanding of a lot of different parts of calculus.</p>
<p>To read ... |
2,491,176 | <p>Is it always true that for small value of c i.e $c\approx 0$<br>
$\int_0^c f(x)dx \approx \int_0^c f1(x)dx $ </p>
<p>where $f1(x)$ is the approximate function for $f(x)$ when x is small?<br>
For an example:<br>
$\int_0^{0.18}(1-\sin x \cos x)dx=0.163974$.<br>
$\int_0^{0.18}(1-x)dx=0.1638$.</p>
<p>In this case bo... | Chappers | 221,811 | <p>The former is using radians, the latter degrees:
$$ \arctan{\left(\frac{19.62}{62.5}\right)} \approx 0.304178 \text{ (radians)} \approx 17.4281^{\circ}. $$</p>
|
355,150 | <p>A. I wonder if every generator of <span class="math-container">$Z({\rm Spin}_n^{\epsilon}(q))$</span> is a square element in <span class="math-container">${\rm Spin}_n^{\epsilon}(q)$</span>?</p>
<p>B. When <span class="math-container">$Z(\Omega_{2m}^{\epsilon}(q))\cong C_2$</span>, is the unique element of order tw... | Yi Wang | 152,963 | <p>With the help of both Professors Robert Guralnick and Frank Lübeck, I get following two answers extracted from their reply to my email.</p>
<ol>
<li>The first one due to Professor Robert Guralnick.</li>
</ol>
<p>Suppose <span class="math-container">$x^2=-1$</span>. Then eigenvalues of <span class="math-container">$x... |
2,011,284 | <p>I know that a singular matrix cannot have an inverse, but I was curious as to whether or not a matrix with a non-zero determinant can have an inverse. And from what I understand the requirement for a matrix to be non-singular is that it has a determinant that is not zero.</p>
| Momo | 384,029 | <p>I will add my two cents to the answer above:</p>
<p>One has:</p>
<p>$\mathbf A\cdot \operatorname{adj}(\mathbf A)=\det(\mathbf A)\cdot\mathbf I$</p>
<p>whre $\operatorname{adj}(\mathbf A)$ is the adjugate of matrix $\mathbf A$</p>
<p>So if the elements of $\mathbf A$ are in a field (like real or complex numbers)... |
633,724 | <p>Prove $e^x, e^{2x}..., e^{nx}$ is linear independent on the vector space of $\mathbb{R} \to \mathbb{R}$</p>
<p>isn't it suffice to say that $e^y$ for any $y \in \mathbb{R}$ is in $\mathbb{R}^+$<br>
Therefore, there aren't $\gamma_1, ...\gamma_n$ such that $\gamma_1e^x+\gamma_2e^{2x}...+\gamma_ne^{nx}=0$.<br>
Theref... | Community | -1 | <p>The exercise is
$$
f\alpha = \left\{
\begin{array}{ll}
\mathbb{R}\rightarrow\mathbb{R} \\
\ t\mapsto e^{\alpha t}
\end{array}
\right.
$$ Prove $(f_\alpha)_{\alpha \in\mathbb{R}} $is linear independent.</p>
<p>Let $(f_{\alpha_k})_{1\leq k \leq n} $ a finite number of vectors as $\alpha_1<... |
1,931,335 | <p>I am trying to find the greatest value of $"c"$ for which $7x + 9y = c$ has exactly six solutions over positive integers.</p>
<p>My approach: Let one of the solution = $(a, b)$</p>
<p>General solution is $(a + 9t, b - 7t)$</p>
<p>For values t = 0, 1, 2, 3, 4, 5 we get 6 solutions.</p>
<p>Also $b - 7t$ should be ... | G Cab | 317,234 | <p>So, if (a,b) is a solution of $7x + 9y = c$
then $(a+n,b+m)$, with $7n + 9m = 0$, is also a solution.<br>
But this approach does not look promising.</p>
<p>A different way is to consider that
$$
\sum\limits_{0\, \leqslant \,c} {N(c)\,z^{\,c} } = \frac{1}
{{\left( {1 - z^{\,7} } \right)\left( {1 - z^{\,9} } \right... |
3,820,373 | <p>I'm reading about The Bayes Problem in textbook <em>A Probabilistic Theory of Pattern Recognition</em> by Devroye et al.</p>
<p><a href="https://i.stack.imgur.com/UELeT.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/UELeT.png" alt="enter image description here" /></a>
<a href="https://i.stack.img... | Elle Najt | 54,092 | <p>Some comments:</p>
<ol>
<li><p>You can get intuition from assuming that the set up is that <span class="math-container">$(X,Y)$</span> is some process where <span class="math-container">$Y$</span> is sampled from a distribution that depends on the realization of <span class="math-container">$X$</span>. For instance,... |
354,661 | <p>Can any one help me to answer this question:</p>
<blockquote>
<p>Assuming <span class="math-container">$$f(x)=\begin{cases} x &\text{if }x\in \mathbb{Q} \\ 0 &\text{if }x\in \mathbb{R}\setminus\mathbb{Q} \end{cases}$$</span></p>
<p>show that <span class="math-container">$f$</span> is continuous only at <span... | André Nicolas | 6,312 | <p>Let $x^2=y$. Note that if $|y|\gt R$, then $\sum a_ny^n$ diverges, and if $|y|\lt R$, then $\sum a_ny_n$ converges.</p>
<p>Thus if $x^2\gt R$, then $\sum a_n x^{2n}$ diverges, and if $x^2\lt R$, then $\sum a_nx^{2n}$ converges.</p>
<p>Thus if $|x|\gt\sqrt{R}$, we have divergence, and if $|X|\lt \sqrt{R}$, we have ... |
3,534,596 | <p>Let <span class="math-container">$B(X)$</span> <span class="math-container">$=$</span> <span class="math-container">$\{$</span> <span class="math-container">$f:X\rightarrow \mathbb{R}$</span> <span class="math-container">$:$</span> <span class="math-container">$f$</span> is bounded <span class="math-container">$\}$<... | Community | -1 | <p>I would like to elaborate on my solution.</p>
<p>Definition: Let <span class="math-container">$n\in\mathbb{N}$</span> be fixed. Given a set <span class="math-container">$X$</span>. An <span class="math-container">$n-tuple$</span> of <span class="math-container">$X$</span> is <strong>defined</strong> by a function <... |
3,100,406 | <p>Let <span class="math-container">$G$</span> be some group, and let <span class="math-container">$S_1$</span> and <span class="math-container">$S_2$</span> be some subgroups. Then <span class="math-container">$S_1$</span> and <span class="math-container">$S_2$</span> are said to be commensurable iff <span class="math... | epi163sqrt | 132,007 | <blockquote>
<p>We obtain
<span class="math-container">\begin{align*}
\frac{n!}{x(x+1)\cdots(x+n)}&=\sum_{k=0}^n\frac{A_k}{x+k}\\
n!&=\sum_{k=0}^nA_k\frac{x(x+1)\cdots(x+n)}{x+k}\tag{1}
\end{align*}</span>
Substituting <span class="math-container">$x=-j,0\leq j\leq n$</span> in (1) we get
<span clas... |
2,346,841 | <p>So I have been trying to solve this question, </p>
<p>$$
\int_{0}^{\pi/2}\left\{\,\cot\left(x\right)\,\right\}\,\mathrm{d}x
$$
where $\left\{\right\}$ means the fractional part.
$\left(\,x - \left\lfloor\,x\,\right\rfloor\,\right)$.</p>
<p>Progress so far, </p>
<ul>
<li><code>MATLAB</code> gives out the following... | Paul LeVan | 187,012 | <p>Supplementing Thomas Andrews' answer, since:
$$
S_N =\ln(N+1) - \sum_{n=1}^N \arctan \left(\frac{1}{n} \right) = \ln(N+1) - \sum_{k=0}^\infty \frac{(-1)^k}{2k+1}\sum_{n=1}^N n^{-2k-1} \\
= \ln(N+1) - H_N - \sum_{k=1}^\infty \frac{(-1)^k}{2k+1}H_{N,2k+1}
$$
Where the $H_{N,i}$ denote the generalized Harmonic Numbers,... |
1,809 | <p>Suppose I have function f:</p>
<pre><code>f[x_, y_] := 50000 + x 30000 + y 35000;
</code></pre>
<p>Now I want to find the x and y when f[]<=200000.
I use Reduce:</p>
<pre><code>Reduce[f[x, y] <= 200000 && 1 <= x <= 8 && 1 <= y <= 8 , {x,y}, Integers]
</code></pre>
<p>The result ... | Rolf Mertig | 29 | <pre><code>50000 + x*30000 + y*35000 /. FindInstance[50000 + x*30000 + y*35000 < 200000 &&
((x == 1 && y == 1) || (x == 1 && y == 2) || (x == 1 && y == 3) ||
(x == 2 && y == 1) || (x == 2 && y == 2) || (x == 3 && y == 1)), {x, y},
Integers, 6]
</... |
1,809 | <p>Suppose I have function f:</p>
<pre><code>f[x_, y_] := 50000 + x 30000 + y 35000;
</code></pre>
<p>Now I want to find the x and y when f[]<=200000.
I use Reduce:</p>
<pre><code>Reduce[f[x, y] <= 200000 && 1 <= x <= 8 && 1 <= y <= 8 , {x,y}, Integers]
</code></pre>
<p>The result ... | rcollyer | 52 | <p>I'd do the following:</p>
<pre><code>res = Reduce[f[x, y] <= 200000 && 1 <= x <= 8 && 1 <= y <= 8, {x, y}, Integers];
vals = Apply[List, res[[All, All, 2]], {0, 1}]
(*
==> {{1, 1}, {1, 2}, {1, 3}, {2, 1}, {2, 2}, {3, 1}}
*)
</code></pre>
<p>which uses <code>Part</code> in the form... |
3,294,874 | <p>How should I solve <span class="math-container">$$y'+2y=x\sin(x)$$</span>
I tried this calculation using integrating factor, but it costs me a lot of calculation. Is there better way to solve this kind of problem?</p>
| Dr. Sonnhard Graubner | 175,066 | <p>Computing an integrating factor <span class="math-container">$$\mu(x)=e^{\int2dx}=e^{2x}$$</span>
So we get (after multiplying the equation by <span class="math-container">$\mu(x$</span>):
<span class="math-container">$$\frac{d}{dx}(e^{2x}y(x))=e^{2x}x\sin(x)$$</span>
Now integrate with respect to <span class="math-... |
3,294,874 | <p>How should I solve <span class="math-container">$$y'+2y=x\sin(x)$$</span>
I tried this calculation using integrating factor, but it costs me a lot of calculation. Is there better way to solve this kind of problem?</p>
| Community | -1 | <p>Consider the complex equation</p>
<p><span class="math-container">$$(ze^{2x})'=xe^{2x}e^{ix}=xe^{(2+i)x}.$$</span></p>
<p>You integrate by parts,</p>
<p><span class="math-container">$$ze^{2x}=x\frac{e^{(2+i)x}}{2+i}-\int \frac{e^{(2+i)x}}{2+i}dx=x\frac{e^{(2+i)x}}{2+i}-\frac{e^{(2+i)x}}{(2+i)^2}+C,$$</span></p>
... |
25,209 | <p>Money is typically denominated in a way that allows for a greedy algorithm when computing a given amount $s$ as a sum of denominations $d_i$ of coins or bills:</p>
<p>$$
s = \sum_{i=1}^k n_i d_i\quad\text{with } n_i > 0, d_i > d_{i-1}
$$</p>
<p>We start with the highest denomination $d$ not exceeding $s$ a... | Prashant Vaidyanathan | 51,571 | <p>I recently came up with 1 solution that seemed to show if the given 2 conditions are satisfied, the greedy algorithm would yield the optimal solution.</p>
<p>a) The G.C.D (All the denominations except 1) = 2nd Smallest denomination.</p>
<p>b) The sum of any 2 consecutive denominations must be lesser than the 3rd c... |
173,781 | <p>This question arises from one of the Berkeley Problems in Mathematics, fall 1981. It asks me to find a counterexample to the following claim:</p>
<p>Suppose $\lim\limits_{x\rightarrow x_{0}}f(x)=a$, $\lim\limits_{t\rightarrow a}g(t)=b$, prove or find a counterexample for the claim $\lim\limits_{x\rightarrow x_{0}}g... | Not a grad student | 36,274 | <p>I think you just made a careless error. In the example provided, the limit of $g$ at 0 is not 0; it is 1, as you said. Here $x_0 = 0$, so that $a = 1$. So lim$_{t\rightarrow a}{g(t)}$ = lim$_{t\rightarrow 1}{g(t)} = 0$. And lim$_{x\rightarrow x_0}{g(f(x))} = $ lim$_{x\rightarrow 0}{g(f(x))} = 1$. This is probably as... |
2,212,968 | <p>I am trying to solve an equation that involve the inner product of 2 vectors and I was wondering if there was a way to solve it.</p>
<p>I know that
$$
\vec{M} \cdot \vec{A} = c
$$
Where $c$ is a constant scalar and vectors are in 2D space, Then how do I compute $\vec{M}$?</p>
| Emilio Novati | 187,568 | <p>In $\mathbb{R}^3$, if $\vec M=(x,y,z)^T$ and $\vec A=(a_1,a_2,a_3)^T$, than $\vec M \cdot \vec A=c$ is the equation of a plane:
$$
a_1x+a_2y+a_3z=c
$$
that is the plane orthogonal to $\vec A$ and whose distance from the origin is $d=\frac{|c|}{\sqrt{a_1^2+a_2^2+a_3^2}}$.</p>
<p>So, given $\vec A$ and $c$ , $\vec M$... |
3,081,997 | <p>I've been stuck on this calc problem for a while:</p>
<p>Let <span class="math-container">$f$</span> be the function given by <span class="math-container">$f(x) =\ln(x+1)$</span> and let <span class="math-container">$g$</span> be the function given by <span class="math-container">$g(x) = x^{-1/2}$</span>. At what v... | Alexander Sanchez | 632,278 | <p>The useful fact here is that perpendicular lines have slopes which are negative reciprocals of one another. To see this, just think about what rise/run means when you rotate the coordinate plane by 90 degrees!</p>
<p>Then, you get <span class="math-container">$ f'(x) = \frac{1}{1+x} $</span> and <span class="math-... |
2,842,589 | <p>It has been a while since my undergraduate introductory analysis course, so I am a little rusty. I know that to show two sets $B, C$ are equal, you first show that $B\subseteq C$ and then $C\subseteq B$ . To show $B\subseteq C$ you take any element of $B$, say $b \in B$ and show that $b \in C$. </p>
<p>The main is... | Suzet | 562,979 | <p>By double inclusion. For the first question, because for any positive integer $n$, we clearly have $(a,b-\frac{1}{n}]\subset(a,b)$, we know that $\bigcup^\infty_{n=1}(a,b-\frac{1}{n}]\subset (a,b)$</p>
<p>Now consider any element $x \in (a,b)$. Then, $x<b$ and because $\lim_{n\rightarrow \infty}b-\frac{1}{n}=b$,... |
64,605 | <p>If I print the following <code>BarChart</code> in black and white, then it is very difficult to tell which bar is "a" and which is "b".</p>
<pre><code>BarChart[{{1, 2}}, ChartLegends -> {"a", "b"}]
</code></pre>
<p><img src="https://i.stack.imgur.com/uzYiB.png" alt="enter i... | Alexey Popkov | 280 | <blockquote>
<p>How can I put a simple pattern or texture (like stripes or dots) on
these bars to differentiate them? I know that the solution will
involve <code>ChartElements</code>, but I can't figure it out.</p>
</blockquote>
<p>May be you need something similar to the following?</p>
<pre><code>opts = Sequen... |
911,722 | <p>Determine</p>
<p>$\int e^{2x} \sqrt{e^x+1}dx$</p>
<p>Is there a multiplication rule for integration or something?</p>
| lab bhattacharjee | 33,337 | <p>HINT: $$\sqrt{e^x+1}=u\implies e^x+1=u^2$$</p>
|
911,722 | <p>Determine</p>
<p>$\int e^{2x} \sqrt{e^x+1}dx$</p>
<p>Is there a multiplication rule for integration or something?</p>
| Adi Dani | 12,848 | <p>$$\int e^{2x} \sqrt{e^x+1}dx=\int (e^x\sqrt{e^x+1})e^xdx$$
$e^x+1=t,e^xdx=dt,e^x=t-1$
$$\int e^{2x}\sqrt{e^x+1}dx=\int(t-1)\sqrt tdt=\int t^{3/2}dt-\int t^{1/2}dt=$$
$$=\frac{t^{3/2+1}}{3/2+1}-\frac{t^{1/2+1}}{1/2+1}+C=\frac{2}{5}(e^x+1)^{5/2}-\frac{2}{3}(e^x+1)^{3/2}+C$$</p>
|
1,380,216 | <p>For $n=1,2,3,\ldots$, let
$$f(x) = (x^2-1)^n .$$
Show that the $n$-th derivative $f^{(n)}$ has distinct real roots in $[-1,1]$.<br>
I have no idea about the problem. Could I have a hint? </p>
| Jack D'Aurizio | 44,121 | <p>$f(x)=(x^2-1)^n$ is an even function with two zeroes of multiplicity $n$ at $x=\pm 1$. It follows that $f'(x)$ has two zeroes of multiplicity $n-1$ at $x=\pm 1$ and a zero at $x=0$. By Rolle's theorem, $f''(x)$ has a zero in $(-1,0)$ and a zero in $(0,1)$. Moreover, there are two zeroes of multiplicity $n-2$ at $x=\... |
2,555,789 | <p>Let $\{u_{n}\}$ be a sequence such as $\forall n \in N; u_{n+2}=\frac{1}{2}u_{n+1}+\frac{1}{2}u_{n}+1$ </p>
<p>and $u_{1}$ and $u_{0}$ are fixed reals</p>
<p>I need to find the general term of this sequence.</p>
<p>I would have known how to find it if not for the "$+1$" there.</p>
<p>I noticed that if I could fi... | Simply Beautiful Art | 272,831 | <p>Hint:</p>
<p>Define another sequence</p>
<p>$$v_{n+1}=av_n+b$$</p>
<p>By self-substitution, see that</p>
<p>$$v_{n+2}=(a-x)v_{n+1}+axv_n+b(x+1)$$</p>
<p>Can you find suitable $a,b,x$ so that this sequence matches up with yours? And can you solve for $v_n$'s simpler form?</p>
|
69,245 | <p>Currently i am developing a game which is based on many computations of random values and therefore i have implemented many algorithms like the Mersenne-Twister etc. Unfortunately, all generators return uniformly distributed values and i want to modify this to make the min/max (and their surrounding) values rarer th... | robjohn | 13,854 | <p>The <a href="http://en.wikipedia.org/wiki/Marsaglia_polar_method" rel="nofollow">Marsaglia Polar Method</a> is an algorithmic improvement on the <a href="http://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform" rel="nofollow">Box-Muller Transform</a>. The Box-Muller Transform uses two uniform random variables, $(u... |
1,286,720 | <blockquote>
<p>to get a license, one's need to pass a 3 stages exam.
<br>
If one's fail in one stage he can not continue to the next stage.
<br> The probability he will pass the first stage is $0.9$, if he passed the first stage, the probability he will pass the second test is $0.8$.
if he passed the first... | Dr. Sonnhard Graubner | 175,066 | <p>we have $$64\equiv 8^{38}\mod 210$$</p>
|
3,408,623 | <p>Definition: A set <span class="math-container">$A$</span> is countable if there exists a bijection <span class="math-container">$f:\mathbb{N}\rightarrow A$</span>.
NOTE: According to the definition I'm using, countable = countably infinite.</p>
<p>Definition: A set is at most countable if it is finite or countable.<... | Greg Martin | 16,078 | <p>With these definitions, the assertion is false as stated: just let <span class="math-container">$A_n=\emptyset$</span> for all <span class="math-container">$n$</span>. Or, more generally, fix a finite set <span class="math-container">$U$</span> and choose every <span class="math-container">$A_n$</span> to be a subse... |
2,421,512 | <p>Let $D\subset \mathbb{R}^2$, $0$ be a limit point of $D$, and $f:D\to \mathbb{R}$ be a function.</p>
<p>How to prove or disprove that $\displaystyle\lim_{x\to 0}f(x,\sin x)=\lim_{x\to 0}f(x, x)$ ?</p>
<p>If it is not true, what are the sufficient and necessary conditions such that $\displaystyle\lim_{x\to 0}f(x,\s... | An aedonist | 143,679 | <p>I wonder if the function $$ f(x,y) = \frac{e^{x -y}-1}{x-y}$$ provides a counterexample.
$f$ is not defined for $(x,x)$, so it does not even make sense to talk about the limit $$lim_{x \to 0} f(x,x) $$
Yet, $$lim_{x \to 0} f(x, \sin x) = 1 $$ seems perfectly defined.</p>
|
2,421,512 | <p>Let $D\subset \mathbb{R}^2$, $0$ be a limit point of $D$, and $f:D\to \mathbb{R}$ be a function.</p>
<p>How to prove or disprove that $\displaystyle\lim_{x\to 0}f(x,\sin x)=\lim_{x\to 0}f(x, x)$ ?</p>
<p>If it is not true, what are the sufficient and necessary conditions such that $\displaystyle\lim_{x\to 0}f(x,\s... | Paramanand Singh | 72,031 | <p>Idea is to analyze the difference $f(x, \sin x) - f(x, x) $ which by mean value theorem is equal to $$(\sin x - x) f_{y} (x, \xi) $$ for some $\xi$ between $x$ and $\sin x$. Now you can see that an obvious condition is that $f_{y} (x, x) $ should not diverge too much. More specifically our job is done if $f_{y} (x,... |
2,656,551 | <p>I am trying to solve the following problem:</p>
<p>Suppose $X$ is a continuous random variable with density $f$ over $[0,1]$. Then show that $(\int_0^1xf(x)dx)(\int_0^1x^3f(x)dx)>(\int_0^1x^2f(x)dx)^2$.</p>
<p>I am trying to apply Fubini's theorem, but am getting equality for the two sides. What mistake am I ma... | Angina Seng | 436,618 | <p>Let $R$ be the set of even integers. This is a ring without unity.</p>
<p>Let $R'$ be the ring of integers modulo $3$. This is a ring with unity.</p>
<p>Let $\phi:R\to R'$ be the "reduction modulo $3$" map. This is a surjective homomorphism of rings.</p>
|
2,656,551 | <p>I am trying to solve the following problem:</p>
<p>Suppose $X$ is a continuous random variable with density $f$ over $[0,1]$. Then show that $(\int_0^1xf(x)dx)(\int_0^1x^3f(x)dx)>(\int_0^1x^2f(x)dx)^2$.</p>
<p>I am trying to apply Fubini's theorem, but am getting equality for the two sides. What mistake am I ma... | fred goodman | 124,085 | <p>Algebras of functions are a source of examples. Consider the restriction homomorphism from $C_0(\mathbb R)$ onto $C[0,1]$</p>
<p><strong>Remark</strong> All of the examples given up to now, maybe with exception of that of Jendrik Stelzner, have the same feel to me. E.g., the example of Lord Shark: associate to... |
2,656,551 | <p>I am trying to solve the following problem:</p>
<p>Suppose $X$ is a continuous random variable with density $f$ over $[0,1]$. Then show that $(\int_0^1xf(x)dx)(\int_0^1x^3f(x)dx)>(\int_0^1x^2f(x)dx)^2$.</p>
<p>I am trying to apply Fubini's theorem, but am getting equality for the two sides. What mistake am I ma... | fred goodman | 124,085 | <p>Another rather different source of examples in analysis would be the algebras of continuous functions on a (continous) compact group, under convolution. These algebras don't have an identity (since the identity wants to be the point mass at the group identity). But the groups, and therefore the convolution algebr... |
31,473 | <p>In a recent post <a href="https://math.meta.stackexchange.com/questions/31425/autofilters-for-hot-network-questions">Autofilters for Hot Network Questions</a> the possibility was raised to created a list of words which when contained in a title of a question would prevent the question from being included in the HNQ ... | Community | -1 | <p>This answer proposes the following word for inclusion in the HNQ block-list:</p>
<p><strong>opinion</strong></p>
<p>Please up-vote if you agree with the inclusion. Down-vote if you disagree with the inclusion.</p>
|
31,473 | <p>In a recent post <a href="https://math.meta.stackexchange.com/questions/31425/autofilters-for-hot-network-questions">Autofilters for Hot Network Questions</a> the possibility was raised to created a list of words which when contained in a title of a question would prevent the question from being included in the HNQ ... | Xander Henderson | 468,350 | <p><a href="/questions/tagged/status-completed" class="post-tag moderator-tag" title="show questions tagged 'status-completed'" rel="tag">status-completed</a></p>
<p>I propose the following word</p>
<p><strong>help</strong></p>
<p>for inclusion in the HNQ block-list.</p>
<p>Please up-vote if you agree with the ... |
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