Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
xsin(1/x) Holder on [0,1] I know $x \sin(1/x)$ is not Lipschitz on $[0,1]$, but some experimentation makes me conjecture that it is $1/2$-Holder. What is a good way to prove this?
| I will sketch the proof that $f(x)=x \sin(x^{-1})$ is 1/2-Holder on $[0,1/2\pi]$.
There are two cases.
*
*If $x,y\in [\frac{1}{2\pi (n+1)} , \frac{1}{2\pi n}]$.
Write $x= \frac{1}{2\pi n+ \xi}$ and $y=\frac{1}{2\pi n+\zeta}$ where $0 \le \xi,\zeta \le 2\pi$.
There exists a constant $0<c_1$ such that
$$| f(x)- f(y... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What's wrong with this argument? (Limits) In order to compute $\displaystyle \lim_{x \to \infty} \sqrt{9x^2 + x} - 3x$ we can multiply by the conjugate and eventually arrive at a limit value $1/6$.
But what about the line of reasoning below, what is wrong with the argument and why? I can't think of a simple explanatio... | The symbol $\approx $ is often misused. It can mean numerical approximation or it can mean asymptotically equal.
For example it is not true that (n+1) 2 $\approx $ n 2 in both senses. This you can see from the difference
${(n + 1)^2} - {n^2} = 1 + 2n$ .
You can observe that the difference is not small. Indeed it t... | {
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"timestamp": "2023-03-29T00:00:00",
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Inconclusive Second derivative test ,Now how shall i proceed While doing maxima and minima questions i have encountered upon question in which i cannot show nature of points
P1 : Given $f(x,y) = 2x^4 - 3x^{2}y + y^{2}$
Doubtful case is as origin .
P2: $f(x,y)$ = $y^{2} + x^{4} +x^{2}y $
Doubtful case is at origin
Than... | P1: The function is $0$ on the curves $y = x^2$ and $y = 2 x^2$, is negative in between them and positive otherwise. So $(0,0)$ is a saddle point since there are positive and negative values of $f$ arbitrarily close to $(0,0)$. The fact that it is a saddle does not follow simply from the discriminant being $0$, for exa... | {
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How do you find the integral, $\int2x\,dx/(1+x)$ Integral: $\displaystyle \int \dfrac{2x}{1+x}dx$. I think I have to use $u$ substitution, but I'm having trouble understanding what to do. Thank you in advance!
| Here's a trick against long division:
$$
\begin{align}
\int \frac{2x}{1 + x}\mathrm dx &= 2\int \frac{(x + 1) - 1}{(x + 1)}\mathrm dx \\
&= 2\left(\int\mathrm dx -\int\frac{\mathrm d(x+1)}{x+1}\right)\\
&= 2\Big(x - \ln|x+1|\Big) \color{grey}{+ \mathcal C}
\end{align}$$
Note that $2x + \ln\left|\frac{1}{x^2 + 2x + 1}\r... | {
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Evaluate $\lim_{n\rightarrow\infty}(1+\frac{1}{n})(1+\frac{2}{n})^{\frac{1}{2}}\cdots(1+\frac{n}{n})^{\frac{1}{n}}$ Evaluate
$$\lim_{n\rightarrow\infty}\left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)^{\frac{1}{2}}\cdots \left(1+\frac{n}{n}\right)^{\frac{1}{n}}$$
solve:
$$ \exp\left\{\lim_{n\rightarrow\infty} \frac... | $$\int_0^1\frac{ln(1+x)}{x}dx$$
$$=Li_2(-1)-Li_2(0)$$
$$=-\sum \frac{0}{n^2}+\sum \frac{(-1)^n}{n^2}$$
$$=\frac{\pi^2}{12}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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'Easy' question on continuity of integral Here's a problem that I recently stumbled upon. It seems pretty easy, and quite intuitive yet every time I try to solve it, I run into some difficulties. Here it goes :
Let $\phi \in BC(\mathbb{R}^2,\mathbb{R})$ (bounded and continuous) be a function such that
$$\forall_{x \in ... | Here is a counterexample:
$$|\phi(x,z)| = x^2e^{-x}|z|e^{-x|z|}\chi_{[0,\infty)}(x).$$
The function is bounded and continuous, but
$$\psi(0) = 0,$$
and for all $x > 0$,
$$\psi(x) = x^2e^{-x}\int_{-\infty}^{\infty}|z|e^{-x|z|}\,dz= 2x^2e^{-x}\int_{0}^{\infty}ze^{-xz}\,dz=2e^{-x}>0.$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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How do I solve this differential equation $y''^2-2y'y''+3=0$ in parametric form? I do have quite no idea about this one.
The obvious substitution $y'=p, y''=p\frac{dp}{dy}$ doesn't make the situation any better
| Note that $2y'y'' = ((y')^2)'$
Then let $g = y'$ so the equation reformulates to:
$$(g')^2 - g^2 + 3 =0 $$ which is much simpler an easier to solve. Then find a solution for $g = g(x) = y' $ and integrate to find $y(x) $
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Factorization of $x^6 - 1$ I started by intuition since I'm familiar with the formula $a^2-b^2 = (a-b)(a+b)$. So in our case $$x^6 - 1 = ({x^3} - 1)({x^3}+1)$$
How should I proceed? I assume there's some sort of algorithm to keep the process till you reach a form of irreducible linear polynomials.
| ${x^3} - 1=(x-1)(x^2+x+1)$
${x^3} + 1=(x+1)(x^2-x+1)$
The last quadratic term is already irreducible in $\mathbb{R}$.
If you mean irreducible in $\mathbb{C}$, you may split it into product of factor $(x-e^{inw})$, where $n=0\cdots 5$,$w=\frac{\pi}{3}$ .
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to evaluate $\lim_{n\to\infty}\sqrt[n]{\frac{1\cdot3\cdot5\cdot\ldots\cdot(2n-1)}{2\cdot4\cdot6\cdot\ldots\cdot2n}}$ Im tempted to say that the limit of this sequence is 1 because infinite root of infinite number is close to 1 but maybe Im mising here something? What will be inside the root?
This is the sequence: ... | It is quite easy to show by induction that:
$$\frac{(2n-1)!!}{(2n)!!}\geq\frac{1}{\sqrt{2n}}\tag{1}$$
since the last line is implied by:
$$\frac{2n+1}{2n+2}\geq\sqrt{\frac{2n}{2n+2}}$$
that is equivalent to:
$$(2n+1)^2\geq 2n(2n+2) = (2n+1)^2-1.$$
Using $(1)$ and the trivial bound $\frac{(2n-1)!!}{(2n)!!}\leq 1$, it fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1024290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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Is a projective $R[G]$-module a projective $R[H]$-module if $H$ is a subgroup of $G$? I have a ring $R$ of characteristic $0$ and a finite group $G$. Let $H$ be a subgroup of $G$.
Question: If $M$ is a projective $R[G]$-module where $R[G]$ is the usual group ring then is $M$ also projective as an $R[H]$-module?
T... | The mentioned assumptions are not necessary. If $R$ is any ring and $H$ is a subgroup of any group $G$, then $R[G]$ is a free $R[H]$-module (both from left and from right). This is basically because $G$ is a free $H$-set. We have $G = \coprod_i H g_i$ for some $g_i \in G$, and hence $R[G] = \bigoplus_i R[H] g_i$.
Since... | {
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Prove that $\gcd(2^a - 1, 2^b - 1) = 2^{\gcd(a,b)} - 1$ I have two questions about a prove that I have to do for my mathematic study. I'm now thinking about it the whole day, but can't find the prove.
Let $a,b \in \mathbb Z_{>0}$.
(a) Prove: $\gcd(2^a - 1, 2^b - 1) = 2^{\gcd(a,b)} - 1$
(b) Is this also true when you r... | Without loss of generality, let us assume that $a>b$.
You can write $2^a-1 = 2^b2^{a-b} - 2^{a-b} + 2^{a-b} - 1 = 2^{a-b}(2^b-1) + 2^{a-b}-1$.
$\therefore gcd(2^a-1,2^b-1) = gcd(2^b-1,2^{a-b}-1).$
You can now continue the proof on the same lines as Euclid's proof for gcd of two integers.
There is no significance of the... | {
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By using just the definition of a Cauchy Sequence, how could I show $s_n=n^2$ is not a Cauchy Sequence? Here's what I have,
Let $\epsilon>0$. Then there exists N such that $m,n>N \implies |n^2-m^2|<\epsilon.$
So now we have to find a $N$ that makes the implication true. So, $|n^2-m^2|<\epsilon=n^2<\epsilon+m^2=n<\sqrt{... | For $n \neq m$, $|n^{2} - m^{2}| \in \mathbb{Z}$, $|n^{2} - m^{2}| \neq 0$. There isn't an integer between $0$ and $1$, so if you pick $\epsilon = \frac{1}{2}$, then $|m^{2} - n^{2}| > \epsilon$ whenever $m \neq n$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Evaluating $I_{\alpha}=\int_{0}^{\infty} \frac{\ln(1+x^2)}{x^\alpha}dx$ using complex analysis Again, improper integrals involving $\ln(1+x^2)$
I am trying to get a result for the integral $I_{\alpha}=\int_{0}^{\infty} \frac{\ln(1+x^2)}{x^\alpha}dx$ - asked above link- using some complex analysis, however, I couldn't f... | Differentiation under the integral sign is another possibility. Let:
$$ I(\alpha,\beta) = \int_{0}^{+\infty}\frac{\log(1+\beta x^2)}{x^\alpha}\,dx. \tag{1}$$
Then assuming $1<\alpha<3$ we have:
$$\frac{\partial}{\partial\beta} I(\alpha,\beta) = \int_{0}^{+\infty}\frac{x^{2-\alpha}}{1+\beta x^2}\,dx = \beta^{\frac{\alph... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1024643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Finite state Markov chain Under what conditions a Markov chain can be considered as finite (and not infinite)?
Thank you!
| The characterization of a Markov Chain as finite or infinite refers to the state space $Ω$ in which the Markov Chain takes it's values. Thus, a finite (infinite) Markov chain is a process which moves among the elements of a finite (infinite) set $Ω$. In other words, if $X_n$ describes the state of the process at time $... | {
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$\text{arg}(i \: \text{conj}(z))=\text{arg}(i)+\text{arg}(\text{conj}(z))$ This is a Complex Analysis question.
Let z be a complex number, i be the imaginary unit, arg be the argument of z, and $\bar{z}$be the complex conjugate of z.
How do I prove that the following equality holds?
$\text{arg}(i \: \bar{z})=\text{arg}... | For $z=a+bi$, use $\arg(z_1z_2)=\arg(z_1)+\arg(z_2),\,\mod(-\pi,\pi]$:
$$
\arg(i\overline{z})=\arg(i)+\arg(\overline{z}).
$$
To see why the identity is true, note that $z=|z|e^{i\arg z}$, so
$$
\arg(z_1z_2)=\arg(|z_1z_2|e^{i(\arg(z_1)+\arg(z_2))})=\arg(z_1)+\arg(z_2),\,\mod (-\pi,\pi].$$
Throughout I am using $\arg$ a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Verification of an R-Module isomorphism between $R^n$ and its dual With one step at a time, I am getting slightly more used to $R$-Modules.
Let $R$ denote a commutative Ring with $\mathbb{1}$ and $n$ a natural number. For the tuple $a:= (a_i)_{i=1}^n \in R^n$ we have the mapping: $$\varphi_a: R^n \to R ,\ (x_i)_{i=1}... | Given $\varphi\colon R^n\to R$ you want to find $a=(a_1,a_2,\dots,a_n)\in R^n$ such that $\varphi=\varphi_a$. In particular, you want that
$$
\varphi(e_1)=\varphi_a(e_1)=a_1
$$
where $e_1=(1,0,\dots,0)$.
Can you go on from here?
A similar idea can be used for injectivity.
The rest is maybe a bit too verbose, but correc... | {
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Compute the limit: $\lim_{x\to \infty}x\ \log\left(\frac{x+c}{x-c}\right)$ Does anyone know how o compute the following limit?
$$\lim_{x\to \infty}x\ \log\left(\frac{x+c}{x-c}\right)$$
I tried to split it up as follows:
$\lim_{x\to \infty}\left[x\ \log(x+c)-x\ \log(x-c)\right]$ but still this is indeterminate form of t... | Hint; Let $t=\frac1x$${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 3
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Do elementary row operations give a similar matrix transformation? So we define two matrices $A,B$ to be similar if there exists an invertible square matrix $P$ such that $AP=PB$. I was wondering if $A,B$ are related via elementary row operations (say, they are connected via some permutation rows for example) then are ... | No. For instance, because row permutations do not preserve the trace.
$$
\begin{pmatrix}
1 & 2 \\
4 & 3
\end{pmatrix}\to
\begin{pmatrix}
4 & 3 \\
1 & 2
\end{pmatrix}
$$you can chenge the first matrix to the second via a permutation of the rows, but the trace of the first one is $4$ and the trace of... | {
"language": "en",
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Diffeomorphism: Unit Ball vs. Euclidean Space In my differential geometry class we are being asked to prove that the open unit ball
$B^n$ = { $x$ $\in$ $\mathbb{R}$$^n$ such that |$x$| < $1$}
is diffeomorphic to $\mathbb{R}$$^n$
I am having a hard time with this as I am brand new not only to differential geometry, b... | Let $\phi \colon [0,1) \to [0, \infty)$ a diffeomorphism with inverse $\psi$. Some possible choices: $t \mapsto \frac{t}{1-t}$, $t \mapsto \tan (\frac{\pi}{2}\cdot t)$.
The map
$$x \mapsto \phi(||x||) \cdot \frac{x}{||x||}$$
is a diffeomorphism from $B^n$ to $ \mathbb{R}^n$ with inverse
$$y \mapsto \psi(||y||) \cdot ... | {
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"timestamp": "2023-03-29T00:00:00",
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Second Degree Polynomial Interpolation, error related We want to create a table of the exponential integral function $$E_{1}(x)=\int_{x}^{\infty}\frac{e^{-t}}{t}dt, x>0$$
over the interval $x \in [1,10]$ with stepsize $h$. How large can $h$ be if a user of your table expects to obtain values at arbitrary x-locations wi... | Hints: I will map it out, please fill in the details.
The error formula for second degree polynomial interpolation is given by:
$$\tag 1 |P_2(x) - f(x)| \le \dfrac{|(x-x_0)(x-x_1)(x-x_2)|}{3!}~\mbox{max}_{a \le x \le b} |f^{(3)}(x)|$$
Since we are using three points, we can use equal spacing and take $x_0 = -h, x_1 = 0... | {
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How to show a particular set $S$ is a basis for $M_{2\times 2}$? I'm given a set $S$ of four $2\times 2$ matrices with numbers in them and need to show they are a basis for $M_{2\times 2}$. all I know is that the linear combination of these matrices $= [0,0,0,0]$ must only have a solution of a scalar * each of them bei... | You need to transform them to coordinate vectors and then your matrix will be converted in form (a,b,c,d) (i,e vector). then procceed with usual method of checking L.I ,by checking determinant or whatever
I highly recommend following for complete understanding of subject
https://www.youtube.com/playlist?list=PLGqzsq0e... | {
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Describe all the complex numbers $z$ for which $(iz − 1 )/(z − i)$ is real.
Describe all the complex numbers $z$ for which $(iz − 1 )/(z − i)$ is real.
Your answer should be expressed as a set of the form $S = \{z \in\mathbb C : \text{conditions satisfied by }z\}$.
I started solving for $((iz − 1 )/(z − i)) = \overl... | We need $i\cdot\dfrac{z+i}{z-i}$ purely real
$\iff\dfrac{z+i}{z-i}$ purely imaginary $=iy$(say) where $y$ is real
$$\frac{z+i}{z-i}=iy\implies z+i=iy(z-i)=z(iy)+y\implies z(1-iy)=y-i\implies z=\frac{y-i}{1-iy}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1025518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Largest "leap-to-generality" in math history? Grothendieck, who is famous inter alia for his capacity/tendency to look for the most general formulation of a problem, introduced a number of new concepts (with topos maybe the most famous ?) that would generalize existing ones and provide a unified, more elegant and more ... | Perhaps Claude Shannon deserves a mention for his work, such as defining Entropy in the 1948 paper "A Mathematical Theory of Communication" and generally spurring the field of information theory. He created very general models for communication and gave some profound results about them. I don't know much about Shanno... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "48",
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Why an ideal in a ring is a submodule of a free module over that ring I know an ideal in a ring is a module over this ring, but I don't know why it's a submodule of a free module, what's the free module? Thank you.
| A fintely generated free $R$-module is isomorphic to $R^n$. Take $n=1$ and you see that $R$ is a free $R$-module. Now it is left to check that an ideal $I\subset R$ defines an $R$-submodule. But this is clear by definition. Is every step and the result clear to you?
| {
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How can we be sure of periodicity by testing some terms? A mod $n$ Fibonacci sequence is simply defined as the Fibonacci sequence, except all terms are in mod $n$.
Now to determine periodicity, the worked solutions computed the first 20 or so terms and then observed that it was periodic with period 16.
My question is: ... | If $F_{16} \equiv F_0 \equiv 0 \text{ modulo }n$ and $F_{17} \equiv F_1 \equiv 1 \text{ modulo }n$ then you can use induction to show $F_{16+k} \equiv F_k \text{ modulo }n$ for all non-negative integer $k$.
This does not guarantee that the period is $16$, but it does ensure the period divides $16$: for example with $n... | {
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Verify the identity $\cos^2x-\sin^2x = 2\cos^2x-1$ I am having problems understanding how to verify this identity. I am quite sure that it is to be solved using the Pythagorean identities but, alas, I'm not seeing what might otherwise be obvious.
I need to verify the identity
$$\cos^2x-\sin^2x = 2\cos^2x-1$$
Thank yo... | Yes, indeed, we can use the Pythagorean Identity:
$$\cos^2 x + \sin^2 x = 1 \iff \color{blue}{\sin^2 x = 1-\cos^2 x}$$
$$\begin{align}\cos^2x-\color{blue}{\sin^2x} & = \cos^2 x - \color{blue}{(1-\cos^2 x)}\\\\
&= \cos^2 x - 1 + \cos^2 x \\ \\
& = 2\cos^2 x - 1\end{align}$$
| {
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How can I evaluate $\lim_{n \to\infty}\left(1\cdot2\cdot3+2\cdot3\cdot4+\dots+n(n+1)(n+2)\right)/\left(1^2+2^2+3^2+\dots+n^2\right)^2$? How can I evaluate this limit? Give me a hint, please.
$$\lim_{n \to\infty}\frac{1\cdot2\cdot3+2\cdot3\cdot4+\dots+n(n+1)(n+2)}{\left(1^2+2^2+3^2+\dots+n^2\right)^2}$$
| Hint:
$$1\cdot 2\cdot 3 + \dots +n(n+1)(n+2) \le n(n+2)^3 \le 8n^4$$
$$(1^2 + 2^2 + \dots + n^2)^2 = \left(\frac{n(n+1)(2n+1)}{6}\right)^2\ge \frac{1}{9}n^6$$
Hence
$$\frac{1\cdot 2\cdot 3 + \dots +n(n+1)(n+2)}{(1^2 + 2^2 + \dots + n^2)^2} \le \frac{8n^4}{1/9\cdot n^6}=\frac{72}{n^2}\to 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1026026",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
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Geometrical Interpretation of Cauchy Riemann equations? Differentiation has an obvious geometric interpretation, and the Cauchy Riemann equations are closely linked with differentiation.
Do the Cauchy Riemann equations have a geometric interpretation?
| The version of the Cauchy-Riemann equations that I prefer is (assuming we are using $x,y$ as coordinates for the real and imaginary part on the domain)
$$
i\partial_x f = \partial_y f
$$
This has a direct geometrical interpretation: the partial derivative with respect to the real part rotated by $\pi/2$ (which is what ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1026134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 7,
"answer_id": 5
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Help to determine a basis for eigenspace Please find a basis for the eigenspace corresponding to eigenvalue=3 for the following matrix:
$$
\pmatrix{3&1&0\\0&3&1\\0&0&3}
$$
[3 1 0]
[0 3 1]
[0 0 3]
I have already calculated [A-(lambda)I] and the result is the following augmented matrix:
x1 x2 x3
[0 1 0 ... | Apparently $\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}\in\ker (A-3I)$ iff $x_2=x_3=0$. So the eigenspace is spanned by $\begin{pmatrix}1\\0\\0\end{pmatrix}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1026210",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Is $x^2-2$ irreducible over R and Q? I'm not sure if it is valid to say that $x^2 - 2$ can be factorised to $2\cdot\left(\frac 12x^2 - 1\right)$ for it to be reducible in Q.
Though I know $(x + \sqrt{2})(x - \sqrt{2})$ works in the reals.
| The polynomial is ofcourse reducible over $\mathbb{R}$ as it can be written as a product of polynomials of (strictly) smaller degrees.
Again, by Eisenstein's criteria it is irreducible over $\mathbb{Q}$
| {
"language": "en",
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Is there a game theory for doing the dishes in a shared living situation? It occurred to me this morning (when I was intentionally not tidying up my flatmate's dishes) that doing the dishes in a shared living situation, such as at an office, or living with housemates, might be subject to a kind of game theory.
The ide... | You could take the approach @Pburg suggests and view it in the repeated games context, but I think it's simpler to view it as a positive externality question. Here's a related question (with solutions) from a game theory class (where improving one's garden, I argue, is comparable to doing the dishes):
Question:
Alice a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1026420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
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Probability of selecting three of the same thing from a collection Question: A collection of 6 items is to be randomly drawn from a bin containing 100 good items and 8
defective items. What is the probability that exactly 3 of the items chosen are defective?
My Attempt: Well we know that in total there are 108 items. ... | The number $X$ of the defective items in the sample followes the hypergeometric distribution with parameters $N=108$ (population size), $K=8$ ("successes" in the population) and $n=6$ (sample size). Thus $$P(X=3)=\frac{\binom{8}{3}\binom{108-8}{6-3}}{\binom{108}{6}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1026532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Integration: u substitution problem I'm trying to integrate the following:
$\int_{0}^{\infty} x^2 e^{-2x}$
I know that $\int_{0}^{\infty} x^n e^{-x} dx = n!$
Feels like a $u$ substitution problem but I'm having trouble making use of the above.
Thanks,
Mariogs
| Look at the more general integral, related to Laplace transforms, of this type:
\begin{align*}
\mathcal{L}\left\{t^n\right\}\left(s\right)&=\int_0^\infty t^n e^{-st}\:dt,
\end{align*}
and you may solve using integration by parts to always get something of the form
\begin{align*}
\int_0^\infty t^n e^{-st}\:dt & = \frac{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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How many even 3 digit numbers contain at least one 7. How many even 3 digit numbers contain at least one 7.
I got 126, but it was not an answer choice for the problem. Can anyone help?
| Count the number of $3$ digit even numbers that do not contain $7$ and subtract from the total number of $3$ digit even numbers.
Let $xyz$ be the $3$ digit even number. Total number of $3$ digit even numbers is $450$, since $x$ has $9$ options, $y$ has $10$ options and $z$ has $5$ options. To obtain the number of $3$ ... | {
"language": "en",
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"source": "stackexchange",
"question_score": "4",
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How to find the integer solutions of $\frac{2^m-1}{2^{m+x}-3^x}=2a+1$? Is there a way to find all integer triplets of $(x, m, a)$ for the following equation.
$$\frac{2^m-1}{2^{m+x}-3^x}=2a+1$$
| @Next did already link to the other Q&A, so it is no more need to discuss this further. But I've looked at this one time with a slightly different focus, and may be the reformulation looks interesting for you for further experimenting.
Let for convenience $2a+1 = k$ and let us express $3^x $ in terms of $2^m$ such th... | {
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Need to draw phase portrait near the equilibrium points of differential equation So, this equation
$$\ddot{x}+3\dot{x}-4x+2x^2 = 0.$$
I can write like a system
\begin{equation}
\left\{ \begin{array}{ll}
\dot{x} = v, \\
\dot{v} = 2x^2 - 4x - 3v.
\end{array} \right.
\end{equation}
Finding the equilibrium points:
\begin{... | Your first calculation is correct and leads to an unstable saddle.
However, you made a slight error in your Jacobian which affected your second calculation (the first was not affected due to the zero value).
For the Jacobian, you should have:
$$J(x,y)=\begin{bmatrix}
0 & 1 \\
4-4х & -3 \\
\end{bmatrix}$$
When y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1026887",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Difference between "limit point" and "points in the closure" Given a topology $(X,T)$, $A\subset X$, $x \in X$ is a limit point of A if $\forall$ open $U$ that contains $x$, $(U\cap A)$\ {$x$} $\neq \emptyset$. $x \in X$ is in $cl(A)$ if $\forall$ open $U$ that contains $x$, $U\cap A$ $\neq \emptyset$. Is there any exa... | consider a finite set A of singletons in $\mathbb{R}$, there are no limit points, but they are al in $cl(A)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1026975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Acceleration from Velocity How would you find the acceleration of an automobile at $t$ seconds from the formula giving velocity $v(t)$ in $ft/sec$:
$$v(t) = \frac{85t}{6t+16}$$
$5$ seconds ? _____$ft/sec^2$
$10$ seconds ? _____$ft/sec^2$
$20$ seconds ? _____$ft/sec^2$
Do I use the formula:
$$a = \frac{v_1 - v_0}{t_1-t_... | I would strongly suggest that instead of simply applying formulas, you strengthen your concepts first.
The first equation you quoted:
$$a = \frac{v_1 - v_0}{t}$$
only applies in the case of a constant acceleration.
The acceleration here is not constant.
To find it you'll need to apply the more general rule:
$$a = \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1027090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can we make a sequence of real numbers such that polynomial of any degree with co-efficients of the sequence has all its roots real and distinct ? Does there exist a sequence of real numbers $(a_n)$ such that $\forall n \in \mathbb N$ , the polynomial $a_nx^n+a_{n-1}x^{n-1}+...+a_o$ has all $n$ real roots ? Can we mak... | Here is an inductive, semi-non-constructive approach: assume I have a polynomial $p(x) = a_0 + \dots + a_n x^n$, with $n$ real roots and which goes to infinity as $x$ goes to infinity. Let M be large enough that all the zeros of $p$ are contained in $[-M,M]$.
Now, pick $a_{n+1}$ positive but so small that $a_{n+1}x^{n+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1027184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Power series expansion involving non integer exponent I'm working on a real and complex analysis course right now and one power series question has me really stumped:
I'm not sure what to do with the non integer in the exponent, as my initial plan of differentiating the power series of 1/(1-x) won't work.
Any help on ... | I do not see what is the problem after Olivier Oloa's answer (use the generalized binomial theorem).
Doing so, the series expansion you look for is, around $x=a$ $$\frac{1}{(1-x)^{3/2}}=\frac{1}{(1-a)^{3/2}}+\frac{3 (x-a)}{2 (1-a)^{5/2}}+\frac{15 (x-a)^2}{8
(1-a)^{7/2}}+\frac{35 (x-a)^3}{16 (1-a)^{9/2}}+O\left((x-a)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1027435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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How can I show this equality between inverses of functions? Let $f:X\to Y$ be a function between metric spaces $X$ and $Y$. Show that for any $B\subset Y$, $f^{-1}(B^\complement)=(f^{-1}(B))^\complement$.
I was able to show that they both map to $B^\complement$, but I know that that's not enough to prove equality. By d... | Your $f^{-1}(B^\complement)\not\subset B$ is true but probably not what you meant, since the lefthand side is a subset of $X$ and the righthand side a subset of $Y$.
In any case you’re making it much too hard: $x\in f^{-1}[Y\setminus B]$ iff $f(x)\in Y\setminus B$ iff $f(x)\notin B$ iff $x\notin f^{-1}[B]$ iff $x\in X\... | {
"language": "en",
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"source": "stackexchange",
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Permutations and school timetable If there are 6 periods in each working day of a school. In how many different ways can one arrange 5 subjects such that each subject is allowed at least one period? I tried this way- One of the six periods can be arranged in 5 ways and the remaining 5 periods in 5 factorial ways. Total... | Lets think how we might pick this time table we can first assign a slot for each of our five subjects then finally pick the subject for the remaining slot.
Subject A has a choice of 6 slots
Subject B has a choice of 5 slots
Subject C has a choice of 4 slots
Subject D has a choice of 3 slots
Subject E has a choice of 2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1027759",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
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Question about derivatives and inequalities Problem: Assume that $f: \mathbb R\rightarrow\mathbb R$ and $g: \mathbb R\rightarrow\mathbb R$ are differentiable and $f(0) = g(0)$ and $f'(0) < g'(0)$. Prove that there exists $h > 0$, so that $f(x) < g(x)$ when $0<x<h$.
I tried to solve this by letting $F(x) = g(x) - f(x)$,... | Since $F(x)/x$ tends to a limit $F'(0)>0$ as $x\to0$, there is $h>0$ so that $F(x)/x>0$ for all $x\in(0,h)$.
Thus $F(x)>0$ for all $x\in(0,h)$, and your result follows.
The condition $F(x)/x\to F'(0)$ only means that $F(x)/x$ is close to $F'(0)$ (or positive) near $0$; the limit "does not see" the values for large $x$.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Derivative relation between two equal functions I am stuck with the following problem. Suppose $g: \Bbb R\rightarrow\Bbb R$ is $C^1$. $f(x,y)=g(x^2+y^2)$. I need to show that $xf_y=yf_x$
My attempt was:
$f_x=g_x \cdot 2x$ (1) and $f_y=g_y\cdot 2y$ (2)
Multiplying (1) by y and (2) by x, I get
$yf_x=g_x \cdot2xy$ and $xf... | Apply the chain rule
$$f_x(x,y)=g'(x^2+y^2)\cdot (2x) \quad\Longrightarrow\quad y\cdot f_x(x,y)=g'(x^2+y^2)\cdot 2xy$$
$$f_y(x,y)=g'(x^2+y^2)\cdot (2y) \quad\Longrightarrow\quad x\cdot f_y(x,y)=g'(x^2+y^2)\cdot 2xy$$
And the equality follows.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Sum of $\sum_{n=0}^{\infty }\frac{1}{4^{(n/3)+1}}$ Find the sum of $$\sum_{n=0}^{\infty }\frac{1}{4^{(n/3)+1}}$$
| prove by induction that for the definite sum holds
$\sum_{n=0}^{m}\frac{1}{4^{\frac{n}{3}+1}}=-1/12\, \left( {2}^{2/3}+2\,\sqrt [3]{2}+4 \right) \left( \left( 1/2
\,\sqrt [3]{2} \right) ^{m+1}-1 \right)
$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1028018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Find all possible combinations of $A, A, A, B, B$ 10 year old daughter has this problem.
She knows that all possible combinations of $A,B,C,D$ are $4! = 24 $
She figured it like this:
If I write down $A$ first, it has $4$ possible places.
If I write down $B$ next, it has $3$ possible places for each of the places of $... | Hint.
First pretend that the $A$'s and $B$'s are actually different - for example, say they're numbered $A_1$, $A_2$, $A_3$, $B_1$, $B_2$. Then this is the same problem you mentioned, and there are $5!$ ways to arrange the letters.
The problem is that you'll obtain many arrangements that differ only because of the num... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$2x = 2y \Rightarrow x = y$ with $x,y \in \mathbb{Z}$ How can I show, $2x = 2y \Rightarrow x = y$ with $x,y \in \mathbb{Z}$, when I can only use elements out of $\mathbb{Z}$? It is $\frac{1}{2} \notin \mathbb{Z}$, so I can't multiply both sides with $\frac{1}{2}$. How can I prove it?
| $$\begin{align}
2x&=2y
\\ \Rightarrow2x-2y&=0
\\ \Rightarrow 2(x-y)&=0
\\ \Rightarrow 2=0&\text{ or }x-y=0
\\ \Rightarrow x&=y.
\end{align}$$
EDIT: This (or rather I) implicitly uses the field axioms of $\mathbb{Q}$. We need this to fix it.
EDIT 2 How about this:
$$\begin{align}
2x&=2y
\\\Rightarrow x+x&=y+y
\\\Rightar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1028193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Stokes’ Theorem to find integration Use Stokes’ Theorem to evaluate integration $c (xy \,dx+ yz\, dy + zx\, dz)$ where and $C$ is the triangle with vertices $(1,0,0),(0,1,0),(0,0,1)$, oriented counter-clockwise rotation as viewed from above.
can anyone please help me with this?
| You are planning to evaluate the path integral
$$\oint_C \vec{X} \cdot d\vec{r}$$
where
$$\vec{X} = \left(\begin{array}{cc}xy\\yz\\zx\end{array}\right) \ .$$
Stokes Theorem gives
$$\oint_C \vec{X} \cdot d\vec{r} = \iint_{\text{Inside} \ C} \nabla\times \vec{X} \cdot d\vec{S}$$
so we start by evaluating (google how to c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1028311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Help with Related Rates problem I'm slowly working my way through a ton of these problems but have come across one that has me stumped.
Here's the problem in full:
Water is leaking out of an inverted conical tank at a rate of
$0.0081$${\frac {m^3}{min}}$. At the same time water is being pumped
into the tank at a ... | We can use similar triangles to get rid of a variable. Since we're given $\frac{dH}{dt}$, we want to eliminate $r$ by expressing it as a function of $H$. To this end, notice that at any point in time, we have that:
$$
\frac{r}{H} = \frac{3/2}{8} \iff r = \frac{3}{16}H
$$
Substituting before differentiating, we obtain:
... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Proofs of Sets and Subsets I have these proof problems that I need some help on, any direction would be great. Thanks
Let A, B, and C be subsets of some universal set U
(a) Prove the following:
IF $A \cap B$ $\subseteq$ C, and $'A \cap B$ $\subseteq$ C, THEN $B \subseteq C$
(b) Either prove the following or provide a c... | Hint. For (a) it is given that
$$A\cap B\subseteq C\ ,\quad A'\cap B\subseteq C$$
and you have to prove $B\subseteq C$. You should know the basic way of proving a subset statement like this: assume $x$ is in the LHS, and use this assumption (and the given facts) to prove that $x$ is in the RHS.
So, let $x\in B$. Con... | {
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"url": "https://math.stackexchange.com/questions/1028466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to decompose a matrix into an antisymmetric matrix plus a multiple of the identity I was given a problem to solve earlier that I couldn't figure out. I don't still have it, but it was basically: Given the invertible matrix $A$, find the invertible matrix $P$, such that $A=P^{-1}CP$. Where $$C=\begin{bmatrix} c & ... | Hint: If two matrices are diagonalizable with the same eigenvalues, then they are similar.
If an $n \times n$ matrix has $n$ distinct eigenvalues, it is necessarily diagonalizable.
The punchline: you should try to find a matrix $C$ that has the same eigenvalues as $A$.
Note that if we have invertible matrices $S,T$ su... | {
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Range conditions on a linear operator While reading though some engineering literature, I came across some logic that I found a bit strange. Mathematically, the statement might look something like this:
I have a linear operator $A:L^2(\Bbb{R}^3)\rightarrow L^2(\Bbb{R}^4)$, that is a mapping which takes functions of t... | I think it always has to be true. It is like mapping points in a flat plane to a 3D space. You get a volume 0 manifold in the 3D space if there is a 1 to 1 mapping.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1028695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Trigonometric substitution in the integral $\int x^2 (1-x^2)^{\frac{9}{2}} \ \mathrm dx$ I'm trying to solve
$$\int_{-1}^{1} x^2(1-x^2)^{\frac{9}{2}} \, dx$$
The hint said to use the substitution $x=\sin y$
I got $$\int_{-\pi/2}^{\pi/2} \sin^2y \cos^{\frac{11}{2}} y \,dy$$
| As said in comments, the antiderivative reduces to the evaluation of $$I=\int \cos^{10}(x)dx-\int \cos^{12}(x)dx=J_{10}-J_{12}$$ with $$J_n=\int \cos^{n}(x)dx$$ The $J_n$ terms can easily be computed since we can easily establish (performing two integrations by parts) the recurrence relation $$J_n=\frac 1n \cos^{n-1}(x... | {
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"timestamp": "2023-03-29T00:00:00",
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Is $\sqrt2$ a tricky notation? When someone asked me how to solve $x^2=9$,I can easily say, $x=3$ or $-3$. But what about $x^2=2$? There is NOT any "ordinary" number to solve this question. It's an irrational number. So we say helplessly, the answer is $\pm\sqrt2$, but what does $\sqrt2$ mean? It's a number, when squar... | Yes, $\sqrt{2}$ only tells you that it is a number which, when squared, yields $2$. It's a whole lot more informative than any other thing you might write for the same number. However, the fact that $\sqrt{2}^2=2$ is really important in certain contexts. For instance, in higher mathematics, we are often less concerned ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1028866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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sum and product of Lipschitz functions I have the following question from my notes, where $f$ and $g$ are Lipschitz functions on $A ⊂ \Bbb R$.
I'm able to show that the sum $f + g$ is also a Lipschitz function, however I'm stuck on trying to show that if $f,g$ are bounded on $A$, then the product $fg$ is Lipschitz
on $... | Suppose $|f| \leq M$ and $|g| \leq M$.
Then
$$ |(fg)(x) - (fg)(y)| \leq |f(x)g(x) - f(x)g(y)| + |f(x)g(y) - f(y)g(y)| \leq M(|g(x) - g(y)|+|f(x)-f(y)|)$$
Consider $f(x) = x \, \forall x \in [0,\infty)$ which is Lipchitz on $[0, \infty)$ but $f^2$ is not.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1028974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 1,
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} |
Closed- form of $\int_0^1 \frac{{\text{Li}}_3^2(-x)}{x^2}\,dx$ Is there a possibility to find a closed-form for
$$\int_0^1 \frac{{\text{Li}}_3^2(-x)}{x^2}\,dx$$
| Focus on the relation
$$\frac{d}{dx}{\rm Li}_k(x)=\frac{1}{x}{\rm Li}_{k-1}(x)$$
Let's look at the derivative of ${\rm Li}_m {\rm Li}_n$ in general:
$$\frac{d}{dx}({\rm Li}_m(-x){\rm Li}_n(-x))=\frac{1}{x}({\rm Li}_{m-1}(-x){\rm Li}_n(-x)+{\rm Li}_m(-x){\rm Li}_{n-1}(-x))$$
Let's define
$$f(m,n)=\int_0^1 \frac{{\rm Li}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1029101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 1,
"answer_id": 0
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Integration with respect to a signed measure Let $\mu$ be a singed measure, $f\in C_c(X)$, I want to show
$$\int fd(c\mu) = c \int fd\mu, \forall c \in \mathbb{R}$$
Since $c\mu$ is also a singed measure, I think by definitionm I need to show $$\int fd(c\mu)^+-\int fd(c\mu)^- = c \int fd\mu^+-c \int fd\mu^-$$
But I thi... | Seems correct, I assume that you used the following line of reasoning: if $c<0$ then
$$
(c\mu)^+ = \frac12(|c\mu|+c\mu) = \frac12|c|(|\mu|-\mu) = |c|\mu^- = -c\mu^-.
$$
Alternatively, you could go via the decomposition of the state space into positive and negative sets.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1029179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Why is indefinite integral called so? Two questions that are greatly lingering on my mind:
1.
Integral is all about area(as written in Wolfram).
But what about indefinite integral? What is the integral about it?? Is it measuring area?? Nope. It is the collection of functions the derivative of which give the original... | A primitive of a function $f$ is another function $F$ such that $F'=f$. If $F$ is a primitive of $f$, so is $F+C$ for any constant $C$, the so called constant of integration. The indefinite integral of $f$ can be thought of as the set of all primitives of $f$:
$$
\int f=F+C.
$$
Why indefinite? Because is there some ind... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1029310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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Is $\nabla$ a vector? The following passage has been extracted from the book "Mathematical methods for Physicists":
A key idea of the present chapter is that a quantity that is properly called a vector must have the transformation properties that preserve its essential features under coordinate transformation; there... | With $f$ a scalar function of the coordinates, $\nabla f$ is a vector called the gradient of $f$.
With $f$ a vector function of the coordinates, $\nabla.f$ is a scalar called the divergence of $f$.
With $f$ a vector function of the coordinates, $\nabla\times f$ is a vector called the curl of $f$.
These three symbols ($... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1029433",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
A fair game with triangular numbers?
Definition 1
A ball game is a state where you have $n$ white balls and $m$ black balls.
The rule is that you remove first one ball from the cup. And without returning the first ball, you pick another.
*
*$P(A)$ is defined as the probability of drawing two balls with opposite colo... | I think the diagram may not generalise in an obviously triangular way. For example with $3$ and $6$ the diagram has $36$ lines ($72$ if you count them in both directions) and the best I could do was something like
I think the most you can say is that this suggests for a fair game you have $$mn = \tfrac12m(m-1) +\tfr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Book recommendation for ordinary differential equations This question has been posted before, but I need book with specific qualifications. I do not need books for engineers, book that is centered around calculations and stuff. I need to find a book that is theoretical, proves the statements and has good presentation o... | I would like to add "Differential Equations With Applications and Historical Notes by George Simmons" to this list. This is a very well written text on ODEs and very approachable and explains concepts very clearly.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1029646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 4,
"answer_id": 2
} |
How to solve the integral equation? How to solve the integral equation
$$ \int_{-20}^{x} \left| \left| \left| \left| \left| \left|
\left| \left| t \right| -1 \right| -1 \right| -1 \right| -1 \right|
-1 \right| -1 \right| -1 \right| \,{\rm d}t={\frac {4027}{2}}?$$
| The main tough part is getting a grasp on the function in the integral. Let
$$
A_0(t) = |t|
$$
and for $n > 0$
$$
A_n(t) = |A_{n-1}(t) - 1|
$$
Then the function integrated is $A_7(t)$ (count the -1's).
Now it is easy to prove (by a two-step induction proof on odd and even $n$) that for all $n \in \Bbb{Z}$
$$
|t| \leq ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1029746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Very basic question about submanifolds. I'm beginning to study differential geometry and I'm a litle confused about the concept of submanifold of a differentiable manifolds.
Can someone provide me an example of how to show that a non-open subset of a manifold is a submanifold and other showing it isn't?
| Here are some very simple examples.
Let $M = \mathbb R^2$. Let $N$ be a straight line in $M$. Then $N$ is not open in $M$ and a submanifold of $M$. For example, you can choose global coordinates in $M$ such that $N$ is the first coordinate axis.
Let $P \subseteq M$ be the union of two straight lines which meet in one p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1029842",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The behavior of BV functions at a point of approximate continuity Given $u\in BV(\mathbb R^N)$, we say $u$ is approximate continuos at $x$ and the approximation limit is $l\in R$ if
$$ \lim_{r\to 0}\frac{\mathcal{L}^N(B(x,r)\cap \{|u-l|>\epsilon\})}{r^N} =0 $$
for all $\epsilon>0$. (I know the approximate continuity ... | (I'll answer the first question; please post the second one separately.)
The answer is negative. In $N\ge 2$ dimensions, let $\xi$ be some unit vector and define
$$
u(x) =\sum_{n=1}^\infty \left(1-4^n |x-2^{-n}\xi|\right)^+
$$
This function is in BV, because the BV norm of the $n$th term (i.e., $L^1$ norm of its gradi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1029991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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An odd function $f$ is differentiable at zero. Prove $f'(0)=0$? I know that $f'$ of an even function is odd function, thus I have $f(x)=f(-x)$.
However I'd no idea how to prove that $f'(0)=0$?
Please answer my question...
| I assume you mean that $f(x)$ is an even function. If so, note that for an even function $f(x)$, it satisfies $f(x)=f(-x)$. Now if a function is differentiable, it satisfies
$$f'(x) =\lim_{h\to0^+}\frac{f(x+h)-f(x)}{h}=\lim_{h\to0^-}\frac{f(x+h)-f(x)}{h}$$
Now note that
$$f'(0)=\lim_{h\to0^+}\frac{f(h)-f(0)}{h}=\lim_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1030166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Evaluate $\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$ using complex analysis How do I compute
$$\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$$
What I am doing is take
$$f(z)=\frac{(\log z)^2}{1+z^2}$$
and calculating
$\text{Res}(f,z=i) = \dfrac{d}{dz} \dfrac{(\log z)^2}{1+z^2}$
which came out to be $\dfrac{\pi}{2}-\df... | For this integral, you want to define you branch cut along the negative x axis and use a contour with a semi circle around the origin.
Also, note we are taking the real Cauchy Principal value.
Then
$$
\int_0^{\infty}\frac{(\ln(z))^2}{1+z^2}dz = \int_{\Gamma}f(z)dz+\int_{\gamma}f(z)dz + \int_{-\infty}^0f(z)dz + \int_0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1030246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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How do I convert this parametric expression to an implicit one I have:
$$x=5+8 \cos \theta$$
$$y=4+8 \sin \theta$$
With $ -\frac {3\pi}4 \le \theta \le 0$
If I wanted to write that implicitly, how would I do it? I get that it's a circle, and I can easily write the circle implicitly, but I'm not sure how to convert the ... | Use that $\exists \theta : (v,w) = (\cos\theta, \sin\theta) \iff v^2 + w^2 = 1$.
You immediately get that the parametric curve is a part of the curve defined by
the implicit equation
$$
\left(\frac{x-5}8\right)^2 + \left(\frac{y-4}8\right)^2 = 1$$
To get only the part that you want, you must also make sure that $-\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1030485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Do solutions of $\dot{x} = \frac{x}{t^2} + t$ exist satisfying $x(0) =0$ Suppose we have the 1-dimensional ODE
\begin{equation}
\dot{x} = \frac{x}{t^2} + t
\end{equation}
Do there exist solution curves with initial condition $x(0)=0$? If you proceed in a standard way then you would get as solution formula
\begin{equa... | The problem is that your original problem is not a Cauchy problem
The usual setup for solving differential equations is that you are given a function $f: [a,b] \times \mathbb R \to \mathbb R$ and a point $(x_0, y_0)$ and you want to find a function $y:[a,b]\to\mathbb R$ which satisfies two conditions:
*
*$f(x,y(x)) ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Showing that normal line passes through a point. I need to show that a line passes through a point. How should I go about doing this? The question is:
Let $L$ be the normal line at $(1,1,1)$ to the level surface of $f(x,y,z) = x^2 - z$ that passes through $(1,1,1)$. Show that $L$ passes through the point $(3,1,0)$.
T... | If the line passes through the two given points, its direction vector is $(3,1,0)-(1,1,1)=(2,0,-1)$, indeed parallel to the gradient.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1030814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Analytic Combinatorics to asymptotically estimate the number of objects of size at most n? I have read some bits of Flajolet's and Sedgewick's book on Analytic Combinatorics.
I am quiet curious as how to asymptotically estimate the number of objects of size at most n.
Suppose for example that I specified a combinatoria... | Indeed, it does the trick!
B(z) = $A(z) \times \sum\limits_n z^n = \sum\limits_{n} \sum\limits_{k = 0}^n a_k z^n$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1030930",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Can the underlying set functor corresponding to an algebraic theory always be viewed as a model of that theory? Let $\mathsf{T}$ denote a Lawvere theory, and let $\mathbf{C}$ denote its category of models in $\mathbf{Set}$. Write $U : \mathbf{C} \rightarrow \mathbf{Set}$ for the underlying set functor. I think that $U$... | In more detail: Recall that a model of a Lawvere theory $\mathcal{T}$ in a category (with finite products) $\mathcal{S}$ is a finite-product-preserving functor $\mathcal{T} \to \mathcal{S}$. Thus, for any category $\mathcal{C}$, a $\mathcal{T}$-model in $[\mathcal{C}, \mathbf{Set}]$ is the same thing as a diagram $\mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1030984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Is there any holomorphic function in a unit ball Is there any holomorphic function in a unit ball such that $f(1/n)=n^{-5/2}$ for $n=2,3,\dots$
Natural candidate is $f(z)=z^{5/2}$ But it isn't holomorphic obviously inside that ball. Can you tell me whatI should check?
| If $f$ is such a function, then $g(z)=f(z)^2$ conincides with $z\mapsto z^5$ on a sequence of points converging to $0$. Hence $g(z)=z^5$ for all $z$. If we write $f(z)=z^kh(z)$ with $k\in\mathbb N_0$, $h$ holomorphic, $h(0)\ne0$, we find that $z^{2k}h(z)^2=z^5$, qea.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1031141",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Subtraction of functions with BigO When trying to assess the Big $O$ of two functions that are added together, we take the max of the two. What happens if there is subtraction instead of addiiton?
for instance: $$f(n) = O(n^3) $$
$$ \text{and} $$ $$g(n) = O(n^3)$$
then
$$ (f-g)(n)$$
| Note that the sign of a function doesn't matter in $O $-notation:
If $f(n)\in O (h(n)) $ then $-f(n)\in O (h(n))$ follows directly from the definition of the $O $-Notation.
For two functions $f (n)\in O (h_1 (n)) $ and $g (n)\in O (h_2 (n))$ you know
$$f(n)+g (n)\in O (\max (h_1 (n), h_2 (n))). $$ where in this case
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1031270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Suppose $f$ is integrable on $\mathbb{R}$, and $g$ is locally integrable and bounded. Then $f*g$ is uniformly continuous and bounded?
Suppose $f$ is integrable on $\mathbb{R}$, and $g$ is locally
integrable and bounded. Then $f*g$ is uniformly continuous and
bounded?
I don't even know where to start proving or di... | Assuming $g$ bounded
$$
|f\ast g(x)|\le\int_{\mathbb{R}}|g(y)|\,|f(x-y)|\,dy\le\|g\|_\infty\|f\|_1.
$$
Next, for $h\in\mathbb{R}$
$$
|f\ast g(x+h)-f\ast g(x)|\le\|g\|_\infty\int_{\mathbb{R}}|f(x+h-y)-f(x-y)|\,dy=\int_{\mathbb{R}}|f(z+h)-f(z)|\,dz.
$$
Sincc $f$ is integrable,
$$
\lim_{h\to0}\int_{\mathbb{R}}|f(z+h)-f(z)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1031372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Certain property of supremum Given real numbers $x_{ni}, n\in \mathbb{N}, i\in I$, does it hold that $$\sup \bigg\{ \sum_{n\in\mathbb{N}}x_{ni}|i\in I\bigg\}=\sum_{n\in \mathbb{N}}\sup\bigg\{x_{ni}|i\in I \bigg\}?$$
Thanks in advance.
| No. A counterexample is $I = \{ \alpha, \beta \}$,
$$x_{n, \alpha} = \frac{1}{n^2}$$
$$x_{1, \beta} = \frac{5}{4}, \ \ x_{n, \beta} = 0 \ \mbox{for $n \geq 2$}$$
Then
$$\sup_{i \in I} \sum_{n \in \mathbb{N}} x_{n,i}= \max \{ \frac{\pi^2}{6} , \frac{5}{4}\} = \frac{\pi^2}{6}$$
while
$$\sum_{n \in \mathbb{N}} \sup_{i \in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1031540",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove this statement: $\binom{r}{r}+\binom{r+1}{r}+\cdots+\binom{n}{r}=\binom{n+1}{r+1}$ Let $n$ and $r$ be positive integers with $n \ge r$. Prove that
Still a beginner here. Need to learn formatting.
I am guessing by induction? Not sure what or how to go forward with this.
Need help with the proof.
| What is the number on the left counting?
The number of subsets of size $r$ of $\{1,2,3\dots r\}$ plus the number of subsets of size $r$ of $\{1,2,3\dots r+1\}$ plus the number of subsets of size $r$ of $\{1,2,3\dots r+1,r+2\}$ and so on up until the number of subsets of size $r$ of the set $\{1,2,3\dots n\}$
What is th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1031651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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$\mathbf R^2-\{\mathbf 0\}$ is homeomorphic to $S^1\times \mathbf R$. I am trying to to prove the following:
$\mathbf R^2-\{\mathbf 0\}$ is homeomorphic to $S^1\times \mathbf R$.
Since $\mathbf R^+=\{x\in \mathbf R:x>0\}$ is homeomorphic to $\mathbf R$, it suffices to show that $S^1\times \mathbf R^+$ is homeomorphic... | $g^{-1}$ sends $re^{i\theta}$ to $(e^{i\theta},r)$. It is enough to show $re^{i\theta}\mapsto e^{i\theta}$ and $re^{i\theta}\mapsto r$ are continuous. The latter is the norm. Hence it is continuous. The first is a quotient of the identity by the norm. It remains only to prove that the quotient (reciprocal + multiplicat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1031729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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A Pigeonhole Principle Question Show that in a party of $n$ people, there are two people having identical number of friends.
I am a beginner at Pigeonhole Principle problems and have produced a solution to this intermediate level question. I am excited about it and I want my solution to be cross-checked and if anything... | Clearly there is a minimum value of $n$ for which this statement holds, and
equally obviously it's not $n=1.$ The statement should be something of the form,
"in a party of $n$ people, where $n$ is at least ..., ... ."
I would strongly recommend being much clearer about the fact that you are going
to examine three sepa... | {
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"url": "https://math.stackexchange.com/questions/1031818",
"timestamp": "2023-03-29T00:00:00",
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Prove that $ \frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!}+\cdots + \frac{n}{(n+1)!} = 1 - \frac{1}{(n+1)!}$ for $n\in \mathbb N$ I want to prove that if $n \in \mathbb N$ then $$\frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!}+ \cdots+ \frac{n}{(n+1)!} = 1 - \frac{1}{(n+1)!}.$$
I think I am stuck on two fronts. First, I don't... | Induction.
*
*Base. $n = 1: \frac{1}{2!} = 1 - \frac{1}{2!}$.
*Step. $n = m$ $-$ true. Let's prove for $m + 1$:
$$
\frac{1}{2!} + \dots + \frac{m}{(m+1)!} + \frac{m+1}{(m+2)!} = 1 - \frac{1}{(m+1)!} = 1 - \frac{m+2}{(m+2)!} + \frac{m+1}{(m+2)!} =
$$
$$
= 1 - \frac{1}{(m+2)!}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1031909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
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$\lim_{x \to \infty} x\sin(\frac{1}{x}) = 1$ (epsilon-delta like condition) So, it is clear in many ways that this limit holds. However, I am interested in proving this with an "epsilon-delta" like condition. The start of my proof:
Let $f(x) = x\,\sin\left(\frac{1}{x}\right)$. We would like to show that for any $\ep... | $\displaystyle \left|x\,\text{sin}\left(\frac{1}{x}\right) - 1\right| \leq \left|x\,\text{sin}\left(\frac{1}{x}\right)\right| + 1$ is true but not helpful, as the left hand side is about $0$ and the right about $2$.
For $x\gt 0$ you have $\dfrac1x \gt \sin\left(\dfrac1x\right) \gt \dfrac1x-\dfrac1{6x^3}$ and so: $$1 \g... | {
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Given $a,b,c$ are the sides of a triangle. Prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$ Given $a,b,c$ are the sides of a triangle. Prove that $\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}<2$.
My attempt:
I could solve it by using the semiperimeter concept. I tried to transform this equation since it is a ... | As these are sides of a triangle, let $a=x+y, b= y+z, c=z+x$, and using homogeneity, set $x+y+z=1$. The inequality is then to show:
$$\frac{1-x}{1+x}+\frac{1-y}{1+y}+\frac{1-z}{1+z} < 2$$
Note that $x \in (0, 1) \implies 1+x > 1 \implies \dfrac{1-x}{1+x}< 1-x$. Sum that across $x, y, z$ to get the above inequality.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Volume of water in a cone Let, slant height of a cone be 6cm, and radius be 3cm and the cone be uniform. Let a uniform & solid sphere of radius 1cm be put in the cone & fill the cone by water. What is the minimum volume of water such that the sphere is under water?
I find that the height of the cone is $\sqrt(27)$cm & ... | You can easily compute the demi-angle $\alpha$ of the cone.
$\cos \alpha=\dfrac {\sqrt{27}}{6}$
Consider that the sphere is in contact with the edge of the cone. Thus the distance between the center of the sphere $O$ with the edge is $1$, radius of the sphere.
Let's call $d$ the distance between the apex of the cone a... | {
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"answer_id": 0
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Combinatorics - pick every object in a set Suppose I have a set of N objects and I pick from this set (with replacement) n>N times. How many permutations are there that include all N objects?
I've tried a number of different things, but am kind of going in circles and expect it should be a pretty simple solution if you... | Assuming the answer in my comments above comes back positive, that we are interested simply in the number of times each object is selected and not the order in which it was selected, suppose there are $N$ objects and you want select $n$ times with replacement, tallying how many times each was selected. Let the $N$ obj... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1032498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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If $M$ is a $R$-module with $R=R_1\times \cdots \times R_n$ then $M\simeq M_1\times \cdots \times M_n$ where $M_i$ is a $R_i$-module? Let $R_1, \ldots, R_n$ be rings and consider the ring $R:=R_1\times \cdots\times R_n$. How can I show every $R$-module $M$ is isomorphic to a product $M_1\times \cdots\times M_n$ where e... | It's easier with just two rings and then you can do induction.
So consider $R=R_1\times R_2$ and the two idempotents $e_1=(1,0)$, $e_2=(0,1)$. If $M$ is an $R$-module, you can consider $M_1=e_1M$ and $M_2=e_2M$.
For all $r\in R$, $e_1r=re_1$, and similarly for $e_2$, so $M_1$ and $M_2$ are $R$-submodules of $M$ and $M=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1032599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Probability: find the probability of event B given that event A occurs My Problem,
Suppose a family has 2 children,
if one children is randomly selected and it is a girl then what is the probability of the second child to be a girl ?
Please help. Thanks
| Let's work by definition of conditional probabilities:
Let A = first child is girl, and B = second child is girl.
$$P(B \mid A) = \frac{P(A \cap B) }{P(A)}= \frac{0.5 \times 0.5}{0.5}=0.5$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1032675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
When can ZFC be said to have been "born"? The "History" section of the Wikipedia article on ZFC isn't particularly helpful. The only thing I have understood from it is that ZFC had appeared after 1922. In what book or paper was ZFC first explicitly formulated and proposed?
| Here are some relevant quotes from Fraenkel, Bar-Hillel, Levy's "Foundations of set theory":
"Zermelo's vague notion of a definite statement did not live up to the standard of rigor customary in mathematics ... In 1921/22, independently and almost simultaneously, two different methods were offered [by Fraenkel and Sko... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1032769",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 5,
"answer_id": 2
} |
One measurable $\lim$ and one theorem How we can prove following theorem?
Let $f_n \ge 0 $ be measurable, $\lim f_n = f $ and $f_n \le f$ for each $n$. Show that $$\int f(x)dx=\lim_n \int f_n(x)dx $$
Any idea would be highly appreciated.
| Fatou's lemma gives $\int f = \int \liminf_n f_n \le \liminf_n \int f_n$ (the values may be infinite).
If $\liminf_n \int f_n < \infty$, then $f$ is integrable and the result follows from the dominated convergence theorem.
Otherwise, we have $\lim_n \int f_n = \infty$, and since $0 \le f_n \le f$, we have
$\int f_n \le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1032875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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At Most Two Distinct Members of A The quantified predicate logic statement that describes at most two distinct members of A, where A, is some arbitrary set is:
$\forall$xyz( (Px $\land$ Py $\land$ Pz) $\Rightarrow$ (x=y $\lor$ x=z $\lor$ y=z) )
I can parse this quantified statement into three cases:
*
*There are No ... | In the case 1 (no elements) the antecedent $(Px ∧ Py ∧ Pz)$ is is false and the implication is true.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1033045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Double Integral $\int\limits_0^a\int\limits_0^a\frac{dx\,dy}{(x^2+y^2+a^2)^\frac32}$ How to solve this integral?
$$\int_0^a\!\!\!\int_0^a\frac{dx\,dy}{(x^2+y^2+a^2)^\frac32}$$
my attempt
$$
\int_0^a\!\!\!\int_0^a\frac{dx \, dy}{(x^2+y^2+a^2)^\frac{3}{2}}=
\int_0^a\!\!\!\int_0^a\frac{dx}{(x^2+\rho^2)^\frac{3}{2}}dy\\
\r... | Both the function that you are integrating as the region over which you are integrating it get unchanged if you exchange $x$ with $y$. Therefore, your integral is equal to$$2\int_0^a\int_0^x\frac1{(a^2+x^2+y^2)^{3/2}}\,\mathrm dy\,\mathrm dx.$$You can compute this integral using polar coordinates: $\theta$ can take val... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1033129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 3
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Trace of nilpotent matrix over a ring Let $R$ be a commutative ring with unity, and $n$ a positive integer.
Let $A\in \mathfrak{M}_n(R)$ such that there exists $m\in \mathbb N$, for which $A^m=0$.
Is it true that there exists $\ell\in \mathbb N$, such that $\bigl(\text{tr}(A)\bigr)^\ell=0$ ?
Remark : It is true for $n=... | If $\mathfrak{p}$ is any prime ideal of $R$, then $A/\mathfrak{p}$ is a domain. We can embed it in its fraction field, then in an algebraic closure. Then your matrix mod $\mathfrak{p}$ is triangularisable, with of course eigenvalues $0$.
Hence, the reduction mod $\mathfrak{p}$ of your matrix has vanishing trace. Since ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1033231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Holder continuity using Sobolev imbedding We assume for any $V\subset \subset U$ and $1<p<\infty$
$||u||_{W^{2,p}(V)}\le C(||\Delta u||_{L^p(U)}+||u||_{L^1(U)})$ for some $C=C(V,U,p)$.
Given, $B=\{x∈R^3,|x|<1/2\}$ and we suppose $u∈H^1(B)$ is a weak solution of $-\Delta u+cu=f$ for some $c(x){\in L^q(B)}, 3/2<q<2$ an... | First notice that, by Hölder's inequality, we have $cu\in L^p$ for $1/p=1/q+1/6$ (since $u\in L^6$ by the Sobolev embedding). Next your inequality gives $u\in W^{2,p}$ for this same $p$. Applying Sobolev embedding again, we get $\nabla u \in L^{p*}$ where as usual $1/p^* =1/p-1/n$. If we substitute we get
$$
\frac{1}{p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1033291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why base of a logrithim function must be greater than one? I'm in domain section of my textbook. It says that for logarithm functions, the base must be greater than one.
I can understand why base shouldn't be one but what is problem with negative numbers?
for example, What's wrong with $\log_{-2} x$ ?
| Because when working only with real numbers, we do not know how to make sense of the expression $(-2)^\sqrt{2}$, for example.
In general the expression $a^x$, with $x \in \mathbb R$, makes sense only if $a > 0$.
Since $a$ is the basis of the logarithm, you can see why we limit ourselves with positive base.
With complex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1033402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
On Galois closure I'm working on this problem in Hungerford: For $\sigma \in Aut_F \bar{F}$, show that every finite extension of $K$, the fixed field of $\sigma$, is cyclic.
For a finite extension $L$ of $K$, let $M$ be a Galois closure of $L$ then I can show that $M/K$ is cyclic so $L/K$ is cyclic.. but there is one p... | In this setting $L/K$ will always be separable.
To see that let $z\in L\setminus K$ be arbitrary. Because $L/K$ is finite, $z$ is algebraic over $K$. Therefore $z$ has a minimal polynomial $m(x)\in K[x]$. Over $\overline{F}$ $m(x)$ splits into linear factors
$$
m(x)=(x-z_1)(x-z_2)\cdots (x-z_n)
$$
with $z_1=z$. The ele... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1033500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to draw or plot illustrative figures? stackexchange users
I would like to plot or draw some illustrative figures for my research paper. I've tried GeoGebra already. But couldn't draw them as I wanted.
So my question is How can I draw them?
Or would you tell me what are their functions so that I can plot them in Mat... | I like the program Asymptote. See a tutorial and a gallery of examples. Also, you can see the drawing in Calculating line integrals via Stokes theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1033579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A question about the properties of the pseudospectrum Assume that $A\in \mathbb{C}^{n\times n}$. The $\epsilon-$pseudospectrum of $A$ is defined by
$$\sigma_{\epsilon}(A)=\{z\in C \quad | \quad \Arrowvert (zI-A)^{-1} \Arrowvert>\frac{1}{\epsilon}\}.$$
Why $\sigma_{\epsilon}(A)$ is a bounded open subset of $\mathbb{C}$?... | Take $x\in \mathbb C^n$, $z\in \mathbb C$. Then it holds
$$
|z|\cdot \|x\| = \|zx\| \le \|(zI-A)x\| + \|Ax\| \le \|(zI-A)x\| + \|A\|\cdot \|x\|,
$$
which implies
$$
\|(zI-A)x\| \ge (|z|-\|A\|)\|x\|.
$$
If $|z| \ge \|A\| + \epsilon$, then this implies
$$
\|(zI-A)^{-1}\| \le \frac1{|z|-\|A\|} \le \frac1\epsilon.
$$
Henc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1033640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Equality condition for convolution's $L^p$ norm. Suppose that $1< p< \infty$, $f\in L^1(R)$, and $g\in L^p(R)$ and that $\|f*g\|_p=\|f\|_1\|g\|_p$. Show that then either $f=0$ a.e. or $g=0$ a.e.
I have solved for $g=0$ a.e. if $\|f\|_1>0$ using the equality condition for Hölder ; $\alpha f^p=\beta g^q$ for some $\alpha... | In general, if you want to prove a statement of the form $A\implies (B\ \text{or}\ C)$, it is logically equivalent to show $(A\ \text{and}\ (\text{not}\ B))\implies C$. The other answer essentially shows how this works in this particular case, but also you could convince yourself of this by making a truth table with co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1033720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
''Differential equation'' with known solution $\sin$ and $\cos$ I am given the following two two equations
$f,g : \mathbb{R} \to \mathbb{R}$ are differentiable on $\mathbb{R}$ and they satisfy $\forall x,y \in \mathbb{R}$ $$f(x+y) = f(x)g(y)+f(y)g(x)\\g(x+y)=g(x)g(y)-f(x)f(y) $$ with $f(-x)=-f(x), \forall x \in \math... | Well, you can note that $$0=f(x-x)=f(x)g(-x)+f(-x)g(x)=f(x)g(-x)-f(x)g(x)=f(x)\bigl[g(x)-g(-x)\bigr].$$ This holds for all $x\in\Bbb R,$ and since $f'(0)\ne 0,$ then $f$ isn't the constant zero function. It follows that $g(x)=g(-x)$ whenever $f(x)\ne 0.$ In particular $f$ is non-zero in some punctured neighborhood abou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1033831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Get the numbers from (0-30) by using the number $2$ four times How can I get the numbers from (0-30) by using the number $2$ four times.Use any common mathematical function and the (+,-,*,/,^)
I tried to solve this puzzle, but I couldn't solve it completely. Some of my results were:
$$2/2-2/2=0$$ $$(2*2)/(2*2)=1$$ $$2/... | $\left\lfloor \exp\left(2^2\right)\right\rfloor+2+2=11$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1034122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 11,
"answer_id": 4
} |
Semi-infinite plate problem when 2 edges are equal temperature and one edge is a function of position Here is the problem posted:
Now here is my solution for a)
I knew to discard the $e^{ky}$ and $\sin(kx)$ due to $T\to 20$ when $y\to\infty$ and $T= 20$ at $x = 0$.
Which leaves me with $\cos(kx)*e^{-ky} = T$.
Now I h... | I suggest you start making $t=T-20$ your new temperature function; recall that the Laplace operator is linear, and if $T$ is a solution so is $t$ and vice versa. The advantage of this new temperature is that it makes 3 out of 4 boundary conditions homogeneous, making the problem easier to handle.
More precisely, if you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1034209",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Secant line and diameter of a circle A secant line incident to a circle at points $A$ and $C$ intersects the circle's diameter at point $B$ with a $45^\circ$ angle. If the length of $AB$ is $1$ and the length of $BC$ is $7$, then what is the circle's radius?
|
We can set up a system of equations for $BE$ and $BD$, then that will give us the radius, which is
$$r=\frac{BE+BD}{2}$$
First, by the chord-chord theorem (AKA the power chord theorem), we can write
$$AB\times BC=BE\times BD$$
$$\Rightarrow BE\times BD=7\quad\quad\quad(1)$$
Next, we can use the cosine law in $\triang... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1034313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
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