Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
determine the number of homomorphisms from $D_5$ to $\mathbb{R^*} $ and from $D_5$ to $S_4$ I'm trying to do these three exercises for my math study:
a) Determine the number of homomorphisms from $D_5$ to $\mathbb{R^*}$
b) Determine the number of homomorphisms from $D_5$ to $S_4$
c) give an injective homomorphism form ... | For a) the answer is two because :
$$Hom(D_5,\mathbb{R}^*)=Hom(D_5/[D_5,D_5],\mathbb{R}^*)=Hom(Z/2Z,\mathbb{R}^*) $$
The cardinal of the last one is $2$.
Edit : another way without abelianization.
Take $r$ a rotation and $s$ a reflection generating $D_5$. Take $f\in Hom(D_5,\mathbb{R}^*)$. You have $f(r)^{5}=1$, the o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1212788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Meaning of correlation In probability books it is sometimes mentioned that correlation is a measure of linearity of a relationship between random variables.
This claim is supported by the observation that $\left| \rho(X,Y)\right|=1 \iff X=b+aY$.
But let consider a less extreme case: $X,Y,Z$ are random variables such t... | It depends on what measure of correlation you use.
The Pearson correlation (the most commonly used one) measures the linearity of a relationship between two random variables.
The Spearman rank correlation however, also measures nonlinear (monotonic) relationships. This is defined as the Pearson correlation coefficient ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1212859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Eigenvalues of the sum of a diagonal matrix and the outer product of two vectors Does a $n \times n$ matrix $M = D + u. v^T$ with $D$ diagonal, $u$ and $v$ two given vectors, and $n$ a positive integer have some interesting properties in term of spectrum (all the real parts of the eigenvalues positive, for example) ?
I... | If $D$ is a positive definite diagonal matrix and $u:=[u_1,\ldots,u_n]^T$ and $v:=[v_1,\ldots,v_n]^T$ are positive vectors, a very useful fact is that $M:=D+uv^T$ is symmetrizable. That is, if
$$\tag{1}
S:=\mathrm{diag}\left(\sqrt{\frac{u_i}{v_i}}\right)_{i=1}^n,
$$
then
$$
S^{-1}MS=D+ww^T, \quad w:=S^{-1}u=Sv,
$$
is s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1212981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Number of elements of $\mathbb{Z}_p$ that satisfy a certain property Let $S(n,p)=\{a\in \mathbb{Z}_p : a^n=1$ (mod $p$)$\}$ where $p\geq3$ is a prime number and $1\leq n\leq p$. I am interested in finding a general formula for cardinality of $S(n,p)$. For example, I know that $|S(1,p)|=1$ and $|S(p-1,p)|=p-1$ (by Ferma... | Already this has been answered. But I'd like to make two points: as $0$ is ruled out we are looking at a subset of the group $\mathbf{Z}_p^*$. And $x\mapsto x^n$ is a homomorphism of this group, and so the solution set $S(n,p)$ you are looking for is the kernel of that homomorphism.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1213106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Take seven courses out of 20 with requirement
To fulfill the requirements for a certain degree, a student can choose to take any 7 out of a list of 20 courses, with the constraint that at least 1 of 7 courses must be a statistics course. Suppose that 5 of the 20 courses are statistics courses.
From Introduction to Pro... | There are $\binom{20}{7}$ possibilities, but $\binom{15}{7}$ of them are not ok, hence the answer is $\binom{20}{7}-\binom{15}{7}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1213185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proving that $\int_0^\infty\frac{J_{2a}(2x)~J_{2b}(2x)}{x^{2n+1}}~dx~=~\frac12\cdot\frac{(a+b-n-1)!~(2n)!}{(n+a+b)!~(n+a-b)!~(n-a+b)!}$
How could we prove that $$\int_0^\infty\frac{J_{2a}(2x)~J_{2b}(2x)}{x^{2n+1}}~dx~=~\frac12\cdot\frac{(a+b-n-1)!~(2n)!}{(n+a+b)!~(n+a-b)!~(n-a+b)!}$$ for $a+b>n>-\dfrac12$ ?
Inspired... | To save from typing a result a highlight will be listed for now.
*
*The integral in question is a reduction of the more general integral
\begin{align}
\int_{0}^{\infty} \frac{J_{\mu}(at) \, J_{\nu}(bt)}{t^{\lambda}} \, dt = \frac{b^{\nu} \Gamma\left( \frac{\mu + \nu - \lambda +1}{2}\right)}{2^{\lambda} \, a^{\nu - ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1213371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 2,
"answer_id": 1
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Prove that $\frac{1}{(1-x)^k}$ is a generating function for $\binom{n-k-1}{k-1}$ On my discrete math lecture there was a fact that:
$\frac{1}{(1-x)^k}$ is a generating function for $a_n=\binom{n-k-1}{k-1}$
I'm interested in combinatorial proof of this fact. Is there any simple proof of such kind available somewhere? (I... | Suppose we have a sequence of values of length $k$ where the values are any non-negative integer. This is has the combinatorial specification
$$\mathfrak{S}_{=k}\left(\sum_{q\ge 0} \mathcal{Z}^q\right).$$
This gives the generating function
$$\frac{1}{(1-z)^k}.$$
On the other hand all such sequences can be obtained by s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1213435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Which of these two factorizations of $5$ in $\mathcal{O}_{\mathbb{Q}(\sqrt{29})}$ is more valid? $$5 = (-1) \left( \frac{3 - \sqrt{29}}{2} \right) \left( \frac{3 + \sqrt{29}}{2} \right)$$
or
$$5 = \left( \frac{7 - \sqrt{29}}{2} \right) \left( \frac{7 + \sqrt{29}}{2} \right)?$$
$\mathcal{O}_{\mathbb{Q}(\sqrt{29})}$ is ... | Note that
$$ \frac{7-\sqrt{29}}2 \cdot \frac{5+\sqrt{29}}2 = \frac{3+\sqrt{29}}2 $$
and
$$ \frac{5+\sqrt{29}}2 \cdot \frac{-5+\sqrt{29}}2 = 1 $$
So the two factorizations are equally good, just related by unit factors. (Remember that even in an UFD, factorizations are only unique up to associatedness).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1213530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 4
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Limit as $x$ tend to zero of: $x/[\ln (x^2+2x+4) - \ln(x+4)]$ Without making use of LHôpital's Rule solve:
$$\lim_{x\to 0} {x\over \ln (x^2+2x+4) - \ln(x+4)}$$
$ x^2+2x+4=0$ has no real roots which seems to be the gist of the issue.
I have attempted several variable changes but none seemed to work.
| An approach without L'Hopital's rule.
$$\lim_{x\to 0} {x\over \ln (x^2+2x+4) - \ln(x+4)}=\lim_{x\to 0} {1\over {1\over x}\ln {x^2+2x+4\over x+4}}=\lim_{x\to 0} {1\over \ln \big ({x^2+2x+4\over x+4}\big)^{1\over x}}$$
but
$$({x^2+2x+4\over x+4}\big)^{1\over x}=({x^2+x+x+4\over x+4}\big)^{1\over x}=\big(1+{x^2+x\over x+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1213655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the limit of this sequence involving logs of binomials? Define $$P_{n} = \frac{\log \binom{n}{0} + \log \binom{n}{1} + \dots +\log \binom{n}{n}}{n^{2}}$$ where $n = 1, 2, \dots$ and $\log$ is the natural log function. Find $$\lim_{n\to\infty} P_{n}$$ Using the property of the $\log$, I am thinking to find the ... | The limit is $1/2$.
Here is a proof by authority:
This is copied from here:
Prove that $\prod_{k=1}^{\infty} \big\{(1+\frac1{k})^{k+\frac1{2}}\big/e\big\} = \dfrac{e}{\sqrt{2\pi}}$
$$\prod_{k=0}^n \binom{n}{k} \sim
C^{-1}\frac{e^{n(n+2)/2}}{n^{(3n+2)/6}(2\pi)^{(2n+1)/4}}
\exp\big\{-\sum_{p\ge 1}\frac{B_{p+1}+B_{p+2}}{p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1213843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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A unique solution Find the sum of all values of k so that the system
$$y=|x+23|+|x−5|+|x−48|$$$$y=2x+k$$
has exactly one solution in real numbers. If the system has one solution, then one of the three $x's$, should be 0. Then, there are 3 solutions, when each of the modulus becomes 0. Where am I going wrong?
| $$y-2x$$ is a piecewise linear function and is convex. The slope is nowhere zero (slope values are $-5,-3,-1,1$) so that it achieves an isolated minimum, which is the requested value of $k$, at the lowest vertex.
$$\begin{align}-23&\to145\\5&\to61\\48&\to\color{green}{18}\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1214058",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
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How does the minus come in Geometric Series I'm looking into the geometric series and can't understand how the 1 - .01 comes in below:
0.272727... = 0.27 + 0.0027 + 0.000027 + 0.00000027 + ...
= 0.27 + 0.27(.01) + 0.27(.01)^2 + 0.27(.01)^3 + ...
= 0.27 / (1-.01)
= 0.27 / 0.99
... | Question is why do solve it using geometric progressions at all. When you use the basic method taught as kids in school, you're basically deriving the sum of an infinite GP unknowingly. Here's the easiest way-
x=0.272727272727...
And 100x=27.272727272727....
Thus, 99x=27
x=27/99
This is basically like deriving the sum ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1214193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Prove $\{n^2\}_{n=1}^{\infty}$ is not convergent Definition of convergent sequence:
There exists an $x$ such that for all $\epsilon > 0$, there exists an $N$ such that for all $n \geq N$, $d(x_n, x) < \epsilon$.
So the negation is that there does not exist an $x$ such that for all $\epsilon > 0$, there exists an $N$ ... | The definition of convergence is
"There exists a $x\in\mathbb{R}$, such that for all $\epsilon>0$, there exists an $N$ such that for all $n\ge N$, $d(x_n,x)<\epsilon$"
The negation is
"There is no $x\in\mathbb{R}$, such that for all $\epsilon>0$, there exists an $N$ such that for all $n\ge N$, $d(x_n,x)<\epsilon$"
Sinc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1214271",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Parametric representation of a plane cut of a sphere at y=5 The sphere is given by $x^2+y^2+z^2=36$
Parametric Form:
$$x=6\sin t\cos u$$
$$y=6\sin t\sin u$$
$$z=6\cos t$$
If the sphere is 'cut' at $z=5$ this problem is trivial. ($0<t<\arccos(5/6),0<u<2\pi$), but although the sphere cut at $y=5$ seems to be just as simp... | Choose for parametrization:
$$\begin{gathered}
x = 6cos(t)cos(u) \hfill \\
y = 6sin(t) \hfill \\
z = 6cos(t)sin(u) \hfill \\
\end{gathered}$$
Rotate half circle around y-axis.
Then to solve
$$y = 6sin(t) = 5$$
is done as before. Choose right parametrization.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1214357",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Complex Number use in Daily LIfe What are the different properties of Complex Numbers. ?
I have doubt on real life use of complex numbers. Where and in what conditions do we use complex numbers in our day to day life.
My main focus is to know apart from Electrical ENgineering where it is used. Daily Life use. which can... | Why are real numbers useful? Most people can think of many reasons they are useful, they allow people to encode information into symbols that most anyone can understand. Its the same case with complex numbers. Most examples give highly specific and niche uses for complex numbers, but in reality, they could be used anyw... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1214446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Linear subspace closed under all special orthogonal matrices
Let $n\in \mathbb N$ and $E$ be an $n$ dimensional vector space over $\mathbb R$.
Let $F$ be a linear subspace of $E$ such that $\forall f\in SO(E), f(F)\subset F$
Prove that $F=\{0\}$ or $F=E$
Although the result is quite intuitive, I haven't been able to ... | To prove the result, it will suffice to show that if $F$ contains some line$~L_0$ through the origin (in other words if $F\neq\{0\}$) then $F$ contains every line through the origin (in other words if $F=E$). Let $L_1$ be an arbitrary line through the origin; one needs to find of $g\in SO(E)$ such that $g(L_0)=L_1$; th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1214524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the notation (if any) for series probability inclusion? In statistics, what is the notation to use for an event $A$ in $B$ in $C$ in $D$, etc., where the series may continue for a large number of events? The following works for a few events:
$$A\cap B\cap C\cap D\cap\cdots$$
...However, this can become long and... | For $A_1\cap\cdots\cap A_n$ you can write $\bigcap_{k=1}^n A_k$. This is coded in MathJax and LaTeX as \bigcap, and in a "displayed" as opposed to "inline" context, it puts the subscripts and superscripts directly below and above the symbol, just as with $\sum$, thus:
$$
\bigcap_{k=1}^n A_k
$$
If you want the subscrip... | {
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"timestamp": "2023-03-29T00:00:00",
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Show that all the intervals in $\mathbb{R}$ are uncountable
Question:
Show that all the intervals in $\mathbb{R}$ are uncountable.
I have already proven that $\mathbb{R}$ is uncountable by using the following:
Suppose $\mathbb{R}$ is countable. Then every infinite subset of $\mathbb{R}$ is also countable. This contra... | Hint: Consider $\phi : \mathbb R \to (-1,1)$ defined by $$\phi(x) = \frac{x}{1 +|x|}$$ and notice that $f: (-1,1) \to (a,b)$ defined by $$f(x) = \frac{1}{2}\bigg((b-a)x + a+ b\bigg)$$
is a bijection.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1215006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
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Is this a valid step in a convergence proof? I'm asked to say what the following limit is, and then prove it using the definition of convergence.
$\lim_{n\rightarrow\infty}$$\dfrac{3n^2+1}{4n^2+n+2}$.
Is it valid to say that the sequence behaves like $\dfrac{3n^2}{4n^2}$ for large n?
| Just to expand a bit on the above solution, just to the left of the first (*) we need $\frac{5}{16n}< ϵ $ which implies that we need $\frac{5}{16ϵ} < n$ in case it wasn't already plainly obvious. Then setting N > $\frac{5}{16ϵ}$ guarantees the result for all n≥N.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1215112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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$\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$? I need to show that the extension $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$, and compute its Galois group.
I am learning Galois theory by myself and got stuck in this exercise. I know the fundamental theorem of Galois theory. Any help would... | Let $\alpha = \sqrt{2 + \sqrt{2}}$.
An extension is Galois if it is separable and normal. Both of these properties can be resolved by considering the minimal polynomial $p$ of $\alpha$ (i.e. $p(x) = 0$).
Now $(\alpha^2 - 2)^2 - 2 = 0$ so the minimal polynomial is $x^4 - 4x^2 + 2$.
The extension is Galois since this pol... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1215186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Minimize Product of Sums of Squared Distances The Question
Given two sets of vectors $S_1$ and $S_2$，we want to find a unit vector $s$ such that
$$\{\sum_{u\in S_1}(\|u\|^2-\langle u, s \rangle^2)\}
\cdot
\{\sum_{v\in S_2}(\|v\|^2 - \langle v, s \rangle^2)\}$$ is minimized,
where $\langle *, * \rangle$ denotes the inn... | I don't think there is any (semi)closed-form solution, but you can simplify the problem quite a bit (when the dimension of the vector space is small). Presumably all the vectors here are real. Let $A=\sum_{u\in S_1}uu^T$ and $B=\sum_{v\in S_2}vv^T$. Since
$$
\|u\|^2-\langle u, s \rangle^2 = u^Tu - (s^Tu)(u^Ts)
= \opera... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1215279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Are all functions that have an inverse bijective functions? To have an inverse, a function must be injective i.e one-one.
Now, I believe the function must be surjective i.e. onto, to have an inverse, since if it is not surjective, the function's inverse's domain will have some elements left out which are not mapped to ... | Yes. Think about the definition of a continuous mapping.
Let $f:X\to Y$ be a function between two spaces. Topologically, a continuous mapping of $f$ is if $f^{-1}(G)$ is open in $X$ whenever $G$ is open in $Y$. In basic terms, this means that if you have $f:X\to Y$ to be continuous, then $f^{-1}:Y\to X$ has to also be ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1215365",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 7,
"answer_id": 5
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Nonlinear discontinuous convex function Let $X$ be a normed vector space. Construct nonlinear discontinuous convex function $f:X \rightarrow \mathbb{R}$.
I tried something with $\frac{-1}{\|x\|}$ but had no success.
| Such example does not always exist. For example, if $\dim X<\infty$, $X$ is isomorphic to $\mathbb{R}^n$ and every convex function $f:\mathbb{R}^n\to \mathbb{R}$ is continuous.
For $\dim X=\infty$, recall that every infinite-dimensional normed space contains a discontinuous linear functional, say $f$. Let $g=f+c$, with... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1215486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Determinant of matrix times a constant. Prove that $\det(kA) = k^n \det(A)$ for and ($n \times n$) matrix.
I have tried looking at this a couple of ways but can't figure out where to start. It's confusing to me since the equation for a determinant is such a weird summation.
| To elaborate on the above answer, the formula for the determinant is
$$\operatorname{det}(A)=\sum_{\sigma\in S_n}sign(\sigma)\Pi_{i=1}^na_{i,\sigma_i}$$
so
$$\operatorname{det}(kA)=\sum_{\sigma\in S_n}sign(\sigma)\Pi_{i=1}^nka_{i,\sigma_i}$$
$$=k^n\operatorname{det}(A)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1215555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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"answer_id": 3
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Problem when $x=\cos (a) +i\sin(a),\ y=\cos (b) +i\sin(b),\ z=\cos (c) +i\sin(c),\ x+y+z=0$ Problem: If $$x=\cos (a) +i\sin(a),\ y=\cos (b) +i\sin(b),\ z=\cos (c) +i\sin(c),\ x+y+z=0$$ then which of the following can be true:
1) $\cos 3a + \cos 3b + \cos 3c = 3 \cos (a+b+c)$
2) $1+\cos (a-b) + \cos (b-c) =0$
3) $\cos 2... | HINT:
For $(1),$
Using If $a,b,c \in R$ are distinct, then $-a^3-b^3-c^3+3abc \neq 0$.,
we have $x^3+y^3+z^3=3xyz$
Now $x^3=e^{i(3a)}=\cos3a+i\sin3a$
and $xy=e^{i(a+b)}=\cos(a+b)+i\sin(a+b),xyz=\cdots$
For $(3),(4)$
$x+y+z=0\implies\cos a+\cos b+\cos c=\sin a+\sin b+\sin c=0$
So, $x^{-1}+y^{-1}+z^{-1}=?$
Now $x^2+y^2+z... | {
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What is the name of a number ending with zero's? I am currently writing a very specific graph of a function implementation. The graph can have min/max values e.g. $134$ and $1876$ respectively.
I'm calculating "nice" numbers. For min/max they are $100$ and $1900$ respectively.
Is there a commonly used name for such a n... | They're exact multiples of powers of ten; they're also rounded to fewer significant figures. The benefit is to rapid human interpretation of annotation.
I haven't heard of an established term for such a pair, but "reduced precision bracket" might be a good descriptor.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1215923",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Dimension of a subspace smaller than dimension of intersection Suppose I have a finite-dimensional vector space $V$, and $U_1, U_2$ are subspaces of $V$, such that $U_1\nsubseteq U_2$.
Is it possible that $\dim{U_1}\leq\dim{U_1 \cap U_2}$?
| Fact: If $X$ is a subspace of the finite dimensional vector space $Y,$ and $\dim X = \dim Y,$ then $X=Y.$ Hence if $\dim U_1 \le \dim U_1\cap U_2,$ then $\dim U_1 = \dim U_1\cap U_2,$ hence $U_1\cap U_2$ equals $U_1.$ This implies $U_1 \subset U_2,$ contradiction.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the Taylor series expansion (centered at $z_{0}=0$) of the function $f(z)=\sin(z^3)$. Find the Taylor series expansion (centered at $z_{0}=0$) of the function $f(z)=\sin(z^3)$.
Use this expansion: a) to find $f^{69}(0)$; b) to compute the integral transversed once in the positive (with respect to the disk) direct... | Use the Taylor series for sine (just plug in $z^3$ wherever a $z$ appears in the original series). Notice what this "does" to the coefficients: the coefficient of $z$ in the original series is now the coefficient of $z^3$, the coefficient of $z^2$ in the original now belongs to $z^6$, etc.
| {
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Find $z$ s.t. $\frac{1+z}{z-1}$ is real I must find all $z$ s.t. $\dfrac{1+z}{z-1}$ is real.
So, $\dfrac{1+z}{1-z}$ is real when the Imaginary part is $0$.
I simplified the fraction to $$-1 - \dfrac{2}{a+ib-1}$$
but for what $a,b$ is the RHS $0$?
| If $z+1 = r(z-1)$, then the complex numbers $0, z-1, z+1$ lie on a line. But the line through $z-1$ and $z+1$ can only contain $0$ if $z$ is real.
| {
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$V=\mathcal{Z}(xy-z) \cong \mathbb{A}^2$. This question is typically seen in the beginning of a commutative algebra course or algebraic geometry course.
Let $V = \mathcal{Z}(xy-z) \subset \mathbb{A}^3$. Here $\mathcal{Z}$ is the zero locus. Prove that $V$ is isomorphic to $\mathbb{A}^2$ as algebraic sets and provide a... | Hint: Use the projection
\begin{align*}
\pi : V &\to \mathbb{A}^2\\
(x,y,z) &\mapsto (x,y) \, .
\end{align*}
Can you find its inverse?
To find the induced maps on the coordinate rings, recall that these maps are just given by preccomposition, e.g., for $\pi : V \to \mathbb{A}^2$,
\begin{align*}
\widetilde{\pi} : k[\mat... | {
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Sequence Word Problem General Question Whenever I do sequence word problems in my math homework I often end up accidentally adding 1 more term than needed, or subtracting 1 more term. Word problems seem ambiguous to me in wording a lot of the time and I don't know whether to do $(n-1)d$, or $nd$. Can anyone give some g... | An answer specific to the example in the comments:
To find the difference between each row we can use the information about rows 3 and 12. We know that row three has 41 boxes, and row 12 has 23 boxes. Since we have an arithmetic progression, this means that
$$
41+(12-3)n=23,
$$
or $n=-2$. Thus when you go up one row,... | {
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What method was used here to expand $\ln(z)$? On Wikipedia's entry for bilinear transform, there is this formula:
\begin{align}
s &= \frac{1}{T} \ln(z) \\[6pt]
&= \frac{2}{T} \left[\frac{z-1}{z+1} + \frac{1}{3} \left( \frac{z-1}{z+1} \right)^3 + \frac{1}{5} \left( \frac{z-1}{z+1} \right)^5 + \frac{1}{7} \left( \fra... | Taylor series show
$$\log((1-z)/(1+z)) = \log(1-z) - \log(1+z) = 2(z+z^3/3 + \cdots)$$
then let $y=(1-z)/(1+z)$.
The math works, though something doesn't jive with my intuition. I guess what's confusing is that the Taylor series already looks like log but it is missing the even terms. With that, as z goes to infinity t... | {
"language": "en",
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"source": "stackexchange",
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How do we know Euclid's sequence always generates a new prime? How do we know that $(p_1 \cdot \ldots \cdot p_k)+1$ is always prime, for $p_1 = 2$, $p_2 = 3$, $p_3 = 5$, $\ldots$ (i.e., the first $k$ prime numbers)?
Euclid's proof that there is no maximum prime number seems to assume this is true.
| That's not true. Euclid's proof does not conclude that this expression is a new prime, it concludes that there exists primes that were not in our original list, that lie between the $k'th$ prime and $p_{1}p_{2}...p_{k} + 1$ (including $p_{1}p_{2}...p_{k} + 1$). Among those extra primes that must exist, $p_{1}p_{2}...p_... | {
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for f(x,y,z) find point on surface nearest to origin $f(x,y,z)=x^2+2y^2-z^2$, $S=\{(x,y,z): f(x,y,z)=1\}$ find point on S nearest to origin.
I thought I would use Lagrange multipliers to solve this problem, but when I use $f(x,y,z)=x^2+2y^2-z^2$ and $g(x,y,z)=x^2+2y^2-z^2-1$, along with,
$$g(x,y,z)=0$$
$$f_x=\lambda g_... | The distance from $(x,y,z)$ to the origin is $\sqrt{x^2+y^2+z^2}$.
We therefore want to minimize $\sqrt{x^2+y^2+z^2}$, or equivalently $x^2+y^2+z^2$, subject to the condition $x^2+2y^2-z^2-1=0$. So $x^2+y^2+z^2$ should be your $f$. Now the usual process should work well.
| {
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About a polynomial with complex coefficients taking integer values for sufficiently large integers Let $f(x)$ be a polynomial with complex coefficients such that $\exists n_0 \in \mathbb Z^+$ such that $f(n) \in \mathbb Z , \forall n \ge n_0$, then is it true that $f(n) \in \mathbb Z , \forall n \in \mathbb Z$ ?
| Proof by induction on the degree of the polynomial. Trivially true for degree zero. Assume true for degree $n$. Let $f$ have degree $n+1$ and be integral for all sufficiently large integer arguments. Then same is true for $g(x)=f(x+1)-f(x)$, a polynomial of degree $n$. By the induction hypothesis, $g$ is integer-valued... | {
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The diophantine equation $x^2+y^2=3z^2$ I tried to solve this question but without success:
Find all the integer solutions of the equation: $x^2+y^2=3z^2$
I know that if the sum of two squares is divided by $3$ then the two numbers are divided by $3$, hence if $(x,y,z)$ is a solution then $x=3a,y=3b$. I have $3a^2+3b... | How about using infinite descent ?
As for any integer $a\equiv0,\pm1\pmod4\implies a^2\equiv0,1$
We have $x^2+y^2\equiv0\pmod3$
So, $3|(x,y)$
Let $x=3X,y=3Y$
and so on
| {
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Show that if we exchange elements in 2 different basis will still give us a basis. I came across a proof of the following theorem:
Let V be a free module over a division ring D. Suppose X and Y are two finite basis of V, then |X| = |Y|.
that uses the fact that if we exchange elements in 2 different basis, we will still... | Let $X=\{x_1, x_2\}$, $Y=\{y_1, y_2\}$ with $y_1=x_2$, $y_2=x_1$. Now $\{x_1, y_2\} = \{x_1\}$ is not a basis in general. Hence, your statement is wrong.
| {
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"source": "stackexchange",
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Is the optimum of this problem unique? I have a convex optimisation problem:
$$\min_{x_i} \sum_{i=1}^N a_i \exp(-x_i/b_i)\quad\text{ s.t. }\sum_{i=1}^N x_i=x\text{ and } x_i\ge 0$$
The KKT conditions are:
$$\lambda=\begin{cases}
a_i/b_i \exp(- x_i/b_i) &\text{ if }a_i/b_i>\lambda \\
a_i/b_i+\mu_i &\text{ else }\end{... | Hint: A strictly convex function has at most one (global) minimum.
| {
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Prove that equation $x^6+x^5-x^4-x^3+x^2+x-1=0$ has two real roots Prove that equation
$$x^6+x^5-x^4-x^3+x^2+x-1=0$$ has two real roots
and $$x^6-x^5+x^4+x^3-x^2-x+1=0$$
has two real roots
I think that:
$$x^{4k+2}+x^{4k+1}-x^{4k}-x^{4k-1}+x^{4k-2}+x^{4k-3}-..+x^2+x-1=0$$
and
$$x^{4k+2}-x^{4k+1}+x^{4k}+x^{4k-1}-x^{... | HINT: you can write this equation:
$x^6+x^5-x^4-x^3=-x^2-x+1$
$y=x^6+x^5-x^4-x^3$
$y=-x^2-x+1$
and draw two functions noting that there are two points where two fnctions intersect.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Geometry - Determine all points along a ray from starting coordinates and direction I am working on a video game. I need to determine each point along a ray with every x interval with the following information:
*
*X, Y, Z coordinates of the starting point of the ray, and,
*X, Y, Z coordinates representing the angle... | Well if $A = (a_1,a_2,a_3)$ is the start point, and $d =(b_1,b_2,b_3)$ is the direction of the reay, you can parametrize the ray by
$$ p = a + td = \begin{pmatrix} a_1+td_1 \\ a_2 + td_2 \\ a_3+td_3\end{pmatrix}$$ for $t \in \mathbb R$, or for $t \in [0,1]$ if you want to get a point $p$ that is in between $a$ and $b$... | {
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"timestamp": "2023-03-29T00:00:00",
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How to prove that $\lfloor a\rfloor+\lfloor b\rfloor\leq \lfloor a+b\rfloor$ We have the floor function $F(x)=[x]$ such that $F(2.5)=[2.5]=2, F(-3.5)=[-3.5]=-4, F(2)=[2]=2$.
How can I prove the following property of floor function:
$$[a]+[b] \le [a+b]$$
| Consider the algebraic representation of $\mathrm{floor}(x)=\lfloor x\rfloor$ (usually the floor of $x$ is represented by $\lfloor x\rfloor$, not $[x]$):
$$
x-1<\color{blue}{\lfloor x\rfloor\leq x}.
$$
Hence, for arbitrary $a$ and $b$, we have
$$
\lfloor a\rfloor\leq a\tag{1}
$$
and
$$
\lfloor b\rfloor\leq b.\tag{2}
$$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Dual of a matrix lie algebra In fact I already calculate the dual space with a formula, but I did'd understand some steps of the formula.
So, I want to calculate the dual space of The lie algebra of $SL(2,R)$. Knowing that $B=\left(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} , \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmat... | This is a general fact from linear algebra. If $e_1, \ldots , e_n$ is a basis of a vector space $V$, then $V^{\ast}$ has the dual basis consisting of linear functions $f_1, \ldots, f_n$ defined by $f_i(e_j) = 0$ for all $i$ and $j$ such that $i\neq j$, and $f_i(e_i) = 1$ for all $i$. If a linear transformation $\phi$ i... | {
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What is wrong with this putative proof? So I've spent about an hour trying to figure out what is wrong with this proof. Could somebody clearly explain it to me? I don't need a counterexample. For some reason I was able to figure that out.
Thanks.
Theorem. $\;$ Suppose $R$ is a total order on $A$ and $B\subseteq A$. The... | The proof shows that for an arbitrary $x$, either $bRx$ or $xRb$. Therefore $\forall x(xRb\lor bRx)$. So far so good.
But $\forall x(P(x)\lor Q(x))$ is not equivalent to $\forall xP(x)\lor\forall x Q(x)$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Finding lines of symmetry algebraically How would you determine the lines of symmetry of the curve $y=f(x)$ without sketching the curve itself?
I was working on the problem of finding the lines of symmetry of the curve given by:
$ x^{4}$+$ y^{4}$ = $u$ where u is a positive real number.
but I had to resort to doing a q... | A line L given by $ax+by+c=0$ is a line of symmetry for your curve, if and only if for any point $P_1(x_1,y_1)$ from the curve, its symmetrical point $P_2(x_2,y_2)$ (with respect to the line L) is also on the curve.
Now:
(1) express $x_2$ and $y_2$ as functions of $x_1, y_1$ (this can be done from the equation of L);
... | {
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Is the sequence of $\mathbb Z$-modules below exact? I came across the following example in my algebra textbook:
Considere the sequence of $\mathbb Z$-modules: $$0\longrightarrow \mathbb Z_2\stackrel{f}{\longrightarrow}\mathbb Z_4\stackrel{g}{\longrightarrow}\mathbb Z_2\longrightarrow 0,$$
where $f$ is the isomorphism o... | It seems to me that the second $\Bbb Z_2$ is actually $2\Bbb Z_4$, that is, it is:
$\{[0]_4,[2]_4\}$ (this is isomorphic to $\Bbb Z_2$).
It should be clear then, that $gf = 0$, as one would expect in an exact sequence.
| {
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Finding the joint unconditional distribution of $X$ and $Y=N-X$ for $X\sim \mathrm{Bin}(N,p)$ and $N\sim \mathrm{Pois}(\lambda)$. The question asks to find the unconditional joint distribution of $X$ and $Y=N-X$, given that
*
*$N$ has a Poisson distribution with parameter $\lambda$, and
*$X$ has a conditional dist... | You have been given the conditional probability mass of $X\mid N$, so why not use it?
$\begin{align}
\mathsf P(X=k, Y=h) & = \mathsf P(X=k, N-X=h)
\\ & = \mathsf P(X=k, N=h+k)
\\ & = \mathsf P(X=k\mid N=h+k)\cdot \mathsf P(N=h+k)
\\ & = \binom{h+k}{k}p^{k}(1-p)^{h}\cdot\frac{\lambda^{h+k}e^{-\lambda}}{(h+k)!}
\end{alig... | {
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Show that a subspace is proper dense in $l^1$ sequence space. L^1 space. Let Y = $L^1 $($\mu$) where $\mu$ is counting measure on N.
Let X = {$f$ $\in$ Y : $\sum_{n=1}^{\infty}$ n|$f(n)$|<$\infty$}, equipped with the $L^1$ norm.
Show that X is a proper dense subspace of Y.
I don't know how to show that X is dense in Y... | Let $e_i :\mathbb N\to\mathbb R$ defined as
$$
e_i(j)=\left\{\begin{array}{lll} 1 &\text{if}&i=j,\\ 0 &\text{if}&i\ne j.\end{array}\right.
$$
Then $e_i\in X$. If $f\in Y$, then setting
$$
f_n=\sum_{i=1}^n f(i)\, e_i\in X,
$$
we observe that
$$
\|f-f_n\|_{L^1}=\sum_{i>n}\lvert\,f(i)\rvert.
$$
Clearly, the right hand si... | {
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Euler sum of a divergent series So I have a series $1+0+(-1)+0+(-1)+0+1+0+1+0+(-1)+...$
Is it correct to rearrange this as $1+0+(-1)+0+1+0+(-1)+0+1+0+(-1)+0...$
The second problem can be done as an Euler sum and the answer is $\frac{1}{2}$. In general, I know that absolutely convergent series can be rearranged but I'm ... | A series converges, by definition, if the sequence of partial sums $S_n$ converges to some number. So, call your individual terms $a_n$, the sequence of partial sums is $\displaystyle \sum _{k=1}^n a_k$. If you look at your sequence of partial sums, you can see it doesn't converge to any number, because it varies... | {
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Why only the numerator is derived? Why the derivative of $y = \frac{x^5}{a+b}-\frac{x^2}{a-b}-x$ is solved by deriving just the numerators?
The solution is $\frac{dy}{dx}=\frac{5x^4}{a-b}-\frac{2x}{a-b}-1$.
| Because the denominator does not depend on $x$. So if we were to formally use $\left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}$ we get by plugging $v'=0$ ($v$ does not depend on $x$) $\left(\frac{u}{v}\right)'=\frac{u'}{v}$
| {
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Second order linear ODE determine coefficient A vertical spring has the spring constant $32$ N/m. A body with the mass 2 kg is attached to its end and is at equilibrium. At the time $t=0$ a variable force $$F(t)=12\sin(\omega t)$$ is applied in the direction of gravity. Describe the motion of the body for different val... | The equation is:
$$y''=6\sin\omega t-16y$$
Let $y=a\sin(\omega t+\theta)$
Now:
$$-a\omega^2\sin(\omega t+\theta)=6\sin\omega t-16a\sin(\omega t+\theta)$$
Or:
$$\underbrace{(16-\omega^2)a}_{\alpha}\sin(\omega t+\theta)=6\sin\omega t$$
Now:
$$(\alpha\cos\theta-6)\sin\omega t+(\alpha\sin\theta)\cos\omega t=0\\
\sin\left(\... | {
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What does the solution of a PDE represent? So I took a course in PDE's this semester and now the semester is over and I'm still having issue with what exactly we solved for. I mean it in this sense, in your usual first or second calculus course you are taught the concept that your second derivative represents accelerat... | There are tons of situations where the solution of a PDE discribes the behaviour over time:
1) pendulum
2) (radio active) decay
3) mass-spring-damper systems
4) inductance-capacitance-resistance systems
5) heat distribution over time
6) water level in a bucket with a leak
7) Temperature of a heated house
8) (electromag... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1218955",
"timestamp": "2023-03-29T00:00:00",
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Quadratic Integers in $\mathbb Q[\sqrt{-5}]$ Can someone tell me if $\frac{3}{5}$, $2+3\sqrt{-5}$, $\frac{3+8\sqrt{-5}}{2}$, $\frac{3+8\sqrt{-5}}{5}$, $i\sqrt{-5}$ are all quadratic integers in $\mathbb Q[\sqrt{-5}]$. And if so why are they in $\mathbb Q[\sqrt{-5}]$.
| Only one of them is. There are a number of different ways to tell, and of these the easiest are probably the minimal polynomial and the algebraic norm.
In a quadratic extension of $\mathbb{Q}$, the minimal polynomial of an algebraic number $z$ looks something like $a_2 x^2 + a_1 x + a_0$, and if $a_2 = 1$, we have an a... | {
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"timestamp": "2023-03-29T00:00:00",
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$\lim_{n\to\infty} \frac{1}{\log(n)}\sum _{k=1}^n \frac{\cos (\sin (2 \pi \log (k)))}{k}$ What tools would you gladly recommend me for computing precisely the limit below? Maybe a starting point?
$$\lim_{n\to\infty} \frac{1}{\log(n)}\sum _{k=1}^n \frac{\cos (\sin (2 \pi \log (k)))}{k}$$
| Let $f(x) = \frac{\cos(\sin(2\pi x))}{x}$, we have
$$f'(x) = - \frac{2\pi\cos(2\pi x)\sin(\sin(2\pi x)) + \cos(\sin(2\pi x))}{x^2}
\implies |f'(x)| \le \frac{2\pi + 1}{x^2}
$$
By MVT, for any $x \in (k,k+1]$, we can find a $\xi \in (0,1)$ such that
$$|f(x) - f(k)| = |f'(k + \xi(x-k))|(x-k) \le \frac{2\pi+1}{k^2}$$
This... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1219170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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$\max(a,b)=\frac{a+b+|a-b|}{2}$ generalization I am aware of an occasionally handy identity: $$\max(a,b)=\frac{a+b+|a-b|}{2}$$
However, I have found I'm unable to come up with a nice similar form for $\max(a,b,c)$. Of course I could always use the fact that $\max(a,b,c)=\max(\max(a,b),c)$ to write
$$\begin{align}\max(a... | Symmetrizing in the obvious way:
$$\frac{a+b+c}3+\frac{|a-b|+|b-c|+|a-c|}{12}+\\\frac{|a+b-2c+|a-b||+|a+c-2b+|a-c||+|b+c-2a+|b-c||}{12}$$
For four variables, I found:
$$\tfrac14\left(a+b+c+d+|a-b|+|c-d|+|a+b-c-d+|a-b|-|c-d||\right)$$
Of course you don't have to use absolute value signs: $$\lim_{m\to\infty}\sqrt[m]{a^m+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1219291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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Maximizing the area of a triangle with its vertices on a parabola. So, here's the question:
I have the parabola $y=x^2$. Take the points $A=(-1.5, 2.25)$ and $B=(3, 9)$, and connect them with a straight line. Now, I am trying find out how to take a third point on the parabola $C=(x,x^2)$, with $x\in[-1.5,3]$, in such a... | Assuming $A=\left(-\frac{3}{2},\frac{9}{4}\right),B=(3,9),C=(x,x^2)$, the area of $ABC$ is maximized when the distance between $C$ and the line $AB$ is maximized, i.e. when the tangent to the parabola at $C$ has the same slope of the $AB$ line. Since the slope of the $AB$ line is $m=\frac{9-9/4}{3+3/2}=\frac{3}{2}$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1219402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Showing that the function $f(x,y)=x\sin y+y\cos x$ is Lipschitz I wanted to show that $f(x,y)=x\sin y+y\cos x$ sastisfy Lipschitz conditions. but I can't separate it to $L|y_1-y_2|$.
According to my lecturer, the Lipschitz condition should be $$|f(x,y_1)-f(x,y_2)|\le L|y2-y1|$$
I was able show that $x^2+y^2$ in ... | Since $\frac {\partial f} {\partial y} =x \cos y+ \cos x$, it follows that
$$\left| \frac {\partial f} {\partial y} \right| = |x \cos y+ \cos x| \le |x \cos y|+|\cos x| = |x| \, |\cos y|+|\cos x| \le a+1$$
since $|x| \le a$, $|\cos y| \le 1$ and $|\sin y| \le 1$. This means that $|f(x,y_1) - f(x,y_2)| \le (a+1) \, |y_1... | {
"language": "en",
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"source": "stackexchange",
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If $X$, $Y$ are independent random variables and $E[X+Y]$ exists, then $E[X]$ and $E[Y]$ exist. I've been trying to show E$|X+Y|$ < $\infty$ $\Rightarrow$ E$|X|$ < $\infty$ by showing E$|X|$ $\leq$ E$|X+Y|$, but I'm stuck and cannot proceed from here.
Someone can help me, please?
-----[added]-----
$$E|X+Y| = \int\int|x... | Your second approach works. If $X$ and $Y$ are independent and $|X+Y|$ is integrable, then
$$
E|X+Y| = \int|x+y|dP_{(X,Y)}(x,y)=\int\left[\int|x+y|dP_X(x)\right]dP_Y(y)
$$
by Fubini's theorem, and moreover the function $$y\mapsto\int|x+y|dP_X(x)=E|X+y|$$ is integrable with respect to $P_Y$, hence finite almost surely. ... | {
"language": "en",
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CY 3-folds are $T^2 \times \mathbb{R}$ fibrations over the base $\mathbb{R}^3$. What does it mean? In this article at section 2. Toric geometry and Mirror Symmetry there is the statement that CY 3-folds are $T^2 \times \mathbb{R}$ fibrations over the base $\mathbb{R}^3$. Now, my questions refers to pages 3 and 4. Altho... | There is a map $F:\mathbb C^3 \to \mathbb R^3$ given by
$$F (z_1, z_2, z_3) = (|z_1|^2 - |z_3|^2, |z_2|^2 - |z_3|^2, Im(z_1z_2z_3))$$
Then one can think of $\mathbb C^3$ as a "fibration" of $\mathbb R^3$:
$$\mathbb C^3 = \bigcup_{x\in \mathbb R^3} F^{-1}(x).$$
One can check that when $x$ does not lie in the "degenerat... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A level Central Limit Theorem question How many times must a fair die be rolled in order for there to be less than 1% chance that the mean of the scores differ from 3.5 by more than 0.1?
The answer is $n≥1936$, but how do you get to this answer?
| Find mean and variance for one fair die. Let $X$ be the number of faces showing on one fair die.
$$\mu = E(X) = (1 + 2 + \cdots + 6)/6 = [6(7)/2]/6 = 7/2 = 3.5,$$
where we have used the formula that the sum of the first $n$ integers
is $n(n+1)/2.$ Also,
$$E(X^2) = (1^2 + 2^2 + \cdots + 6^2)/6 = \frac{6(7)(13)/6}{6} = 9... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Classifying a set closed under two unary operators Suppose we have a single square $S$ with distinct colors at its four vertices. We define two operators: the rotation operator $R$, which rotates the square $90^\circ$ clockwise, and the transpose operator $T$, which flips the square along its main diagonal (from the to... | $R$ and $T$ are actually elements of a group and the group they generate is the group of symmetries of the square. There are 8 elements of this group including the identity which leaves the entire square fixed. The binary operation here is function composition, which is to say, $R*T$ is the motion that results in apply... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to compute $(a+1)^b\pmod{n}$ using $a^b\pmod{n}$? As we know, we can compute $a^b \pmod{n}$ efficiently using Right-to-left binary method Modular exponentiation.
Assume b is a prime number .
Can we compute directly $(a+1)^b\pmod{n}$ using $a^b\pmod{n}$?
| In general no, but yes in some special cases, e.g. if $\ n\mid \pm a^k\! +\! a\! +\! 1\,$ for small $\,k\,$ then
${\rm mod}\ n\!:\,\ \color{#c00}{a\!+\!1\,\equiv\, \pm a^k}\,\Rightarrow\, (\color{#c00}{a\!+\!1})^b \equiv (\color{#c00}{\pm a^k})^b\equiv (\pm1)^b (a^b)^k$
so you need only raise the known result $\,a^b\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1220014",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Infinite Product - Seems to telescope Evaluate
$$\left(1 + \frac{2}{3+1}\right)\left(1 + \frac{2}{3^2 + 1}\right)\left(1 + \frac{2}{3^3 + 1}\right)\cdots$$
It looks like this product telescopes: the denominators cancel out (except the last one) and the numerators all become 3.
What would my answer be?
| we have the following identity (which affirms that the product telescopes):
$$\left (1+\frac{2}{3^n+1}\right)=3\cdot\frac{3^{n-1}+1}{3^n+1}=\frac{1+3^{-(n-1)}}{1+3^{-n}}$$
(as denoted in the comment by Thomas Andrews)and as a result:
$$\prod_{k=1}^n \left (1+\frac{2}{3^k+1}\right)=\frac{2\cdot 3^n}{3^n+1}=\frac{2}{1+3^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find circumcenter when distance between ABC points of triangle with two points's ratio given The complete problem is:
I am having three points A,B,C whose ratio of the distances from points (1,0) and (-1,0) is 1:3 each. Then I need the coordinates of the circumcenter of the triangle formed by points A,B & C.
Can anybo... | I'd proceed like this:
*
*Prove that, in 2D, the set of all points defined by a fixed ratio
$\rho\neq1$ of their distances to two distinct given points
(which I will call poles for reasons not explained here)
is a circle $K$.
(I'd use equations with cartesian coordinates for that,
but if you find a coordinate-free r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1220215",
"timestamp": "2023-03-29T00:00:00",
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How many arrangements of letters in REPETITION are there with the first E occurring before the first T? The question is:
How many arrangements of letters in REPETITION are there with the first E occurring before the first T?
According the book, the answer is $3 \cdot \frac{10!}{2!4!}$, but I'm having trouble understand... | First we'll arrange the letters like this:
$$\text{E E T T}\quad\text{R P I I O N}$$
Now replace $\text{EETT}$ with $****$:
$$\text{* * * *}\quad\text{R P I I O N}$$
The number of arrangements is $$\frac{10!}{4!2!}$$ because four symbols are the same, two other symbols are the same, and all the rest are different.
Now ... | {
"language": "en",
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The probability that the ratio of two independent standard normal variables is less than $1$ Let the independent random variables $X,Y\sim N(0,1)$. Prove that $P(X/Y < 1) = 3/4. $
Could anyone help me prove this analytically? Thanks.
Progress: My first thought was to integrate the joint density function: $\dfrac{e^{-\... | It is well-known that the random variable $X/Y$ has the two-sided Cauchy density $\frac{1}{\pi(1+x^2)}$ for $-\infty<x<\infty$. Thus
$P(X/Y<1)=\int_{-\infty}^{1}\frac{1}{\pi(1+x^2)}dx=0.5+\int_{0}^{1}\frac{1}{\pi(1+x^2)}dx$
and so
$P(X/Y<1)=0.5+\frac{1}{\pi}\hbox{arctg}(1)=0.5+0.25=0.75$.
Note: A much simpler way i... | {
"language": "en",
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Has anyone seen this combinatorial identity involving the Bernoulli and Stirling numbers? Does anyone know a nice (combinatorial?) proof and/or reference for the following identity?
$$\left( \frac{\alpha}{1 - e^{-\alpha}} \right)^{n+1} \equiv \sum_{j=0}^n \frac{(n-j)!}{n!} |s(n+1, n+1-j)| \alpha^j \bmod \alpha^{n+1}.$... | The coefficients $B_j^{(r)}$ defined by$$\sum_{j = 0}^\infty B_j^{(r)} {{x^j}\over{j!}} = \left({x\over{e^x - 1}}\right)^r$$are usually called higher order Bernoulli numbers, so your identity is a formula for $B_j^{(r)}$ for $j < r$.
Let $c(n, k) = |s(n, k)| = (-1)^{n - k}s(n, k)$. This is a fairly standard notation, u... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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limits involving a piecewise function, Prove that if $c \ne 2$, then f does not have a limit at $x = c$. $$f(x) = \begin{cases} (x-2)^3 & \text{if $x$ is rational } \\
(2-x) & \text{if $x$ is irrational }
\end{cases}$$
(i) Prove that if $c \ne 2$, then f does not have a limit at $x = c$.
(ii) Prove that $\lim_{x\to2} ... | For part $(i)$ you're on the right track, but note that we are concerned with $x_n, y_n$ converging to any arbitrary $c$. Note that your $x_n, y_n$ both converge to $0$ (also that $y_n$ is necessarily even a sequence of irrationals - consider $n = 4$). I'm guessing you are to come up with the sequences yourself, and I ... | {
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Binomial Expansion without inifinty series Variable here is "a" and "b".The question is to simplify the $\sqrt [3] { (\frac{a}{\sqrt{b^3} })\times {\frac {\sqrt{a^6b^2}}{a} }+ \frac{a}{b^2}}$
So these are my steps
=$\left(\frac {a^3b} {b^\frac {3} {2} }+ \frac {a} {b^2}\right)^\frac {1} {3}$
=$\left({a^3b^{1-\frac {3} ... | In general,
$$(x+y)^{1/3}\ne x^{1/3}+y^{1/3}$$
For example, take $x=y=1$. If this weren't true, we'd have $\sqrt[3]2=2$.
In fact, if $x$ and $y$ are positive, we have:
$$(x+y)^{1/3}<x^{1/3}+y^{1/3}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Calculation of real root values of $x$ in $\sqrt{x+1}-\sqrt{x-1}=\sqrt{4x-1}.$
Calculation of x real root values from $ y(x)=\sqrt{x+1}-\sqrt{x-1}-\sqrt{4x-1} $
$\bf{My\; Solution::}$ Here domain of equation is $\displaystyle x\geq 1$. So squaring both sides we get
$\displaystyle (x+1)+(x-1)-2\sqrt{x^2-1}=(4x-1)$.
$... | For $x\ge1$ we have $$\sqrt{4x-1}\ge \sqrt {3x} $$
and $$\sqrt{x+1}\le \sqrt {2x}$$
hence
$$\sqrt{x+1}-\sqrt{x-1}\le \sqrt{2x}<\sqrt{3x}\le\sqrt{4x-1} $$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The number of zeros of a polynomial that almost changes signs Let $p$ be a polynomial, and let $x_0, x_1, \dots, x_n$ be distinct numbers in the interval $[-1, 1]$, listed in increasing order, for which the following holds:
$$
(-1)^ip(x_i) \geq 0,\hspace{1cm}i \in \{0, 1, \dots, n\}
$$
Is it the case that $p$ has no fe... | Counting the zeros with multiplicity, the answer is still yes. If $p$ is a (real) polynomial, and $x_0 < x_1 < \dotsc < x_n$ are points such that $(-1)^i p(x_i) \geqslant 0$ for $0 \leqslant i \leqslant n$, then $p$ has at least $n$ zeros in the interval $[x_0,x_n]$ counted with multiplicity.
We prove that by induction... | {
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Why $\max \left\{ {{x^T}Ax:x \in {R^n},{x^T}x = 1} \right\}$ is the largest real eigenvalue of A? Let $A \in {M_n}(R)$ and A is symmetric.Why $\max \left\{ {{x^T}Ax:x \in {R^n},{x^T}x = 1} \right\}$ is the largest real eigenvalue of A?
| So, here's an intuitive and somewhat geometric explaination of what is happening.
$1:$ For $\lambda$ to be an eigenvalue, we need to be able to find a vector $v$ such that $Av=\lambda v$
$2:$ We observe that $v$ is an eigenvector iff $v\over{||v||}$ is an eigenvector. So, we reduce the search space to the unit spher... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proving real polynomials of degree greater than or equal to 3 are reducible this is the proof I'm given:
Example 1.20. Let $f \in \mathbb{R}[x]$ and suppose that $\deg(f) > 3$. Then $f$ is reducible.
Proof. By the Fundamental theorem of algebra there are $A \in C$ such that $$f (x) = (x - \lambda_1) \dots (x - \lambda... | For every 2 complex numbers $a, b \in \mathbb{C}$ it's true that
$$\overline{a+b} = \overline{a} + \overline{b}$$
$$\overline{ab} = \overline{a} \overline{b}$$
And of course $a = \overline{a}$ for every $a \in \mathbb{R}$. So if $f(x) \in \mathbb{R}[x]$, then $\overline{f(x)} = f(\overline x)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Floating point number,Mantissa,Exponent In this computer, numbers are stored in $12$-bits. We will also assume
that for a floating point (real) number, $6$ bits of these bits are reserved for
the mantissa (or significand) with $2^{k-1}-1$ as the exponent bias (where
$k$ is the number of bits for the characteristic).
$0... | Usually the mantissa is considered to have a binary point after the first bit, so your mantissa would be $1.1100_2=\frac 74=1.75_{10}$. Sometimes a leading $1$ is assumed, so your mantissa would be $(1).11100_2=\frac{15}8=1.875_{10}$ This gives one more bit of precision. To find the exponent, you subtract the offset f... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Let $\pi$ be a prime element in $\Bbb{Z}[i]$. Then $N(\pi)=2$ or $N(\pi)=p$ s.t. $p$ is prime and $p\equiv 1\pmod 4$
Let $\pi$ denote a prime element in $\Bbb{Z}[i]$ such that $\pi\not\in \Bbb{Z},i\Bbb{Z}$. Prove that $N(\pi)=2$ or $N(\pi)=p$, where $p$ is a prime number $\equiv 1\pmod4$. Give a complete classificatio... | It's easy to see that $1+i$ and $1-i$ are irreducible in $\mathbb{Z}[i]$ and the only elements $x\in\mathbb{Z}[i]$ such that $N(x)$ are those two, up to multiplication by invertible elements (that is, $1$, $-1$, $i$ and $-i$). In particular $2$ is not irreducible in $\mathbb{Z}$.
Suppose $p>2$ is a prime integer such t... | {
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Why does this approximation of square roots using derivatives work this way? I came up with this way to estimate square roots by hand, but part of it doesn't seem to make sense.
Consider how $f(n) = \sqrt{n^2+\varepsilon} \approx n$ when $\varepsilon$ is small. Therefore, using the tangent line with slope $f'(n) = \fra... | This seems like an improper application of the tangent line approximation. The usual approximation is
$$f(x+\varepsilon)\approx f(x)+\varepsilon f'(x)\tag{*}
$$
for $\varepsilon$ small. Your choice of $f$ doesn't match up with $(*)$. But using $f(x)=\sqrt x$ gives
$$
\sqrt{x+\varepsilon}=f(x+\varepsilon)\approx\sqrt x+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1221391",
"timestamp": "2023-03-29T00:00:00",
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Find basis and dimension of a subspace Problem: Let V be the subspace of all 2x2 matrices over R, and W the subspace spanned by:
\begin{bmatrix}
1 & -5 \\
-4 & 2 \\
\end{bmatrix}
\begin{bmatrix}
1 & 1 \\
-1 & 5 \\
\end{bmatrix}
\begin{bmatrix}
2 & -4 \\
-5 & 7 \\
\end{bmatrix}
\begin{bmatrix}
1 & -7 \\
-5 & 1 \\
\end{b... | You can consider each matrix to be a vector in $\mathbb{R}^4$.
The only pivots are in the first two columns, so the first two matrices are linearly independent and form a basis for the subspace. The last two are linear combinations of the first. Notice that $M_3=M_1+M_2$, and $M_4=\frac{4}{3} M_1-\frac{1}{3}M_2$
| {
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"timestamp": "2023-03-29T00:00:00",
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A finite abelian group has order $p^n$, where $p$ is prime, if and only if the order of every element of $G$ is a power of $p$ Suppose that G is a finite Abelian group. Prove that G has order $p^n$, where p is prime, if and only if the order of every element of G is a power of p.
I tried the following route, but got s... | Assume $p$ divides the order, and $q$ is some other prime that also divides the order. By Cauchy's theorem there is an element of order $q$.
Therefore, if the order is not a power of a prime then all the elements can't be of order a power of the same prime.
Assume that the order of the group is $p^n$. Then the order o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1221570",
"timestamp": "2023-03-29T00:00:00",
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Asymptotic expansion of exp of exp I am having difficulties trying to find the asymptotic expansion of $I(\lambda)=\int^{\infty}_{1}\frac{1}{x^{2}}\exp(-\lambda\exp(-x))\mathrm{d}x$ as $\lambda\rightarrow\infty$ up to terms of order $O((\ln\lambda)^{-2})$. How does $(\ln\lambda)^{-1}$ appear as a small parameter? Pleas... | If we substitute $e^{-x} = y$ and integrate by parts the integral becomes
$$
\begin{align}
\int_1^\infty x^{-2} \exp(-\lambda e^{-x})\,dx &= \int_0^{1/e} y^{-1} (\log y)^{-2} e^{-\lambda y}\,dy \\
&= e^{-\lambda/e} - \lambda \int_0^{1/e} (\log y)^{-1} e^{-\lambda y}\,dy. \tag{1}
\end{align}
$$
It was proved by Erdélyi ... | {
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"answer_id": 0
} |
The geometry of unit vectors that have specific angle with a given vector It is easy to see that for $S^2$ this space is nothing but a circle that is the intersection of a cone with aperture $2\alpha$ (where $\alpha$ is the predifined specific angle), and $S^2$. My question is that is this observation extendable to hig... | General spherical coordinates are given by
$$ \begin{align*}
x_1 &=r\sin{\phi_1}\sin{\phi_2}\dotsm\sin{\phi_{n-2}}\sin{\phi_{n-1}} \\
x_2 &=r\sin{\phi_1}\sin{\phi_2}\dotsm\sin{\phi_{n-2}}\cos{\phi_{n-1}} \\
x_3 &=r\sin{\phi_1}\sin{\phi_2}\dotsm\cos{\phi_{n-2}} \\
&\vdots\\
x_{n-1} &=r\sin{\phi_1}\cos{\phi_2}\\
x_n &= ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1221780",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
When does equality hold in this case? Give example of two vectors $x$ and $y$ such that $$||x+y||_2^2 = ||x||_2^2+||y||_2^2$$
and
$$<x,y>\neq0$$
I can't seem to find any two vectors $x$ and $y$ that satisfied both conditions at the same time.
| In $\mathbb{C}$ as $\mathbb{C}$-space, with $(z,w)=z\overline{w}$
$$|1+i|^2=2=1+1=|1|^2+|i|^2$$
and $$(1,i)=1\overline{i}=-i\neq0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1221886",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Clarification of the statement of the Pumping Lemma In class we were told that Pumping Lemma states:
"Let A be a regular language over $\Sigma$. Then there exists k such that for any words $x,y,z\in\Sigma^{*}$, such that $w=xyz\in A$ and $\lvert y\rvert\ge k$, there exists words $u,v,w$ such that $y=uvw$, $v\neq\epsilo... | Your understanding is correct. If you want to apply the lemma to a language you know (or assume) is regular, you can choose $x$, $y$, and $z$ any which way you fancy, as long as you make sure $y$ is long enough. (But since you get $k$ from the lemma, what you actually need in the usual scenario is to provide is a proce... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1221968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
the number of inversions in the permutation "reverse" Known, that number of inversions is $k$ in permutation:
$$\begin{pmatrix}
1& ...& n& \\
a_1& ...& a_n&
\end{pmatrix}$$
Find number of inversions in permutation (let's call it "reverse"):
$$\begin{pmatrix}
1& ...& n& \\
a_n& ...& a_1&
\end{pmatrix}
... | Hints.
*
*Think of some permutation $ι$ on $[n]$ so that for any permutation $σ$ on $[n]$, “$σ$-reverse” is $σι$.
*An inversion of any permutation $σ$ on $[n]$ is a pair $(i,j)$ with $i < j$ such that $\frac{σ(i) - σ(j)}{i-j} < 0$.
*For permutations $σ$, $τ$ on $[n]$ you have $\frac{(στ)(i) - (στ)(j)}{i - j} = \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1222158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Proving irreducibility of Markov chain I have a Markov chain:
*
*state: a permutation of n cards
*transition: taking the top-most card and randomly choose one of the n possible positions for the card
I know it is obviously irreducible because we can arrive at any permutation states from any starting permutation s... | Look at the 'reverse' move, in which you pick a card from within the deck and move it to the top. These reverse moves can get you from starting state $(a_{\pi(1)},\ldots,a_{\pi(n)})$ to final state $(a_1,\ldots,a_n)$ by first locating card $a_n$ and moving it to the top, then locating card $a_{n-1}$ and moving it to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1222356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that $R(G)\cong K_0(F[G])$ Let $F$ be a field of characteristic $0$, $G$ a finite group and let $R(G)$ be the additive group of functions $G\to F$ generated by characters of $G$ of degree $1$.
Question: How can we show that $R(G)\cong K_0(F[G])$ where $K_0(F[G])$ is the Grothendieck group of finitely generated p... | If $G$ is a finite group and $F$ is a field of characteristic zero, Then the group algebra $F[G]$ is semisimple by maschke's theorem (this is a fact that is demonstrated in any book of representation theory, i.e Fulton Representation Theory) with one matrix ring factor for each conjugacy class of $G$, so $K_0(F[G])\con... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1222436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
The Euler-Poisson equation $$\int_{0}^\pi (x''^2+4x^2) dt$$
$$ x(0)=x'(0)=0; x(\pi)=0;x'(\pi)=\sinh(\pi)$$
This is The Euler-Poisson equation, i found:
$$\frac {\partial f}{\partial x}-\frac {d}{dt} \frac{\partial f}{\partial x'} +\frac {d^2}{d^2t} \frac{\partial f}{\partial x''}=0$$
$$2x''''=8x$$
$$x''''(t)=4x;$$
deci... | $$x^{(4)}=-4x$$
$$x(t)=C_1 e^{-t} sin(t)+C_2 e^{t} sin(t)+C_3 e^{-t} cos(t)+C_4 e^{t} cos(t)$$
$$C_1=1/2; C_2=-1/2; C_3=C_4=0$$
$$x(t)=-sin(t) sinh(t)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1222504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If $f$ is a real continuously differentiable function defined on $[0,b]$ then $\int f^2\leq\int f'^2$
Suppose $f$ is a continuously differentiable real valued function defined on an interval $[0,b]$ with $f(0)=f(b)$. Prove that$$\int_0^bf'(x)^2dx\geq\int_0^bf(x)^2dx$$
I do have some doubts about this because if I con... | Suppose that $f(a)=f(b)=0$.
We have $$f(x) = \int_a^x f'(s)ds = - \int_x^bf'(s)ds,$$hence
$$|f(x)|\le \min\left(\int_a^x |f'(s)|ds,\int_x^b|f'(s)|ds\right)\le \frac 12 \int_a^b|f'(s)|ds.$$
We integrate with respect to $x$ to get
$$\int_a^b f^2(x)dx\le \frac 14 (b-a)\left(\int_a^b|f'(s)|ds\right)^2\le \frac{(b-a)^2}{4}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1222641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Sum of geometric and Poisson distribution Suppose I have $X \sim \mathrm{Geom}(p)$ and $Y=\mathrm{Pois}(\lambda)$.
I want to create $Z = X + Y$, where the $X$ begins at $0$ rather than $1$.
Is this possible? Then I would calculate the mean and variance.
| Something that I thought of to compute the pmf, then the expectation, is:
$Pr(X=k)= p(1-k)^{k}$
$Pr(Y=k)=\frac{\lambda^k e^{\lambda}}{k!}$
Then:
$Pr(Z=k)=p(1-p)^{k} \frac{\lambda^k e^{\lambda}}{k!}$
Finally, the expected value of $Z$ would be
$E[Z] = \sum_{k=0}^{\infty}k p(1-p)^k \frac{\lambda^k e^{\lambda}}{k!} $
$E[... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1222718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
vector field question Consider the vector field $$F(x,y,z)=(zy+\sin x, zx-2y, yx-z)$$ (a) Is there a scalar field $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ whose gradient is $F$?
(b) Compute $\int _C F\cdot dr \neq 0$ where the curve $C$ is given by $x=y=z^2$ between $(0,0,0)$ and $(0,0,1)$.
I have no idea how to do the ... | The way to approach this is to consider $F(x,y,z) = F_x \hat{x} + F_y \hat{y} + F_z \hat{z}$, and compute the following integrals:
$$
\int F_x(x,y,z)dx \hspace{3pc} \int F_y(x,y,z) dy \hspace{3pc} \int F_z(x,y,z)dz
$$
I'll compute the first one for you: $\int (yz + \sin x)dx = xyz - \cos x + c(y,z)$ where we observe th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1222801",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Show that $\ln (x) \leq x-1 $ Show that $\ln (x) \leq x-1 $
I'm not really sure how to show this, it's obvious if we draw a graph of it but that won't suffice here. Could we somehow use the fact that $e^x$ is the inverse? I mean, if $e^{x-1} \geq x$ then would the statement be proved?
| Yes, one can use $$\tag1e^x\ge 1+x,$$ which holds for all $x\in\mathbb R$ (and can be dubbed the most useful inequality involving the exponential function). This again can be shown in several ways.
If you defined $e^x$ as limit $\lim_{n\to\infty}\left(1+\frac xn\right)^n$, then $(1)$ follows from Bernoullis inequality:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1222872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 3
} |
A generalization of the anti-automorphisms of a group Recently, there was a question about anti-automorphisms of a group $G$. In this case, the inversion map $\iota : G \rightarrow G : x \mapsto x^{-1}$ is an anti-automorphism and $\iota \mathrm{Aut}(G)$ is the coset of all anti-automorphisms.
Let $K$ be a permutation... | An example of a $\langle (1\,2\,3)\rangle$-automorphism is $f(x)=ex$, where $e$ is some central involution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1222972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What's the supremum of the following set $\{ n + \frac{(-1)^n}{n} : n \in \mathbb{N}\}$ What's the supremum of the following set $\{ n + \frac{(-1)^n}{n} : n \in \mathbb{N}\}$?
I know that the infimum is $0$, but what about the supremum? I have calculated with Maxima the first $1000$ terms, and it seems that the numera... | To see that it isn't bounded above, observe that for a positive integer $k$
$$2k + \frac{(-1)^{2k}}{2k} = 2k + \frac1{2k}>2k. $$
Given $M>0$, choose $k$ such that $2k>M$. Then $2k + \frac1{2k}$ is an element of the bracketed set which is greater than $M$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1223160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Interpreting $\begin{bmatrix} 1 & 2 & 3 \end{bmatrix} \cdot h $ as a scalar or matrix multiplication As part of another problem I am working on, I have the following product to work out.
$\begin{bmatrix} 1 & 2 & 3 \end{bmatrix} \cdot h $
where $h$ is a scalar. My question is, if I commute the row vector and the scalar... | As a reminder, if we have $A$ and $B$ be $n\times m$ and $m \times p$ matrices (respectively) then $AB$ is a $n \times p$ matrix. Matrix multiplication will work if the number of columns of the first matrix matches the number of rows of the second matrix. This is also the reason why matrix multiplication is not commuta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1223209",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Find all the complex numbers $z$ satisfying
Find all the complex numbers $z$ satisfying
$$
\bigg|\frac{1+z}{1-z}\bigg|=1
$$
So far I´ve done this:
$$
z=a+bi \\
\bigg|\frac{(1+a)+bi}{(1-a)-bi}\bigg|=1 \\
\mathrm{expression*conjugated} \\
\bigg|\frac{1+2bi-(a^2)-(b^2)}{1+2a+a^2+b^2}\bigg|=1
$$
| Hint: $|c/d|=|c|/|d|$ and your numerator and denominator then represent the distances from ??????
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1223297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
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. 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.
Proof: Case 1: Consider $x_{... | 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 differentiability... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1223398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Orbit-Stabiliser Theorem Applied "The group S6 acts on the group Z6 via σ([a]) = [σ(a)], for σ ∈ S6 and a∈{1,...,6}.
A permutation that is also an isomorphism is called an automorphism. The set G of
automorphisms of Z6 is a group. Use the orbit-stabiliser theorem to find its order."
I am considering the element [1], an... | Hint: Recall that for any homomorphism $\pi$, the value of $\pi(g)$ determines the value of $\pi$ on all of $\langle g \rangle$. What does that mean in your situation?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1223623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to differentiate $y=\sqrt{\frac{1+x}{1-x}}$? I'm trying to solve this problem but I think I'm missing something.
Here's what I've done so far:
$$g(x) = \frac{1+x}{1-x}$$
$$u = 1+x$$
$$u' = 1$$
$$v = 1-x$$
$$v' = -1$$
$$g'(x) = \frac{(1-x) -(-1)(1+x)}{(1-x)^2}$$
$$g'(x) = \frac{1-x+1+x}{(1-x)^2}$$
$$g'(x) = \frac{2... | $$ y^2=\frac{1+x}{1-x}=\frac2{1-x}-1\implies Derivative =\frac2{(1-x)^2} $$
$$ 2 y y ^{'} = RHS $$
$$y'=\sqrt\frac{1-x}{1+x}\cdot\frac1{(1-x)^2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1223727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 4
} |
Proof of sum results I was going through some of my notes when I found both these sums with their results
$$
x^0+x^1+x^2+x^3+... = \frac{1}{1-x}, |x|<1
$$
$$
0+1+2x+3x^2+4x^3+... = \frac{1}{(1-x)^2}
$$
I tried but I was unable to prove or confirm that these results are actually correct, could anyone please help me conf... | we have $\sum_{i=0}^nx^i=\frac{x^{n+1}-1}{x-1}$ and the limit for $n$ tends to infinity exists if $|x|<1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1223811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 3
} |
Is there an operation that takes $a^b$ and $a^c$, and returns $a^{bc}$? I know that multiplying exponents of the same base will give you that base to the power of the sum of the exponents ($a^b \times a^c = a^{b+c}$), but is there anything that can be done with exponents that will give you some base to the power of the... | The function as initially described fails to exist by the definition of a function: for any given input (or input "vector") $x$, there must exist a single output $f(x)$. Here, the quickest counterexample for any such function $f(a^b,a^c)=a^{bc}$ is to note that $2^4=4^2$ and then any combination of these produces mult... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1223920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Is there any other constant which satisfy Euler formula? Every body knows Euler Formula
$e^{ix}=\cos x +i\sin x$
Is there any other constant beside $i$ which satisfies the above equation?
| If I understand correctly your question you ask about the
equation exp(ax) = cos(x) + asin(x) ; a ≠ i, not for equality of functions but for
equality of numbers. We see the problem from the two viewpoints.
For real functions it is clear that it is impossible... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1224081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
differentiation of a matrix function In statistics, the residual sum of squares is given by the formula
$$ \operatorname{RSS}(\beta) = (\mathbf{y} - \mathbf{X}\beta)^T(\mathbf{y} - \mathbf{X}\beta)$$
I know differentiation of scalar functions, but how to I perform derivatives on this wrt $\beta$? By the way, I am tryi... | First you can remove the transposition sign from the first bracket:
$RSS=(\mathbf{y}^T - \beta ^T \mathbf{X} ^T)(\mathbf{y} - \mathbf{X}\beta)$
Multiplying out:
$RSS=y^Ty-\beta ^T \mathbf{X} ^Ty-y^TX\beta+\beta^TX^T X\beta $
$\beta ^T \mathbf{X} ^Ty$ and $y^TX\beta$ are equal. Thus
$RSS=y^Ty-2\beta ^T \mathbf{X} ^Ty+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1224163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
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