Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Factorial number of digits Is there any neat way to solve how many digits the number $20!$ have? I'm looking a solution which does not use computers, calculators nor log tables, just pen and paper.
| I will use log as the base b logarithm and ln as the natural log.
Then number of digits of x in base b is given by one more than the floor of the log(x).
log(n!)=sum(log(k)) for k=1,2,...,n
We can interpret this as the Riemann sum for the integral from 1 to n of log(x) dx. This integral is actually a lower bound. The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/136831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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"answer_id": 3
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What is $\limsup\limits_{n\to\infty} \cos (n)$, when $n$ is a natural number? I think the answer should be $1$, but am having some difficulties proving it. I can't seem to show that, for any $\delta$ and $n > m$, $|n - k(2\pi)| < \delta$. Is there another approach to this or is there something I'm missing?
| You are on the right track. If $|n-2\pi k|<\delta$ then $|\frac{n}{k}-2\pi|<\frac \delta k$. So $\frac{n}{k}$ must be a "good" approximation for $2\pi$ to even have a chance.
Then it depends on what you know about rational approximations of irrational numbers. Do you know about continued fractions?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/136897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to find $\int{\frac{x}{\sqrt{x^2+1}}dx}$? I started simplifying $$\int{\dfrac{x}{\sqrt{x^2+1}}dx}$$
but I always get this:
$$\int{x(x^2+1)^{-1/2}dx}.$$
But I don't know how to follow by that way.
| Personally, I dislike the use of variable substitution, which is sort of mechanical, for problems that can be solved by applying concept. Not to mention that changing variables is always taken extremely lightly, as if we can just plug in any expression for $x$ and presto! new variable! For example $u = x^2+1$ is clearl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/136960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is the Connect Four gaming board 7x6? (or: algorithm for creating Connect $N$ board) The Connect Four board is 7x6, as opposed to 8x8, 16x16, or even 4x4. Is there a specific, mathematical reason for this? The reason I'm asking is because I'm developing a program that will be able to generate Connect $N$ boards, ... | So it seems that a 7x6 board was chosen because it's "the smallest board which isn't easily shown to be a draw". In addition, it was also speculated that there should probably be an even amount of columns. Therefore, it seems that the dimensions of a Connect $N$ board are a function of $N$. I see two possible functi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/137103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why do engineers use the Z-transform and mathematicians use generating functions? For a (complex valued) sequence $(a_n)_{n\in\mathbb{N}}$ there is the associated generating function
$$
f(z) = \sum_{n=0}^\infty a_nz^n$$
and the $z$-Transform
$$
Z(a)(z) = \sum_{n=0}^\infty a_nz^{-n}$$
which only differ by the sign of th... | Given a sequence of numbers $\{x[n] \colon n \in \mathbb Z\}$ the $z$-transform
is defined as
$$X(z) = \sum_n x[n]z^{-n}$$ which when evaluated at $z = \exp(j\omega)$
(where $j = \sqrt{-1}$ is what electrical engineers typically use for
what mathematicians denote by $i$) gives
$${X}(\exp(j \omega)) = \sum_n x[n] \exp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/137178",
"timestamp": "2023-03-29T00:00:00",
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An infinite finitely generated group contains an isometric copy of $\mathbb{R}$, i.e., contains a bi-infinite geodesic The question is: prove that an infinite finitely generated group $G$ contains an isometric copy of $\mathbb{R}$, i.e., contains a bi-infinite geodesic ($G$ is equipped with the word metric).
I do not e... | I'm just going to focus on what you've said you are confused about, namely:
"How could there be an isometry between a subgraph of the Cayley graph of G and the real line $\mathbb{R}$?".
We can extend the word metric on $G$ to a metric on the Cayley graph in a natural way, with each edge being an isometric copy of a uni... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/137245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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compactness property I am a new user in Math Stack Exchange. I don't know how to solve part of this problem, so I hope that one of the users can give me a hand.
Let $f$ be a continuous function from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$ with the following properties:$A\subset \mathbb{R}^{n}$ is open then $f(A)$ is open... | Take $y \in \overline{f(\mathbb{R}^n)}$.
Let $B_\varepsilon = \{x | d(x,y) \leq \varepsilon\}$.
Now,
$\emptyset \neq B_\varepsilon \cap f(\mathbb{R}^n) = f\left(f^{-1}(B_\varepsilon)\right)$.
Because $f^{-1}(B_\varepsilon)$ is compact, $B_\varepsilon \cap f(\mathbb{R}^n)$,
as the image of a compact by $f$, is a decreas... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/137314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to find perpendicular vector to another vector? How do I find a vector perpendicular to a vector like this: $$3\mathbf{i}+4\mathbf{j}-2\mathbf{k}?$$
Could anyone explain this to me, please?
I have a solution to this when I have $3\mathbf{i}+4\mathbf{j}$, but could not solve if I have $3$ components...
When I google... | The vectors perpendicular to $(3,4,-2)$ form a two dimensional subspace, the plane $3x+4y-2z=0$, through the origin.
To get solutions, choose values for any two of $x,y$ and $z$, and then use the equation to solve for the third.
The space of solutions could also be described as $V^{\perp}$, where $V=\{(3t,4t,-2t):t\in\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Successive Lottery Drawings and Choosing Winning Number Consider the following scenario:
Suppose on some date $D1$ the number $N$ is a winning number in a fair lottery where a "play" is considered the selection of a finite set of numbers. By "fair" I mean that the winning number will be selected at random. At some late... | Just adding to what André Nicolas said, he's accurate. Some prize tiers are usually shared between all people who got a winning combination so this has an effect.
For example, in 2005 A record 110 players won the second prize tier of 500,000 and 100,000 dollar prizes (depending on Powerplay) in a single Powerball drawi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/137412",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Sex distribution Suppose there are N male and N female students. They are randomly distributed into k groups.
Is it more probable for a male student to find himself in a group with more guys and for female student to find herself in a group with more girls?
The question is motivated by an argument with my mother. She c... | This problem also gives the answer why you are "always" in the longer queue at the supermarket.
If $k=1$ the answer is trivial: All groups are gender balanced.
Therefore we shall assume that $k>1$.
Assume Samuel and Samantha were ill the day the groups were originally formed.
If the two Sams are assigned to the groups ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/137568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find equation of a plane that passes through point and contains the intersection line of 2 other planes Find equation of a plane that passes through point P $(-1,4,2)$ that contains the intersection line of the planes
$$\begin{align*}
4x-y+z-2&=0\\
2x+y-2z-3&=0
\end{align*}$$
Attempt:
I found the the direction vector ... | Consider the family of planes $u(4x-y+z-2)+(1-u)(2x+y-2z-3)=0$ where $u$ is a parameter. You can find the appropriate value of $u$ by substituting in the coordinates of the given point and solving for $u$; the value thus obtained can be substituted in the equation for the family to yield the particular member you need.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding subgroups of index 2 of $G = \prod\limits_{i=1}^\infty \mathbb{Z}_n$ I looked at this question and its answer. The answer uses the fact that every vector space has a basis, so there are uncountable subgroups of index 2 if $n=p$ where $p$ is prime.
Are there uncountable subgroups of index 2 if $n$ is not prime ?... | If $n$ is odd, $G$ has no subgroups of index $2$. Indeed, if $H$ is a subgroup of index dividing $2$, and $g\in G$, then $2g\in H$ (since $G/H$ has order $2$, so $2(g+H) = 0+H$). Since every element of $G$, hence of $H$, has order dividing $n$, and $\gcd(2,n)=1$, then $\langle 2g\rangle = \langle g\rangle$, so $g\in\la... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/137713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Primitive roots as roots of equations.
Take $g$ to be a primitive root $\pmod p$, and $n \in \{0, 1,\ldots,p-2\}$ write down a necessary sufficient condition for $x=g^n$ to be a root of $x^5\equiv 1\pmod p$ . This should depend on $n$ and $p$ only, not $g$.
How many such roots $x$ of this equation are there? This ans... | Hint. In any abelian group, if $a$ has order $n$, then $a^r$ has order $n/\gcd(n,r)$.
(Your idea is fine, except that you got the wrong congruence: it should be $5n\equiv p-1\pmod{p-1}$, not modulo $p$; do you see why?)
For the second part, you'll need to see what you get from the first part. That will help you figure... | {
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Symmetric and exterior power of representation Does there exist some simple formulas for the characters
$$\chi_{\Lambda^{k}V}~~~~\text{and}~~~\chi_{\text{Sym}^{k}V},$$
where $V$ is a representation of some finite group?
Thanks.
| This is not quite an answer, but Fulton & Harris, §2.1 on page 13 gives a Formula for $k=2$:
$$\chi_{\bigwedge^2 V}(g) = \frac{1}{2}\cdot\left( \chi_V(g)^2 - \chi_V(g^2)\right)$$
as well as, in the Exercise below,
$$\chi_{\mathrm{Sym}^2(V)}(g) = \frac{1}{2}\cdot\left( \chi_V(g)^2 + \chi_V(g^2)\right)$$
Maybe you can lo... | {
"language": "en",
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"source": "stackexchange",
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is it possible to get the Riemann zeros since we know that the number of Riemann zeros on the interval $ (0,E) $ is given by $ N(E) = \frac{1}{\pi}\operatorname{Arg}\xi(1/2+iE) $
is then possible to get the inverse function $ N(E)^{-1}$ so with this inverse we can evaluate the Riemann zeros $ \rho $ ??
i mean the Riem... | No, your formula is wrong. $N(E)= \frac{1}{\pi} Arg \xi (1/2+iE) $ + a nonzero term coming from the integration along the lines $\Im s =E$ (you are applying an argument prinicple).
Besides, any function $N: \mathbb{R} \rightarrow\mathbb{Z}$ can't be injective for cardinality considerations.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What does it really mean for something to be "trivial"? I see this word a lot when I read about mathematics. Is this meant to be another way of saying "obvious" or "easy"? What if it's actually wrong? It's like when I see "the rest is left as an exercise to the reader", it feels like a bit of a cop-out. What does this ... | It can mean different things. For example:
Obvious after a few moments thought.
Clear from a commonly used argument or a short one line proof.
However, it is often also used to mean the most simple example of something. For example, a trivial group is the group of one element. A trivial vector space is the space {0}.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/138112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "34",
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"answer_id": 1
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Factoring over a finite field Consider $f=x^4-2\in \mathbb{F}_3[x]$, the field with three elements. I want to find the Galois group of this polynomial.
Is there an easy or slick way to factor such a polynomial over a finite field?
| The coefficients are reduced modulo 3, so
$$
x^4-2=x^4-3x^2+1=(x^4-2x^2+1)-x^2=(x^2-1)^2-x^2=(x^2+x-1)(x^2-x-1).
$$
It is easy to see that neither $x^2+x-1$ nor $x^2-x-1$ have any roots any $F_3$. As they are both quadratic, the roots are in $F_9$. Therefore the Galois group is $Gal(F_9/F_3)$, i.e. cyclic of order two.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Probability of components to fail I want to verify my reasoning with you.
An electronic system contains 15 components. The probability that a component might fail is 0.15 given that they fail independently. Knowing that at least 4 and at most 7 failed, what is the probability that exactly 5 failed?
My solution:
$X... | You already know the answer is $a=p_5/(p_4+p_5+p_6+p_7)$ where $p_k=\mathrm P(X=k)$. Further simplifications occur if one considers the ratios $r_k=p_{k+1}/p_k$ of successive weights. To wit,
$$
r_k=\frac{{n\choose k+1}p^{k+1}(1-p)^{n-k-1}}{{n\choose k}p^{k}(1-p)^{n-k}}=\frac{n-k}{k+1}\color{blue}{t}\quad\text{with}\ \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/138224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Inscrutable proof in Humphrey's book on Lie algebras and representations This is a question pertaining to Humphrey's Introduction to Lie Algebras and Representation Theory
Is there an explanation of the lemma in §4.3-Cartan's Criterion? I understand the proof given there but I fail to understand how anybody could have ... | This doesn't entirely answer your question but the key ingredients are (1) the rationals are nice in that their squares are non-negative (2) you can get from general field elements to rationals using a linear functional f (3) getting a handle on x by way of the eigenvalues of s.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 2,
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Algebrically independent elements
Possible Duplicate:
Why does K->K(X) preserve the degree of field extensions?
Suppose $t_1,t_2,\ldots,t_n$ are algebrically independent over $K$ containing $F$.
How to show that $[K(t_1,\ldots,t_n):F(t_1,\ldots,t_n)]=[K:F]$?
| Using the answer in link provided by Zev, your question can be answered by simple induction over $n$. For $n=1$ we proceed along one of the answers shown over there. Assume we have shown the theorem for some $n$. Then we have $[K(t_1,\ldots,t_n):F(t_1,\ldots,t_n)]=[K:F]$, and by the theorem for $n=1$ we have also $[K(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/138465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Showing a series is a solution to a differential equation I am attempting to show that the series $y(x)\sum_{n=0}^{\infty} a_{n}x^n$ is a solution to the differential equation $(1-x)^2y''-2y=0$ provided that $(n+2)a_{n+2}-2na_{n+1}+(n-2)a_n=0$
So i have:
$$y=\sum_{n=0}^{\infty} a_{n}x^n$$
$$y'=\sum_{n=0}^{\infty}na_{n}... | You are correct.
Only you need to go on and observe that the lhs of your last equation factorizes as: $$(n+1)[(n+2)a_{n+2}-2n a_{n+1}+(n-2)a_n]$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Intuition and derivation of the geometric mean I've run through a bunch of searches, especially here on SO, but I simply couldn't find something that answers a question that has been on my mind lately. How was the geometric mean derived? What is the intuition behind it. Most simply use the final equation as a justifica... | To expand on Domagoj Pandža's great answer, the "distance" function for the geometric mean is:
$$
d(n) = \sum_i \left( \ln x_i - \ln n \right)^2.
$$
The geometric mean $n = \prod_i x_i^{1/k}$ can be derived as the function that minimizes the value of this distance function.
Proof
Expand:
$$
d(n) = \sum_i (\ln x_i)^2 - ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/138589",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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Finding a constant to make a valid pdf Let $f(x) = c\cdot 2^{-x^2}$. How do I find a constant $c$ such that the integral evaluates to $1$?
| Hint: Rewrite
$$f(x) = c \,[e^{\ln(2)}]^{-x^2} = c\, e^{-x^2\ln(2)}$$
and try to exploit the following integral together with some change of variable:
$$
\int^{\infty}_0 e^{-x^2} \,dx = \frac{\sqrt{\pi}}{2}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/138652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Central Limit Theorem/Markov's inequality Here is the question:
Chip dies are manufactured in a facility where it was observed that
the width of the die is normally distributed with mean 5mm and
standard deviation $\sigma$. The manufacturer wants to guarantee that no more
than 1 out of 100 dies fall outside the ... | Assume, without much justification except that we were told to do so, that the width $X$ of the die has normal distribution with mean $5$ and variance $\sigma^2$.
The probability that we are within $k\sigma$ of the mean $5$ (formally, $P(5-k\sigma\le X \le 5+k\sigma)$) is equal to the probability that $|Z|\le k$, wher... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/138704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Lowenheim-Skolem theorem confusion This Wikipedia entry on the Löwenheim–Skolem theorem says:
In mathematical logic, the Löwenheim–Skolem theorem, named for Leopold Löwenheim and Thoralf Skolem, states that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a mode... | Each model has a set of individuals. The size of the model is the cardinality of this set.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 0
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Proof of the Schwarz Lemma I have a question which is (the Schwarz Lemma):
Suppose that $f:\mathbb{D}\rightarrow\mathbb{D}$ is holomorphic and suppose that $f(0)=0$, show that $\lvert f(z)\rvert \leq \lvert z \rvert \forall{z}\in\mathbb{D}$
and the solution is:
Let $g(z)=\frac{f(z)}{z}$ for $z\neq0$ and $g(0)=f'(0)$. T... | $f$ is a function of the unit disk into itself. This means that $|f(z)| < 1$ for all $z \in \mathbb{D}$, and in particular this is true for all $z$ in the boundary of the disk $\mathbb{D}(0,r)$ , $r<1$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Advection diffusion equation The advection diffusion equation is the partial differential equation $$\frac{\partial C}{\partial t} = D\frac{\partial^2 C}{\partial x^2} - v \frac{\partial C}{\partial x}$$ with the boundary conditions $$\lim_{x \to \pm \infty} C(x,t)=0$$ and initial condition $$C(x,0)=f(x).$$ How can I t... | We could simply apply the chain rule, to avoid some confusions we let $ C(x,t) = C(x^* + vt,t^*) = C^*(x^*,t^*)$:
$$
\frac{\partial C}{\partial x} = \frac{\partial C^*}{\partial x^{\phantom{*}}}= \frac{\partial C^*}{\partial x^*} \frac{\partial x^*}{\partial x^{\phantom{*}}} + \frac{\partial C^*}{\partial t^*} \frac{\p... | {
"language": "en",
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Self-Dual Code; generator matrix and parity check matrix
Hi !
I have a parity check matrix $H$ for a code $C$
0 0 0 1 1 1 1 0
0 1 1 0 0 1 1 0
1 0 1 0 1 0 1 0
1 1 1 1 1 1 1 1
I am allowed to assume that
1) the dual of an $(n,k)$-code is an $[n,n-k]$-code
2) $(C^{\perp})^{\perp} = C$ (Here $\perp$ denotes ... | The rows of $H$ generate $C^\perp$.
By definition of the parity check, $xH^\mathrm{T}=0$ iff $x\in C$.
What can you conclude from the fact that $HH^\mathrm{T}=[0]$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/138984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Commutativity between diagonal and unitary matrices? Quick questions:
*
*if you have a diagonal matrix $A$ and a unitary matrix $B$. Do $A$ and $B$ commute?
*if $A$ and $B$ are positive definite matrices. if $a$ is an eigenvalue of $A$ and $b$ is an eigenvalue of $B$, does it follow that $a+b$ is an eigenvalue of ... | For the first question, the answer is no, an explicit example is given by $A:=\pmatrix{1&0\\ 0&2}$ and $B=\pmatrix{1&1\\ -1&1}$. An other way to see it's not true is the following: take $S$ a symmetric matrix, then you can find $D$ diagonal and $U$ orthogonal (hence unitary) such that $S=U^tDU$ and if $U$ and $D$ commu... | {
"language": "en",
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Independent, Normally Distributed R.V. Working on this:
A shot is fired at a circular target. The vertical and the horizontal
coordinates of the point of impact (with the origin sitting at the
target’s center) are independent and normally distributed with $\nu(0, 1)$. Show that the distance of the point of impact ... | Let $r\ge 0$; put $R = \sqrt{X^2 + Y^2}$, where $X$ and $Y$ are the coordinates of the shot. Then
$$P(R\le r) = {1\over 2\pi} \mathop{\int\!\!\int}_{B_r(0)} \exp\{(x^2 + y^2)/2\}\,dx\,dy.$$
Change to polars to get
$$P(R\le r) = {1\over 2\pi}\int_0^r \int_0^{2\pi} \exp(r^2/2)r\,d\theta\, dr
=\int_0^r r\exp\{r^2/2\}\,d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/139144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Area under a curve, difference between dy and dx I am trying to find the area of $ y = 1 $ and $y = x^\frac{1}{4}$ from 0 to 1 and revolving around $ x = 1$
In class we did the problem with respect to y, so from understanding that is taking the "rectangles" from f(y) or the y axis. I was wondering why not just do it wi... | I expect you have drawn a picture, and that it is the region below $y=1$, above $y=x^{1/4}$, from $x=0$ to $x=1$ that is being rotated about $x=1$. When you rotate, you get a cylinder with a kind of an upside down bowl carved out of it, very thin in the middle. You have asked similar questions before, so I will be br... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving identities using Pythagorean, Reciprocal and Quotient Back again, with one last identity that I cannot solve:
$$\frac{\cos \theta}{\csc \theta - 2 \sin \theta} = \frac{\tan\theta}{1-\tan^2\theta}$$.
The simplest I could get the left side to, if at all simpler, is $$\frac{\cos\theta}{\csc^2\theta-2}$$
As for the... | HINT:
$$\begin{align*}
\frac{\tan\theta}{1-\tan^2\theta}&=\frac{\sin\theta\cos\theta}{\cos^2\theta-\sin^2\theta}
\end{align*}$$
$$\begin{align*}
\frac{\cos\theta}{\csc\theta-2\sin\theta}&=\frac{\sin\theta\cos\theta}{1-2\sin^2\theta}
\end{align*}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/139263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Question on conditional independence Consider four random vectors $X, Z, C$ and $W$ in which
$Z_i = W_i+N(0,\sigma)$: iid Gaussian noise for each element of $W$
If $X$ is conditionally independent of $Z$ given $C,$ will X be conditionally independent of $W$ given $C$?
Thank you very much.
| Not necessarily, given the conditions as stated. We can work in one dimension. Let $\eta, \xi$ be two iid $N(0,1)$ random variables. Set $X = \eta - \xi$, $W = \eta$, $Z = \eta + \xi$, and $C=0$ (so conditional independence given $C$ is just independence). Then $X$ and $Z$ are independent (they are jointly Gaussian... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The Set of All Subsequential Limits Given $\{a_n\}_{n=0}^\infty$ and $\{b_n\}_{n=0}^\infty$ bounded sequences; show that if $\lim \limits_{n\to \infty}a_n-b_n=0$ then both sequences have the same subsequential limits.
My attempt to prove this begins with: Let $E_A=\{L|L$ subsequential limit of $a_n$}
and $E_B=\{L|L$ su... | I think you may want to prove that:
*
*Consider two sequences $\{x_n\}$ and $\{y_n\}$ such that $x_n-y_n \to l$ and $y_n \to y$, then, $$x_n \to l+y$$
*Given a convergent sequence $\{x_n\}$ that converges to $x$, all its subsequences converge to the same limit, $x$.
Do you see how that would pay here?
*
*Let $r ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/139438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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In the history of mathematics, has there ever been a mistake? I was just wondering whether or not there have been mistakes in mathematics. Not a conjecture that ended up being false, but a theorem which had a proof that was accepted for a nontrivial amount of time before someone found a hole in the argument. Does this ... | Well, there have been plenty of conjectures which everybody thought were correct, which in fact were not. The one that springs to mind is the Over-estimated Primes Conjecture. I can't seem to find a URL, but essentially there was a formula for estimating the number of primes less than $N$. Thing is, the formula always ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solve $\frac{\cos x}{1+\sin x} + \frac{1+\sin x}{\cos x} = 2$ I am fairly good at solving trig equations yet this one equation has me stumped. I've been trying very hard but was unable to solve it. Can anyone help please? Thank you.
$$\frac{\cos x}{1+\sin x} + \frac{1+\sin x}{\cos x} = 2$$
solve for $x$ in the range of... | HINT: $$\begin{align*}
\frac{\cos x}{1+\sin x} + \frac{1+\sin x}{\cos x}&=\frac{\cos^2 x+(1+\sin x)^2}{\cos x(1+\sin x)}\\
&=\frac{\cos^2 x+\sin^2x+1+2\sin x}{\cos x(1+\sin x)}\;;
\end{align*}$$
now use a familiar trig identity and find something to cancel.
| {
"language": "en",
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"source": "stackexchange",
"question_score": "6",
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How to find the minimum variance portfolio? I am doing some revision questions on my Portfolio Theory module, and have come across the following question:
Consider an investor who has constructed a risky portfolio from N securities. The investment opportunity set is described by the equation:
$$\sigma^2 = 10 - 5{\times... | We want to minimize $\sigma^2$ as a function of $E(r)$. We also likely want $E(r) \ge 0$, but this won't be necessary.
$$0 =\frac{d \sigma^2}{dE(r)}=-5+E(r).$$
Now, looking at the second derivative
$$\frac{d^2 \sigma^2}{dE(r)^2} = 1>0$$
It is convex at every point, so we only need to check minima. And $E(r)=5$ is the m... | {
"language": "en",
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"source": "stackexchange",
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Conditional probability: Why can't I make this assumption? Here is the question
:If we are told that among the 10 coins, at least three have landed on heads. What is the probability that there are at most five heads?
In other words,
P( at most five head | at least three have landed).
The solution says,
$P( X\leq 5 | ... | Your calculation of $P(\text{at most $2$ heads out of $7$ tosses})$ is wrong because you must include $X=0$ and $X=1$ along with $X=2$. The correct result is $\approx 0.227$.
But that is not the main point. You know that three of the ten tosses are heads, but you don't know which ones. For instance, if you are given t... | {
"language": "en",
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What are some examples of a mathematical result being counterintuitive? As I procrastinate studying for my Maths Exams, I want to know what are some cool examples of where math counters intuition.
My first and favorite experience of this is Gabriel's Horn that you see in intro Calc course, where the figure has finite v... | I think a puzzle at calculus level is the following: Given a real number $x$ and a conditionally convergent series, the series can be re-arranged so that its sum is $x$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "133",
"answer_count": 45,
"answer_id": 0
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What are some examples of a mathematical result being counterintuitive? As I procrastinate studying for my Maths Exams, I want to know what are some cool examples of where math counters intuition.
My first and favorite experience of this is Gabriel's Horn that you see in intro Calc course, where the figure has finite v... | Here's a counterintuitive example from The Cauchy Schwarz Master Class, about what happens to cubes and spheres in high dimensions:
Consider a n-dimensional cube with side length 4, $B=[-2,2]^n$, with radius 1 spheres placed inside it at every corner of the smaller cube $[-1,1]^n$. Ie, the set of spheres centered at co... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "133",
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"answer_id": 32
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If $b^2$ is the largest square divisor of $n$ and $a^2 \mid n$, then $a \mid b$. I am trying to prove this:
$n$, $a$ and $b$ are positive integers. If $b^2$ is the largest square
divisor of $n$ and $a^2 \mid n$, then $a \mid b$.
I want to prove this by contradiction, and I don't want to go via the fundamental theor... | Let $\operatorname{lcm}(a,b)=\frac{ab}{\gcd(a,b)}$. Since the $\gcd$ divides both $a$ and $b$, it's clear from the definition that the $\operatorname{lcm}$ is an integer divisible by both $a$ and $b$. And if $a$ does not divide $b$, then the $\operatorname{lcm}$ is strictly greater than $b$, since $a\neq \gcd(a,b)$. By... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Understanding the Leontief inverse What I remember from economics about input/output analysis is that it basically analyses the interdependencies between business sectors and demand. If we use matrices we have $A$ as the input-output matrix, $I$ as an identity matrix and $d$ as final demand. In order to find the final ... | The equation you are concerned with relates total output $x$ to intermediate output $Ax$ plus final output $d$, $$ x = Ax + d $$.
If the inverse $(I - A)^{-1}$ exists, then a unique solution to the equation above exists. Note that some changes of $a_{ij}$ may cause a determinate system to become indeterminate, meanin... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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How to prove $\left\{ \omega|X(\omega)=Y(\omega)\right\} \in\mathcal{F}$ is measurable, if $X$ and $Y$ are measurable? Given a probability space $(\Omega ,\mathcal{F} ,\mu)$. Let $X$ and $Y$ be $\mathcal{F}$-measurable real valued random variables. How would one proove that $\left\{ \omega|X(\omega)=Y(\omega)\right\} ... | $$[X\ne Y]=\bigcup_{q\in\mathbb Q}\left([X\lt q]\cap[Y\geqslant q]\right)\cup\left([X\geqslant q]\cap[Y\lt q]\right)=\bigcup_{q\in\mathbb Q}\left([X\lt q]\Delta[Y\lt q]\right)
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is 1100 a valid state for this machine? A room starts out empty. Every hour, either 2 people enter or 4 people leave. In exactly a year, can there be exactly 1100 people in the room?
I think there can be because 1100 is even, but how do I prove/disprove it?
| The process can be rephrased as making a required motion of $+2$ every time, with an optional additional motion of $-6$ at some steps. The forward part of that motion will reach $+2N$ in $N$ steps, plus some backward motion divisible by $6$ to attain the final total motion of $T=+1100$. Here $N$ and $T$ are given and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/139931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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A characterization of open sets
Let $(M,d)$ be a metric space. Then a set $A\subset M$ is open if, and only if, $A \cap \overline X \subset \overline {A \cap X}$ for every $X\subset M$.
This is a problem from metric spaces, but I think that it only requires topology. I don't know how to do it.
| Suppose first that $A$ is open. Let $X\subseteq M$, and suppose that $x\in A\cap\operatorname{cl}X$. Let $U$ be any open set containing $x$; then $A\cap U$ is an open neighborhood of $x$, and $x\in\operatorname{cl}X$, so $U\cap X\ne\varnothing$. Thus, $U\cap(A\cap X)\ne\varnothing$, and since $U$ was an arbitrary open ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/140012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove here exists a function $f \in L^1(X,\mu)$ with $f>0$ $\mu$-a.e. iff $\mu$ is $\sigma$-finite. How to show that let $(X,\mathcal{M},\mu)$ be a measurable space, there exists a function $f \in L^1(X,\mu)$ with $f>0$ $\mu$-a.e. iff $\mu$ is $\sigma$-finite.
Can you please help me out? Thank you.
| *
*If such a function exists, then put $A_n:=\{x\mid f(x)\geq \frac 1n\}$, $A_n$ is measurable and of finite measure since $f$ is integrable. Let $N:=\{x,f(x)\leq 0\}$. Then $X=N\cup\bigcup_{n\geq 1}A_n$.
*Conversely, if $(X,\mathcal M,\mu)$ is $\sigma$-finite, let $\{A_n\}$ a sequence of pairwise disjoint sets such... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Determinant of a 3x3 matrix with 6 unknowns given the determinants of two 3x3 matrices with same unknowns? Given:
$$
det(A) = 3 \\ det(B) = -4
$$
$$
A =
\begin{pmatrix}
a & b & c \\
1 & 1 & 1\\
d & e & f
\end{pmatrix} \\
B =
\begin{pmatrix}
a & b & c \\
1 & 2 & 3 \\
d & e & f
\end{pmatrix} \\
C =
\begin{pmatrix}
a & b ... | copper.hat's answer is a lovely answer, which uses very fundamental attributes of the determinant. Notice that your answer is an algebraic way of saying the same thing. They are really equivalent, just that you have an algebraic error in your working; your augmented matrix should look like (I tried to preserve the colu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/140144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Minimum tiles for a grid given a few conditions Today, I came across an exercise in Problem Solving Strategies by Johnson and Herr which I was not sure was the best way to solve it. The problem given was:
Below I drew up a quick sketch of a diagram.
Note that each row and column should be the same height and width, w... | Here's the general formula I came up with:
Given $m$ horizontal lines and $n$ vertical lines, the number of tiles needed to cover the grid is
$\lceil\frac{m}{2}\rceil\times\lceil\frac{n}{2}\rceil$
So with $m=12$ and $n=9$
$\lceil\frac{12}{2}\rceil\times\lceil\frac{9}{2}\rceil=\lceil6\rceil\times\lceil4.5\rceil=6\times5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/140191",
"timestamp": "2023-03-29T00:00:00",
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Can every continuous function that is curved be expressed as a formula? By "curved", I mean that there is no portion of a graph of a function that is a straight line. (Let us first limit the case into 2-dimensional cases.. and if anyone can explain the cases in more than 2-dimensional scenarios I would appreciate.. but... | No. First, because when you say "whatever formula" you probably include the elementary operations, maybe fractional power, and a few known functions (sine, cosine, exponential, say); there are many other functions, not only continuous but infinitely differentiable that are known to be impossible to express in terms of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/140240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Diagonal in projective space This is exercise $2.15$ from Harris book "Algebraic Geometry: A First Course". Show that the image of the diagonal in $\mathbb{P}^{n} \times \mathbb{P}^{n}$ under the Segre map is isomorphic to the Veronese variety $v_{2}(\mathbb{P}^{n})$.
Would the idea is just map everything to itself and... | You have exactly the right idea. I would formulate it slightly differently. If we continue with your example, we can write down a map $\mathbf P^2 \to \mathbf P^3$ as
$$ [x:y:z]\mapsto [x:y:y:z].$$
If we restrict this map to the Veronese embedded copy of $\mathbf P^1$, then we get an isomorphism onto the image of the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/140291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Linear algebra question about definition of basis From Wikipedia:
"A basis $B$ of a vector space $V$ over a field $K$ is a linearly independent subset of $V$ that spans (or generates) $V$.(1)
$B$ is a minimal generating set of $V$, i.e., it is a generating set and no proper subset of B is also a generating set.(2)
$B$ ... | The proof looks good (appart form the obvious mix up in the numbering). One thing which is not totally precise:
In your second proof you write
Let $v\in V$ be arbitrary. $B$ is maximal hence $B\cup\{v\}$ is linearly dependent. i.e. $\exists b_i\in B,k_i\in K: \sum_i b_ik_i=v$, i.e. $B$ is spanning.
To be precise you ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/140359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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a function as a character I meet difficulty in Problem 4.5 in the book "Representation theory of finite group, an introductory approach" of Benjamin Steinberg :
For $v=(c_1,\cdots,c_m)\in(\mathbb{Z}/2\mathbb{Z})^m$, let $\alpha(v)=\{i : c_i=[1]\}$. For each $Y\subseteq\{1,\cdots,m\}$, define a function $\chi_Y : (\math... | I think here is the solution : Looking the second question, it suggests us to prove that $\chi_Y$ is a representation, i.e., to prove that $\chi_Y$ is a group homomorphism.
Let $v_1,v_2\in(\mathbb{Z}/2\mathbb{Z})^m$ so
$\alpha(v_1+v_2)=\alpha(v_1)+\alpha(v_2)-\alpha(v_1)\cap\alpha(v_2)$, and so
$|\alpha(v_1+v_2)\cap Y|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/140397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can someone walk me through this differential problem? I'm having a little difficulty understanding how to do the .05 using differentials. I'm just hoping someone can walk me through, step by step, and explain why they are during it that way.
$$\sqrt[3] {27.05}$$
Edit
Apologies to all, I wasn't very clear on what I was... | The general philosophy when using differentials:
You are trying to find the value of a function at a difficult point, but there is a point, very close to the difficult point, where the function is easy to evaluate. Then one can use the formula
$$f(a+\Delta x) = f(a)+\Delta y \approx f(a)+dy = f(a)+f'(a)dx=f(a)+f'(a)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/140454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
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Determining the dimension and a basis for a vector space I have the following problem:
Let $W$ be a vector space of all solutions to these homogenous equations:
$$\begin{matrix} x &+& 2y &+& 2z &-& s &+& 3t &=& 0 \\
x &+& 2y &+& 3z &+& s &+& t &=& 0 \\
3x &+& 6y &+& 8z &+& s &+& 5t &=& 0\end{matrix}$$
Find the... | Subtracting the first row $4$ times from the third, we get the matrix
$$\left(\begin{matrix} 1 & 0 & -1 & 0 & 0 \\ 0 & 5 & 2 & 0 & 0 \\ 0 & 10 & 4 & 0 & 0\end{matrix}\right)$$
Subtracting the second row $2$ times from the third, we get the matrix
$$\left(\begin{matrix} 1 & 0 & -1 & 0 & 0 \\ 0 & 5 & 2 & 0 & 0 \\ 0 & 0 &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/140532",
"timestamp": "2023-03-29T00:00:00",
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About the localization of a UFD
I was wondering, is the localization of a UFD also a UFD?
How would one go about proving this? It seems like it would be kind of messy to prove if it is true.
If it is not true, what about localizing at a prime? Or what if the UFD is Noetherian?
| One slick way is via Kaplansky's characterization: a domain is a UFD iff every nonzero prime ideal contains a nonzero prime. This is easily seen to be preserved by localization, hence the proof. Alternatively, proceed directly by showing that primes stay prime in the localization if they survive (don't become units). T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/140584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Using modular arithmetic, how can one quickly find the natural number n for which $n^5 = 27^5 + 84^5 + 110^5 + 133^5$? Using modular arithmetic, how can one quickly find the natural number n for which $n^5 = 27^5 + 84^5 + 110^5 + 133^5$?
I tried factoring individual components out, but it seemed really tedious.
| If there is such an $n$, it must be a multiple of 6 and 1 less than a multiple of 5, and it must exceed 133 but not by a whole lot, so my money's on 144.
| {
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Parametric equation for a line which lies on a plane Struggling to begin answering the following question:
Let $L$ be the line given by $x = 3-t, y= 2+t, z = -4+2t$. $L$ intersects the plane $3x-2y+z=1$ at the point $P = (3,2,-4)$. Find parametric equations for the line through $P$ which lies on plane and is perpendic... | You know that the line you want is perpendicular to the line L, which has direction vector $\langle -1,1,2\rangle$, and that the line you want lies in the given plane, which has normal vector $\langle 3, -2, 1\rangle$. So the line you want is orthogonal to both $\langle -1,1,2\rangle$ and $\langle 3, -2, 1\rangle$ and... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to tell if a quartic equation has a multiple root. Is there any way to tell whether a quartic equation has double or triple root(s)?
$$x^4 + a x^3 + b x^2 + c x + d = 0$$
| Maybe I misunderstood the question, but I think this might be useful for you.
Let $f(x)\in F[x]$ be a polynomial and $f'(x)$ be the formal derivative of $f(x)$ where $F$ is any field.
Then $f(x)$ has multiple roots if and only if $\gcd(f(x),f'(x))\ne 1$.
If $F$ is a field of characteristic zero then you know more: If ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Vector derivative with power of two in it I want to compute the gradient of the following function with respect to $\beta$
$$L(\beta) = \sum_{i=1}^n (y_i - \phi(x_i)^T \cdot \beta)^2$$
Where $\beta$, $y_i$ and $x_i$ are vectors. The $\phi(x_i)$ simply adds additional coefficients, with the result that $\beta$ and $\phi... | Vector differentiation can be tricky when you're not used to it. One way to get around that is to use summation notation until you're confident enough to perform the derivatives without it.
To begin with, let's define $X_i=\phi(x_i)$ since it will save some typing, and let $X_{ni}$ be the $n$th component of the vector ... | {
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When the group of automorphisms of an extension of fields acts transitively Let $F$ be a field, $f(x)$ a non-constant polynomial, $E$ the splitting field of $f$ over $F$, $G=\mathrm{Aut}_F\;E$. How can I prove that $G$ acts transitively on the roots of $f$ if and only if $f$ is irreducible?
(if we suppose that $f$ does... | I think what you needed as a hypothesis was that the extension is Galois. By this, I mean that the polynomial $f(x)$ splits into distinct linear factors in an extension, so I am just assuming the negation of the problem about having repeated roots that other folks were suggesting. In this case, the transitive action on... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/140927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
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Lebesgue measurable set that is not a Borel measurable set exact duplicate of Lebesgue measurable but not Borel measurable
BUT! can you please translate Miguel's answer and expand it with a formal proof? I'm totally stuck...
In short: Is there a Lebesgue measurable set that is not Borel measurable?
They are an order o... | Let $\phi(x)$ be the Cantor function, which is non-decreasing continuous function on the unit interval $\mathcal{U}_{(0,1)}$. Define $\psi(x) = x + \phi(x)$, which is an increasing continuous function $\psi: [0,1] \to [0,2]$, and hence for every $y \in [0,2]$, there exists a unique $x \in [0,1]$, such that $y = \psi(x)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What did Gauss think about infinity? I have someone who is begging for a conversation with me about infinity. He thinks that Cantor got it wrong, and suggested to me that Gauss did not really believe in infinity, and would not have tolerated any hierarchy of infinities.
I can see that a constructivist approach could in... | Here is a blog post from R J Lipton which throws some light on this question.
Quoting from a letter by Gauss:
... so protestiere ich zuvörderst gegen den Gebrauch einer unendlichen Größe als einer vollendeten, welcher in der Mathematik niemals erlaubt ist. Das Unendliche ist nur eine façon de parler, indem man eigentl... | {
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Highest power of a prime $p$ dividing $N!$ How does one find the highest power of a prime $p$ that divides $N!$ and other related products?
Related question: How many zeros are there at the end of $N!$?
This is being done to reduce abstract duplicates. See
Coping with *abstract* duplicate questions. and List of Gener... | Here's a different approach I found while thinking in terms of relating the sum of digits of consecutive numbers written in base $p$. Part of the appeal of this approach is you might have at one point learned that there is a connection but don't remember what, this gives a quick way to reconstruct it.
Let's consider $s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "70",
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Showing that $\cos(x)$ is a contraction mapping on $[0,\pi]$ How do I show that $\cos(x)$ is a contraction mapping on $[0,\pi]$? I would normally use the mean value theorem and find $\max|-\sin(x)|$ on $(0,\pi)$ but I dont think this will work here.
So I think I need to look at $|\cos(x)-\cos(y)|$ but I can't see what ... | To show that $\cos(x)$ is a contraction mapping on $[0,1]$ you just need to show that it is Lipschitz with a Lipschitz constant less than $1$. Because $\cos(x)$ is continuously differentiable, there is a maximum absolute value of the derivative on each closed interval, and the mean value theorem can be used to show tha... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What is the difference between regression and classification? What is the difference between regression and classification, when we try to generate output for a training data set $x$?
| Regression and classification can work on some common problems where the response variable is respectively continuous and ordinal.
But the result is what would make us choose between the two.
For example, simple/hard classifiers (e.g. SVM) simply try to put the example in specific class (e.g. whether the project is "p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/141381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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When a field extension $E\subset F$ has degree $n$, can I find the degree of the extension $E(x)\subset F(x)?$ This is not a problem I've found stated anywhere, so I'm not sure how much generality I should assume. I will try to ask my question in such a way that answers on different levels of generality could be possib... | My proof is concerned with commutative algebra. Probably there is also field-theoretic one. I prove that: if $E \subseteq F$ is a finite field extension, then $[F(x) : E(x)] = [F \colon E]$.
This follows from: if $E \subseteq F$ is an algebraic field extension, then $F(x) \simeq E(x) \otimes_E F$ as $E(x)$-vector space... | {
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How do I figure out the speed of a jet of water in this example? I should know how to do this but I don't. I'm not very familiar with vectors. Perhaps I will be after this.
So I have a stream of water falling out of a pipe. It obviously forms a parabola of the form $f(x) = -ax^2+h$, where $h$ is the height of the pi... | If the water exits horizontally (you seem to assume that) the height of the water is $y=h-\frac {gt^2}2$. The time to reach the ground comes from setting $y=0, t=\sqrt {\frac{2h}{g}}$The horizontal position is $x=vt$ where $v$ is the velocity on exit from the pipe. Let $d$ be the horizontal distance where the water h... | {
"language": "en",
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Show that $\|f \chi_F\|_p < \epsilon$ Let $ f \in L_p(X, \mathcal{A}, \mu)$, $1 \leq p < \infty$, and let $\epsilon > 0$. Show that there exists a set $E_\epsilon \in \mathcal{A}$ with $\mu(E_\epsilon) < \infty$ such that if $F \in \mathcal{A}$ and $F \cap E_\epsilon = \emptyset$, then $\|f \chi_F\|_p < \epsilon$.
I wa... | Your estimates are fine since everything is finite, now you must exhibit the set $E_\epsilon$. For this consider $E_{\frac{1}{n}} = \{ x\in X : |f(x)|^p \geq \frac{1}{n} \}$, now prove that $g_n=|f\chi_{E_{\frac{1}{n}}}|^p$ gives a monotone increasing sequence, with $|g_n|\leq |f|^p$ and $g_n(x)\to |f(x)|^p$ everywhere... | {
"language": "en",
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Result of the product $0.9 \times 0.99 \times 0.999 \times ...$ My question has two parts:
*
*How can I nicely define the infinite sequence $0.9,\ 0.99,\ 0.999,\ \dots$? One option would be the recursive definition below; is there a nicer way to do this? Maybe put it in a form that makes the second question easier t... | By looking at the decimal representation, it appears that:
$$
\prod_{i=1}^\infty\left(1-\frac1{10^i}\right)=
\sum_{i=1}^\infty
\frac{8 +
\frac{10^{2^i-1}-1}{10^{2i-1}} +
\frac1{10^{6i-2}} +
\frac{10^{4i}-1}{10^{12i-2}}
}{10^{(2i-1)(3i-2)}}
$$
I don't have a proof, but the pattern is so regular that I'm ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
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First-order logic advantage over second-order logic What is the advantage of using first-order logic over second-order logic? Second-order logic is more expressive and there is also a way to overcome Russell's paradox...
So what makes first-order logic standard for set theory?
Thanks.
| First order logic has the completeness theorem (and the compactness theorem), while second-order logic does not have such theorems.
This makes first-order logic pretty nice to work with. Set theory is used to transform other sort of mathematical theories into first-order.
Let us take as an example the natural numbers w... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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The maximum number of nodes in a binary tree of depth $k$ is $2^{k}-1$, $k \geq1$. I am confused with this statement
The maximum number of nodes in a binary tree of depth $k$ is $2^k-1$, $k \geq1$.
How come this is true. Lets say I have the following tree
1
/ \
2 3
Here the depth of the tree is 1. So accord... | We assume that the root of a binary tree is on level 1, so in your mentioned tree, the depth is 2 not 1, so (2 to the power 2 ) - 1 = 3 nodes.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
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Looking for equation that matches graph (image inside) I've been trying to come up with a one-variable function that roughly matches this graph:
The closest I've gotten is $ f(d) = \dfrac{\log(c \cdot (1 - \frac{d}{c}))}{\log c} $, but this approaches $y=0$ at $x=c-1$, and steadily declines from $x=0$, instead of sha... | Let's try it in polar coordinates (as suggested by John).
We will start with a four petals flower getting the polar expression
$$\rho=\frac{m-1+\cos(4\theta)}m$$
$m$ is a parameter and $m\approx 5$ seems appropriate giving :
In your case the graph will be obtained by
$$x=C\rho \cos(\theta),\ y=\rho \sin(\theta)$$
T... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Limit of derivative If $\displaystyle \lim_{x \to \infty} f(x) = a$, and knowing that $\displaystyle \lim_{x \to \infty} xf'(x)$ exists , how would I find that limit?
| I think I've seen this question before, but I've thought of a pleasant little proof.
$\lim f(x) < \infty$, so $\lim \dfrac{f(x)}{\ln x} = 0$
But $\lim \dfrac{f(x)}{\ln x} = \lim \dfrac{f'(x)}{\frac{1}{x}} = \lim x f'(x)$ if $\lim x f'(x)$ exists.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Divergent series and $p$-adics If we naïvely apply the formula $$\sum_0^\infty a^i = {1\over 1-a}$$ when $a=2$, we get the silly-seeming claim that $1+2+4+\ldots = -1$. But in the 2-adic integers, this formula is correct.
Surely this is not a coincidence? What is the connection here? Do the $p$-adics provide general... | This is not so much an answer as a related reference. I wrote a short expository note "Divergence is not the fault of the series," Pi Mu Epsilon Journal, 8, no. 9, 588-589, that discusses this idea and its relation to 2's complement arithmetic for computers.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is my Riemann Sum correct (Example # 2)?
Possible Duplicate:
Is my Riemann Sum correct?
This is my second attempt, the answer seems rather odd so I thought I would have it checked as well.
For the integral: $$\int_{-5}^{2} \left( x^{2} -4 \right) dx$$
My calculations:
$$\begin{align*}\Delta x &= \frac7n\\\\
x_i &= -... | The preliminary computations are fine. That means that the $n$th right hand Riemann sum will be:
$$\begin{align*}
\text{RHS} &= \sum_{i=1}^n f(x_i)\Delta x\\
&= \sum_{i=1}^n\left(21 - \frac{70i}{n} +\frac{49i^2}{n^2}\right)\frac{7}{n}\\
&= \frac{7(21)}{n}\sum_{i=1}^n1 - \frac{7(70)}{n^2}\sum_{i=1}^n i + \frac{7(49)}{n^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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what is the difference between functor and function? As it is, what is the difference between functor and function? As far as I know, they look really similar.
And is functor used in set theory? I know that function is used in set theory.
Thanks.
| A simpler explanation:
Functions map arguments to values while functors map arguments and functions defined over the arguments to values and functions defined over the values, respectively.
Moreover, the functor mappings preserve function composition over the functions on arguments and values.
Briefly, functions map el... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "39",
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Does this polynomial evaluate to prime number whenever $x$ is a natural number? I am trying to prove or disprove following statment:
$x^2-31x+257$ evaluates to a prime number whenever $x$ is a natural number.
First of all, I realized that we can't factorize this polynomial using its square root like
$$ax^2+bx+c=a... | If $x$ is divisible by $257$ then so is $x^2 - 31 x + 257$.
More generally, if $f(x)$ is any polynomial $f(x)$ is divisible by $f(y)$ whenever $x-y$ is divisible by $f(y)$. So there are no non-constant polynomials that produce primes for all positive integers.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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how evaluate $\int_{-1}^{1} z^{\frac{1}{2}}\, dz$? How can evaluate $\int_{-1}^{1} z^{\frac{1}{2}}\, dz$ with the main branch of $z^{\frac{1}{2}}$?
Thanks for your help
| This is an expansion on anon's comment above.
Caveat: I'm not 100% certain what the "main branch" is supposed to do to the negative real axis, but I am going to assume it maps to the positive imaginary axis.
To integrate from $0$ to $1$, that's no problem, that's an old-school integral of a real-valued function on the... | {
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When $X_sWhen I have shown, for $s\le t$ and for two continuous stochastic process an inequality:
$$ X_s \le Y_t$$ P-a.s.
How can I deduce that this P-a.s. simultaneously for all rational $s\le t$ ?
Thank you for your help
EDIT: According to Ilya's answer, I see that we have $$P(X_s\le Y_t\text{ simultaneously for all ... | This follows from the fact that the complement of the event $[\forall s\leqslant t,\,X_s\leqslant Y_t]$ is the event
$$
\left[\exists s\leqslant t,\,X_s\gt Y_t\right]\ =\left[\exists n\in\mathbb N,\,\exists s\in\mathbb Q,\,\exists t\in\mathbb Q,\,s\leqslant t,\,X_s\geqslant Y_t+\frac1n\right],
$$
hence
$$
\left[\exists... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove this inequality about $e$?
Possible Duplicate:
Proving $(1 + 1/n)^{n+1} \gt e$
How to prove this:
$$
\left(\frac{x}{x-1}\right)^x \geq e
\qquad\text{for}\qquad
x \in \mathbb{N}^*
$$
$e$ is the base of the natural logarithm.
and I think the equal satisfies if $x$ converges to infinity.
Thank you!
| First off, $\frac{x}{x-1} > 0$ iff $x < 0$ or $x > 1$,
so we can't take the natural logarithm if $x\in[0,1]$.
My answer addresses the inequality for real-valued $x$, as in the original post
(proving it for $x > 1$ and disproving it for $x < 0$).
Now
$$
e\le
\left(\frac{x}{x-1}\right)^{x}=
\left(1-\frac1x\right)^{-x}
\t... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is the function "signomial"? Function $f:(0, \infty)\longrightarrow \mathbb{R}$ is called $\textbf{signomial}$, if
$$
f(x)=a_0x^{r_0}+a_1x^{r_1}+\ldots+a_kx^{r_k},
$$
where $k \in \mathbb{N}^*:=\{0,1,2, \ldots\}$, and $a_i, r_i \in \mathbb{R}$, $a_i\neq 0$, $r_0<r_1<\ldots<r_k$, and $x$ is a real variable with $x>0$.
... | This is a non-rigorous derivation of an expansion of the function in inverse powers of $p$. I asked a question here about a rigorous justification for it. It turns out that a) the expansion was known, b) it can be rigorously justified and c) it appears to be only an asymptotic expansion, not a convergent series. Howeve... | {
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Prove that [ContextFreeLanguage - RegularLanguage] is always a context free language, but the opposite is false Let L be a context-free grammar and R a regular language. Show that L-R is always context-free, but R-L is not. Hint: try to connect both automata)
The above hint did not help me :(
| Hints: express $R-L$ more basically in set-theoretic terms. Notice anything about what you get in terms of things you know about CFLs? Try some very simple $R$ (always a good tactic, at least to start).
| {
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"timestamp": "2023-03-29T00:00:00",
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eigenvector computation Given a full-rank matrix $X$, and assume that the eigen-decomposition of $X$ is known as $X=V \cdot D \cdot V^{-1}$, where $D$ is a diagonal matrix.
Now let $C$ be a full-rank diagonal matrix, now I want to calucate the eigen-decomposition of $C \cdot X$, that is to find a matrix $V_c$ and a di... | There is no simple relation between the eigen-decompositions of $C$, $X$ and $C X$. In fact, $C X$ does not even have to be diagonalizable. About all you can say is that $\text{det}(CX) = \det(C) \det(X)$, so the product of the eigenvalues for $CX$ (counted by algebraic multiplicity) is the product for $C$ times the ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why $H\neq N_G(H)$? Let $K$ be a field, $f(x)$ a separable irreducible polynomial in $K[x]$. Let $E$ be the splitting field of $f(x)$ over $K$. Let $\alpha,\beta$ be distinct roots of $f(x)$. Suppose $K(\alpha)=K(\beta)$. Call $G=\mathrm{Gal}(E/K)$ and $H=\mathrm{Gal}(E/K(\alpha))$. How can I prove that $H\neq N_G(H)$?... | Since every $\tau\in H$ by definition fixes $\alpha$, one has $\sigma\tau\sigma^{-1}(\beta)=\sigma\tau(\alpha)=\sigma(\alpha)=\beta$, so $\sigma\tau\sigma^{-1}$ fixes $\beta$ and therefore $K(\beta)=K(\alpha)$, whence $\sigma\tau\sigma^{-1}\in H$. So indeed $\sigma\in N_G(H)\setminus H$. This was too easy.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is there any good example about Lie algebra homomorphisms? My textbook gave an example of the trace, but I think to get a better comprehension, more examples are still needed.
Any example will be helpful ~
| A good source of examples is the free Lie algebra $\mathcal L(n)$, on generators $X_1,\ldots, X_n$. This is defined as the vector space with basis given by all formal bracketing expressions of generators, such as $[X_1,X_2]$, $[X_3+2X_4,[X_5,X_7]]$, etc. One the takes the quotient by relators representing antisymmetry ... | {
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Question on y axis of Gamma probability distribution Afternoon.
I'm looking into using the Gamma (Erlang) distribution for a certain quantity that I need to model. I noticed by plugging in some values for the distribution parameters that the y axis values which represent the probability that a random value from the x a... | A probability density function can easily be greater than $1$ in some interval, as long as the total area under the curve is $1$.
As a simple example, suppose that the random variable $X$ has uniform distribution on the interval $[0,1/4]$. Then the appropriate density function is $f_X(x)=4$ on $[0,1/4]$, and $0$ elsewh... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Every embedded hypersurface is locally a regular surface? I'm reading do Carmo's Riemannian Geometry, in ex6.11 d) he wrote that"every embedded hypersurface $M^{n} \subset \bar{M}^{n+1}$ is locally the inverse image of a regular value". Could anyone comment on how to show this?
To be more specific, let $\bar{M}^{n+1}$ ... | By choosing good local coordinates, you can assume that $M = \mathbb{R}^n\subset\mathbb{R}^{n+1} = \overline{M}$. Specifically, assume that $M = \{x\in \mathbb{R}^{n+1} : x_{n+1} = 0\}$. Then $M = f^{-1}(0)$, where $f\colon \mathbb{R}^{n+1}\to \mathbb{R}$ is the map $f(x) = x_{n+1}$. Since $0$ is a regular value for $f... | {
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} |
How to sketch $y=2\cos\,2x+3\sin\,2x$ , $x$ for $[-\pi,\pi]$.
Use addition of ordinate to sketch the graph of $y=2\cos\,2x+3\sin\,2x$ , $x$ for $[-\pi,\pi]$.
I know that there will be three line in graph from the example it show that
$x=0$, $x=\frac{\pi}{4}$, $x=\frac{\pi}{2}$ and something like that I haven't no c... | You probably know the graph of $y=\cos(\theta)$ and of $y=\sin(\theta)$ on $[-2\pi,2\pi]$.
The graph of $y=\cos(2\theta)$ on $[-\pi,\pi]$ is obtained from the graph of $y=\cos(\theta)$ on $[-2\pi,2\pi]$ by performing a horizontal compression by a factor of $2$ (we are making the change from $y=f(x)$ to $y=f(2x)$).
Lik... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/142862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
$\lim_{x \to 0}(f+g)$ if $\lim_{x \to 0}g$ does not exist Let $f$ such that $\lim_{x\to 0}f(x)=\infty$ and let $g(x)=\sin(\frac{1}{x})$. I know that $g$ does not have a limit at $x=0$, but what about $\lim_{x\rightarrow 0}(f(x)+g(x))$?
Thanks!
| Always limit is infinity considering your problem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/142978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Can the product of non-zero ideals in a unital ring be zero? Let $R$ be a ring with unity and $0\neq I,J\lhd R.$ Can it be that $IJ=0?$
It is possible in rings without unity. Let $A$ be a nontrivial abelian group made a ring by defining a zero multiplication on it. Then any subgroup of $S$ of $A$ is an ideal, because f... | Take $n=ab \in \mathbb Z$ and the ideals $(a)$ and $(b)$ in $\mathbb Z/(n)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/143036",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Coproduct in the category of (noncommutative) associative algebras For the case of commutative algebras, I know that the coproduct is given by the tensor product, but how is the situation in the general case? (for associative, but not necessarily commutative algebras over a ring $A$). Does the coproduct even exist in g... | The following is a link to an article which provides a partial answer, namely it gives (on page 8, without proof) the coproduct of two non-commutative algebras (over a field rather than a ring, I don't know the ring case) http://www.google.co.uk/url?q=http://citeseerx.ist.psu.edu/viewdoc/download%3Fdoi%3D10.1.1.6.6129%... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/143098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 4,
"answer_id": 2
} |
What is the method to compute $\binom{n}{r}$ in a recursive manner? How do you solve this?
Find out which recurrence relation involving $\dbinom{n}{r}$ is valid, and thus prove that we can compute $\dbinom{n}{r}$ in a recursive manner.
I appreciate any help. Thank You
| There are many recurrence relations for $\dbinom{n}{r}$. One of the most commonly used one is the following. $$\binom{n+1}{r} = \binom{n}{r} + \binom{n}{r-1}.$$ There are many ways to prove this and one simple way is to look at $\displaystyle (1+x)^{n+1}$. We know that $$(1+x)^{n+1} = (1+x)^{n} (1+x).$$ Now compare the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/143150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Derivative of a random variable w.r.t. a deterministic variable I'm reading about time series and I thought of this procedure: can you differentiate a function containing a random variable.
For example:
$f(t) = a t + b + \epsilon$
where $\epsilon \sim N(0,1)$. Then:
$df/dt = \lim\limits_{\delta t \to 0} {(f(t + \delta ... | A random variable is a function from sample space to the real line. Hence $f(t)$ really stands for $f(t,\omega) = a t + b + \epsilon(\omega)$. This function can be differentiated with respect to $t$, for fixed $\omega$, of course. The resulting derivative, being a function of $\omega$, is a random variable. In this cas... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/143186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
How can one solve the equation $\sqrt{x\sqrt{x} - x} = 1-x$? $$\sqrt{x\sqrt{x} - x} = 1-x$$
I know the solution but have no idea how to solve it analytically.
| Just writing out Robert's manipulation:
$$\eqalign{
& \sqrt {x\sqrt x - x} = 1 - x \cr
& x\sqrt x - x = {\left( {1 - x} \right)^2} \cr
& x\left( {\sqrt x - 1} \right) = {\left( {1 - x} \right)^2} \cr
& \sqrt x - 1 = \frac{{{{\left( {1 - x} \right)}^2}}}{x} \cr
& \sqrt x = \frac{{{{\left( {1 - x}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/143248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 2
} |
$f$ continuous iff $\operatorname{graph}(f)$ is compact The Problem: Let $(E,\tau_E)$ be a compact space and $(F,\tau_F)$ be a Hausdorff space. Show that a function $f:E\rightarrow F$ is continuous if and only if its graph is compact.
My Work: First assume $(E,\tau_E)$ compact and $(F,\tau_F)$ a Hausdorff space . Assu... | $\textbf{Attention Mathstudent:}$ I think you need to assume that $E$ is Hausdorff.
Here are some preliminary thoughts on your problem . I think we are ready to prove the other direction. Recall that if you can show that given any closed set $B$ in $Y$, the preimage under $f$ is also closed then you have proven that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/143306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
} |
How does composition affect eigendecomposition? What relationship is there between the eigenvalues and vectors of linear operator $T$ and the composition $A T$ or $T A$? I'm also interested in analogous results for SVD.
| Friedland has proved the following over the complex field:
If the principal minors of $A$ are not zero, then for every set of $n$ numbers $\lambda_1,\dots,\lambda_n$ there exist a diagonal matrix $B$ such that $BA$ has $\lambda_i$'s as eigenvalues.
Later Dias da Silva extended it to any arbitrary algebraically closed ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/143362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
how to evaluate this integral by considering $\oint_{C_{(R)}} \frac{1}{z^{2}+1}$ Consider the integral $I=\int_{-\infty}^{\infty} \frac{1}{x^{2}+1}\, dx$. Show how to evaluate this integral by considering $\oint_{C_{(R)}} \frac{1}{z^{2}+1}, dz$ where $C_{R}$ is the closed semicircle in the upper half plane with endpoin... | See @anon's answer.
For completion's sake we will examine the function $f(z) = \frac{1}{z^2 + 1}$ parametrized by $z = z_0 + Re^{i \theta}$. If we center the contour at $z_0 = 0$ then the expansion $z^2 = R^2 e^{2it}$ so $f(z) = \frac{1}{R^2 e^{2it} + 1}$.
Given the line integral:
$$\oint_{C}{f(z)\ \mathrm{d}z} = \in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/143472",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
$N$ submodule of $M$ and $N \cong M$ does not necessarily imply that $M=N$ Let $M, N$ be $A$-modules with $A$ being a commutative ring. Suppose that
$N$ is a submodule of $M$ and also that $N$ is isomorphic to $M$. According
to my understanding this does not necessarily imply that $M=N$. Is this statement
accurate? If... | To answer the half about "When can we expect this?":
A module is called cohopfian if every injective endomorphism is surjective. A cohopfian module $M$ will not have any proper submodules isomorphic to $M$.
$M$ will be cohopfian if it is any of the following:
*
*finite
*Artinian
*Noetherian and injective
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/143523",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
There exists a unique function $u\in C^0[-a,a]$ which satisfies this property The problem:
Let $a>0$ and let $g\in C^0([-a,a])$. Prove that there exists a unique function $u\in C^0([-a,a])$ such that $$u(x)=\frac x2u\left(\frac x2\right)+g(x),$$ for all $x\in[-a,a]$.
My attempt At first sight I thought to approach t... | Your approach will work if $a<2$, in the general case, write
$$u(x)=\frac x2u\left(\frac x2\right)+g(x)$$ as
$$u(x)=x^k\alpha_ku\left(\frac x{2^k}\right)+F_k(g)(x)$$
where $F_k$ is a functional of $g$ and $\alpha_k$ satisfy the recurrence relation
$\alpha_{k+1}=\frac{\alpha_k}{2^{k+1}}$. Therefore, $a^{k+1}\alpha_k$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/143587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
Is a closed subset of isolated points in a compact set necessarily finite? If I have a compact set $A$ and a closed subset $\Sigma \subset A$ which only contains isolated points (that is, none of them is a limit point). Does the compactness of $A$ then force $\Sigma$ to have finite cardinality ?
Here is my attempt at a... | Compactness does not mean that sequences have convergent subsequences in a general topological space. See sequential compactness at Wikipedia.
However, your statement is still true. Just stick more closely to the definition of compactness.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/143647",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
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