Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Prove that the eigenvalues of a real symmetric matrix are real I am having a difficult time with the following question. Any help will be much appreciated.
Let $A$ be an $n×n$ real matrix such that $A^T = A$. We call such matrices “symmetric.” Prove that the eigenvalues of a real symmetric matrix are real (i.e. if $\l... | Consider the real operator $$u := (x \mapsto Ax)$$ for all $x \in \mathbb{R}^{n}$ and the complex operator $$\tilde{u} := (x \mapsto Ax) $$ for all $x \in \mathbb{C}^{n}$. Both operators have the same characteristic polynomial, say $p(\lambda) = \det(A - \lambda I)$. Since $A$ is symmetric, $\tilde{u}$ is an hermitian... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/354115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "45",
"answer_count": 9,
"answer_id": 8
} |
Help in proving that $\nabla\cdot (r^n \hat r)=(n+2)r^{n-1}$
Show that$$\nabla \cdot (r^n \hat r)=(n+2)r^{n-1}$$ where $\hat r$ is the unit vector along $\bar r$.
Please give me some hint. I am clueless as of now.
| You can also use Cartesian coordinates and using the fact that $r \hat{r} = \vec{r} = (x,y,z)$.
\begin{align}
r^n \hat{r} &= r^{n-1} (x,y,z) \\
\nabla \cdot r^n \hat{r} & = \partial_x(r^{n-1}x) + \partial_y(r^{n-1}y) + \partial_z(r^{n-1}z)
\end{align}
Each term can be calculated:
$\partial_x(r^{n-1}x) = r^{n-1} + x (n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/354179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Is there a continuous function with these properties (not piecewise) In short, I'm wondering if I can find a continuous non-piecewise function with these properties. I've found one that was close but not perfect. It's actually really useful to have something like this to scale it. Sorry about the lack of formatting (ED... | After playing around with various compositions and games, here's an analytic one:
$$f(x) = \frac 2 \pi \frac {e^x} {e^x + 1} x \arctan x + e^{2x - (2 + \frac 1 \pi)x^2}$$
The first part gives you the limits at $\pm \infty$ and the second part makes adjustments at $0$.
So what was this good for anyway? :)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/354245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Condition for $\det(A^{T}A)=0$ Is it always true that
$\det(A^{T}A)=0$, $\hspace{0.5mm}$ for $A=n \times m$ matrix with $n<m$?
From some notes I am reading on Regression analysis, and from some trials, it would appear this is true.
It is not a result I have seen, surprisingly.
Can anyone provide a proof?
Thanks.
| From the way you wrote it, the product is size $m.$ However, the maximum rank is $n$ which is smaller. The matrix $A^T A$ being square and of non-maximal rank, it has determinant $0.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/354323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Proof that the sum of the cubes of any three consecutive positive integers is divisible by three. So this question has less to do about the proof itself and more to do about whether my chosen method of proof is evidence enough. It can actually be shown by the Principle of Mathematical Induction that the sum of the cube... | Your solution is fine, provided you intended to prove that the sum is divisible by $3$.
If you intended to prove divisibility by $9$, then you've got more work to do!
If you're familiar with working $\pmod 3$, note @Math Gems comment/answer/alternative. (Though to be honest, I would have proceeded as did you, totally... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/354384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 8,
"answer_id": 0
} |
If $J$ is the $n×n$ matrix of all ones, and $A = (l−b)I +bJ$, then $\det(A) = (l − b)^{n−1}(l + (n − 1)b)$ I am stuck on how to prove this by induction.
Let $J$ be the $n×n$ matrix of all ones, and let $A = (l−b)I +bJ$. Show that $$\det(A) = (l − b)^{n−1}(l + (n − 1)b).$$
I have shown that it holds for $n=2$, and I'... | I think that it would be better to use $J_n$ for the $n \times n$ matrix of all ones, (and similarly $A_n, I_n$) so it is clear what the dimensions of the matrices are.
Proof by induction on $n$ that $\det(A_n)=(l-b)^n+nb(l-b)^{n-1}$:
When $n=1, 2$, this is easy to verify. We have $\det(A_1)=\det(l)=l=(l-b)^1+b(l-b)^0$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/354453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Units (invertibles) of polynomial rings over a field
If $R$ is a field, what are the units of $R[X]$?
My attempt: Let $f,g \in R[X]$ and $f(X)g(X)=1$. Then the only solution for the equation is both $f,g \in {R}$. So $U(R[X])=R$, exclude zero elements of $R$.
Is this correct ?
| If $f=a_0+a_1x+\cdots+a_mx^m$ has degree $m$, i.e. $a_m\ne 0$, and $fg=1$ for some $g=b_0+\cdots+b_n x^n$ (and $b_n\ne 0$), then observe that $$0=\deg(1)=\deg(fg)=\deg(a_0b_0+\cdots+a_mb_nx^{n+m})=m+n$$ as $a_mb_n\ne 0$. Hence $m+n=0$ and so $m=n=0$ as $m,n\ge 0$. Hence $f\in R^*$ and $R[x]^*=R^*$ (the star denotes the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/354505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 1,
"answer_id": 0
} |
Euler's proof for the infinitude of the primes I am trying to recast the proof of Euler for the infinitude of the primes in modern mathematical language, but am not sure how it is to be done. The statement is that:
$$\prod_{p\in P} \frac{1}{1-1/p}=\prod_{p\in P} \sum_{k\geq 0} \frac{1}{p^k}=\sum_n\frac{1}{n}$$
Here $P$... | It might be instructive to see the process of moving from a heuristic argument to a rigorous proof.
Probably the simplest thing to do when considering a heuristic argument involving infinite sums (or infinite products or improper integrals or other such things) is to consider its finite truncations. i.e. what can we do... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/354645",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
If a group has 14 men and 11 women, how many different teams can be made with $6$ people that contains exactly $4$ women?
A group has $14$ men and $11$ women.
(a) How many different teams can be made with $7$ people?
(b) How many different teams can be made with $6$ people that contains exactly $4$ women?
Answer ... | Hint:
For the first one, number of ways you can choose $k$ woman and $m$ men= $11\choose k$+ $14\choose m$, such that $k+m=7$.
For the second one, number ways you can choose $4$ women= $14\choose4$
AND number of choosing $2$ men= $11 \choose 2$, when you have an AND, you gotta multiply them.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/354735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Another Series $\sum\limits_{k=2}^\infty \frac{\log(k)}{k}\sin(2k \mu \pi)$ I ran across an interesting series in a paper written by J.W.L. Glaisher. Glaisher mentions that it is a known formula but does not indicate how it can be derived.
I think it is difficult.
$$\sum_{k=2}^\infty \frac{\log(k)}{k}\sin(2k \mu \pi) =... | It suffices to do these integrals:
$$
\begin{align}
\int_0^1 \log(\Gamma(s))\;ds &= \frac{\log(2\pi)}{2}
\tag{1a}\\
\int_0^1 \log(\Gamma(s))\;\cos(2k \pi s)\;ds &= \frac{1}{4k},\qquad k \ge 1
\tag{1b}\\
\int_0^1 \log(\Gamma(s))\;\sin(2k \pi s)\;ds &= \frac{\gamma+\log(2k\pi)}{2k\pi},\qquad k \ge 1
\tag{1c}
\\
\int_0^1 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/354791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 2,
"answer_id": 0
} |
Function design: a logarithm asymptotic to one? I want to design an increasing monotonic function asymptotic to $1$ when $x\to +\infty $ that uses a logarithm.
Also, the function should have "similar properties" to $\dfrac{x}{1+x}$, i.e.:
*
*increasing monotonic
*$f(x)>0$ when $x>0$
*gets close to 1 "quickly",
... | How about
$$
f(x)=\frac{a\log(1+x)}{1+a\log(1+x)},\quad a>0?
$$
Notes:
*
*I have used $\log(1+x)$ instead of $\log x$ to avoid issues near $x=0$ and to make it more similar to $x/(1+x)$.
*Choose $a$ large enough to have $f(10)>0,8$.
*You can see the graph of $f$ (for $a=1$) compared to $x/(1+x)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/354852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If $z$ is a complex number of unit modulus and argument theta If $z$ is a complex number such that $|z|=1$ and $\text{arg} z=\theta$, then what is $$\text{arg}\frac{1 + z}{1+ \overline{z}}?$$
| Multiplying both numerator and denominator by $z$, we get:
$$\arg\left(\frac{1+z}{1+\bar{z}}\right)=\arg\left(\frac{z+z^{2}}{z+1}\right)=\arg\left(\frac{z(1+z)}{1+z}\right)=\arg\left(z\right)$$
We are told that $\arg(z)=\theta$, therefore:
$$\arg\left(\frac{1+z}{1+\bar{z}}\right)=\theta$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/354922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
Is the gradient of a function in $H^2_0$ in $H^1_0$? Suppose we have $f\in H^2_0(U)$, so $f$ is the limit of some sequence $(g_n)$ of smooth compactly supported functions on $U\in\mathbb{R}^n$ (assume bounded & smooth boundary) and $f$ is in the Sobolev space $H^2(U)$. Does this imply that $\frac{\partial f}{\partial x... | If $g_n$ converges towards $f$ in $H^2(U)$, you have
$$ \sum_{|\alpha|\le 2}\left\| \partial^\alpha f - \partial^\alpha g_n \right\|^2_{L^2(U)} \to 0,$$
where the sum ranges over all multiindices $\alpha$, with $|\alpha| \le 2$.
In order to prove $\partial g_n / \partial x_i \to \partial f / \partial x_i$ in $H^1(U)$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/354987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Let $A$ be any uncountable set, and let $B$ be a countable subset of $A$. Prove that the cardinality of $A = A - B $ I am going over my professors answer to the following problem and to be honest I am quite confused :/ Help would be greatly appreciated!
Let $A$ be any uncountable set, and let $B$ be a countable subset... | $|A-B|\leq |A|$ is obvious.
For the reverse inequality,
Use $|A|=|A\times A|$.
Denote the bijection $A\rightarrow A\times A$ by $\phi$, then $\phi(B)$ embeds into a countable subset of $A\times A$.
Then consider the projection $\pi_1$ onto the first coordinate, of the set $\phi(B)$.
Namely $\pi_1 \phi(B)$.
Since $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/355049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
} |
$X(n)$ and $Y(n)$ divergent doesn't imply $X(n)+Y(n)$ divergent. Please, give me an example where $X(n)$ and $Y(n)$ are both divergent series, but $(X(n) + Y(n))$ converges.
| Try $x_n = n, y_n = -n$. Then both $x_n,y_n$ clearly diverge, but $x_n+y_n = 0$ clearly converges.
Or try $x_n = n, y_n = \frac{1}{n}-n$ if you want something less trivial. Again both $x_n,y_n$ diverge, but $x_n+y_n = \frac{1}{n}$ converges.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/355082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Alternate proof for $2^{2^n}+1$ ends with 7, n>1. I have a proof by induction that $2^{2^n}+1$ end with 7. I've been trying to prove that within the theory of rings and ideals, but haven't achieved it yet. The statement is equivalent to $2^{2^n}-6$ ends with zero, so
Prove that for $$ e \in \mathbb{N} : e=2^n \Rightar... | Here is an alternate proof, it is basically the same idea as TA or Marvis, but probably presented in a "elementary" way.
For $n \geq 2$ then
$$2^{2^n}-16=16^{2^{n-2}}-16=16[16^{\alpha}-1]=16(16-1)(\mbox{junk})$$
Since 16 is even and 15 is divisible by 5, it follows that $2^{2^n}-16$ is a multiple of 10....
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/355148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Order of infinite dimension norms I know that
$$\|{f}\|_{L^1(0,L)}\leq\|{f}\|_{L^2(0,L)}\leq\|{f}\|_{\mathscr{C}^1(0,L)}\leq\|{f}\|_{\mathscr{C}^2(0,L)}\leq\|{f}\|_{\mathscr{C}^{\infty}(0,L)}$$
But I don't know where to put in this chain this norm:
$\|{f}\|_{L^{\infty}}=\inf\{C\in\mathbb{R},\left|f(x)\right|\leq{C}\tex... | If the norm $\mathcal C^1$ is defined by $\sup_{0<x<L}|f(x)|+\sup_{0<x<L}|f'(x)|$, then we have
$$\lVert f\rVert_{L^2}\leqslant \sqrt L\lVert f\rVert_{L^{\infty}}\leqslant \sqrt L\lVert f\rVert_{\mathcal C^1}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/355206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
indexed family of sets, not pairwise disjoint but whole family is disjoint I've seen this problem before, but can't remember how to finish it:
Define an indexed family of sets $ \{A_i : i \in \mathbb{N} \}$ in which for any $m,n\in \mathbb{N}, A_m \cap A_n \not= \emptyset$ and $\bigcap A_i = \emptyset$. The closest I c... | How about $A_i =\{n\in\mathbb N\mid n>i\}$? But your choice, more properly written $A_i=(0,1/i)$, is also fine.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/355265",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Given a dense subset of $\mathbb{R}^n$, can we find a line that intersects it in a dense set? I have some difficulties in the following question.
Let $S$ be a dense subset in $\mathbb{R}^n$. Can we find a straight line $L\subset\mathbb{R}^n$ such that $S\cap L$ is a dense subset of $L$.
Note. From the couterexample o... | For each $n>1$ it’s possible to construct a dense subset $D$ of $\Bbb R^n$ such that every straight line in $\Bbb R^n$ intersects $D$ in at most two points.
Let $\mathscr{B}=\{B_n:n\in\omega\}$ be a countable base for $\Bbb R^n$. Construct a set $D=\{x_n:n\in\omega\}\subseteq\Bbb R^n$ recursively as follows. Given $n\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/355339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Defining a metric space I'm studying for actuarial exams, but I always pick up mathematics books because I like to challenge myself and try to learn new branches. Recently I've bought Topology by D. Kahn and am finding it difficult. Here is a problem that I think I'm am answering sufficiently but any help would be gr... | There is someing wrong in 4), just as Brian comments. Here I offered a proof for you:
Proof: Notice that $f(x)=\frac{x} {1+x}$ is increasing on $\mathbb R^+$: to see this, let $g(x)=\frac{1}{x+1}$. It is easily to see that $g(x)$ is descreasing on $\mathbb R^+$. And note that $f(x)+g(x)=1$. Therefore, $f(x)$ is a incr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/355493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
How to read this expression? How can I read this expression :
$$\frac{1}{4} \le a \lt b \le 1$$
Means $a,b$ lies between $\displaystyle \frac{1}{4}$ and $1$?
Or is $a$ less the $b$ also less than equal to $1$?
So $a+b$ won't be greater than $1$?
| Think of the number line. The numbers $\{\tfrac{1}{4}, a, b, 1\}$ are arranged from left to right. The weak inequalities on either end indicate that $a$ could be $\tfrac{1}{4}$ and $b$ could be $1$. However, the strict inequality in the middle indicates that $a$ never equals $b$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/355584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
What does it mean to have a determinant equal to zero? After looking in my book for a couple of hours, I'm still confused about what it means for a $(n\times n)$-matrix $A$ to have a determinant equal to zero, $\det(A)=0$.
I hope someone can explain this to me in plain English.
| Take a 2 x 2 matrix, call it A, plot that in a coordinate system.
A= [[2,1],[4,2]] . --> Numpy notation of a matrix
Following two vectors are written from A
x=[2,4]
y=[1,2]
If you plot that, you can see that they are in the same span. That means x and y vectors do not form an area. Hence, the det(A) is zero. Det refer... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/355644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "117",
"answer_count": 11,
"answer_id": 5
} |
How can I calculate the expected number of changes of state of a discrete-time Markov chain? Assume we have a 2 state Markov chain with the transition matrix:
$$
\left[
\begin{array}
(p & 1-p\\
1-q & q
\end{array}
\right]
$$
and we assume that the first state is the starting state. What is the expected number of state ... | 1. Let $s=(2-p-q)^{-1}$, then $\pi_0=(1-q)s$, $\pi_1=(1-p)s$, defines a stationary distribution $\pi$. If the initial distribution is $\pi$, at each step the distribution is $\pi$ hence the probability that a jump occurs is
$$
r=(1-p)\pi_0+(1-q)\pi_1=2(1-p)(1-q)s.
$$
In particular, the mean number of jumps during $T$ p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/355829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Proving that $L = \{0^k \mid \text{$k$ is composite}\}$ is not regular by pumping lemma
Suppose $L = \{0^k \mid \text{$k$ is composite}\}$. Prove that this language is not regular.
What bugs me in this lemma is that when I choose a string in $L$ and try to consider all cases of dividing it into three parts so that in... | You can’t assume that the pumping constant is even. If you want to start with a word of the form $0^{2k}$ for some $k$, that’s fine, but you can’t take $2k$ to be the pumping constant $p$; you can only assume that $2k\ge p$. But trying to use the pumping lemma directly to prove that $L$ is not regular is going to be a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/355900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Let $a$ be a prime element in PID. Show that $R/(a)$ is a field. Let $a$ be a prime element in PID. Show that $R/(a)$ is a field.
My attempt: Since $a$ is prime, $(a)$ is a prime ideal of $R$. Since $R$ is a PID, every nonzero prime ideal of $R$ is maximal. This implies that $(a)$ is maximal and hence $R/(a)$ is a fie... | Almost. But, you should just snip out ", $R$ is also a UFD and hence". The fact that every non-zero prime is maximal is a fact true about PIDs but NOT about UFDs. Indeed, consider that $\mathbb{Z}[x]/(x)$ is an integral domain but not a field.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/355968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Additive set function properties I am reading an introduction to measure theory, which starts by defining $\sigma$-rings then additive set functions and their properties which are given without proof. I was able to prove two of them which are very easy $\phi(\emptyset)=0$ and $\phi(A_1\cup \ldots\cup A_n)=\displaystyle... | I was able to use the third proposition to prove the first.
Let $A \subset B$. We have $(A \setminus B) \cap B = \emptyset$ then since $\phi$ is additive we have $\phi(A \setminus B) + \phi(B)=\phi((A-B) \cup B)=\phi(A) $ . Then $\phi(A \setminus B)=\phi(A)-\phi(B)$, this proves the second one.
Now for the first : $A \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/356058",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Logarithm calculation result I am carrying out a review of a network protocol, and the author has provided a function to calculate the average steps a message needs to take to traverse a network.
It is written as $$\log_{2^b}(N)$$
Does the positioning of the $2^b$ pose any significance during calculation? I can't find... | The base of the logarithm is $2^b$. You want to find an $x$ such that $(2^b)^x = N$, i.e. $2^{bx} = N$.
You can rewrite that as $$x = \dfrac{\log N}{b}$$ if you take the $\log$ to base-2.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/356125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
An ambulance problem involve sum of two independent uniform random variables An ambulance travels back and forth at a constant speed along a road of length $L$. At a certain moment of time, an accident occurs at a point uniformly distributed on the road.[That is, the distance of the point from one of the fixed ends of ... | Here is a rough sketch of the integration region:
The $x$ and $y$ axes goes between $0$ and $L$. The "shaded" (for lack of a better word) region represents those $X$ and $Y$ such that $|X-Y| \le d$.
The integration region is split in 3 pieces, which I hope you can see from this admittedly crude diagram:
$$P(|X-Y| \le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/356287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Principal axis of a matrix I try to find the definition of the main axis of a matrix.
I saw this phrase in some exercise:
Let $A$ be a positive matrix, $f:G\longrightarrow \mathbb{R}$ a smooth function, $G$ an open set in $\mathbb{R}^n$. I need to find the orthogonal coordinate transformation $y=Px$ such that the main... | Often, principal axes of a matrix refer to its eigenvectors. With this diagonalization, $P$ is the matrix of eigenvectors.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/356361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Examples of 2D wave equations with analytic solutions I need to numerically solve the following wave equation$$\nabla^2\psi(\vec{r},t) - \frac{1}{c(\vec{r})^2}\frac{\partial^2}{\partial t^2}\psi(\vec{r},t) = -s(\vec{r},t)$$ subject to zero initial conditions $$\psi(\vec{r},0)=0, \quad \left.\frac{\partial}{\partial t}\... | Solution using Green functions and using Sommerfeld radiation condition, in cylindrical coordinates.
\begin{eqnarray}
u_s(\rho, \phi, t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \{ -i \pi H_0^{(2)}\left(k | \sigma - \sigma_s| \right) F(\omega) e^{i\omega t} d\omega\}
\end{eqnarray}
Where $s$ denotes the source positio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/356447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Find the area of the surface obtained by revolving $\sqrt{1-x^2}$ about the x-axis? So I began by choosing my formulas:
Since I know the curve is being rotated around the x-axis I choose $2\pi\int yds$
where $y=f(x)=\sqrt{1-x^2}$
$ds=\sqrt{1+[f'(x)]^2}$
When I compute ds, I find that $ds=\sqrt{x^6-2x^4+x^2+1}$
Therefor... | No.
$$f'(x) = -\frac{x}{\sqrt{1-x^2}} \implies 1+f'(x)^2 = \frac{1}{1-x^2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/356511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Does Seperable + First Countable + Sigma-Locally Finite Basis Imply Second Countable? A topological space is separable if it has a countable dense subset. A space is first countable if it has a countable basis at each point. It is second countable if there is a countable basis for the whole space. A collection of su... | For example:
Helly Space;
Right Half-Open Interval Topology;
Weak Parallel Line Topology.
These space are all separable, first countable and paracompact, but not second countable.
Note that a paracompact is the union of one locall finite collection.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/356595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
Proof of Proposition/Theorem V in Gödel's 1931 paper? Proposition V in Gödel's famous 1931 paper is stated as follows:
For every recursive relation $ R(x_{1},...,x_{n})$ there is an n-ary "predicate" $r$ (with "free variables" $u_1,...,u_n$) such that, for all n-tuples of numbers $(x_1,...,x_n)$, we have:
$$R(x_1,...,x... | I'm not completely familiar with Gödel's notation, but I think this is equivalent to theorem 60 in Chapter 2 of The Logic of Provability by George Boolos, which has fairly detailed proofs of this sort of thing (all in chapter 2).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/356674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
} |
Permutation for finding the smallest positive integer Let $\pi = (1,2)(3,4,5,6,7)(8,9,10,11)(12) \in S_{12}$. Find the smallest positive integer $k$ for which $$\pi^{(k)}=\pi \circ \pi \circ\ldots\circ \pi = \iota$$
Generalize. If a $\pi$'s disjoint cycles have length $n_1, n_2,\dots,n_t$, what is the smallest integer ... | $\iota$ represents the identity permutation: every element in $\{1, 2, ..., 12\}$ is mapped to itself:
$\quad \iota = (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)$.
Recall the definition of the order of any element of a finite group.
In the case of a permutation $\pi$ in $S_n$, the exponent $k$ in your question represents... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/356734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How do i prove that $\det(tI-A)$ is a polynomial? In wikipedia, it's said "$\det(tI-A)$ can be explicitly evaluated using exterior algebra", but i have not learned exterior algebra yet and i just want to know whether it is polynomial, not how it looks like.
How do i prove that $\det(tI-A)$ is a polynomial in $\mathbb{F... | To prove that $\det (tI-A)$ is a polynomial you must know some definition, or some properties, of the determinant. The most straightforward and least mystical approach is to use Laplace's formula: http://en.wikipedia.org/wiki/Laplace_expansion
This would give a rather quick way of proving that $\det (tI-A)$ is a polyn... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/356875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Find minimum in a constrained two-variable inequation I would appreciate if somebody could help me with the following problem:
Q: find minimum
$$9a^2+9b^2+c^2$$
where $a^2+b^2\leq 9, c=\sqrt{9-a^2}\sqrt{9-b^2}-2ab$
| Maybe this comes to your rescue.
Consider $b \ge a \ge 0$
When you expansion of $(\sqrt{9-a^2}\sqrt{9-b^2}-2ab)^2=(9-a^2)(9-b^2)+4a^2b^2-4ab \sqrt{(9-a^2)(9-b^2)}$
This attains minimum when $4ab \sqrt{(9-a^2)(9-b^2)}$ is maximum.
Applying AM-GM :
$\dfrac{9-a^2+9-b^2}{2} \ge \sqrt{(9-a^2)(9-b^2)} \implies 9- \dfrac{9... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/357035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
$f^{-1}(U)$ is regular open set in $X$ for regular open set $U$ in $Y$, whenever $f$ is continuous. Let $f$ be a continuous function from space $X$ to space $Y$. If $U$ is regular open set in $Y$, it it true that $f^{-1}(U)$ is a regular open set in $X$?
| Not necessarily. Consider the absolute value function $x \mapsto | x |$, and the inverse image of $(0,1)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/357098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 0
} |
Quicksort analysis problem This is a problem from a probability textbook, not a CS one, if you are curious. Since I'm too lazy to retype the $\LaTeX$ I will post an ugly stitched screenshot:
This seems ridiculously hard to approach, and it doesn't help that all the difficult problems have no solutions in the textbook... | The question is already broken into pieces in order to help you out.
a) This is the law of total expectation, using the fact that the pivot is chosen randomly.
b) Once we have a pivot, we need to split the remaining $n-1$ numbers into $2$ groups (one comparison each), and then solve the two sub-problems; one of of size... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/357203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How can I prove that $xy\leq x^2+y^2$? How can I prove that $xy\leq x^2+y^2$ for all $x,y\in\mathbb{R}$ ?
| $$x^2+y^2-xy=\frac{(2x-y)^2+3y^2}4=\frac{(2x-y)^2+(\sqrt3y)^2}4$$
Now, the square of any real numbers is $\ge0$
So, $(2x-y)^2+(\sqrt3y)^2\ge0,$ the equality occurs if each $=0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/357272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "38",
"answer_count": 25,
"answer_id": 0
} |
Linear Recurrence Relations I'm having trouble understanding the process of solving simple linear recurrence relation problems. The problem in the book is this:
$$
0=a_{n+1}-1.5a_n,\ n \ge 0
$$
What is the general process, and purpose, of solving this? Unfortunately there is a very large language barrier between my pro... | The general solution to the equation
$$a_{n+1} = k a_n$$
is
$$a_n = B \cdot k^n$$
for some constant $B$, which is related to an initial condition.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/357297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Examples of $\kappa$-Fréchet-Urysohn spaces. We say that
A space $X$ is $\kappa$-Fréchet-Urysohn at a point $x\in X$, if whenever $x\in\overline{U}$, where $U$ is a regular open subset of $X$, some
sequence of points of $U$ converges to $x$.
I'm looking for some examples of $\kappa$-Fréchet-Urysohn space. I guess ... | What examples are you looking for?
I think, for some exotic examples you should search Engelking’s “General Topology” (on Frechet-Urysohn spaces) and part 10 “Generalized metric spaces”
of the “Handbook of Set-Theoretic Topology”.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/357384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
An analytic function is onto All sets are subsets of $\mathbb{C}$. Suppose $f: U \to D$ is analytic where $U$ is bounded and open, and $D$ is the open unit disk.
Now suppose we can continuously extend $f$ to $\bar{f}: \bar{U} \to \bar D$, such that $\bar{f}(\partial U) \subseteq \partial D$. To show that $f$ is onto, ... | Hint: If $f$ is not onto then $D$ containts a point $w$ which is on the boundary of $f(U)$. Take a sequence in $U$ with $f(z_k)\to w$, and use compactness.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/357452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Learning Combinatorial Species. I have been reading the book conceptual mathematics(first edition) and I'm also about halfway through Diestel's Graph theory (4th edition). I was wondering if I was able to start learning about combinatorial species. This is very interesting to me because I love combinatorics and it make... | For an easy to understand introduction,
http://dept.cs.williams.edu/~byorgey/pub/species-pearl.pdf
seems to be nice. But it leans more towards the computer science applications.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/357508",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Distribution of a random variable
$X_1$, $X_2$, $X_3$ are independent random variables, each with an exponential distribution, but with means of $2.0, 5.0, 10.0$ respectively. Let $Y$= the smallest or minimum value of these three random variables. Derive and identify the distribution of $Y$. (The distribution function... | The wiki on exponential distribution has an answer to that. The answer of course is exponential distribution.
http://en.wikipedia.org/wiki/Exponential_distribution#Distribution_of_the_minimum_of_exponential_random_variables
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/357581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Density of sum of two independent uniform random variables on $[0,1]$ I am trying to understand an example from my textbook.
Let's say $Z = X + Y$, where $X$ and $Y$ are independent uniform random variables with range $[0,1]$. Then the PDF
is
$$f(z) = \begin{cases}
z & \text{for $0 < z < 1$} \\
2-z & \text{for $1 \le z... | By the hint of jay-sun, consider this idea, if and only if $f_X (z-y) = 1$ when $0 \le z-y \le 1$. So we get
$$ z-1 \le y \le z $$
however, $z \in [0, 2]$, the range of $y$ may not be in the range of $[0, 1]$ in order to get $f_X (z-y) = 1$, and the value $1$ is a good splitting point. Because $z-1 \in [-1, 1]$.
Cons... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/357672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "86",
"answer_count": 6,
"answer_id": 1
} |
Find an orthogonal vector to 2 vector I have the following problem:
A B C D are the 4 consecutive summit of a parallelogram, and have the following coordinates
A(1,-1,1);B(3,0,2);C(2,3,4);D(0,2,3)
I must find a vector that is orthogonal to both CB and CD.
How? Is there some kind of formula?
Thanks,
| The cross product of two vectors is orthogonal to both, and has magnitude equal to the area of the parallelogram bounded on two sides by those vectors. Thus, if you have:
$$\vec{CB} = \langle3-2, 0-3, 2-4\rangle = \langle1, -3, -2\rangle$$
$$\vec{CD} = \langle0-2, 2-3, 3-4\rangle = \langle-2, -1, -1\rangle$$
Compute t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/357746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Understanding the fundamentals of pattern recognition I'm learning now about sequences and series: patterns in short. This is part of my Calc II class. I'm finding I'm having difficulty in detecting all of the patterns that my text book is asking me to solve. My question at this point isn't directly about a homework... | The suggested expressions weren’t found one at a time: they’re synthesized from the whole pattern. We don’t seriously consider $4-3$ for the first one, for instance, because it doesn’t fit nicely with anything else. I’d describe the thought process this way. First we calculate the first few partial sums:
$$\begin{array... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/357874",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to generate random symmetric positive definite matrices using MATLAB? Could anybody tell me how to generate random symmetric positive definite matrices using MATLAB?
| The algorithm I described in the comments is elaborated below. I will use $\tt{MATLAB}$ notation.
function A = generateSPDmatrix(n)
% Generate a dense n x n symmetric, positive definite matrix
A = rand(n,n); % generate a random n x n matrix
% construct a symmetric matrix using either
A = 0.5*(A+A'); OR
A = A*A';
% Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/357980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "43",
"answer_count": 5,
"answer_id": 1
} |
What is the real life use of hyperbola? The point of this question is to compile a list of applications of hyperbola because a lot of people are unknown to it and asks it frequently.
| Applications of hyperbola
Dulles Airport, designed by Eero Saarinen, has a roof in the
shape of a hyperbolic paraboloid. The hyperbolic paraboloid is a three-dimensional
surface that is a hyperbola in one cross-section, and a parabola in another cross section.
This is a Gear Transmission. Greatest application of a pai... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/358056",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 8,
"answer_id": 2
} |
factor group is cyclic. Prove that a factor group of a cyclic group is cyclic.
I didn't understand last two lines of proof ..
Therefore $gH=(aH)^i$ for any coset $gH$.
so $G/H$ is cyclic , by definition of cyclic groups.
How $gH=(aH)^i$ of any coset $gH$.
proves factor group to be cyclic.
Please explain.
| I just wanted to mention that more generally, if $G$ is generated by $n$ elements, then every factor group of $G$ is generated by at most $n$ elements:
Let $G$ be generated by $\{x_1,\ldots x_n\}$, and let $N$ be a normal subgroup of $G$. Then every coset of $N$ in $G$ can be expressed as a product of the cosets $Nx_1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/358114",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Simplifying $\sum\limits_{k=1}^{n-1} (2k + \log_2(k) - 1)$ I'm trying to simplify the following summation:
$$\sum_{k=1}^{n-1} (2k + \log_2(k) - 1)$$.
I've basically done the following:
$$\sum_{k=1}^{n-1} (2k + \log_2(k) - 1) \\
=
\sum_{k=1}^{n-1} 2k + \sum_{k=1}^{n-1} \log_2(k) - \sum_{k=1}^{n-1} 1\\
=
\frac{n(n-1)}{2... | (Just so people know this has been answered)
You right
$$\sum_{k=1}^{n} \log k = \log (n!)$$
If you want to justify it formally, you can try using induction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/358180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Find $x,y$ such that $x=4y$ and $1$-$9$ occur in $x$ or $y$ exactly once.
$x$ is a $5$-digits number, while $y$ is $4$-digits number. $x=4y$, and they used up all numbers from 1 to 9. Find $x,y$.
Can someone give me some ideas please? Thank you.
| Here are the $x,y$ pairs a quick bit of code found.
15768 3942
17568 4392
23184 5796
31824 7956
No insight to offer at the moment I'm afraid..
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/358286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Whats the percentage of somebody getting homework in class there is a 25% chance you get homework in one class
there is a 40% chance you get homework in another class
what is the probability you get homework in both classes
| it is between 0 and 0.25, based on the information you have given. However, if you assume they are independent. i.e. they are two teachers who do not disccuss whether they should give home work on the same day or not
then P(A and B) = P(A)P(B) = 0.25 * 0.4 = 0.1
A is getting work from the 1st class, B is getting work f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/358354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluating $\lim \limits_{n\to \infty}\,\,\, n\!\! \int\limits_{0}^{\pi/2}\!\! \left(1-\sqrt [n]{\sin x} \right)\,\mathrm dx$
Evaluate the following limit:
$$\lim \limits_{n\to \infty}\,\,\, n\!\! \int\limits_{0}^{\pi/2}\!\! \left(1-\sqrt [n]{\sin x} \right)\,\mathrm dx $$
I have done the problem .
My method:
First... | You can have a close form solution; infact if $Re(1/n)>-1$ you have that the integral collapse in:
$$\int_{0}^{\pi/2}\left[1-(\sin(x))^{1/n}\right]dx=\frac{1}{2} \left(\pi -\frac{2 \sqrt{\pi } n \Gamma
\left(\frac{n+1}{2 n}\right)}{\Gamma
\left(\frac{1}{2 n}\right)}\right)$$
So we define:
$$y(n)=\frac{n}{2} \left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/358405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 6,
"answer_id": 4
} |
dealing with sum of squares (1) I need to be able to conclude that there are $a, b \in \Bbb Z$, not 0, such that $|a| < √p,\ |b| < √p$
and $$a^2 + 2b^2 ≡ 0\ (mod\ p)$$
I'm not sure how to go about this at all. But apparently it is supposed to help me show (2) that
there are $a, b \in \Bbb Z$, such that either $$a^2 +... | Hint $\ $ Apply the following result of Aubry-Thue
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/358462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
integrate $\int_0^{2\pi} e^{\cos \theta} \cos( \sin \theta) d\theta$ How to integrate
$ 1)\displaystyle \int_0^{2\pi} e^{\cos \theta} \cos( \sin \theta) d\theta$
$ 2)\displaystyle \int_0^{2\pi} e^{\cos \theta} \sin ( \sin \theta) d\theta$
| Let $\gamma$ be the unitary circumference positively parametrized going around just once.
Consider $\displaystyle \int _\gamma \frac{e^z}{z}\,dz$.
On the one hand $$\begin{align}
\int _\gamma \frac{e^z}{z}\mathrm dz&=\int \limits_0^{2\pi}\frac{e^{e^{i\theta}}}{e^{i\theta}}ie^{i\theta}\mathrm d\theta\\
&=i\int _0^{2\pi}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/358548",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
} |
How to expand $(a_0+a_1x+a_2x^2+...a_nx^n)^2$? I know you can easily expand $(x+y)^n$ using the binomial expansion. However, is there a simple summation formula for the following expansion?
$$(a_0+a_1x+a_2x^2+...+a_nx^n)^2$$
I found something called the multinomial theorem on wikipedia but I'm not sure if that applies ... | To elaborate on the answer vonbrand gave, perhaps the following will help:
$$(a + b)^2 \;\; = \;\; a^2 + b^2 + 2ab$$
$$(a + b + c)^2 \;\; = \;\; a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$$
$$(a + b + c + d)^2 \;\; = \;\; a^2 + b^2 + c^2 + d^2 + 2ab + 2ac + 2ad + 2bc + 2bd + 2cd$$
In words, the square of a polynomial is the sum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/358604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
How to solve $(1+x)^{y+1}=(1-x)^{y-1}$ for $x$? Suppose $y \in [0,1]$ is some constant, and $x \in [y,1]$. How to solve the following equation for $x$:
$\frac{1+y}{2}\log_2(1+x)+\frac{1-y}{2}\log_2(1-x)=0$ ?
Or equivalently $1+x = (1-x)^{\frac{y-1}{y+1}}$?
Thanks very much.
| If we set $f = \frac{1+x}{1-x}$ and $\eta = \frac{y+1}{2}$ some manipulation yields the following:
$$2f^{\eta} - f - 1 = 0$$
For rational $\eta = \frac{p}{q}$, this can be converted to a polynomial in $f^{\frac{1}{q}}$, and is likely "unsolveable" exactly, and would require numerical methods (like Newton-Raphson etc, w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/358685",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Ways to fill a $n\times n$ square with $1\times 1$ squares and $1\times 2$ rectangles I came up with this question when I'm actually starring at the wall of my dorm hall. I'm not sure if I'm asking it correctly, but that's what I roughly have:
So, how many ways (pattern) that there are to fill a $n\times n:n\in\mathbb... | We can probably give some upper and lower bounds though. Let $t_n$ be the possible ways to tile an $n\times n$ in the manner you described. At each square, we may have $5$ possibilities: either a $1\times 1$ square, or $4$ kinds of $1\times 2$ rectangles going up, right, down, or left. This gives you the upper bound $t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/358755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 4,
"answer_id": 0
} |
Give an example of a simply ordered set without the least upper bound property. In Theorem 27.1 in Topology by Munkres, he states "Let $X$ be a simply ordered set having the least upper bound property. In the order topology, each closed interval in $X$ is compact."
(The LUB property is if a subset is bounded above, th... | According to the strict definitions given by the OP, the null set fails to have a Least Upper Bound while still being simply ordered.
The Least Upper Bound of a set, as defined at the Wikipedia page he links to requires that it be a member of that set. The null set, having no members, clearly lacks a LUB.
However, the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/358814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 7,
"answer_id": 4
} |
The Radical of $SL(n,k)$ For an algebraically closed field $k$, I'd like to show that the algebraic group $G=SL(n,k)$ is semisimple. Since $G$ is connected and nontrivial, this amounts to showing that the radical of $G$, denoted $R(G)$, is trivial. $R(G)$ can be defined as the unique largest normal, solvable, connect... | The fact is that the quotient $\mathrm{PSL}_n(k)$ of $\mathrm{SL}_n(k)$ by its center is simple. Since the center of $\mathrm{SL}_n(k)$ consists, as you say, of the $n$th roots of unity, this shows that there are no nontrivial connected normal subgroups of $\mathrm{SL}_n(k)$.
The fact that the projective special linear... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/358841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
$\mathbb Q/\mathbb Z$ is an infinite group I'm trying to prove that $\mathbb Q/\mathbb Z$ is an infinite abelian group, the easiest part is to prove that this set is an abelian group, I'm trying so hard to prove that this group is infinite without success.
This set is defined to be equivalences classes of $\mathbb Q$,... | Another hint: prove that for
$$n,k\in\Bbb N\;,\;\;n\neq k\;,\;\;\;\frac{1}{n}+\Bbb Z\neq\frac{1}{k}+\Bbb Z$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/358907",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 8,
"answer_id": 4
} |
How can the graph of an equivalence relation be conceptualized? Consider a generic equivalence relation $R$ on a set $S$. By definition, if we partition $S$ using the relation $R$ into $\pi_S$, whose members are the congruence classes $c_1, c_2...$ then
$aRb \text{ iff a and b are members of the same congruence class i... | The domain is $S\times S$ and codomain is {true, false}. Said another way, you can just think of any relation as a subset of $S\times S$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/358986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Dimensions: $\bigcap^{k}_{i=1}V_i \neq \{0\}$ Let $V$ be a vector space of dimension $n$ and let $V_1,V_2,\ldots,V_k \subset V$ be subspaces of
$V$. Assume that
\begin{eqnarray}
\sum^{k}_{i=1} \dim(V_i) > n(k-1).
\end{eqnarray}
To show that $\bigcap^{k}_{i=1}V_i \neq \{0\}$, what must be done? Also, could there be an ... | Hint: take complements. That is, pick vector spaces $W_i$ with $\dim(W_i) + \dim(V_i) = n$ and $W_i \cap V_i = \{0\}$.
EDIT: thanks, Ted
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/359052",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
Which is the function that this sequence of functions converges Prove that $$ \left(\sqrt x, \sqrt{x + \sqrt x}, \sqrt{x + \sqrt {x + \sqrt x}}, \ldots\right)$$ in $[0,\infty)$ is convergent and I should find the limit function as well.
For give a idea, I was plotting the sequence and it's look like
| Note that $f(x)$ is limited by $\sqrt{x}+1$ as can be shown easily by $induction$.
For the limit $y=f(x)$ we have the following :
$\sqrt{x+y}=y$
$x+y=y^2$
$0=y^2-y-x$
$\Delta= 1+4x $
$ So $ , $y=f(x)=\frac{1+\sqrt{1+4x}}{2}$ for $x>0$ and $f(0)=0 $
.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/359145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Are there any other constructions of a finite field with characteristic $p$ except $\Bbb Z_p$? I mean, $\Bbb Z_p$ is an instance of $\Bbb F_p$, I wonder if there are other ways to construct a field with characteristic $p$?
Thanks a lot!
| Just to supplement the other answers: As stated in the other answers, for every prime power $p^r$, $r>0$, there is a unique (up to isomorphism) field with $p^r$ elements. There are also infinite fields of characteristic $p$, for instance if $F$ is any field of characteristic $p$ (e.g., $\mathbb Z_p$), the field $F(t)$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/359212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 0
} |
Can you find function which satisfies $f(ab)=\frac{f(a)}{f(b)}$? Can you find function which satisfies $f(ab)=\frac{f(a)}{f(b)}$? For example $log(x)$ satisfies condition $f(ab)=f(a)+f(b)$ and $x^2$ satisfies $f(ab)=f(a)f(b)$?
| Let us reformulate the question as classify all maps $f : G \rightarrow H$ which need not be group morphism that satisfies the condition $f(ab)=f(a)f(b)^{-1}$
A simple calculation shows that $f(e)=f(x)f(x^{-1})^{-1}= f(x^{-1})f(x)^{-1}$ or we have $f(x)=f(e)^{-1}f(x^{-1})=f(e)f(x^{-1})$ or $f(e)^{-1}=f(e)$ Now $f(x)=f(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/359277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Why is this function Lipschitz? Let $f:A \to B$ where $A$, $B \subset \mathbb{R}^n$.
Suppose
$$\lVert f(y_1) - f(y_2)\rVert_{\ell_\infty} \geq C\lVert y_1 - y_2 \rVert_{\ell_\infty}$$
This tells us that $f$ is one to one and that the inverse is Lipschitz.
I am told that $f$ is bi-Lipschitz; so $f$ is also Lipschitz, b... | I was inaccurate: in the general settings of your problem, where the only thing we know on $A$ is that $A \subset \mathbb{R}^n$, it is not true that even if $f \in W^{1,\infty}(A)$ then $f \in \text{Lip}(A)$. Neverthless what is true is the following:
$\textbf{Theorem:}$ Leu $U$ be open and bounded, with $\partial U$ o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/359342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Simplicial cohomology of $ \Bbb{R}\text{P}^2$ I've managed to confuse myself on a simple cohomology calculation. I'm working with the usual $\Delta$-complex on $X = \mathbf{R}\mathbf{P}^2$ and I've computed the complex as $\newcommand{Z}{\mathbf{Z}}$
$$ 0 \to \Z \oplus \Z \stackrel{\partial^0}{\to} \Z \oplus \Z \oplus... | Assuming that you have computed your cochain complex correctly, the problem with your first approach is that it is not true that
$$(A \oplus B)/(C \oplus D) \cong A/B \oplus C/D.$$
Instead to calculate $H^2(X)$ you will need to work with generators and relations. Define $a := (1,0)$ and $b:= (0,1)$, your basis vectors ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/359418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Number of conjugacy classes of the reflection in $D_n$.
Consider the conjugation action of $D_n$ on $D_n$. Prove that the
number of conjugacy classes of the reflections are
$\begin{cases} 1 &\text{ if } n=\text{odd} \\ 2 &\text{ if }
n=\text{even} \end{cases} $
I tried this:
Let $σ$ be a reflection. And $ρ$ be t... | Hint. The Orbit-Stabilizer theorem gives you that $[G:C_G(g)]$ is the size of the conjugacy class containing $g$. When $n$ is odd, a reflection $g$ commutes only with itself (why?), so $g$ has $[G:C_G(g)]=|G|/2$ elements, which are easily identified as the other reflections. Now, use this same technique to figure out... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/359511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How to evaluate powers of powers (i.e. $2^3^4$) in absence of parentheses? If you look at $2^{3^4}$, what is the expected result? Should it be read as $2^{(3^4)}$ or $(2^3)^4$? Normally I would use parentheses to make the meaning clear, but if none are shown, what would you expect?
(In this case, the formatting gives a... | In the same way that any expressions in brackets inside other brackets are done before the rest of the things in the brackets, I'd say that one works from the top down in such a case.
i.e. because we do $(a*d)$ first in$((a*b)*c)*d$, I'd imagine it'd be the expected thing to do $x^{(y^z)}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/359577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 1
} |
Reversing the Gram Matrix Let $A$ be a $M\times N$ real matrix, then $B=A^TA$ is the gramian of $A$. Suppose $B$ is given, is $A$ unique? Can I say something on it depending on $M$ and $N$.
| $A$ will definitely not be unique without some pretty serious restrictions. The simplest case to think about might be to consider $M\times 1$ 'matrices', i.e. column vectors. Then, $A^TA$ is simply the norm-squared of $A$, so for instance $A^TA=1$ would hold for any vector with norm $1$ (i.e. the unit sphere in $\Bbb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/359635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Condition number question Please, help me with this problem:
Let $A$ a matrix of orden $100$,
$$A\ =\ \left(\begin{array}{ccccc}
1 & 2 & & & \\
& 1 & 2 & & \\
& & \ddots & \ddots & \\
& & & 1 & 2\\
& & & & 1
\end{array}\right).$$
Show that $\mbox{cond}_2... | We consider a general order $n$. Calculate $\|Ax\|/\|x\|$ with $x=(1,1,\ldots,1)^T$ to get a lower bound $p=\sqrt{\frac{9(n-1)+1}{n}}$ for $\sigma_1(A)$. Compute $\|Ax\|/\|x\|$ for $x=\left((-2)^{n-1},(-2)^{n-2},\,\ldots,\,-2,1\right)^T$ to get an upper bound $q=\sqrt{\frac{3}{4^n-1}}$ for $\sigma_n(A)$. Now $\frac pq$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/359684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Standard Young Tableaux and Bijection to saturated chains in Young Lattice I'm reading Sagan's book The Symmetric Group and am quite confused.
I was under the assumption that any tableau with entries weakly increasing along a row and strictly increasing down a column would be considered standard Young tableau, e.g.
$$1... | Actually your Young tableau corresponds to the chain $$\begin{array}{cccccc}\emptyset & \prec & \bullet & \prec & \bullet & \bullet & \prec & \bullet & \bullet \\ & & & & \bullet & & & \bullet & \bullet\end{array}$$
that is, $\emptyset \prec (1,0) \prec (2,1) \prec (2,2)$, which is not saturated.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/359744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
The Best of Dover Books (a.k.a the best cheap mathematical texts) Perhaps this is a repeat question -- let me know if it is -- but I am interested in knowing the best of Dover mathematics books. The reason is because Dover books are very cheap and most other books are not: For example, while something like Needham's Vi... | Though it lacks any treatment of cardinal functions, Stephen Willard’s General Topology remains one of the best treatments of point-set topology at the advanced undergraduate or beginning graduate level. Steen & Seebach, Counterexamples in Topology, is not a text, but it is a splendid reference; the title is self-expla... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/359784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "142",
"answer_count": 23,
"answer_id": 14
} |
Why aren't logarithms defined for negative $x$? Given a logarithm is true, if and only if, $y = \log_b{x}$ and $b^y = x$ (and $x$ and $b$ are positive, and $b$ is not equal to $1$)[1], are true, why aren't logarithms defined for negative numbers?
Why can't $b$ be negative? Take $(-2)^3 = -8$ for example. Turning that i... | For the real, continuous exponentiation operator -- the used in the definition of the real, continuous logarithm -- $(-2)^3$ is undefined, because it has a negative base.
The motivation stems from continuity: If $(-2)^3$ is defined, then $(-2)^{\pi}$ and $(-2)^{22/7}$ should both be defined as well, and be "close" in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/359855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 2
} |
On the weak closedness I have some difficulties in this question.
Let $X$ be a nonreflexive Banach space and $K\subset X$ be a nonempty, convex, bounded and closed in norm. We consider $K$ as a subset of $X^{**}$. I would like to ask whether $K$ is closed w.r.t the topology $\sigma(X^{**}, X^*)$.
Thank you for all comm... | No, this fails in every nonreflexive space for $K=\{x\in X:\|x\|\le 1\}$, the closed unit ball of $X$.
Indeed, any neighborhood of a point $p\in X^{**}$ contains the intersection of finitely many "slabs" $\{x^{**}\in X^{**}: a_j< \langle x^{**}, x^*_j\rangle < b_j \}$ for some $x^*_j\in X^*$and $a_j,b_j\in \mathbb R$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/359933",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding the limit of a weird sequence function? Do you know any convergent sequence of continuous functions, that his limit function is a discontinuous function in infinitely many points of its domain (?)
| Let $$f_n(x) = \begin{cases} nx & \hat{x}\in[0,\frac{1}{n}) \\ 1 & \hat{x}\in[\frac{1}{n},\frac{n-1}{n}) \\ n-nx & \hat{x}\in(\frac{n-1}{n},1)\end{cases},$$ where $\hat{x}$ is the fractional part of $x$. It is not to hard to see that $f_n$ is continuous for every $n\in\mathbb{N}$, but $\lim_{n\to\infty} f_n = f_\infty$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/360007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Meaure theory problem no. 12 page 92, book by stein and shakarchi Show that there are $f \in L^1(\Bbb{R}^d,m)$ and a sequence $\{f_n\}$ with $f_n \in L^1(\Bbb{R}^d,m)$ such that $\|f_n - f\|_{L^1} \to 0$, but $f_n(x) \to f(x)$ for no $x$.
| Let us see the idea for $\mathbb{R}$ . Let $f_1$ be a characteristic function of $[0,0.5]$, $f_2$ --- of [0.5,1], $f_3$ of [0,0.25] and so on. They form the desired sequence on $L^1([0,1])$. The rest is an exercise because $\mathbb{R}$ can be covered by a countable number of intervals.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/360069",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
About injectivity of induced homomorphisms on quotient rings Let $A, B$ be commutative rings with identity, let $f: A \rightarrow B$ be a ring homomorphism (with $f(1) = 1$), let $\mathfrak{a}$ be an ideal of $A$, $\mathfrak{b}$ an ideal of $B$ such that $f(\mathfrak{a}) \subseteq \mathfrak{b}$. Then there is a well-de... | Let $\bar f$ is injective. Then $f(a)\in \mathfrak{b} $ implies $a\in \mathfrak{a} $. This means $\mathfrak{a} \supseteq \mathfrak{b}^c$ hence $\mathfrak{a}=\mathfrak{b}^c$ .
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/360142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
$\mathbb{R^+}$ is the disjoint union of two nonempty sets, each closed under addition. I saw Using Zorn's lemma show that $\mathbb R^+$ is the disjoint union of two sets closed under addition. and have a question related to the answer (I'm not sure if this is the right place to post it);
Why don't we just take
$\mathca... | There is a reason why you shouldn't try to explicitly construct such a decomposition. If $\mathbb{R}^{+}$ is a disjoint union of $A$ and $B$, where both $A$ and $B$ are closed under addition then neither one of $A$ and $B$ is either Lebesgue measurable or has Baire property since if $X$ is either a non meager set of re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/360208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Proving $\sqrt{2}\in\mathbb{Q_7}$? Why does Hensel's lemma imply that $\sqrt{2}\in\mathbb{Q_7}$?
I understand Hensel's lemma, namely:
Let $f(x)$ be a polynomial with integer coefficients, and let $m$, $k$ be positive integers such that $m \leq k$. If $r$ is an integer such that $f(r) \equiv 0 \pmod{p^k}$ and $f'(r) \... | This is how I see it and, perhaps, it will help you: we can easily solve the polynomial equation
$$p(x)=x^2-2=0\pmod 7\;\;(\text{ i.e., in the ring (field)} \;\;\;\Bbb F_7:=\Bbb Z/7\Bbb Z)$$
and we know that there's a solution $\,w:=\sqrt 2=3\in \Bbb F_{7}\,$
Since the roots $\,w\,$ is simple (i.e., $\,p'(w)\neq 0\,$) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/360282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 3
} |
Series with complex terms, convergence Could you tell me how to determine convergence of series with terms being products of real and complex numbers, like this:
$\sum_{n=1} ^{\infty}\frac{n (2+i)^n}{2^n}$ , $ \ \ \ \ \sum_{n=1} ^{\infty}\frac{1}{\sqrt{n} +i}$?
I know that $\sum (a_n +ib_n)$ is convergent iff $\sum a_n... | We have
$$\left|\frac{n(2+i)^n}{2^n}\right|=n\left(\frac{\sqrt{5}}{2}\right)^n\not\to0$$
so the series $\displaystyle\sum_{n=1}^\infty \frac{n(2+i)^n}{2^n}$ is divergent.
For the second series we have
$$\frac{1}{\sqrt{n}+i}\sim_\infty\frac{1}{\sqrt{n}}$$
then the series $\displaystyle\sum_{n=1}^\infty \frac{1}{\sqrt{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/360345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Shooting game - a probability question
In a shooting game, the probability for Jack to hit a target is 0.6.
Suppose he makes 8 shots, find the probabilities that he can hit the
target in more than 5 shots.
I find this question in an exercise and do not know how to solve it. I have tried my best but my answer is d... | Total shots = n = 8
Success = p = The probability for Jack to hit a target is 0.6
Failure $= q = (1-p) =$ The probability for Jack to NOT Hit a target is $(1- 0.6) = 0.4$
P( He can hit the target in more than 5 shots, i.e. from 6 to 8 ) $= [ P(X = 6) + P(X = 7) + P(X = 8) ] =$ $
[\binom{8}{6}.(.6)^{6}.(.4)^{8-6} + \b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/360402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Polynomials not dense in holder spaces How to prove that the polynomials are not dense in Holder space with exponent, say, $\frac{1}{2}$?
| By exhibiting a function that cannot be approximated by polynomials in the norm of $C^{1/2}$, such as $f(x)=\sqrt{x}$ on the interval $[0,1]$. The proof is divided into steps below; you might not need to read all of them.
Let $p$ be a polynomial.
$(p(x)-p(0))/x\to p'(0)$ as $x\to 0^+$
$|p(x)-p(0)|/x^{1/2}\to 0$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/360537",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Graph Concavity Test I'm studying for my final, and I'm having a problem with one of the questions. Everything before hand has been going fine and is correct, but I'm not understanding this part of the concavity test.
$$f(x) = \frac{2(x+1)}{3x^2}$$
$$f'(x) =-\frac{2(x+2)}{3x^3}$$
$$f''(x) = \frac{4(x+3)}{3x^4}$$
For th... | If I got your question correctly, you are working on the function $f(x)$ as above and want to know the concavity of it. First of all note that as the first comment above says; $x=0$ is also a critical point for $f$. Remember what is the definition of a critical point for a function. Secondly, you see that $x=0$ cannot ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/360604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
transformation of integral from 0 to infinity to 0 to 1 How do I transform the integral $$\int_0^\infty e^{-x^2} dx$$ from 0 to $\infty$ to o to 1 and. I have to devise a monte carlo algorithm to solve this further, so any advise would be of great help
| Pick your favorite invertible, increasing function $f : (0,1) \to (0,+\infty)$. Make a change of variable $x = f(y)$.
Or, pick your favorite invertible, increasing function $g : (0,+\infty) \to (0,1)$. Make a change of variable $y = g(x)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/360675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Prove that $\tan(75^\circ) = 2 + \sqrt{3}$ My (very simple) question to a friend was how do I prove the following using basic trig principles:
$\tan75^\circ = 2 + \sqrt{3}$
He gave this proof (via a text message!)
$1. \tan75^\circ$
$2. = \tan(60^\circ + (30/2)^\circ)$
$3. = (\tan60^\circ + \tan(30/2)^\circ) / (1 - \ta... | You can rather use $\tan (75)=\tan(45+30)$ and plug into the formula by Metin. Cause: Your $15^\circ$ is not so trivial.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/360747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 2
} |
Suggest an Antique Math Book worth reading? I'm not a math wizard, but I recently started reading through a few math books to prepare myself for some upcoming classes and I'm starting to really get into it. Then I noticed a few antique math books at a used bookstore and bought them thinking that, if nothing else, they ... | For a book that is not going to teach you any new math, but will give you a window into how a mathematical personality might think or act, I would recommend I Want to be a Mathematician by Paul Halmos. Quite a fun read, full of all of the joys and nuisances of being a high class working mathematician.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/360801",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 11,
"answer_id": 9
} |
Brent's algorithm
Use Brent's algorithm to find all real roots of the equation
$$9-\sqrt{99+2x-x^2}=\cos(2x),\\ x\in[-8,10]$$
I am having difficulty understanding Brent's algorithm. I looked at an example in wikipedia and in my book but the examples given isn't the same as this question. Any help will be greatly ap... | The wikipedia entry you cite explains Brent's algorithm as a modification on other ones. Write down all algorithms that are mentioned in there, see how they go into Brent's. Perhaps try one or two iterations of each to feel how they work.
Try to write Brent's algorithm down as a program in some language you are familia... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/360881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find the coordinates of any stationary points on the curve $y = {1 \over {1 + {x^2}}}$ and state it's nature I know I could use the quotient rule and determine the second differential and check if it's a max/min point, the problem is the book hasn't covered the quotient rule yet and this section of the book concerns ex... | As stated in the comments below, you can check whether a "stationary point" (a point where the first derivative is zero), is a maximum or minimum by using the first derivative. Evaluate points on each side of $x = 0$ to determine on which side it is decreasing (where $f'(x)$ is negative) and which side it is increasing... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/360957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
The angle between two unit vectors is not what I expected Ok imagine a vector with only X and Z components that make a 45 degree angle with the positive X axis. It's a unit vector. Now also imagine a unit vector that has the same direction as the positive x axis. Now imagine rotating both of these around the Z axis.
I... | If you start with the vectors $(1,0,0)$ and $(1/\sqrt{2},0,1/\sqrt{2})$, and rotate both by $45^\circ$ about the $z\text{-axis}$, then you end up with $(1/\sqrt{2},1/\sqrt{2},0)$ and $(1/2,1/2,1/\sqrt{2})$. The second point is not $(1/\sqrt{3},1/\sqrt{3},1/\sqrt{3})$ as you imagined. If you think about it, the $z\tex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/361036",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Show that $(2+i)$ is a prime ideal
Consider the set Gaussian integer $\mathbb{Z}[i]$. Show that $(2+i)$ is a prime ideal.
I try to come out with a quotient ring such that the set Gaussian integers over the ideal $(2+i)$ is either a field or integral domain. But I failed to see what is the quotient ring.
| The quotient ring you are after must be $\Bbb Z[i]/I$ where $I=(2+i)$, otherwise it would not tell you much about the status is the ideal $I$. You must know that multiplication by a complex number is a combination of rotation and scaling, and so multiplication by $2+i$ is the unique such operation that sends $1\in\Bbb ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/361099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 4
} |
First Order Logic: Formula for $y$ is the sum of non-negative powers of $2$ As the title states, is it possible to write down a first order formula that states that $y$ can be written as the sum of non-negative powers of $2$.
I have been trying for the past hour or two to get a formula that does so (if it is possible)... | In the natural numbers, the formula $\theta(x) \equiv x = x$ works. Think about binary notation.
More seriously, once you have developed the machinery to quantify over finite sequences, it is not so hard to write down the formula. Let $\phi(x)$ define the set of powers of 2. The formula will look like this:
$$
(\exist... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/361190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Directional derivative of a scalar field in the direction of fastest increase of another such field Suppose $f,g : \mathbb{R}^n \rightarrow \mathbb{R}$ are scalar fields. What expression represents the directional derivative of $f$ in the direction in which $g$ is increasing the fastest?
| The vector field encoding the greatest increase in $g$ is the gradient of $g$, so the directional derivative of $f$ in the direction of $\text{grad}(g)$ is just $\text{grad}(f)\bullet \text{grad}(g)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/361249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Neatest proof that set of finite subsets is countable? I am looking for a beautiful way of showing the following basic result in elementary set theory:
If $A$ is a countable set then the set of finite subsets of $A$ is
countable.
I proved it as follows but my proof is somewhat fugly so I was wondering if there is a... | A proof for finite subsets of $\mathbb{N}$:
For every $n \in \mathbb{N}$, there are finitely many finite sets $S \subseteq \mathbb{N}$ whose sum $\sum S = n$.
Then we can enumerate every finite set by enumerating all $n \in \mathbb{N}$ and then enumerating every (finitely many) set $S$ whose sum is $\sum S = n$.
Since ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/361320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 13,
"answer_id": 8
} |
G is a group, H is a subgroup of G, and N a normal group of G. Prove that N is a normal subgroup of NH. So I have already proved that NH is a subgroup of G. To prove that N is a normal subgroup of NH I said the we need to show $xNx^{-1}$ is a subgroup of NH for all $x\in NH$
Or am i defining it wrong?
| For any subgroup $K$ of $G$ with $N \subset K \subset G$, you can show that $N$ is a normal subgroup of $K$. Also, you can easily show $N \subset NH$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/361380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Uniform convergence for $x\arctan(nx)$ I am to check the uniform convergence of this sequence of functions : $f_{n}(x) = x\arctan(nx)$ where $x \in \mathbb{R} $. I came to a conclusion that $f_{n}(x) \rightarrow \frac{\left|x\right|\pi}{2} $. So if $x\in [a,b]$ then $\sup_{x \in [a,b]}\left|f_n(x)- \frac{\left|x\right|... | Hint: Use the fact that
$$\arctan t+\arctan\left(\frac 1t\right)=\frac{\pi}2$$
for all $t>0$ and that $f_n$ are even.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/361439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How to prove that $\det(M) = (-1)^k \det(A) \det(B)?$ Let $\mathbf{A}$ and $\mathbf{B}$ be $k \times k$ matrices and $\mathbf{M}$ is the block matrix
$$\mathbf{M} = \begin{pmatrix}0 & \mathbf{B} \\ \mathbf{A} & 0\end{pmatrix}.$$
How to prove that $\det(\mathbf{M}) = (-1)^k \det(\mathbf{A}) \det(\mathbf{B})$?
| Here is one way among others:
$$
\left( \matrix{0&B\\A&0}\right)=\left( \matrix{0&I_k\\I_k&0}\right)\left( \matrix{A&0\\0&B}\right).
$$
I assume you are allowed to use the block diagonal case, which gives $\det A\cdot\det B$ for the matrix on the far right. Just in case, this follows for instance from Leibniz formula. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/361552",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 0
} |
Time dilation by special relativity When reading about special relativity and time dilation I encounter a problem;
Here is a link: Time dilation in GPS
On page 1 under header "2. Time dilation by special relativity." It says: Since $(1 – x)^{-1/2} ≈ 1 + x /2$ for small $x$, we get...
How is $(1 – x)^{-1/2} ≈ 1 + x /2$... | That is a mathematical approximation that is valid when $x\ll 1$ (meaning $x$ much less than one). It can be easily demonstrated if you know calculus: for any function $f(x)$ that is defined and derivable at $x=0$, $f(x)=f(0)+f'(0)x+\ldots$ (it is called a Taylor series expansion). The first two terms are enough if $x$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/361657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.