Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
In a polynomial of $n$ degree, what numbers can fill the $n$? Until now, I've seen that the $n$ could be filled with the set $\mathbb{N}_0$ and $-\infty$ but I still didn't see mentions on other sets of numbers. As I thought that having 0 and $-\infty$ as degrees of a polynomial were unusual, I started to think if it ... | The degree of a polynomial can only take the values that you've specified. For that, let's revisit the definition of a polynomial. Personally, I was taught that a polynomial (in one variable) is an algebraic expression which can be written in the form $$a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$$ where $n$ is a non-negativ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/185810",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How do you take the product of Bernoulli distribution? I have a prior distribution,
$$p(\boldsymbol\theta|\pi)=\prod\limits_{i=1}^K p(\theta_i|\pi).$$
$\theta_i$ can equal $0$ or $1$, so I am using a Bernoulli distribtion so that
$$p(\boldsymbol\theta|\pi)=\prod\limits_{i=1}^K \pi^{\theta_i}(1-\pi)^{1-\theta_i}.$$
I th... | The equation you have can be represented as follows:
$$p(\boldsymbol x|\theta)=\prod\limits_{i=1}^K \theta^{x_i}(1-\theta)^{1-x_i}=\theta^{\sum_i x_i}(1-\theta)^{K-\sum_i x_i}$$
We have the Bayes rule
$$p(\theta|x)=\frac{p(x|\theta)p(\theta)}{p(x)}$$
as $\theta$ is known, we have the joint density $p(x,\theta)=p(\theta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/185860",
"timestamp": "2023-03-29T00:00:00",
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Prove that if $n$ is a positive integer then $2^{3n}-1$ is divisible by $7$. I encountered a problem in a book that was designed for IMO trainees. The problem had something to do with divisibility.
Prove that if $n$ is a positive integer then $2^{3n}-1$ is divisible by $7$.
Can somebody give me a hint on this proble... | Hint: Note that $8 \equiv 1~~~(\text{mod } 7)$.
So,
$$2^{3n}=(2^3)^n=8^n\equiv \ldots~~~(\text{mod } 7)=\ldots~~~(\text{mod } 7)$$
Try to fill in the gaps!
Solution:
Note that $8 \equiv 1~~(\text{mod } 7)$. This means that $8$ leaves a remainder of $1$ when divided by $7$. Now assuming that you are aware of some basi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/185916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Convergence of a sequence of non-negative real numbers $x_n$ given that $x_{n+1} \leq x_n + 1/n^2$. Let $x_n$ be a sequence of the type described above. It is not monotonic in general, so boundedness won't help. So, it seems as if I should show it's Cauchy. A wrong way to do this would be as follows (I'm on a mobile de... | Note that
$$
\lim_{n\to\infty}\sum_{k=n}^\infty\frac1{k^2}=0
$$
and for $m\gt n$,
$$
a_m\le a_n+\sum_{k=n}^\infty\frac1{k^2}
$$
First, take the $\limsup\limits_{m\to\infty}$:
$$
\limsup_{m\to\infty}a_m\le a_n+\sum_{k=n}^\infty\frac1{k^2}
$$
which must be non-negative. Then take the $\liminf\limits_{n\to\infty}$:
$$
\li... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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A puzzle with powers and tetration mod n A friend recently asked me if I could solve these three problems:
(a) Prove that the sequence $ 1^1, 2^2, 3^3, \dots \pmod{3}$ in other words $\{n^n \pmod{3} \}$ is periodic, and find the length of the period.
(b) Prove that the sequence $1, 2^2, 3^{3^3},\dots \pmod{4}$ i.e. $\... | For $k\ge 1$ we have
$$\begin{align*}
{^{k+2}n}\bmod5&=(n\bmod5)^{^{k+1}n}\bmod5=(n\bmod5)^{^{k+1}n\bmod4}\bmod5\\
{^{k+1}n}\bmod4&=(n\bmod4)^{^kn}=\begin{cases}
0,&\text{if }n\bmod2=0\\
n\bmod4,&\text{otherwise}\;,
\end{cases}
\end{align*}$$
so
$${^{k+2}n}\bmod5=\begin{cases}
(n\bmod5)^0=1,&\text{if }n\bmod2=0\\
(n\bm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/186023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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Theorems with an extraordinary exception or a small number of sporadic exceptions The Whitney graph isomorphism theorem gives an example of an extraordinary exception: a very general statement holds except for one very specific case.
Another example is the classification theorem for finite simple groups: a very general... | How about the Big Picard theorem? http://en.wikipedia.org/wiki/Picard_theorem
If a function $f:\mathbb{C}\to \mathbb{C}$ is analytic and has an essential singularity at $z_0\in \mathbb{C}$, then in any open set containing $z_0$, $f(z)$ takes on all possible complex values, with at most one possible exception, infinitel... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/186103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "102",
"answer_count": 40,
"answer_id": 20
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Is a right inverse of a surjective linear map linear? On a finite dimensional vector space, the answer is yes (because surjective linear map must be an isomorphism). Does this extend to infinite dimensional vector space? In other words, for any linear surjection $T:V\rightarrow V$, AC guarantees the existence of right ... | No. Let $V = \text{span}(e_1, e_2, ...)$ and let $T : V \to V$ be given by $T e_1 = 0, T e_i = e_{i-1}$. A right inverse $S$ for $T$ necessarily sends $v = \sum c_i e_i$ to $\sum c_i e_{i+1} + c_v e_1$ but $c_v$ may be an arbitrary function of $v$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/186146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Limit of $(x_n)$ with $0
Let $0 < x_1 < 1$ and $x_{n + 1} = x_n - x_n^{n + 1}$ for $n \geqslant 1$.
Prove that the limit exists and find the limit in terms of $x_1$.
I have proved the existence but cannot manage the other part.
Thanks for any help.
| Note that $x_2=x_1(1-x_1)$ with $0\lt x_1\lt1$ hence $0\lt x_2\lt1/4$, that $(x_n)_{n\geqslant1}$ is decreasing, in particular $(x_n)_{n\geqslant1}$ converges to some value $\ell(x_1)$ in $[0,x_1)$, and that $x_n\leqslant x_2$ for every $n\geqslant2$. Hence, for every $n\geqslant2$, $x_{n+1}\geqslant x_n-x_2^{n+1}$, wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/186293",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 2
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$\operatorname{Aut}(\mathbb Z_n)$ is isomorphic to $U_n$. I've tried, but I can't solve the question. Please help me prove that:
$\operatorname{Aut}(\mathbb Z_n)$ is isomorphic to $U_n$.
| (If you know about ring theory.) Since $\mathbb Z_n$ is an abelian group, we can consider its endomorphism ring (where addition is component-wise and multiplication is given by composition). This endomorphism ring is simply $\mathbb Z_n$, since the endomorphism is completely determined by its action on a generator, and... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Integer combination i want write a module to find the integer combination for a multi variable fomula. For example
$8x + 9y \le 124$
The module will return all possible positive integer for $x$ and $y$.Eg. $x=2$, $y=12$.
It does not necessary be exactly $124$, could be any number less or equal to $124$. Must be as clos... | Considering the equal case, if you analyze some of the data points that produce integer solution to the original inequality:
$$...,(-34,44), (-25,36), (-16,28), (-7,20),(2,12),(11,4),...$$
you can see that:
$$x_{i}=x_{i-1}+9$$
to get $y$ values, re-write the inequality as follows (and use the equal part):
$$y = \frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/186395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Optimization problem involving step function I've got to optimize the following function with respect to $\phi$:
$q(\phi, x) = \frac{1}{n} \sum_{i=1}^{n}{H(y_i)}$
where
$y_i = k - \phi l - x_i$
and $H(.)$ denotes the Heaviside function. $k$ and $l$ are constants, and $x$ follows either (1) a continuous uniform distribu... | The step functions all add up in the same direction, since $\phi$ has the same sign in all $y_i$ – thus $q$ is minimal for all $\phi$ such that all step functions are $0$, which occurs for $\phi\lessgtr(k-x)/l$, where the inequality is $\lt$ or $\gt$ and $x$ is the greatest or least of the $x_i$, depending on whether $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/186441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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natural solutions for $9m+9n=mn$ and $9m+9n=2m^2n^2$ Please help me find the natural solutions for $9m+9n=mn$ and $9m+9n=2m^2n^2$ where m and n are relatively prime.
I tried solving the first equation in the following way: $9m+9n=mn \rightarrow (9-n)m+9n=0 $
$\rightarrow m=-\frac{9n}{9-n}$
Thanks in advance.
| $$mn=9n+9m \Rightarrow (m-9)(n-9)=81$$
This equation is very easy to solve, just keep in mind that even if $m,n$ are positive, $m-9,n-9$ could be negative. But there are only 6 ways of writing 81 as the product of two integers.
The second one is trickier, but if $mn >9$ then it is easy to prove that
$$2m^2n^2> 18mn > 9... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/186519",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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If $T$ is bounded and $F$ has finite rank, what is the spectrum of $T+F$? Suppose that $T$ is a bounded operator with finite spectrum. What happens with the spectrum of $T+F$, where $F$ has finite rank? Is it possible that $\sigma(T+F)$ has non-empty interior? Is it always at most countable?
Update: If $\sigma(T)=\{0\... | It is always true if $T$ is self-adjoint. Here is a theorem that you might be interested:
If $T$ is self-adjoint, a complex number is in the spectrum of $T$ but not in its essential spectrum iff it is an isolated eigenvalue of $T$ of finite multipliticity.
The result can be found on page 32 of Analytic K-homology by N... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/186567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 2,
"answer_id": 1
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Finding a paper by John von Neumann written in 1951 There's a 1951 article by John von Neumann, Various techniques used in connection with random digits, which I would really like to read. It is widely cited, but I can't seem to find an actual copy of the paper, be it free or paying.
Is there a general strategy to find... | One of the citations gives the bibliographic info,
von Neumann J, Various Techniques Used in Connection with Random Digits, Notes by G E
Forsythe, National Bureau of Standards Applied Math Series, 12 (1951) pp 36-38. Reprinted in von
Neumann's Collected Works, 5 (1963), Pergamon Press pp 768-770.
That should be en... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/186626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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"answer_id": 1
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about the differentiability : the general case Let $U$ be an open set in $\mathbb{R}^{n}$ and $f :U \subset \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be a given function. We say that $f$ is differentiable at $x_{0}\in U$ if the partial derivatives of $f$ existi at $x_{0}$ and if
$$\displaystyle \lim_{x \rightarrow x_... | Let me compute for the example, $f(x,y)=(x^2+y^2, x+y)$. We write $f_1(x,y)=x^2+y^2$ and $f_2(x,y)=x+y$. Then
$$
\frac{\partial f_1}{\partial x}(x,y) = 2x,\quad
\frac{\partial f_1}{\partial y}(x,y) = 2y,\quad
\frac{\partial f_2}{\partial x}(x,y) = 1,\quad
\frac{\partial f_2}{\partial y}(x,y) = 1.
$$
Since these four fu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/186696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How many numbers less than $x$ are co-prime to $x$ Is there a fast way , or a direct function to give the count of numbers less than $x$ and co-prime to $x$ .
for example if $x$ = 3 ; then $n = 2$ and if $x$ = 8 ; then $n = 4$.
| Yes there is. First of all, you have to prime factorize your $x$, any put it in exponential form. Suppose you have the number $x = 50$. The prime factorization is $5^2 * 2^1$. Now take each number seperately. Take the bases and subtract 1 from all of them. $5-1=4$. $2-1=1$. Now evaluate the each base/exponent combina... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Nonsingularity of Euclidean distance matrix Let $x_1, \dots, x_k \in \mathbb{R}^n$ be distinct points and let $A$ be the matrix defined by $A_{ij} = d(x_i, x_j)$, where $d$ is the Euclidean distance. Is $A$ always nonsingular?
I have a feeling this should be well known (or, at least a reference should exists), on the o... | I think it should be possible to show that your distance matrix is always nonsingular by showing that it is always a Euclidean distance matrix (in the usual sense of the term) for a non-degenerate set of points. I don't give a full proof but sketch some ideas that I think can be fleshed out into a proof.
Two relevant p... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the best way to solve a problem given remainders and divisors? $x$ is a positive integer less than $100$. When $x$ is divided by $5$, its remainder is $4$. When $x$ is divided by $23$, its remainder is $7$. What is $x$ ?
| So, $x=5y+4$ and $x=23z+7$ for some integers $y,z$
So, $5y+4=23z+7=>5y+20=23z+23$ adding 16 to either sides,
$5(y+4)=23(z+1)=>y+4=\frac{23(z+1)}{5}$, so,$5|(z+1)$ as $y+4$ is integer and $(5,23)=1$
$=>z+1=5w$ for some integer $w$.
$x=23(5w-1)+7=115w-16$
As $0<x<100$ so,$x=99$ putting $w=1$
Alternatively, We have $5y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/186881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Infinite descent This Wikipedia article of Infinite Descent says:
We have
$ 3 \mid a_1^2+b_1^2 \,$.
This is only true if both $a_1$ and $b_1$ are divisible by $3$.
But how can this be proved?
| Suppose $a_1 = 3 q_1 + r_1$ and $b_1 = 3 q_2 + r_2$, where $r_1$ and $r_2$ is either $-1$, $0$ or $1$. Then
$$
a_1^2 + b_1^2 = 3 \left( 3 q_1^2 + 3 q_2^2 + 2 q_1 r_1 + 2 q_2 r_2 \right) + r_1^2 + r_2^2
$$
For $a_1^2 + b_1^2$ to be divisible by $3$, we should have $r_1^2 + r_2^2 = 0$, since $0\leqslant r_1^2+r_2^2 < 3... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Linear algebraic dynamical system that has complex entries in the matrix Suppose that there is a dynamical system that has the form of $\mathbb{x}_{k+1} = A\mathbb{x}_k$.
Suppose that one eigenvalue of $A$ matrix is complex number, the form of $a-bi$.
We then convert $\mathbb{x}_k = P\mathbb{y}_k$ where matrix $P$ is t... | Note that if $Av=\lambda v$ then $Cw=\lambda w$ where $w=P^{-1}v$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/187045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Dedekind Cuts in Rudin' analysis - Step 4 This question refers to the construction of $\mathbb{R}$ from $\mathbb{Q}$ using Dedekind cuts, as presented in Rudin's "Principles of Mathematical Analysis" pp. 17-21.
More specifically, in the last paragraph of step 4, Rudin says that for $\alpha$ a fixed cut, and given $v \i... | If $v\in 0^*$, then $v<0$, so $w>0$. Let $\gamma=\sup \alpha$. The Archimedean property says that for any $\gamma$ there is some integer $m$ such that $mw\geq\gamma$. Since $\mathbb{N}$ is well-ordered, there is a smallest $m$ such that $mw\geq\gamma$, call this $n+1$. This means that $nw<\gamma$, and hence $nw\in \al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/187108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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$H\vartriangleleft G$ and $|H|\not\equiv 1 (\mathrm{mod} \ p)$ then $H\cap C_{G}(P)\neq1$
Let $G$, a finite group, has $H$ as a proper normal subgroup and let $P$ be an arbitrary $p$-subgroup of $G$ ($p$ is a prime). Then $$|H|\not\equiv 1 (\mathrm{mod} \ p)\Longrightarrow H\cap C_{G}(P)\neq1$$
What I have done:
I ... | Let $P$ act on $H$; the number of fixed points is the number of elements in $C_G(P)\cap H$. Now use the easy fact that the number of fixed points is congruent to $|H|\pmod{p}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solving this linear system based on the combustion of methane, has no constants I recently discovered that I could solve a chemical reaction using a linear system. So I thought I would try something simple like the combustion of methane.
where x y z and w are the moles of each molecule
x $CH_4$ + y $O_2$ = z $H_2$O + w... | Each of your equations can be rearranged to give
$$x-w=0$$
$$2y-z-2w=0$$
$$4x-2z=0$$
Your idea of using Gaussian elimination is then a very good one. Or you can solve directly. From the first equation, you get $$x=w.$$ From the third equation, you get $$z=\frac{x}{2}=\frac{w}{2}.$$ Then from the second equation you get... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/187241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Why are two vectors that are parallel equivalent? Why are two parallel vectors with the same magnitude equivalent?
Why is their start point irrelevant?
How can a vector starting at $\,(0, -10)\,$ going to $\,(10, 0)\,$ be the same as
a vector starting at $\,(10, 10)\,$ and going to $\,(20, 20)\,$?
| Components of a vector determine the length and the direction of the vector, but not it's basepoint. Therefore 2 vectors are equivalent if and only if they have the same components, but this gives the idea that any 2 parallel vectors are equal though they have different base points. To avoid this confusion, we take tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/187304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Inequality with two absolute values I'm new here, and I was wondering if any of you could help me out with this little problem that is already getting on my nerves since I've been trying to solve it for hours.
Studying for my next test on inequalities with absolute values, I found this one:
$$ |x-3|-|x-4|<x $$
(I pre... | $|x-3|-|x-4|< x$, I write in $|x-3|< x+|x-4|$
but remember: $|r|< s \implies -s < r < s$.
So I write the equation in the form: $-x-|x-4| < x-3 < x+|x-4|$.
From this inequality I obtain 2 equations:
(a) $-x-|x-4| < x-3$
(b) $x-3 < x+|x-4|$.
Remember too : $|r| > s \implies r > s \text{ or } r < -s $.
So this... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/187343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 6,
"answer_id": 4
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Existence of an Infinite Length Path I came across the following simple definition
A path $\gamma$ in $\mathbb{R}^n$ that connects the point $a \in \mathbb{R}^n$ to the point $b \in \mathbb{R}^n$, is a continuous $\gamma : [0, 1] \to \mathbb{R}^n$ such that $\gamma(0) = a$ and $\gamma(1) = b$. We denote by $\ell(\gamm... | Yes, for instance Koch snowflake is a such example. Let's do the following construction:
*
*Start with the segment $A_0 = [0,1]$.
*Subdivise $A_0$ in three equal pieces.
*Replace the middle third by an equilateral triangle with base $[\frac13, \frac23]$.
*Suppress the base of that triangle. You get the path $A_1$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/187402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
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Generating integer solutions to $4mn - m^2 + n^2 = ±1$ How can I generate positive integer solutions to $m$ and $n$ that satisfy the equation:
$4mn - m^2 + n^2 = ±1$,
subject to the constraints that $m$ and $n$ are coprime, $m-n$ is odd and $m > n$.
| Hint: Completing the square yields an equation of the form: $$x^2-Dy^2=\pm 1$$
for a particular $D$.
There's actually a simple recursion that generates all solutions.
Let $a_0=0$, $a_1=1$, and $a_{k+2}=4a_{k+1}+a_{k}$. Then the general solution is $(m,n)=(a_{k+1},a_{k})$.
This gives the positive solutions. The solutio... | {
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"timestamp": "2023-03-29T00:00:00",
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"answer_id": 0
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Sufficiency to prove the convergence of a sequence using even and odd terms Given a sequence $a_{n}$, if I know that the sequence of even terms converges to the same limit as the subsequence of odd terms:
$$\lim_{n\rightarrow\infty} a_{2n}=\lim_{n\to\infty} a_{2n-1}=L$$
Is this sufficient to prove that the $\lim_{n\t... | If you are familiar with subsequences, you can easily prove as follows. Let $a_{n_k}$ be the subsequence which converges to $\limsup a_n$. it is obviously convergent and contain infinitely many odds or infinitely many evens, or both. Hence, $\limsup a_n = L$. The same holds for $\liminf a_n$, hence the limit of the who... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/187525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 0
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What is exactly "Algebraic Dynamics"? Could somebody please give a big picture description of what exactly is the object of study in the area of Algebraic Dynamics? Is it related to Dynamical Systems? If yes in what sense? Also, what is the main mathematical discipline underpinning Algebraic Dynamics? Is it algebraic g... | The Wiki article states that it is a combination of dynamical systems and number theory. I know it's a redirect, but WP's information on this point is probably reliable enough :)
(Are you checking here because you are not comfortable with WP info? It is a serious question which I'm curious about.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/187564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
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Is independence preserved by conditioning? $X_1$ and $X_2$ are independent. $Y_1|X_1\sim\mathrm{Ber}\left(X_1\right)$, $Y_2|X_2\sim\mathrm{Ber}\left(X_2\right)$. Are $Y_1$ and $Y_2$ necessarily independent? (Assume $\mathrm{P}\left(0<X_1<1\right)=1$, $\mathrm{P}\left(0<X_2<1\right)=1$)
| No. Let $X_1,X_2$ be independent uniform (0,1) random variables, and then define $$Y_1=1_{(X_1\geq X_2)}\,\mbox{ and }\, Y_2=1_{(X_2> X_1)}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/187633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Cauchy nets in a metric space Say that a net $a_i$ in a metric space is cauchy if for every $\epsilon > 0$ there exists $I$ such that for all $i, j \geq I$ one has $d(a_i,a_j) \leq \epsilon$. If the metric space is complete, does it hold (and in either case why) that every cauchy net converges?
| Consider a Cauchy net:
$$\forall \lambda,\lambda'\geq\lambda_n:\quad d(x_\lambda,x_\lambda')<\frac{1}{n}$$
Extract a Cauchy sequence:
$$x_n:=x_{\lambda(n)}\quad\lambda(n):=\lambda_1\wedge\ldots\wedge\lambda_n$$
Apply completeness:
$$d(x_\lambda,x)\leq d(x_\lambda,x_{n_0})+d(x_{n_0},x)<\frac{N}{2}+\frac{N}{2}\leq\epsilo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/187703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 2,
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Cardinality of the set of all pairs of integers The set $S$ of all pairs of integers can be represented as $\{i \ | \ i \in \mathbb{Z} \} \times \{j\ | \ j \in \mathbb{Z}\}$. In other words, all coordinates on the cartesian plane where $x, y$ are integers.
I also know that a set is countable when $|S|\leq |\mathbb{N}^+... | Define $\sigma: \Bbb Z \times \Bbb Z \to \Bbb Z \times \Bbb Z$ by
$$
\sigma(m,n) = \left\{\begin{array}{lr}
(1,-1), & \text{for } (m,n) = (0,0)\\
(m, n+1), & \text{for } m \gt 0 \, \land \, -m \le n \lt m\\
(m-1, n), & \text{for } n \gt 0 \, \land \, -n \lt m \le n\\
(m, n-1), & ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/187751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
Solving improper integrals and u-substitution on infinite series convergent tests This is the question:
Use the integral test to determine the convergence of $\sum_{n=1}^{\infty}\frac{1}{1+2n}$.
I started by writing:
$$\int_1^\infty\frac{1}{1+2x}dx=\lim_{a \rightarrow \infty}\left(\int_1^a\frac{1}{1+2x}dx\right)$$
I th... | $$u=1+2x\Longrightarrow du=2dx\Longrightarrow dx=\frac{1}{2}du$$
Remember, not only you substitute the variable and nothing more: you also have to change the $\,dx\,$ and the integral's limits:
$$u=1+2x\,\,,\,\text{so}\,\, x=1\Longrightarrow u=1+2\cdot 1 =3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/187959",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
$\mathbb{Z}$ has no torsion? What does is mean to say that $\mathbb{Z}$ has no torsion?
This is an important fact for any course?
Thanks, I heard that in my field theory course, but I don't know what it is.
| @d555, you might want to know that the notion of torison is extremely important, for example if $A$ is finitely generated and abelian group, then it can be written as the direct sum of its torsion subgroup $T(A)$ and a torsion-free subgroup (but this is not true for all infinitely generated abelian groups). $T(A)$ is u... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Trying to find angle at which projectile was fired. So let's say I have a parabolic function that describes the displacement of some projectile without air resistance. Let's say it's
$$y=-4.9x^2+V_0x.$$
I want to know at what angle the projectile was fired. I notice that
$$\tan \theta_0=f'(x_0)$$ so the angle should... | From what you are saying, it looks like you are thinking of $f(x)$ as a real valued function, and in effect you are only considering linear motion. (This would be fine, if the cannon were firing straight up or straight down or straight up.)
However, you are interested in knowing the interesting two-dimensional trajecto... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding the value of y in terms of x. Is it possible to get the value of $y$ in terms of $x$ from the below equation? If so please give give me a clue how to do that :)
$$y \sqrt{y^2 + 1} + \ln\left(y + \sqrt{y^2 + 1}\right) = \frac{a}{x^2}.$$
| Since $\ \rm{asinh}(y)=\ln\left(y + \sqrt{y^2 + 1}\right)\ $ let's set $\ y:=\sinh(u)\ $ and rewrite your equation as :
$$\sinh(u) \sqrt{\sinh(u)^2 + 1} + u = \frac{a}{x^2}$$
$$\sinh(u) \cosh(u) + u = \frac{a}{x^2}$$
$$\sinh(2u) + 2u = 2\frac{a}{x^2}$$
After that I fear you'll have to solve this numerically (to get $u$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to prove $ \int_0^\infty e^{-x^2} \; dx = \frac{\sqrt{\pi}}2$ without changing into polar coordinates? How to prove $ \int_0^\infty e^{-x^2} \; dx = \frac{\sqrt{\pi}}2$ other than changing into polar coordinates? It is possible to prove it using infinite series?
| $$ \int_{-\infty} ^\infty e^{-x^2} \; dx =I$$
$$ \int_{-\infty}^\infty e^{-y^2} \; dx =I$$
$$\int_{-\infty}^\infty e^{-x^2}\times e^{-y^2} \; dx =I^2$$
$$\int_{-\infty}^\infty e^{-(x^2+y^2)}\; dx =I^2$$
Polar Coordinates: $x^2+y^2=R^2,-\pi\leqslant \theta\leqslant +\pi$
$$\int_{-\pi}^{+\pi} \int_{0}^ \infty e^{-(R^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/188241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Chess tournament, graph
Problem. In chess tournament each player, from all $n$ players, played one game with every another player. Prove that it is possible to number all players with numbers from $1$ to $n$ in such way that every player has unique number and none of them lost the game against player with number great... | You can prove this by induction. The root is - according to taste - a game with just one or two players.
So let's assume we have found a numbering for $n$ players and along comes the $n+1^{st}$ player and plays against every other player.
If he looses against the $n^{th}$ player, we give him the number $n+1$. If he win... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/188288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Is any Mersenne number $M_p$ divisible by $p+2$? More precisely, does there exist a natural number $p$ such that $(2^p-1)/(p+2)$ is also a natural number? It seems to me that this is a really simple problem (with the answer "no"), but I couldn't find anything on the web. There are some facts known about division by $p+... | Another proof is as follows. First for $p=2$, it follows because $2^p-1=3<4=p+2$.
For odd primes $p$, there is a theorem (see Mersenne Prime for proof) that states that every factor of $2^p-1$ is of the form $2kp+1$ for some integer $k\geq0$. Thus we want a $p$ for which $p+2=2kp+1$. There is no such prime because for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/188422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Random Sequence Generator function I want to find out a function or algorithm, whichever is suitable, which can provide me a random sequence. Like
Input: 3
Output: {1,2,3} or {1,3,2} or {2,1,3} or {2,3,1} or {3,1,3} or {3,2,1}
Same as if I will enter a number N, output will be a random permutation of the set {1,2,...N}... | The first $O(n)$ shuffle or random permutation generator was published by Richard Durstenfeld in 1964. This algorithm came to the notice of programmers because it was included in Knuth's TAOCP, Vol 2, 1969, page 125, as Algorithm P. A succinct statement of the algorithm is:
\begin{equation}
\text{for }\ k \leftarrow... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove the boundedness of the solutions of a nonlinear differential equation I have the following differential equation:
$$
\ddot{x} = -\log(x) - 1
$$
and I need to prove that every solution of this equation is a bounded function. From the phase plane portrait, it is obvious that this is true:
How can I construc... | $x_1=x$, $x_2=\dot{x}$
$\dot{x}_1=x_2$
$\dot{x}_2=-\log(x_1)-1$
we know that to have the solution for the ODE we need $x_1>0$
consider the following function
$$V=(x_1\log(x_1)+0.5*x_2^2)$$
the derivative of this function along the system trajectories is
$$\dot{V}=x_2\log(x_1)+x_2+x_2(-\log(x_1)-1)=0$$
Therefore, the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/188533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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how can one find the value of the expression, $(1^2+2^2+3^2+\cdots+n^2)$
Possible Duplicate:
Proof that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$?
Summation of natural number set with power of $m$
How to get to the formula for the sum of squares of first n numbers?
how can one find the value of the expressio... | The claim is that
$\sum_{i = 1}^n i^2 = \frac{n(n + 1)(2n + 1)}{6}$
We will verify this by induction.
Clearly $n = 1$ holds.
Suppose the formula holds for $n$. Lets verify it holds for $n + 1$.
$$\sum_{i = 1}^{n + 1} i^2 = \sum_{i = 1}^n i^2 + (n + 1)^2 = \frac{n(n + 1)(2n + 1)}{6} + (n + 1)^2
\\
= \frac{n(n + 1)(2n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/188602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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Why is $a^n - b^n$ divisible by $a-b$? I did some mathematical induction problems on divisibility
*
*$9^n$ $-$ $2^n$ is divisible by 7.
*$4^n$ $-$ $1$ is divisible by 3.
*$9^n$ $-$ $4^n$ is divisible by 5.
Can these be generalized as
$a^n$ $-$ $b^n$$ = (a-b)N$, where N is an integer?
But why is $a^n$ $-$ $b^n$$ ... | Another proof:
Denote $r=b/a$. We know that the sum of a geometric progression of the type $1+r+r^2+\ldots+r^{n-1}$ is equal to $\frac{1-r^n}{1-r}$. Thus, we have
\begin{align}
1-r^n&=(1-r)(1+r+r^2+\ldots+r^{n-1}),\quad\text{substituting $r=b/a$ gives:}\\
a^n-b^n &= (a-b)\color{red}{(a^{n-1}+a^{n-2}b+\ldots+b^{n-1})}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/188657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "45",
"answer_count": 8,
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Limsups of nets The limsup on sequences of extended real numbers is usually taken to be either of these two things, which are equivalent:
*
*the sup of all subsequential limits.
*The limit of the sup of the tail ends of the sequence.
For the situation with nets, the same arguments guarantee the existence of the a... | As far as I can say, the more usual definition of limit superior of a net is the one using limit of suprema of tails:
$$\limsup x_d = \lim_{d\in D} \sup_{e\ge d} x_e = \inf_{d\in D} \sup_{e\ge d} x_e.$$
But you would get an equivalent definition, if you defined $\limsup x_d$ as the largest cluster point of the net. Thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/188722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Fréchet derivative I have been doing some self study in multivariable calculus.
I get the geometric idea behind looking at a derivative as a linear function, but analytically how does one prove this?
I mean if $f'(c)$ is the derivative of a function between two Euclidean spaces at a point $c$ in the domain... then is i... | It is not true that $f'(c_1+c_2)=f'(c_1)+f'(c_2)$ in general.
However, the derivative $f'(c)$ is the matrix of the differential $df_c$ and the expression which you write shows why $df_c(h) = f'(c)h$ is linear in the $h$-variable. It is simply matrix multiplication and all matrix multiplication maps are linear. The subt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/188774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Module Theory for the Working Student Question:
What level of familiarity and comfort with modules should someone looking to work through Hatcher's Algebraic Topology possess?
Motivation:
I am taking my first graduate course in Algebraic Topology this coming October. The general outline of the course is as follows:
... | Looks like you'll definitely want to know the homological aspects: projective, injective and flat modules. There are lots of different characterizations for these which are useful to know. (Sorry, don't know the insides of D&F very well, perhaps it is covered.)
The Fundamental theorem of finitely generated modules over... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/188853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Evaluating $\int_0^{\frac\pi2}\frac{\ln{(\sin x)}\ \ln{(\cos x})}{\tan x}\ dx$ I need to solve
$$
\int_0^{\Large\frac\pi2}\frac{\ln{(\sin x)}\ \ln{(\cos x})}{\tan x}\ dx
$$
I tried to use symmetric properties of the trigonometric functions as is commonly used to compute
$$
\int_0^{\Large\frac\pi2}\ln\sin x\ dx = -\frac... | Rewrite the integral as
$$
\int_0^{\Large\frac\pi2}\frac{\ln{(\sin x)}\ \ln{(\cos x})}{\tan x}\ dx=\int_0^{\Large\frac\pi2}\frac{\ln{(\sin x)}\ \ln{\sqrt{1-\sin^2 x}}}{\sin x}\cdot\cos x\ dx.
$$
Set $t=\sin x\ \color{red}{\Rightarrow}\ dt=\cos x\ dx$, then we obtain
\begin{align}
\int_0^{\Large\frac\pi2}\frac{\ln{(\sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/188921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "37",
"answer_count": 5,
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What am I losing if I decide to perform all math by computer? I solve mathematical problems every day, some by hand and some with the computer. I wonder: What will I lose if I start doing the mathematical problems only by computer? I've read this text and the author says that as techonology progress happens, we should... | Yes you can solve most of the problems by computer but you will lose your critical thinking ability, you are going to be an operator not a creator eventually!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/188945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 6,
"answer_id": 0
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Which separation axiom? Let $X$ be a topological space. Assume that for all $x_1,x_2 \in X$ there exist open neighbourhoods $U_i$ of $x_i$ such that $U_1 \cap U_2 = \emptyset$. Such a space, as we all know, is called Hausdorff. What would we call a space, and which separation axioms would the space satisfy, if $\overli... | Such a space is known as $T_{2\frac{1}{2}}$ or Urysohn according to Wikipedia.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/189042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving inequality $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{3\sqrt[3]{abc}}{a+b+c} \geq 4$ I started to study inequalities - I try to solve a lot of inequlites and read interesting .solutions . I have a good pdf, you can view from here . The inequality which I tried to solve and I didn't manage to find a solution can... | Write $$\frac ab+\frac ab+\frac bc\geq \frac{3a}{\sqrt[3]{abc}}$$ by AM-GM.
You get $$\operatorname{LHS} \geq \frac{a+b+c}{\sqrt[3]{abc}}+n\left(\frac{\sqrt[3]{abc}}{a+b+c}\right).$$
Set $$z:=\frac{a+b+c}{\sqrt[3]{abc}}$$ and then notice that for $n\leq 3$, $$z+\frac{3n}{z}\geq 3+n.$$ Indeed the minimum is reached for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/189143",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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How to construct a one-to one correspondence between$\left [ 0,1 \right ]\bigcup \left [ 2,3 \right ]\bigcup ..$ and $\left [ 0,1 \right ]$ How can I construct a one-to one correspondence between the Set $\left [ 0,1 \right ]\bigcup \left [ 2,3 \right ]\bigcup\left [ 4,5 \right ] ... $ and the set $\left [ 0,1 \right ... | For $x\in{(k,k+1)}$ with $k\geq 2$ and $k$ even define $f(x)=\frac{1}{2x-k}$; for $x\in(0,1]$ define $f(x)=\frac{x+1}{2}$; set $f(0)=0$. Now it remains to map $A=\{2,3,4,5,..\}$ bijectively to $\{1/2,1/4,1/6,1/8..,\}$ to do this define $f(x)=\frac{1}{2(x-1)}$ on $A$. This should give you the desired bijection. Please... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/189191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Closed form representation of an irrational number Can an arbitrary non-terminating and non-repeating decimal be represented in any other way? For example if I construct such a number like 0.1 01 001 0001 ... (which is irrational by definition), can it be represented in a closed form using algebraic operators? Can it h... | Since $0.1 = \frac{1}{10}$, $0.001 = \frac{1}{10^3}$, $0.0000001 = \frac{1}{10^6}$. Making a guess that $n$-th term is $10^{-n(n+1)/2}$ the sum, representing the irrational number becomes
$$
0.1010010001\ldots = \sum_{k=0}^\infty \frac{1}{10^{\frac{k(k+1)}{2}}} = \left.\frac{1}{2 q^{1/4}} \theta_2\left(0, q\right)-1\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/189250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Prime elements in $\mathbb{Z}/n\mathbb{Z}$ I'm tring to determine the prime elements in the ring $\mathbb{Z}/n\mathbb{Z}$.
| Every ideal of $\mathbb{Z}/n\mathbb{Z}$ is of the form $m\mathbb{Z}/n\mathbb{Z}$, where $m$ is a divisor of $n$. And its residue ring is isomorphic to $\mathbb{Z}/m\mathbb{Z}$.
Hence a prime ideal of $\mathbb{Z}/n\mathbb{Z}$ is of the form $p\mathbb{Z}/n\mathbb{Z}$, where $p$ is a prime divisor of $n$.
Hence every prim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/189394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Future Lifetime Distribution Suppose the future lifetime of someone aged $20$ denoted by $T_{20}$ is subject to the force of mortality $\mu_x = \frac{1}{100-x}$ for $x< 100$. What is $\text{Var}[\min(T_{20},50)]$?
So we have: $$E[\min(T_x,50)|T_{20} > 50] = 50$$ $$\text{Var}[\min(T_{20},50)|T_{20} > 50] = ?$$ $$E[\min(... | Hint: The random variable $\min(T_{20},50)|T_{20} > 50)$ doesn't vary much!
If you then want to use a formula, call the above random variable $Y$. We want $E((Y-50)^2)$. How did you decide earlier that $E(Y)=50$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/189512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What branch of the Math can help me with this? I would love to focus on the branches of the Math that can help me with:
*
*generation of entropy, i suppose that most of the works are based on statistic since even a big part of the cryptographic world starts from this and is tested with the help of the statistic. But... | 1) Dynamical systems, more precisely: Ergodic theory. See Introduction to ergodic theory by Ya. G. Sinai or Ya. B. Pesin's books.
2) Fractal geometry. You can refer to Fractal Geometry by Kenneth Falconer.
You can also see Chaos and Fractals: New Frontiers of Science (more elementary) by Heinz-Otto Peitgen, Hartmut Jür... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/189583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Intuition behind gradient VS curvature In Newton's method, one computes the gradient of a cost function, (the 'slope') as well as its hessian matrix, (ie, second derivative of the cost function, or 'curvature'). I understand the intuition, that the less 'curved' the cost landscape is at some specific weight, the bigger... | Intuitively, curvature is how fast the slope is changing: a greater rate of change of the slope means it is more curved.
So it is related to the derivative of the slope, i.e. the derivative of the derivative or the second derivative.
| {
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Proving that one quantity *eventually* surpasses the other I want to prove the following; $$\forall t>0\ \ \forall m\in \mathbb{N} \ \ \exists N \in \mathbb{N} \ \ \forall n\geq N: \ (1+t)^n > n^m.$$
For readers who hate quantifiers, here's the version in words: "$(1+t)^n$ eventually surpasses $n^m$ for some $t>0$ and ... | 1) Yes. The inequality is equivalent to $((1+t)^{\frac{1}{m}})^n>n$ by taking the $\frac{1}{m}$th power of both sides which reduces to your original problem to the problem with $m=1$ since $(1+t)^{\frac{1}{m}}$ is of the form $1+u$ with $u$ a strictly positive real.
So we'll assume $m=1$. We can expand $(1+t)^n$ using ... | {
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Does $(\mathbf A+\epsilon \mathbf I)^{-1}$ always exist? Why? Does $(\mathbf A+\epsilon \mathbf I)^{-1}$ always exist, given that $\mathbf A$ is a square and positive (and possibly singular) matrix and $\epsilon$ is a small positive number? I want to use this to regularize a sample covariance matrix ($\mathbf A = \Sigm... | Yes, if $\mathbf A$ is any $n \times n$ matrix, then $\mathbf A+\epsilon \mathbf I$ is invertible for sufficiently small $\epsilon > 0$. This is because $\det (\mathbf A + \epsilon \mathbf I)$ is a polynomial in $\epsilon$ of degree $n$, and so it has a finite number of zeroes.
| {
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Physical meaning of spline interpolation I remember that when I took my Numerical Analysis class, the professor said the spline interpolation take its name from a kind of wood sticks used to draw curved lines. Also Wikipedia say that the name is due to those elastic rulers:
Elastic rulers that were bent to pass throug... | The equations of cubic splines are derived from the physical laws that govern bending of thin beams. For example, see http://stem2.org/je/cs.pdf.
The spline equation is an approximate solution of the minimum energy bending equation, valid when the amount of bending is small.
Generally, in computer-aided geometric desig... | {
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What's the meaning of $C$-embedded? What's the meaning of $C$-embedded? It is a topological notion. Thanks ahead.
| A set $A \subset X$ ($X$ is a topological space) is $C$-embedded in $X$ iff every real-valued continuous function $f$ defined on $A$ has a continuous extension $g$ from $X$ to $\mathbb{R}$ (so $g(x) = f(x)$ for all $x \in A$).
A related notion of $C^{\ast}$-embedded exist where continuous real-valued functions are (in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/189922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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solving for a coefficent term of factored polynomial.
Given: the coefficent of $x^2$ in the expansion of $(1+2x+ax^2)(2-x)^6$ is $48,$ find the value of the constant $a.$
I expanded it and got
$64-64\,x-144\,{x}^{2}+320\,{x}^{3}-260\,{x}^{4}+108\,{x}^{5}-23\,{x}^{
6}+2\,{x}^{7}+64\,a{x}^{2}-192\,a{x}^{3}+240\,a{x}^{4... | It would be much easier to just compute the coefficient at $x^2$ in the expansion of $(1+2x+ax^2)(2-x)^6$. You can begin by computing:
$$ (2-x)^6
= 64 - 6 \cdot 2^5 x + 15 \cdot 2^4 x^2 + x^3 \cdot (...)
= 64 - 192 x + 240 x^2 + x^3 \cdot (...) $$
Now, multiply this by $(1+2x+ax^2)$. Again, you're only interested in ... | {
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Image of a morphism According to Wikipedia, image of a morphism $\phi:X\rightarrow Y$ in a category is a monomorphism $i:I\rightarrow Y$ satisfying the following conditions:
*
*There is a morphism $\alpha:X\rightarrow I$ such that $i\circ\alpha=\phi$.
*If $j:J\rightarrow Y$ is a monomorphism and $\beta:X\rightarrow... | The term "image" suggests that this concept is modeled on the image of a map, a morphism in the category of sets. In that case, $I$ can be any set equipotent with the image (in the conventional sense) of $\phi$, and $\alpha$ is generally neither unique, nor an epimorphism (a surjective map). For uniqueness, note that y... | {
"language": "en",
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How to check if a point is inside a rectangle? There is a point $(x,y)$, and a rectangle $a(x_1,y_1),b(x_2,y_2),c(x_3,y_3),d(x_4,y_4)$, how can one check if the point inside the rectangle?
| Given how much attention this post has gotten and how long ago it was asked, I'm surprised that no one here mentioned the following method.
A rectangle is the image of the unit square under an affine map. Simply apply the inverse of this affine map to the point in question, and then check if the result is in the unit ... | {
"language": "en",
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Determining whether or not spaces are separable I've been going over practice problems, and I ran into this one. I was wondering if anyone could help me out with the following problem.
Let $X$ be a metric space of all bounded sequences $(a_n) \subset \mathbb{R}$ with the metric defined by
$$d( (a_n), (b_n)) = \sup... | Suppose that $A=\{\alpha_n:n\in\Bbb N\}$ is a countable subset of $X$, where $\alpha_n$ is the sequence $\langle a_{n,k}:k\in\Bbb N\rangle$. Note that for any $x\in\Bbb R$ there is always a $y\in[-1,1]$ such that $|x-y|\ge 1$. Thus, we can construct a sequence $\beta=\langle b_k:k\in\Bbb N\rangle$ such that $b_k\in[-1,... | {
"language": "en",
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Integer solutions of $p^2 + xp - 6y = \pm1$ Given a prime $p$, how can we find positive integer solutions $(x,y)$ of the equation:
$$p^2 + xp - 6y = \pm1$$
| If $p=2$ or $p=3$, you can't.
Otherwise, the extended Euclid algorithm produces a solution $(x_0,y_0)$ of $x_0 p - 6 y_0 = \pm 1$ (in fact, one for $+1$ and one for $-1$).
Then $x=x_0+6k-p$ and $y=y_0+pk$ is a solution of $p^2+xp-6y=\pm 1$. Since both $x$ and $y$ grow with $k$, we find infinitely many positive solution... | {
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Are these two quotient rings of $\Bbb Z[x]$ isomorphic? Are the rings $\mathbb{Z}[x]/(x^2+7)$ and $\mathbb{Z}[x]/(2x^2+7)$ isomorphic?
Attempted Solution:
My guess is that they are not isomorphic. I am having trouble demonstrating this. Any hints, as to how i should approach this?
| Suppose they are isomorphic. Then
$\left(\mathbb{Z}[x]/(x^2+7)\right)/(2) \cong \left(\mathbb{Z}[x]/(2x^2+7)\right)/(2)$. Ravi helpfully pointed out that considering ideals in either ring in terms of $x$ will often give us different ideals, but we do not suffer from this problem when using ideals generated by integers ... | {
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Find $\lim\limits_{n\to+\infty}(u_n\sqrt{n})$ Let ${u_n}$ be a sequence defined by $u_o=a \in [0,2), u_n=\frac{u_{n-1}^2-1}{n} $ for all $n \in \mathbb N^*$
Find $\lim\limits_{n\to+\infty}{(u_n\sqrt{n})}$
I try with Cesaro, find $\lim\limits_{n\to+\infty}(\frac{1}{u_n^2}-\frac{1}{u_{n-1}^2})$ then we get $\lim\limits_{... | If ever $u_N\le 0$, then all $-1/n\le u_n\le0$ for all $n>N$, hence $u_n\sqrt n\to 0$.
Therefore we may assume for the rest of the argument that $u_n>0$ for all $n$.
Let $e_n = n+2-u_n$. Then $0<e_0<2$. Using the recursion formula for $e_n$ show that the assumption that $e_n\le2$ for all $n$ leads to $e_n\ge2^n e_0$. T... | {
"language": "en",
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How to prove that the Torus and a Ball have the same Cardinality How to prove that the Torus and a Ball have the same Cardinality ?
The proof is using the Cantor Berenstein Theorem.
I now that they are subsets of $\mathbb{R}^{3}$ so I can write $\leq \aleph$ but I do not know how to prove $\geqslant \aleph$.
Thanks
| Hint:
Show that a circle is equipotent with $[0,2\pi)$ by fixing a base point, and sending each point on the circle to its angle, where $0$ is the base point.
Since both the torus and the ball contain a circle, both have at least $\aleph$ many elements.
| {
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If $G$ is a finite group and $g \in G$, then $O(\langle g\rangle)$ is a divisor of $O(G)$ Does this result mean:
*
*Given any finite group, if we are able to find a cyclic group out of it (subgroup), then the order of the cyclic group will be a divisor of the original group.
If I am right in interpreting it, can o... | You have a finite group $G$ and you take any element $g\in G$. Then $\langle g \rangle$ is a subgroup of $G$. Then, as mentioned in the comment by anon, you can apply Lagrange's theorem to get the conclusion that you want.
As an example of this, you could consider the symmetric group $S_5$. You pick a random element $\... | {
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Is $x^y$ - $a^b$ divisible by $z$, where $y$ is large? The exact problem I'm looking at is:
Is $4^{1536} - 9^{4824}$ divisible by $35$?
But in general, how do you determine divisibility if the exponents are large?
| You can use binomial theorem to break up the individual bases into multiples of the divisor and then you can expand binomially to check divisibilty. This is in a general case of doing such a problem . There may be more methods of doing such a problem.
| {
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Every point closed $\stackrel{?}{\Rightarrow}$ space is Hausdorff If a topological space is Hausdorff, then every point is closed.
Is the converse true?
Edited: Let $G$ be a topological group and $H$ the intersection of all
neighborhoods of zero. Since every coset of $H$ is closed, every point
of $G/H$ will be closed. ... | Notice that $H$ is a closed normal subgroup of $G$, for that see e.g. this for proof that it is a closed subgroup (equal to $\operatorname{cl} \{e\}$), and for normality just notice that conjugation preserves the neighbourhoods of identity (as a set), so it does preserve intersection as well.
From that we see that $G/H... | {
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Finding disjoint neighborhoods of two points in $\Bbb R$ Let $x$ and $y$ be unique real numbers. How do you prove that there exists a neighborhood $P$ of $x$ and a neighborhood $Q$ of $y$ such that $P \cap Q = \emptyset$?
| Hint: open intervals of length $e$ centered at $x$ and $y$ are such neighborhoods, and if $e$ is small enough, they will not intersect. How small does $e$ have to be for this to happen?
| {
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find the eigenvalues of a linear operator T Let $A$ be $m*m$ and B be $n*n$ complex matrices, and consider the linear operator $T$ on the space $C^{m*n}$ of all $m*n$ complex matrices defined by $T(M) = AMB$.
-Show how to construct an eigenvector for $T$ out of a pair of column vectors $X, Y$, where $X$ is an eigenvec... | There is a standard way to do this kind of exercise. Firstly assume that $ A$ and $ B$ are diagonal. Then a short calculation shows that the eigenvalues are as given in previous solutions, i.e., the pairwise products of those of these matrices. The result then holds for diagonalisable matrices by a suitable choice o... | {
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What's the answer of (10+13) ≡? As to Modulus operation I only have seen this form:
(x + y) mod z ≡ K
So I can't understand the question, by the way the answers are :
a) 8 (mod 12)
b) 9 (mod 12)
c) 10 (mod 12)
d) 11 (mod 12)
e) None of the above
| The OP may be taking a Computer Science course, in which if $b$ is a positive integer, then $a\bmod{b}$ is the remainder when $a$ is divided by $b$. In that case $\bmod$ is a binary operator. That is different from the $x\equiv y\pmod{m}$ of number theory, which is a ternary relation (or, for fixed $m$, a binary rela... | {
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Determinant of matrices along a line between two given matrices The question, with no simplifications or motivation:
Let $A$ and $B$ be square matrices of the same size (with real or complex coefficients). What is the most reasonable formula one can find for the determinant $$\det((1-t)A + tB)$$ as a function of $t \i... | I think I have an answer to the last case I mentioned ($A=I$, all $(1-t)I + tB$ invertible). The key is to write
$$\begin{aligned}
\int_0^t c(\tau) \; d\tau &= \operatorname{trace} \int_0^t ((1-\tau)I + \tau B)^{-1} (B-I) \; d\tau \\
&= \operatorname{trace} \log ((1-t)I + tB)
\end{aligned}
$$
using that $\frac{d}{dt}((... | {
"language": "en",
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Proof of the Hardy-Littlewood Tauberian theorem Can someone point me to a proof of the Hardy-Littlewood Tauberian theorem, that is suitable enough to be shown to high school students? (with knowledge of calculus, sequences and series of course)
| Have you looked at the presentation in Titchmarsh's Theory of Functions (Section 7.5)? The only non-elementary part of the argument is Weierstrass's approximation theorem, which you can probably assume as a fact. The preliminary material given also include an "easy" special case where the exposition certainly can be un... | {
"language": "en",
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Evaluating $ \lim\limits_{n\to\infty} \sum_{k=1}^{n^2} \frac{n}{n^2+k^2} $ How would you evaluate the following series?
$$\lim_{n\to\infty} \sum_{k=1}^{n^2} \frac{n}{n^2+k^2} $$
Thanks.
| We have the following important theorem,
Theorem: Let for the monotonic function $f$ ,$\int_{0}^\infty f(x)dx$ exists and we have $\lim_{x\to\infty}f(x)=0$ and $f(x)>0$ then
we have
$$\lim_{h\to0^+}h\sum_{v=0}^\infty f(vh)=\int_{0}^\infty f(x)dx$$
It is enough to take $h^{-1}=t$ and $f(x)=\frac{2}{1+x^2}$, then we... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
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"answer_id": 5
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is it true that the infinity norm can be bounded using the $L_2$ norm the following way? Let $v \in \mathbb{R}^k$, and let $A \in \mathbb{R}^{m \times k}$ and let $B \in \mathbb{R}^{m \times n}$ such that each column of $B$, $B_i$, has $$||B_i||_2 \le 1.$$
Is it true that:
*
*$||v A^{\top} B||_{\infty} \le ||v A^{\t... | 1) Cauchy-Schwarz says $$|(v A^T B)_i| = |v A^T B_i| \le \|v A^T\|_2 \|B_i\|_2 \le \|v A^T\|_2$$
2) Yes because the spectral norm is the operator norm corresponding to the $2$-norm on vectors, and $\|A\|_{\text{spectral}} = \|A^T\|_{\text{spectral}}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/191011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Evaluate $\int\frac{dx}{\sin(x+a)\sin(x+b)}$ Please help me evaluate:
$$
\int\frac{dx}{\sin(x+a)\sin(x+b)}
$$
| The given integral is: $$\int\frac{dx}{\sin(x+a)\sin(x+b)}$$
The given integral can write:
$$\int\frac{dx}{\sin(x+a)\sin(x+b)}=\int\frac{\sin(x+a)}{\sin(x+b)}\cdot\frac{dx}{\sin^2(x+a)}$$
We substition
$$\frac{\sin(x+a)}{\sin(x+b)}=t$$
By the substition of the above have:
$$\frac{dx}{\sin^2(x+a)}=\frac{dt}{\sin(a-b)}$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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calculate $\int_{-\infty}^{+\infty} \cos(at) e^{-bt^2} dt$ Could someone please help me to calculate the integral of:
$$\int_{-\infty}^{+\infty} \cos (at) e^{-bt^2} dt.$$
a and b both real, b>0.
I have tried integration by parts, but I can't seem to simplify it to anything useful. Essentially, I would like to arrive at... | Hint:
Use the fact that
$$\int_{-\infty}^\infty e^{iat- bt^2}\,dt = \sqrt{\frac{\pi}{b}} e^{-a^2/4b} $$
which is valid for $b>0$.
To derive this formula, complete the square in the exponent and then shift the integration contour a bit.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/191125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Representations of integers by a binary quadratic form Let $\mathfrak{F}$ be the set of binary quadratic forms over $\mathbb{Z}$.
Let $f(x, y) = ax^2 + bxy + cy^2 \in \mathfrak{F}$.
Let $\alpha = \left( \begin{array}{ccc}
p & q \\
r & s \end{array} \right)$ be an element of $SL_2(\mathbb{Z})$.
We write $f^\alpha(x, y) ... | Lemma 1
Let $f = ax^2 + bxy + cy^2 \in \mathfrak{F}$.
Let $\alpha = \left( \begin{array}{ccc}
p & q \\
r & s \end{array} \right)$ be an element of $SL_2(\mathbb{Z})$.
Then $f^\alpha(x, y) = f(px + qy, rx + sy) = kx^2 + lxy + my^2$,
where
$k = ap^2 + bpr + cr^2$
$l = 2apq + b(ps + qr) + 2crs$
$m = aq^2 + bqs + cs^2$.
Pr... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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derive the formula for the left rectangle sum $f(x)=x^2+1$ from $0$ to $3$ Simply that, derive the formula for the left rectangle sum $f(x)=x^2+1$ from $0$ to $3$
This is when you use like rectangles and Riemann sums to approximate an integral. Not really sure what this means to derive the formula ?
| Can you follow the very similar example on this web site?
http://www2.seminolestate.edu/lvosbury/CalculusI_Folder/RiemannSumDemo.htm
It is important for you to learn what is going on here.
I would strongly recommend you use all three (left, right and midpoint) to find the integral.
Of course, you know what the answer ... | {
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Sum of angles in $\mathbb{R}^n$ Given three vectors $v_1,v_2$ and $v_3$ in $\mathbb{R}^n$ with the standard scalar product the follwing is true
$$\angle(v_1,v_2)+\angle(v_2,v_3)\geq \angle(v_1,v_3).$$
It tried to substitute $\angle(v_1,v_2) = cos^{-1}\frac{v_1 \cdot v_2}{\Vert v_1 \Vert \Vert v_2 \Vert}$ but I could no... | You can reduce the problem to $\mathbb{R}^3$, and there wlog v2=(0,0,1)
then one gets an easy to prove inequality if one writes everything in polar coordinates.
| {
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"url": "https://math.stackexchange.com/questions/191306",
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"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Example of a sequence with countable many cluster points Can someone give a concrete example of a sequence of reals that has countable infinite many cluster points ?
| First fix some bijection $f : \mathbb{N} \to \mathbb{N} \times \mathbb{N}$. For $n \in \mathbb{N}$ let $g(n)$ denote the first coordinate of $f(n)$ and let $h(n)$ denote the second corrdinate.
Then define a sequence $\{ x_n \}_{n=1}^\infty$ by $$x_n = g(n) + 2^{-h(n)}.$$
Then every natural number is a cluster point o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/191358",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
A formal proof that a sum of infinite series is a series of a sum? I feel confused when dealing with ininities of any kind. E.g. the next equation is confusing me.
$$\displaystyle\sum^\infty_{n} (f_1(n) + f_2(n)) = \displaystyle\sum^\infty_{n_1=1} f_1(n_1) + \displaystyle\sum^\infty_{n_2=1} f_2(n_2)$$
How do people dea... | You have not quite stated a result fully. We state a result, and then write down the main elements of a proof. You should at least scan the proof, and then go to the final paragraph.
We will show that if $\sum_{i=1}^\infty f(i)$ and $\sum_{j=1}^\infty g(j)$ both exist, then so does $\sum_{k=1}^\infty (f(k)+g(k))$, and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/191422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Locally integrable functions Formulation:
Let $v\in L^1_\text{loc}(\mathbb{R}^3)$ and $f \in H^1(\mathbb{R}^3)$ such that
\begin{equation}
\int f^2 v_+ = \int f^2 v_- = +\infty.
\end{equation}
Here, $v_- = \max(0,-f)$, $v_+ = \max(0,f)$, i.e., the negative and
positive parts of $v=v_+ - v_-$, respectively.
Question:... | When $v \in L^1_{loc}(\mathbb{R}^3)$, then $v$ is the density of an absolutely continuous signed measure $\mu$. Take a Hahn decomposition of $\mathbb{R}^3$ in two measurable sets, so that $\mathbb{R}^3$ is disjoint union of say $P$ and $N$, $P$ is positive for $\mu$ and $N$ is negative for $\mu$, defined as $\mu$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/191461",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Random walking and the expected value I was asked this question at an interview, and I didn't know how to solve it. Was curious if anyone could help me.
Lets say we have a square, with vertex's 1234. I can randomly walk to each neighbouring vertex with equal probability. My goal is to start at '1', and get back to '1'.... | By symmetry, the unique invariant probability measure $\pi$ for this Markov chain is uniform on the four states. The expected return time is therefore $\mathbb{E}_1(T_1)=1/\pi(1)=4.$
This principle is easy to remember and can be used to solve other interesting problems.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/191518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Prove that $i^i$ is a real number According to WolframAlpha, $i^i=e^{-\pi/2}$ but I don't know how I can prove it.
| This would come right from Euler's formula. Let's derive it first.
There are many ways to derive it though, the Taylor series method being the most popular; here I’ll go through a different proof.
Let the polar form of the complex number be equal to $z$ .
$$\implies z = \cos x + i\sin x$$
Differentiating on both sides ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/191572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "83",
"answer_count": 6,
"answer_id": 3
} |
Evaluating $\int_0^\infty\frac{\sin(x)}{x^2+1}\, dx$ I have seen $$\int_0^\infty \frac{\cos(x)}{x^2+1} \, dx=\frac{\pi}{2e}$$ evaluated in various ways.
It's rather popular when studying CA.
But, what about $$\int_0^\infty \frac{\sin(x)}{x^2+1} \, dx\,\,?$$
This appears to be trickier and more challenging.
I found th... | Mellin transform of sine is, for $-1<\Re(s)<1$:
$$
G_1(s) = \mathcal{M}_s(\sin(x)) = \int_0^\infty x^{s-1}\sin(x) \mathrm{d} x =\Im \int_0^\infty x^{s-1}\mathrm{e}^{i x} \mathrm{d} x = \Im \left( i^s\int_0^\infty x^{s-1}\mathrm{e}^{-x} \mathrm{d} x \right)= \Gamma(s) \sin\left(\frac{\pi s}{2}\right) = 2^{s-1} \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/191639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 4,
"answer_id": 1
} |
Help evaluating a limit I have the following limit:
$$\lim_{n\rightarrow\infty}e^{-\alpha\sqrt{n}}\sum_{k=0}^{n-1}2^{-n-k} {{n-1+k}\choose k}\sum_{m=0}^{n-1-k}\frac{(\alpha\sqrt{n})^m}{m!}$$
where $\alpha>0$.
Evaluating this in Mathematica suggests that this converges, but I don't know how to evaluate it. Any help wou... | I would start even more simple-mindedly by replacing the inner sum
with its infinite $n$ value of $e^{\alpha \sqrt{n}}$.
This cancels out the outer expression, so we are left with
$\lim_{n\rightarrow\infty}\sum_{k=0}^{n-1}2^{-n-k} {{n-1+k}\choose k}$.
Doing some manipulation,
$\sum_{k=0}^{n-1}2^{-n-k} {{n-1+k}\choose k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/191738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
Showing $H=\langle a,b|a^2=b^3=1,(ab)^n=(ab^{-1}ab)^k\rangle$.
Let $G=\langle a,b|a^2=b^3=1,(ab)^n=(ab^{-1}ab)^k \rangle$. Prove that $G$ can be generated with $ab$ and $ab^{-1}ab$. And from there, $\langle(ab)^n\rangle\subset Z(G)$.
Problem wants $H=\langle ab,ab^{-1}ab \rangle$ to be $G$. Clearly, $H\leqslant G$ an... | The question is answered in the comments. However so that this question does not remain listed as unanswered forever, I will provide a solution. I will also give the details for the first part of the question.
Part 1 Show $G= \langle ab, ab^{-1}ab \rangle$
Let $H=\langle ab, ab^{-1}ab \rangle $. It is clear that $ H ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/191791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
point outside a non-convex shape I have a non-convex shape (object) in black on the figure on the link. At the beginning, All red points are outside the shape. Next, I apply a random transformation on some points. This create a new shape (yellow).
What I want is to fill the outside of the shape with a specific color b... | In that case, use a point in polygon algorithm. Then you can test each point if it is in your new polygon or not.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/191822",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Showing pass equivalence of cinquefoil knot According to C.C. Adams, The knot book, pp 224, "every knot is either pass equivalent to the trefoil knot or the unknot".
A pass move is the following:
Can someone show me how to show that the Cinquefoil knot is pass equivalent to unknot or trefoil? Been trying on paper but ... | The general method is demonstrated in Kauffman's book On Knots. Put the knot into a "band position" So that the Seifert surface is illustrated as a disk with twisted and intertangled bands attached. Then the orientations match those of your figure. You can pass one band over another. Your knot is the braid closure of $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/191894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A question about a closed set Let $X = C([0; 1])$. For all $f, g \in X$, we define the metric $d$ by
$d(f; g) = \sup_x |f(x) - g(x)|$. Show that $S := \{ f\in X : f(0) = 0 \}$ is closed in $(X; d)$.
I am trying to show that $X \setminus S$ is open but I don't know where to start showing that.
I wanna add something more... | I'd try to show that $S$ contains all its limit points. To this end let $f$ be a limit point of $S$. Let $f_n$ be a sequence in $S$ converging to $f$ in the sup norm.
Now we show that $f$ is also in $S$:
By assumption, for $\varepsilon > 0$ you have that $\sup_{z \in [0,1]}|f_n(z) - f(z)| < \varepsilon$ for $n$ large ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/191936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Graph for which certain induced subgraphs are cycles Let us call a graph G $nice$ if for any vertex $v \in G$, the induced subgraph on the vertices adjacent to $v$ is exactly a cycle.
Is there anything that we can conclude about nice graphs? In particular, can we find a different (maybe simpler) but equivalent formulat... | Wrong Answer
Given a finite connected "nice" graph, $G$, you can take all triples $\{a,b,c\}$ of nodes with $\{a,b\}$,$\{b,c\}$, and $\{a,c\}$ edges in the graph.
Take these as $2$-simplexes, and stitch them together in the obvious way.
The fact that $G$ is nice means that each edge must be on exactly two triangles. Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/191986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Quadratic equations that are unsolvable in any successive quadratic extensions of a field of characteristic 2
Show that for a field $L$ of characteristic $2$ there exist quadratic equations which cannot be solved by adjoining square roots of elements in the field $L$.
In $\mathbb{Z_2}$ adjoining all square roots we o... | If $L$ is a finite field of characteristic two, then consider the mapping
$$
p:L\rightarrow L, x\mapsto x+x^2.
$$
Because $F:x\mapsto x^2$ respects sums: $$F(x+y)=(x+y)^2=x^2+2xy+y^2=x^2+y^2=F(x)+F(y),$$ the mapping $p$ is a homomorphism of additive groups. We see that $x\in \mathrm{Ker}\ p$, if and only if $x=0$ or $x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/192061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Solve $\sqrt{x-4} + 10 = \sqrt{x+4}$ Solve: $$\sqrt{x-4} + 10 = \sqrt{x+4}$$
Little help here? >.<
| Square both sides, and you get
$$x - 4 + 20\sqrt{x - 4} + 100 = x + 4$$
This simplifies to
$$20\sqrt{x - 4} = -92$$
or just
$$\sqrt{x - 4} = -\frac{92}{20}$$
Since square roots of numbers are always nonnegative, this cannot have a solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/192125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
} |
limit at infinity $f(x)=x+ax \sin(x)$ Let $f:\Bbb R\rightarrow \Bbb R$ be defined by $f(x)= x+ ax\sin x$.
I would like to show that if $|a| < 1$, then $\lim\limits_{x\rightarrow\pm \infty}f(x)=\pm \infty$.
Thanks for your time.
| We start by looking at the case when $x$ is (large) positive. The idea is that if $|a|\lt 1$, then since $|\sin x|\lt 1$, the term $ax\sin x$, even if it happens to be negative, can't cancel out the large positiveness of the front term $x$. We now proceed more formally.
Note that $|\sin x|\le 1$ for all $x$, so $x|a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/192261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
The product of all elements in $G$ cannot belong to $H$
Let $G$ be a finite group and $H\leq G$ be a subgroup of order odd such that $[G:H]=2$. Therefore the product of all elements in $G$ cannot belong to $H$.
I assume $|H|=m$ so $|G|=2m$. Since $[G:H]=2$ so $H\trianglelefteq G$ and that; half of the elements of the... | Consider the image of the product under the quotient map $G\to G/H\cong C_2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/192311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 3
} |
Internal Direct Sum Questiom I'm posed with the following problem. Given a vector space $\,V\,$ over a field (whose characteristic isn't $\,2$), we have a linear transformation from $\,V\,$ to itself.
We have subspaces
$$V_+=\{v\;:\; Tv=v\}\,\,,\,\, V_-=\{v\;:\; Tv=-v\}$$
I want to show that $\,V\,$ is the internal... | Another approach:
Lemma: In arbitrary characteristic, if $V$ is a $K$-vector space and $P:V\to V$ is an endomorphism with $P^2=\lambda P$, $\lambda\ne 0$, then $V=\ker P\oplus\ker (\lambda I-P)$.
Proof: If $v\in \ker P\cap\ker (\lambda I-P)$, then $\lambda v = (\lambda I-P)v+Pv = 0$, hence $v=0$. Thus $\ker P\cap\ker ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/192368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
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