Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Let X~geometric (1/3), and let Y=|X-5|. Find the range n and PMF of Y. Let X~geometric (1/3), and let Y=|X-5|. Find the range n and PMF of Y.
Here is my trial
If $x=0$, $P(Y=|0-5|)=P(Y=5)=\left(\frac{2}{3}\right)^5 \frac{1}{3}$
If $x=1$, $P(Y=|1-5|)=P(Y=4)=\left(\frac{2}{3}\right)^4 \frac{1}{3}$
If $x=2$, $P(Y=|2-5|)=... | Range of $Y$ is $(0,\infty)$.
$$P(Y=k) \;=\; (2/3)^{k-1} * (1/3).$$
| {
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How does this basic probability question solution make sense? I am just having some trouble understanding why it is valid to answer the following question in a certain way.
Suppose we have iid Bernoulli trials, with $p=0.7$,
We do trials until we have 4 success or 4 fail, if we get 4 success first, we win, else we lose... |
Now what I dont understand is my, one a work page I had seen, the professor had wrote that we can solve it by noting that it is binomial with 7 trials, and we win if we have 4 or more success. But I dont understand why this is valid because how can we have more then 4 successes, ie , that game is over at that point.
... | {
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Proving that $R^2\to 1$ as the degree of a polynomial $k\to \infty$ for a least squares regression. Recently, I thought of the following interesting problem. Given a set of data, I noticed that as the degree of a polynomial increases, in general the $R^2$ value tends to increase too.
I will define the $R^2$ value as th... | A $n-1$ degree polynomial can perfectly fit through $n$ points $(x_i,y_i)$ by choosing the right coefficients. Therefore, for big enough degree of the polynomial, the sum will disappear and thus the $R^2$ will equal one.
| {
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Why Use Arbitrary Unions and Finite Intersections in Topology? Why the definition of a topological space defined under finite intersection and arbitrary union
What if we change the conditions by arbitrary intersection and finite union?
| General topology arose in large part as a generalization of real analysis. On the real line with the usual topology, an arbitrary union or finite intersection of open intervals is open.
We cannot push to even countably infinite intersections, however, since
$$
\bigcap_{n=1}^\infty \left(-\frac{1}{n}, \frac{1}{n} \righ... | {
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Any counter example for $P(A|C)=P(A),P(B|C)=P(B)$ but $P(A\cap B|C)\neq P(A\cap B)$? Let $A,B,C$ be three events, what would be an example that $P(A|C)=P(A)$ and $P(B|C)=P(B)$ do not imply $P(A\cap B|C)= P(A\cap B)$?
| Consider a dice with 8 faces with numbers from $1$ to $8$ on them respectively. We roll that dice. Denote $X$ as the resulting face.
Let $A=[X=1,X=2]$, $B = [X=2,X = 3]$, $C=[X~\text{is even}]$. Then we definitely have $$P(A|C)=P(X=2\vert C)=\frac14=P(A)$$ and $$P(B|C)=P(X=2\vert C)=\frac14=P(B).$$
But $$P(A\cap B|C) ... | {
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On the sum: $\sum\limits_{n=0}^{\infty}\left[\,\sum\limits_{k=1}^{a}\frac{1}{an+k}-\sum\limits_{k=1}^{b}\frac{1}{bn+k}\,\right]$ How to prove:
$$ \sum_{n=0}^{\infty}\left[\,\sum_{k=1}^{a}\frac{1}{a\,n+k}-\sum_{k=1}^{b}\frac{1}{b\,n+k}\,\right]=\log\left(\frac{a}{b}\right)\qquad\colon\,a\,,b\in\mathbb{N}^{+}\tag{1}$$ ... | For starters, let us give an integral representation to the general term of the outer sum:
$$ \sum_{k=1}^{a}\frac{1}{an+k}-\sum_{k=1}^{b}\frac{1}{bn+k} = \int_{0}^{1}\left(x^{an}\sum_{k=1}^{a}x^{k-1}-x^{bn}\sum_{k=1}^{b}x^{k-1}\right)\,dx \tag{1}$$
turning the whole sum into:
$$ \lim_{N\to +\infty}\sum_{n= 0}^{N}\int_{... | {
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How to show that $\lim\limits_{x \to 0} \frac{3x-\sin 3x}{x^3}=9/2$? $\lim\limits_{x \to 0} \frac{3x-\sin 3x}{x^3}$
I need to prove that this limit equals to $\frac{9}{2}$.
Can someone give me a step by step solution?
EDIT: I am sorry. The $x$ goes to $0$, not $1$.
| Using l'Hôpital's rule;$$\lim_{x\to0}\frac{3x-\sin(3x)}{x^3}=\lim_{x\to0}\frac{3-3\cos(3x)}{3x^2}=\lim_{x\to0}\frac{9\sin(3x)}{6x}=\lim_{x\to0}\frac{27\cos(3x)}{6}=\frac{9}{2}$$
| {
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Solving Trigonometric Derivatives I have a function $F(θ) = \sin^{−1} \sqrt{\sin(11θ)}$
I derived the following answer using basic trigonometric and quotient rules.
$\dfrac{11\csc \left(\left(11\theta \right)^{\left(\frac{1}{2}\right)}\right)\cos \left(\left(11\theta \right)\right)}{2\sqrt{1-\sin \left(11\theta \right)... | We have $\sin F(x)= (\sin 11x)^{1/2}.$ so by the Chain Rule, $$(1)....(\cos F(x)\;)\cdot F'(x)=(1/2)(\sin 11x)^{(1/2-1)})(11\cos 11 x)=\frac {11\cos 11x}{2\sqrt {\sin 11x}}.$$ Assuming that we are using the main (or usual) branch of $\sin^{-1}$, which takes values in $[-\pi /2,\pi /2],$ we have $F(x)\in [-\pi /2,\pi ... | {
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Error of Taylor Polynomial a) Construct the Taylor polynomials of degree 4 at $x_0 = 0$ for the following functions:
$$f(x) = \frac{1}{2+x}$$$$ f(x)=\sin\left(\frac{x}{3}\right)$$
b) Find a bound for the error terms for $x\in [-1,1].$
I have the solution to the first part as
$$\frac{1}{2+x} \approx \frac{1}{2}-\frac{x... | Recall the remainder term is bounded by $\displaystyle{Mr^{n+1}\over (n+1)!}$ where $M=\displaystyle\sup_{a\le x\le b}|f^{(n+1)}(x)|$ and $r={b-a\over 2}$ is the radius of the interval in question. In your case you know the derivatives well enough to see that
$$|f^{(n+1)}(x)|=\begin{cases}\displaystyle{n!\over (2+x)^{n... | {
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Translate linear motion to polar coordinates I'm a math noob, so even though I want to understand a general solution to my problem, I don't even know how to articulate it except as an example. So:
The setup:
Suppose I want to draw a segment from (1,2) to (0,1) on the Cartesian plane.
Suppose I get to do so by tracing ... | The transformation from Cartesian coordinates to polar coordinates is
$$
x = r\cos\theta, \quad y=r\sin\theta.
$$
Since the equation of the line on which your two points lie is $y=x+1$, its polar equation is
$$
r\sin\theta=r\cos\theta+1,
$$
which you can solve for $r$ to obtain
$$
r = \frac1{\sin\theta - \cos\theta}.
$... | {
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Finding $\lim_{z\to i} \frac{\arctan(1+z^2)^2}{\sin^2(1+z^2)}$ without using Taylor expansions I'm stuck trying to compute this limit:
$$\lim_{z\to i} \frac{\arctan(1+z^2)^2}{\sin^2(1+z^2)}$$
I tried to use the logarithm form of $\arctan$ and the exponential form of $\sin$ but the formula got too complicated.
| $$\lim_{z\to i} \frac{\arctan(1+z^2)^2}{\sin^2(1+z^2)}=\lim_{w\to 0} \frac{\dfrac{\arctan w^2}{w^2}}{\dfrac{\sin^2 w}{w^2}}=1.$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Find the equation of the tangent to the curve Find the equation of the tangent to the curve:
$$y = e^x +1$$
At the point:
$$(1, e+1)$$
My process:
$$Gradient: y' = e^x$$
Tangent: $$y-(e+1) = e^x(x-1)$$
$$y= xe^x-e^x+e+1$$
I don't understand where my mistake is. I found the derivative (gradient) and then put it into the... | At $(1, e+1)$,
$$y' = e^{1} = e$$
Then,
$$y - (e + 1) = e(x - 1)$$
$$y = ex + 1$$
I think it is a careless mistake.
| {
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Triangle Inequality Issue Suppose that $\left| a \right| \leq 1$ and $\left| b \right| \leq 1$, is there a nice way, other than a proof by cases, to show that $$\left| \left| a \right|^n - \left| b \right|^n \right| \leq 1?$$ I'm obviously aware of the triangle inequality $\left| \left| a \right| - \left| b \right| \ri... | From $|a|^n \le 1$ we get $|a|^n \le 1+|b|^n$, hence $|a|^n-|b|^n \le 1$
In the same way we show: $|b|^n-|a|^n \le 1$
| {
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"timestamp": "2023-03-29T00:00:00",
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Laplace transform of Bessel function of order zero I'm trying to prove that the Laplace transform of the function
$$
J_0(a\sqrt{x^2+2bx})
$$
is
$$
\frac{1}{\sqrt{p^2+a^2}} \exp\left\{bp- b\sqrt{p^2+a^2} \right\}
$$
as asserted in the eqworld. This formula can also be found in the book "Tables of Integral Transforms" pa... | By the substitution $x\mapsto bx$, the problem is equivalent to finding the Laplace transform of $f_c(x)=J_0(c\sqrt{x^2+x})$ or the inverse Laplace transform of
$$ g_c(p)=\frac{1}{\sqrt{p^2+1}}\,\exp\left(\frac{-c}{p+\sqrt{p^2+1}}\right). \tag{1}$$
It is useful to recall that $\mathcal{L}(J_n(x))(p) = \frac{1}{\sqrt{1... | {
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Well Ordering Principle, Zorn's Lemma, Induction, etc. I know there are plenty of proofs online that show Induction implies well ordering, well ordering implies induction, zorn's lemma implies induction, etc. and what I seem to be getting is that in most courses professors introduce these by assuming one and proving th... | All three are variations of the Axiom of Choice: https://en.wikipedia.org/wiki/Axiom_of_choice
There is no ultimate proof because it is an independent axiom.
| {
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How can we prove this inequality involving e? $$ \frac{1}{1+|x|} \le \frac{e^x - 1}{x} \le1 + |x|(e - 2) $$ where $$x \in [-1,0)\cup(0,1]$$
How can we prove this inequality? My text used it to prove the limit of the middle function when $x→0$ which is $1$ using sandwich theorem.
I have no idea how to begin. I know I c... | It sounds awkard to use these inequalities to prove the limit you want.
The limit $\lim_{x\to 0} (e^x-1)/x=1$ follows by L'Hospital's rule as you said, or by using the fact that $(e^x-1)/x=1+\sum_{n\ge 1} x^n/(n+1)!$ and that the power series $\sum_{n\ge 1} x^n/(n+1)!$ is absolutely convergent and hence the sum converg... | {
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Induction - request for help in proving lemma Could anyone help with proving the following lemma, please?
Let: $n\in \mathbb{N}$, $Z_{n}^{*}:=\{k\in\mathbb{N}: k\in\{1,\dots,n\} \wedge \space GCD(k,n)=1\}$. Then: $\forall n\in \mathbb{N} \space \forall p \in \mathbb{P}: |Z_{p^{n}}^{*}|=p^{n}-p^{n-1}$
I tried to pro... | You may be overthinking it. ${\rm gcd \ }(k,p^{n}) = 1$ holds if and only if $k$ is not divisible by $p$.
So you need to count the number of elements in $\mathbb Z_{p^n}$ that are not divisible by $p$.
Ask yourself:
(1) How many elements are there in $\mathbb Z_{p^n}$ in total?
(2) What fraction of these elements are d... | {
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Conjecture about the digits of $\pi$ Consider irrational numbers between 1 and 9.
Lets call a specific one $a$.
Let $n>0$ be an integer.
In decimal consider the first $n$ digits of $a$.
Call that string $A(a,n)$.
Now consider the next $n$ digits of $a$ after the first $n$. Call that string $B(a,n)$.
Define $a$ has the ... | I believe your conjecture is open, but here are some partial results:
Let $a$ be defined to have the infinitely-often-repeating-digits property (IORDP) if there are infinitely many values of $n$ such that $A(a,n) = B(a,n)$.
If $a$ has IORDP, it has irrationality measure at least $2$. That's because in the repeat-digit... | {
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Probability of winning after adding coin condition Assume that we have two games $X, Y$ with probability of winning $P(X)$ and $P(Y)$ respectively. Now I create a new game $Z$: I throw a coin and if I get heads I play $X$. Otherwise I play $Y$. What is the probability of winning in $Z$? Do we need more info on $X, Y$ i... | Let $P(Z)$ be the probability of winning game $Z$. Then we have that $P(Z) = \frac{1}{2}P(X) + \frac{1}{2}P(Y)$. If you draw a probability tree you can visualise why this is true. If you flip a coin - you either get heads or tails, each with probability $\frac{1}{2}$. Then if you get heads, the probability of winning i... | {
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Calculate the image and a basis of the image (matrix)
What's the image of the matrix? What's the basis of the image?
$M=\begin{pmatrix}
-1 & 1 & 1\\
-2 & -3 & 6\\ 0 & -1 & 1 \end{pmatrix}$
First transposed the matrix:
$M^{T}=\begin{pmatrix}
-1 & -2 & 0\\
1 & -3 & -1\\
1 & 6 & 1
\end{pmatrix}$
Now we u... | The image of a matrix is the same as its column space. To find column space, you first find the row echelon form of the given matrix (do not transpose it). The definition of row-echelon form is:
*
*Rows with all zero's are below any nonzero rows
*The leading entry in each nonzero row is a one
*All entries below ... | {
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$\arcsin x- \arccos x= \pi/6$, difficult conclusion I do not understand a conclusion in the following equation. To understand my problem, please read through the example steps and my questions below.
Solve:
$\arcsin x - \arccos x = \frac{\pi}{6}$
$\arcsin x = \arccos x + \frac{\pi}{6}$
Substitute $u$ for $\arccos x$... | Hint:
if $u=\arccos (x)$ than
$\cos u=x$
and
$\sin u=\pm\sqrt{1-\cos^2u}=\pm \sqrt{1-x^2}$
substitute and square your equation and you have
$
\frac{3}{4}(1-x^2)=\frac{1}{4}x^2
$
that can be solved, (with care to improper solutions).
| {
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Why is the Axiom of Choice not needed when the collection of sets is finite? According to Wikipedia:
Informally put, the axiom of choice says that given any collection of
bins, each containing at least one object, it is possible to make a
selection of exactly one object from each bin. In many cases such a
select... | This is a rider to the excellent answers given by Clive and Dustan. To address the case of a single pair of socks: assume you are given an unordered pair of socks $\{x, y\}$ such that the socks $x$ and $y$ have no distinguishing features. Then $x$ and $y$ are the same sock. In this case, you should take the sock back t... | {
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Prove ideal of matrix is that set Given the set of matrices:
$M=\left\{
\begin{pmatrix}
a & 0 \\
0 & b
\end{pmatrix}
\mid
a,b \in \Bbb Z
\right\}$.
It is easily seen that $M$ is a commutative subring of $M_{2,2}(\Bbb Z)$. If K is ideal of M prove that exits $m,n \in \Bbb Z$ to $K=\left\{
\begin{pmatrix}
ma & 0 \\
0 & ... | Let $A = \left\{\begin{pmatrix} a & 0 \\ 0 & 0\end{pmatrix} |\; a \in \mathbb{Z}\right\}$. Clearly $A$ is an ideal of $M$, and $A$ is a ring isomorphic to $\mathbb{Z}$. Since $K$ is an ideal of $M$, we know that $K\cap A$ is also an ideal of $M$. Moreover, we see that $K\cap A$ is an ideal of $A$, and therefore $K\c... | {
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Convergence of the following sum Does the following sum converge? $$\sum_{n=1}^{\infty}\frac{\sin^2(n)}{n}$$
I tried the ratio test and got that $\rho=0$ which means that the series converges absolutely. However, Mathematica and Wolfram Alpha do not give a result when trying to find its convergence. Am I wrong?
| $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newc... | {
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"answer_id": 1
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A team of 11 players with at least 3 bowlers and 1 wicket keeper is to be formed from 16 players; 4 are bowlers; 2 are wicket keepers.
Out of 16 players in a cricket team, 4 are bowlers and 2 are wicket keepers. A team of 11 players with at least 3 bowlers and 1 wicket keeper is to be formed. Find the number of ways t... | Your solution counts some solutions more than once, namely if you have four bowlers or two wicket keeps you count them four times and twice respectively. The book however counts each in a different place: Each term on the LHS is one of four possibilities, depending on how many bowlers/wicket keepers there are.
| {
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Product of functions is $0$ but none of the functions is identically $0$?
Given that $f(x) \cdot g(x) = 0$ for all $x$ is it true that at least one of the functions is $0$ for all $x$?
The correct answer is this doesn't necessarily hold true. Can you give such example? I have a feeling this had something to do with p... | One continuous example: $f(x)=|x|+x, g(x)=|x|-x$.
For any $x$, at least one of the functions must be $0$, but there is nothing stopping them from "sharing" the $x$-axis between them.
| {
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"url": "https://math.stackexchange.com/questions/2168152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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One implication in $f'(x) \ge 0 \iff f \ \text{is monotonically increasing}$ I am trying to understand why $f'(x) \ge 0 \iff f \ \text{is monotonically increasing}$ with the usual set of assumptions. To do this I am trying to prove the two implications. It is relatively easy to get why $\impliedby$ holds since an incre... | Yes, by the fundamental theorem of calculus. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be differentiable with $f'(x) \geq 0 \forall x \in \mathbb{R}$. Taking $x \in \mathbb{R}$ and $h \in \mathbb{R}_{\geq 0}$.
\begin{align*}
f(x+h) - f(x) = \int_{x}^{x+h} f'(t) dt \geq 0
\end{align*}
Rearranging, $f(x + h) \geq f(x)$... | {
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$Why$ is the axis of symmetry of a parabola $-{b\over 2a}$ and ${not}$ ${b\over 2a}$? I'm working on a lesson plan for my students regarding completing the square for a parabola, and I've done the following:
$$\begin{align}ax^2+bx+c &= a\left(x^2+{b\over a}x\right) + c \\ & = a\left(x^2+{b\over a}x+{b^2\over 4a^2}-{b^... | First show that the parabola
$y=ax^2$ is symmetrical about the $y-$axis ($x=0$).
Displacing it by $c$ vertically gives
$y=ax^2+c$
which is also symmetrical about $x=0$.
Moving it by $h$ to the left gives
$y=a(x-h)^2+c$
which is symmetrical about $x-h=0$ or $x=h$.
From the equation given
$$ax^2+bx+c=a\left(x+\frac ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2168412",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
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Can a multiple of $15$ and a multiple of $21$ differ by $1$? I know a solution to this question having to do with the fact that the $\gcd(15, 21) = 3$, so the answer is no.
But I can't figure out what is the reasoning behind this. Any help would be really appreciated!
| No: you can prove that $\text{gcd}(a, a +b) = \text{gcd}(a,b)$. Therefore, we have that $\text{gcd}(a,a+1) = \text{gcd}(a,1) =1$.
If we now consider multiples of $15$ and $21$, say $k \cdot 15, n \cdot 21$ with $k, n \in \mathbb{Z}$, such that $n \cdot 21 = k \cdot 15 + 1$, then we find that $3$ divides $\text{gcd}(k \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
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Is a surjection from the natural numbers enough to show that a set is countable? We have all seen how Cantor showed that the rational numbers are countable with his zig-zag method, but I want to show the same thing without the zig-zag, so here is my approach, does it work?
We can list ALL the rational numbers.
$\frac{1... | There are two common definitions of countability. One is more properly called "countably infinite" where $X$ is countably infinite if it can be put in bijection with $\mathbb{N}$. The other, weaker definition of countability is exactly what you said, i.e. that we can map $\mathbb{N}$ onto $X$. So the latter would encom... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2168594",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that $\cos 20^{\circ} + \cos 100^{\circ} + \cos {140^{\circ}} = 0$ Assume $A = \cos 20^{\circ} + \cos 100^{\circ} + \cos 140^{\circ}$ . Prove that value of $A$ is zero.
My try : $A = 2\cos 60^{\circ} \cos 40^{\circ} + \cos 140^{\circ}$ and I'm stuck here
| $\cos 20^{\circ} + \cos 100^{\circ} = \cos (60^{\circ}-40^{\circ}) + \cos (60^{\circ}+40^{\circ}) = \cos 40^{\circ}$ by using the addition formulae for $\cos$.
Then $\cos 140^{\circ} = \cos (180^{\circ} - 40^{\circ}) = -\cos (40^{\circ})$. So $$\cos 20^{\circ} + \cos 100^{\circ} + \cos 140^{\circ} = \cos 40^{\circ} - ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2168706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Why is $\mathbb{R}$ not a linear subspace of $\mathbb{R}^3$? I thought that $\mathbb{R}$ would just be any 1-dimensional line in $\mathbb{R}^3$ and that as long as you multiply or add two vectors on that line together, you'd still be in $\mathbb{R}$.
| Every one-dimensional subspace of $\mathbb R^3$ is isomorphic to $\mathbb R$ as $\mathbb R$ vector spaces. For many people, isomorphism is enough to call that copy "$\mathbb R$", but one has to remember that it isn't special and that there are many other such copies.
Someone might object to saying that $\mathbb R$ is a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2168812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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second statement of GAGA GAGA theorem 2 states that:
If $\mathscr{F}$ and $\mathscr{G}$ are two coherent algebraic sheaves on X, every analytic homomoprhsim of $\mathscr{F}^h$ into $\mathscr{G}^h$ comes from a unique algebraic homomorphism of $\mathscr{F}$ into $\mathscr{G}$.
What are the homomorphisms between $\mathsc... | If $X$ is a scheme over $\mathbb{C}$ then an algebraic homomorphism between $\mathcal{F}$ and $\mathcal{G}$ is a $\mathcal{O}_{X}$-linear homomorphism of the sheaves $\mathcal{F}$ and $\mathcal{G}$ which is given by a compatible collection of homomorphisms between $\mathcal{G}(U)$ and $\mathcal{F}(U)$ (or if you wish, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$\int_{0}^{1}f(x)\cdot g(x)dx=\int_{0}^{1}f(x)dx\cdot \int_{0}^{1}g(x)dx$ Let $f:[0,1]\rightarrow \mathbb{R} $ be a continuous function so that $\int_{0}^{1}f(x)\cdot g(x)dx=\int_{0}^{1}f(x)dx\cdot \int_{0}^{1}g(x)dx$, for any $g:[0,1]\rightarrow \mathbb{R}$, continuous and not differentiable.
Prove that $f$ is constan... | As @RobertIsrael pointed out, the non-differentiable continuous functions are dense in $C[0,1]$ with respect to the uniform norm, and therefore
$$\int_0^1 f(x) g(x) \, dx = \int_0^1 f(x) \, dx \int_0^1 g(x) \, dx$$
holds for any continuous function $g$. If we choose $f=g$, then we find
$$\int_0^1 f(x)^2 \, dx = \left( ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2169154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that $x<2^x$
How can we prove that given that $x\leq 2$, then $2^x>x$?
I think that this seems to be intuitively correct but I don't know how to prove it. Can it be proven without calculus?
| Note: Hazem Orabi's solution is really the one I prefer personally but since i had fun messing with a more convoluted and ultimately inconclusive approach I'd just like to share it.
Solution
Let's do this for rational numbers $x = p/q$ and then just assume it extends properly to real numbers.
Let $p,q$ be relatively p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2169275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Closed form expression for modulo function? I wonder if there is a closed form expression that returns the values of modulo function for integers $(n \mod m)$? I mean, the modulo operation is not really analytic since one chops off the number after division. But maybe there is a continuous periodic function that is equ... | Remark that for reals variables $\quad\displaystyle{n \mod m=n-\lfloor \frac nm\rfloor\times m}$
So for $m$ fixed the function $x:\mathbb R\to \mathbb R$ with $f(x)=x\mod m$ is piecewise continuous since the $floor$ function is periodic and continuous inside its period interval.
This is the blue curve below ($m=5$).
Yo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Why is a dot product called a dot product? I have just started learning about vectors at school, and some of the applications of vectors are still a bit confusing to me. I'm hoping that finding out the etymology behind the word dot product can help me better understand what a dot product is. In other words, its... | The dot product of two vectors will return a scalar. Algebraically, it is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. The name "dot product" is derived from... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can every possible 'imaginary number' be expressed in terms of $i$? The domain of the function $f(x)=\sqrt{x}$ can be extended to all real numbers by introducing a new number, $i=\sqrt{-1}$.
Can this be done for any function, say $\arcsin{x}$, or $\log{x}$?
What is $\arcsin{2}$? Or $\log{-3}$?
| Euler's formula tells us that $e^{ix} = \cos(x)+i\sin(x)$. Therefore, (in a sense) $\ln (-3) = \ln3+\pi i$, since
$$e^{\ln{3}+\pi i} = e^{\ln 3}e^{\pi i} = 3(\cos \pi + i\sin \pi) = -3$$
(note: $\ln 3+k\pi i$ would have worked for any odd integer $k$)
Similarly, if we let $z = \frac{\pi}{2}-i\ln(2+\sqrt 3)$, then
$$\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2169814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove or disprove that closure of union of infinitely many sets is equivalent to union of infinitely many closure. Let $\overline{A},\overline{B}$ denote the closure of a set $A,B\subseteq \mathbb{R}$ respectively.
Prove or disprove that
\begin{align*}
\bigcup_{n=1}^{\infty}\overline{A_{n}} = \overline{\bigcup_{n=1}^{... | Induction is a little trickier than that, I actually made the same mistake when I first learned it.
Induction says that a statement is true for all $n\in \Bbb N$, but is says nothing about an infinite number. For example $A\cap B$ is open for open $A$ and $B$, so by induction any finite intersection of open sets is ope... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding the remainder to an awful division
If $x=(11+11) \times (12+12) \times (13+13) \times\cdots \times
(18+18) \times (19+19)$, what will be the remainder if $x$ is divided by $100$?
I tried simplifying the expression like this:
$$\frac{2(11)\times2(12)\times2(13)\cdots\times2(19)}{100}$$
$$\frac{2^8(19!\div10!)}... | $$
x=(20+2) \times (20+4) \times (20+6) \times\cdots \times
(20+16) \times (20+18)
$$
After the multiplication is expanded, we get many terms. But the remainder is driven by the last two terms, namely
$$
S = S_1 + S_2 = \left(2\cdot4\cdot\dots\cdot18\right) + \left(20\cdot2\cdot4\cdot\dots\cdot18\left(\frac12+\dots+\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2170064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Digits of irrationals I've been studying floating point arithmetic and I've read somewhere that numbers with infinitely many decimal digits without recursion are irrational.
But since we can't know all the digits of such a number then how did we come to the conclusion that its digits have no recursion? Does it have any... | The simpler (and ancient) way to know if a number $a$ is irrational is to explicitly show that it cannot be expressed as a quotient $\frac{n}{m}$ of two integers $n,m$.
But there are numbers, as the number $\pi+e$, for which we don't know if they are rational or irrational.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2170210",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Real Analysis question comparing functions I am trying to compare the functions $\lim f'_n$ and $(\lim f_n)'$with the following sequence,
$$f_n(x)=\frac{x^n}{n}$$
for $x\in [0,1]$
So I have already $limf_n'$ by calculating the derivative with respect to x and got the following,
$$f_n'(x) = x^{n-1}$$
So $f_n'(x)$ conver... | Note that
$$\lim_{n\to \infty}f_n(x)=\lim_{n\to \infty}\frac{x^n}{n}=0$$
for all $x\in [0,1]$.
What is the derivative of $0$?
The point of the exercise is to show that the derivative of the limit of a sequence of functions (i.e, $(\lim_{n\to \infty}f_n(x))'$) is not always equal to the limit of the derivative of tha... | {
"language": "en",
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Prove: $\text{$n$ is even} \iff n^n\equiv 1\mod{(n+1)}$
Prove: $\text{$n$ is even} \iff n^n\equiv 1\mod{(n+1)}$
where $n\in\mathbb{N}$.
First, to prove $n^n\equiv 1\mod{(n+1)}\implies\text{$n$ is even}$, I supposed $n^n\equiv 1\mod{(n+1)}$ is true.
It goes like this:
The supposed proposition could be rewriten in the ... | The reverse direction is false for the counterexample of $n=1$. It happens to be true however for all other values of $n\geq 2$.
Note that $1^1\equiv 1\pmod{1+1}$ however $1$ is not even.
To prove the reverse direction is true for all $n\geq 2$ we can do the same as when we prove the forward direction except this time... | {
"language": "en",
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$\lim_{x\to0}\,(a^x+b^x-c^x)^{\frac1x}$
Given $a>b>c>0$, calculate$\displaystyle\,\,\lim_{x\to0}\,(a^x+b^x-c^x)^{\frac1x}\,$
I tried doing some algebraic manipulations and squeeze it, but couldn't get much further.
| $$\lim_{x\to0}(a^x+b^x-c^x)^\frac{1}{x}=[1^\infty]=\exp\lim_{x\to 0}(a^x+b^x-c^x-1)\frac{1}{x}\boxed=\\(a^x+b^x-c^x-1)\frac{1}{x}=(a^x-c^x+b^x-1)\frac{1}{x}=c^x\cdot\frac{\left(\frac{a}{c}\right)^x-1}{x}+\frac{b^x-1}{x}\\ \boxed =\exp \lim_{x\to 0}\left(c^x\cdot\frac{\left(\frac{a}{c}\right)^x-1}{x}+\frac{b^x-1}{x} \ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2170630",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What does "the sum of every third element in the $n$-th row of Pascal's triangle" mean? I am looking at the following problem. I don't want to know how it's done, I would just like to see the problem reworded in less confusing terms if possible:
| Let ${n\choose k}$ be the $k^{th}$ entry of the $n^{th}$ row of the triangle. (starting at $k=0$)
$S_{n,0} = {n\choose 0}+{n\choose 3}+{n\choose 6}+\cdots\\
S_{n,1} = {n\choose 1}+{n\choose 4}+{n\choose 7}+\cdots\\
S_{n,2} = {n\choose 2}+{n\choose 5}+{n\choose 8}+\cdots$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2170707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Group theory - composition vs. multiplication In my professor's lecture notes, I've noticed that sometimes the $\circ$ symbol is used when operations are combined, but at other times multiplication is referred to , for example in the definition of a homomorphic relationship:
$$\phi:G\rightarrow H,\,\ \ \ g,h\in G, \ \... | The notation distinguishes between the binary operation for each group. $\star$ is the operation in $G$ and $\cdot$ is the operation in $H$. It stresses that $g \star h$ is in $G$ while $\phi(g) \cdot \phi (h)$ is in $H$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2170872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Two different expansions of $\frac{z}{1-z}$ This is exercise 21 of Chapter 1 from Stein and Shakarchi's Complex Analysis.
Show that for $|z|<1$ one has $$\frac{z}{1-z^2}+\frac{z^2}{1-z^4}+\cdots +\frac{z^{2^n}}{1-z^{2^{n+1}}}+\cdots =\frac{z}{1-z}$$and
$$\frac{z}{1+z}+\frac{2z^2}{1+z^2}+\cdots \frac{2^k z^{2^k}}{1+z^{... | Since minimalrho has explained how to proceed with the given hint, I'll give an alternative method. The $k$th summand of the first series can be written
$$\frac{z^{2^k}}{1 - z^{2^{k}}} - \frac{z^{2^{k+1}}}{1-z^{2^{k+1}}}$$
and the $k$th summand of the second series can be written
$$\frac{2^kz^{2^k}}{1 - z^{2^k}} - \fr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How do I add the terms in the binomial expansion of $(100+2)^6$? So, I stumbled upon the following question.
Using binomial theorem compute $102^6$.
Now, I broke the number into 100+2.
Then, applying binomial theorem
$\binom {6} {0}$$100^6(1)$+$\binom {6} {1}$$100^5(2)$+....
I stumbled upon this step. How did they add... | That number is $1\; 12 \; 6\; 16 \; 24\;192\; 64$ (space added for emphasis). Notice a relationship with the coefficients of the powers of $10$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2171078",
"timestamp": "2023-03-29T00:00:00",
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Show that if $ab \equiv ac$ mod $n$ and $d=(a,n)$, then $b \equiv c$ mod $\frac{n}{d}$ What I know so far:
We know by the definition of congruence that $n$ divides $ab-ac$. So, there exists an integer $k$ such that $a(b-c)=kn$, and since $d=(a,n)$ we know that $a=ds$ and $n=dt$ from some integers $s$ and $t$. Then su... | Since $\gcd(a,n)=d$, we get
$$\gcd\bigg(\frac{a}{d},\frac{n}{d}\bigg)=1.$$ Write
$$r=\frac{a}{d}\quad\text{and}\quad s=\frac{n}{d}.$$ Then $$\gcd(r,s)=1\quad\text{and}\quad \frac{a}{n}=\frac{r}{s}.$$ Also, we get
$$n\big|(ab-ac).$$ Hence,
$$\frac{ab-ac}{n}\in\Bbb Z.$$ But,
$$\begin{align}
\frac{ab-ac}{n}&=\frac{a(b-c)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2171194",
"timestamp": "2023-03-29T00:00:00",
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Evaluating the right limit of: $1+\sin(\frac{2\pi}{x}\sqrt{x})$ at $0$ I have a limit as $$\lim_{x\rightarrow 0^{+}} \left((1+\sin\left(\frac{2\pi}{x}\right))\sqrt{x}\right).$$ I am planning to use Squeeze Theorem, so I say that
$-1 \leq \sin(x) \leq 1 \implies -1 \leq \sin(\frac{2\pi}{x}) \leq 1 \implies 0 \leq 1 + \s... |
Is there any problem?
Your derivation is fine. Using the squeeze theorem is a good idea since
$$
\lim_{x \rightarrow 0^{+}}\sin\left(\cfrac{2\pi}{x}\right)
$$does not exist. Observe that
$$
\lim_{n\rightarrow \infty}\sin\left(\cfrac{2\pi}{1/n}\right)=\lim_{n\rightarrow \infty}\sin(2n\pi)=0
$$ and that
$$
\lim_{n\righ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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True or false: Every real homogeneous linear system of equation which has more than one solution has infinite solutions This is a task from a test exam you can find here (in German):
http://docdro.id/QRtdXkM
Is the following statement true or false?
Every real homogeneous linear system of equation that has more than
... | Indeed, this is even true for non-homogeneous linear systems. Consider the system $Ax=b$, and assume $x_0$ and $x_1$ are solutions. Then for any $x_\lambda = (1-\lambda)x_0+\lambda x_1$ you get
$$Ax_\lambda = A((1-\lambda)x_0 + \lambda x_1) = (1-\lambda)A x_0 + \lambda A x_1 = (1-\lambda) b + \lambda b = b$$
Therefore ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2171371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Help with summation I have been working on the following summation, which is a part of some bigger problem.
$\sum_{i=1}^{n-2} \frac{i}{2(n-2)}$
Now I am stuck, because non of the formulas, that I know, seem to be suitable.
I tried to solve it in so many ways, but I get a wrong answer-- compared to what the professo... | Hint:
If you know this formula:
$$\sum_{k=1}^ak=\frac{a(a+1)}2$$
Then you know this:
$$\sum_{i=1}^{n-2}\frac i{2(n-2)}=\frac1{2(n-2)}\sum_{i=1}^{n-2}i=\frac1{2(n-2)}\frac{(n-2)(n-1)}2=\frac{n-1}4$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2171494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Dimension of tracefree matrix subspace. Let $P\in M_n(\Bbb R)$ be an invertible matrix.
Find the dimension of the following subspace:
$$L = \{ X \in M_n(\Bbb R)| tr(PX)=0\}$$
Don't know where to start. Any help?
| The map
$f:X \rightarrow tr(PX)$ is linear from $M_n(\mathbb{R})$ to $\mathbb{R}$.
Your subspace $L$ is the kernel of $f$.
Since the image of $f$ has dimension 1 (take for example $X=P^{-1}$ to see that $f\neq0$).
You deduce that $\mathrm{dim}(L)=\mathrm{dim}(M_n(\mathbb{R}))-\mathrm{dim}(\mathbb{R})=n^2-1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2171577",
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Notation for a placeholder function that maps to other functions I'm trying to write a very general equation to calculate some $Value$ which relies on a context-dependent function $g$.
How do I concisely communicate that $g$ maps to different functions under different contexts?
So far I have:
\begin{gather}
&\text{Val... | The Iverson brackets could be useful in this case. Let $P$ be a proposition. We write
\begin{align*}
[[P]]=
\begin{cases}
1&\qquad \text{if $P$ is true}\\
0&\qquad \text{if $P$ is false}
\end{cases}
\end{align*}
This way we can write
\begin{align*}
g(a,b)=A(a,b)[[X(a,b)]]+B(a,b)[[Y(a,b)]]+C(a,b)[[Z(a,b)]... | {
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} |
Uniqueness of the remainder and quotient in an Euclidean domain
Let $R$ be a Euclidean ring with Euclidean norm $N$. Let $a,b\in R\setminus\{0\}$ and let $q,r\in R$ such that $a=bq+r$ with $r=0$ or $N(r)<N(b)$.
Prove that $r$ and $q$ are unique if and only if $N(a+b)\le\max\{N(a),N(b)\}$.
I do not see how to relat... | We can assume that $N$ satisfies the property $N(a)\le N(ab)$ for all $a,b\in R\setminus \{0\}$ (for details check this).
Using the above property it's easy to prove that $ N(x)=N (x') $ if $ x $ and $ x'$ are associates.
$\Longrightarrow$) We're going to prove it by contrapositive. Let's suppose that $N(a+b)>\max\{N(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2171776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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What are the chances? There is a class with 20 students.
We pick 4 lucky students who can go to the cinema.
What are the chances? Andrew can go to the cinema, but his best friend John can't go with them.
| The probability that Andrew will be picked out is (as for every student) $\frac4{20}=\frac1{5}$. Under the condition that Andrew is picked out the probability that John will not be picked out is $\frac{16}{19}$.
So:$$\Pr(\text{Andrew lucky}\wedge\text{John unlucky})=$$$$\Pr(\text{Andrew lucky})\Pr(\text{John unlucky}\m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2171866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Research papers in algebraic number theory for undergraduates I am an undergraduate. I am interested in algebraic number theory.
My Background:
(1) I have read the first 5 chapters of the book Number Fields by Daniel A Marcus.
(2) I have read the first and third chapter of Koblitz's book on $p$-adic numbers.
(3) I ha... | I think that the most accessible research-level number theory for an undergraduate with your background is where number theory intersects combinatorics. Two topics that I know professors who use number theory in their combinatorics work work on the arithmetic properties of finite fields and questions about Latin square... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2171989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Prove that $x^p\equiv 1$ (mod $p$) has only one solution. I know that said solution is $x\equiv 1$ (mod $p$). However, I'm having difficulty proving this result.
So far, I've tried $x^p\equiv 1$ (mod $p$) $ \Rightarrow $ $p\mid (x^p-1) \Rightarrow p\mid(x-1)(x^{p-1} + x^{p-2} + \cdots + x + 1)$.
From here, it's clear... | $(x-1)^p \equiv (x^p - 1) \mod p$ by binomial theorem. So, if $p$ divides $(x^p - 1)$, it certainly divides $(x-1)^p$. Therefore, by fermat's last theorem, $p$ divides $x-1$. Hence, you get the desired result.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is the tangent sheaf $\mathscr{T}_{\mathbf{P}^2}$ a direct sum of line bundles? It is well known (theorem of Grothendieck) that every vector bundle on $\mathbf{P}^1$ is a direct sum of line bundles. What about $\mathbf{P}^2$? I figure the answer must be no, but is the tangent sheaf $\mathscr{T}_{\mathbf{P}^2}$ a counte... | The total Chern class of $T_{\mathbb P^2}$ is $1 + 3h + 3h^2$ which is not a product of linear polynomials in $h$, so it cannot be the sum of two lines bundles.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2172159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find the number of generators of the cyclic group $\mathbb{Z}_{p^r}$ Let $p$ be a prime number. Find the number of generators of the cyclic group $\mathbb{Z}_{p^r}$, where $r \in \mathbb{Z} \geq 1$
I'm trying to understand the question and am experimenting with $p=5$ and $r=1,2,3$.
When $r=1$ it generates $\mathbb{Z_5}... | You have the right idea, but remember it's $p^r$, not $pr$. So for instance, for $p=5$ and $r=2$, you get $\mathbb{Z}_{25}$, not $\mathbb{Z}_{10}$.
This also makes the question easier to answer: you just have to count how many integers between $1$ and $p^r$ are relatively prime to $p^r$. An integer is relatively prim... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Poisson distribution formula How is the Poisson distribution formula obtained? according to the theory it represents, for instance, the number of cars that go through some fixed route during a certain time. But again, how is the formula of the Poisson distribution derived?
| Suppose:
*
*The probability of an event occurring in a time interval of length $\Delta t$ is $\lambda \Delta t$, where $\Delta t$ is a small parameter (held fixed for the moment).
*Events in disjoint intervals are independent
*Only one event can occur in each interval.
The first two assumptions are natural; the ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Summation, capital pi properties I know that for summation,
$$\sum_{i = 1}^{x} a_{i} + \sum_{i = x+1}^{n} a_{i} = \sum_{i = 1}^{n} a_{i}$$
Does the same concept apply for capital pi? Such as:
$$\prod_{i = 1}^{x} a_{i} \cdot\prod_{i = x+1}^{n} a_{i} = \prod_{i = 1}^{n} a_{i}$$
I am trying to show that
$$\prod_{i = 1}... | The notation $\displaystyle\prod_{i=n}^m a_i$ can also be written as $\displaystyle\prod_{i\in\{n,..,m\}}a_i$.
This notation can be generalised as $\displaystyle \prod_{i\in A}a_i$ for any set of indices, $A\subseteq\Bbb N^+$.
We do want the following to hold for any disjoint sets of indices, $A, B$.
$$\prod_{i\in A\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Showing that linear subset is not a subspace of the Vector space $V$ I am given the following
$V = \mathbb R^4$
$W = \{(w,x,y,z)\in \mathbb R^4|w+2x-4y+2 = 0\}$
I have to prove or disprove that $W$ is a subspace of $V$.
Now, my linear algebra is fairly weak as I haven't taken it in almost 4 years but for a subspace to ... | No, it is not a subspace.
It is because you have to verify your first point: "$0\in W$".
But $(0,0,0,0)\notin W$ because
$$0+2\times 0-4\times 0+2\ne 0.$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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How can I solve for mod to get a value? I want to solve the following part involving mod:
1 = -5(19) (mod 96)
Apparently, this mod in brackets (mod 96) here is different from the mod that I know e.g. its not the remainder value that you get by dividing.
What of kind of mod is it and how can I solve it step by step to ... | $1 \equiv -5(19) \mod 96$ is equivalent to $1 \equiv -95 \mod 96$ by multiplying out the brackets.
For your question it's equivalent to going round a clock with $96$ hours on it - each time you reach $96$th hour it goes back to $0$ and starts again.
This can also be extended to negative numbers, and adding multiples ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is every $T_1$ topological space homeomorphic to a subspace of a separable $T_1$ topological space (with the same cardinality)? Let $X$ be a $T_1$ topological space , then is it true that $X$ is homeomorphic to a subspace of a separable $T_1$ topological space with the same cardinality of $X$ ? If not then what if we d... | We can define $\overline X = X \sqcup\{1,2,3,\ldots\}$ with the topology generated by these two flavours of open sets.
*
*Sets of the form $U \sqcup \{n,n+1,\ldots\}$ for some $n \in \mathbb N$ and open $U \subset X$.
*Sets of the form $\varnothing \sqcup W$ for any subset $W \subset \{1,2,3,\ldots\}$
Observe that... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Propositional logic: Proof question (p∧q)→r⊢(p→q)→r Am I correct to assume that there is no proof for $$(p∧q)→r ⊢ (p→q)→r$$
I´ve spent hours trying to figure it out, by now I suspect there might have been a mistake in the exercise. I have been able to proof
$$(p∧q)→r ⊢ p→(q→r)$$ (using Fitch notation), so it seems unl... | There is no proof because the tableau of $\lnot (((p\land q)\to r)\to ((p\to q)\to r)))$ is not closed.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2172977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Closed Bounded Intervals and Uniform Continuity For this Theorem, the proof on my course notes is like:
Suppose, for a contradiction,that $f: [a,b] \to \mathbf R$ is continuous but not uniformly continuous on $[a,b]$.
Choose $\epsilon \gt 0$ so that for all $\delta \gt 0$ there exisit $x,y \in [a,b]$ such that $|x... |
Please see Paramanand Singhs answer for a more detailed and correct exposition.
Necessary for the application of the Bolzano-Weierstraß Theorem is that the sequence of question is bounded. But this can only happen if $[a,b]$ is a closed interval, making this condition necessary too (otherwise there might be unbound... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find the minimum value of $f=x^2+y^2+x(1-y)+y(1-x)$, holds on $x,y$ are integers. Let $x$ and $y$ are integers such that $-2\le x\le3$ and $-8\le y\le4$
Find the minimum value of $f=x^2+y^2+x(1-y)+y(1-x)\tag{1}$
From $(1)$, I get $f=(x-y)^2+x+y$ and I don't know what I should do next.
I had tried to use differential of... | This is a kind of solving :
you can write a program in matlab to find minimum
a=zeros(6,13);
for x=-2:3
for y=-8:4
a(x+3,y+9)=(x-y)^2+x+y
end;
end;
if you run this ,you will have
$f(x,y)=(x-y)^2+x+y$
as you can see max $f(x,y)=116 $ ,min $f(x,y)=-4$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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f is analytic mapping of the unit disk into itself such that f(a) = 0. Show $|f(z)| \leq |\frac{z-a}{1-\bar{a}z}|$ $f$ is analytic mapping of the unit disk into itself such that $f(a) = 0$. Show $|f(z)| \leq |\frac{z-a}{1-\bar{a}z}|$
I considered $F(z) =f(\phi_{-a}(z))$ where $\phi_a(z) = \frac{z-a}{1-\bar{a}z}$ maps u... | You're almost there. Now, let $\xi=\phi_{-a}(z)$. We get $F(z)=f(\phi_{-a}(z))=f(\xi)$, for So the inequality would be, in this terms,
$$|f(\xi)|<|\phi_{a}(\xi)|=\left|\frac{\xi-a}{1-\bar a\xi}\right|$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Complication in summation How do I deal with this summation?
$$\sum_{j=n}^{2n-1} \frac{1}{n+j}$$
Do I just substitute all j's with n? That seems to be too easy.
*edit: This is a part of a larger problem, which is to find the limit of the summation as n tends to infinity.
| Hint #1: In general, simplifying the sum of unit fractions is not easy.
Hint #2: Evaluating a sum given a summation notation is easy if you know the index of summation, the lower and upper limits for the index, and the general formula for the summands.
| {
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"timestamp": "2023-03-29T00:00:00",
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How to simplify ${(1+2i)}^6$? How to simplify ${(1+2i)}^6$ using De Moivre's formula?
I have found that $r=\sqrt 5$ and $\tan x=2$ but I can't find the exact value of $x$.
| $$|1+2i|=\sqrt5\;,\;\;\arg(1+2i)=\arctan 2\implies (1+2i)^6=5^3e^{6\arctan 2\cdot i} $$
and now:
$$\begin{cases}&5^3=125\\{}\\
&e^{6\arctan2\cdot i}=0.936+0.352\,i\end{cases}\implies(1+2i)^6=125(0.936+0.352\,i)=117+44i$$
But this looks weird and, anyway, is way simpler first calculating the third power and then squarin... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Are the following convergent or divergent? Are the following convergent or divergent? Justify
$1.\sum_{n=2}^\infty (-1)^n\frac{1}{\ln(n)}$
$2. \left(\frac{1}{ln(n)}\right)^\infty _{n=2}$
$3.\sum_{n=1}^\infty 2^{-n/4}\cos(\pi n/100)$
My thoughts are:
$1.$ is covergent because for $\sum_{n=2}^\infty \frac{1}{\ln(n)}$, $ ... | For the first - don't only state that $|\frac{1}{\ln(n)}|$ is decreasing - the fact that $\lim_{n\to\infty}|\frac{1}{\ln(n)}|=0$ is certainly relevant.
The alternating series test states:
if $|a_n|$ decreases monotonically, all $a_n$ are positive or all negative, and $\lim_{n\to\infty}a_n$, then the alternating series... | {
"language": "en",
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Gamma distribution and pdf Let $X \sim \mathsf{Gamma}(2,3)$ and $Y = 2X.$
Find the pdf of $Y$ at $y=13.5.$
Attempt: $f_X(x)= 2*[1/9*\Gamma(2)]*x*e^{-x/3}.$
Do I have to integrate now?
| $X\sim \text{Gamma}(2,3)\implies f_X(x)=\frac{1}{9\Gamma(2)}xe^{-x/3}$ for $x>0$. The method of CDF yields (for $y>0$):
$$\begin{align}F_Y(y)&=P(Y\leq y)\\&=P\left(X\leq\frac{y}{2}\right)\\&=\int_0^{y/2}\frac{1}{9}xe^{-x/3}dx\\&=\frac{-1}{3}\left[(x+3)e^{-x/3}\right]^{y/2}_0\\&=1-\frac{y+6}{6}e^{-y/6}\end{align}$$
Usin... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Derivative of $ \sin^x(x) $ I was trying to find the derivative for $\sin^x(x)$
I followed two methods, to get to different answers and after comparing the answer with Wolfram Alpha, I found the one which was correct and which was wrong, however I am unable to reconcile why the one which was wrong is incorrect.
The met... | You can use the following trick: take every instance of the variable ($x$) that appears in the expression, temporarily consider the others as constants, and differentiate. The desired derivative is the sum of all contributions.
$$\sin^x(x)$$ yields
$$(\sin^xc)'=\log\sin c\cdot\sin^xc\text{ (derivative of an exponential... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2174117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Do there exists a number with $r$ repeats which forms a perfect square? Take a number $x=\overline{a_1a_2a_3\cdots a_n}$. Its repeated form is like $\overline{\underbrace{a_1a_2a_3\cdots a_n}_{\text{x}}\underbrace{a_1a_2a_3\cdots a_n}_{\text{x}}}$
And $\exists$ $s$ such that the repeated form is a perfect square, that ... | Let $r\ge2$ be the number of repetitions. We look for $n$-digit numbers $a$ such that
$$
\frac{10^{rn}-1}{10^n-1}\,a=\square.
$$
Let $s$ be the square free part of $(10^{rn}-1)/(10^n-1)$, that is
$$
\frac{10^{rn}-1}{10^n-1}=s\,m^2
$$
and $s$ has no square divisors (other than $1$.) Then we must have
$$
a=s\,k^2\quad\te... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2174295",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof by induction the divisiblity Proof by induction, that
$$x_n=10^{(3n+2)} + 4(-1)^n\text{ is divisible by 52, when n}\in N $$
for now I did it like that:
$$\text{for } n=0:$$
$$10^2+4=104$$
$$104/2=52$$
$$\text{Let's assume that:}$$
$$x_n=10^{(3n+2)} + 4(-1)^n=52k$$
$$\text {so else}$$
$$4(-1)^n=52k-10^{3n+2}$$
... | For $n+1$ you have:
$$10^{(3n+2)+3}+4(-1)^{n+1}=10^3\cdot10^{3n+2}+4(-1)^n\cdot(-1)=\\
=10^3\cdot[52k-4(-1)^n]-4(-1)^n=10^3\cdot52k-(-1)^n(4004)$$
but $4004=52\cdot 77$, then
$$10^{(3n+2)+3}+4(-1)^{n+1}=52[10^3k-77(-1)^n]$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2174413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Does zero divide zero I wanted to know is zero divisible by zero? I've read that division by zero is not allowed in mathematics, but for instance in Apostol's Introduction to analytic number theory, it states that only $0$ divides $0$, and I've seen problems in form $0\mid f(x)$, wanting the possible amounts of $x$ (in... | This depends on your context.
In the context of fields, like the rational numbers or the real numbers, $0$ does not divide anything, since division is given by multiplying by the multiplicative inverse (which exists from the axioms).
However, in the context of the natural numbers we define the divisibility relation as ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2174535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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if the atlas of a topological manifold contain only one chart if I have an atlas $\mathcal{A}$ for a topological $m$-manifold $\mathcal{M}$ which consists of only one chart; a homeomorphism $\phi:U \subseteq \mathbb{R}^m \to \mathcal{M}$.
is it then true that $\mathcal{M}$ is necessarily smooth? I think so, because the... | It doesn't make sense to ask whether a topological manifold is "smooth" or not. Smoothness is not a property that topological manifolds may or may not possess; it's an additional structure that must be added -- either a maximal atlas of smoothly compatible charts, or an equivalence class of such atlases, depending on y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2174704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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if $\sum_0^\infty a_n x^n = (\sum_0^\infty x^n )(\sum_0^\infty x^{2n})$ what is $a_n$? if $$\sum_0^\infty a_n x^n = (\sum_0^\infty x^n )(\sum_0^\infty x^{2n})$$
what is $a_n$?
Here is my approach
let $b_n= 1$ and $c_n= x^n$
Then by forming / relating to cauchy product we can conlude that the product is equal to :
$$\... | Using the geometric summation formula $\displaystyle\sum x^n=\dfrac1{1-x}$,
$$A=\frac1{1-x}\frac1{1-x^2}=\frac1{4(1+x)}+\frac1{4(1-x)}+\frac1{2(1-x)^2}\\
=\frac1{4(1+x)}+\frac1{4(1-x)}+\frac12\frac d{dx}\frac x{1-x}$$ so that
$$a_n=\frac14(-1)^n+\frac14+\frac12(n+1).$$
This is a generating function approach.
| {
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"url": "https://math.stackexchange.com/questions/2174818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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Show that $\varphi:\mathbb{R}→Gl_2 (\mathbb{R})$ defined by $\varphi(a)=\begin{pmatrix} 1 & a \\ 0 & 1\end{pmatrix}$ is not an isomorphism $\varphi:\mathbb{R}→Gl_2 (\mathbb{R})$ defined by the matrix $\varphi(a)=\begin{pmatrix} 1 & a \\ 0 & 1\end{pmatrix}$
An isomorphism is a homomorphism that is also bijective. $\v... | Take $A:=\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$; then $A\in GL_2(\Bbb R)$ but $\varphi(a)\neq A$ for every $a\in\Bbb R$, thus $\varphi$ is NOT surjective, and in particular it cannot be an isomorphism.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Inner Product on the space $P_2$ I would just like some confirmation on a problem and a small hint when starting the next. The initial question is:
Show that $⟨p,q⟩=p(0)q(0)+p(1)q(1)+p(2)q(2)$ defines an inner product on the space $P_2$ of polynomials of degree at most 2.
Here's my solution.
$$\langle p,q \rangle... | How to perform the Gram-Schmidt algorithm:
Our new orthonormal basis will be $\{b_1, b_2, b_3\}$. But first let's just find an orthogonal basis $\{c_1, c_2, c_3\}$ and then we'll normalize it.
Start with $c_1 = 1$. OK, the first one was easy.
Now for $c_2$, we want to take the second basis vector in $\{1, x, x^2\}$ a... | {
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"timestamp": "2023-03-29T00:00:00",
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Bessel function at large order I'm trying to expand a modified Bessel function such that
$$K_n \left(\sqrt{n} \left(a_0 + a_1 \frac{1}{n} + a_2 \frac{1}{n^2} + \ldots \right)\right) = A(n) \left( b_0 + b_1 \frac{1}{n} + b_2 \frac{1}{n^2} + \ldots \right) $$
in the large $n$ limit and $A(n)$ is some function of $n$ that... | Unless I misunderstand, there's something missing here. It looks to me like $K_n(\sqrt{n})$ blows up rather rapidly as $n \to \infty$. For example,
for $n=10$ I get approximately $1413.798936$, for $n = 20$ approximately $4.796177691 \times 10^9$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2175241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Can $H_n(A) \cong H_n(X)$, where $(X,A)$ is a simply-connected $CW$ pair, always be induced by the inclusion map? Let $(X,A)$ be a simply connected $CW$ pair such that $H_n(A)\cong H_n(X)$ for some $n$. I wonder if the isomorphism can be induced by inclusion $i:A\hookrightarrow X$ in this case.
Remark:
Note that if th... | No. For instance, let $X=D^3\vee S^2$ and let $A=\partial D^3\subset D^3\subset X$. Then $A$ and $X$ have isomorphic homology: $A$ is homeomorphic to $S^2$, and $X$ is homotopy equivalent to $S^2$. But the inclusion $A\to X$ is nullhomotopic, and in particular induces the $0$ map on $H_2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2175329",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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} |
Proof of a trigonometric inequality in $(0,\pi/2)$ I want to show
$$f(x)=-512\sin\frac{4x}7+1048\sin\frac{6x}7-800\sin\frac{8x}7+216\sin\frac{10x}7>0\quad x\in(0,\pi/2)$$
This trigonometric inequality has been verified by Mathematica using the Plot commend. However, I cannot give a rigorous proof of it. Any suggestion,... | Let $\cos\frac{2x}{7}=t$.
Hence, $$f'(x)=\frac{16}{7}(t-1)^2(270t^3+140t^2-131t-34)$$
By the way, $t=\cos\frac{2x}{7}>\cos\frac{\pi}{7},$ which says that $f'(x)>0$
and $f(x)>f(0)=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2175443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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How to prove floor function inequality $\sum\limits_{k=1}^{n}\frac{\{kx\}}{\lfloor kx\rfloor }<\sum\limits_{k=1}^{n}\frac{1}{2k-1}$ for $x>1$
Let $x>1$ be a real number. Show that for any positive $n$
$$\sum_{k=1}^{n}\dfrac{\{kx\}}{\lfloor kx\rfloor }<\sum_{k=1}^{n}\dfrac{1}{2k-1}\tag{1}$$
where $\{x\}=x-\lfloor x\r... | Some thoughts:
Looking at several plots indicates that $$f_n(x):=\sum_{k=1}^n{\{kx\}\over\lfloor kx\rfloor}$$
is largest immediately to the left of $x=2$. Now for $x=2-\epsilon$ with $0<\epsilon\ll1$ one has
$$\lfloor kx\rfloor=2k-1,\quad\{kx\}=1-2k\epsilon$$
and therefore
$$f_n(x)=\sum_{k=1}^n{1-2k\epsilon\over2k-1}<\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2175624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 3,
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Answering questions for higher dimensions This isn't a question about how to visualize higher dimensions, or how intuit them, or how unintuitive they are.
Rather, it's a hypothetical question about the kinds of questions that might be easier to answer (not necessarily prove, but to suggest) if we could visualize $n$ d... | When I asked this question, multiple commenters were surprised to learn that there don't exist three orthogonal lines with latices coordinates and length $2$ in $3$D space. Multidimension geometry in general tends to be unintuitive because of how used to $2$D we are. In the same vein of thought, manifolds would be a lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2175782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Real life situation for an implicit function What could be an example of a real life situation for which an implicit function may arouse?
In real life, while plotting a value against the other, wouldn't it be the case that the function would not be implicitly defined?
This relates to the understanding
Why is the... | You have a circular curve of which you want to know the diameter. You have a measuring wheel to get the arc length and a meter to get the chord. (This is a true real-life situation in road management.)
Then geometry tells you that
$$\sin\frac ad=\frac cd.$$
Denoting the unknown $y:=a/d$ and the independent variable $x:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2175943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Integral of $\frac1x$ When integrating $\frac1x$, you would get $\ln|x|+c$. Working under that assumption, any given derivative of $\ln|ax|+c$ would give you the same answer as any derivative of $\ln|x|+c$. Given this, after deriving and then integrating, $\ln|2x|+c$ and $\ln|100x|+c$ would have the same solutions of ... | $$\ln|ax|+C = \ln|a| + \ln|x| + C = \ln|x| + C$$
Note that $\ln|a|$ is constant, so we can simply wrap it into $C$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2176091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Compute $\int_{0}^{\infty} \frac{x^\alpha}{x^2+x+1}dx$ So I am asked to integrate:
$\int_{0}^{\infty} \frac{x^{\alpha}}{x^{2}+x+1}dx$, where $-1<\alpha<1$
As this is done in the Complex Analysis course I tried to consider the function
$f(z)=\frac{z^{\alpha}}{z^{2}+z+1}$
and to integrate it over the contour:
$\gamma_{... | In order that $\frac{x^{\alpha}}{x^2+x+1}$ is integrable over $\mathbb{R}^+$ we need $-1<\text{Re}(\alpha)<1$. With such assumption
$$ I(\alpha)=\int_{0}^{+\infty}\frac{x^{\alpha}}{x^2+x+1}\,dx = \int_{0}^{+\infty}\frac{x^{\alpha+1}-x^{\alpha}}{x^3-1}\,dx \tag{0}$$
can be written (by splitting the integration range and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2176217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve the initial value problem $9y''(t)-2y'(t)+y(t)=t\cdot e^{-t/4}; y(0)=0, y'(0)=1$ using Laplace transformations Solve the initial value problem $9y''(t)-2y'(t)+y(t)=t\cdot e^{-t/4}; y(0)=0, y'(0)=1$ using Laplace transformations
I rearranged for $\bar y(s)=\dfrac{1}{(s+1/4)^2(9s^2-2s+1)}+\dfrac{9}{(9s^2-2s+1)}$
Fo... | Noting
$$ \dfrac{s}{9s^2-2s+1}=\frac19\frac{s}{(s-\frac19)^2+(\frac{2\sqrt2}{9})^2},\dfrac{1}{9s^2-2s+1}=\frac19\frac{1}{(s-\frac19)^2+(\frac{2\sqrt2}{9})^2}$$
one has
\begin{eqnarray}
&&L^{-1}(\dfrac{s}{9s^2-2s+1})\\
&=&\frac19L^{-1}(\frac{s}{(s-\frac19)^2+(\frac{2\sqrt2}{9})^2})\\
&=&\frac19L^{-1}(\frac{s-\frac19}{(s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2176283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find all possible positive integers $n$ such that $3^{n-1} + 5^{n-1} \mid 3^n + 5^n $. Proof explanation Question: Find all possible positive integers $n$ such that $3^{n-1} + 5^{n-1} \mid 3^n + 5^n $.
Solution: Let's suppose that $3^{n-1}+5^{n-1}\mid 3^n+5^n$, so there is some positive integer $k$ such that $3^n+5^n=... | Perhaps an easier way to look at it: the equation
$$3^n+5^n=k(3^{n-1}+5^{n-1})$$
can be rewritten
$$5^{n-1}(5-k)=3^{n-1}(k-3)\ .$$
Therefore $5-k$ and $k-3$ must have the same sign. They can't both be negative as then $k$ would be simultaneously less than $3$ and greater than $5$, and they clearly can't both be zero, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2176378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Limit of an indeterminate form with a quadratic expression under square root The problem is:
$$ \lim_{x\to0}\frac{\sqrt{1+x+x^2}-1}{x} $$
So far, I've tried substituting $\sqrt{1+x+x^2}-1$ with some variable $t$, but when $x\to0$, $t\to\sqrt{1}$.
I have also tried to rationalize the numerator, and applied l'hospital... | The binomial expansion for a square root is simple:
$$\sqrt{a+b}=(a+b)^{1/2}=a^{1/2}+\frac12b\cdot a^{-1/2}+\mathcal O(b^2\cdot a^{-3/2})$$
Thus,
$$\sqrt{1+x+x^2}=1^{1/2}+\frac12(x+x^2)1^{-1/2}+\mathcal O(x^2)$$
$$\begin{align}\frac{\sqrt{1+x+x^2}-1}x&=\frac{1+\frac12(x+x^2)-1+\mathcal O(x^2)}x\\&=\frac{\frac12(x+x^2)+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2176506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 3
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How can I 'prove' the derivative of this function? Consider the function
$$
f: (-1, 1) \rightarrow \mathbb{R}, \hspace{15px} f(x) = \sum_{n=1}^{\infty}(-1)^{n+1} \cdot \frac{x^n}{n}
$$
I am required to show that the derivative of this function is
$$
f'(x) = \frac{1}{1+x}
$$
I have attempted to do this using the element... | HINT: if you knows that
$$\int_0^x\frac{\mathrm dt}{1+t}=\ln(1+x),\quad x\in (-1,1)$$
then the question is equivalent to prove that
$$\sum_{k=1}^\infty(-1)^{k+1}\frac{x^k}k=\ln(1+x),\quad x\in(-1,1)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2176594",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Are the fundamental groups of $X$ and $X/A$ isomorphic when $A$ is contractible? Let $X$ be a topological space, $A$ a contractible subspace of $X$, and $f : X \rightarrow X/A$ the quotient map. I want to say that $f$ always induces an isomorphism between fundamental groups, or at least that it does under certain assum... | There is a result which states:
If the map $i:(A,x_0)\hookrightarrow (X,x_0)$ is a cofibration, and $A$ is contractible then the projection $p:(X,x_0)\to (X/A,\ast)$ is a homotopy equivalence.
In particular, if $(X,A,x_0)$ is a relative CW-complex, then the inclusion $i:(A,x_0)\hookrightarrow (X,x_0)$ is a cofibratio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2176683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Prove that (X,d) is complete Let $X=\mathbb{R}$ and $d(x,y)=\min(1, |x-y|)$
I am trying to prove that this metric space is compete. I know that means that every Cauchy sequence converges. So when |x-y| < 1, that would just be the standard metric on R which I know is complete. I'm a bit confused for when d(x,y)=1 though... | First, some intuition: Convergence is a property that only care about "really big" values of $N$. If you change the behavior of the beginning of a sequence, that's never going to change the convergence behavior. Similarly, it only cares about "really small" values of $\epsilon$. Your metric only differs from the usual ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2176790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Why does factoring out eigenvector and result in the identity scaled by the eigenvalue? In deriving a method to find the eigenvalues (and corresponding eigenvectors of a linear mapping), we start with the definition:
$$Av = \lambda v$$
$$Av - \lambda v = 0$$
$$(A - \lambda I)v = 0$$.
I am confused about how to arrive a... | Say $V$ is a vector space over a field $\mathbb F$ and $A, B : V \to V$ are linear maps. Then, for any $\lambda,\mu \in \mathbb F$, the linear map $\lambda A + \mu B : V \to V$ is defined by
$$(\lambda A + \mu B) (v) = \lambda A(v) + \mu B(v)$$
for all $v \in V$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2176890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Prove the following: For n ≥ 4, n ^2 ≤ 2^n I have been asked to prove the following:
For n ≥ 4, n$^2$ ≤ 2$^n$
I will argue by induction the statement P(k): for n ≥ 4, n$^2$ ≤ 2$^n$.
First, consider the base case P(4) = 16 ≤ 16 which is true, so we assume P(n) holds and consider P(n+1).
2$^{n+1}$ = 2$^n$2
We know that 2... | Hint For $n\ge 4$ you know that $(n+1)/n = 1 + 1/n \le 5/4$ so $(n+1)^2/n^2 \le 25/16 \le 2.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2176978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Integral inequality possibly related to probability theory
Let $f$ be Riemann integrable such that $\int_a^b f(t) \ dt = 1$ and $f \geq 0$ on $[a,b]$. If $\sigma \in C^2$ is convex, show
$$\sigma\left(\int_a^b tf(t) \ dt\right)\leq\int_a^b f(t)\sigma(t) \ dt $$
and, in addition, discuss when equality holds given the t... | Of course it's just Jensen's inequality. Since you don't want to use that, here is a direct proof by integrating by parts several times.
By integration by parts, $$\int_a^b t f(t) \, \mathrm{d}t = b - \int_a^b F(t) \, \mathrm{d}t$$ where $F(t) = \int_a^t f(s) \, \mathrm{d}x.$
Consider the function $$g(x) = \sigma \Big... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2177151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Let $\{ x_n \}_{n=1}^{\infty}$ such that $x_{n+1}=x_n-x_n^3$ and $0Let $\{ x_n \}_{n=1}^{\infty}$ such that $x_{n+1}=x_n-x_n^3$ and $0<x_1<1, \ \forall n\in \mathbb{N}.$
Prove:
*
*$\lim_{n \rightarrow \infty} x_n=0$
*Calculate$\ \lim_{n \rightarrow \infty} nx_n^2$
*Let $f$ be a differentiable in $\mathbb{R}$ such ... | $\quad$ $\bullet$ For the first one, we recall $t \in ]0,1[$ then $t > t^3 > 0$. So from $x_1 \in ]0,1[$ and $x_{n+1} = x_n - x_n^3, \forall n \geq 2$, we can prove that $ \forall n \in \mathbb{N}, x_n > 0$ and $$x_1 > x_2 > x_3 > \ldots > 0\ .$$
Hence $\{x_n\}_{n \in \mathbb{N}}$ converges. Suppose that $\lim\limits_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2177262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
What are the trailing number of the zeroes in the given integer
Problem Statement:- The number of zeroes at the end of the integer
$$100!-101!+\ldots-109!+110!$$
I am having a bit of a trouble in thinking how do I proceed. A little push in the right direction would be appreciated.
And if you are posting a full sol... | The number of zero digits at the end of $n!$ can be found by looking at the factors, and decomposing them into $2$ and $5$ factors. You need one $2$ and one $5$ to produce a trailing zero.
How to combine this with that alternating sum, I have no idea yet.
BTW: I explained it to my friend Ruby and she tells me the sum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2177408",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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