Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Example of strict inclusion in continuity condition $f(\overline{A})\subseteq \overline{f(A)}$ One definition of continuity is the condition
$$f(\overline{A})\subseteq \overline{f(A)},$$
for all $A\subseteq X$.
To understand this condition better, I tried to find an example of a real-valued function $f\colon\mathbb{R}\... | Yes. $\;\;\;$ $\hspace{.04 in}f(x) = \dfrac1{1\hspace{-0.05 in}+\hspace{-0.04 in}\left(\hspace{-0.02 in}x^2\hspace{-0.04 in}\right)} \:$ and $\: A = \mathbb{R}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1358951",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Prove that $\sum_{i=1}^{i=n} \frac{1}{i(n+1-i)} \le1$ $$f(n)=\sum_{i=1}^{i=n} \dfrac{1}{i(n+1-i)} \le 1$$
For example, we have $f(3)=\dfrac{1}{1\cdot3}+\dfrac{1}{2\cdot2}+\dfrac{1}{3\cdot1}=\dfrac{11}{12}\lt 1$
If true, it can be used to prove:
Proving $x\ln^2x−(x−1)^2<0$ for all $x∈(0,1)$
Also, can you prove $f(n)\g... | For second part
$f(n)-f(n+1)= \sum\limits_{i=1}^n [\frac{1}{i(n+1-i)}-\frac{1}{i(n+2-i)}] -\frac{1}{n+1}$
$= \sum\limits_{i=1}^n [\frac{1}{i(n+1-i)}-\frac{1}{i(n+2-i)} -\frac{1}{n(n+1)}]$
$= \sum\limits_{i=1}^n [\frac{1}{i(n+1-i)}-\frac{1}{i(n+2-i)} -(\frac{1}{n}-\frac{1}{n+1})]$
$= \sum\limits_{i=1}^n [(\frac{1}{i(n+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1359030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Why is $-i^3 = i$? Why is the value of $-i^3$ equal to $i$?
After experimenting, I got this result -
$-i^3=-i^2\cdot -i=1 \cdot -i=-i$
What is the error in my proof?
EDIT
Here is the original proof -
$$-i^3=\left(\frac1i\right)^3=\frac{1}{i^3}=\frac{1}{-i}=-(-i)=i$$
| Wondering if you meant $(-i)^3$ or $-(i^3)$?
For the former, $(-i)^3=-i \times -i \times -i=-(i^3)=-[(i^2)(i)]=-[(-1)(i)]=i$. Note that $i^2=-1$.
For the latter, $-(i^3)=-[(i^2)(i)]=-[(-1)(i)]=i$.
So, both are the same; i.e., $(-i)^3=i=-(i^3)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1359206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
Prove that $\mathbb Z^n$ is not isomorphic to $\mathbb Z^m$ for $m\neq n$
Prove that $\mathbb Z^n$ is not isomorphic to $\mathbb Z^m$ for $m\neq n$.
My try:
Let $\mathbb Z^n\cong \mathbb Z^m $. To show that $m=n$.
Case 1: Let $m>n$. Now that $\mathbb Z^m$ has $m$ generators whereas $\mathbb Z^n$ has $n$ generators a... | $\mathbb{Z}^n \cong \mathbb{Z}^m$ implies $(\mathbb{Z}/2)^n \cong \mathbb{Z}^n / 2 \mathbb{Z}^n \cong \mathbb{Z}^m / 2 \mathbb{Z}^m \cong (\mathbb{Z}/2)^m$. By comparing the number of elements, we get $2^n=2^m$, i.e. $n=m$. (No linear algebra is necessary here!)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1359326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 1
} |
An "apparent" contradiction for eigenvalues signs of $A=\left( \begin{array}{cc} a & -a \\ a-1 & 1-a \\ \end{array} \right)$. The following matrix
$$A=\left(
\begin{array}{cc}
a & -a \\
a-1 & 1-a \\
\end{array}
\right)$$
has eigenvalues $\lambda=0,1$ $\forall a\in\mathbb R$. Therefore $\lambda\geq 0$.
So I... | Checking that all eigenvalues are non-negative does not imply that a matrix is positive semidefinite. This test only works if the matrix is Hermitian (symmetric in the real case), which the original matrix is not.
For a concrete counterexample, take $a=0, x=10, y=1$, and your quadratic form is negative.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1359457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Differentiate the Function: $y=\frac{ae^x+b}{ce^x+d}$ $y=\frac{ae^x+b}{ce^x+d}$
$\frac{(ce^x+d)\cdot [ae^x+b]'-[(ae^x+b)\cdot[ce^x+d]'}{(ce^x+d)^2}$
numerator only shown (') indicates find the derivative
$(ce^x+d)\cdot(a[e^x]'+(e^x)[a]')+1)-[(ae^x+b)\cdot (c[e^x]'+e^x[c]')+1]$
$(ce^x+d)\cdot(ae^x+(e^x))+1)-[(ae^x+b)\cd... | $\frac{ae^x+b}{ce^x+d}=\frac{a}{c}-\frac{ad-bc}{c}(ce^x+d)^{-1}$ implies $f'(x)=(ad-bc)e^x(ce^x+d)^{-2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1359520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Help with this Trigonometry integral I've to find this integral:
$$\int_{\frac{\pi}{365}}^{\frac{365} {\pi}}\left(\frac{4^\pi}{\tan(x)}+\tanh^{-1}(x)-4\sec^2(x)\right)dx$$
So far as I know to go:
$$\int_{\frac{\pi}{365}}^{\frac{365} {\pi}}\left(\frac{4^\pi}{\tan(x)}+\tanh^{-1}(x)-4\sec^2(x)\right)dx=$$
$$\int_{\frac{\p... | Hints:
$$\begin{align}&\int\frac{dx}{\tan x}=\int\frac{\cos x}{\sin x}dx=\log|\sin x|+C\\{}\\
&\int\arctan x\;dx\stackrel{\text{by parts}}=x\arctan x-\frac12\log\left(1+x^2\right)+C\\{}\\
&\int\sec^2x\;dx=\tan x+C\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1359590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Is it possible to put an equilateral triangle onto a square grid so that all the vertices are in corners? In the following collection of problems - arXiv:1110.1556v2 [math.HO] - the following question is posed:
Is it possible to put an equilateral triangle onto a square grid so that all the vertices are in corners?
T... | No, because that would imply an infinite sequence of smaller and smaller triangles with the same property:
$\hspace{90pt}$
The key to the proof below is this property:
For any point $(x,y) \in \mathbb{Z}^2$ we have that $(-y,x)$ and $(y,-x)$ are its ccw and cw rotations around $(0,0)$ by $\frac{\pi}{2}$.
This implies... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1359762",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 5,
"answer_id": 3
} |
Show $|\exp(-x/2) - \exp(-y/2)| \leq |x-y|/4$ for $x,y\geq 0$. I am trying to show this inequality:
$$
\left|e^{-x/2} - e^{-y/2}\right| \leq \frac{|x-y|}{4}
$$
for $x,y\geq 0$.
I've gotten stuck and could use some kind assistance.
Many thanks in advanced!
| This is incorrect as written. E.g., for $x = 0, y = 1$, $$LHS = 1 - e^{-1/2} \approx 0.393 \color{red}{>} \frac{1}{4} = RHS$$
However it is true that
$$\left|e^{-x/2} - e^{-y/2}\right| \leq \frac{|x-y|}{2} \quad\text{ for all } x, y \geq 0$$
To see this, write $f(x) = e^{-x/2}$. Then by the Mean Value Theorem, for al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1359859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Integral of divergence equal to divergence of integral? Just as the heading reads...is the integral of the divergence of a vector field equal to the divergence of the integral of a vector field?
$\int\nabla\cdot\vec U dz = 0$
same as
$\frac\partial\partial_x \int u(x,y,z) dz +\frac\partial\partial_y \int v(x,y,z) dz... | For terms like $\frac{\partial}{\partial x} \int u(x,y,z) \mathrm dz$ you could apply the Leibniz integral rule, see https://en.wikipedia.org/wiki/Leibniz_integral_rule.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1360119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Showing f(0) is bound above by geometric mean of supremum over intervals? So I am working on the following problem.
Suppose that $f$ is entire and $n$ is a fixed positive integer. If
$$I_k:=\left[\frac{2(k-1)\pi}{n},\frac{2k\pi}{n}\right],$$
for $k=1,2,\dots,n$ and $$\alpha_k:=\sup_{\theta\in I_k}|f(e^{i\theta})|,... | If $f(0)=0$ there is nothing to prove. From now on assume $f(0)\ne0$.
Let $a_1,\dots,a_m$ be the zeroes of $f$ in $\{|z|<1\}$. Suppose first that $f(z)\ne0$ if $|z|=1$. By Jensen's formula
$$
\log|f(0)|=\sum_{i=1}^m\log|a_k|+\frac{1}{2\,\pi}\int_0^{2\pi}\log|f(e^{i\theta})|\,d\theta\le\frac1n\sum_{k=1}^n\log|\alpha_k|,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Computing the shape operator I am trying to compute the shape operator and Gaussian curvature for some smooth zero sets of polynomials $f$ in $\mathbb{R}^n$, oriented by $N = \nabla f / || \nabla f||$
The approach I thinking about is this (which I didn't learn from a text, so maybe there is a problem with it):
*
*Co... | Okay - I figured out my mistake. Pretty silly - I was jumping from 2 to 3 as if my choice was of an orthonormal basis, but of course it was not.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1360332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$|\mathbb N^{\mathbb N}| = |2^\mathbb N|$ by finding a bijection I was trying to find a direct proof that $|\mathbb N^{\mathbb N}| = |2^\mathbb N|$, by finding a bijection between the two sets. The idea that came to mind was to start with the sequence of natural numbers. Each natural number would be 'translated' in a s... | I am thinking of this mapping:
for $A = (a_1,a_2,\cdots,a_n,\cdots) \in \mathbb{N}^\mathbb{N}$,
1) if $a_1 > 0$, then map $A$ to:
$$ (a_1 \mbox{ copies of } 1, \, a_2 \mbox{ copies of } 0, \, \cdots , a_n \mbox{ copies of } (n \mod 2), \,), \cdots$$
2) if $a_1 = 0$, and there exists a $k \in \mathbb{N}$ such that $a_k ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1360428",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Is $1 : 7 = 1 / 8$ or is it $1/7$? In a certain (non-mathematical) Stack Exchange, when I wrote $n : m = n / m$ where $n$ and $m$ are positive integers, one of the moderators said "No! $n : m$ is usually the notation for "$n$ parts in $(n + m)$ parts vs. $m$ parts in $(n + m)$ parts, thus it means $n / (m + n)$." And m... | If an amount of money is shared among A and B in the ratio $3:5$ then A gets ${3\over 8}$ of the total, but ${3\over5}$ as much as B.
In my view an expression of the form $a:b$ is NAN (not a number) but a way of talking to be parsed in real time. Mathematically the pair $(a,b)$ can be considered as homogeneous coordina... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1360562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 4
} |
Summation stuck under radical sign I am trying to evaluate the following sum, but I'm unable to solve it in any general way.
$$S=\sum_{k=1}^n\sqrt{1+\frac{1}{(k)^2}+\frac{1}{(k+1)^2} }$$
How can I do it?
| The expression under the square root is $$\frac {k^4+2k^3+3k^2+2k+1} {k^2 (k+1)^2} = \frac {(1+k+k^2)^2} {k^2 (k+1)^2}$$ So your sum becomes $$\sum \limits _{k=1} ^n \frac {1+k+k^2} {k (k+1)} = \sum \limits _{k=1} ^n \left(1 + \frac 1 {k (k+1)} \right) = n + \sum \limits _{k=1} ^n \left(\frac 1 k - \frac 1 {k+1}\right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1360643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
How to show $ \Big\vert \frac{\sin(x)}{x} \Big\vert $ is bounded by $1$? This may be a silly question, but I cannot figure it out. I want to prove that
$ \Big\vert \frac{\sin(x)}{x} \Big\vert \leq 1 $ for $x\in[-1,0)\cup(0,1]$,
but I don't even know where to start.
| If you draw a picture, it looks like we can use the triangle with points $A =(\cos(x),\sin(x))$, $B = (1,0)$, $C = (\cos(x),0)$. $AC$ has length $\sin(x)$, and $AB$ is the hypotenuse. So $AC$ is longer than $\sin(x)$. But then, since the shortest path connecting two points is a straight line, we must have $|AB| \leq ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1360747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Given that $2\cos(x + 50) = \sin(x + 40)$ show that $\tan x = \frac{1}{3}\tan 40$ Given that:
$$
2\cos(x + 50) = \sin(x + 40)
$$
Show, without using a calculator, that:
$$
\tan x = \frac{1}{3}\tan 40
$$
I've got the majority of it:
$$
2\cos x\cos50-2\sin x\sin50=\sin x\cos40+\cos x\sin40\\
$$
$$
\frac{2\cos50 - \sin40}... | Where you have left of using $\cos(90^\circ-x)=\sin x,\sin(90^\circ-x)=\cos x$
$$\frac{2\cos50^\circ - \sin40^\circ}{2\sin50^\circ + \cos40^\circ}=\frac{2\sin40^\circ- \sin40^\circ}{2\cos40^\circ + \cos40^\circ}=?$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1360841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Minimum 6-connected graph on 200 vertices
Find the samllest number of edges in 6-vertex-connected graph on 200 vertices.
I think that the answer is 600 , using the fact that $\delta(G) \geq \kappa(G)$.
But the smallest 6-vertex-connected graph that I could come up with is $K_{6,194}$ (the complete bipartite graph wit... | As JMoritz noted :
I would think that the graph with adjacencies defined as $v_i$ is adjacent to $v_j$ with $j=((−1+i+n) \ mod \ 200)+1,$ $n∈\{−3,−2,−1,1,2,3\}$. (I.e. $v_4$ is adjacent to each of $v_1,v_2,v_3,v_5,v_6,v_7$ whereas $v_1$ is adjacent to $v_{198},v_{199},v_{200},v_2,v_3,v_4$). Each vertex has exactly 6 e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1360928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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In dual basis, why the functions are of the form $\sum_{i=1}^{n}a_ix_i$? My book says (Linear Algebra - Lipschutz): Let V vector space where $dim(V) = n$. Any functional $\phi$ of $V*$ has the representation $\phi(x_1, x_2, ..., x_n) = a_1x_1 + a_2x_2 + ... + a_nx_n$. Why?
I also need to study the topic "Dual basis" in... | Let $X$ be a vector space and $X^*$ be the dual space. Then $X^*$ consists of linear functions $T:X\rightarrow\mathbb{R}$. Let $\{e_i\}_{i=1}^n$ be a basis of $X$. Define $a_i:=T(e_i)$. Any vector $x\in X$ can be written as $x=\sum_{i=1}^n x_ie_i$. Thus by linearity $T(v)=\sum_{i=1}^n a_i x_i$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1361040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding all groups of order $7$ up to isomorphism? I'm learning group theory but I didn't learn any concepts of building groups.
I know that there exists the identity group $\{e\}$, and the group with 2 elements: $\{e,a\}$.
If I try to create a group with 3 elements, let's say: $\{e,a,b\}$ then we would have:
$ea = a, ... | Up to isomorphism, there is only one group of prime order $p$.
Any group of prime order $p$ isomorphic to $C_p$: the cyclic group of order $p$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1361137",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
Associated matrix with respect of a basis Given the following linear transformation:
$$f : \mathbb{R}^2 \to \mathbb{R}^3 | f(1; 0) = (1; 1; 0), f(0; 1)=(0; 1; 1)$$
find the associated matrix of $f$ with respect of the following basis:
$R = ((1; 0); (0; 1))$ of $\mathbb{R}^2$
and
$R^1 =
((1; 0; 0); (1; 1; 0); (0; 1; 1))... | Assuming your associated Matrix is with respect to $R$ and the standart basis of $\mathbb{R}^3$, it is correct. But your associated Matrix is always depending on both bases of your vector spaces.
As $f(1;0)=0\cdot(1;0;0)+1\cdot(1;1;0)+0\cdot (0;1;1)$
and $f(0;1)=0\cdot(1;0;0)+0\cdot(1;1;0)+1\cdot(0;1;1)$
your Matri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1361223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Doubly stochastic matrix proof
A transition matrix $P$ is said to be doubly stochastic if the sum
over each column equals one, that is $\sum_i P_{ij}=1\space\forall i$.
If such a chain is irreducible and aperiodic and consists of $M+1$
states $0,1,\dots,M$ show that the limiting probabilities are given by
$$\p... | Proof:
We first must note that $\pi_j$ is the unique solution to $\pi_j=\sum \limits_{i=0} \pi_i P_{ij}$ and $\sum \limits_{i=0}\pi_i=1$.
Let's use $\pi_i=1$. From the double stochastic nature of the matrix, we have $$\pi_j=\sum_{i=0}^M \pi_iP_{ij}=\sum_{i=0}^M P_{ij}=1$$
Hence, $\pi_i=1$ is a valid solution to the fi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1361358",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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If $\lim\limits_{n \to \infty} \frac{a_n}{b_n}=1 \rightarrow \sum_{n=1}^\infty a_n = \sum_{n=1}^\infty b_n$ If $\lim\limits_{n \to \infty} \frac{a_n}{b_n}=1 \rightarrow \sum_{n=1}^\infty a_n = \sum_{n=1}^\infty b_n$
How may I prove this? Or give an example where it doesn't apply
Taking into account that $a_n > 0, b_n >... | this doesn't work in general, your series don't even have to exist, take
$a_n=b_n=2$ then $\sum_{n=1}^{\infty}a_n$ doesn't exists.
Or another example $a_n=b_n=2+(-1)^n$, the same problem with none existing sums...so you need at the very least that the sequences $a_n,b_n$ are converging to $0$ for having the series to b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1361441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How to use definition of limit to compute the derivative of |x| Using definition of limit, I need to show
$$\lim_{\epsilon \to 0} \frac {|x + \epsilon| - |x|}{\epsilon} = \frac {x}{|x| }, x \neq 0$$
How should I proceed to get out of the absolute value signs?
| HINT:
Multiply and divide by $$|x+\epsilon|+|x|$$
SPOILER ALERT:
$$\begin{align}\lim_{\epsilon \to 0}\left(\frac{|x+\epsilon|-|x)}{\epsilon}\right)&=\lim_{\epsilon \to 0}\left(\left(\frac{|x+\epsilon|-|x|}{\epsilon}\right)\left(\frac{|x+\epsilon|+|x|}{|x+\epsilon|+|x|}\right)\right)\\\\&=\lim_{\epsilon \to 0}\left(\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1361513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
How to find the integral $\int \frac{\sqrt{1+x^{2n}}\left(\log(1+x^{2n}) -2n \log x\right)}{x^{3n+1}}dx$? How to evaluate the integral :
$$\int \frac{\sqrt{1+x^{2n}} \, \left(\ln(1+x^{2n}) -2n \, \ln x \right) \, dx}{x^{3n+1}}$$
I have attempted an evaluation, but I am at a loss as to a useful result. Thanks for any a... | By using the change of variable $$u=\dfrac1{x^{2n}},\quad \log u = -2n \log x, \quad \dfrac{du}u=-2n\dfrac{dx}x,$$ then an integration by parts, one gets
$$
\begin{align}
&\int \frac{\sqrt{1+x^{2n}}\{\log(1+x^{2n}) -2n \log x\}}{x^{3n+1}}dx\\\\
&= \int \frac{\sqrt{1+x^{2n}}\:\log\left(1+\frac1{x^{2n}}\right)}{x^{3n+1}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1361575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
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Can this be the class-equation of a finite group $G$ of order 10? Can this be the class-equation of a finite group $G$ of order 10?
$10=1+1+2+2+2+2$
I know that if conjugacy class of an element has order one then it must belong to $Z(G)$ and vice versa. Here $o(Z(G))=2$ .Also order of the conjugacy class must divide ... | $G/Z(G)$ is cyclic group of order $5$ and hence it implies that $G$ is abelian see, which means $G=Z(G)$ but class equation tells us $|Z(G)|=2$. Hence not possible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1361686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Maximum and minimum Expected values when taking colored balls We have a sack with $60$ balls.
From them $15$ balls are red, $15$ green, $15$ blue and $15$ yellow.
We take $30$ balls from the sack.
What's the expected number of balls of the color from which the most balls had been taken? And from the color from which th... | Comment. This plot shows maximum and minimum values from a million runs
of this experiment. Points are randomly 'jittered' $\pm 0.3$
to prevent massive 'overplotting'. A few very rare, but possible
combinations of values at upper-left of the plot did not occur in this particular
simulation. From computations, the respe... | {
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"url": "https://math.stackexchange.com/questions/1361789",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Expanding a function into a series I am trying to follow a proof in QFT notes, however I am unable to follow this step - it's basically Laurent/Taylor expansion but I have very little experience with it.
It's claimed that:
$$\frac{\pi}{2d}\frac{e^{a\pi/d}}{(e^{a\pi/d}-1)^2}=\frac{d}{2\pi a^2} - \frac{\pi}{24d} + O(a^2)... | $a<<d\iff \dfrac ad<<1, \dfrac{a\pi}d<<1, $ let $\dfrac{a\pi}d=2x$
$$F=\dfrac{e^{2x}}{(e^{2x}-1)^2}=\dfrac1{(e^x-e^{-x})^2}$$
Use $e^y=\sum_{r=0}^\infty\dfrac{y^r}{r!}$
$$F=\dfrac1{4\left[x+\dfrac{x^3}{3!}+O(x^5)\right]}=\dfrac1{4x^2\left[1+\dfrac{x^2}6+O(x^4)\right]}$$
As $1+\dfrac{x^2}6+O(x^4)\approx1+\dfrac{x^2}6,$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1361968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Problem in proof of Chinese remainder theorem, and applying it. Please don't mark it as duplicate. First read the whole question.
So Chinese Remainder Theorem states that,:
Let $n_1,n_2,...,n_k$ be $k$ positive integers which are pairwise relatively prime. If $a_1,a_2,...,a_k$ are such that $(a_j,n_j)=1$ for $j=1,2,.... | Here
we have a unique solution $x\equiv m_j(\mod n_j)$. $(2)$
it should say instead $x\equiv m_j'(\mod n_j)$. The $m_j'$ is the solution of $m_jx\equiv 1(\mod n_j)$. To use the notation $m_j'$ is somewhat common, but it is not crucial either. You could just as well say let $u_j$ be the solution of $m_jx\equiv 1(\m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1362078",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Limit of a product is the product of the limits - when? The limit of the product of two functions should be equal to the product of the limits:
$$\lim_{x\to\infty}f(x)g(x) = \lim_{x\to\infty}f(x) \lim_{x\to\infty}g(x)$$
Now, the limit of $\frac{(x-1)3}{4x}$ = $\frac{3}{4}$
But the limit of $\frac{x-1}{4}$ = $\infty$ an... | The two expressions are equal when both of the limits on the right side exist and are finite numbers.
Likewise $\displaystyle \lim_{x\to a}\frac{f(x)}{g(x)} = \frac{\lim_{x\to a} f(x)}{\lim_{x\to b} g(x)}$ if the limits in the numerator and denominator exist and are finite numbers and the limit in the denominator is no... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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find the integral using Integration by partial Fractions Here is my work for this problem...just wanted a check over and see if i missed anything
Original Problem: $\int$ $\frac6{x^3-3x^2}$
F 6/x^3-3x^2= F 6/x^2(x-3)
6/x^2(x-3)= Ax+B/x^2+C/(x-3)
C/x^2(x-3)= (A+C)x^2+(B-3A)x-3B
B=-2
A=-2/3
C=1/3
F C/x^2-3x^2 dx= F(-2/3)... | We have the expansion
$$\frac{1}{x^3-3x^2}=\frac{A+Bx}{x^2}+\frac{C}{x-3}\tag 1$$
Multiplying both sides of $(1)$ by $x-3$ and letting $x\to 3$ reveals that $C=1/9$.
Multiplying both sides of $(1)$ by $x^2$ and letting $x\to 0$ reveals that $A=-1/3$.
Multiplying both sides of $(1)$ by $x^2$, taking a derivative with ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculate 2000! (mod 2003) Calculate 2000! (mod 2003)
This can easily be solved by programming but is there a way to solve it, possibly with knowledge about finite fields? (2003 is a prime number, so mod(2003) is a finite field) .
As much details as possible please, I want to actually understand.
| For any odd prime $p$ we have $\left(p-1\right)!\equiv p-1\,\left(\text{mod}\,p\right)$ and $\left(p-2\right)\left(p-3\right)\equiv 2\,\left(\text{mod}\,p\right)$ so $\left(p-1\right)!\equiv \frac{p-1}{2}\,\left(\text{mod}\,p\right)$. The case $p=2003$ gives $\frac{p-1}{2}=1001$.
| {
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"timestamp": "2023-03-29T00:00:00",
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The limit as $n$ approaches infinity of $n\left(a^{1/n}-1\right)$ I need to know how to calculate this without using l'hospitals rule:
limit as $x$ approaches infinity of: $$x\left(a^{1/x}-1\right)$$
I saw that the answer is $\log(a)$, but I want to know how they got it.
The book implies that I should be able to find i... | The exponential function is a convex function, hence $x<y$ gives:
$$ e^x \leq \frac{e^y-e^x}{y-x} \leq e^{y} \tag{1}$$
and assuming $a>1$ we have:
$$ e^{0}\leq \frac{e^{x\log a}-e^0}{x\log a-0}\leq e^{x\log a}\tag{2}$$
or:
$$ \log a \leq \frac{a^x-1}{x}\leq a^x \log a \tag{3}$$
hence the claim follows by squeezing.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1362452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Binary Integer Programming I need to form teams. There are 8 projects and 60 students.
Each project has different requirements.
For example, out of 5 total requirements, project 1 has 2 requirements: must have a programmer and must have an analyst
Project 2 has 1 requirement: must have a programmer
Project 3 has 5 requ... | You will need 480 variables of the form $P01S01$ to $P08S60$.
These variables must be integer variables equal to 0 or 1.
$P03S40=1$ means that student 40 is allocated to project 40.
$P03S40=0$ means that student 40 is not allocated to project 40.
You will need up to 5 constraints for each project.
So if project 14 need... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1362553",
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"source": "stackexchange",
"question_score": "1",
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Simple Puzzle: A Matter Of Time I am trying to solve a simple puzzle:
Fifty Minutes ago if it was four times as many minutes past three O'clock, how many minutes is it to six O'clock.
I tried solving it:
Let x be the minutes past 5,
then 120 + x - 50 = 4x
which gives the wrong answer.
The correct solution has the form... | Because the time until $6$ is not the same as the time past $5$, treating $x$ this way results in a solution that is based on a false assumption. To remedy this, you would want to re-write the RHS to match with how you have written the LHS.
In order to avoid algebraic acrobatics, consider the problem as such: the "goal... | {
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"source": "stackexchange",
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What am I doing wrong? Partial Fraction Decomp. $$\int\frac{1-v}{(v+1)^2}dv$$
I think I am supposed to do PFD, but solving for A and B I get zero for both.
$$(1-v) = A(v+1) + B(v+1)$$
let $v = -1$
$$A = \frac{2}{0}, B = \frac{2}{0}$$
So this is undefined? (or infinity?)
| If you’re interested in the integration rather than the abstract partial fraction problem, you should make the substitution $u=v+1$, giving
$$
\int\frac{(2-u)du}{u^2}=\int\Big(2u^{-2}-\frac1u\Big)du\,,
$$
then make the backward substitution after doing your simple integration.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Which statement "must be false"? Given a function $f$ continuous on $[-4, 1]$ with its maximum at $(-3, 5)$ and its minimum at $(1/2, -6)$, is it not correct to say that both statements (B) and (D) must be false?
(A) The graph of $f$ crosses both axes.
(B) $f$ is always decreasing on $[-4, 1]$.
(C) $f(-2)=0$,
(D) $f(-1... | If you assume that decreasing means $x<y \Rightarrow f(x)> f(y)$ then $(B)$ "must" be false
If you assume that decreasing means $x\leq y \Rightarrow f(x)\geq f(y)$ then $(B)$ might be true
If you assume that $f$ has a global maximum in the point $(-3,5)$ then $(D)$ "must" be false
If you assume that $f$ has a local ma... | {
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"timestamp": "2023-03-29T00:00:00",
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Find $\lim_{x\to \frac\pi2}\frac{\tan2x}{x-\frac\pi2}$ without l'hopital's rule. I'm required to find $$\lim_{x\to\frac\pi2}\frac{\tan2x}{x-\frac\pi2}$$ without l'hopital's rule.
Identity of $\tan2x$ has not worked.
Kindly help.
| et $x=\frac\pi2 + h$
then as $x\to \frac\pi2$ then $h\to 0$
Therefore
$$\lim_{x\to \frac\pi2}\frac{tan2x}{x-\frac\pi2}\\
=\lim_{h\to 0}\frac{tan2(\frac\pi2+h)}{\frac\pi2+h-\frac\pi2}\\
=\lim_{h\to 0}\frac{tan(\pi+2h)}{h}\\
=\lim_{h\to 0}\frac{tan2h}{h}\\
=\lim_{h\to 0}\frac{(2h) + (2h)^3/3 + ....}{h}\\
=2$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 10,
"answer_id": 9
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Can $x\pi$ be rational? When I was solving a math test, I came across this problem -
Let $x$ be an irrational number. What type of number is $x\pi$?
a) Rational only
b) Irrational only
c) Could be rational or irrational
I was surprised to see that the answer was option c.
Can anyone tell me for what value of $x$ is $... | If $x=\frac{1}{\pi}$ then they multiply to give $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1362954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A question from Titchmarsh's zeta function book. On page 30, he writes that $\xi(0)=-\zeta(0)=1/2$, but on page 16 he writes that:
$\xi(s)=1/2 s(s-1)\pi^{-1/2s}\Gamma(1/2s)\zeta(s)$ in eq.(2.1.12); so if I plug into this equation $s=0$ then I get that it should vanish, shouldn't it?
What's wrong here?
https://books.goo... | For the functional equation
$$
\xi(s) = \xi(1 - s)
$$
we have the definition
$$
\xi(s) = \frac{1}{2}\pi^{-s/2}s(s-1)\Gamma\left(\frac{s}{2}\right)\zeta(s).\!
$$
The functional equation just gives $\xi(0)=\xi(1)$. The function $Z(s)=\frac{1}{2}\pi^{-s/2}\Gamma(\frac{s}{2})\zeta(s)$ has a meromorphic continuat... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Technique to solve limits I was making list of limits exercises, I can't use L' hôpital to solve, I have to solve using only the properties of limits. The only techniques that I know are: I. trying to replace x by the number II. divide III. multiply under the terms of the conjugate The following limit is solved by divi... | You know these things by experience. Because when we start solving questions, we start to get the hang of it. Now when you do this example, you divide it by the expression, because you would have done factorising exercises before and you remember them.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1363166",
"timestamp": "2023-03-29T00:00:00",
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Evaluate the Integral: $\int_0^2 \frac{dx}{e^{\pi x}}$
Evaluate the definite integral: $$\int_0^2 \frac{\mathrm{d}x}{e^{\pi x}}$$
My attempt:
$u=e^{\pi x}$
$du=\pi e^{\pi x}\ dx$
$\int_0^2 \frac{1}{u}\ dx$
So at this point do I divide $\pi e^{\pi x}$ by dx?
Thus, $\frac{du}{\pi\ e^{\pi x}}=dx$
and $\int_0^2 \frac{1}... | note that in $\int_0^2 \frac{1}{u}\cdot \frac{du}{\pi\ e^{\pi x}}$ we can substitute $\pi\ e^{\pi x}$ with $\pi u$. Thus $$\int_0^2 \frac{1}{u}\cdot \frac{du}{\pi\ e^{\pi x}} = \int_0^2 \frac{1}{u}\cdot \frac{du}{\pi\ u} = \int_0^2 \frac{du}{\pi u^2} = \frac{1}{\pi}\left[ -\frac{1}{u} \right]^{u=2}_{u=0}$$.
However you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1363258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The ratio of jacobi theta functions Let $q=e^{2\pi i\tau}$. If $\theta_2$ and $\theta_3$ are jacobi theta functions , is it true that the ratio of the two functions can be expressed as a continued fraction of the form
$$
\frac{\theta_2(q^2)}{\theta_3(q^2)}=2q^{1/2}\prod_{n=1}^\infty \frac{(1+q^{4n})^2}{(1+q^{4n-2})^2}=... | The answer is yes. Given the nome $q = \exp(i\pi\tau)$, elliptic lambda function $\lambda(\tau)$, Dedekind eta function $\eta(\tau)$, Jacobi theta functions $\vartheta_n(0,q)$, and Ramanujan's octic cfrac, the following relations are known,
$$\begin{aligned}
u(\tau) & = \big(\lambda(\tau)\big)^{1/8} = \frac{\sqrt{2}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1363353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the order of an element of finite group Let $G$ be a finite group and $g,h\in G-\{1\}$ such that $g^{-1}hg=h^2$.
In addition $o(g)=5$ and $o(h)$ is an odd integer. Find $o(h)$.
I know from a previous exercise that if there exists a natural number $i$ such that $g^{-1}hg=h^i$ then for all $n\in \mathbb{N}$, $g^{-n}... | Square both sides
$$
g^{-1}h^2 g = h^4
$$
Now replace $h^2$
$$
g^{-1}(g^{-1} h g) g = g^{-2} h g^2 = h^4
$$
Again square and expand
$$
g^{-3}hg^3 = g^{-2}h^2g^2 = h^8.
$$
By repeating this process we find
$$
g^{-k} h g^k = h^{2^k}
$$
for $k>0$
so in particular
$$
h = g^{-5}hg^5 = h^{32}.
$$
Hence $h^{31} = 1$. Thus $O... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1363522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Indentifying $\sin(mx) = 2\cos(x)\sin\left[(m-1)x\right] - \sin\left[(m-2)x\right]$ I encountered in a work of Joseph Fourier's the identity:
$$\sin(mx) = 2\cos(x)\sin\left[(m-1)x\right] - \sin\left[(m-2)x\right]$$
which holds for all real $m$ and $x$.
I had trouble, however, locating this in common lists of trigonomet... | or just observe:
$$
\sin ((m-1)x \pm x) = \sin(m-1)x \cos x \pm \sin x \cos(m-1)x
$$
and add the two equations together
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find lim$_{x \to 0}\left(\frac{1}{x} - \frac{\cos x}{\sin x}\right).$ $$\lim_{x \to 0} \left(\frac{1}{x} - \frac{\cos x}{\sin x}\right) = \frac{\sin x - x\cos x}{x\sin x}= \frac{0}{0}.$$
L'Hopital's: $$\lim_{x \to 0} \frac{f'(x)}{g'(x)} = -\frac{1}{x^2} + \frac{1}{\sin ^2x} = \frac{0}{0}.$$
Once again, using L'Hopital... | $$\lim_{x\to 0} \frac{1}{x}-\frac{\cos(x)}{\sin(x)}=\lim_{x\to 0} \frac{1}{x}-\frac{1}{\tan(x)}=\lim_{x\to 0} \frac{1}{x}-\frac{1}{x+\frac{x^3}{3}+O(x^5)}=\lim_{x\to 0}\frac{1}{x}-\frac{1}{x}=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1363719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 5
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Bounds for $\frac{x-y}{x+y}$ How can I find upper and lower bounds for $\displaystyle\frac{x-y}{x+y}$? So I do see that
$$\frac{x-y}{x+y} = \frac{1}{x+y}\cdot(x-y) = \frac{x}{x+y} - \frac{y}{x+y} > \frac{1}{x+y} - \frac{1}{x+y} = \frac{0}{x+y} = 0$$
(is it correct?) but I don't get how to find the upper bound.
| Suppose $\dfrac{x-y}{x+y} = c$, where $c \neq -1$.
Then
$$\begin{eqnarray}
x - y &=& c(x + y) \\
x - cx &=& y + cy \\
(1 - c)x &=& (1 + c)y
\end{eqnarray}$$
Therefore $y = \dfrac{1-c}{1+c} x$.
But if $\dfrac{x-y}{x+y} = -1$, then $x - y = -1(x + y)$,
and from this we conclude that $x = 0$ and $y$ can be anything you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1363835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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$A$ and $B$ similar if $A^2=B^2=0$ and dimension of range $A$ and $B$ are equal Suppose $A$ and $B$ are linear transformations on finite dimensional vector space $V$,s.t. $A,B\neq 0$ and $A^2=B^2=0$. Suppose the dimension of range $A$ and $B$ are equal, can $A$ and $B$ be similar?
| If $A^2 = 0$, then $A$ satisfies the polynomial $x^2$, and and similarly for $B$. So then the minimal polynomial for $A$ and $B$ both divide $x^2$. But since $A\neq 0$, $B \neq 0$, then $x^2$ must be the minimal polynomial for both $A$ and $B$.
Then the Jordan Canonical Form of $A$ and $B$ will include at least one $2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1363902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Messaging probabilities I am part of a large family - we have twenty-four people who send texts back and forth, in various configurations. What would be the total number of possible message threads? All the different one-on-ones - the three persons groups, four, etc., up to all twenty-four of us on one thread.
| If we encode:
$(0110\cdots)$ as 1st member (alphabetical ordering say) not included, 2nd and 3rd included, 4th not, so forth...
Then there are $2^{24}-25$ ways.
$2^{24}$ is the number of such binary strings, and 24 of those are messages to oneself, which we discount, as well as empty threads (messages to no one)
It is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1363998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Generalizing the Fibonacci sum $\sum_{n=0}^{\infty}\frac{F_n}{10^n} = \frac{10}{89}$ Given the Fibonacci, tribonacci, and tetranacci numbers,
$$F_n = 0,1,1,2,3,5,8\dots$$
$$T_n = 0, 1, 1, 2, 4, 7, 13, 24,\dots$$
$$U_n = 0, 1, 1, 2, 4, 8, 15, 29, \dots$$
and so on, how do we show that,
$$\sum_{n=0}^{\infty}\frac{F_n}{10... | The difference equations given by the suggested series are:
\begin{align}
F_{n+2} &= F_{n+1} + F_{n} \\
T_{n+3} &= T_{n+2} + T_{n+1} + T_{n} \\ \tag{1}
U_{n+4} &= U_{n+3} + U_{n+2} + U_{n+1} + U_{n}
\end{align}
and so on. In general they take on the form
\begin{align}\tag{2}
\phi_{n+m} = \sum_{k=0}^{m-1} \phi_{n+m-k-1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1364092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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The maximum and minimum values of the expression Here is the question:find the difference between maximum and minimum values of $u^2$ where $$u=\sqrt{a^2\cos^2x+b^2\sin^2x} + \sqrt{a^2\sin^2x+b^2\cos^2x}$$
My try:I have just normally squared the expression and got
$u^2=a^2\cos^2x+b^2\sin^2x + a^2\sin^2x+b^2\cos^2x +2... | Write
$$\cos^2x=\frac{1+\cos 2x}{2}$$
and
$$\sin^2x=\frac{1-\cos 2x}{2}$$
Then, we have
$$u=\sqrt{A+B\cos 2x}+\sqrt{A-B\cos 2x}\tag 1$$
where
$$A=\frac{a^2+b^2}{2}$$
$$B=\frac{a^2-b^2}{2}$$
Taking the derivative of u in $(1)$ and setting the derivative equal to zero reveals
$$\frac{-B\sin 2x}{\sqrt{A+B\cos 2x}}+\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1364170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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What's wrong in my thinking about Bézout's theorem? First, I know that every hypersurface of degree $d$ defined in $\mathbb{CP}^n$ is diffeomorphic. By using this fact, I wanted to calculate the Euler characteristic of hypersurface of degree $d$.
To begin, let $\chi_n^d$ be an Euler characteristic of projective complex... | The correct statement is that every smooth hypersurface of degree $d$ in $\mathbb{CP}^n$ is diffeomorphic. The union of a smooth hypersurface of degree $d_1$ and a smooth hypersurface of degree $d_2$ is a singular hypersurface of degree $d_1 + d_2$.
Consider in particular the case $n = 2, d_1 = d_2 = 1$. A smooth hype... | {
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"url": "https://math.stackexchange.com/questions/1364463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How to solve $\log(x -1) + \log(x - 2) = 2?$ I'm doing this exercise:
$$\log(x - 1) + \log(x - 2) = 2$$
My steps:
Step 1:
$$\log(x-1)(x - 2) = 2$$
Step 2:
$$(x - 1)(x - 2) = 10^2$$
Step 3:
$$x^2 - 3x + 2 = 100$$
Step 4:
$$x^2 = 98 + 3x$$
But I don't know what else I can do. In fact, I've doubts about the execution of ... | You've done everything correctly so far, from $$x^2 = 3x +98 \iff x^2 - 3x - 98 = 0$$ which gives us (using the quadratic formula) $$x = \frac{3 \pm \sqrt{401}}{2}.$$
You'll have to discard the negative solution given the implicit impositions placed on $x$ through the logarithm. Now this doesn't match any of your optio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1364532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What does it mean when two sets are "adjoined" in a metric space? I encountered the word "adjoined" in Baby Rudin, Chapter 2 concerning basic topology on Euclidean space. It appeared in the proof to Theorem 2.35
Theorem$\quad$ Closed subsets of compact sets are compact
Proof$\quad$ Suppose $F\subset K\subset X$, $F$ i... | The adjoining simply means taking a union of $F^C$ and $\\{V_{\alpha}\\}$.
Since $F$ is closed, $F^C$ is open. Also we assumed $K$ is compact, hence, $K$ has a finite open cover, say $\\{V_{\alpha}\\}$ which covers $K$. Adjoining $F^C$, or more simply put, taking union of $F^C$ and $\\{V_{\alpha}\\}$ will also cover $K... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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Keep video for certain time frames based upon percentages? (85/13/2 breakdown) I'm trying to build an equation for an Excel spreadsheet that is used to calculate storage requirements for video retention over a given policy. Currently it works great for policies that only have one retention period where everything is ke... | The second one is correct. In the first one you didn't take the extra leap day into account, but you did in the second.
| {
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Prove that $\cos^2(\theta) + \cos^2(\theta +120^{\circ}) + \cos^2(\theta-120^{\circ})=3/2$
Prove that $$\cos^2(\theta) + \cos^2(\theta +120^{\circ}) + \cos^2(\theta-120^{\circ})=\frac{3}{2}$$
I thought of rewriting $$\cos^2(\theta +120^{\circ}) + \cos^2(\theta-120^{\circ})$$ as $$\cos^2(90^{\circ}+ (\theta +30^{\circ... | Since $\cos(2x)=2\cos^2(x)-1$, we have $$\cos^2(x)=\frac {\cos(2x)+1}2$$
Therefore, $$\cos^2(\theta)=\frac {\cos(2\theta)+1}{2}$$ $$\cos^2(\theta+120)=\frac{\cos(2\theta+240^\circ)+1}{2}$$ $$\cos^2(\theta-120)=\frac{\cos(2\theta-240^\circ)+1}{2}$$
So the original equation become: $$\frac{\cos(2\theta)+1+cos(2\theta+240... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Extra theorems of Peano Arithmetic with Omega Rule Are there any theorems that Peano Arithmetic with the infinitary inference rule "If $P(0)$, $P(S0)$, $P(SS0)$, $P(SSS0)$, etc all hold, then $\forall x P(x)$ holds" can prove, that regular Peano arithmetic can't? And can someone give me an example, if there is one.
| Sure - the consistency of (the usual version of) $PA$! (I'll write "$PA_\omega$" for "$PA$ plus the $\omega$-rule"; I believe this is standard.)
$Con(PA)$ is the statement "there is no proof of "$0=1$" from the axioms of $PA$;" when properly encoded (via Godel numbering), this is a statement of the form $\forall x\varp... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Fixed points of diffeomorphisms: eigenvalues of the pushforward I want to answer this question: Let $M$ be a smooth manifold, and let $f: M \rightarrow M$ be a diffeomorphism. Let $\mathrm{Fix}_f$ be the fixed points of $f$, and suppose that $x \in \mathrm{Fix}_f$ is not isolated. Show that $df_x$ has an eigenvalue of ... | My friend helped me solve this problem. His solution uses more analysis than I would like, but here it is. Choose a chart for $M$ on an open set $U$ identifying $x$ with the origin, and then choose $V \subset U$ so that $f(V) \subseteq U$. Now we may view $f$ as a diffeomorphism in a neighborhood of the origin in $\mat... | {
"language": "en",
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"source": "stackexchange",
"question_score": "3",
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Pullbacks and homotopy equivalences Say I have a map between pullback squares $(Y \rightarrow Z \leftarrow X) \to (Y' \rightarrow Z' \leftarrow X')$. If the maps $X \to X'$, $Y \to Y'$ and $Z \to Z'$ are homotopy equivalences, does it follow that the induced map $X \times_Z Y \to X' \times_{Z'} Y'$ between the pullbac... | I'll answer the question in the pushout case.
If one of the maps of each pushout square is a cofibration, the induced map will be a homotopy equivalence, see Proposition 5.3.4 of Tammo tom Dieck's "Algebraic Topology". You don't need any further conditions on the spaces for this, not even that they are compactly genera... | {
"language": "en",
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Trigonometric Integrals $\int \frac{1}{1+\sin^2(x)}\mathrm{d}x$ and $\int \frac{1-\tan(x)}{1+\tan(x)} \mathrm{d}x$ Any idea of calculating this two integrals $\int \frac{1}{1+\sin^2(x)}\,dx$ and $\int \frac{1-\tan(x)}{1+\tan(x)} \mathrm{d}x$?
I found a solution online for the first one but it requires complex numbers w... | For the first one you may use that trig. 1 $$\frac{1}{1+\sin^{2}(x)} = \frac{1}{\cos^{2}(x)+2\sin^{2}(x)}= \frac{1}{\cos^{2}(t)}\frac{1}{1+2\tan^{2}(x)}$$ Now it is pretty clear that the change of variable $\tan(x)=t$ reduces to a standard arctanget integral $\int{\frac{dt}{1+2t^{2}}}$
For the second one, notice that $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding $F(x)$ from $F(kx),$ where $F(x)$ is the antiderivative of the function $f(x)$. I have that $F(e^{x}x) = e^{x}x^{2} - e^{x}x + e^{x} - 1$, and I would like to find $F(x)$.
Attempt
Since $F(e^{x}x) = e^{x}x^{2} - e^{x}x + e^{x} - 1,$ $F(t) = \alpha_{1}t^{\beta_{1}} + \alpha_{2}t^{\beta_{2}} + \alpha_{3}t^{\beta_... | I am not sure I can understand your question. I hope my answer will help you. We can rephrase your question in the following way:
Let $F$ be a function such that $F(e^{x}x) = e^{x}x^{2} - e^{x}x + e^{x} - 1$ for all $x \in \mathbb{R}$. What is the expression of $F$?
We can start by considering the map $x \mapsto y=x ... | {
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"source": "stackexchange",
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Reciprocal relations in Roulette /glissette rollings If a catenary rolls on a straight line its focus traces out a parabola and vice versa.
Is it true? Are there more such examples and how are they co-related?
In case of a circle rolling on a fixed straight line we have a cycloid trace for a point on circle periphery ... | It is an old problem that goes back to James Gregory 1668 in "Geometriae pars universalis". He invented a transformation between polar and orthonormal coordinates
$$ y=\rho, \,\, x= \int \rho\, d \theta$$
There is identity of arc length between the polar curve and $(x,y) $ curve by a rolling motion and the pole runs... | {
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Integrating $\frac{\sec^2\theta}{1+\tan^2\theta \cos^2(2\alpha)}$ with respect to $\theta$ I'm having some issues with the following integral
$$\int_{\frac{-\pi}{2}}^\frac{\pi}{2}\frac{\sec^2\theta}{1+\tan^2\theta \cos^2(2\alpha)}d\theta$$
My attempt is as follows, substitute $u=\tan\theta$(but this gives infinite boun... | Notice,
$\color{blue}{\int_{-a}^{a}f(x) dx=2\int_{0}^{a}f(x) dx\iff f(-x)=f(x)}$, Now we have
$$\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}\frac{\sec^2\theta d\theta}{1+\tan^2\theta\cos^22\alpha}$$
$$=2\int_{0}^{\frac{\pi}{2}}\frac{\sec^2\theta d\theta}{1+\tan^2\theta\cos^22\alpha}$$ Now, let $\tan \theta=t \implies \sec^2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Why are these following variance and expected value computations legitimate? I spent over an hour of my exam's given time to calculate the variances and expected values as given here: Let $p,q\in (0,1)$. The number of costumers entering a supermarket is a r.v. $X$ with geometric distribution with parameter $q$. Every c... | There are $X$ customers and each buys with probability $p$. So the total number of buys is $X\cdot p$.
The Geometric Distribution has expected value $E(X)=\frac{1}{q}$ and $p$ is constant (hence independent).
So the expected number of buys is $$E(Xp)=E(X)\cdot p=\frac{p}{q}.$$
Since $p$ is fixed, the variance is $$V(Xp... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Branching Paths Problem I was drawing some shapes during class, and I came across the following problem. If one takes steps of constant length, and one must deviate a constant angle $\alpha$ from one's previous step either left or right on the next step, for an angle $\alpha$ what is the set of all possible points I ca... | NOTE: This is perhaps assuming you don't have to turn every time, i.e. you can just travel one unit in the current direction. It depends on the angle. If the angle is $2 \pi, \pi, \pi / 2, 3 \pi / 2$,$2 \pi / 3$,$4 \pi / 3$, $\pi / 3$ or $5 \pi / 3$ then the set of points you can reach will form what's called a "lattic... | {
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"source": "stackexchange",
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When does :$\sigma(\sigma(2n))=\sigma(\sigma(n))$ and $\sigma(n)$ is sum divisors of the positive integer $n$? Is there someone who can show me When does: $$\sigma(\sigma(2n))=\sigma(\sigma(n))$$ where : $\sigma(n)$ denotes the sum of divisors of the positive integer $n$ ?
Note (1) : I came across this problem when I r... | $\sigma(2n)=3\times\sigma(n)$
if $m=\sigma(n)$
we should find $\sigma(3 m)=\sigma(m)$
$\sigma(3m)=4\times\sigma(m)$
then $4\sigma(m)=\sigma(m)$
contradiction so there is no solution.
| {
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How many positive, three digit integers contain atleast one 7? This is the Question: How many positive, three digit integers contain atleast one 7?
For these kind of questions I have always followed a technique of first taking care of the restriction provided in the question. The Restriction is contain atleast one 7
Th... | I think the analogy with the permutations of letters is making this problem more complicated than it needs to be.
Using the restriction that the number has at least one seven, you can first find the numbers that have exactly one $7$, then the numbers that have two $7$s, and then the number that has three $7$s and then... | {
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Angle of intersection of the given curves. What is the angle of intersection of $$[|\sin x| + |\cos x|]$$ And the curve $$ x^2 + y^2 = 5 $$ where $[n]$ denotes greatest integer function.
This is a homework question. I have tried to find the intersection of these two curves but i am unable to do so. In the solution book... | 1) From the above hint it is proved that [|sinx+cosx|]=1...(eq i)
Again,if we find the local extremum for the above equation(eq i) [0<=x<=(pi/2)],
we get the maximum value as (root 2)=1.414(approx), minimum value=1.
So,in this way we can also get that [|sinx+cosx|]=1.
Thus y=[|sinx|+|cosx|]=1
2) Now, find the point of ... | {
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Nature of the roots of quadratic equation Here is the problem that I need to prove:
If $x$ is real and $\displaystyle{\ p = \frac{3(x^2+1)}{(2x-1)}}$, prove that $\ p^2-3(p+3) \geq 0$
Here is what I did:
\begin{align*}
p(2x-1)=3(x^2+1) \\
3x^2 - 2px + (p+3)=0 \\
b^2 - 4ac = 4(p^2-3(p+3))
\end{align*}
By inspection I c... | You have $$3x^2 - 2px + (p+3) = 0.$$ Given that $x$ is real the quadratic needs to have a discriminant $\Delta \ge 0$. So $$\Delta = 4p^2 - 12(p+3) \geq 0.$$
Dividing by $4$ yields $$\bbox[10px, border: blue 1px solid]{p^2 - 3(p+3) \ge 0.}$$ as required.
| {
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How to solve $\sin78^\circ-\sin66^\circ-\sin42^\circ+\sin6^\circ$ Question:
$ \sin78^\circ-\sin66^\circ-\sin42^\circ+\sin6° $
I have partially solved this:-
$$ \sin78^\circ-\sin42^\circ +\sin6^\circ-\sin66^\circ $$
$$ 2\cos\left(\frac{78^\circ+42^\circ}{2}\right) \sin\left(\frac{78^\circ-42^\circ}{2}\right) + 2\cos\le... | $\sin(5\cdot78^\circ)=\sin(360^\circ+30^\circ)=\sin30^\circ$
$\sin\{5(-66^\circ)\}=\sin(-360^\circ+30^\circ)=\sin30^\circ$
If $\sin5x=\sin30^\circ\implies5x=n180^\circ+(-1)^n30^\circ$ where $n$ is any integer
$\implies x=n72^\circ+6^\circ$ where $n\equiv-2,-1,0,1,2\pmod5$
Again, $\sin5x=16\sin^5x-20\sin^3x+5\sin x$
So,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$T^2 = T$ and $T$ is normal implies $T$ is hermitian
Let $T:V\to V$ ($V$ is finite dimensional), a normal linear-operator such that $T^2=T$. Show that $T$ is hermitian.
So I know that if $T$ is normal then $T$ is hermitian iff the roots of $f_T(x)$ are real.
I also figured out that $T^2 = T$ implies $T(c)\in\mathbb{R... | If $T^2=T$, then the only eigenvalues of $T$ are $0$ and/or $1$. If $T$ is normal, then it is unitarily diagonalizable. If $T$ is unitarily diagonalizable with real eigenvalues, then it is Hermitian.
This might be an overkill proof, but it gets the job done. Note that this requires $V$ to be finite dimensional.
| {
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"timestamp": "2023-03-29T00:00:00",
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Confused about the Arrow Category If we use the definition of the Arrow category and the notation from here.
$$
\require{AMScd}
\begin{CD}
A @>h>> C \\
@VVfV @VVgV \\
B @>k>> D
\end{CD}
$$
I think I can understand how the object $f$ from the $C^2$ category gets transformed using $h$ and $k$ into object $g$ if the star... | $C^{2}$ has objects that are arrows $f:a\to b$ of $C$.
A morphism $\phi :f\to g$ is a pair $(h,k)$ such that $k\circ f=g\circ h$
$\tag 1 \begin{matrix}
\operatorname a & \xrightarrow{{f}} & \operatorname b \\
\left\downarrow h\vphantom{\int}\right. & & \left\downarrow k\vphantom{\int}\right.\\
\operatorname c& \xrigh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1366720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Confusion about the centre of a p-group If a non-cyclic group $G$, non-commutative also has order $p^{3}$ does that mean for every $x\in G$ , $x^{p}$ is in $Z(G)$? I am trying to solve a problem from $p$-groups and at this point I am stuck.
| Let $G$ be non-commutative of order $p^3$, and $Z$ be its center.
Then for any $g ∈ G$ we have $g^p ∈ Z$, since $G/Z\simeq \mathbb{Z}/(p) × \mathbb{Z}/(p)$ (this follows since $G/Z$ cyclic would imply $G$ abelian).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Ratio of CDF to PDF increasing? Let $\Phi(x)$ be a cumulative normal distribution function and $\phi(x)$ the associated probability density function.
Is the ratio $\frac{\Phi(x)}{\phi(x)}$ increasing in x?
Numerically it seems to be true. Is there any ways to prove it analytically?
Thanks
| Maybe there is a simpler way, but here is a thought. Use the relations $ \Phi(x)' = \phi(x)$ and $\phi(x)'= -x\phi(x)$
$$\left(\frac{\Phi(x)}{\phi(x)}\right)' = \frac{\Phi(x)'\phi(x) - \Phi(x)\phi(x)'}{\phi(x)^2} = \frac{\phi^2(x) + x \Phi(x)\phi(x)}{\phi(x)^2} = 1 + x \frac{\Phi(x)}{\phi(x)}$$
If $\left(\frac{\Phi(x)... | {
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"timestamp": "2023-03-29T00:00:00",
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Find the asymptotes of the Folium of Descartes ($x^3+y^3-3xy=0$) I'm trying to find the asymptotes of the Folium of Descartes, which has the equation $$x^3+y^3-3xy=0$$
I was also told to find the curve length in the first quadrant, and to do so I parametrized it by finding the intersection between the curve and the lin... | I notice that in your parametrization
$$ x+y+1 = \frac{3t}{1+t^3} + \frac{3t^2}{1+t^3}+1 = \frac{(1+t)^2}{1-t+t^2}$$
so $x+y+1=0$ is the asymptote.
| {
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"timestamp": "2023-03-29T00:00:00",
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What is the cardinality of the set of all non-measurable sets in $\Bbb R^n$? The cardinality of the set of all measurable sets in $\Bbb{R}^n$ can be shown to be the same as the power set of $\Bbb{R}$ by looking into Cantor set.
Denote $M=$$\{$$Ω⊆\Bbb{R}^n:Ω$ is measurable$\}$, then $card(M)≤card(2^\Bbb{R})$. The Canto... | Given that $(0,1)$ has a non-measurable subset, adding points in $(1,2)$ can't make it measurable. There are $2^{\mathfrak c}$ subsets of $(1,2)$, so take your non-measurable subset of $(0,1)$ union each subset of $(1,2)$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Integral of Sinc times Exponent of Squared variable I would like to integrate this in my research:
$$\int\limits_{-\infty}^\infty{\frac{e^{i b x^2}\sin{(a x)}}{x}}dx$$
where a and b are both real and greater than zero. If possible, I would like to take this a step further and integrate
$$\int\limits_{-\infty}^\infty{\f... | For first, get rid of the extra parameter by setting $c=\frac{b}{a^2}$:
$$I= \int_{\mathbb{R}}e^{ibx^2}\frac{\sin(ax)}{x}\,dx = \int_{\mathbb{R}}e^{icx^2}\frac{\sin x}{x}\,dx=\text{Im PV}\int_{\mathbb{R}}e^{icx^2+ix}\frac{dx}{x} $$
then translate the $x$ variable in order to get:
$$ I = \text{Im}\left(e^{-\frac{i}{4c}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1367275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Problem with a physical equation When I solved a physics problem, I found a little problem of math calculation at :
$$E=E_{c}+E_{p}= \frac{ms'^2}{2}+ \frac{mgs^2}{8R} ~~~~\text{(1)}$$
(this is the equation where I met the problem for solving it). The problem said that are no neconservative force, so our $E$ will be zer... | Just from glancing at your problem, it looks like the differential equation will have periodic solutions of the form
$$
s (t) = A \sin \sqrt{ \frac{g}{4R} } t + B \cos \sqrt{ \frac{g}{4R} } t .
$$
In which case, the frequency of oscillation is given by the quantity $\sqrt{ \frac{g}{4R} }$. I don't know if that answers... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1367402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Constructing $\mathbb{C}$ from $\mathbb{R}$ I'm having difficulty grasping the notion that you can define the complex numbers as $\mathbb{C}=\mathbb{R}[t]/\langle t^2+1\rangle$.
As far as I understand, $\mathbb{R}[t]$ is the set of all polynomials in $t$ whose coefficients are real.
When we take the quotient, we are f... | The complex number $a+bi$ corresponds to the equivalence class of polynomials that contains the polynomial $bt+a$.
For example $2+3i$ corresponds to the equivalence class
$$ \{3t+2, t^2+3t+3, -t^2+3t+1, t^3+4t+2, \pi t^4+\pi t^2+3t+2,\ldots \} $$
You should be able to check that addition and multiplication of polynomia... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1367497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 0
} |
$M= \{ A \in Mat_{2 \times 2}{\mathbb{R}}| \det(A)=1 \}$ is homeomorphic to $S^{1} \times \mathbb{R}^{2}$ Let's consider a group $M$ (under multiplication) of all matrices $A$ of size $2 \times 2$ over $\mathbb{R}$ so that $\det(A)=1$. How to show that the group is homeomorphic to the $S^{1} \times \mathbb{R^{2}}$?
Top... | The group $M$, which I am going to rename $G$, acts transitively on $\mathbb{R}^2 \setminus \{(0, 0)\}$ which is clearly homeomorphic to $\mathbb{R} \times S^1$. The stabilizer of the vector
$$
\begin{pmatrix} 1 \\ 0 \end{pmatrix}
$$
is the set of matrices of the form
$$
\begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix}
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1367557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
show that $X$ is homeomorphic to the $n$ dimensional (real) projective space. Let $D^n=\{(x_1,...,x_n)\in\mathbb{R}^n: \Sigma_{i=1}^{n}x_i^2\leq 1\}$.
*
*Let $X=D^n\times\{0\}\cup D^n\times\{1\}$ and let $Y$ be the quotient of $X$ obtained by identifying $(x,0)$ and $(x,1)$ for all $x\in\partial D^n$. Show that $Y$... | The $\phi$ you defined has codomain $X,$ not $Y.$ You want to take the composition of this map with the quotient map $X \to Y.$ Also, in order to conclude that $\phi$ is a homeomorphism you need to show that $Y$ is Hausdorff. This is easy enough; I'll let you fill in the details. (Alternatively, you could construct the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1367754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Is there a function $f''(0)$ exists, $f'$ is not continuous on $(-\delta,\delta)$ Is there a function $f\colon(-\delta,\delta)\to\Bbb R$ satisfying the folowing conditions(real number $ \delta\gt0$)?
(i) $f$ is differentiable on $(-\delta,\delta)$;
(ii) the second derivative of $f$ exists at $0$, that is $f''(0)$ e... | The standard way to get a function that is differentiable everywhere (with bounded derivative) but whose derivative has a discontinuity point is
$$ g(x) = \begin{cases} 0 & x=0 \\ x^2\sin(1/x) & x\ne 0 \end{cases} $$
Now select your $x_n$s and define
$$ f(x) = x^2 \sum_{k=1}^\infty \frac{g(x-x_k)}{2^k} $$
This places a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1367817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Expectation and waiting time
There are three jobs that need to be processed, with the processing
time of job $i$ being exponential with rate $\mu_i$. There are two
processors available, so processing on two of the jobs can immediately
start, with processing on the final job to start when one of the
initial one... | This is not a full solution. I simply try to explain how the order of jobs starting affects the min you are after:
if jobs $1$ and $2$ start first the the probability that job $1$ finishes before job $2$ is $$p_{12}=\int_0^{\infty}\int_0^{t_2}\mu_1\mu_2e^{-\mu_1t_1-\mu_2t_2}dt_1dt_2=\frac{\mu_1}{\mu_1+\mu_2}$$ therefo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1368009",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Evaluating and proving $\lim\limits_{x\to\infty}\frac{\sin x}x$ I've just started learning about limits. Why can we say $$ \lim_{x\rightarrow \infty} \frac{\sin x}{x} = 0 $$ even though $\lim_{x\rightarrow \infty} \sin x$ does not exist?
It seems like the fact that sin is bounded could cause this, but I'd like to see ... | We have $$\lim\frac{f}{g}=\frac{\lim f}{\lim g} $$
if all three limits exist and $\lim g$ is not zero. This doesn't mean that the existence of th elimit on the left would imply the existence of both limits on the right!
Similarly, note for the application of l'Hopital that some conditions must be met: The limit $\lim\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1368086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 2
} |
Separability of $l^{p}$ spaces How can I prove that the space $l^{p}$ equipped with the norm (for $x=(x_{n}) \in l^{p})$:
$\|x\|_{p}=(\displaystyle\sum_{n}|x_{n}|^{p})^{1/p}$
Is a separable space? (i.e. showing that there is a countable dense set in $l_{p}$). I saw a proof in which they started with first letting $x=(x... | The idea here is to see that the set of finite sequences is dense in $l^p$ (for $p < \infty$)
So we will approach $x$ by the sequence $x^{(n)}$, with $x^{(n)}_i = x_i$ if $i\leq n$ and $x^{(n)}_i = 0$ if $i > n$
This gives you
$$\| x- x^{(n)} \|_p^p = \sum_{k>n} |x_k|^p$$
As the serie $\sum |x_k|^p$ converge, the t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1368174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 1,
"answer_id": 0
} |
Is there a simpler way to compute the residue of a function at a pole of order 3? The function $$\frac {1}{z^2(e^{i2\pi z}-1)}$$ has a triple pole at z = 0. To compute the residue of f at z = 0, I can compute the Laurent expansion of f about z = 0, and then read off the coefficient of the 1/z term. But the fraction r... | Suppose $f$ has a pole of order $n$ at $z_0$, i.e.
$$
f(z)=\frac{a_{-n}}{(z-z_0)^n}+\frac{a_{-n+1}}{(z-z_0)^{n-1}}+\dots+\frac{a_{-1}}{z-z_0}+g(z)
$$
in a neighborhood $U\ni z_0$, where $g(z)$ is holomorphic. Thus
$$
(z-z_0)^nf(z)=a_{-n}+a_{-n+1}(z-z_0)+\dots+a_{-1}(z-z_0)^{n-1}+g(z)(z-z_0)^n
$$
$$
\left(\frac{d}{dz}\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1368245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Card Distribution: Expected Value Percy shuffles a standard $52$-card deck and starts turning over cards one at a time, stopping as soon as the first spade is revealed.
What is the expected number of cards that Percy turns over before stopping (including the spade)?
(Note: There are $13$ spades in a deck.)
I am complet... | The expected number of cards is the sum over all cards $k$ of the probability $p_k$ that card $k$ will be turned. Card $k$ will be turned if no previous card was a spade. Thus
$$
p_k=\frac{\binom{39}k}{\binom{52}k}=\frac{39!}{52!}\frac{(52-k)!}{(39-k)!}
$$
and
$$
\sum_{k=0}^{39} p_k= \frac{39!}{52!}\sum_{k=0}^{39}\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1368352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
General Solution of $\sin(mx)+\sin(nx)=0$ Problem:
Find the general solution of $$\sin(mx)+\sin(nx)=0$$
My attempt:
$$$$
$$\sin(mx)=-\sin(nx)$$
$$=\cos\left(\dfrac{\pi}{2}-mx\right)=\cos\left(\dfrac{\pi}{2}+nx\right)$$
Using $\cos\theta=\cos\alpha\Rightarrow \theta=2n\pi\pm \alpha,$$$$$
$$\text{CASE } 1:\theta=2n\pi+... | Hint
Why not to use $$\sin(x)+\sin(y)=2\sin(\frac{x+y}2)\cos(\frac{x-y}2)$$ So, you just have a product to consider which will make life quite easier, I guess.
I am sure that you can take from here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1368424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Minor mistake computing $\int \frac{1}{x^3+2x^2-3x} \; dx$? I'm trying to compute:
$$\int \frac{1}{x^3+2x^2-3x} \; dx$$
Until now, I did the following: Factoring:
$$x^3+2x^2-3x=x(x-1)(x+3)$$
To obtain the parcial fractions:
$$\frac{a}{x}+\frac{b}{x+3}+\frac{c}{x-1}=\frac{1}{x(x-1)(x+3)}$$
$$a(x-1)(x+3)+bx(x-1)+cx(x+3)=... | It should be $2a-b+3c=0$, not $2a-b-3c=0$.
As a side note, if you want to solve $a(x-1)(x+3)+bx(x-1)+cx(x+3)=1$, then it's a lot easier to plug in $x=0, 1, -3$ instead of expanding.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1368512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Indefinite integral $\int{3x^2\over (x^3+2)^4}dx$ Question:
How to solve this indefinite interal $$\int{3x^2\over (x^3+2)^4}dx$$
Attempt:
$3\int x^2{1\over (x^3+2)^4}dx$
Let $u=x^2$ then $du={1\over (x^3+2)^4}dx$.
Am I on the right track or going about it the wrong way?
ok so I make $du=3x^2dx$ but what do I do with... | Your option $u=x^2$ seems to be a bad one. Indeed, $du=2x\,dx$ doesn't mate very well with the given numerator, and the denominator will turn to the unappetizing $(\sqrt u^3+2)^4$.
It is much more appealing to notice that $3x^2$ is the derivative of $x^3$, so that a substitution $u=x^3$ will yield $du$ at the numerator... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1368670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Given that $p$ is an odd prime, is the GCD of any two numbers of the form $2^p + 1$ always equal to $3$? I have checked it for some numbers and it appears to be true. Also I am able to reduce it and get the value $3$ for specific primes $p_1$, $p_2$ by using the Euclidean algorithm but I am not able to find a general a... | In general, the following statement is true. Let $a,b$ be positive integers and $m\ge 2$ an integer. Then
$$
\gcd(m^a-1,m^b-1)=m^{\gcd(a,b)}-1.
$$
This can be proved by considering the Euclidean algorithm in base $m$,
see the references here.
Edit: As pointed out by Daniel, one can show in a similar way that
$$
\gcd(m^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1368747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
why one of eigen values of a covariance matrix is zero? let say you have a square matrix A, you calculate covariance of it, then calculate eigenvalues and eigen vectors. It follows that one of eigen values is equal to zero. why is it so? what does it mean in terms of interpretation of eigen vectors?
Thanks
| It is because $n$ points in $n$-dimensional space are always situated on a codimension (at least) one affine subspace. The variance in a orthogonal direction to this subspace will be zero and this variance is expressed by an eigenvalue of the covariance matrix of these $n$ points.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1368852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
show that $f$ does not have a zero in the disc $\{z:|z|<|a|\}$.
Consider the unit disc $D$ and an analytic function $f:D\to D$. If $f(0)=a\not=0$ then show that $f$ does not have a zero in the disc $\{z:|z|<|a|\}$.
My Try:
Consider $g(z)=f(z)-a$. Then $g:D\to D$ is analytic and $g(0)=0$. Now apply Schwarz lemma on $g... | You are correct to think of the Schwarz lemma.
Define $g(z) = \dfrac{z - a}{1 - \bar a z}$ so that $g : D \to D$ is analytic and $g(a) = 0$. If $h(z) = g(f(z))$ then $h : D \to D$ and $h(0) = 0$ so by the Schwarz lemma $|h(z)| \le |z|$ for all $z \in D$. Thus $$\left| \frac{f(z) - a}{1 - \bar a f(z)} \right| \le |z|$$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1368929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Can anyone give an example of a closed set contains no interval but with finite non-zero Lebesgue measure? Can anyone give an example of a closed set $F$ of $\Bbb{R}$ such that $0<|F|<+\infty$ and $F$ contains no open interval? Thank you!
| Enumerate the rational points of $[0,1]$ as a sequence $(r_n)_{n\in\mathbb N}$. Then, choose a sequence of positive numbers $(\varepsilon_n)$ such that $\sum_1^\infty 2\varepsilon_n<1$ and set $K:=[0,1]\setminus \bigcup_1^\infty ]r_n-\varepsilon_n, r_n+\varepsilon_n[$. This is a closed set with finite measure; it conta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1369001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
Show that the sequence given by $x_{n+1}=x_n+\frac{\sqrt {|x_n|}}{n^2}$ is convergent
My Try:
It is clear that $x_n$ is monotonically increasing. If we assume that the sequence converges to $a$ then $\displaystyle a=a+\frac{\sqrt{|a|}}{n^2}$. Hence $a=0$. So, I was going to prove that the sup of the sequence is $0$. ... | No, the limit is not $0$ in general.
Hint:
$$x_n - x_1 = \sum_{j=1}^{n-1} (x_{j+1} - x_j) = \sum_{j=1}^{n-1} \dfrac{\sqrt{|x_j|}}{j^2} $$
Find $M$ such that $|x_1| < M^2$ and $M \sum_{j=1}^\infty 1/j^2 < M^2 - x_1$, and show that
all $|x_j| \le M^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1369120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Prove that $(a,b)^2=(a^2,b^2,ab)$ I am trying to prove that $(a,b)^2=(a^2,b^2,ab)$ and was told that this follows from some very basic $\gcd$ laws. What am I not seeing?
| Let $(a,b) = c$, then $a = cp, b = cq$ with $(p,q) = 1$. Thus: $(a^2,b^2,ab) = (c^2p^2, c^2q^2,c^2pq) = c^2(p^2,q^2,pq)$. We need to prove: $(p^2,q^2,pq) = 1$. We have: $(p^2,q^2,pq) = d \Rightarrow d \mid p^2, d\mid q^2 \Rightarrow d \mid (p^2,q^2) = 1$ since $(p,q) = 1 \Rightarrow d = 1$. Thus the problem is solved.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1369260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Determine if a basis consists of eigenvectors So, this might be a silly question, but here it is. I am doing a couple of problems computing $[T]_\beta$, and determining whether $\beta$ is a basis consisting of eigenvectors of $T$. My problem is this: I understand that if $[T]_\beta$ is a diagonal matrix, then clearly $... | Of course, we have
$$
T \beta_j = \sum_{i=1}^n \alpha_i \beta_i
$$
For some scalars $\alpha_i$. In particular, $\alpha_i$ is the $i$th entry of the column of $[T]_\beta$ corresponding to $\beta_j$ (the $j$th column, in particular).
If $[T]_\beta$ is not diagonal, then for some $i \neq j$, we have $\alpha_j \neq 0$.
Bec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1369319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to divide certain polynomials? Can somebody help me with this question?
$$\frac{15p^3+16p^2+46}{3p+5}$$
For some reason I can't wrap my head around the process used to divide polynomials, I can do long division but every time somebody explains the long division of polynomials to me I can't understand it whatsoever.... | $\color{red}{\rm Reds}$ are the terms we want to kill, $\color{blue}{\rm blues}$ are pieces of our answer.
Consider $\color{red}{15p^3}+16p^2 + 46$. If I multiply $3p+5$ by $\color{blue}{5p^2}$, we get $\color{red}{15p^3}+25p^2$.
Subtracting it, we're left with $\color{red}{-9p^2}+46$. If I multiply $3p+5$ by $\color{b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1369380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
$y(x)$ be a continuous solution of the initial value problem $y'+2y=f(x)$ , $y(0)=0$
Let , $y(x)$ be a continuous solution of the initial value problem $y'+2y=f(x)$ , $y(0)=0$ , where, $$f(x)=\begin{cases}1 & \text{ if } 0\le x\le 1\\0 & \text{ if } x>1\end{cases}$$Then, $y(3/2)$ equals to
(A) $\frac{\sinh(1)}{e^3}$
... | `Solving the Part (1)
$y′+2y=1 , y(0)=0$
We get
$$y(x) = \frac{(1-e^{-2x}) }{2}...........................................(A)$$
Consider the Part(2),
$y′+2y=0 $
$y(x)=Ke^{-2x}$
$y(1)=1/2*(1-e^{-2})$ By the continuity find $y(1)$ from equation $(A)$
$ K=y(1)e^2$
hence
$y(x)=y(1)e^2e^{-2x}$
$y(3/2)=sinh(1)/(e^2)$
$So... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1369451",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Sum of square root of non perfect square positive integers is always irrational? Let $S$ be a set of positive integers such that no element of $S$ is a perfect square. Is it true that $\sum_{s_i \in S} \sqrt{s_i}$ is always irrational?
Motivation. Suppose the length of the circumference of a polygon whose nodes are loc... | The answer is "yes"; see here (page 87)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1369525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
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