Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Cutting a Möbius strip down the middle Why does the result of cutting a Möbius strip down the middle lengthwise have two full twists in it? I can account for one full twist--the identification of the top left corner with the bottom right is a half twist; similarly, the top right corner and bottom left identification co... | Observe that the boundary of a Möbius strip is a circle. When you cut, you create more boundary; this is in fact a second circle.
During this process, the Möbius strip loses its non-orientability. Make two Möbius strips with paper and some tape. Cut one and leave the other uncut. Now take each and draw a line down t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/67542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 3,
"answer_id": 2
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How to calculate point y with given point x of a angled line I dropped out of school to early I guess, but I bet you guys can help me here.
I've got a sloped line starting from point a(0|130) and ending at b(700|0).
I need an equation to calculate the y-coordinate when the point x is given, e.g. 300. Can someone help m... | You want the two point form of a linear equation. If your points are $(x_1,y_1)$ and $(x_2,y_2)$, the equation is $y-y_1=(x-x_1)\frac{y_2-y_1}{x_2-x_1}$. In your case, $y=-\frac{130}{700}(x-700)$
| {
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"url": "https://math.stackexchange.com/questions/67602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is the zero factorial one i.e ($0!=1$)?
Possible Duplicate:
Prove $0! = 1$ from first principles
Why does 0! = 1?
I was wondering why,
$0! = 1$
Can anyone please help me understand it.
Thanks.
| Answer 1: The "empty product" is (in general) taken to be 1, so that formulae are consistent without having to look over your shoulder. Take logs and it is equivalent to the empty sum being zero.
Answer 2: $(n-1)! = \frac {n!} n$ applied with $n=1$
Answer 3: Convention - for the reasons above, it works.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What does the exclamation mark do? I've seen this but never knew what it does. Can any one let me in on the details? Thanks.
| For completeness:
Although in mathematics the $!$ almost always refers to the factorial function, you often see it in quasi-mathematical contexts with a different meaning.
For example, in many programming languages it is used to mean negation, for example in Java the expression !true evaluates to false. It is also comm... | {
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"url": "https://math.stackexchange.com/questions/67801",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 1
} |
How would I solve $\frac{(n - 10)(n - 9)(n - 8)\times\ldots\times(n - 2)(n - 1)n}{11!} = 12376$ for some $n$ without brute forcing it? Given this equation:
$$
\frac{(n - 10)(n - 9)(n - 8)\times\ldots\times(n - 2)(n - 1)n}{11!} = 12376
$$
How would I find $n$?
I already know the answer to this, all thanks toWolfram|Al... | $n(n-1)\cdots(n-10)/11! = 2^3 \cdot 7 \cdot 13 \cdot 17$. It's not hard to see that $11! = 2^8 3^3 5^2 7^1 11^1$; this is apparently known as de Polignac's formula although I didn't know the name. Therefore
$n(n-1) \cdots (n-10) = 2^{11} 3^3 5^2 7^2 11^1 13^1 17^1$.
In particular 17 appears in the factorization but 19 ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Square root of differential operator If $D_x$ is the differential operator. eg. $D_x x^3=3 x^2$.
How can I find out what the operator $Q_x=(1+(k D_x)^2)^{(-1/2)}$ does to a (differentiable) function $f(x)$? ($k$ is a real number)
For instance what is $Q_x x^3$?
| It probably means this: Expand the expression $(1+(kt)^2)^{-1/2}$ as a power series in $t$, getting
$$
a_0 + a_1 t + a_2 t^2 + a_3 t^3 + \cdots,
$$
and then put $D_x$ where $t$ was:
$$
a_0 + a_1 D_x + a_2 D_x^2 + a_3 D_x^3 + \cdots
$$
and then apply that operator to $x^3$:
$$
a_0 x^3 + a_1 D_x x^3 + a_2 D_x^2 x^3 ... | {
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Does taking closure preserve finite index subgroups? Let $K \leq H$ be two subgroups of a topological group $G$ and suppose that $K$ has finite index in $H$. Does it follow that $\bar{K}$ has finite index in $\bar{H}$ ?
| The answer is yes in general, and here is a proof, which is an adaptation of MartianInvader's:
Let $K$ have finite index in $H$, with coset reps. $h_1,\ldots,h_n$.
Since multiplication by any element of $G$ is a homeomorphism from $G$ to
itself (since $G$ is a topological group), we
see that each coset $h_i \overline{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/68034",
"timestamp": "2023-03-29T00:00:00",
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Condition For Existence Of Phase Flows I am a bit confused about the existence of one-parameter groups of diffeomorphisms/phase flows for various types of ODE's. Specifically, there is a problem in V.I. Arnold's Classical Mechanics text that asks to prove that a positive potential energy guarantees a phase flow, and al... | How does Arnold define "phase flow"? As far as I know, part of the definition of a flow requires the solutions to exist for all $t > 0$. If they go to $\infty$ in a finite time, they don't exist after that. On the other hand, I don't see why not having a $C^\infty$ inverse would be an issue.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/68077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Expectation of supremum Let $x(t)$ a real valued stochastic process and $T>0$ a constant. Is it true that:
$$\mathbb{E}\left[\sup_{t\in [0,T]} |x(t)|\right] \leq T \sup_{t\in [0,T]} \mathbb{E}\left[|x(t)|\right] \text{ ?}$$
Thanks for your help.
| Elaboration on the comment by Zhen, just consider $x(t) = 1$ a.s. for all $t$ and $T = 0.5$
| {
"language": "en",
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Notations involving squiggly lines over horizontal lines Is there a symbol for "homeomorphic to"? I looked on Wikipedia, but it doesn't seem to mention one? Also, for isomorphism, is the symbol a squiggly line over an equals sign? What is the symbol with a squiggly line over just one horizontal line? Thanks.
| I use $\cong$ for isomorphism in a category, which includes both homeomorphism and isomorphism of groups, etc. I have seen $\simeq$ used to mean homotopy equivalence, but I don't know how standard this is.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/68241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Determining the truth value of a statement I am stuck with the following question,
Determine the truth value of each of the following statments(a
statement is a sentence that evaluates to either true or false but you
can not be indecisive).
If 2 is even then New York has a large population.
Now I don't get what d... | If X Then Y is an implication. In other words, the truth of X implies the truth of Y. The "implies" operator is defined in exactly this manner. Google "implies operator truth table" to see the definition for every combination of values. Most importantly, think about why it's defined in this manner by substituting in pl... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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summation of x * (y choose x) binomial coefficients What does this summation simplify to?
$$ \sum_{x=0}^{y} \frac{x}{x!(y-x)!} $$
I was able to realize that it is equivalent to the summation of $x\dbinom{y}{x}$ if you divide and multiply by $y!$, but I am unsure of how to further simplify.
Thanks for the help!
| Using generating function technique as in answer to your other question:
Using $g_1(t) = t \exp(t) = \sum_{x=0}^\infty t^{x+1} \frac{1}{x!} = \sum_{x=0}^\infty t^{x+1} \frac{x+1}{(x+1)!} = \sum_{x=-1}^\infty t^{x+1} \frac{x+1}{(x+1)!} = \sum_{x=0}^\infty t^{x} \frac{x}{x!}$ and $g_2(t) = \exp(t)$.
$$
\sum_{x=0}^{y} ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is this map a homeomorphism? A few hours ago a user posted a link to this pdf:
There was a discussion about Proposition 3.2.8. I read it, and near the end, there is a map given
$$
\bigcap_{i_1,\dots,i_n,\dots\in\{0,1\}}X_{i_1,\dots,i_n,\dots}\mapsto (i_1,\dots,i_n,\dots).
$$
And it says this is a homeomorphism. ... | If you examine the construction of $C$, you’ll see that each set $Y_{i_1,\dots,i_n}$ is the closure of a certain open ball; to simplify the notation, let $B_{i_1,\dots,i_n}$ be that open ball. The map in question is a bijection that takes $B_{i_1,\dots,i_n}\cap C$ to $$\{(j_1,j_2,\dots)\in\{0,1\}^{\mathbb{Z}^+}: j_1=i_... | {
"language": "en",
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Is there a rule of integration that corresponds to the quotient rule? When teaching the integration method of u-substitution, I like to emphasize its connection with the chain rule of integration. Likewise, the intimate connection between the product rule of derivatives and the method of integration by parts comes up i... | I guess you could arrange an analog to integration by parts, but making students learn it would be superfluous.
$$ \int \frac{du}{v} = \frac{u}{v} + \int \frac{u}{v^2} dv.$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
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"answer_id": 1
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Gaussian Elimination Does a simple Gaussian elimination works on all matrices? Or is there cases where it doesn't work?
My guess is yes, it works on all kinds of matrices, but somehow I remember my teacher points out that it doesn't works on all matrices. But I'm not sure, because I have been given alot of methods, and... | Gaussian elimination without pivoting works only for matrices all whose leading principal minors are non-zero. See http://en.wikipedia.org/wiki/LU_decomposition#Existence_and_uniqueness.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/68613",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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On the GCD of a Pair of Fermat Numbers I've been working with the Fermat numbers recently, but this problem has really tripped me up. If the Fermat theorem is set as $f_a=2^{2^a}+1$, then how can we say that for an integer $b<a$, the $\gcd(f_b,f_a)=1$?
| Claim. $f_n=f_0\cdots f_{n-1}+2$.
The result holds for $f_1$: $f_0=2^{2^0}+1 = 2^1+1 = 3$, $f_1=2^{2}+1 = 5 = 3+2$.
Assume the result holds for $f_n$. Then
$$\begin{align*}
f_{n+1} &= 2^{2^{n+1}}+1\\
&= (2^{2^n})^2 + 1\\
&= (f_n-1)^2 +1\\
&= f_n^2 - 2f_n +2\\
&= f_n(f_0\cdots f_{n-1} + 2) -2f_n + 2\\
&= f_0\cdots... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/68653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 3
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The chain rule for a function to $\mathbf{C}$ Let $f:U\longrightarrow \mathbf{C}$ be a holomorphic function, where $U$ is a Riemann surface, e.g., $U=\mathbf{C}$, $U=B(0,1)$ or $U$ is the complex upper half plane, etc.
For $a$ in $\mathbf{C}$, let $t_a:\mathbf{C} \longrightarrow \mathbf{C}$ be the translation by $a$, i... | The forms will be different if $a\not=0$, namely if $\mathrm{d} f = w(z) \mathrm{d}z$ locally, then $\mathrm{d}\left( t_a \circ f\right) = w(z-a) \mathrm{d} z$.
Added: Above, I was using the following, unconventional definition for the composition,
$(t_a \circ f)(z) = f(t_a(z)) = f(z-a)$.
The conventional definition, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/68768",
"timestamp": "2023-03-29T00:00:00",
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Solve $t_{n}=t_{n-1}+t_{n-3}-t_{n-4}$? I missed the lectures on how to solve this, and it's really kicking my butt. Could you help me out with solving this?
Solve the following recurrence exactly.
$$
t_n = \begin{cases}
n, &\text{if } n=0,1,2,3,
\\ t_{n-1}+t_{n-3}-t_{n-4}, &\text{otherwise.}
\end{cases}
$$
E... | Let's tackle it the general way. Define the ordinary generating function:
$$
T(z) = \sum_{n \ge 0} t_n z^n
$$
Writing the recurrence as $t_{n + 4} = t_{n + 3} + t_{n + 1} - t_n$,
the properties of ordinary generating functions give:
$$
\begin{align*}
\frac{T(z) - t_0 - t_1 z - t_2 z^2 - t_3 z^3}{z^4}
&= \frac{T(z) -... | {
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"timestamp": "2023-03-29T00:00:00",
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Sum of a series of minimums I should get sum of the following minimums.Is there any way to solve it?
$$\min\left\{2,\frac{n}2\right\} + \min\left\{3,\frac{n}2\right\} + \min\left\{4,\frac{n}2\right\} + \cdots + \min\left\{n+1, \frac{n}2\right\}=\sum_{i=1}^n \min(i+1,n/2)$$
| If $n$ is even, your sum splits as
$$\sum_{i=1}^{\frac{n}{2}-2} \min\left(i+1,\frac{n}{2}\right)+\frac{n}{2}+\sum_{i=\frac{n}{2}}^{n} \min\left(i+1,\frac{n}{2}\right)=\sum_{i=1}^{\frac{n}{2}-2} (i+1)+\frac{n}{2}+\frac{n}{2}\sum_{i=\frac{n}{2}}^{n} 1$$
If $n$ is odd, you can perform a similar split:
$$\sum_{i=1}^{\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/68873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $\cos(x)$ is identically zero using integration by parts Consider $$\int\cos(t-x)\sin(x)dx,$$ where $t$ is a constant.
Evaluating the integral by parts, let
\begin{align}
u = \cos(t-x),\ dv = \sin(x), \\
du = \sin(t-x),\ v = -\cos(x),
\end{align}
so
$$
\int\cos(t-x)\sin(x)dx = -\cos(t-x)\cos(x) - \int\sin(t... | A standard trigonometric identity says that
$$\sin(t-x)\sin(x)-\cos(t-x)\cos(x)$$
is equal to
$$
-\cos((t-x)+x)
$$
and that is $-\cos t$. As a function of $x$, this is a constant, i.e. since there's no "$x$" in this expression, it doesn't change as $x$ changes. Since the "dazzling new identity", if stated correctly... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/68926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Primes sum ratio Let
$$G(n)=\begin{cases}1 &\text{if }n \text{ is a prime }\equiv 3\bmod17\\0&\text{otherwise}\end{cases}$$
And let
$$P(n)=\begin{cases}1 &\text{if }n \text{ is a prime }\\0&\text{otherwise.}\end{cases}$$
How to prove that
$$\lim_{N\to\infty}\frac{\sum\limits_{n=1}^N G(n)}{\sum\limits_{n=1}^N P(n)}=\... | The first sum follows from Siegel-Walfisz_theorem
Summation by parts on the second sum should yield for large $N$:
$$\frac{\sum\limits_{n=1}^N n\,G(n)}{\sum\limits_{n=1}^N n\,P(n)}=\frac{(N\sum\limits_{n=1}^N G(n))-\sum\limits_{n=1}^{N-1}\sum\limits_{k=0}^{n} G(k)}{\sum\limits_{n=1}^N n\,P(n)}=\frac{(N\sum\limits_{n=1}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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Confusion about a specific notation In the following symbolic mathematical statement $n \in \omega $, what does $\omega$ stand for? Does it have something to do with the continuum, or is it just another way to denote the set of natural numbers?
| The notation of $\omega$ is coming from ordinals, and it denotes the least ordinal number which is not finite.
The von Neumann ordinals are transitive sets which are well ordered by $\in$. We can define these sets by induction:
*
*$0=\varnothing$;
*$\alpha+1 = \alpha\cup\{\alpha\}$;
*If $\beta$ is limit and all $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/69030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What is a good book for learning math, from middle school level? Which books are recommended for learning math from the ground up and review the basics - from middle school to graduate school math?
I am about to finish my masters of science in computer science and I can use and understand complex math, but I feel lik... | It depends what your level is and what you're interested in. I think a book that's not about maths but uses maths is probably more interesting for most people. I've noticed this in undergraduates as well: give someone a course using the exact same maths but with the particulars of their subject area subbed in, and they... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/69060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "63",
"answer_count": 15,
"answer_id": 11
} |
Equation of a rectangle I need to graph a rectangle on the Cartesian coordinate system. Is there an equation for a rectangle? I can't find it anywhere.
| I found recently a new parametric form for a rectangle, that I did not know earlier:
$$
\begin{align}
x(u) &= \frac{1}{2}\cdot w\cdot \mathrm{sgn}(\cos(u)),\\
y(u) &= \frac{1}{2}\cdot h\cdot \mathrm{sgn}(\sin(u)),\quad (0 \leq u \leq 2\pi)
\end{align}
$$
where $w$ is the width of the rectangle and $h$ is its height. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/69099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 9,
"answer_id": 1
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Integrating $\int \frac{1}{1+e^x} dx$ I wish to integrate
$$\int_{-a}^a \frac{dx}{1+e^x}.$$ By symmetry, the above is equal to
$$\int_{-a}^a \frac{dx}{1+e^{-x}}$$ Now multiply by $e^x/e^x$ to get $$\int_{-a}^a \frac{e^x}{1+e^x} dx$$
which integrates to
$$\log(1+e^x) |^a_{-a} = \log((1+e^a)/(1+e^{-a})),$$
which is ... | Both answers are equal. Split your answer into $\log(1+e^a)-\log(1+e^{-a})$, and write this as
$$\begin{align*}\log(e^a(1+e^{-a}))-\log(e^{-a}(1+e^a))&=\log e^a+\log(1+e^{-a})-\log e^{-a}-\log(1+e^a)\\
&=2a+\log((1+e^{-a})/(1+e^a))\end{align*}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/69179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
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How does one prove if a multivariate function is constant? Suppose we are given a function $f(x_{1}, x_{2})$. Does showing that $\frac{\partial f}{\partial x_{i}} = 0$ for $i = 1, 2$ imply that $f$ is a constant? Does this hold if we have $n$ variables instead?
| Yes, it does, as long as the function is continuous on a connected domain and the partials exist (let's not get into anything pathological here).
And the proof is the exact same as in the one variable case (If there are two points whose values we want to compare, they lie on the same line. Use the multivariable mean v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/69294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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The law of sines in hyperbolic geometry What is the geometrical meaning of the constant $k$ in the law of sines, $\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c}=k$ in hyperbolic geometry? I know the meaning of the constant only in Euclidean and spherical geometry.
| As given by Will Jagy, k must be inside the argument:
$$ \frac{\sin A}{\sinh(a/k)} = \frac{\sin B}{\sinh(b/k)} = \frac{\sin C}{\sinh(c/k)} $$
This is the Law of Hyperbolic trigonometry where k is the pseudoradius, constant Gauss curvature $K= -1/k^2$. Please also refer to " Pan-geometry", a set of relations mirrored ... | {
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"timestamp": "2023-03-29T00:00:00",
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Why are samples always taken from iid random variables? In most mathematical statistic textbook problems, a question always ask: Given you have $X_1, X_2, \ldots, X_n$ iid from a random sample with pdf:(some pdf). My question is why can't the sample come from one random variable such as $X_1$ since $X_1$ itself is a ra... | A random variable is something that has one definite value each time you do the experiment (whatever you define "the experiment" to be), but possibly a different value each time you do it. If you collect a sample of several random values, the production of all those random values must -- in order to fit the structure o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/69406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How many different combinations of $X$ sweaters can we buy if we have $Y$ colors to choose from?
How many different combinations of $X$ sweaters can we buy if we have
$Y$ colors to choose from?
According to my teacher the right way to think about this problem is to think of partitioning $X$ identical objects (sweat... | The classical solution to this problem is as follows:
Order the $Y$ colors. Write $n_1$ zeroes if there are $n_1$ sweater of the first color. Write a single one. Write $n_2$ zeroes where $n_2$ is the number of sweaters of the second color. Write a single one, and so on.
You get a string of length $X+Y-1$ that has e... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Closed form for a pair of continued fractions What is $1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cdots}}}$ ?
What is $1+\cfrac{2}{1+\cfrac{3}{1+\cdots}}$ ?
It does bear some resemblance to the continued fraction for $e$, which is $2+\cfrac{2}{2+\cfrac{3}{3+\cfrac{4}{4+\cdots}}}$.
Another thing I was wondering: can all tr... | I don't know if either of the continued fractions can be expressed in terms of common functions and constants. However, all real numbers can be expressed as a continued fractions containing only integers. The continued fractions terminate for rational numbers, repeat for a quadratic algebraic numbers, and neither term... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is wrong with my reasoning regarding finding volumes by integration? The problem from the book is (this is Calculus 2 stuff):
Find the volume common to two spheres, each with radius $r$, if the center of each sphere lies on the surface of the other sphere.
I put the center of one sphere at the origin, so its equ... | The analytic geometry of $3$-dimensional space is not needed to solve this problem. In particular, there is no need for the equations of the spheres. All we need is some information about the volumes of solids of revolution.
Draw two circles of radius $1$, one with center $(0,0)$, the other with center $(1,0)$. ... | {
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Proving two lines trisects a line A question from my vector calculus assignment. Geometry, anything visual, is by far my weakest area. I've been literally staring at this question for hours in frustrations and I give up (and I do mean hours). I don't even now where to start... not feeling good over here.
Question:
In ... | Note that EBP and EDA are similar triangles. Since 2BP=AD, it follows that 2EB=ED, and thus 3EB=BD. Which is to say, AP trisects BD.
| {
"language": "en",
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"source": "stackexchange",
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What is an easy way to prove the equality $r > s > 0$ implies $x^r > x^s$? I have been using simple inequalities of fractional powers on a positive interval and keep abusing the inequality for $x>1$. I was just wondering if there is a nice way to prove the inequality in a couple line:
Let $x \in [1,\infty)$ and $r,s \... | If you accept that $x^y\gt 1$ for $x\gt 1$ and $y \gt 0$, then $x^r=x^{r-s}x^s \gt x^s$ for $x\gt 1$ and $r \gt s$.
| {
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"source": "stackexchange",
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Is the class of cardinals totally ordered? In a Wikipedia article
http://en.wikipedia.org/wiki/Aleph_number#Aleph-one
I encountered the following sentence:
"If the axiom of choice (AC) is used, it can be proved that the class of cardinal numbers is totally ordered."
But isnt't the class of ordinals totally ordered (in ... | If I understand the problem correctly, it depends on your definition of cardinal. If you define the cardinals as initial ordinals, then your argument works fine, but without choice you cannot show that every set is equinumerous to some cardinal. (Since AC is equivalent to every set being well-orderable.)
On the other h... | {
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How to prove that proj(proj(b onto a) onto a) = proj(b onto a)? How to prove that proj(proj(b onto a) onto a) = proj(b onto a)?
It makes perfect sense conceptually, but I keep going in circles when I try to prove it mathematically. Any help would be appreciated.
| If they are vectors in ${\mathbb R}^n$, you can do it analytically too. You have
$$proj_{\bf a}({\bf b}) = ({\bf b} \cdot {{\bf a} \over ||{\bf a}||}) {{\bf a} \over ||{\bf a}||}$$
So if ${\bf c}$ denotes $proj_{\bf a}({\bf b})$ then
$$proj_{\bf a}({\bf c}) = ({\bf c} \cdot {{\bf a} \over ||{\bf a}||}) {{\bf a} \over |... | {
"language": "en",
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"source": "stackexchange",
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Counting Number of k-tuples Let $A = \{a_1, \dots, a_n\}$ be a collection of distinct elements and let $S$ denote the collection all $k$-tuples $(a_{i_1}, \dots a_{i_k})$ where $i_1, \dots i_k$ is an increasing sequence of numbers from the set $\{1, \dots n \}$. How can one prove rigorously, and from first principles,... | We will show that the number of ways of selecting a subset of $k$ distinct objects from a pool of $n$ of them is given by the binomial coefficient
$$
\binom{n}{k} = \frac{n!}{k!(n-k)!}.
$$
I find this proof easiest to visualize. First imagine permuting all the $n$ objects in a sequence; this can be done in $n!$ ways... | {
"language": "en",
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ODE question: $y'+A(t) y =B(t)$, with $y(0)=0, B>0$ implies $y\ge 0$; another proof? I am trying to prove that the solution for the different equation
$$y'+A(t) y =B(t)$$
with initial condition $y(0)=0$ and the assumption that $B\ge 0$, has non-negative solution for all $t\ge 0$, i.e. $y(t)\ge 0$ for all $t\ge 0$.
The... | A natural approach is to start from the special case where $A(t)=0$ for every $t\geqslant0$. Then the ODE reads $y'(t)=B(t)$ hence $y'(t)\geqslant0$ for every $t\geqslant0$ hence $y$ is nondecreasing. Since $y(0)\geqslant0$, this proves that $y(t)\geqslant0$ for every $t\geqslant0$.
One can deduce the general case from... | {
"language": "en",
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Functions with subscripts? In the equation:
$f_\theta(x)=\theta_1x$
Is there a reason that $\theta$ might be a subscript of $f$ and not either a second parameter or left out of the left side of the equation altogether? Does it differ from the following?
$f(x,\theta)=\theta_1x$
(I've been following the Machine Learning ... | As you note, this is mostly notational choice. I might call the $\theta$ a parameter, rather than an independent variable. That is to say, you are meant to think of $\theta$ as being fixed, and $x$ as varying.
As an example (though I am not sure of the context you saw this notation), maybe you are interested in des... | {
"language": "en",
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"source": "stackexchange",
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Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$ Given the geometric series:
$1 + x^2 + x^4 + x^6 + x^8 + \cdots$
We can recast it as:
$S = 1 + x^2 \, (1 + x^2 + x^4 + x^6 + x^8 + \cdots)$, where $S = 1 + x^2 + x^4 + x^6 + x^8 + \cdots$.
This recasting is possible only because there is an infin... | The $n$th partial sum of your series is
$$
\begin{align*}
S_n &= 1+x^2+x^4+\cdots +x^{2n}= 1+x^2(1+x^2+x^4+\cdots +x^{2n-2})\\
&= 1+x^2S_{n-1}
\end{align*}
$$
Assuming your series converges you get that
$$
\lim_{n\to\infty}S_n=\lim_{n\to\infty}S_{n-1}=S.
$$
Thus $S=1+x^2S$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Not homotopy equivalent to a 3-manifold w/ boundary Let $X_g$ be the wedge of $g$ copies of the circle $S^1$ where $g>1$. Prove that $X_g \times X_g$ is not homotopy equivalent to a 3-manifold with boundary.
| If it is a homotopy equivalent to a $3$-manifold $M$, looks at the homology long exact sequence for the pair $(M,\partial M)$ with $\mathbb Z_2$-coefficients. By Poincare duality, $H_i(M)\cong H_{3-i}(M,\partial M)$. You also know the homology groups of $M$, since you know those of $X$. If $\partial M$ has $c$ componen... | {
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"source": "stackexchange",
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Bijections from $A$ to $B$, where $A$ is the set of subsets of $[n]$ that have even size and $B$ is the set of subsets of $[n]$ that have odd size Let $A$ be the set of subsets of $[n]$ that have even size, and let $B$ be the set of subsets of $[n]$ that have odd size. Establish a bijection from $A$ to $B$. The follo... | For $n$ an odd positive integer, there is a natural procedure. The mapping that takes any subset $E$ of $[n]$ with an even number of elements to its complement $[n]\setminus E$ is a bijection from $A$ to $B$.
Dealing with even positive $n$ is messier. Here is one way. The subsets of $[n]$ can be divided into two type... | {
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Question regarding upper bound of fixed-point function The problem is to estimate the value of $\sqrt[3]{25}$ using fixed-point iteration. Since $\sqrt[3]{25} = 2.924017738$, I start with $p_0 = 2.5$. A sloppy C++ program yield an approximation to within $10^{-4}$ by $14$ iterations.
#include <cmath>
#include <iostream... | If I understand this right, $p_n$ converges to a fixed point of $g$. Taking $g(x)=\sqrt5/x$ as you have done, the fixed point of $g$ is not the $\root3\of{25}$ that you are after, but rather it is $\root4\of5$. So it's no wonder everything is going haywire.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Proving an inequality between functions: are the bounds sufficient if both strictly increase and are concave? I would like to show that
$$f(n) > g(n)$$ for all $n$ within a certain range.
If I can show that both $f(n)$ and $g(n)$ are strictly increasing with $n$, and that both are strictly concave, and that $f(n) > g(... | No. Consider, for example, $f(x)=1+12x-x^2$ and $g(x)=20x-10x^2$ between $0$ and $1$.
Plotted by Wolfram Alpha.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding inverse cosh I am trying to find $\cosh^{-1}1$ I end up with something that looks like $e^y+e^{-y}=2x$. I followed the formula correctly so I believe that is correct up to this point. I then plug in $1$ for $x$ and I get $e^y+e^{-y}=2$ which, according to my mathematical knowledge, is still correct. From here I... | start with
$$\cosh(y)=x$$
since
$$\cosh^2(y)-\sinh^2(y)=1$$ or $$x^2-\sinh^2(y)=1$$
then
$$\sinh(y)=\sqrt{x^2-1}$$
now add $\cosh(y)=x$ to both sides to make
$$\sinh(y)+\cosh(y) = \sqrt{x^2-1} + x $$
which the left hand side simplifies to : $\exp(y)$
so the answer is $$y=\ln\left(\sqrt{x^2-1}+x\right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/70500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
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Span of permutation matrices The set $P$ of $n \times n$ permutation matrices spans a subspace of dimension $(n-1)^2+1$ within, say, the $n \times n$ complex matrices. Is there another description of this space? In particular, I am interested in a description of a subset of the permutation matrices which will form a ... | As user1551 points out, your space is the span of all "magic matrices" -- all $n\times n$ matrices for which every row and column sum is equal to the same constant (depending on the matrix). As an algebra this is isomorphic to $\mathbb{C} \oplus M_{n-1}(\mathbb{C})$.
You can think of this as the image in $\operatornam... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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Connected components of subspaces vs. space If $Y$ is a subspace of $X$, and $C$ is a connected component of $Y$, then C need not be a connected component of $X$ (take for instance two disjoint open discs in $\mathbb{R}^2$).
But I read that, under the same hypothesis, $C$ need not even be connected in $X$. Could you pl... | Isn't it just false? The image of a connected subspace by the injection $Y\longrightarrow X$ is connected...
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Countable subadditivity of the Lebesgue measure Let $\lbrace F_n \rbrace$ be a sequence of sets in a $\sigma$-algebra $\mathcal{A}$. I want to show that $$m\left(\bigcup F_n\right)\leq \sum m\left(F_n\right)$$ where $m$ is a countable additive measure defined for all sets in a $\sigma$ algebra $\mathcal{A}$.
I think I ... | Given a union of sets $\bigcup_{n = 1}^\infty F_n$, you can create a disjoint union of sets as follows.
Set $G_1 = F_1$, $G_2 = F_2 \setminus F_1$, $G_3 = F_3 \setminus (F_1 \cup F_2)$, and so on. Can you see what $G_n$ needs to be?
Using $m(\bigcup_{n = 1}^\infty G_n)$ and monotonicity, you can prove $m(\bigcup_{n = ... | {
"language": "en",
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A ring element with a left inverse but no right inverse? Can I have a hint on how to construct a ring $A$ such that there are $a, b \in A$ for which $ab = 1$ but $ba \neq 1$, please? It seems that square matrices over a field are out of question because of the determinants, and that implies that no faithful finite-dime... | Take the ring of linear operators on the space of polynomials. Then consider (formal) integration and differentiation. Integration is injective but not surjective. Differentiation is surjective but not injective.
| {
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"source": "stackexchange",
"question_score": "30",
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normal groups of a infinite product of groups I have a question regarding the quotient of a infinite product of groups.
Suppose $(G_{i})_{i \in I}$ are abelian groups with $|I|$ infinite and each $G_i$ has a normal subgroup $N_i$. Is it true in general that
$$\prod_{i \in I} G_i/ \prod_{i \in I} N_i \cong \prod_{i\in I... | Here is a slightly more general statement.
Let $(X_i)$ be a family of sets, and $X$ its product. For each $i$ let $E_i\subset X_i^2$ be an equivalence relation. Write $x\ \sim_i\ y$ for $(x,y)\in E_i$. Let $E$ be the product of the $E_i$. There is a canonical bijection between $X^2$ and the product of the $X_i^2$. Thu... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Cardinality of Borel sigma algebra It seems it's well known that if a sigma algebra is generated by countably many sets, then the cardinality of it is either finite or $c$ (the cardinality of continuum). But it seems hard to prove it, and actually hard to find a proof of it. Can anyone help me out?
| It is easy to prove that the $\sigma$-algebra is either finite or has cardinality at least $2^{\aleph_0}$.
One way to prove that it has cardinality at most $2^{\aleph_0}$, without explicitly using transfinite recursion, is the following.
It is easy to see that it is enough to prove this upper bound for a "generic" $\si... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What Implications Can be Drawn from a Binomial Distribution? Hello everyone I understand how to calculate a binomial distribution or how to identify when it has occurred in a data set. My question is what does it imply when this type of distribution occurs?
Lets say for example you are a student in a physics class and ... |
Lets say for example you are a student in a physics class and the professor states that the distribution of grades on the first exam throughout all sections was a binomial distribution. With typical class averages of around 40 to 50 percent. How would you interpret that statement?
Most likely the professor was talkin... | {
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Lesser-known integration tricks I am currently studying for the GRE math subject test, which heavily tests calculus. I've reviewed most of the basic calculus techniques (integration by parts, trig substitutions, etc.) I am now looking for a list or reference for some lesser-known tricks or clever substitutions that are... | When integrating rational functions by partial fractions decomposition, the trickiest type of antiderivative that one might need to compute is
$$I_n = \int \frac{dx}{(1+x^2)^n}.$$
(Integrals involving more general quadratic factors can be reduced to such integrals, plus integrals of the much easier type $\int \frac{x \... | {
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Consequences of the Langlands program I have been reading the book Fearless Symmetry by Ash and Gross.It talks about Langlands program, which it says is the conjecture that there is a correspondence between any Galois representation coming from the etale cohomology of a Z-variety
and an appropriate generalization of a ... | There are many applications of the Langlands program to number theory; this is why so many top-level researchers in number theory are focusing their attention on it.
One such application (proved six or so years ago by Clozel, Harris, and Taylor) is the Sato--Tate conjecture, which describes rather precisely the deviati... | {
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Proof of dividing fractions $\frac{a/b}{c/d}=\frac{ad}{bc}$ For dividing two fractional expressions, how does the division sign turns into multiplication? Is there a step by step proof which proves
$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}=\frac{ad}{bc}?$$
| Suppose $\frac{a}{b}$ and $\frac{c}{d}$ are fractions. That is, $a$, $b$, $c$, $d$ are whole numbers and $b\ne0$, $d\ne0$. In addition we require $c\ne0$.
Let $\frac{a}{b}\div\frac{c}{d}=A$. Then by definition of division of fractions , $A$ is a unique fraction so that $A\times\frac{c}{d}=\frac{a}{b}$.
However, $(... | {
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Proof that series diverges Prove that $\displaystyle\sum_{n=1}^\infty\frac{1}{n(1+1/2+\cdots+1/n)}$ diverges.
I think the only way to prove this is to find another series to compare using the comparison or limit tests. So far, I have been unable to find such a series.
| This answer is similar in spirit to Didier Piau's answer.
The following theorem is a very useful tool:
Suppose that $a_k > 0$ form a decreasing sequence of real numbers. Then
$$\sum_{k=1}^\infty a_k$$ converges if and only if $$\sum_{k=1}^\infty 2^k a_{2^k}$$ converges.
Applying this to the problem in hand we are reduc... | {
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On the meaning of being algebraically closed The definition of algebraic number is that $\alpha$ is an algebraic number if there is a nonzero polynomial $p(x)$ in $\mathbb{Q}[x]$ such that $p(\alpha)=0$.
By algebraic closure, every nonconstant polynomial with algebraic coefficients has algebraic roots; then, there will... | Let $p(x) = a_0+a_1x+\cdots +a_{n-1}x^{n-1} + x^n$ be a polynomial with coefficients in $\overline{\mathbb{Q}}$. For each $i$, $0\leq i\leq n-1$, let $a_i=b_{i1}, b_{i2},\ldots,b_{im_i}$ be the $m_i$ conjugates of $a_i$ (that is, the "other" roots of the monic irreducible polynomial with coefficients in $\mathbb{Q}$ th... | {
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Questions about cosets: "If $aH\neq Hb$, then $aH\cap Hb=\emptyset$"? Let $H$ be a subgroup of group $G$, and let $a$ and $b$ belong to $G$. Then, it is known that
$$
aH=bH\qquad\text{or}\qquad aH\cap bH=\emptyset
$$
In other words, $aH\neq bH$ implies $aH\cap bH=\emptyset$. What can we say about the statement
"If ... | It is sometimes true and sometimes false.
For example, if $H$ is a normal subgroup of $G$, then it is true.
If $H$ is the subgroup generated by the permutation $(12)$ inside $G=S_3$, the symmetric group of degree $3$, then $(123)H\neq H(132)$, yet $(13)\in(123)H\cap H(132)$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The limit of locally integrable functions If ${f_i} \in L_{\rm loc}^1(\Omega )$ with $\Omega $ an open set in ${\mathbb R^n}$ , and ${f_i}$ are uniformly bounded in ${L^1}$ for every compact set, is it necessarily true that there is a subsequece of ${f_i}$ converging weakly to a regular Borel measure?
| Take $K_j$ a sequence of compact sets such that their interior grows to $\Omega$.
That is, $\mathrm{int}(K_j) \uparrow \Omega$.
Let $f_i^0$ be a sub-sequence of $f_i$ such that $f_i^0|_{K_0}$
converges to a Borel measure $\mu_0$ over $K_0$.
For each $j > 0$, take a sub-sequence $f_i^j$ of $f_i^{j-1}$ converging
to a Bo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/71405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Large $n$ asymptotic of $\int_0^\infty \left( 1 + x/n\right)^{n-1} \exp(-x) \, \mathrm{d} x$ While thinking of 71432, I encountered the following integral:
$$
\mathcal{I}_n = \int_0^\infty \left( 1 + \frac{x}{n}\right)^{n-1} \mathrm{e}^{-x} \, \mathrm{d} x
$$
Eric's answer to the linked question implies that $\mathc... | Interesting. I've got a representation
$$
\mathcal{I}_n = n e^n \int_1^\infty t^{n-1} e^{- nt}\, dt
$$
which can be obtained from yours by the change of variables $t=1+\frac xn$. After some fiddling one can get
$$
2\mathcal{I}_n= n e^n \int_0^\infty t^{n-1} e^{- nt}\, dt+o(\mathcal{I}_n)=
n^{-n} e^n \Gamma(n+1)+\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/71447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 3,
"answer_id": 1
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moment-generating function of the chi-square distribution How do we find the moment-generating function of the chi-square distribution? I really couldn't figure it out. The integral is
$$E[e^{tX}]=\frac{1}{2^{r/2}\Gamma(r/2)}\int_0^\infty x^{(r-2)/2}e^{-x/2}e^{tx}dx.$$
I'm going over it for a while but can't seem to... | In case you have not yet figure it out, the value of the integral follows by simple scaling of the integrand. First, assume $t < \frac{1}{2}$, then change variables $x = (1-2 t) y$:
$$
\int_0^\infty x^{(r-2)/2} \mathrm{e}^{-x/2}\mathrm{e}^{t x}\mathrm{d}x =
\int_0^\infty x^{r/2} \mathrm{e}^{-\frac{(1-2 t) x}{2}} \, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/71516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
A couple of problems involving divisibility and congruence I'm trying to solve a few problems and can't seem to figure them out. Since they are somewhat related, maybe solving one of them will give me the missing link to solve the others.
$(1)\ \ $ Prove that there's no $a$ so that $ a^3 \equiv -3 \pmod{13}$
So I need... | HINT $\rm\ (2)\quad\ mod\ 7\!:\ \{\pm 1,\:\pm 2,\:\pm3\}^3\equiv\: \pm1\:,\:$ so squaring yields $\rm\ a^6\equiv 1\ \ if\ \ a\not\equiv 0\:.$
$\rm(3)\quad \ mod\ 7\!:\ \ if\ \ a^2\equiv -b^2\:,\:$ then, by above, cubing yields $\rm\: 1\equiv -1\ $ for $\rm\ a,b\not\equiv 0\:.$
$\rm(1)\quad \ mod\ 13\!:\ \{\pm1,\:\pm... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$11$ divisibility We know that $1331$ is divisible by $11$.
As per the $11$ divisibility test, we can say $1331$ is divisible by $11$. However we cannot get any quotient. If we subtract each unit digit in the following way, we can see the quotient when $1331$ is divided by $11$.
$1331 \implies 133 -1= 132$
$132 \implie... | HINT $\ $ Specialize $\rm\ x = 10\ $ below
$$\rm(x+1)\ (a_n\ x^n +\:\cdots\:+a_1\ x + a_0)\ =\ a_n\ x^{n+1}+ (a_n+a_{n-1})\ x^{n}+\:\cdots\:(a_1+a_0)\ x+ a_0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/71638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Prove there is no element of order 6 in a simple group of order 168 Let $G$ be a simple group of order 168. Let $n_p$ be the number of Sylow $p$ subgroups in $G$.
I have already shown: $n_7 = 8$, $n_3 = 28$, $n_2 \in \left\{7, 21 \right\}$
Need to show: $n_2 = 21$ (showing there is no element of order 6 In $G$ will suf... | If there is an element of order 6, then that centralizes the Sylow $3$-subgroup $P_3$ generated by its square. You have already shown that $|N(P_3)|=168/n_3=6$. Therefore the normalizer of any Sylow $3$-subgroup would have to be cyclic of order 6, and an element of order 6 belongs to exactly one such normalizer. Thus y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/71768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
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Correlation Coefficient between these two random variables Suppose that $X$ is real-valued normal random variable with mean $\mu$ and variance $\sigma^2$. What is the correlation coefficient between $X$ and $X^2$?
| Hint: You are trying to find:
$$\frac{E\left[\left(X^2-E\left[X^2\right]\right)\left(X-E\left[X\right]\right)\right]}{\sqrt{E\left[\left(X^2-E\left[X^2\right]\right)^2\right]E\left[\left(X-E\left[X\right]\right)^2\right]}}$$
For a normal distribution the raw moments are
*
*$E\left[X^1\right] = \mu$
*$E\left[X^2\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/71832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Why is the math for negative exponents so? This is what we are taught:
$$5^{-2} = \left({\frac{1}{5}}\right)^{2}$$
but I don't understand why we take the inverse of the base when we have a negative exponent. Can anyone explain why?
| For natural numbers $n$, $m$, we have $x^nx^m=x^{n+m}$. If you want this rule to be preserved when defining exponentiation by all integers, then you must have
$x^0x^n = x^{n+0} = x^n$, so that you must define $x^0 = 1$. And then, arguing similarly, you have $x^nx^{-n} = x^{n-n}=x^0=1$, so that $x^{-n}=1/x^n$.
Now, you ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
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Representation of this function using a single formula without conditions Is it possible to represent the following function with a single formula, without using conditions? If not, how to prove it?
$F(x) = \begin{cases}u(x), & x \le 0, \ v(x) & x > 0 \end{cases}$
So that it will become something like that: $F(x) = G(x... | Note: This answers the original question, asking whether a formula like $F(x)=G(u(x),v(x))$ might represent the function $F$ defined as $F(x) = u(x)$ if $x \leqslant 0$ and $F(x)=v(x)$ if $x > 0$.
The OP finally reacted to remarks made by several readers that another answer did not address this, by modifying the ques... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/71951",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A graph with less than 10 vertices contains a short circuit? Lately I read an old paper by Paul Erdős and L. Pósa ("On the maximal number of disjoint circuits of a graph") and stumbled across the following step in a proof (I changed it a bit to be easier to read):
It is well known and easy to show that every (undirecte... | Because every vertex have degree $\ge 2$, there must be at least one cycle.
Consider, therefore, a cycle of minimal length; call this length $n$. Because each vertex in the cycle has degree $\ge 3$, it is connected to at least one vertex apart from its two neighbors in the cycle. That cannot be a non-neighbor member of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/72076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Express this curve in the rectangular form
Express the curve $r = \dfrac{9}{4+\sin \theta}$ in rectangular form.
And what is the rectangular form?
If I get the expression in rectangular form, how am I able to convert it back to polar coordinate?
|
what is the rectangular form?
It is the $y=f(x)$ expression of the curve in the $x,y$ referential (see picture). It can also be the implicit form $F(x,y)=F(x,f(x))\equiv 0$.
Steps:
1) transformation of polar into rectangular coordinates (also known as Cartesian coordinates) (see picture)
$$x=r\cos \theta ,$$
$$y=r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/72123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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General Lebesgue Dominated Convergence Theorem In Royden (4th edition), it says one can prove the General Lebesgue Dominated Convergence Theorem by simply replacing $g-f_n$ and $g+f_n$ with $g_n-f_n$ and $g_n+f_n$. I proceeded to do this, but I feel like the proof is incorrect.
So here is the statement:
Let $\{f_n\}_... | You made a mistake:
$$\liminf \int (g_n-f_n) = \int g-\limsup \int f_n$$
not
$$\liminf \int (g_n-f_n) = \int g-\liminf \int f_n.$$
Here is the proof:
$$\int (g-f)\leq \liminf \int (g_n-f_n)=\int g -\limsup \int f_n$$
which means that
$$\limsup \int f_n\leq \int f$$
Also
$$\int (g+f)\leq \liminf \int(g_n+f_n)=\int g + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/72174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "47",
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Existence of least squares solution to $Ax=b$ Does a least squares solution to $Ax=b$ always exist?
| If you think at the least squares problem geometrically, the answer is obviously "yes", by definition.
Let me try to explain why. For the sake of simplicity, assume the number of rows of $A$ is greater or equal than the number of its columns and it has full rang (i.e., its columns are linearly independent vectors). Wit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/72222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 7,
"answer_id": 0
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How to show that $\frac{f}{g}$ is measurable Here is my attempt to show that $\frac{f}{g}~,g\neq 0$ is a measurable function, if $f$ and $g$ are measurable function. I'd be happy if someone could look if it's okay.
Since $fg$ is measurable, it is enough to show that $\frac{1}{g}$ is measurable.
$$
\left\{x\;\left|\;... | Using the fact that $fg$ is a measurable function and in view of the identity $$\frac{f}{g}=f\frac{1}{g}$$ it suffices to show that $1/g$ (with $g\not=0$) is measurable on $E$. Indeed, $$E\left(\frac{1}{g}<\alpha\right)=\left\{\begin{array}{lll} E\left(g<0\right) & \quad\text{if }\alpha=0\\ E\left(g>1/\alpha\right)\cup... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/72323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Cyclic group with exactly 3 subgroups: itself $\{e\}$ and one of order $7$. Isn't this impossible?
Suppose a cyclic group has exactly three subgroups: $G$ itself, $\{e\}$, and a subgroup of order $7$. What is $|G|$? What can you say if $7$ is replaced with $p$ where $p$ is a prime?
Well, I see a contradiction: the or... | Hint: how can a number other than $7$ not have other factors?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is it ever $i$ time? I am asking this question as a response to reading two different questions:
Is it ever Pi time?
and
Are complex number real?
So I ask, is it ever $i$ time? Could we arbitrarily define time as following the imaginary line instead of the real one?
(NOTE: I have NO experience with complex number... | In the Wikipedia article titled Paul Émile Appell, we read that "He discovered a physical interpretation of the imaginary period of the doubly periodic function whose restriction to real arguments describes the motion of an ideal pendulum."
The interpretation is this: The real period is the real period. The maximum de... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/72510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Generating coordinates for 'N' points on the circumference of an ellipse with fixed nearest-neighbor spacing I have an ellipse with semimajor axis $A$ and semiminor axis $B$. I would like to pick $N$ points along the circumference of the ellipse such that the Euclidean distance between any two nearest-neighbor points,... | I will assume that $A$, $B$ and $N$ are given, and that $d$ is unknown.
There is always a solution. Let $L$ be the perimeter of the ellipse. An obvious constraint is $N\,d<L$. Take $d\in(0,L/N)$. As explained in Gerry Myerson's answer, pick a point $P_1$ on the ellipse, and then pick points $P_2,\dots,P_N$ such that $P... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/72555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Simple use of a permutation rule in calculating probability I have the following problem from DeGroot:
A box contains 100 balls, of which 40 are red. Suppose that the balls are drawn from the box one at a time at random, without replacement. Determine (1) the probability that the first ball drawn will be red. (2) the p... | The answers are (a) $40/100$; (b) $40/100$; (c) $40/100$.
(a) Since balls tend to roll around, let us imagine instead that we have $100$ cards, with the numbers $1$ to $100$ written on them. The cards with numbers $1$ to $40$ are red, and the rest are blue.
The cards are shuffled thoroughly, and we deal the top card. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to get apparent linear diameter from angular diameter Say I have an object, whose actual size is 10 units in diameter, and it is 100 units away.
I can find the angular diameter as such: $2\arctan(5/100) = 5.725\ $ radians.
Can I use this angular diameter to find the apparent linear size (that is, the size it appear... | It appears you are using the wrong angular units: $2\;\tan^{-1}\left(\frac{5}{100}\right)=5.7248$ degrees $=0.099917$ radians.
The formula you cite above is valid for a flat object perpendicular to the line of sight. If your object is a sphere, the angular diameter is given by $2\;\sin^{-1}\left(\frac{5}{100}\right)=5... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove for which $n \in \mathbb{N}$: $9n^3 - 3 ≤ 8^n$ A homework assignment requires me to find out and prove using induction for which $n ≥ 0$ $9n^3 - 3 ≤ 8^n$ and I have conducted multiple approaches and consulted multiple people and other resources with limited success. I appreciate any hint you can give me.
Thanks i... | Let $f(n)=n^3-3$, and let $g(n)=8^n$. We compute a little, to see what is going on.
We have $f(0) \le g(0)$; $f(1)\le g(1)$; $f(2) > g(2)$; $f(3) \le g(3)$; $f(4) \le g(4)$. Indeed $f(4)=573$ and $g(4)=4096$, so it's not even close.
The exponential function $8^x$ ultimately grows incomparably faster than the polynomia... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Real elliptic curves in the fundamental domain of $\Gamma(2)$ An elliptic curve (over $\mathbf{C}$) is real if its j-invariant is real.
The set of real elliptic curves in the standard fundamental domain of $\mathrm{SL}_2(\mathbf{Z})$ can be explicitly described.
In the standard fundamental domain $$F(\mathbf{SL}_2(\ma... | Question answered in the comments by David Loeffler.
I'm not sure which fundamental domain for $\Gamma(2)$ you consider to be "standard"? But whichever one you go for, it'll just be the points in your bigger domain whose $SL_2(\Bbb{Z})$ orbit contains a point of the set you just wrote down.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Order of finite fields is $p^n$ Let $F$ be a finite field.
How do I prove that the order of $F$ is always of order $p^n$ where $p$ is prime?
| A slight variation on caffeinmachine's answer that I prefer, because I think it shows more of the structure of what's going on:
Let $F$ be a finite field (and thus has characteristic $p$, a prime).
*
*Every element of $F$ has order $p$ in the additive group $(F,+)$. So $(F,+)$ is a $p$-group.
*A group is a $p$-grou... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "68",
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Variance of sample variance? What is the variance of the sample variance? In other words I am looking for $\mathrm{Var}(S^2)$.
I have started by expanding out $\mathrm{Var}(S^2)$ into $E(S^4) - [E(S^2)]^2$
I know that $[E(S^2)]^2$ is $\sigma$ to the power of 4. And that is as far as I got.
| Maybe, this will help. Let's suppose the samples are taking from a normal distribution. Then using the fact that $\frac{(n-1)S^2}{\sigma^2}$ is a chi squared random variable with $(n-1)$ degrees of freedom, we get
$$\begin{align*}
\text{Var}~\frac{(n-1)S^2}{\sigma^2} & = \text{Var}~\chi^{2}_{n-1} \\
\frac{(n-1)^2}{\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/72975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "101",
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Find $a$ and $b$ with whom this expression $x\bullet y=(a+x)(b+y)$ is associative I need to find a and b with whom this expression is associative: $$x\bullet y=(a+x)(b+y)$$
Also known that $$x,y\in Z$$
So firstly I write: $$(x\bullet y)\bullet z=x\bullet (y\bullet z)$$
Then I express and express them and after small al... | Assuming that you still want the element $1 \in \mathbb{Z}$ to be a multiplicative identity element with your new multiplication, here is an alternative way of quickly seeing that $a=b=0$.
For an arbitrary $y\in \mathbb{Z}$ we shall have:
$y=1 \bullet y=(a+1)(b+y)=ab+ay+b+y=(a+1)b+(a+1)y$
This is satisfied if and only ... | {
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Calculating the exponential of a $4 \times 4$ matrix
Find $e^{At}$, where $$A = \begin{bmatrix} 1 & -1 & 1 & 0\\ 1 & 1 & 0 & 1\\ 0 & 0 & 1 & -1\\ 0 & 0 & 1 & 1\\ \end{bmatrix}$$
So, let me just find $e^{A}$ for now and I can generalize later. I notice right away that I can write
$$A = \begin{bmatrix} B & I_{2}... | Consider $M(t) = \exp(t A)$, and as you noticed, it has block-diagonal form
$$
M(t) = \left(\begin{array}{cc} \exp(t B) & n(t) \\ 0_{2 \times 2} & \exp(t B) \end{array} \right).
$$
Notice that $M^\prime(t) = A \cdot M(t)$, and this results in a the following differential equation for $n(t)$ matrix:
$$
n^\prime(t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/73112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Index notation for tensors: is the spacing important? While reading physics textbooks I often come across notation like this;
$$J_{\alpha}{}^{\beta},\ \Gamma_{\alpha \beta}{}^{\gamma}, K^\alpha{}_{\beta}.$$ Notice the spacing in indices. I don't understand why they do not write simply $J_{\alpha}^\beta, \Gamma_{\alpha... | It's important to keep track of the ordering if you want to use a metric to raise and lower indices freely (without explicitly writing out $g_{ij}$'s all the time).
For example (using Penrose abstract index notation), if you raise the index $a$ on the tensor $K_{ab}$, then you get $K^a{}_b (=g^{ac} K_{cb})$, whereas if... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/73171",
"timestamp": "2023-03-29T00:00:00",
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Solve for equation algebraically Is it possible to write the following function as $H(x)=$'some expresssion` ?
$$D(x) = H(x) + H(x-1)$$
Edit:
Hey everyone, thanks for all the great responses, and just to clarify H(x) and D(x) are always going to be polynomials, I wasn't sure if that made too big of a difference so I di... | $H(x)$ could be defined as anything on $[0,1)$, then the relation
$$
D(x) = H(x) + H(x-1)\tag{1}
$$
would define $H(x)$ on the rest of $\mathbb{R}$. For example, for $x\ge0$,
$$
H(x)=(-1)^{\lfloor x\rfloor}H(x-\lfloor x\rfloor)+\sum_{k=0}^{\lfloor x\rfloor-1}(-1)^kD(x-k)\tag{2}
$$
and for $x<0$,
$$
H(x)=(-1)^{\lfloor x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/73222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Limiting distribution of sum of normals How would I go about solving this problem below? I am not exactly sure where to start. I know that I need to make use of the Lebesgue Dominated Convergence theorem as well. Thanks for the help.
Let $X_1, X_2, \ldots, X_n$ be a random sample of size $n$ from a distribution that i... | Another way to show $P(Z_n\le z)=0$ for all $z$ as Did suggests:
Note that $Z_n\sim N(n\mu, n\sigma^2)$, so for some fixed real number $z$, $P(Z_n\le z)=\frac{1}{2}\biggr[1+erf\biggr(\frac{z-n\mu}{\sqrt{2n\sigma^2}}\biggr)\biggr]\to0$ as $n\to\infty$,
since $\frac{z-n\mu}{\sqrt{2n\sigma^2}}\to-\infty$ and $\lim_{x\to-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/73259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to integrate this trigonometry function? The question is $ \displaystyle \int{ \frac{1-r^{2}}{1-2r\cos(\theta)+r^{2}}} d\theta$.
I know it will be used weierstrass substitution to solve but i did not have any idea of it.
| There's a Wikipedia article about this technique: Weierstrass substitution.
Notice that what you've got here is $\displaystyle\int\frac{d\theta}{a+b\cos\theta}$. The factor $1-r^2$ pulls out, and $a=1+r^2$ and $b=-2r$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/73350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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How to prove $\lim_{n \to \infty} \sqrt{n}(\sqrt[n]{n} - 1) = 0$? I want to show that $$\lim_{n \to \infty} \sqrt{n}(\sqrt[n]{n}-1) = 0$$ and my assistant teacher gave me the hint to find a proper estimate for $\sqrt[n]{n}-1$ in order to do this. I know how one shows that $\lim_{n \to \infty} \sqrt[n]{n} = 1$, to do th... | The OP's attempt can be pushed to get a complete proof.
$$
n = (1+x_n)^n \geq 1 + nx_n + \frac{n(n-1)}{2} x_n^2 + \frac{n(n-1)(n-2)}{6} x_n^3 > \frac{n(n-1)(n-2) x_n^3}{6} > \frac{n^3 x_n^3}{8},
$$
provided $n$ is "large enough" 1. Therefore, (again, for large enough $n$,) $x_n < 2 n^{-2/3}$,
and hence $\sqrt{n} x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/73403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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Fractional cardinalities of sets Is there any extension of the usual notion of cardinalities of sets such that there is some sets with fractional cardinalities such as 5/2, ie a set with 2.5 elements, what would be an example of such a set?
Basically is there any consistent set theory where there is a set whose cardina... | One can extend the notion of cardinality to include negative and non-integer values by using the Euler characteristic and homotopy cardinality. For example, the space of finite sets has homotopy cardinality $e=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\dotsi$. The idea is to sum over each finite set, inversely weighted ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/73470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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"answer_id": 3
} |
A number when successively divided by $9$, $11$ and $13$ leaves remainders $8$, $9$ and $8$ respectively A number when successively divided by $9$, $11$ and $13$ leaves remainders $8$, $9$ and
$8$ respectively.
The answer is $881$, but how? Any clue about how this is solved?
| First when the number is divided by 9. the remainder is 8.
So N = 9x+8.
Similarly, next x = 11y+9, and y=13z+8.
So N = 99y+89 = 99(13z+8)+89 = 1287z+792+89 = 1287z+881.
So N is of the form, 1287*(A whole Number)+881.
If you need to find the minimum number, then it would be 881.
Hope that helps.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/73532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
existence and uniqueness of Hermite interpolation polynomial What are the proofs of existence and uniqueness of Hermite interpolation polynomial?
suppose $x_{0},...,x_{n}$ are distinct nodes and $i=1 , ... ,n$ and $m_{i}$ are in Natural numbers. prove exist uniqueness polynomial $H_{N}$ with degree N=$m_{1}+...+m_{n}$-... | I think you've got your indices mixed up a bit; they're sometimes starting at $0$ and sometimes at $1$. I'll assume that the nodes are labeled from $1$ to $n$ and the first $m_i$ derivatives at $x_i$ are determined, that is, the derivatives from $0$ to $m_i-1$.
A straightforward proof consists in showing how to constru... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/73617",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Norm of adjoint operator in Hilbert space Suppose $H$ is a Hilbert space and let $T \in B(H,H)$ where in our notation $B(H,H)$ denotes the set of all linear continuous operators $H \rightarrow H$.
We defined the adjoint of $T$ as the unique $T^* \in B(H,H)$ such that $\langle Tx,y \rangle = \langle x, T^*y\rangle$ for ... | Why don't you look at what is $T^{**}$ ...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/73670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
"answer_id": 0
} |
Does Zorn Lemma imply the existence of a (not unique) maximal prolongation of any solution of an ode? Let be given a map $F:(x,y)\in\mathbb{R}\times\mathbb{R}^n\to F(x,y)\in\mathbb{R}^n$.
Let us denote by $\mathcal{P}$ the set whose elements are the solutions of the ode $y'=F(x,y)$, i.e. the differentiable maps $u:J\to... | Yes, Zorn's Lemma should be all you need. Take the set of partial solutions that extend your initial solution, and order them by the subset relation under the common definition of a function as the set of pairs $\langle x, f(x)\rangle$. Then the union of all functions in a chain will be another partial solution, so Zor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/73760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Prove an inequality by Induction: $(1-x)^n + (1+x)^n < 2^n$ Could you give me some hints, please, to the following problem.
Given $x \in \mathbb{R}$ such that $|x| < 1$. Prove by induction the following inequality for all $n \geq 2$:
$$(1-x)^n + (1+x)^n < 2^n$$
$1$ Basis:
$$n=2$$
$$(1-x)^2 + (1+x)^2 < 2^2$$
$$(1-2x+x^... | The proof by induction is natural and fairly straightforward, but it’s worth pointing out that induction isn’t actually needed for this result if one has the binomial theorem at hand:
Corrected:
$$\begin{align*}
(1-x)^n+(1+x)^n &= \sum_{k=0}^n\binom{n}k (-1)^kx^k + \sum_{k=0}^n \binom{n}k x^k\\
&= \sum_{k=0}^n\binom{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/73783",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 2
} |
Solving a recurrence using substitutions I have to solve this recurrence using substitutions:
$(n+1)(n-2)a_n=n(n^2-n-1)a_{n-1}-(n-1)^3a_{n-2}$ with $a_2=a_3=1$.
The only useful substitution that I see is $b_n=(n+1)a_n$, but I don't know how to go on, could you help me please?
| So if $b_n=(n+1)a_n$, then $b_{n-1}=na_{n-1}$, and $b_{n-2}=(n-1)a_{n-2}$, and you equation becomes $$(n-2)b_n=(n^2-n-1)b_{n-1}-(n-1)^2b_{n-2}$$ which is a little simpler than what you started with, though I must admit I can't see offhand any easy way to get to a solution from there. Are you working from a text or some... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/73844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Is $\sum_{n=0}^{\infty}2^n$ equal to $-1$? Why?
Possible Duplicate:
Divisibility with sums going to infinity
From Wikipedia and Minute Physics i see that the sum would be -1. I find this challenging to understand, how does a sum of positive integers end up being negative?
| It's all about the principle of analytic continuation. The function $f(x)=\sum_{n=0}^\infty z^n$ defines an analytic function in the unit disk, equal to the meromorphic function $g(z)=1/(1-z)$. Note that the equality $f\equiv g$ holds only in the disk, where $f$ converges absolutely. Despite this, if we naively want... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/73907",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Why is $(0, 0)$ not a minimum of $f(x, y) = (y-3x^2)(y-x^2)$? There is an exercise in my lists about those functions:
$$f(x, y) = (y-3x^2)(y-x^2) = 3 x^4-4 x^2 y+y^2$$
$$g(t) = f(vt) = f(at, bt); a, b \in \mathbf{R}$$
It asks me to prove that $t = 0$ is a local minimum of $g$ for all $a, b
\in \mathbf{R}$
I did it eas... | Draw the set of points in the $xy$-plane where $f(x,y) = 0$. Then look at the regions of the plane that are created and figure out in which of them $f(x,y)$ is positive and in which $f(x,y)$ is negative. From there, you should be able to prove that $(0,0)$ is neither a local maximum nor a local minimum point. (Hint:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/73949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 2
} |
Statistics: Maximising expected value of a function of a random variable An agent wishes to solve his optimisation problem: $ \mbox{max}_{\theta} \ \ \mathbb{E}U(\theta S_1 + (w - \theta) + Y)$, where $S_1$ is a random variable, $Y$ a contingent claim and $U(x) = x - \frac{1}{2}\epsilon x^2$.
My problem is - how to I '... | Expanding the comment by Ilya:
$$\mathbb{E}\,U(\theta S_1 + (w - \theta) + Y)
=\mathbb{E} (\theta S_1 + (w - \theta) + Y) - \frac{\epsilon}{2} \mathbb{E} \left((\theta S_1 + (w - \theta) + Y)^2\right)
$$
is a quadratic polynomial in $\theta $ with negative leading coefficients. Its unique point of maximum is found by s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/74035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Are any two Cantor sets ; "Fat" and "Standard" Diffeomorphic to each Other? All:
I know any two Cantor sets; "fat" , and "Standard"(middle-third) are homeomorphic to each other. Still, are they diffeomorphic to each other? I think yes, since they are both $0$-dimensional manifolds (###), and any two $0$-dimensional man... | I think I found an answer to my question, coinciding with the idea in Ryan's last paragraph: absolute continuity takes sets of measure zero to sets of measure zero. A diffeomorphism defined on [0,1] is Lipshitz continuous, since it has a bounded first derivative (by continuity of f' and compactness of [0,1]), so that i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/74077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Explicit formula for Fermat's 4k+1 theorem Let $p$ be a prime number of the form $4k+1$. Fermat's theorem asserts that $p$ is a sum of two squares, $p=x^2+y^2$.
There are different proofs of this statement (descent, Gaussian integers,...). And recently I've learned there is the following explicit formula (due to Gauss)... | Here is a high level proof. I assume it can be done in a more elementary way. Chapter 3 of Silverman's Arithmetic of Elliptic Curves is a good reference for the ideas I am using.
Let $E$ be the elliptic curve $y^2 = x^3+x$. By a theorem of Weyl, the number of points on $E$ over $\mathbb{F}_p$ is $p- \alpha- \overline{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/74132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 4,
"answer_id": 3
} |
what's the relationship between a.s. continuous and m.s. continuous? suppose that X(t) is a s.p. on T with $EX(t)^2<+\infty$. we give two kinds of continuity of X(t).
*
*X(t) is continuous a.s.
*X(t) is m.s. continuous, i.e. $\lim\limits_{\triangle t \rightarrow 0}E(X(t+\triangle t)-X(t))^2=0$.
Then, what's the ... | I don't know if there is a clear relation between both concepts.
For example if you take the Brownian Motion it satisfies 1 and 2 but if you take a Poisson process then it only satisfies 2 (although it satisfies a weaker form of condition 1 which is continuity in probability).
The question is what do you want to do ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/74203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
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